Hand Book Of Mathematics

Preface’ The present volume is an outgrowth of a Conference on Mathematical Tables held at Cambridge, Mass., on September 15-16, 1954, under the auspices of the National Science Foundation and the Massachusetts Institute of Technology. The purpose of the meeting was to evaluate the need for mathematical tables in the light of the availability of large scale computing machines. It was the consensus of opinion that in spite of the increasing use of the new machines the basic need for tables would continue to exist. Numerical tables of mathematical functions are in continual demand by scientists and engineers. A greater variety of functions and higher accuracy of tabulation are now required as a result of scientific advances and, especially, of the increasing use of automatic computers. In the latter connection, the tables serve mainly forpreliminarysurveys of problems before programming for machine operation. For those without easy access to machines, such tables are, of course, indispensable. Consequently, the Conference recognized that there was a pressing need for a modernized version of the classical tables of functions of Jahnke-Emde. To implement the project, the National Science Foundation requested the National Bureau of Standards to prepare such a volume and established an Ad Hoc Advisory Committee, with Professor Philip M. Morse of the Massachusetts Institute of Technology as chairman, to advise the staff of the National Bureau of Standards during the ~course of its preparation. In addition to the Chairman, the Committee consisted of A. Erdelyi, M. C. Gray, N. Metropolis, J. B. Rosser, H. C. Thacher, Jr., John Todd, C. B. Tompkins, and J. W. Tukey. The primary aim has been to include a maximum of useful information within the limits of a moderately large volume, with particular attention to the needs of scientists in all fields. An attempt has been made to cover the entire field of special functions. To carry out the goal set forth by tbe Ad Hoc Committee, it has been necessary to supplement the tables by including the mathematical properties that are important in computation work, as well as by providing numerical methods which demonstrate the use and extension of the tables. The Handbook was prepared under the direction of the late Milton Abramowitz, Its success has depended greatly upon the cooperation of and Irene A. Stegun. Their efforts together with the cooperation of the Ad HOC many mathematicians. Committee are greatly appreciated. The particular contributions of these and other individuals are acknowledged at appropriate places in the text. The sponsorship of the National Science Foundation for the preparation of the material is gratefully recognized. It is hoped that this volume will not only meet the needs of all table users but will in many cases acquaint its users with new functions. ALLEN V. ASTIN, L?imctor. Washington,

D.C.

Preface

to the Ninth

Printing

The enthusiastic reception accorded the “Handbook of Mathematical Functions” is little short of unprecedented in the long history of mathematical tables that began when John Napier published his tables of logarithms in 1614. Only four and one-half years after the first copy came from the press in 1964, Myron Tribus, the Assistant Secretary of Commerce for Science and Technology, presented the 100,OOOth copy of the Handbook to Lee A. DuBridge, then Science Advisor to the President. Today, total distribution is approaching the 150,000 mark at a scarcely diminished rate. The successof the Handbook has not ended our interest in the subject. On the contrary, we continue our close watch over the growing and changing world of computation and to discuss with outside experts and among ourselves the various proposals for possible extension or supplementation of the formulas, methods and tables that make up the Handbook. In keeping with previous policy, a number of errors discovered since the last printing have been corrected. Aside from this, the mathematical tables and accompanying text are unaltered. However, some noteworthy changes have been made in Chapter 2: Physical Constants and Conversion Factors, pp. 6-8. The table on page 7 has been revised to give the values of physical constants obtained in a recent reevaluation; and pages 6 and 8 have been modified to reflect changes in definition and nomenclature of physical units and in the values adopted for the acceleration due to gravity in the revised Potsdam system. The record of continuing acceptance of the Handbook, the praise that has come from all quarters, and the fact that it is one of the most-quoted scientific publications in recent years are evidence that the hope expressed by Dr. Astin in his Preface is being amply fulfilled. LEWIS M. BRANSCOMB,

Director

National Bureau of Standards November 1970

Foreword This volume is the result of the cooperative effort of many persons and a number of organizations. The National Bureau of Standards has long been turning out mathematical tables and has had under consideration, for at least IO years, the production of a compendium like the present one. During a Conference on Tables, called by the NBS Applied Mathematics Division on May 15, 19.52, Dr. Abramowitz of t,hat Division mentioned preliminary plans for such an undertaking, but indicated the need for technical advice and financial support. The Mathematics Division of the National Research Council has also had an active interest in tables; since 1943 it has published the quarterly journal, “Mathematical Tables and Aids to Computation” (MTAC),, editorial supervision being exercised by a Committee of the Division. Subsequent to the NBS Conference on Tables in 1952 the attention of the National Science Foundation was drawn to the desirability of financing activity in table production. With its support a z-day Conference on Tables was called at the Massachusetts Institute of Technology on September 15-16, 1954, to discuss the needs for tables of various kinds. Twenty-eight persons attended, representing scientists and engineers using tables as well as table producers. This conference reached consensus on several cpnclusions and recomlmendations, which were set forth in tbe published Report of the Conference. There was general agreement, for example, “that the advent of high-speed cornputting equipment changed the task of table making but definitely did not remove the need for tables”. It was also agreed that “an outstanding need is for a Handbook of Tables for the Occasional Computer, with tables of usually encountered functions and a set of formulas and tables for interpolation and other techniques useful to the occasional computer”. The Report suggested that the NBS undertake the production of such a Handbook and that the NSF contribute financial assistance. The Conference elected, from its participants, the following Committee: P. M. Morse (Chairman), M. Abramowitz, J. H. Curtiss, R. W. Hamming, D. H. Lehmer, C. B. Tompkins, J. W. Tukey, to help implement these and other recommendations. The Bureau of Standards undertook to produce the recommended tables and the To provide technical guidance National Science Foundation made funds available. to the Mathematics Division of the Bureau, which carried out the work, and to provide the NSF with independent judgments on grants ffor the work, the Conference Committee was reconstituted as the Committee on Revision of Mathematical Tables of the Mathematics Division of the National Research Council. This, after some changes of membership, became the Committee which is signing this Foreword. The present volume is evidence that Conferences can sometimes reach conclusions and that their recommendations sometimes get acted on. V

,/”

VI

FOREWORD

Active work was started at the Bureau in 1956. The overall plan, the selection of authors for the various chapters, and the enthusiasm required to begin the task were contributions of Dr. Abramowitz. Since his untimely death, the effort has continued under the general direction of Irene A. Stegun. The workers at the Bureau and the members of the Committee have had many discussions about content, style and layout. Though many details have had t’o be argued out as they came up, the basic specifications of the volume have remained the same as were outlined by the Massachusetts Institute of Technology Conference of 1954. The Committee wishes here to register its commendation of the magnitude and quality of the task carried out by the staff of the NBS Computing Section and their expert collaborators in planning, collecting and editing these Tables, and its appreciation of the willingness with which its various suggestions were incorporated into the plans. We hope this resulting volume will be judged by its users to be a worthy memorial to the vision and industry of its chief architect, Milton Abramowitz. We regret he did not live to see its publication. P. M. MORSE, Chairman. A. ERD~LYI M. C. GRAY N. C. METROPOLIS J. B. ROSSER H. C. THACHER. Jr. JOHN TODD ‘C. B. TOMPKINS J. W. TUKEY.

Handbook

of Mathematical

Functions

with

Formulas, Edited

Graphs, by Milton

1.

and Mathematical Abramowitz

and Irene A. Stegun

Introduction

The present Handbook has been designed to provide scientific investigators with a comprehensive and self-contained summary of the mathematical functions that arise in physical and engineering problems. The well-known Tables of Funct.ions by E. Jahnke and F. Emde has been invaluable to workers in these fields in its many editions’ during the past half-century. The present volume ext,ends the work of these authors by giving more extensive and more accurate numerical tables, and by giving larger collections of mathematical properties of the tabulated functions. The number of functions covered has also been increased. The classification of functions and organization of the chapters in this Handbook is similar to that of An Index of Mathematical Tables by A. Fletcher, J. C. P. Miller, and L. Rosenhead. In general, the chapters contain numerical tables, graphs, polynomial or rational approximations for automatic computers, and statements of the principal mathematical properties of the tabulated functions, particularly those of computa-

2.

Tables

Accuracy

The number of significant figures given in each table has depended to some extent on the number available in existing tabulations. There has been no attempt to make it uniform throughout the Handbook, which would have been a costly and laborious undertaking. In most tables at least five significant figures have been provided, and the tabular’ intervals have generally been chosen to ensure that linear interpolation will yield. fouror five-figure accuracy, which suffices in most physical applications. Users requiring higher 1 The most recent, the sixth, with F. Loesch added as cc-author, was published in 1960 by McGraw-Hill, U.S.A., and Teubner, Germany. 2 The second edition, with L. J. Comrie added as co-author, was published in two volumes in 1962 by Addison-Wesley, U.S.A., and Scientific Computing Service Ltd., Great Britain.

tional importance. Many numerical examples are given to illustrate the use of the tables and also the computation of function values which lie outside their range. At the end of the text in each chapter there is a short bibliography giving books and papers in which proofs of the mathematical properties stated in the chapter may be found. Also listed in the bibliographies are the more important numerical tables. Comprehensive lists of tables are given in the Index mentioned above, and current information on new tables is to be found in the National Research Council quarterly Mathematics of Computation (formerly Mathematical Tables and Other Aids to Computation). The ma.thematical notations used in this Handbook are those commonly adopted in standard texts, particularly Higher Transcendental Functions, Volumes 1-3, by A. ErdBlyi, W. Magnus, F. Oberhettinger and F. G. Tricomi (McGrawHill, 1953-55). Some alternative notations have also been listed. The introduction of new symbols has been kept to a minimum, and an effort has been made to avoid the use of conflicting notation.

of the Tables precision in their interpolates may obtain them by use of higher-order interpolation procedures, described below. In certain tables many-figured function values are given at irregular intervals in the argument. An example is provided by Table 9.4. The purpose of these tables is to furnish “key values” for the checking of programs for automatic computers; no question of interpolation arises. The maximum end-figure error, or “tolerance” in the tables in this Handbook is 6/& of 1 unit everywhere in the case of the elementary functions, and 1 unit in the case of the higher functions except in a few cases where it has been permitted to rise to 2 units.

IX /-

.

INTRODUCTION

X

3.

Auxiliary

Functions

One of the objects of this Handbook is to provide tables or computing methods which enable the user to evaluate the tabulated functions over complete ranges of real values of their parameters. In order to achieve this object, frequent use has been made of auxiliary functions to remove the infinite part of the original functions at their singularities, and auxiliary arguments to co e with infinite ranges. An example will make t fi e procedure clear. The exponential integral of positive argument is given by

4.

775 ;:; E

: 89608 89717

4302 8737

d0 g. I

ze*El . 89823 .89927 90029

(z) 7113 7306 7888

8: 4 ix

:.90227 90129

4695 60”3

[ 1 ‘453

The numbers in the square brackets mean that the maximum error in a linear interpolate is 3X10m6, and that to interpolate to the full tabular accuracy five points must be used in Lagrange’s and Aitken’s methods. 8 A. C. Aitken On inte elation b iteration of roportional out the use of diherences, ‘Brot Edin i: urgh Math. 8 oc. 3.6676

recludes direct interThe logarithmic singularity polation near x=0. The Punctions Ei(x)-In x and x-liEi(ln x-r], however, are wellbehaved and readily interpolable in this region. Either will do as an auxiliary function; the latter was in fact selected as it yields slightly higher accuracy when Ei(x) is recovered. The function x-‘[Ei(x)-ln x-r] has been tabulated to nine decimals for the range 05x<+. For +<x12, Ei(x) is sufficiently well-behaved to admit direct tabulation, but for larger values of x, its exponential character predominates. A smoother and more readily interpolable function for large x is xe-“Ei(x); this has been tabulated for 2 <x510. Finally, the range 10 <x_<m is covered by use of the inverse argument x-l. Twenty-one entries of xe-“Ei(x), corresponding to x-l = .l(- .005)0, suffice to produce an interpolable table.

Interpolation

The tables in this Handbook are not provided with differences or other aids to interpolation, because it was felt that the space they require could be better employed by the tabulation of additional functions. Admittedly aids could have been given without consuming extra space by increasing the intervals of tabulation, but this would have conflicted with the requirement that linear interpolation is accurate to four or five figures. For applications in which linear interpolation is insufficiently accurate it is intended that Lagrange’s formula or Aitken’s method of iterative linear interpolation3 be used. To help the user, there is a statement at the foot of most tables of the maximum error in a linear interpolate, and the number of function values needed in Lagrange’s formula or Aitken’s method to interpolate to full tabular accuracy. As an example, consider the following extract from Table 5.1. zez El (2) . 89268 7854 : 89384 89497 6312 9666

and Arguments

parts, with. (1932).

Let us suppose that we wish to compute the value of xeZ&(x) for x=7.9527 from this table. We describe in turn the application of the methods of linear interpolation, Lagrange and Aitken, and of alternative methods based on differences and Taylor’s series. (1) Linear interpolation. The formula for this process is given by jp= (1 -P)joSPfi where jO, ji are consecutive tabular values of the function, corresponding to arguments x0, x1, respectively; p is the given fraction of the argument interval p= (x--x0>/(x1-~0> and jP the required instance, we have

interpolate.

jo=.89717

ji=.89823

4302

In the present 7113

p=.527

The most convenient way to evaluate the formula on a desk calculating machine is.to set o and ji in turn on the keyboard, and carry out t d e multiplications by l-p and p cumulatively; a partial check is then provided by the multiplier dial reading unity. We obtain j.6z,E.‘;9;72;&39717

4302)+.527(.89823

7113)

Since it is known that there is a possible error of 3 X 10 -6 in the linear formula, we round off this result to .89773. The maximum possible error in this answer is composed of the error committed

INTRODUCTION

by the last roundingJ that is, .4403X 10m5, plus 3 X lo-‘, and so certainly cannot exceed .8X lo-‘. (2) Lagrange’s formula. In this example, the relevant formula is the 5-point one, given by

The numbers in the third and fourth columns are the first and second differences of the values of xezEl(x) (see below) ; the smallness of the second difference provides a check on the three interpolations. The required value is now obtained by linear interpolation :

f=A-,(p)f_z+A-,(p)f-1+Ao(p>fo+A,(p)fi +A&)fa Tables of the coefficients An(p) are given in chapter 25 for the range p=O(.Ol)l. We evaluate the formula for p=.52, .53 and .54 in turn. Again, in each evaluation we accumulate the An(p) in the multiplier register since their sum is unity. We now have the following subtable. x m=&(x) 7.952

.89772

9757

7.953

.89774

0379

fn=.3(.89772

.;

&

-2

1 2 3 4 5

7.9 8.1 7.8 8.2 7.7

0999

Yn=ze”G@) : : . .

89823 89717 89927 89608 90029 89497

7113 4302 7888 8737 7306 9666

Yo.I 89773 :89774

Yo, 1.2. I

Yo. 1, (I

44034 48264 2 90220 4 98773 2 35221

0379)

In cases where the correct order of the Lagrange polynomial is not known, one of the prelimina interpolations may have to be performed witT polynomials of two or more different orders as a check on their adequacy. (3) Aitken’s method of iterative linear interpolation. The scheme for carrying out this process in the present example is as follows:

10620 .89775

9757)+.7(.89774

= 239773 7192.

10622 7.954

XI

X,-X

Yo.1.a.s.n

.0473 0527 . 1473 -. 1527 . 2473 -. 2527

-. .89773

71499 2394 1216 2706

. 89773

71938 89773 ii

71930 30

Here 20-x 1 Yo x,-x x.--20 Yn x,-x 1 Yo.1 Yo.1 ,n=x,-x G--z1 l/O.”

S2fl

yo,n=-

Yo.

1.

. .

., m--l.m.n--

1 ~n-%n

safz wa

l/0.1. . . ., n-1.98 Yo.1. . . -, m-1.n

x,-x x,-x

1

If the quantities Z.-X and x~--5 are used as multipliers when forming the cross-product on a desk machine, their accumulation (~~-2) -(x,-x) in the multiplier register is the divisor to be used at that stage. An extra decimal place is usually carried in the intermediate interpolates to safeguard against accumulation of rounding errors. The order in which the tabular values are used is immaterial to some extent, but to achieve the maximum rate of convergence and at the same time minimize accumulation of rounding errors, we begin, as in this example, with the tabular argument nearest to the given argument, then take the nearest of the remaining tabular arguments, and so on. The number of tabular values required to achieve a given precision emerges naturally in the course of the iterations. Thus in the present example six values were used, even though it was known in advance that five would suffice. The extra row confirms the convergence and provides a valuable check. (4) Difference formulas. We use the central difference notation (chapter 25),

Here Sf1l2=f1-f0, 8f3/a=fz-f1, . . . ,, a2/1=sf3ia-afiia=fa-2fi+fo ~af3~~=~aja-~aj~=fa-3j2+3fi-k 8'fa=~aj~fsla-6~3~2=f4-~f~+~ja-4f~+fo

and so on. In the present example the relevant part of the difference table is as follows, the differences being lace of the written in units of the last decimal function, as is customary. The sma Bness of the high differences provides a check on the function values 7:9 8.0

Applying, formula

xe=El(x) .89717 4302 . 89823 7113

for example,

j~=(l-P)fo+E2(P)~*jo+E4(P)~4jo+

-2 -2

SY 2754 2036

Everett’s

. . . +Pfl+F2(P)~afl+F4(P)~4fl+

S4f -34 -39

interpolation . . * *

and takin the numerical values of the interpolation toe flf cients Es(p), E4 ), F,(p) and F,(p) from Table 25.1, we find t!l at

,,/

INTRODUCTION

XII 10Qf.6,=

.473(89717 4302) + .061196(2 2754) - .012(34) + .527(89823 7113) + .063439(2 2036) - .012(39) = 89773 7193.

We may notice in passing that Everett’s formula shows that the error in a linear interpolate is approximately mPwfo+

F2(P)wl=

m(P)

can be used. We first compute as many of the derivatives ftn) (~0) as are significant, and then evaluate the series for the given value of 2. An advisable check on the computed values of the derivatives is to reproduce the adjacent tabular values by evaluating the series for z=zl and x1. In the present

+ ~2(P)lk?f0+wJ

f’(z)=(l+Z-‘)f(Z)-1 f”(2)=(1+2-‘)f’(Z)--Z-Qf(2) f”‘(X) = (1 -i-z-y’(2)

Inverse

With linear interpolation there is no difference in principle between direct and inverse interpolation. In cases where the linear formula rovides an insufficiently accurate answer, two met fl ods are available. We may interpolate directly, for example, by Lagrange’s formula to prepare a new table at a fine interval in the neighborhood of the approximate value, and then apply accurate inverse linear interpolation to the subtabulated values. Alternatively, we may use Aitken’s method or even possibly the Taylor’s series method, with the roles of function and argument interchanged. It is important to realize that the accuracy of an inverse interpolate may be very different from that of a direct interpolate. This is particularly true in regions where the function is slowly varying, for example, near a maximum or minimum. The maximum precision attainable in an inverse interpolate can be estimated with the aid of the formula

AxmAj/df dx in which Aj is the maximum possible error in the function values. Example. Given xe”Ei(z) = .9, find 2 from the table on page X. (i) Inverse linear interpolation. The formula for v is In the present ‘=.90029

example, we have

.9 - .89927 7888 72 2112 7306.89927 7888=101=‘708357’

p:‘(xo)/k!

i ; 3

. . .

5.

-22~Qf’(5)

+22-y(2).

With x0=7.9 and x-x0= .0527 our computations are as follows: an extra decimal has been retained in the values of the terms in the series to safeguard against accumulation of rounding errors.

~(x,=~(xo)+(x-x,,~~+(x-xo,~~~ +(~-x,)q$+

we have

f(x) =xeZEt(x)

Since the maximum value of IEz(p)+Fz(p)I in the range O
(5) Taylor’s series. In cases where the successive derivatives of the tabulated function can be computed fairly easily, Taylor’s expansion

example,

(x--so) y’k’(x0)/k!

.89717

4302

.89717

4302

- .00113 .01074 .00012

0669 7621 1987

-.ooooo .00056 .ooooo .a9773

6033 3159 7194 0017

53 9

Interpolation The desired z is therefore z=zQ+p(z,--2,,)=8.1+.708357(.1)=8.17083

57

To estimate the possible error in this answer, we recall that the maximum error of direct linear interpolation in this table is Aj=3X lOwe. An approximate value for dj/dx is the ratio of the first difference to the argument interval (chapter 25), in this case .OlO. Hence the maximum error in x is approximately 3XlO-e/(.OlO), that is, .0003. (ii) Subtabulation method. To improve the ap roximate value of x just obtained, we interpo Pate directly for p=.70, .7l and .72 with the aid of Lagrange’s 5-point formula, xe=El (x)

X

8. 170

.

89999 -.-_

8.171

.

90000

6

QQ

1 0151 3834

-2 1 0149

8. 172

Inverse gives

90001

linear

Hencex=8.17062 An estimate

3983

interpolation

in the new

table

23. of the maximum

error in this result

is df~1x10-8_1x10-7 Ajl z .OlO (iii) Aitken’s method. This is carried out in the same manner as in direct interpolation.

INTRODUCTION n 0

yn=xeeZE1(x) . 90029 7306

2, 8. 2

4 3

: 89927 90129 . 89823

6033 7888 7113

8. 31 8. 0

8. 17083 17023 8. 17113

5712 1505 8043

%

: 90227 89717

4302 4695

7. 49 8.

8. 16992 17144

0382 9437

Z0.n

QJ.98 8. 1706,l

8. 17062

21 8142 7335

6. Bivariate Bivariate interpolation is generally most simply performed as a sequence of univariate interpolations. We carry out the interpolation in one direction, by one of the methods already described, for several tabular values of the second argument in the neighborhood of its given value. The interpolates are differenced as a check, and

Generation

a.1 2.n

9521 2 5948

The estimate of the maximum error in this result is the same as in the subtabulation method. An indication of the error is also provided by the

7.

XIII

of Functions

Many of the special mathematical functions which depend on a parameter, called their index, order or degree, satisfy a linear difference equation (or recurrence relation) with respect to this parameter. Examples are furnished by the Legendre function P,(z), the Bessel function Jn(z) and the exponential integral E,(x), for which we have the respective recurrence relations

zo.l.2.3.74

2244 231 415

8. 17062

discrepancy in the highest case xo .I ,2.3 A, and ZLI .2.8 .s.

7306

-. . 00072 00129 -. 00176

2112 6033 2887

-. .00227 00282

5G98 4695

interpolates,

in this

I

Interpolation interpolation is then carried out in the second direction. An alternative procedure in the case of functions of a complex variable is to use the Taylor’s series expansion, provided that successive derivatives of the function can be computed without much difficulty.

from

Recurrence

Relations

(iii) the direction in which the recurrence applied. Examples are as follows. Stability-increasing Pm(x),

p:(2)

Qnb),

Q:(x)

y&9,

KG)

J-n-&), &Cd

P”(X), nE,+,+xE,,=e-=.

is being

n

(x
z-t44 (n
Stability-decreasing

Jn+*-~Jn+J.-l=O

Particularly for automatic work, recurrence relations provide an important and powerful computing tool. If the values of P&r) or Jn(z) are known for two consecutive values of n, or E',(z) is known for one value of n, then the function may be computed for other values of n by successive applications of the relation. Since generation is carried out perforce with rounded values, it is vital to know how errors may be propagated in the recurrence process. If the errors do not grow relative to the size of the wanted function, the process is said to be stable. If, however, the relative errors grow and will eventually overwhelm the wanted function, the process is unstable. It is important to realize that st,ability may depend on (i) the particular solution of the difference equation being computed; (ii) the values of x or other parameters in the difference equation;

2318 265

l/n-u .00029

P.,(z)

7t @
Qnh), Q:(x) J&4, Jn+Hcd Em(z) F,,(t,

Z.@) , Zn+&) (n >r) p) (Coulomb

wave

function)

Illustrations of the generation of functions from their recurrence relations are given in the pertinent chapters. It is also shown that even in cases where the recurrence process is unstable, it may still be used when the starting values are known to sufficient accuracy. Mention must also be made here of a refinement, due to J. C. P. Miller, which enables a recurrence process which is stable for decreasing n to be applied without any knowledge of starting values for large n. Miller’s algorithm, which is wellsuited to automatic work, is described in 19.28,

Example

1.

INTRODUCTION

XIV

8.

Acknowledgments

The production of this volume has been the result of the unrelenting efforts of many persons, all of whose contributions have been instrumental in accomplishing the task. The Editor expresses his thanks to each and every one. The Ad Hoc Advisory Committee individually and together were instrumental in establishing the basic tenets that served as a guide in the formation of the entire work. In particular, special thanks are due to Professor Philip M. Morse for his continuous encouragement and support. Professors J. Todd and A. Erdelyi, panel members of the Conferences on Tables and members of the Advisory Committee have maintained an undiminished interest, offered many suggestions and carefully read all the chapters. Irene A. Stegun has served eff ectively as associate editor, sharing in each stage of the planning of the volume. Without her untiring efforts, completion would never have been possible. Appreciation is expressed for the generous cooperation of publishers and authors in granting permission for the use of their source material. Acknowledgments for tabular material taken wholly or in part from published works are iven on the first page of each table. Myrtle R. Ke Yilington corresponded with authors and publishers to obtain formal permission for including their material, maintained uniformity throughout the

bibliographic references and assisted in preparing the introductory material. Valuable assistance in the preparation, checkin and editing of the tabular material was receive IFi from Ruth E. Capuano, Elizabeth F. Godefroy, David S. Liepman, Kermit Nelson, Bertha H. Walter and Ruth Zucker. Equally important has been the untiring cooperation, assistance, and patience of the members of the NBS staff in handling the myriad of detail necessarily attending the publication of a volume of this magnitude. Especially appreciated have been the helpful discussions and services from the members of the Office of Technical Information in the areas of editorial format, graphic art layout, printing detail, preprinting reproduction needs, as well as attention to promotional detail and financial support. In addition, the clerical and typing stafI of the Applied Mathematics Division merit commendation for their efficient and patient production of manuscript copy involving complicated technical notation. Finally, the continued support of Dr. E. W. Cannon, chief of the Applied Mathematics Division, and the advice of Dr. F. L. Alt, assistant chief, as well as of the many mathematicians in the Division, is gratefully acknowledged. M.

ABRAMOWITZ.

1.

Mathematical DAVID

Constants

S. LIEPMAN ’

Contents Page

Mathematical

Table 1.1.

+i,nprime

Constants


20s.

...............

2

..................

2

Some roots of 2, 3, 5, 10, 100, 1000, e, 20s ..........

2

e*n, n=l(l)lO,

....................

2

20s ....................

2

25s

e*tns, n=l(l)lO, eas, e*‘,

20s

.......................

ln n, log,, n, n=2(1)10, In 7~,In&,

na, n=1(1)9, a*“, n=l(l)lO,

........

24D

r(4), l/r($),

3 3

25s ....................

of T, powers and roots involving 26s

3

T,

25s .......

..................

3

.......................

3

15D .....................

Bureau of Standards.

3 3

24D. .................

r(2),l/r(z),lnr(2),2~3,a,g,q,~,g,g,~,

2 3

25s .....................

lo, l’, 1” in radians,

1 National

26, 25s

25s ...................

1 radian in degrees,

~,lny,

primes
logI, ?r, log,, e, 25s ...............

n In 10, n=1(1)9,

Fractions

2

3

15D.

........

3

MATHEMATICAL

TABLE

1.1.

1 4142 i7320 2.2360 2..6457 3.3166 3.6055 4.1231 4.3588 4.7958 5.3851 5.5677 6.0827 6.4031 6.5574 6.8556 7.2801 7.6811 7.8102 8.1853 8.4261 8.5440 8.8881 9.1104 9.4339 9.8488

13562 50807 67977 51311 24790 51275 05625 98943 31523 64807 64362 62530 24237 38524 54600 09889 45747 49675 52771 49773 03745 94417 33579 81132 57801

37;: 56887 49978 06459 35539 46398 61766 54067 31271 13450 83002 29821 43284 30200 40104 28051 86860 90665 87244 i7635 31753 31558 14429 05660 79610

:E: 19221 96890 86865 06523 41249 82711 81758 43941 99700 86306 11679 88501 88819 38113 47217

1) lj 2) 2) 3) 3) 3) 4)

2.7182 7.3890 2.0085 5.4598 1.4841 4.0342 1.0966 2.9809 8.1030 2.2026

81828 56098 53692 15003 31591 87934 33158 57987 83927 46579

45:04 93065 31876 31442 02576 92735 42845 04172 57538 48067

52353 02272 67740 39078 60342 12260 85992 82747 40077 16516

1) 2) 4) 5) 6) 8j 9) LO) 12) 13)

2.3140 5.3549 1.2391 2.8675 6.6356 1.5355 3.5533 8.2226 1.9027 4.4031

69263 16555 64780 13131 23999 29353 21280 31558 73895 50586

enr 27792 24764 79166 36653 34113 95446 84704 55949 29216 06320

69006 73650 97482 29975 42333 69392 43597 95275 12917 29011

26224 72417

14792 99019

64190 79852

47180 12288 94361 37912 59469 10149 41541 24577 85092 95272 49357 13344 38979 94215 95829 87204 17912 72066 00115

In n 55994 66810 11989 43410 22805 05531 67983 33621 99404 79837 46153 05621 16644 92914 98647 48514 64422 70430 69356

0.6931 1.0986 1.3862 1.6094 1.7917 1.9459 2.0794 2.1972 2.3025 2.3978 2.5649 2.8332 2.9444 3.1354 3.3672 3.4339 3.6109 I: %

CONSTANTS

MATHEMATICAL

50488 72935 96964 05905 98491 92931 05498 35522

CONSTANTS 1O’fi

3.1622

77660

16837

93320

1O’fl 10"'

2.1544 1.7782 1.5848

79410 34690 93192

03892 03188 46111

%X 34853

4.6415 2.5118 5.6234 3.9810 1.2599 1.4422 1.1892 1.3160 7.0710 5.7735 4.4721

88833 86431 13251 71705 21049 49570 07115 74012 67811 02691 35954

61277 50958 90349 53497 89487 30740 06275 95249 86547 89625 99957

FKd 08040 25077 31648 83823 20667 24608 52440 76451 93928

4.8104 2.1932 11 2.0787 li 4.5593 1.6487 1) 6.0653 1.3956 1) 7.1653 -I

~-

77380 80050 95763 81277 21270 06597 12425 13105

96535 73801 50761 65996 70012 12633 08608 73789

16555 54566 90855 23677 81468 42360 95286 25043

1) 1) 2) 2j 3) 3) 4) 4) 4) 5)

3.6787 1.3533 4.9787 1.8315 6.7379 2.4787 9.1188 3.3546 1.2340 4.5399

94411e-“71442 52832 36612 06836 78639 63888 87341 46999 08546 52176 66635 19655 54516 26279 02511 98040 86679 92976 24848

32159 69189 42979 80293 70966 84230 20800 83882 54949 51535

- 2) - 3) - 5) - 6) - 7) - 9i -1oj -11) -13) -14j

4.3213 1.8674 8.0699 3.4873 1.5070 6. 5124 2.8142 1.2161 5.2554 2.2711

91826 42731 51757 42356 17275 12136 68457 55670 85176 01068

37722 70798 03045 20899 39006 07990 48555 94093 00644 32409

49774 88144 99239 54918 46107 07282 27211 08397 85552 38387

-

6.5988 5.6145

03584 94835

53125 66885

37077 16982

0102 7712 0205 9897 7815 4509 0308 5424 0000 0413 1139 2304 2787 3617 4623 4913 1. 5682 1.6127 1.6334

99956 12547 99913 00043 12503 80400 99869 25094 00000 92685 43352 48921 53600 27836 97997 61693 01724 83856 68455

101’5 1OOlfl loo”5 1000"' lOOO"5

2lB 3’B

2114

3114

2-m 3-m 5-‘fi

(--

(-i.

((-

60287 30427 92853 11026 11156 83872 63720 43592 09997 95790

-

1) 1) lj

~--~

--~-

-----

* *

*

-----

55238 39995 34242 71802 36048 45167 31361 13891 76367 59152

e--nr

2) 1)

log10

53094 96913 xz::: 50008 33051 59282 93827 56840 05440 67360 60802 04600 96908 40271 62459 44443 78038 24234

172321 952452 344642 007593 124774 053527 516964 904905 179915 619436 534874 495346 090274 067528 832720 291643 680957 667634 728425 *See page xx.

12

63981 l!id62 27962 36018 83643 14256 91943 39324 00000 15822 30683 37827 95282 01759 89895 83427 06699 71973 57958

19521 43729 39042 80478 63250 83071 58564 87459 00000 50407 67692 39285 89615 %Ei 26796 49968 54945 65264

37389 50279 74778 62611 87668 22163 12167 ~:~:: 50200 ::%i X%f 32847 66704 08451 09412 05088

MATHEMATICAL

TABLE

1.1.

CONSTANTS

MATHEMATICAL

3.8501 3. 9702 4.0775 4.1108 4. 2046 4.2626 4. 2904 4.3694 4.4188 4.4886 4. 5747

47601 91913 37443 73864 92619 79877 59441 47852 40607 36369 10978

In n 71005 55212 90.571 17331 39096 04131 14839 46702 79659 73213 50338

85868 18341 94506 12487 60596 54213 11290 14941 79234 98383 28221

209507 444691 160.504 513891 700720 294545 921089 729455 754722 178155 167216

(-1)

1. 1447 9. 1893

29885 85332

84940 04672

01741 74178

43427 03296

( ( ( ( (

1) 1) 1) 1) 1)

2.3025 4.6051 6.9077 9.2103 1. 1512 1. 3815 1. 6118 1. 8420 2. 0723

85092 70185 55278 40371 92546 51055 09565 68074 26583

nln 10 99404 98809 98213 97618 49702 79642 09583 39523 69464

56840 13680 70520 27360 28420 74104 19788 65472 11156

17991 35983 53974 71966 08996 10795 12594 14393 16192

1) 1) 2) 2j 3) 3) 4) 4)

3.1415 9.8696 3. 1006 9. 7409 3.0601 9.6138 3.0202 9.4885 2.9809 9.3648

92653 04401 27668 09103 96847 91935 93227 31016 09933 04747

58979 08935 02998 40024 85281 75304 77679 07057 34462 60830

32384 86188 20175 37236 45326 43703 20675 40071 11666 20973

62643 34491 47632 44033 27413 02194 14206 28576 50940 71669

1. 5707 1. 0471 7.8539 1. 7724 1.4645 1.3313 2. 1450 2.3597 5. 5683 2. 2459 2. 5066 1. 2533 2. 2214

96326 97551 81633 53850 91887 35363 29397 30492 27996 15771 28274 14137 41469

79489 19659 97448 90551 56152 80038 11102 41469 83170 83610 63100 31550 07918

66192 77461 30961 60272 32630 97127 56000 68875 78452 45473 05024 02512 31235

31322 54214 56608 98167 20143 97535 77444 78474 84818 42715 15765 07883 07940

57. 2957 0. 0174

79513 53292

08232 51994

08767 32957

98155’ 69237r

CONSTANTS-Continued log10

loglog

(-1) (-1)

logl0e

( ( ( ( ( (

( ( C ( ( ( ( (

(-1)

t:

I-, -, r (714)

In In In ln

r(i/3) r(2/3) r(lj4) r(3/4)

1)

97857 75869 52011 29835 74802 58348 22860 27091 78092 90006 71734

93571 60078 64214 01076 70082 71907 12045 29044 37607 64491 f ?624

74644 90456 41902 70338 64341 52860 59010 14279 39038 27847 48517

14219 32992 60656 85749 49132 92829 74387 94821 32760 23543 84362

4.9714 4.3429

98726 44819

94133 03251

85435 82765

12683 11289

92653 85307 77960 37061 96326 55592 14857 74122 33388

5s”9”19 17958 76937 43591 79489 15387 51285 87183 23081

32384 64769 97153 72953 66192 59430 52669 45907 39146

62643 25287 87930 85057 31322 77586 23850 70115 16379

3. 1415 6. 2831 !a. 4247 1) 1. 2566 1) 1.5707 1) 1. 8849 1) 2. 1991 1)2.5132 1) 2.8274

n

7P

0. 5772

15664

90153

1. 7724 2. 6789 1. 3541 3. 6256 1. 2254 0.8929 0. 9027 0. 9064 0. 9190 0. 9854 0.3031 1.2880 0.2032

53850 38534 17939 09908 16702 79511 45292 02477 62526 20646 50275 22524 80951

905516 707748 426400 221908 465178 569249 950934 055477 848883 927767 147523 698077 431296

28606

06512

; 3 4 x 7 i 10 s-ii ;h;

J2

r-1 13 ?r-l/4 +f3 ,-%I4 *-3/Z r--c (2r)-'12 (a/,)"2 2'f2/?r

n

1.6720 1.7242 1.7708 1.7853 1.8260 1.. 8512 1. 8633 I.. 8976 1. 9190 1. 9493 1. 9867

*-”

-1) -1)

3. 1830 1.. 0132 3.2251 1. 0265 3.2677 1. 0401 3. 3109 1. 0539 3.3546 1. 0678

98861 11836 53443 98225 63643 61473 36801 03916 80357 27922

83790 42337 31994 46843 05338 29585 77566 53493 20886 68615

67153 77144 89184 35189 54726 22960 76432 66633 91287 33662

77675 38795 42205 15278 28250 89838 59528 17287 39854 04078

4.7123 4.1887 4.4428 (-1) 5. 6418 (- 1) 6. 8278 (-1) 7. 5112 (-1) 4. 6619 (- 1) 4. 2377 (-1) 1. 7958 ( -2) 4.. 4525 (-1) 3. 9894 (-1) 7. 9788 (-1) 4.5015

88980 90204 82938 95835 40632 55444 40770 72081 71221 26726 22804 45608 81580

38468 78639 15836 47756 55295 64942 35411 23757 25166 69229 01432 02865 78533

98576 09846 62470 28694 68146 48285 61438 59679 56168 06151 67793 35587 03477

93965 16858 15881 80795 70208 87030 19885 10077 90820 35273 99461 98921 75996

0. 0002 0. 0000

90888 04848

20866 13681

57215 10953

96154r 59936r

48223

37662

-2j -2) -3) -31

-4j -4) -5) -5)

1’ 1” In Y

-0.

l/~U/‘4 l/Ul/3) l/r(2/3) mm) mw m4/3) mw) mw uw4 In r(4/3) In r(5/3) In r(5/4) in r(7/4)

. *See page II.

0. 0. 0. 0. 0. :l. l. l. :l. -0. -0. -0. -0.

5495

39312

98164

5641 3732 7384 2758 8160 1198 1077 1032 0880 1131 1023 0982 0844

89583 82173 88111 15662 48939 46521 32167 62651 65252 91641 14832 71836 01121

547756 907395 621648 830209 098263 722186 432472 320837 131017 740343 960640 421813 020486

*

i

2. Physical

Constants

and Conversion

A. G. MCNISH

Factors

1

Contents Table Table Table Table Table Table

2.1. 2.2. 2.3. 2.4. 2.5. 2.6.

Common Units and Conversion Factors . . . . . . . . . Names and Conversion Factors for Electric and Magnetic Units . . . . . . . . . . . . . . . . . . . . . . . Adjusted Values of Constants . . . . . . . . . . . . . Miscellaneous Conversion Factors. . . . . . . . . . . . Conversion Factors for Customary U.S. Units to Metric Units . . . . . . . . . . . . . . . . . . . . . . . Geodetic Constants . . . . . . . . . , . . . . . . . .

* National Bureau of Standards.

Page 6 6 7 8 8 8

2. Physical

Constants

and

Table

Quantity

and

Conversion

Common

Factors

for

Electric

and

=

SI

emu name

name

I

Magnetic

=

force

Magnetomotive force Magnetic flu* Magnetic flux density Electric displacement

ampere coulomb volt ohm henry farad amp. turns/ meter amp. turns weber tesla

I tbampere 1tbcoulomb

abvolt abohm centimeter

Conversion

statampere statcoulomb statvolt statohm

10-l LO-’ 108 100 100 10-g 4*x

centimeter gil bert maxwell gauss --._-_-______

Units

=

SI unit/ esu unit

-

oersted

-

Example: If the value assigned to a current *Divide this number by 4?r if unrationalized 6

and

SI unit/ emu unit

esu name

-

Current Charge Potential Resistance Inductance Capacitance Magnetizing

Units Factors

The SI unit of electric current is the ampere defined by the equation 2r,,,Z1ZJ4~= F giving the force in vacua per unit length between two infinitely long parallel conductors of infinitesimal cross-section. If F is in newtons, and rrn has the numerical value 477 X lo-‘, then I1 and Zr are in amperes. The customary equations define the other electric and magnetic units of SI such as the volt, ohm, farad, henry, etc. The force between electric charges in a vacuum in this system is given by Q, Qn/4nrerg= F, re having the numerical value 10r/4nc2 where c is the speed of light in meters per second (r,= 8.854 x 10-12). The CGS unrationalized system is obtained by deleting 4n in the denominators in these equations and expressing F in dynes, and r in centimeters. Setting r,,, equal to unity defines the CGS unrationalized electromagnetic system (emu), re then taking the numerical value of 1/c2. Setting re equal to unity defines the CGS unrationalized electrostatic system (esu), r,,, then taking the numerical value of l/cz.

Mass-the kilogram -fixed by the international kilogram at S&vres, France. Time-the second- fixed as l/31,556,925.9747 of the tropical year 1900 at 12” ephemeris time, or the duration of 9,19‘2,631,770 cycles of the hyperfine transition frequency of cesiurn 133. Temperature-the degree-fixed on a thermodynamic basis by taking the temperature for the triple point of natural water as 273.16 “K. (The Celsius scale is obtained by adding -273.15 to the Kelvin scale.) Other units are defined in terms of them by assigning the value unity to the proportionThe ality constant in each defining equation. entire system, including electricity units, is called the Systi.?me International d’unitds (SI). Taking the l/100 part of the meter as the unit of length and the l/1000 part of the kilogram as the unit of mass, similarly, gives =

2.1.

often used in physics

~

(1 meter - 1650763.73h).

2.2. Names

Factors

rise to the CGS system, and chemistry.

The tables in this chapter supply some of the more commonly needed physical constants and conversion factors. All scientific measurements in the fields of mechanics and heat are based upon four international arbitrarily adopted units, the magnitudes of which are fixed by four agreed on standards: Lengththe meter -fixed by the vacuum wavelength of radiation corresponding to the transition 2Plu-5Da of krypton 86

Table

Conversion

-3x 100 -3 x 109 -(1/3)X 10-Z -(1/9)X 10-u %(1/9)X 10-l’ -9x 10” -3 x loo*

IO-3*

4rX lo-I* 108 10’

__---___----__-______-_____

-3/10** -(1/3)X 10-z -(1/3)X 10-B -3x 105*

10-J*

I_-..____..____

-

-

is 100 amperes its value in abamperes is 100X10-‘=lO. system is involved; other numbers are unchanged.

3. Elementary

Analytical

MILTON

Methods

ABRAMOWITZ l

Contents Elementary 3.1.

3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.3. 3.9. 3.10.

Analytical

Methods

Page 10

.................

Binomial Theorem and Binomial Coefficients; Arithmetic and Geometric Progressions; Arithmetic, Geometric, Harmonic and Generalized Means ............... Inequalities ...................... Rules for Differentiation and Integration ......... Limits, Maxima and Minima .............. Absolute and Relative Errors .............. Infinite Series ..................... Complex Numbers and Functions ............ Algebraic Equations .................. Successive Approximation Methods ........... Theorems on Continued Fractions ............

Numerical 3.11.

Methods

Table

3.1.

19 19 23

............................

Powers and Roots

13 14 14 16 17 18 19 19

.......................

Use and Extension of the Tables ............ 3.12. Computing Techniques. ................

References

10 10 11

. . . . . . . . . . . . . . . . . .

24

n’“, k=l(l)lO, 24, l/2, l/3, l/4, l/5 n=2(1)999, Exact or 10s

The author acknowledges the assistance of Peter J. U’Hara the preparation and checking of the table of powers and roots.

1 National

Bureau of Standards.

(Deceased.)

and Kermit

C. Nelson in

3. Elementary

Analytical

3.1. Binomial Theorem and Binomial Coefficients; Arithmetic and Geometric Progressions; Arithmetic, Geometric, Harmonic and Generalized Means Binomial

Methods

3.1.9 Sum

a”-w+

. . . +b”

Coefficients

of Geometric

(see chapter

integer)

s,=a+ar+a?+

n =nc7t=n(n-l)

24)

. .k., (n--k+l)=

Arithmetic

Mean

Geometric

3.1.12

A

. - . +a. Mean

of

Mean

n Quantities

G

(at>O,k=1,2,.

G= (a,&. . . a,,)l’” Harmonic

. .,n)

H

of n Quantities

3.1.13

3.1.5

.+$) 3.1.6

1+c)+@+.

(ah>O,k=1,2,.

1-G)+@-

3.1.7

of Binomial

Coefficients

0

M(t)=O(t
lim M(t) =max. t+m 3.1.17 t&M(t)=min. 3.1.16

1 1 1 1 1

7---8---9---lo_--_ ll---12----

2 3 4 5 6

1 7 1 8 1 9 1 10 1 11 1 12

; 1 6 4’ 1 10 10 5 15 20 15

1 6

10

II.

a2,

(a,,%,.

* * *, a,) =mtLx. a

. .,a,)=min.a

M(l)=A

3.1.20

M(-l)=H 3.2. Inequalities

1 8 36

1 9 45

Relation 1 10 55

For a more extensive table see chapter 24. page

3.1.19

1

21 35 35 21 7 28 56 70 56 28 36 84126126 84 45120210252210120 55165330462462330165 66220495792924792495220

@I,

liiM(t)=G

3.1.18 3---4---5..--6-e-m

some ak zero)

z

3.1.8

2---I

Mean

M(t) ==(i g a:yf

3.1.14

. . . +(--I)“@=0

3.1.15 Table

. .,n)

. . +c)=2” General&d

*See

(--l
n

(k-;-l)

3.1.4

a(l-P) l--P

n Quantities

of

&h+az+

3.1.11 @=(n:k)=(-l)k

n Terms

to

lim s.=a/(l-r) n-t-

n! (n-k)!k!

0k

3.1.3

Progreamion

. . . +a+‘=---

3.1.2 *

(a+z))

3.1.10

(n a positive Binomial

7+-l)&;

last term in series=Z=a+(n-1)d

an-%2 Sum

+C)

n Terms

to

. . . +(a++-114

=na+;

Theorem

a”-lb+@

Progression

a-t-b+d)+b+24+

3.1.1 (a+b)“=a”+c)

of Arithmetic

Between and

Arithmetic, Generalized

Geometric, Means

Harmonic

3.2.1 A> G>H,

3.2.2

equality

if and only if al=az=

min. a<M(t)<mnx.

a

. . . =a,,

ELEMENTARY

3.2.3

min. a
3.2.5

Minkowski’s

If p>l

Inequalities

Inequality

for

Sums

and &, bk>O for all x:,

3.2.12

if t<s unless all ak are equal, or s
11

METHODS

a

holds if all ak are equal, or t
M(t)<&!(s)

3.2.4

ANALYTICAL

(&

(ak+bk)p~‘p<(&

a;)“‘+($

holds if and only

t?qUdity

i-f

bi$“j (c=con-

bk=C&

stant>O).

Iall-la21_
Minkowski’s

3.2,6

Inequality

5or Integrals

If P>l, Chehyshev’s

Inequality

If alla2>a,z

. . . >a,

b,>b,>b,>

. . . >bn

3.2.7

n 5

akbk>

-(kc,

k=l

Hiilder’s

If ;++,p>1,

2

3.2.13

(Jb a

‘> (&

Inequality

for

I?(~)+9(z)I~~s)llp~(~b

a

a

lJ~(z)l%iz)l’p +(Jb

“)

a

Iscd lqp

equality holds if and only if g(z) =cf(x) stant>O).

Sums

(c=con-

fj>l 3.3. Rules

for

Differentiation

and

Integration

Derivatives

equality holds if and only if jbkl=cIuEIP-’ stant>O). If p=q=2 we get Cauchy’s

(c=con-

$; (cu) =c $9 c constant

3.3.1

3.3.2

Inequality

-g (u+v)++g

3.2.9 [&

akb,]2<&

a; &

b: (equality

for &=Cbk,

& (uv) =u E+v

3.3.3

2

c constant). Hiilder’s 4

Inequality

for

Integrals

3.3.4

& (u/v)=

vdu/dx-udv/dx v2 -

,

1r;++=1,p>1,

q>l

.g u(v) =g

3.3.5

gi

3.2.10

& (u”) =uD e %+1.n U 2)

3.3.6 equality holds if and only (c=constant>O). If p=p=2 we get Schwa&s

3.2.11

Inequality

if

jgCs)I=clflr)Ip-’

Leibniz’s

Theorem

for

Differentiation

of an Integral

3.3.7 b(c)

d &

s

a(c)

f

(2,

wx b(c)

3

a(cj

s

b ,J(x,c)dx+f@,

4

$--f

(a,

4

2

12

ELEMENTARY

Leibniz’s

Theorem

for

Differentiation

ANALYTICAL

The following

of a Product

3.3.8 s

METHODS

formulas

S

P(x)dx

n>l

is an integer.

where P(x)

(ux”+ fJx+cy

(t&g

u+(y)

g

g

+(g

ds

g

are useful for evaluating is a polynomial

and

3.3.16 +...+~)d~rg+...+cg

dx 2 (ax2+br+c)=(4c&c-bz)~

S dX -&=1g

3.3.9

(b2--4uc
a

2az+b-

3.3.17

I 2azfbf

#x -d2y dy -3 dy2=dr2 0 zc

3.3.10

$=

-B

g-3

Integration

3.3.12

(g--j

($)-"

-2

S

by Parts

S

3.3.20

jidu=w+dti

(a+ bx$c+dxj=k

jkudx=(jidx) Integrals

v-s(judx)

of Rational

(Integration

Algebraic

2 dx

dx 1 =- arctan E! u2+b2;C2 ub U

Functions

3.3.22

const,ants are omitted)

S(ux+f,)"dx=(ux+6)"+1 4n+1> S S S (n#-1)

3.3.24

3.3.15

$&In

)ax+b)

S

t(u+bx)

of Irrational

dx =h2 (c+dx)11’2

3.3.27

=+

3.3.28

=h2

3.3.29 3.3.30

(x2;1-“a2)2=&

arctan

~+20~(xf+U2)

3.3.25

Integrals

3.3.26

c+dx bc In I-a+bx I

dx (a+bx)“P(c+dx)=[d(bc~ud)]1~2

S

=[d(ad&]1/2

Algebraic

arctan arcsin

(ad # bc)

S___ lnb2+ b2x21 S S

3.3.21

3.3.23

3.3.14

(P-4ac=O)

3.3.19

S 3.8.13

(b2-4m>O)

=2az+b

3.3.18 3.3.11

(b2--4uc)t (P-4acy

Functions

-d(a+

1 W
bx) 1’2

C b(c+dx) 2bdx+ad+ bc-ad

bc >

ln J[bd(n+ bx)]1/2+ b(c+dx)1/21 arctan ~~~~-J” d(u+bx)1’2-[d(ad-bc)]1’2 In d(a+bx)1’2+[d(ud-bc)]“2

(b>O, dO)

(d(u&-bc)
I

@b-J--4>O)

ELEMENTARY

If zn=un+ivn, 3.7.23

then ~~+l:=u,+,+iv~+~

u,+~=xu~-~v,;

ANALYTICAL

where

3.8. Algebraic Solution

v,+,=xv,+yu,

9?z” and 92” are called harmonic

17

METHODS

polynomials.

3.7.24

Equations

of Quadratic

3.8.1

Given az2+ bz+c=O,

21,2=-

0-

2”, j-k

gf, p= b2-4ac, .zi+,za= -b/u,

3.7.25

If a>O, two real roots, p=O, two equal roots, a
Roots

3.7.26

z*=&=rte+rs=r+

cos @+iri

sin $0

Solution

If --?r<~< ?r this is the principal root. The other root has the opposite sign. The principa: root is given by 3.7.27

d=[+(r+x)]+&-i[$(r--x)]*=ufiv

where sign is taken tc

2uv=y and where the ambiguous be the same as the sign of y. 3 . 7 . 28

I

l&l-I221

I

Let sl=[r-+(q3+r2)q+,

Catwhy-Riemann

Equations

au -=-,

av

au -=--

av

a~

by

by

ax

;,

; f$=-$

a

If z=Tefff, 3.7.31

sz=[T’-((p3+?3*]*

then

f(z)=f(x+iy)=u(x,yy)+iv(x,y)whereu(x,y),v(z,y aA real, is unaly& at those points z=z+$ which 3.7.30

Equations

one real root and a pair of complex c.onjugate roots, $+9=0, all roots real and at least two are equal, . p3+r2<0, all roots real (irreducible case).

_<1z1~~2111211+I~21

Functions,

roots.

$+P>O,

root if - ?r<0 5 7r) (k=l,2,3, a . ., n-1)

Inequalities

Complex

If

of Cubic

Z~Z~=C/U

Given Z3+a2z2+ulz+a0==0, let

3.8.2

Zl/n,Tl/nefe/n , (principal

Other roots are Pet(B+2rn’ln

3.7.29

Equations

If zl, z2, z3 are the roots of the cubic equation g=;

Z~+Z~+Z~=-CIC~

Laplace’s

~~~2+~$,+&~,:=~~

Equation

The functions U(X, y) and v(x, y) are callec harmonic functions and satisfy Laplace’s equation Cartesian

Polar

3.7.33

r ;

(r g)+$=r

of Quartic

Equations

Given 24+~323+a3~~+~1l~+ag=O, real root u1 of the cubic equation

3.8.3

Coordinates

r

i!?+$E~2+g2=o

3.7.32

Solution

U3

-

a2U2

i-

(~23

-

4ao)U

-

(a; j- Uoa$ - kW3)

find the =

and determine the four roots of the quartic solutions of the two quadratic equations

Coordinates

;

(r g+g=o

0

as

18

ELEMENTARY

ANALYTICAL

If all roots of the cubic equation arc real, USC the value of U, which gives real coefficients in the *quadratic equation and select signs so that if

METHODS Method

of Iteration

then

-a3,

4142==0.

Convergence

of

an

fcxk)

xk

2, . . .).

Approximution

Process

(n=l,

bk)

will converge quadratically to x=5: (if instead of the condition (2) above), (1) Monotonic convergence, f(zO)r’(zo) >0 and f’(s), j”(z) do not change sign in the interval (Q, t), or (2) Osdato y conwgence, f(xJf” (x0)<0 and f’(s), f”(z) do not change sign in the interval (x0, x1), xo<E<xl. Newton’s

3.9.6 x=N’l”

3.9.2 Let zl, z2, x3, . . . be an infinite sequence of approximations to a number f. Then, if 1%n+~-~I<&n-tlk,

Rule

f’

Then, if f’(z)>0 and the constants cn are negative and bounded, the sequence x,, converges monotonically to the root [. If c,,=c=constantO, then the process converges but not necessarily monotonically. of

Approximations

Methods

3.9.1 Let z=zl be an approximation to x=[ where f(t) =0 and both x1 and [ are in the interval a$r
Degree

of Successive

xk+l=

(n=l,

b.

If z=zk is an approximation to the solution z= I of f(z) =0 then the sequence

Comments

GI+1=G+C&n)

for aLzSb,

Newton’s

Approximation

General

Method

will

3.9.5

-&,

z1z2z3z4=ao.

Czj2,=u2,

3.9. Successive

Newton’s

z2 j2,2t=

IF’(s)J
(2) a<xo &‘F(~)~xo’~ -

If zl, z2, z3, z4 are the roots, z2 j=

Substitution)

3.9.4 The iteration scheme Q+~=F(z~) converge to a zero of z=F(z) if (1)

pl+p2=a2,plp2+pl+q2=a2,p~q2+p2q~=al,

(Successive

Method

Applied

to Real

nth

Roots

Given x”=N, if zk is an approximation then the sequence xk+l=-

;

[$i+(n-l)xk]

2, . . .) will converge quadratically

to a.

where A and k are independent of n, the sequence is said to have convergence of at most the kth degree (or order or index) to [. If k=l and A<1 the convergence is linear; if k=2 the convergence is quadratic. Regula

Falsi

(False

Position) Aitken’e

3.9.3 Given y=f(z) to find 5 such that f(.$)=o, choose ~0 and x1 such that f(rO) and f(zl) have opposite signs and compute x*=x,

-Hi

f,=

f 1~o-JoX, jlVfO

for

Acceleration

of

Sequences

3.9.7 If 2k, &+I, zri+2 are three successive iterates in p, sequence converging with an error which is approximately in geometric progression, then

* &=xk-

Then continue with x2 and either of x0 or x1 for which f(;ro) or j(zl) is of opposite sign to f(zl). Regula falsi is equivalent to inverse linear interpolation,

G-Process

(5k--k+1)*=;tk~k+2-2:+1.

A*&

is an improved

estimate

OGtk) then Z=s+O(P),

A*Xk

’

of x. Ix\<~.

In fact, if zk”x+*

ELEMENTARY

ANALYTICAL

METHODS

* 3.10.

Theorems

on Continued

Fractions

A,B,_l-A,-lB,=(-l)n-’

(4)

(5)

Definitions

kiI al;

For every n>O, j,=b,

1

claI

ClC&

c2c3a3

&I-lW%

c,bl+ czbz+ caba+ ’ * * c,b,’ (6)

l+b,+b,b,+

. . . +bzb3. . . b, bz =-- 1 l- b,+l-

=b,,+&e&.

..

. . . +;=-&

d+$+ If the number of terms is finite, j is called a ternlinating continued fraction. If the number of ternls is infinite, j is called an infinite cont’inued fraction and the terminating fraction

1 --a0

x+A ...l

_- b3 b,+l-

--&

I

. . . $yu

1

2

t(-1,n----5

aoGa2

aof

1

=-

aox

n1

n

_ . . . a, _- a12

___ al-x+

uo+

b, * ’ ‘--b,+l

I12-xf

%-1X

* . . +un-2

is called the nth convergent of j. A (2) If lim -A exists, the infinite continued fracIt-+- 88 If uf= 1 and the tion j is said to be convergent. bt are integers there is always convergence. Theorems (1)

If

and br are positive

at

then j2n<j2n+2,

fin-1 >f*n+, . (2)

If j.=+

n A,=b,A,-~+a,A,-2 Bn=bnBn-l+anBn-2

where A-1=l,

A,,=bo, B-1=0,

B,=l.

0

.2

.4

.6

.8

FIGURE 3.1 1 i

y:=xn* 2, 5.

*n=0,,5t 29 1,

Numerical 3.11. Example

using Table

Use and

Extension

1. Computti 3.1.

of the Tables

xl9 and x4’ for x=29

Methods Linear

I

3p=x9. x10 =6.10326 1248. 102’ = (1.25184 9008. 1036)2/29

in

Table

Repetition

for fourth

yields the same result.

gives

3.1

roots with

1 4 ~7~3+3(5.507144)-]=5.50714 [ .

= (1.45071 4598. 1013)(4.207072333. 1014) x4’= (x*4)2/x

interpolation

(919.826)“4-5.507144. By Newton’s method N=919.826,

3845

Thus,

~“~=5.50714 3845/10$=1.74151. 1796, ~-~“=zt/x=.18983

05683.

=5.40388 2547. lO6* 3.12. Example

2.

(9.19826)“‘=

Compute

x-3’4 for x=9.19826.

(919.826/100)1’4= (919.826)1’4/10t

Computing

Techniques

Example 3. Solve the quadratic equation x2- 18.2x+.056 given the coeflicients as 18.2 f .l, *see

page

II.

..

20

ELEMENTARY

.056f

.OOI.

ANALYTICAL

<

METHODS

Example 5. Solve the cubic equation x3- 18.12 -34.8=0. To use Newton’s method we first form the table of f(z)=23-1S.1r-34.8

From 3.8.1 the solution is

z=4(18.2f-[(18.2)2-4(.05B)]:) =3(18.2~[:J31]t)=3(18.2~18.~) = 18.1969, .OOJ

4” -43.2 f(x)

The smaller root may be obtained more accurately from * .05fi/18.1969= .0031& .OOOl. Example

5 6 7

Compute (-3 + .0076i)i.

4.

From 3.7.26, (-3+.0976i)~=u+iv Y r!y u=2G? I,-= (

.3 72.6 181.5

We obtain by linear inverse interpolation:

where

O-(-.3) 72.6-(-.3)=5’oo4’

x,=5+

*, j”= (t”+y’)t >

Thus

Using Newton’s method, f’(x) =3x2-

r=[(-3)2+(.0076)2]~=(9.00005776)~=3.00000

1

Ij= 3.00000 9627- (-3) 2

f=

9627

21 =zo-f&J/f’

.73205 2196

We note that the principal square root has been computed. Example

Solve the quartic equation

6.

~‘-2.37752

Into

Quadratic

(22 + p12 + qd w

by Inverse

00526 '

'

Repetition yields x1=5.00526 5097. Dividing f(x) by x-5.00526 5097 gives x2+5.00526 5097x i-6.95267 869 the zeros of which are -2.50263 2549 f.83036

8OOi.

We seek that value of y, for which y(nJ =O. Inverse interpolation in ~(a,) gives ~(a,) =O for pl -2.003. Then,

Factors

+ p2x + 92)

Interpolation

Starting with the trial value pI = 1 we compute successively

QI

42

p1= a’--am 42-pll

pz=an-p1

9. 4. 053 526

--2. 1. 093 543

2. 2

4. 115

-3.

- 1.. 284 165

106

QI

Inverse interpolation

-2.

550

172

between qI=2.2 :md thus,

- 2: E

Qz

P2

PI

4. 51706 4.,51684 4. 51661

7640 2260 6903

-2. -2. -2.

55259 55282 55306

257 851 447

17506 . 17530 . 17553

Y h)

765 358 955

.00078 . 00001 -. 00075

552 655 263

gives q,=2.00420 2152, and we get finally,

_. 2. 00420

4. 520

Inverse interpolation 2.003 gives ql=2.0041,

5. 383

. 729

2. 0041 2. 0042 2. 0043

P2

Ykll)

. 011

Y(Qd=ql-t92+p*P2 - a2

_____

::

171

~~----

2.003

q2=;

C--.07215 9936jz5 57.020048

I

4922x3+6.07350 5741.x’ -11.17938 023s+9.05265 5259=0.

Resolution

18.1 we get

(d

=5.004-

.0076 u=&=2(1.73205 21g6)=.00219 392926

PI

-

.

2152

Qz 4. 51683

-

PI 7410

-2.

55283

Y (Ql)

P2 358

17530

8659

-.

00000

0011

and pl=

4

ELEMENTARY Double

Precision

Multiplication Desk Calculator

and

Division

ANALYTICAL on

a

Example 7. MultiplyM=20243 97459 71664 32102 by m=69732 82428 43662 95023 on a 10X10X20 desk calculating machine. Let MO=20243 97459, Ml=71664 32102, mO= 69732 82428, ml=43662 95023. Then Mm= M0m0102’+ (Mom,+Mlmo) 101o+M~ml. (1) Multiply ,W1m1=31290 75681 96300 28346 and record the digits 96300 28346 appearing in positions 1 to 10 of the product dial. (2) Transfer the digits 31290 75681 from positions 11 to 20 of the product dial to positions 1 to 10 of the product dial. (3) Multiply cumulatively M,mo+Mom,+31290 75681=58812 67160 12663 25894 and record the digits 12663 25894 in positions 1 to 10. (4) Transfer the digits 58812 67160 from positions 11 to 20 to positions 1 t,0 10. (5) Multiply cumulatively Mom,+58812 67160 =14116 69523 40138 17612. The results as obtained are shown below, 9630028346 1266325894 14116695234013817612 141166952340138176121266325894963~?28346

If the product Mm is wanted to 20 digits, only the result obtained in step 5 need be recorded. Further, if the allowable error in the 20th place is a unit’, the operation MImI may be omitted. When either of the factors M or m contains less than 20 digits it is convenient to position the numbers as if they both had 20 digits. This multiplication process may be extended to any higher accuracy desired. 8. Divide N=14116 69523 40138 17612 by d=20243 97459 71664 32102. Method (1)--linear interpolation.

Method @)--If N and d are numbers each not more than 19 digits let N=N1+NolOQ, d=dI+ dolO where No and do contain 10 digits and N, and dl not more than 9 digits. Then N NolOQ+N, zs- 1 [.N-y] d=,lOQ+d, dolO Here N= 14116 69523 40138 1761, d=20243# 97459 71664 3210 No= 14116 69523, do=20243 97459, d,=71664 3210 (1) NodI= 10116 63378 42188 8830 (productdial). (2) (Nod,)/do=49973 55504 (quotient dial). (3) N- (N&/d,= 14116 69!;22 90164 62106 (product dial). (4) [N- (NodI)/do]/dolOQ= .69732 82428=first 10 digits of quotient in quotient dial. Remainder =r=O8839 11654, in positions 1 to 10 of product dial. (5) r/(d010Q)=.43662 9502.10-“O=next 9 digits of quotient. N/d=.69732 82428 43662 9502. This method may be modified to give the quotient of 20 digit numbers. Method (1) may be extended to quotients of numbers containing more than 20 digits by employing higher order interpolation. 9. Sum the series S= l-&+*-i to 5D using the Euler transform. + The sum of the first 8 terms is .634524 to 6D. If u,=ljn we get Example

n 9

%7 . 111111

10

. 100000

Difference X.71664 32102=24685 64402&10-*O (note this is an 11 X 10 multiplication). Quotient= (69732 82430 90519 39054-246856 44028).10-20 =.69732 82428 43662 95028 There is an error of 3 units in the 20th place due to neglect of the contribution from second differences.

A*u,

Au,

A3u,

A%,,

-11111

Example

N/20243 97459.101’= .69732 82430 90519 39054 N/20243 97460.10”= .69732 82427 46057 26941 Difference=3 44462 12113.

21

METHODS

2020 -505

-9091 11

. 090909 -7576

12

156

1515

. 083333

-349 1166

-6410 13

. 076923

From 3.6.27 we then obtain SC 634524+.111111 -_ 2

(-.011111)+.002020 23 22 -(-- .000505) +.000156 24 26

= .634524+ .055556+ .002778+ .000253 -+ .000032+ .000005 = .693148 (S=ln 2=.6931472 to 7D).

22

ELEMENTARY

Example

10.

Evaluate

s0

- F =&

dx=g

s,.

$?jk-‘=glI k-‘+l& (k+10)-2

transform.

(Icfl)= y

s,’ sin;;;;t)

METHODS

m sin 2 dx s Cl J:

the integral

=- G to 4D using the Euler

ANALYTICAL

dx 1 +jY&p-

dt+%

Evaluating the integrals in numerical integration we get

(-l)f

g

dt.

where f(k)

the last

sum

by

$, k-2=1.54976 s

= (k+10)-2.

...

Thus,

7731+.1

- .005 + .00016 6667 - .OOOOO0333 = 1.64493 4065,

Ic

as compared

1.85194

Example

.43379

12.

A

A2

A3

A4

4067.

Compute

arctanx=$3+g7+

.25661 . 18260

with $=1.64493

to 5D for x= .2. Here n>l, &,=O, b,=2n-1,

x2 4x2 9x2 al=x, A-l=l,

. . .

an=(n-l)2x2 for Bdl=O, A,,=O,

. 14180

-2587 .11593

799 - 1788

-321

478

.09805

-1310

- 168

.08495

A0 -= Bo ’

153

310 - 1000

A -r,*g

.07495 The sum to k=3 is 1.49216. Applying Euler transform to the remainder we obtain

Bl

the

A=.197368

f (.14180)-h

(-.02587)+&

B2

(.00799)

A3

-;

(-.00321)+$

= .07090 + .00647 + .00100+ .00020 + .00005 = .07862 We obtain the value of the integral compared with 1.57080. Example

11.

as 1.57018 as

Sum the series $I kep==f P the Euler-Maclaurin summation formula. From 3.6.28 we have for n= a,

B=.197396

(.00153)

using

3

[II A4

3.032

Bq = 15.36

Note that in carrying out the recurrence method for computing continued fractions the numerators A, and the denominators B, must be used as originally computed. The numerators and denominators obtained by reducing An/B, to lower terms must not be used.

ELEMENTARY

ANALYTICAL

23

METHODS

References Texts

[3.1] R. A. Buckingham, Numerical methods (Pitman Publishing Corp., New York, N.Y., 1957). [3.2] T. Fort, Finite differences (Clarendon Press, Oxford, England, 1948). [3.3] L. Fox, The use and construction of mathematical tables, Mathematical Tables, vol. 1, National Physical Laboratory (Her Majesty’s Stationery Office, London, England, 1956). [3.4] G. H. Hardy, A course of pure mathematics, 9th ed. (Cambridge Univ. Press, Cambridge, England, and The Macmillan Co., New York, N.Y., 1947). [3.5] D. R. Hartree, Numerical analysis (Clarendon Press, Oxford, England, 1952). [3.6] F. B. Hildebrand, Introduction to numerical analysis (McGraw-Hill Book Co., Inc., New York, N.Y., 1956). [3.7] A. S. Householder, Principles of numerical analysis (McGraw-Hill Book Co., Inc., New York, N.Y., 1953). [3.8] L. V. Kantorowitsch and V. I. Krylow, Naherungsmethoden der Hoheren Analysis (VEB Deutscher Verlag der Wissenschaften, Berlin, Germany, 1956; translated from Russian, Moscow, U.S.S.R., 1952). [3.9] K. Knopp, Theory and application of infinite series (Blackie and Son, Ltd., London, England, 1951). [3.10] Z. Kopal, Numerical analysis (John Wiley & Sons, Inc., New York, N.Y., 1955). [3.11] G. Kowalewski, Interpolation und genaherte Quadratur (B. G. Teubner, Leipzig, Germany, 1932). [3.12] K. S. Kuns, Numerical analysis (McGraw-Hill Book Co., Inc., New York, N.Y., 1957). [3.13] C. Lanczos, Applied analysis (Prentice-Hall, Inc., Englewood Cliffs, N.J., 1956). [3.14] I. M. Longman, Note on a method for computing infinite integrals of oscillatory functions, Proc. Cambridge Philos. Sot. 52, 764 (1956). [3.15] S. E. Mikeladze, Numerical methods of mathematical analysis (Russian) (Gos. Izdat. TehnTeor. Lit., Moscow, U.S.S.R., 1953). [3.16] W. E. Milne, Numerical calculus (Princeton Univ. Press, Princeton, N.J., 1949). [3.17] L. M. Milne-Thomson, The calculus of finite differences (Macmillan and Co., Ltd., London, England, 1951).

[3.18] H. [3.19]

(3.201 [3.21]

[3.22] [3.23]

[3.24] [3.25]

Mineur, Techniques de calcul numerique (Librairie Polytechnique Ch. B&anger, Paris, France, 1952). National Physical Laboratory, Modern computing methods, Notes on Applied1 Science No. 16 (Her Majesty’s Stationery Office, London, England, 1957). J. B. Rosser, Transformations to speed the convergence of series, J. Research NBS 46, 56-64 (1951). J. B. Scarborough, Numerical mathematical analysis, 3d ed. (The Johns Hopkins Press, Baltimore, Md.; Oxford Univ. Press, London, England, 1955). J. F. Steffensen, Interpolation (Chelsea Publishing Co., New York, N.Y., 1950). H. S. Wall, Analytic theory of continued fractions (D. Van Nostrand Co., Inc., New York, N.Y., 1948). E. T. Whittaker and G. Robinson, The calculus of observations, 4th ed. (Blackie and Son, Ltd., London, England, 1944). R. Zurmtihl, Praktische M.athematik (SpringerVerlag, Berlin, Germany, 1953).

Mathematical

Tables

and

Collections

of Formulas

[3.26] E. P. Adams, Smithsonian mathematical formulae and tables of elliptic functions, 3d reprint (The Smithsonian Institution, Wa,shington, D.C., 1957). [3.27] L. J. Comrie, Barlow’s tables of squares, cubes, square roots, cube roots a,nd reciprocals of all integers up to 12,500 (Chelmical Publishing Co., Inc., New York, N.Y., 1954). [3.28] H. B. Dwight, Tables of integrals and other mathematical data, 3d ed. (The Macmillan Co., New York, N.Y., 1957). [3.29] Gt. Britain H.M. Nautical Almanac Office, Interpolation and allied tables (Her Majesty’s Stationery Office, London, England, 1956). [3.30] B. 0. Peirce, A short table of integrals, 4th ed. (Ginn and Co., Boston, Mass., 1956). [3.31] G. Schulz, Formelsammlung zur praktischen Mathematik (de Gruyter and Co., Berlin, Germany, 1945).

24

ELEMENTARY

ANALYTICAL

POWERS

Table 3.1

AND ROOTS nk

k : 3 4 : 7 fz

10 24

nl=

for use

See Examples 1-5

T&L n3, n‘L 5 12= .L

16

n7==

128

of the table. Floating decimal notation: 910=3486784401 = (9)3.4867 84401

s 8

l/3 l/4 l/5

1024

167

77216

&L

1.4142 13562

nl/3= .&4= 7$1/5=

1.2599 I.1892 1.1486

2 i 190 24

l/2 l/3 l/4 l/5

(16)5.9604

64478

2.2360 1.7099 1.4953

66176

1.8171 20593 1.5650 84580

1.9158 2.6457 1.9129 1.6265

12314 51311 31183 76562

1.4309

1.4757

73162

(18)4.7383

81338

2.4494

1.3797 29662

24

w v3 l/4 v5

10000 00000 00000 00000 00000 00000 00000 00000

3.1622 2.1544

77660 34690

1.5848

69081

93192

1 17 194 2143 9)2.3579

( (10)2.5937

2:: 3375 50625 7 59375 113 90625 1708 59375 11)5.7665 24

l/2 l/3 l/4 l/5

(28)1.6834

79

85984

358 4299 9)5.1597

31808 81696 80352

10)6.1917 (25)7.9496

84720

3.4641 2.2894 1.8612 1.6437

01615 28485 09718 51830

14 -.241 41U3 9j6.9757

2% 4913 83521 19857 _37569-38673 57441

16 256 4096 65536

10 48576 167 77216

36422

35456 67296 47674 11628

(llj1.1858

03906

2684 914.2949 (lOj6;8719 (12)1.0995

(12)2.0159

93900

11220

(28)7.9228

16251

(29)3.3944

86713

3.8729 2.4662

83346 12074

1.7187

71928

1.9679 89671

00000

4.0000 2.5198 2.0000 1.7411

42100

00000

01127

1.4142 1.3195

1342 17728

(21)4.7223 2.8284 2.0000 1.6017 1.5157

I

4.1231 2.5712 2.0305

78765

05626 81591 43185

1.7623 40348

13562 07911

9

7%

6561 59049 5 31441 47 82969 430 46721 3874 20489 9)3.4867 84401

41824

(

66483

(22)7.9766

27125

00000

92831 16567

44308

00000

3.0000 2.0800 1.7320 1.5518

83823 50808 45574

13

1’946

169 --_

2197 28561 3 71293 48 26809 627 48517 8157 30721

-. __._

24790 80091 60287 94266

I

49570 74013 30940

9)1.0737

1728 20736 2 48832

61051 71561 87171 58881 47691 32676

3.3166 2.2239 1.8211 1.6153

(

12

1:: 1331 14641

42460

(24)9.8497

1.7782 79410

(20)

89743

1’0: 1000

1 10 100 1000 ( 9 11.0000 (10 1.0000 (24) 1.0000

23543 64801 53607 75249

49167

00000 1.5874 01052 2.0000

516; 4096 32168 2 62144 20 97152 167 77216

1 17649 8 57 403 2824

(14)2.8147

8

3:; 2401 16807

100 77696

61977 75947 48781

1.4422 1.3160 1.2457 7

2:: 1296 7716 46656 2 79936 16 79616

604

1024 4096 16384 65536 2 62144 10 48516

(11)2.8242 95365 1.7320 50808

21050 07115 98355

6 2: 125 625 3125 15625 78125 3 90625 19 53125 97 65625

: 3 4

2:; 729 2187 6561 19683 59049

25%

.lO= $4,

27

2:

nL ngss=

l/2

METHODS

10)1.0604 11)1.3785

2744 38416 5 37824 75 29536 1054 13504

49937 84918

(26)5.4280

07704

3.6055 2.3513

51275 34688

1.6702

71652

(27)3.2141

1.8988 28922

1 18 340 6122

4.2426 2.6207 2.0597 1.7826

3:: 5832 04976 89568 12224 20032

58845

1

24 470 8938

(30)4.8987

40687 41394 67144 02458

57381 42264

1.9343 36420 1.6952 18203

10)1.1019 96058 11 1.9835 92904 12 1 3.5704 67227 (30)1.3382

99700

3.7416 2.4101

6859 30321 76099 45881 71739

62931

4.3588 2.6684 2.0877 1.8019

98944 01649 97630 83127

23

1

24

w l/3 l/4 l/5

42000 8000 60000

1 94481

32 640 9) 1.2800 10)2.5600 (11)5.1200 (13)1.0240

00000 00000 00000 00000 00000 00000

( 9)1.8010 (10)3.7822 (11)7.9428 (13)1.6679

(31)1.6777

21600

(31)5.4108

4.4721 2.7144 2.1147 1.8205

35955 17617 42527 64203

4:: 10648 2 34256 51 53632 1133 79904

441 9261

40 857

4.5825 2.7589 2.1406

84101 66121 88541

5:: 13824 3 31776 79 62624 1911 02976

85936 00466 88098

19838 75695 24176 95143

1.8384 16287

(32)1.6525 4.6904 2.8020 2.1657

10926 15760 39331 36771

1.8556 00736

(32)4.8025 4.7958 2.8438 2.1899 1.8721

07640 31523 66980 38703 71231

(33)1.3337 4.8989 2.0844 2.2133 1.8881

35777 79486 99141 63839 75023

ELEMENTARY

ANALYTICAL

POWERS

6::

(33)3.5527 13679 5.0000 00000 2.9240 17738 2.2360 67977 1.9036 53939

26 676 17576 4 56976 118 81376 3089 15776 9 8.0318 10176 11 2.0882 70646 12 5.4295 03679 14 I 1.4116 70957 33)9.1066 85770 5.0990 19514 2.9624 96068 2.2581 00864 1.9186 45192

9:: 27000 6 10000 243 00000 7290 00000 10 2.1870 00000 11 6.5610 00000 13 1.9683 00000 Ii 14 5.9049 00000 (35)2.8242 95365 5.4772 25575 3.1072 32506 2.3403‘47319 1.9743 50486

31 961 29791 23521 29151 03681 61411 10374 62216 82870 26610 64363 80652 11062 40755

15625 3 90625 97 65625 2441 40625

35 1225 42875 15 00625 525 21075

24

l/2 l/3 l/4 l/5

15)2.7585 47354 (37)1.1419 13124 5.9160 79783 3.2710 66310 2.4322 99279 2.0361 68005 40 1600 64000 25 60000 1024 00000

24

l/2 l/3 l/4 l/5

6 7

9”

10 24

l/2 l/3 l/4 l/5

(16)1.0485 (38)2.8147 6.3245 3.4199 2.5148 2.0912

76000 49767 55320 51893 66859 79105

45 2025 91125 41 00625 1845 28125 ( 9)8.3037 65625 (11)3.7366 94531 (13 1.6815 12539 (14 7.5668 06426 (I6 I 3.4050 62892 (39)4.7544 50505 6.7082 03932 3.5568 93304 2.5900 20064 2.1411 27368

9 286 0675 10 2.7512 11 8.5289 13 2.6439 14 8.1962 (35)6.2041 5.5677 3.1413 2.3596 1.9873

II

46656 16 79616 604 66176

15)3.6561 (37)2.2452 6.0000 3.3019 2.4494 2.0476

28 1158 9)4.7501 11)1.9475 12)7.9849 (38)5.0911 6.4031 3.4482 2.5304 2.1016

58440 25771 00000 27249 89743 72511 16G 68921 25761 56201 04241 42739 25229 10945 24237 17240 39534 32478

25

METHODS

AND ROOT!3

nk

Table

21

5 143 3674 10 1.0460 11 2.8242 12 7.6255 14 I 2.0589 34)2.2528 5.1961 3.0000 2.2795 1.9331

729 19683 31441 48907 20489 35320 95365 97405 11321 39954 52423 00000 07057 82045

32768 10 48576 335 54432

(36)1.3292 5.6560 3~1748 2.3704 2.0000

27996 54249 02104 14230 00000

37 1369 50653 18 74161 693 43957

(37)4.3335 6.0827 3.3322 2.4663 2.0589

25711 62530 21852 25715 24137

17:: 74088 31 11696 1306 91232

(38)9.0778 6.4807 3.4760 2.5457 2.1117

49315 40698 26645 29895 85765

1 48 2293 (10 1.0779 11 5.0662 13 2.3811 15 1.1191 ii16 5.2599 (40)1.3500 6.8556 3.6088 2.6183 2.1598

22d9 03823 79681 45007 21533 31205 28666 30473 13224 46075 54600 26080 30499 30012

6 172 4818 10 1.3492 11 3.7780 13 1.0578 II 14 2.9619 (34)5.3925 5.2915 3.0365 2.3003 1.9472

7:: 21952 14656 10368 90304 92851 19983 45595 61661 32264 02622 88972 26634 94361

10;; 35937 11 85921 391 35393 9 1.2914'67969 '10 4.2618 44298 '12 1.4064 08618 113I 4.6411 48440 15)1.5315 78985 (36)2.7818 55434 5.7445 62647 3.2075 34330 2.3967 El727 2.0123 46617 30 1444 54672 20 85136 792 35168 9 3.0109 36384 11 1.1441 55826 ~12 4.3477 92138 14 1.6521 61013 15 I 6.2782 11848 (37)8.2187 60383 6.1644 14003 3.3619 75407 2.4828 23796 2.0699 35054

34 1470 9 6.3213 11 I 2.7181 13 1.1688 14 1 5.0259 16)2.1611 39)1.5967 6.5574 3.5033 2.5607 2.1217

18:; 79507 18801 08443 63049 86111 20028 26119 48231 72093 38524 98060 49602 47461

1 53 2548 10)1.2230 11 5.8706 13 12.8179 1.3526

23:: 10592 08416 03968 59046 83423 28043 05461

40)2.2376 6.9282 3.6342 2.6321 2.1689

37322 03230 41186 48026 43542

8::

24389 7 07281 2105 11149 5948 23321

(35)1.2518 5.3851 3.0723 2.32!05 1.9610

49008 64807 16826 95787 09057

34 1156 39304 13 36336 454 35424 9)1.5448 04416

(36)5.

6950 5.8309 3.2396 2.41147 2.0;!43

03680 51895 11801 36403 97459 19

15% 59319 23 13441 902 24199

(38)1.5330 6.2449 3. 3912 2.4989 2.0807

29700 97998 11443 99399 16549

19;: 85184 37 48096 1649 16224

(39)2.7'724 6.6332 3.5303 2.5'755 2.1'315

53276 49581 48335 09577 25513

47

21;: 97336 44 77456 2059 62976

(39)8.0572 6.7823 3.5830 2.6042 2.1505

70802 29983 47871 90687 60013

_ _._..

57 64801

(40)3.6‘703 7.0000 3.6!593 2.6457 2.1'779

The numbers in square brackets at the bottom of the page mean that the maximum error in a linear interpolate is a X 10-P (p in parentheses), and that to interpolate to the full tabular accuracy 11~ poJnts must be used in Lagrange's and Aitkens methods for the respective functions W'. *See page xx.

3.1

36822 00000 05710 51311 06425

26

ELEMENTARY

Table

3.1

ANALYTICAL

POWERS

AND ROOTS

k : : 2 7 t 10 24

l/2 l/3 :::

24

l/2 l/3 l/4 l/5

1 62 3125 10 1.5625 11 7.8125 13 3.9062 15 1.9531 16 I 9.7656 (40)5.9604 1.0710

3.6840 2.6591 2.1867

25:: 25600 50000 00000 00000 00000 50000 25000 25000 64478 67812 31499 47948 24148

30:: 1 66375 91 50625 5032 84375 10 2.7680 64063 12 1.5224 35234 '13 8.3733 93789 '15 I 4.6053 66584 :17)2.5329 51621 (41)5.8708 98173 7.4161 98487 3.8029 52461 2.7232 69815 2.2288 07384

1 67 3450 10 I 1.7596 11 8.9741 13 4.5167 1.1904 40)9.5870 7.1414 3.1084 2.6723 2.1954

7176

24

l/2 l/3 l/4 l/5

24

l/2 l/3 l/4 l/5

(42)4.7383 7.7459 3.9148 2.7831 2.2679

45118 01897

67858

1.4833

14774

3.8258 2.7355 2.2368

62366 64800 53829

60 3600 2 16000 129 60000 10)4.6656

29769

48342 02551 11157 49614 44515

1.7488

74104

(41i2.4133 7.2801 3.1562 2.6981 2.2123

53110 09889

24

l/2 l/3 l/4 l/5

(44)1.9158 8.3666 4.1212 2.8925 2.3389

12314 00265 85300 07608 42837

.1[(-,,9]

67876 56822

(41)3.7796 7.3484 3.7197 2.7108 2.2206

38253 69228 63150 06011 43035

57

(42)1.3835 7.5498 3.8485 2.1476 2.2447

55344 34435 01131 96205

86134

34:: 2 05379 121 17361 7149 24299

(42)2.1002 7.6157 3.8708 2.7596 2.2526

54121 73106 76641 69021 07878

(42)3.1655 7.6811 3.8929 2.7714 2.2603

43453 45748 96416 88002 22470 64

(42)

7.0455

68477

7.8102

49676

3.9364

97183

2.7946 2.2754

(43)

82393 43032

1.0408 7.8740 3.9578

2.8060 2.2828

66 4356 2 81496

189

(43)4.6671

8.1240 4.0412 2.8502 2.3115

79722 07874

; 43; 1.5281

75339

91610 66263 55056

7.9372

53933

3.9790 2.8173 2.2901

57208 13241 72049

44:; 3 00763

74736

78950 38405 40021 69883 79249

39% 2 50047 157 52961 9974 36547._ .._. ___

38:: 2 38328 147 76336 9161 32832

201

(43)6.6956 8.1853 4.0615

2.8610 2.3185

88867

(43)9.5546

52172

8.2462

48100 05553 41963

4.081'6 2.8716 2.3254

254 ( 9 I 1.8042 (11 1.2810 I 14 12 9.0951 6.4515 (16 I 4.5848 (18 3.2552 (44)2.6927 8.4261 4.1408 2.9027 2.3455

11681 29351 02839

20158 35312 50072 43551 76876 49773 17749 83108 81669

,i[(-;)4]

3 73248 268 13856

( 9I (11 13 14

1.9349 1.3931 1.0030 1.2220

17632 40695 61300 41363

[ :86{ :: :I;: (44)3.7668 8.4852 4.1601 2.9129 2.3521

i:::: 63772 81374 67646 50630 58045

(43)2.2300 8.0000 4.0000 2.8284 2.2973

30685 11251 55102 21711 22030

(44)1.3563 8.3066 4.1015 2.8821 2.3322

70007 23863 65930 21417 21626

54:: 4 05224 299 86576

3 89017 283 98241 -71593 9 2.0730

11 1.5133 13 1.1047 14 I 8.0646

74520 00000 00000 27125 96710

4766; 3 28509 226 67121

73 5329

5184

50:: 3 57911

40% 2 62144 167 77216

462648 3 14432 213 81376

51121

72

49:: 3 43000 240 10000

85154

54 2916 1 51464 85 03056 4591 65024

3244 1 85193 105 56001 6016 92057

00000

4226: 2 74625 178 50625 ( 9)1.1602 90625 10)7.5418 89063 12)4.9022 27891 3.1864 48129 I ('l8)1.3462 74334 (43)3.2353 44710 8.0622 51148 4.0207 25159 2.8394 11514 2.3045 31620

53 2809 48877 90481 95493 36113 11140 69041 63592

il17 (41)1.5278 7.2111 3.7325 2.6853 2.2039

37;: 2 26981 138 45841 8445 96301

00000

81338 66692 67641 57684 33155

1 78 4181 2.2164 1.1747 6.2259 3.2997

10 12 13 15

94451

24238 33090 28429

nk

52 2704 1 40608 73 11616 3802 04032

26;: 32651 65201 25251 28780 06779

56 3136 1 75616 98 34496 5507 31776

(41)9.0471

METHODS

42263 39852 00919

16 5 2976 1814: 8871 58671 25830

;44;5.2450

,i[(-55131

8.5440 4.1793 2.9230 2.3586

38047 03745 39196 12786 55818

(44)7.2704 8.6023 4.1983 2.9329 2.3650

“:[(-55)2]

49690 25261 36454 72088 82169

75 5625 4 21875 316

l/2 l/3 l/4 l/5

(45)1.0033 8.6602 4.2171 2.9428 2.3714

ANALYTICAL

POWERS

AND ROOTS

76 5776 4 38976 333 62176

40625

91278 54038 63326 30956 40610

ELEMENTARY

(45)1.3788

79182

8.7177

97887

4.2358 2.0525 2.3777

23584 91724 30992

nk

Table

59:; 4 56533 351 53041

(45)1.8870 8.7749 4.2543 2.9622 2.3839

23915 64387 20865

6078: 4'74552 370 15056

55503

8.8317 4.:2726 2.'3718 2. 3901

67;: 5 51368 452 12176

474

56638

27

METHO:DS

60866

58682 27866 15677

3.1

62:; 4 93039 389 50081

(45)3.4918 8.8881 4;2908 2.9813 2.3962

06676

94417 40427 07501 12991

81 b56i

6480:

40:

l/2 l/3 l/4 l/5

(45)4.7223 8.9442 4.3088 2.9906 2.4022

522

24

(19)1.2157

66483

(45)6.X626

71747

(46)2.6789 9.:!736 4.41140 3.0452 2.4372

8190:

9 11 I 13 15 24

l/2 l/3 l/4 l/5

24

l/2 l/3 l/4 l/5

65: 5.9049 5.3144 4.7829 4.3046

46)7.9766 9.4868 4.4814 3.0800 2.4595

8 814 9 7.7378 11 I 7.3509 13 6.9633 15 I b.6342 17 6.3024 19 5.9873 (47)2.919.3 9.7467 4.5629 3.1219 2.4862

(45) a.5414

66801

9.0553

85138 81486

6 36056 547 00816

00625

44457 29672 70277 53252

85441

00000 48711 00000 24685 86 7396

72;: 6 14125

39031 18495 04962 61646

47818

29000

82:: 53571

10000 00000

74961 21451

10000

92520 10194 25276 98001 6118i 04400 92014

69000

72100 04890 84401 44308

(47)

32981 04747 70288 09486

1.0399 9.!;393 4.4979 3.0885 2.4649

41445

90619 50932

96 9216 8 04736

90;: 57375 50625 09375 18906 72961

849

34656

02635

85641 49570

(19)6.6483 (47)3. 7541 9.?979 4.!,788 3.I.301 2.4914

4-55151

70::

5 92704 497 87136

4.3444 3.0092 2.4141

16698 41771 87

7569 6 58503 572 89761 9)4.9842 09207

(46)3.5355 9.3273 4.4310 3.0540 2.4428

91351 79053 47622 75810 89656

92 8464 7 78688 716 39296

(46)1.1425 9. 1104 4. 3620 3. ~0183 2A200

a5726 63047 57435 41015 44749

97 9P"9 .-, 9 12673 885 29281

26360 32467 58971 56970 69160 61879

4-y]

(17)7.6023 (19)7.3742 (47)4.8141 9.8488 4.5947 3.1382 2.4966

10587 41269 72219 57802 00892 a8993 30932 1

;[(-y]

(46)1.5230 9.1651 4.3795 3.0274 2.4258

(46)4.6514 9.:3808 4.4479 3.13628 2.4484

04745 31520 60181 14314 79851

(46)6.1004 9.4339 4.4647 3.0714 2.4540

25945 81132 45096 78656

19455 94 8836

86:; a 04357 748 05201 9)6.9568 83693 11 6.4699 01834 13 6.0170 08706 17 5.2041 19 4.8398 47)1:7522 9.6436 4. 5306 3.1054 2.4756

10388 51390 19140 00104 04834

89 7921 7 04969 627 42241

.,

15 5.!5958

(47)1.3517 9.5916 4.5143 3.0970 2.4703

47375 33579 70671 49479 01407

88 -I"44 6 81472 599 69536

a 30584 780 74896

18097

10830 23072 28603 5Oi'bl 54896 22799 91866

9d48 9 41192 922 36816 9)9.0392 07868

(47)2.2650 9.6953 4.5468 3.1137 2.4809

960

01461 59715 35944 37258 93182 98:; 9 70299 59601

11 8.8584 13 1 8.6812 15)8.5076 8.3374

23E109 55932 30226 77621

I 119 I 9.5099 9.4148 I 13 15 9.3206 9.2274

00499 01494 46944 53479

(47)6.1578 9.8994 4.0104 3.1463 2.5017

03265 945'37

(47)7.8567 9.9498 4.6260 3.1543 2.5068

81408 74371 65009 42146 42442

I::{;:z: 2383%

04313

94097 69392 90243 94345

611:;

5 71787 58321

66546

9.0000 4.3267 3.0000 2.4082

97562 48868

9.2195 4.3968 3.0363 2.4315

l/3 l/4 l/5

5 31441 430 46721

71910 69380

(46)2.0232

l/2

12000 60000 00000 40000 52000 21600 77280 41824

36292

46284 57527

,,a[,-;)91

28

ELEMENTARY

Table

ANALYTICAL

POWERS

3.1

METHODS

AND ROOTS

nk

k

101 10201 10 30301 1040 60401

: :

5

103 10609

102 10404 10 61200 1002 43216

10 92121 1125 50001

1nr __.

10816 11 24864 1169 85856

7" : 10 24

l/2 l/3 l/4 l/5

(48)1.0000 ( 1)1.0000 4.6415 3.1622 2.5110

00000 00000 00034 11660 66432

48) 1.2691

2.5168

105 11025 11 57625 1215 50625

24

l/2 l/3 l/4 l/5

Iii10 14 1.2762 12 1.3400 1.4071

95641 00423 81563

16 1.4774 18 1.5513 20 1.6280 (48)3.2250 ( 1)1.0246 4.7116 3.2010 2.5365

55444 28216

24 l/3 l/4 l/5

(48)9.8497 ( 1)1.0408 4.7914 3.2385 2.5602

(4Q4.0489 ( 1)1.0295 4.1326 3.2086 2.5413

l/2

l/3 l/4 l/5

17 2073 10)2.4883 12)2.9059

24

w

l/3 l/4 l/5

(20)6.1917 (49)7.9496 ( 1)1.0954 4.9324 3.3097 2.6051

4.6123 3.1719 2.5218

20720 71028 54546

(48)2.0327 1)1.0140 4.6075 3.1857 2.5267

107 11449 12 25043 1310 79601

32676 08048 19857 31040 21376

17619 80529 44131 22111

90170

34641 63014 23491 80436 30642

(48)5.0723 ( 1)1.0344 4.1414 3.2162 2.5461

111 12321 13 67631 1518 07041

(49)1.2239 ( 1) 1.0535 4.0058 3.2458 2.5640

115 13225

(49)2.0625 ( 1)1.0723 4.0629 3.2147 2.5830

37249

15658 65375 95534 67180 65499

__._-

120 14400 20000 60000 20000 04000

lb 1873

(49)4.3297 ( 1)1.0016 4.8909 3.2008 2.5920

14641 17 71561 2143 50001

22::

36422 04720

45115 24149 50920 71085

(49)9.7017 ( 1)1.1000 4.9460

23378 07443 00000

3.3166 2.6094

24790 90635

11”[(-45)3]

,@‘]

59396

21453 07613

b2893 f3;;:

3:2531 2.5694

15 60096 1810 63936

20)4.4114 35079 (49)3.5236 41704 ( 1)1.0770 32961 4.0769 90961 3.2810 18035 2.5075,66964

66953 00043

112 12544 14 04928 1573 51936

(49)1.5170 ( 1):';;;;

(50)1.1820 ( 1y;; 3:3234 2.6137

53123 70314

56186 97660

03156

09795 49001

lb30

41361

(49)1.8708 ( 1)1.0630 4.8345 3.2603 2.5740

09051 14581 00127 90439 42354

01613 88721

122 14884 15048 33456 00163 03959 10830 07213 02000 31415 50242 ;;a;;

108 11664 59712 40896 28077 74323 24269 30210 04627 24997 80737 30405

lb 1930

(49)5.3109 ( l)l.O862 4.9040 3.2950 2.5964

118 13924 43032 71716

00621

70049 60131 73252 20703

123 15129 18 60061 2280 06641 I I (50)1.4370 ( 1)1.1090 4.9731 3.3302 2.6100

,4-;V3]

80104 53651 89033 45713 68602

46q6)5]

04165 03903 69375 36868 67508

40)2.5633

1)1.0190 4.7026 3.1934 2.5316

109 11801 12 95029 1411 58161

(48)7.9110 ( 1)1.0440 4.7768 3.2311 2.5555

113 12769 14 42097

__-_,

20675 65303 73246 68168 12982

94106 89157 40148 32501 80003

12 1360 10 1.4693 12 1.5868 14 1.7138 16 1.8509 18 I 1.9990 20)2.1589 4Q6.3411 1)1.0392 4.lb22 3.2237 2.5500

94627

99944 95077 93980 85873 17482

15 20075 1749 00625

24

90229

(48)1.6084

106 11236 11 91016 1262 41696

110 12100 13 31000 1464 10000

w

34649

83175 30651 56181 46315 55391 114 12996 14 01544

(49)2.3212 ( 1)1.0677 4.0480 3.2675 2.5705

lb 2005

(49)6.5031 ( 1)1.0900 4.9106 3.3028 2.6000

20685 07825 07586 79071

02140 119 14161 85159 33921

99444 71211 64734 33952 14507 124 15376

19 2364 10 2.9316 12 3.6352 14 4.5076 16 I 5.5095 10)6.9309 (20)8.5944 (50)1.7463 ( 1)1.1135 4.9066 3.3369 2.6223

06624

21376 25062 15077 66696 06703 80312 25506 06393

52073 30952 93965 11047

ELEMENTARY

ANALYTICAL

POWERS 125 15625 19 53125 2441 40625

10 3.0517 12 3.8146 14 4.7683 16 I 5.9604 18) 7.4505

57813

(10 1 3.1757 (12 4.0015

96938 04141

(14 I 5.0418

95218

20)9.3132

91266 71582 64478 80597 25746

24

50)2.1175

82368

(50)2.5638

52774

l/2

1)1.1180

33989

( 1)1.1224

97216

5.0000 3.3431 2.6265

l/3 l/4 l/5

(16

6.3527

(18 (21)

8.0045

30000 09000 51700 07210

1.3785

84918

131

(50)5.4280

07704

(50)6.

( 1)1.1401

75425

( 1)1.1445

97019 48375

5.0787 3.3831 2.6512

11681

24

1.3427 ( 1)1.1618

97252 95004

5.1299 3.4086 2.6672

27840 58099 68608

(51)

l/3 l/4 l/5

(10)5.3782

40000

(12)7.5295 (15)1.0541 (17) 1.4757

36000 35040

(19 24 l/2 l/3 l/4 l/5

( 1)1.1832 5.1924 3.4397 2.6867

1 2 3 4 5 6 7 8 1'0 24

l/2

l/3 l/4 l/5

2.0661

(21 I 2.8925 (51)3.2141

89056

51)1.6030

01028

1)1.1661

90379

99700

51)3.8129

15957 94102 90628 39790

( 1) 1.1874

145 21025 30 48625 4420 50625 0 6.4097 34063 2 1 9.2941 14391 15)1.3476 46587 17)1.9540 87551 19)2.8334 26948 21)4.1084 69075 51)7.4616 01544

( 1)1.1489

02016 87418 18888 25688 37894 59535 56968

19 2.2027 21)3.1059

04678 46550

5.2048 3.4459 2.6905

5.0916 3.3895 2.6553

(51)1.9111 ( 1)1.1704 5.1551 3.4212 2.6751

63181 52970 OR461 141 19881 03221 54161 83670 41975 84764

1)1.2041 5.2535 3.4701 2.7056

59458 87872 00082 62363

13625

(

0,1597 31428 67602 84058

1)1.2083 5.2656 3.4760 2.7093

Table 128 16384 20 97152 2684 35456

44882 69991 36735 13222 25206

19 2.3474 21 3.3333 (51) (

129 16641 21. 46689 276’) 22881

91621

(50)3.7414

44192

(50)4.5097

56022

( 1)1.1313

70850

( 1)1.135;'

81669

5.0396 3.3635 2.6390

23 3129 10)4.1615 12)5.5349 14 7.3614

84200 85661 15822

5.0527 3.37O:L 2.6431.

133 17689 52637 00721 79589 00854 18136

134 17956 24 3224

I 21)1.7318 19)1.3021 16 1 9.7906

86120 61254 74468

(50)9.3851

10346

( 1)1.1532

56259

4.3204 5.7893 7.7577 l.O39!j 1.3920 1.8665 (’ 51) 1.1233 1)1.1575

68722 62690 18337

5.1172 3.4023 2.6633

5.1044 3.3959 2.6593

74347 36005 26458

10 I 12 14 17 I 19 21

06104 17936

00342 36459 10855 33255 74561 85912 50184 83690 29947 28159 05339

139 19321 26 85619 373'3 01041

26 28072 3626 73936

(51)2.2756

11258

( 51)2.7061

70815

( 1)1.1747

34012

1)1.178')

82612

5.1676 3.4274 2.6790

49252 39296 19145

_-1

5.18O:l 3.4336 2.6828

01467 31623 90577

144 20736 29 85984 4299 81696

.I

29 24207 4181 61601 10) 5.9797 10894

43237 69396

4.5177

29930

1)1.1916 5.2171 3.4520 2.6943

37529 03446 10326 72696

(51)5.3464 ( 1)1.1958 5.2293 3.4580 2.6981

141 21609 31 76523 4669 48881 10)6.8641 48551

21)4.7116 (52)1.0366

53533 11527

(

35565 32088 04545 85417

1)1.2124 5.2776 3.4820 2.7130

3.1

(21) 1.1805

_“_I.

146 21316 31 12136 4543 71856

(51)8.7997

43370 61224 07280

28 63288 4065 86896

513518 8,1510 26159

28871 34209 27863 16727 67070

12529

nk

25 71353 3522 75361 (10)4.8261 72446 12)6.6118 56251

ii i

25695 96823 79413

5>!314

3421 10 4.6525 12 1 6.3275 14 8.6054 17 1.1703 19 1.5916 21 I 2.1646

28 3952 10 5.5730 12 7.8580 15 1.1079 17 1.5622

42767

57088 53078 23282 71840

5.1425 3.4149 2.6712

140 19600 27 44000 3841 60000

5.0265 3.3569 2.6348

127 16129 48383 44641 36941 72915 58602 23424 54749 33853 83316

132 17424 22 99968 3035 95776 10)4.0074 64243 12)5.2898 52801 14 1 6.9826 05697 39521 I 16 9.2170 49217 76966 (50)7.8302 26935

136 18496 25 15456

135 18225 24 60375 3321 50625

l/2

5239

1)1.1269

97935 68959 16865

17161

24

5.0657 3.3766 2.6472

68619

22 48091 2944 99921

l/2 l/3 l/4 l/5

12848

5.0132 3.3503 2. 6307

130 16900 21 97000 2856 10000

20 2601 10)3.3038 12 4.1958 14 5.3287 16 6.7675 18 8.5947 21 1.0915 50)3.0994

87975

1.0085

00000 01525 27804

10)3.7129 12)4.8268 14 6.2748 16 1 8.1573

AND ROOTS

126 15876 20 Of3376 2520 47376

29

METHODS

42484 26074 21532 71824 56943

51)6.319.7 1) 1.2000 5.2414 3.4641 2.7019

148 21904 32 41792 4797 85216

4'3715

00000 82788 01615 20077

149 22201 3'3 07949 49213 84401

(52)1.2197

79049

(52)1.4337

40132

( 1)1.2165

52506

( 1)1.2206

55562

5.2895 3.4879 2.7167

72473 11275 66686

5. 3014 3.4937 2.7204

59192 88147 28110

30

ELEMENTARY Table

ANALYTICAL

POWERS

3.1

AND

METHODS

ROOTS

nk

k 150 22500 33 75000 5062 50000

: 3 5"

151 22801 34 42951 5198 85601 1.1853

b

91159

; 1:

24

l/2

1.6834 ( 1)1.2247

11220 44871

5.3132 3.4996 2.7240

92846

(52)

l/3 l/4 l/5

35512 69927

155 24025 37 23875 5772 00625 I :;I

24

l/2

;: T%

47627

(52)4.3150

94990

( 1)1.2449

89960

( 1)1.2489 5.3832

99600 12612

3.5341 2.7455

18843 21947

5.3716 3.5284 2.7419

85355 41525 92987

(52)7.9228 ( 1)1.2649 5.4288 3.5565 2.7594

l/3 ::;

44 7412 11 L2229 13 I 2.0179 15)3.3295 17 5.4937 19 1 9.0647 (22) 1.4956

60000

35233 58820 59323

165 27225 92125 00625 81031 18702 65858 83665 43047 82603

(53)1.6581

15050

(

1)1.2845 5.4848

23258 06552

3.5840 2.7764

24634 94317

170 28900 49 13000 8352 10000

6.9757 1.1858 24 l/3 l/4 l/5

(53)3.3944 ( 1) 1.3038 5.5396 3.6108 2.7931

I 6.5831 I1921 4.3310 ( 1)1.2328 ( 17 2.8493 52)2.3133

5.3368 3.5112 2.7312

6718

10 9.5388

5.4401

3.5621 2.7629

I

40965 82267 69056

I

75387

(52)2.7076

61312

(52)3.lb59

82801

( 1)1.2369

31688

( 1)1.2409

13

99256 07183 43278

1 3.6914 19 5.7955 21 9.0990

51946 79555 59901

5.3946 3.5397 2.7490

96409 90712 68931 32856

74186

40481 58257 73137 21220

(52)5.8582 ( 1)1.2569 5.4061

79483 80509 20176

3.5453 2.7525

92093 25920

162

163 26569 43 30747 7059 11761

47536

1.8075 I1311 11.1157

71008 49033

I (

2.9282 4.7437 7.6848

29434 31683 45327

I 1.2449 53)1.0674 ( 1)1.2727

44943 81480

21825 0296b 00056

5.4513 3.5676 2.7663

92206 61778

21345 23734

76411 09873 64660 42676 51635

53)2.2140 1)1.2922 5.5068 3.5948 2.7831

90189 84798 78446 36294

92813

(53) 1.2373 ( 1)1.2767 5.4625 3.5731 2.7697

(53)3.9075 ( 1)1.3076 5.5504 3.6161 2.7963

68945 69683 99103 71571 99540

78329 14533 55571 14235 30547

7964:

(53)2.5551 ( 1)1.2961 5.5178 3. b002 2.7865

87425 48140 48353 05744

18023

-._

50 8752

2232.2661 (53)4.4945 ( 1)l

3114 5'3:6214 5612 2.7996

51 8957 1.5496 2.6808

29929 77717 45041 38921 75333

13878

1.3880 2.4013 (53)5.1654

81379 80785 29935

87705

( %?::;

;%46

97766 46817

62559

08411 36670 45765

159 25281 40

6391

19679

28961

(52)b.8160

22003

( 1)1.2609

52021

5.4175 3.5509 2.7560

01515 88625 01343

__.

164 26896 44 10944 7233 94816

(53)1.4330 1)l. 2806 5.4737 3.5785 2.7731

168 28224 41632 94176 78216 07402 56436 22812 64632 88583

173

171 29241 50 00211 8550 36081

5.3601 3.5227 2.7304

I

(11)1.3382

(53)1.9168 ( 1)1.2884 5.4958 3.5894 2.7798

00782 67365

I

167 Z-r&i9 46 57463 7777 96321

27556 45 74296 7593 33136

81241 03963 80069

158 24964 39 44312 6232 01296

26244

57441 78765 86713

5.3484 3.5170 2.7348

42 51528 6887

154 23716 36 52264 5624 48656

79543 84905 81203

03297 43086 95679

1.4976 I(1517 2.3512 I (52)5.0302 ( 1)1.2529

98241

03274 57754

94816

157

15 17 19 22

(52)9.2007 ( 1)1.2688

153 23409 35 81577 5479 81281

24649 38 69893 6075 73201

161 25921 41 73281

67296

16251 11064

5337

13 1.2332 15)1.8745 10 I 8.1136

156 24336 37 96416 5922 40896 (10)9.2389 57978

82490 05615 22977

17)4.2949

l/2

74022 53712 92374

(52)3.6979

6553

24

5.3250 3.5054 2.7276

2"E

160 25600 40 96000

l/2

52704 20573

I 17)3.3316 15)2.1494 8.0041

l/3 l/4 l/5

24

(52)1.9744 ( 1)1.2288

152 23104 15 11808

3:bZbb 2.8029

99110 10436

20335 24847 03675 81908

20b81 lb9

28561 48 26809 8157 30721

(53)2.9463 ( 1)1.3000 5.5267 3.6055 2.7898

26763

00000 74814 51275 27436

174 30276 52 68024 9166 36176

(53)5.9317 ( 1)1.3190 5.5827 3. b319 2.8061

37979 90596 70172 28683

43329

ELEMENTARY

ANALYTICAL

POWERS

31

METHODS

AND ROOTS

nk

Table

3.1

k 175 30625 53 59375 9378 60625

: : 2 ; 1: 24

l/2

l/3 l/4 l/5

(53)6.8063

32613

( 1) ;. 5';;;

y;;

3:6371 2.8093

35763 61392

180 32400 58 32000 1.0497 60000 1.8895 68000

24

l/2

l/3 l/4 l/5

1.1019

96058

(54)1.3382 ( 1)1.3416 5.6462 3.6628 2.8252

58845 40786 16173 41501 34501

179 32041 9595 11 I 1.6887 13 2.9721 15 5.2310 17 I 9.2066 20 1.6203 22 2.8518 53)7.8037

12576 42134 a6155 47634 43835 69315 49994 62212

9aii

06241

(53)8.9404

29702

1)1.3266 5.6040

49916 78661

( 1)1.3304 5.6146

13470 72408

(54)1.0234 ( 1)1.3341 5.6252

81638 66406 26328

3.6423 2.0125

20574 64777

3.6474 2.8157

63337 53634

3.6526 2.8189

24271 28111

181 32761 59 29741 9)1.0732 a3121

(54)1.5285 ( l)l.3453 5.6566 3.6679 2.8283

185 34225 63 31625

24

l/2 l/3 l/4 l/5

(54)2.5829 ( 1)1.3601 5.6980 3.6880 2.8407

82606 47051 19215 17151 58702

71637 62405 52026 16217 66697 186 34596 64 34056

: 5

l/3 l/4 l/5

62931 04875 97079 07530 50791

24 l/3

(54)9.1375 ( l)l.3964 5.7988 3.7368 2.8708

69069 24004 89998 75706 26340

n”[‘-9’]

(54)2.2679 ( l)l.3564 5.6877 3.6830 2.8376

laa 35344 66 44672 9)1.2491 98336

51775

(54)3.3434

78670

(54)3.8000

41874

18170 67473

( 1)1.3674 5.7184

79065 79433

( 1)l 5.7,a6 3711

54316 30920

3.6929 2.8430

90888 23174

3.6979 2.0468

44609 74493

3:7il20 2.8499

78502 12786

(54)7.1346 ( 1)1.3892 5.7789 3.7212 2.8649

E7 88001 51842 54055 30326 22953 a7299 90487 95065 44399 96565 56899 13156

54496 92632 49854

54)5.5564 l';.:;;; 3:7175

93542 ff;;;; 63041

(54)6.2983 ( l)l.3856 3.7224 5.7689

::::2" 74059 a6193 89130 40646 19436 98281

2.0589

50746

2.8619

38162

196 38416 75 29536

55)1.0331 ( ";.;;;; 3: 1416 2.8737

197 38809 76 45313

07971

(55)1.1673

18660

it;;; 57367

( 1)1.4035 3.7464 5.0186

66885 20805 47867

3.0772 5.8159 (54)4.3160 ( 1)1.3747 5.7307 3.7077 2.8529

64756

2.8766

91203

j[(-751

,a[,-,31

49187 24728 76683 66123 05790

4,,,2]

03640 14881 la526 72700 93540 92751 38178 194 37636 73 01384

(54)8.0768 (*)1.3928 5.7869 3.7320 2.0670

40718 38828 60372 75599 75844 199 _.

198 39204 77 62392 ( 9)1.5369 53616

(55)1.3181 ( 1)1.4071 5.8284 3.7511 2.8796

20111 65997 33960 23210 80950 189 35721 67 51269

193 63361 49020 22627 84218 97266 86816 04818

(54)4.8987 ( 1)1.3784 5.7408 3.7126 2.8559

76639 74926 11371 08871 89786

54)2.9397

195 38025 74 14875

l/2

65 39203

(54)1.9898 ( 1)1.3527 5.6774 3.6780 2.8345

( 1)1.3638 5.7082

7"

w

70074 73756 51108 73940 85080

73122 08816 40794 43509 88352 184 33856 62 29504

33489 61 28487

190 _.. 36100 68 59000

:

: 10 24

lR1 --_

182 33124 60 28568

(54)1.7446 ( 1)1.3490 5.6670 3.6729 2.8314

(54)1.1707 ( 1)1.3379 5.6357 3.6577 2.8220

39601 78 80599

(55)1.4875 ( 1)1.4106 5.8382 3.7558 2.8825

57746 73598 72461 93499 08624

32

ELEMENTARY

Table

3.1

POWERS 201 40401 20601 40801

40000 80 00000 1.6000 00000

l/2 l/3

6.4000

00000

(55)1.6777 ( 1)1.4142 5.8480 3.7606 2.8853

21600 13562 35476 03093 99812

l/2 l/3

:,/:

%Z 77620 10032 (55)1.8910 ( l);.;&; 3:7652

:::t: 60303 2;;;; 95059

2.8882

79450

205 42025 a6 15125

: : 2 i9 10 24

ANALYTICAL

AND

METHODS

ROOTS

202 40804 a2 42408 1.6649 66416 3.3632 32160 6.1937 20964 1.3723 33251 2.7721 13166 5.5996 68596 1.1311 33056 (55)2,1302 61246 ( 1)1.4212 67040 5.0674 64308 3.1699 69549 2.8911 47666

206 42436 a7 41816 14096 77038 34698 38548

n-4 203 41209 83 65427

(55)2.3983 ( l)l.4247 5.8771 3.7746 2.8940

07745 80685 30659 26716 04537

__-

2na

207 42849 88 69743

43264 89' 98912

:EE 70372 (55)3.0345 ( 1)1.4317 5.8963 3.7838 2.8996

38594 82106 68540 a9674 84668

(55)3.4104 ( ";.,$$ 3:7804

62581 :;;;; 95756

2.9025

08125

210 44100 92 61000

: :

(55)3.8307 ( 1)1.4387 5.9154 3.7930 2.9053

211 44521 93 93931

89523 49457 a1700 a5099 20638

(55)4.3005 ( 1)1.4422 5.9249 3.7976 2.9081

212 44944 95 isiia

10765 20510 92137 57844 22302 213 45369 96 63597

2 1 :

10 24

l/2 l/3

$5"

(55)5.4108 ( 1)1.4491 5.9439 3.8067 2.9136

19838 37675 21953 54096 93459 215 46225 99 38375

(55)6.0642 ( 1)1.4525 5.9533

75557 a3905 41813

3.8112 2.9164

77876 63134

216 46656 100 71696

(55)6.7929 ( 1)1.4560 5.9621 3.8157 2.9192

102

a5105 21978 31958 85604 22328 217 47089 18313

(55)7.6051 ( 1)1.4594 5.9720 3.8202 2.9219

97251 51952 92620 77414 71130

218 47524 103 60232

204 41616 84 89664

(23)1.2482 (55)2.6985 ( 1)1.4282 5.8867 3.7792 2.8968

50286 09916 85686 65317 66709 50171

209 43681 91 29329 9 1.9080 29761 11 3.9877 82200 13 a.3344 64799 16 1.7419 03143 18 3.6405 77569 20 i 7.6088 07119 23)1.5902 40688 (55)4.8251 50531 ( 1)1.4456 83229 5.9344 72140 3.8022 14131 2.9109 13212 214 45756 98 00344 9)2.0972 73616 11)4.4881 65538 13)9.6046 74252 20)9.4129 (23)2.0143 (55)8.5100 ( 1)1.4628 5.9814 3.8247 2.9247

11168 62990 19601 73884 24030 53435 09627 219

105

47961 03459

10

24

l/2 l/3 l/4 l/5

(55)9.5175 03342 ( 1)1.4662 a7830 5.990f26415 3.8292 13796 2.9274 37906

1 :

4

106

220 48400 48000

56)l.o63a ";.;;;; 3:8336

73589 ;;;;; 58625

2.9301

56052

107

(56)1.1885 ( 1)1.4730 6.0092 3.8380 2.9328

94216 91986 45007 88048 64149

(56)2.0533 ( 1)1.4899 6.0550 3.8600 2.9462

222 49284 41048 12656 86096 65313 84996 16690 14905 67090 89736 66443 48947 08345 56780

221 48841 93861

5 6 7

:

10 24

l/2 :s: l/5

(56)1.6525 ( 1)1.4832 6.0368 3.8512 2.9409

10926 39697 10737 85107 28975

(56jl.8425 ( l';;;;$ 3:8556 2.9435

3oo"3 i;'$; 54127 97699

i56j1.3272 ( 1)1.4764 6.0184 3.8425 2.9355

59512 82306 61655 02187 62280

(56)2.2872 ( 1)1.4933 6.0641 3.8643

223 49729 89567 73441 30773 84962 20466 97640 78274 25550 66205 18452 26994 47878

(56)1.4813 ( 1)1.4798 6.0276 3.8469 2.9382

112

(56)2.5465 ( 1)1.4966 6.0731 3.8686

53665 64859 50160 01167 50529 224 50176 39424

51362 62955 77944 72841

ELEMENTARY

ANALYTICAL

METHODS

AND

nk

POWERS

ROOTS

33 Table

3.1

k : 113

:

225 50625 90625

115

226 51076 4317b

116

227 51529 97083

118

229 52441 1213 08989

5i352

2 i 1: 24

l/2 :s: l/5

(56)2.0338

73334

(56)3.1521

18526

( 1)1.5000 6.0822 3.8729 2.9541

00000 01996 83346 76939

(

29638 99349 79501 90210

121

24

l/2

1)1.5033 6.0911 3.0772 2.9567

230 52900 67000

123

56) 3.5044 1)l.

5066 6.1001 3.8815 2.9594

231 53361 26391

124

5568b

(56)4.8025

07640

(56)5.3295

12896

(56)5.9116

89798

75089 25615 22905 91438

( 1)1.5198 b..1357 3.8985 2.9697

68415 92440 48980 b7129

(

;;;$;

if: l/5

5165 6.1269 3.8943 2.9bll

129

235 55225 77675

131

"lb.;::; 3:9027 2.9723

236 55696 44256

133

1)1.5099 6.1091 3.0050 2.9620

232 53824 87168

(

1)l.

56)3.0943

51917 70200 61435 10235

126

62082

( 56) 4.325'b

51908

bb887 14744 29230 130b2

( 1’;. y;

y;;

233 54209 49337

56) 6.5545 ( l)l.5264

38267 33152

6.1534 3.9069 2.9148

49494 60138 91866

61357 33915 237 56169 12053

3:89OQ 2.964b

128

(56) (

1.2640

1';

:'6;; 3:9111 2.9774

238 134

i%::

136

83026 06713 234 54756 12904

79321 g;:; 45426 41049 239 57121 51919

42736 31712

(56)9.861'3

93410

(57)1.0910

:x 74649 49664 56201 55818

"k:;;;

"b;;i;

(

24862

._..

24

l/2 l/3 l/4 l/5

(56)8.0469 ( l)l.5329 6.1710 3.9153 2.9799

01671 70972 05793 17320 81531

(56) 8.9102 ( 1) 1.5362 6.1791 3.9194 2.9825

- .-

138

24

l/2 l/3 l/4 l/5

24 l/3 l/4 l/5

139

(57)1.3337

35777

(57)l.

( 1)1.5491 6.2144 3.9359 2.9925

93338 b5012 79343 55740

(

147

l/2

57600 24000

(57)2.1876 ( 1);

;;;; 3: 9563 3.0049

4736

l)l.5524 6.2230 3.9400 2.9950

245 b0025 06125

148

12697 29150 46606 15921 13380

3: 9236 2.9850

241 58081 97521

141

99791

57)l.b276

17470 84253 12930 45390

1';;;;; 3:9441 2.9975

246 60516 86936

150

21327 36660

(57)l.

;;6';:

(

53798 26190

7910

1y;

62509

(51)2.6590

52293

(

38714 26556 51896 11096

( l)l.Sllb b.2743 3.9643 3:0098

23365 05351 70523 12147

54435 53635 51438 243 59049 48907 84401 86094 11321 54510 66546 12701 79617 10300 pm;

3:9482 3.0000

247 b1009 69223

(57)2.4123

20998 22094 1

143

79087

y"b

5684 6.2658 3.9603 3.0073

6.1971 3.9277 2.9075

242 58564 72480

91225

1)l.

1)1.5427

152

(57)2.9298

(57)1.2065

61943

(

62483 21795 72942 57776

1)1.5459 6.2058 3.9318 2.9900

_

145

?AA 59536 26704

23)7.4799

42569

(57)1.9831 ( l)l.5620

51223 49935 99710 71742 b5081

22039 00000

6.2487 3.9522 3.0024

248 61504 52992 42016 00200 37650 13371 13716 bbOlb 91719

154

15956

(57)

01515 b1305 76966 45305

( 1';.

3.2268 ;;;I) 3.9723

3.0146

249 b2001 36249

91251 ;g 71312 70627

34

ELEMENTARY

Table

3.1

ANALYTICAL

POWERS

METHODS

AND ROOTS

nk

k : 156

250 62500 25000

251 63001

158

13251

160

252 63504 03008

253 64009 94211 52081 19416 46016 41511 65511 23159 91011

: 2 ; 1: 24

w

(51) (

3.5527

1'2.

l/3 l/4 l/5

;g;

319763 3.0170

165

13679

(51)

y9"

( 1)1.5842

3.9099

97952

6.3079 3.9803 3.0194

53644 88168

65025 81315

33001

161

93549 24041 97986

(57)4.3014 ( 1)1.5874 6.3163 3.9042 3.0219

256 65536 17216

169

31119 50181 59598 02604 00136

(51) 4.7303

41643

( 1’;.

97372 03543 29397 94671

;2’:; 3:9882 3.0242

257 66049 74593

6.3330 3.9921 3.0266

258 171

%:E 66096 37653 95144 81472 68820 75555 63693

113

11018

(57)6.2771

01735

(57)6.8927

88615

(51)7.5661

15089

(57)8.3022

1)1.5968

11942 25705 88015 61117

(

ii;;;

( 1)1.6031

21954

( 1)1.6062

37840

(

6.3413 3.9960 3.0290

l/3 l/4 l/5

175

24

57)9.1066

85770

( 1)1.6124

51550

6.3825 4.0155 3.0408

l/3 l/4 l/5

186

1';;:;; 4:oooo 3.0314

00000 33133

4.0039 6.3578 3.0337

260 61600 16000

l/2

00541 61180 91748

4.0077 6.3660 3.0361

262 68644 84128 98736 43669 04412 01560

04299 34273 41703 265 70225 09625

1.4384 ( 1)1.6278 6.4231 4.0341 3.0524

(58)

196

181 91447

1.1993 ( 1)1.6217

27914

(58)1.3136

94086

41406

27474

(

6.3988 4.0232

34278 27910

6.4069 4.0270

58577 67760

3.0431

83226

3.0455

11602

3.0418

32879

07681 68660 90325 47105

188

266 70756 21096

190

261 71289 34163

(58)

192 9 5.1586 12 1.3825 14 I 3.7051

58) 1.8846

83974 68868

1)1.6340 6.4392 4.0422 3.0570

13464 16696 93240 47961

1) 1.6310 6.4413 4.0460 3.0593

10554 05721 72854 34462

6.4312 4.0385 3.0547

27591 02994 54599

39954 61673 04070 00464 87063

6.4633 4.0536 3.0638

40472

58289 02045 54329

(58)

199

(58)2.4618 ( 1)1.6462 6.4712 4.0573 3.0661

02511

51891 01763 13627 48596 53254

201

272 73984 23648

(5Q2.6893

89450

( 1)1.6492

42250

6.4792 4.0610 3.0684

23603 86370 12165

203

(58)2.9369 ( 1) 1.6522 6.4811 4.0648 3.0706

1)1.6248 6.4150 4.0308 3.0501

268 11824 48832 86916 28110 15334

1.1229

(

( 1)1.6431

183

264 69696 99144

( 1)1.6186

50643

(58)2.2528

11088 67601 04982

(58)1.0945

82060

24

47694

6.3743 4.0116 3.0385

96760 89716 55014

49442 76528 89807

(58)

m

21920

54265

60235

270 72900 83000

259 67081 13979

6155 6.3906 4:0193

1.5145 ( 1)1.6309

10548

31145 25531 64507 01647

%; 98010 38372

57)9.9‘355 ( I)1

254 64516 81064

1) 1.6093

263 69169

1.9113

l/3 l/4 l/5

( 1)1.5937

51) 5.7143

l/2

24

163

194

(58)2.0608 (

91176 11164 54111 13851 65640

89564

1) 1.6401

6.4553 4.0498 3.0616

273 14529 46411

205

(58)

269 72361 65109

21947 14811 41906 14141 274 15iC 70024

3.2063

69049

3.0129

11923

ELEMENTARY

ANALYTICAL

POWERS

35

METHODS

AND

ROOTS

212 9 5.8873 12 1.6307 14 4.5172 17 1 1.2512 19 3.4660 21 9.6010 24 I 2.6594 58)4.1640 1)1.6643 6.5186 4.0796 3.0796

277 76719 53933 39441 93025 96680 91180 76569 32097 85891 35828 31698 a3915 22161 11650

nk

Table

3.1

k : 201

:

275 75625 96875

276 210

16116

24576

2 i 1: 24

l/2 l/3 l/4 l/5

(58)3.4993 ( 1)1.6583 6.5029 4.0722 3.0751

219

24

l/2 :;: l/5

(58)5.3925 ( 1)1.6733 6.5421 4.0906 3.0862

(58)3.8178 ( 1)1.6613 6.5108 4.0759 3.0773

28001 12395 57234 38199 51657 280 78400 52000

32264 20053 32620 23489 53577

221

(58)5.8742 ( 1)1.6763 63499 4.0942 3.0884

al225 231 49125

24

l/2 l/3 l/4 l/5

(5a)a.2466 ( 1)1.6aal 6.5808 4.1087 3.0971

243

24

l/2 l/3 l/4 l/5

l/3 l/4 l/5

32480 94302 44365 64171 98013 290 84100 R9000

9)7.1708

246

39885 05461 11620 70950 54901 81796 93656

48566 10900 42039 53453 32215 63618 68441 291 84681 42171 71761

15721 64428

( 1) ;. y'9;

;g:;

( 1) t ‘6;;;

;g;;;

:: ::::

4:1302 3.1101

67707 89906 295 a7025 72375 ;::::

I 17 19 I 6.5907 14 1.9442 5.7355 1.6919

63969 58973 91371 08381

(59)1.8868 ( 1)1.7175 6.6569 4.1443 3.1186

10930 56404 30232 41207 33956

1

20588 30396

259 ( 9 I 7.6765 (12 2.2722

296 a7616 34336 63456 62783

I :;j;:

2%

z"o"8

(59)2.0464 ( 1)1.7204 6.6644 4.1478 3.1207

(58)6.3970 ( 1)1.6792 6.3576 4.0979 3.0906

236

25168

33126 a5562 72186 08689 49967 287 82369 39903

226

58)6.9642 1)1.6822 6.5654 4.1015 3.0928

238

01230 33200 10079 99156 31992

217

(58)4.9488

11121

( 1) ;. ;;g

;g;

4:0869 3.0840

283 a0089 65187

51599 60384 14427 36766 38815 288 a2944 a7872

279 77841 17639

229

66245 45954 80656 06304

(58)7.5794

93086

( 1) ;. y;

$94;;

4:1051 3.0950

55240 21484

241

289 83521 37569

935’39

4.3544 (59)1.3596

411266 3.1079

19524 224

(58)4.5402 ( 1)1.6673 6.5265 4.0832 3.0818

278 77284 a4952

a5616 31410 07486 72583

49008

[J

24

III 12 149 5.4726 1.9135 6.6905 17 1.5651 19 4.4163 22 1.2802 24 3.6615 (58)8.9690 ( 1)1.6911 6.5085 4.1123 3.0993

281 78961 a8041

(59)1.2518

256

l/2

233

42160 24773 30071 35196 a4885

214

49657 65053 43703 48904 45423 1

(58)9.7536 ( 1)1.6941 6.5962 4.1159 3.1015

13040 07435 02284 53637 32807

292 85264 748 97088 49696 25311 49909 05773 16858 a3323 a7302 (59)1.4763 46962 ( 1"6'W3; ;;I;; 411337 3.1122

64325 65011 297 88209

(59)2.2189 ( 1)1.7233 6.6719 4.1513 3.1228

a7131 68794 40272 47726 51191 1

(59)1.0602 ( 1)1.6970 6.6038 4.1195 3.1036

251

(59)1.6025 ( 1)1.7117 6.6418 4.1372 3.1143

7A4 --.

(59)2.4054 ( 1)1.7262 6.6794 4.1548 3.1249

77893 56275 54498 34288 91148

24)4.0642 (59)1.1522 ( 1)1.7(100 6.6114 4.1231 3.1058

293 a5849 53757

91698 24277 52195 98970 93785 298 88804 63592 50416 72824 17015 58671 36838 02776 42278 16789 67650 20032 37723 51295

31407 54005 00000 89018 05626 43502 294 06436

(59)1.7391 ( 1)1.7146 6.6493 4.1408 3.1165

45550 42820 99761 24580 16755

299 a9401 :267 3089s

(59)2.6068 ( 1)1.7?91 6.6868 4.1'583 3.1270

04847 61647 83077 la947 45768

36

ELEMENTARY

Table

ANALYTICAL

POWERS

3.1

METHODS

AND

ROOTS

301 90601 270

(59)2.8242

00000

95365

l/2

l/3 l/4 3.1291

l/5

34645

283

272

70901

215

nk

302 91204 43608 9)8.4288 12)2.5539

303 91809 18127 92481 54422

24)6.5226

83188

270

_-.

92416 280 9)8.5407

59)3.0591

15639

(59)3.3125

81949

(59)3.5861

05682

(59)3.8811

1)1.7349

35157

(

14720 72852 10496 95743

( 1)1.7406

89519

( 1)1.7435

6.7017 4.1652 3.1312

93025 72625

59395 55283 17958

286

l)i.7378 6.7091 4.1687 3.1332

306 93636 52616

289

6.7165 4.1721 3.1353

69962 57138 68030

99856 59577

6.7239 4.1155 3.1314

50814 95260

34853

308

307 94249 34443

94864 292 9 8.9991 12 2.7717 14 8.5369 17 2.6293 19 1 8.0985

i

18112

94464 17056

295

309 95481 03629

18496

46977 80688 90052 21360

2.4943

44579

24

('X9)4.1994

86063

(59)4.5427

01868

(59)4.9127

08679

(59)5.3115

00125

(59)5.7412

10972

l/2

( 1)1.7464

24920

( 1)1.7492

85568

(

9";;;;

( 1)1.7549

92877

( 1y.;;;;

;'4;;;

l/3 l/4

6.7313 4.1790 3.1394

l/5

15497 24910 96244

6.7386 4.1824 3.1415

64101 46136 52236

91000

301

10000

( 9)9.2352

4: 1858 3.1436

80231

(59)6.2041

26610

(59)6.7026

93132

( 1)1.7606

81686

( 1)1.7635

19209

: : 2 i9

99452 41767 22833

312

02859

303

2.8015

6.7678 4.1960 3.1497

58988

311 96721

96100 297

1) ;. y;;

6.7751 4.1994 3.1517

315 99225 55875

(59)7.2395

04640 28072

13417 63512 48146

306 9.5979 3.0041 9.4029 2.9431 9.2120 2.8833 9.0249

313 97969 64297 24961 50513 91105 36216 16356 61119 20304

9 12 I 14 17 19 I 22 24 (59)7.8174

( 1)1.7691

68952 27591 52295

315

312 97344 71328

6.7533 4.1892 3.1456

3.1531

316 99856 54496

6.7896 4.2061 3.1557

76544

317 1 00489 318 55013

411926 3.1476

59756 88127

309

31800

59)8.4393

80601

l)l.7720

61336 62861 95609

99655

04515

6.7968 4.2095 3.1578

84386 18398 09519

318

1 01124 321 57432

314 98596 59144

324

319 1 01761 61759

10 24

l/2

(59)9.1086

34822

( 1)1.7748

23935

1) 1.7716

92116 65931 18306

6.8112 4.2162 3.1618

6.8040 4.2128 3.1598

l/3 l/4 l/5

(59)9.8285

320

24

v2 l/3 l/4 l/5

68000

1.3292 ( 1)1.7888

27996 54382

6.8399 4.2294 3.1697

03787 85054 86385

(60)

84208 49381

6.8184 4.2195 3.1638

61941 37156 20622

(60) 1.4325 ( 1)1.7916 6.8470 4.2327 3.1717

76161

86248 41287 21278 85474 65030

(60)1.1435

38734

( 1)1.7832

55450

6.8256 4.2228 3.1658

322

1 03041 330

I

1.0602 ( 1)1.7804

(60)

321

1 02400 327

62028 38883 84608 05502 21997

21862 35844

6.8541 4.2360 3.1137

24002 78192 38149

6.8327 4.2261 3.1678

71452 76889 02787 324

1 04325 336 10)1.0884 12)3.5157 15 1.1355 17 3.6679 20 1.1847 22 3.8266 Ii 25 1.2360

98267 54024 06498 73199 01432 32163 84885 19218

1.6628 ( 1)1.7972

78568 20076

6.8612 4.2393 3.1757

12036 63249 07571

86248

(60) 1.5436 ( 1)1.7944

37808 57110

323

1 03684 333

24197 60938 14209

1.2330 ( 1)1.7860

(60)

(60)

1 04976 340

12224

1.7909 ( 1)1.8000

36736 00000

6.8682 4.2426 3.1776

85455 40687 11523

(60)

ELEMENTARY

ANALYTICAL

POWERS

37

METHODS

AND ROOTS

nk

Table

3.1

k 325

3

343

10 1.1156 12 I 3.6259 15 1.1784

: 6

17 3.8298 20 1 1.2447 22 4.0452 25) 1.3147 60)1.9284

; 9 10 24

10

326

1 05625

:

64063 08203 20166

6.8753 4.2459 3.1796

l/3 l/4 l/5

(60)2.2343

23554

47009

( 1)1.8083

14132

( 1)1.8110

88750 72871 84924

08618 78985 48440

( Q1.8165 6.9104 4.2621 3.1893

90212 23230 47595 54454

55434

60)2.9913

81825

3.1912

85058

335

95315

(60)3.9909

41565

(60)4.2868

( 1) ;. ;:5";

y',

( 1y;;

y;"o

9

4:2782 3.1989

332

(60)3.2159

84959

01166

4:2813 3.2008

61118

90286 68669

340

( 1)1.8357

55975

4.2845 6.9589 3.2027

24

(60)5.6950

03680

(60)6.lloa

98859

(60)6.5558

l/2

( 1)1.8439

08891

( 1);;;;;

;"8;;;

( 1)1.8493

24201

: 2 7 i

10

6.9795 4.2940 3.2084

1/3 $2

32047 76026 53751

412972 3.2103

29958 38860

345

._

I I

l/3

90657 76961 19552

417

I(1215 I 4.1565 1.3882 17 4.6363 20 1.5487 I(2225 I 5.1727 1.1276

10 1.2444

74114 43539 85542 73711 15819

66688

6.9313 4.2718 3.1951

00768 01446 32308

6.9382 4.2750 3.1970

338 1 14244 386 14472

25) 1.9461 (60)4.9431

3.2046

08291 16051

(60)5.3063

11693

( 1)1.8411

95264

6.9726 4.2909 3.2065

70186

_._ II II

1.6284

53607

1 18336 407

(1012 17 1.4003 15 4.8171 5.7004 1.6571

49439 07395 72660 40890

25 1.9609 22 20 2.3205 6.7456

15244 54607 83848

(60)7.0316

76479

(60)7.5405

43015

(

g;;;;

( 1)1.8547

23699

1) +. i;'o; 4:3035 3.2140

7.0067 4.3066 3.2159

17071 95850 348

1 21801 425

48816 76738

25 2.2009 22 20 7.6811 2.6807

37377 95921 15737

30203 54169 80617 14906 62636

Ii

(60)9.2876

83235

(60)9.9518

04932

( 1)1.8574 7.0135 4.3097 3.2178

17562 79083 76748 35355

( 1)1.8601

07524

( 1)1.8627

93601

( 1)1.8654

75811

1.0661 ( 1)1.8681

49656 15431 1222,6

7.0405 4.3222 3.2252

4[(-46)5]

4-46)2]

+j)l]

08549

I 15 17 1.8069 6.3063

53376

7.0338 4.3191 3.2234

96121 50321 67776 349

1 21104 421 44192

81923

05788 09269 57557

07584

13598

(60)8.6661

7.0271 4.3160 3.2215

82649 15128 64201 144

343

1 17649 403

32074 04899 49006

339 1 14921 389 58219

95243

48952 96386 98608

85420 26935

(60)8.0845

7.0203 4.3128 3.2196

10837

( 1)1.8275

1 20409

21736 92066 44547 60213 30338 39497 76659 13924

334 1 11556 37.x 59704

28759

347

1 19716

414

I

l/2

4.3003 6.9931 3.2122

346

1 19025 410 63625

24

40;

35942 15020 19165

( 1)1.8248

43337 72295 71684 342 16964 01688 57730 57435 35043 61047 81952 10274 77114 12822

39:

26037

35715

(60)3.7146

12753

12427

4.258') 3.187,4

34894

99320

337

(60)4.6038

( I I( (60)2.5864 ( 1)1.813#3 6. 903,1

(60)3.4566

1 135bs

341 16281 51821 27096 53398 66909 30158 47684 24602 04689

:

369

11001

3.1932

34481 75061 79164

329 1 08241 3515 11289 10)1.1716 11408

1 10889

94368

382

10

6.8964 4.2556 3.1854

1 10224 365

37: %E 90290 16737 60238 79440 50918 72309 93134

l/3 l/4 l/5

18774 27697 34426

336 12896

1 12225

: 8

l/2

6.8894 4.2524 3.1835

331 09561 64691 61272 95811 27813 73062 67184 70378 28495

37000

2

24

77028

76350

I 20 25 I 1.4064 22 1.5315 4.6411 (60)2.7818

375

15994 11326 93150 48153 45942 47069 09169

( 1)1.8055

21000 44298 67969 39300

:

31706

(60)2.0759 6.8823 4.2491 3.1815

87552

10 1.1574

75638

Ii~17 15 12 10 1.1859 4.2618 1.2914 3.9135

1

352

15722 44335 10547 30632

359

1 07584

65183

21030

330

l/2

349

12 3.7963 15 1.2452 II 17 4.0842 20 1.3396 22 4.3940 I 25 I 1.4412 (60)2.4042

1 08900

24

1 06929

45976

65540 06300 95476

( l)l.8027

1/3 1/4 l/5

,346

328

327

1 06276

28125

(61)

4647)63

ELEMENTARY

38 Table

POWERS

3.1

k : : z ii 9 10 24

350 1 22500 428 75000 25000 87500 65625

351 1 23201 43551 432 48640 48727 04703

:%f 63867 47354 13124 28693 98732 07727 08809

lb:::: 64375 84096 43263 99400 04063 93928 50768

(61)1.1419

l/2

( "y;;

l/3

413253 3.2271

:::

(61)1.2228

( y:;1 4:3283 3.2289

z 6

l/2 l/3

i//2

(61)1.6050 ( 1)1.8841 7.0806 4.3406 3.2362

20092 44368 98751 73183 76880

360 1 29600 466 56000

24

25771

;61;2.2452

AND ROOTS

(61)1.3092 ( l)l.8761 7.0606 4.3314 3.2307

56304

(61)2.3997

87825

rzk

54042 66304 96671 73541 88532

(61)1.8366 ( 1)1.8894 7.0939 4.3467 3.2399

95605 44363 70945 73933 15199

17652

l/2 l/3 l/4 l/5

3.2453

24

l/2 ::: l/5

42223

365 1 33225 486 27125 (10 1.7748 90063 12 6.4783 48728 15 2.3645 97286 80093 I 17 8.6307 (20 3.1502 34734 (23 1.1498 35678 (25 1 4.1969 00224 (61)3.1262 86296 ( 1)1.9104 97317 7.1465 69499 4.3709 23607 3.2543 07394

50; (10)1.8741

370 36900 53000 61000 95700 26409 87713 79454

3.2471

43191

3.2489

361

490 10)1.7944

61)3.3384 1)1.9131 7.1530 4.3739 3.2560

1 34689 494 30863

1 33956 27896 20994

59019 12647 90095 14319 88625

(61)3.5643 ( 1)1.9157 7.1595 4.3768 3.2578

92671 24406 98825 98909 65967

( 1)1.9287 7.1919 4.3917 3.2666

372 38384 78848 13146 48902 71791 67064 75348 26429 22317 85051 30152 66348 31039 95001

371 1 37641 510 64811

lg% 24 :/,: l/4 l/5

(61)4.3335 ( 1)1.9235 7.1790 4.3858 3.2631

25711 38406 54352 16237 74848 J[(-q5]

40172

(61)4.6235 ( 1)1.9261 7.1855 4.3887 3.2649

31606 36028 16151 76627 36822

4646121

+-p]

(61)1.4999 ( 1)1.8814 7.0740 4.3376 3.2344

358 1 28164 458 82712

(61)1.9642 ( 1)1.8920 7.1005 4.3498 3.2417

362 1 31044 474 37928

(61)2.5645

354 1 25316 443 61864

353 1 24609 439 86971 40288 73217 54146 35134 02402 38480 25983 99442 (61)1.4014 29423 ( 1q.g; 76615 46600 4:3345 3.2326 22125

357 1 27449 454 99293

361 1 30321 10)1.6983

METHODS

352 1 23904 436 14208

356 1 26736 451 18016 10 1.6062 01370 12 5.7180 76876 15 2.0356 35368 17 7.2468 61909 20 i 2.5798 82840 22 9.1843 82909 25 3.2696 40316 (61)1.7171 17251 ( 1)1.8867 96226 7.0873 41061 4.3437 26771 3.2380 98084

355 1 26025 447 38875

: 3

.li 9 10 24

ANALYTICAL

1 498 10 1.8339 12 6.7489 15 2.4836 17 1 9.1397 20 3.3634 23 1.2377 25 I 4.5548 (61)3.8049 ( 1)1.9183 7.1660 4.3798 3.2596

359 1 2asa1 462 68279

(61)2.1002 ( 1)1.8947 7.1071 4.3528 3.2435

31355 88793 88459 14700 28247

363 1 31769 478 32147

51)2.7400 1)1.9052 7.1334 4.3649 3.2507

55202 88772 43955 13137 51567

53237 55888 92490 23697 33187 368 35424 36032 65978 94798 30086 58715 31207 42684 93078 38558 32609 95742 77406 39439

29556 29532 93661 49104 37249

364 1 32496 482 28544

i61)2.9270 ( 1)1.9078 7.1400 4.3679 3.2525

70667 78403 36982 26743 22254

369 1 36161 502 43409

(61)4.0609 ( 1)1.9209 7.1725 4.3828 3.2614

98114 37271 80900 49839 09059

374 1 39876 523 13624

( 1)1.9313 7.1984 4.3946 3.2684

20792 04996 79501 49404 $-;'5]

(61)5.6094 ( 1)1.9339 7.2048 4.3976 3.2702

26383 07961 32147 22040 00047

ELEMENTARY

ANALYTICAL

POWERS

39

METHODS

AND ROOTS

nk

Table

3.1

1 544 10 2.0632 12 7.8198

379 43641 39939 73686 07278

k :

375 1 40625 527 34375

1 531 10 1.9987 12 7.5151 15 I 2.8257 16 1.0624 20 1 3.9948

: 2 i

9 10 24

l/2 l/3

:/,:

(61)5.9806 ( 1)1.9364 7.2112 4.4005 3.2719

78067 91673 47852 58684 46950

380 1 44400 548 72000

(61)8.2187 ( 1)1.9493 7.2431 4.4151 3.2806

60383 58869 56443 54436 25976

: : 6' i 9 10 24

l/2 l/3 l/4 l/5

: :

53901 41687 06349 06853 14120

I 593 10 I 2.3134 12 9.0224 15 1 3.5187 18 1.3723 20 I 5.3520 23 2.0872 25 8.1404 (62)1.5330 ( 1)1.9748 7.3061 4.4439 3.2977

390 52100 19000 41000 19900 43761 10067 09260 83612 06085 29700 41766 43574 19178 13494

377 1 42129 535 82633

5.6477 (61)6.3754

88819 12334

(15)5.7998 (61)6.7950

06469 46060

( 1)1.9390 7.2176 4.4034 3.2736

71943 52160 89461 90130

( l';.;;:;

‘I;;;:

414064 3.2754

14397 29605

54;

(61)7.2410 ( 1)1.9442 7.2304 4.4093 3.2771

(61)9.3222

49236

;61;9.9259

15535

( 1)1.9519 7.2495

22130 04524

( 1)l 7.2558 9544

41507 82029

4.4180 3.2823

56280 50807

( li1.9570 7.2621 4.4230 3.2857

38579 67440 42876 09631

575

(62)1.1970 ( 1)1.9646 7.2810 4.4324 3.2909

386 1 48996 12456

03202 88270 79420 80423 21030

382 1 45924 557 42960

378 42884 10152 03746 86558 96519 62484 64160 28264 96838 77507 22210 26792 33520 65392

381 1 45161 553 06341 10 I 2.1071 71592 12 8.0283 23766 15 1 3.0587 91355 18 1.1653 99506 20 4.4401 72119 23 I 1.6917 05577 25 6.4453 90249 (6Q8.7538 56362

385 1 48225 570 66625

(62)1.1247 ( 1)1.9621 7.2747 4.4296 3.2892

376 41376 57376 17338 77109 06623 65690 70996

414209 3.2840

52418 72019

387 1 49769 579 60603 10 2.2430 75336 12 I 8.6807 01551 15 3.3594 31500 18 I 1.3000 99991 20 5.0313 66963 23 1 1.9471 46755 (25)7.5354 57941 (62)1.2736 88303 l)$.%;; 4:4353 3.2926

391 1 52881 597 76471

)b;;;: 48416 24406

1 50544 584 11072

;62;1.3550

69013

( lj1.9697 7.2936 4.4302 3.2943

71560 33030 10856 24265

392 1 53664 602 36208

9.1386

86663

25)8.3515 (62)1.6302 ( 1':;;;; 414467

59392 04837 ;;f;; 65109

(62)1.7332 ( 1)1.9798 7.3186 4.4496

67559 98987 05586 11420

3.2994

02898

3.3010

88848

_._

395 1 56025 616 29875

1 46689 561 81887

393 1 54449 606 98457

(62)1.8425 ; lil.9824 7.3248 4.4524 3.3027

397 _..

1 56816 620 99136 10)2.4591 25786

58176 22760 29445 40634 71361

398 1 58404 630 44792

1 57609 625 70773

i 18 25Y6.1149 23)1.6134 20)4.2570 15 I 1.1232 2.9637 (61)7.7150 ( 1)1.9467 7.23b7 4.4122 3.2788

38586 40260 06958 98312 44931 90756 92233 97216 46'358 97510

1 47456 566 23104

(62)1.0566 ( !)1.9595 7.2684 4.4267 3.2075

94349 91794 82371 21679

03659

389 1 51321 588 63869

(25)7.9340 (62)1.4414 ( 1)1.9723 7.2998 4.4410 3.2960

611

(62)1.9584 ( 1)1.9849 7.331Cl 4.4552 3.3044

69734 19629 08292 93662 67768 20622 394 1 55236 62984

35730 43324 36930 70277 50453

_..

1 59201 635, 21199

2 i 9 10 24

l/2 l/3

62)2.0812 1)1.9874 7.3372 4.4580 3.3061

78965 60691 33921 94536 26138 I

g

(62)2.2114

87364

(62)2.3494

82217

i62i2.4957

07762

( 1)1.9899 7.3434 4.4609 3.3077

74874 20462 13443 98433

( 1)1

85885 96597 27013 67354

; ljl.9949 7.3557 4.4665 3.3111

93734 62368 35273 32914

[ 41 (-(94

9924 7.3495 4: 4637 3.3094

(62)2.6506 ( 1)1.9974 7.3619 4.4693 3.3127

32365 98436 17821 38246 95131

40

ELEMENTARY

Table

3.1

ANALYTICAL

POWERS

k 1

400 1 60000 640 00000

: 4

644

METHODS

AND ROOTS

401 1 60801 81201

402

2 9"

l/2

l/3 l/4 l/5

(62)2.8147 ( 1)2.0000

49767 00000

7.3680 4.4721 3.3144

(62)2.9885 ( 1)2,0024

62997 35955

7.3741 4.4149 3.3161

54017

.-1 64025 664 30125

24

l/2

l/3 l/J l/5

(62)3.7924

56055

(62)4.0236

( 1)2.0124 7.3986 4.4860 3.3226

61180 36223 46344 99030

(

24

l/2 l/3 l/4 l/5

( 1)2.0248 7.4289

4.4998 3.3308

pm;

414777 3.3177

( 1)2.0074 7.3064

31295 85990

15674

4.4004

91129

61862

3.3194

10850

.-.

406 1 64836 23416

674

04703

24 l/3 l/4 l/5

(62)6.8101

( 1)2.0371 7.4590 4.5134

3.3389

4.4832 3.3210

409 1 67281

1 66464 679 17312

684

06980

(62)4.5273

48373

(62)4.8013

( 1)2 7.4107 0174

95055 24100

( 1)2.0199 7.4168

00988

4.4088 3.3243

12948

14446

4.4943 3.3276

( 1)2.0223 7.4229 4.4970 3.3292

65185

20)8.1420

414915

38251 411 1 68921 26531

3.3259

1 699 10 I 2.8813 13 1.1870 15 4.8908 18)2.0150 20 I 8.3019 23 3.4203

(62)5.3976

37632

( l';.;;;; 4:5025

-M; 69814

3.3324

86236

719

73375

26 1.4091 7218 ( 1)2.0297 7.4410 4.5053 (62)s.

3.3341

416 1 73056 91296

74245

412 69744 34528 02554 96652 38207 25341 04405 84615 98461 06738 78313 18861 06108 06308

59539 30860 07026

06073 74842 14120 82211 36609

413 1 70569 704

44997

414 1 71396 10)2.9376

709

3.5727

1.4791 87127 40143

( 62)6.064i 1)2.0322 7.4470 4.5080 3.3357

417 1 73889 725 11713

( 62)6.4269

57944

58882

53630

20006 98328

1)2.0346 7.4530 4.5107 3.3373

34238 31426 23237

418 1 74724 730

17929

98995

39914 63768 37037

419 1 75561

34632

735

60059

47618

16532 43334 45063 03984 50201

13045 54879 35926

85215 47722

19812 49315 90153 72387 19056 54883

26518 48631

(62)7.2150

59801

(62)7.6430

25690

(62)8.0952

59269

( 1)2.0396 7.4650 4.5162 3.3405

07805 22314 01729 55305

( 1)2

57786 99115 13349 59799

( 1)2.0445 7.4769 4.5216 3.3437

04830 66370 20097 61218

420 1 76400 740 88000 3.1116 96000

26)1.7080 (62)9.0778 ( 1)2.0493 7.4880 4.5270 3.3469

75124 41792 74611 56568

1.3925

(62)4.2684

3.7311 26)1.5521

l/2

85306

( 1)2.0099

408

1 65649 19143

10)3.0528 il 10 18 2.9661 20 15 13 8.7980 5.1084 2.1200 1.2309

(62)3.5739

92707

415 1 722% 714

1ym;

694

25229

19344 65931 10945 45673 58841 20522 63008

91940 28423

;62;3.3676

20630 44168

410 1 68100 21000 61000 62010 04241 42739

20 7.9849 23 3.2738 26 I 1.3422 (62)5.0911

62)3.1728

;:8628; 00315 72718

404 1 63216 659 39264 10)2.6639 46266

1)2.0149 7.4047

689

10 2.6257 13 1 1.1585 15 4.7501 18 1.9475

80393 98439 09590

669

403 1 62409 654 50827

t%: 85282 57283 26278 11364

1

10 24

nk

0420 7.4709 4:5u9 3.3421

421 1 77241 746 18461

422 78084 51448 91106 27047 40136 46338 72154 :9":::

(62)9.6110

38126

(63)1.0174

16609

( 1)2.0518 7.4948

28453 11226

( 1)2.0542 7 5oo7

63858 40668

4.5297 3.3485

11307 47155

4[(-46)4]

+-y

415323 3.3501

96767 36405

+-;‘7]

(62)8.5730

1)2.0469 7.4829

4.5243 3.3453

423 78929 06967

21 1.0249

762

73581 48949 24114 21992 59575 424 1 79776 25024

16974 62680 97814

I $2 I :: z (63)1.0768

83734

( 1)2.0566 7.5066 4.5350 3.3517

96380 60749 81455 22644

(63)1.1396 ( 1)2.0591 7.5125 4.5371 3.3533

.[(-47141

73784 26028 71508 59390 05887

IGLEMENTARY

ANALYTICAL

POWERS

AND

41

METHODS

ROOTS

Table

d

3.1

k 425 1 80625 767 65625

: :

I 773 10 3.2933 1'13 11.4029 115 I 5.9166 t 18 2.5460 I 21 1.0846 123 4.6204 I 26 1.9683 (63)1.2759 / 1)2.0639 7.5243 4.5431 3.3564

2 i 1'0 24

(63)1.2059

63938

w it: l/5

3.3548

86145

427 1 82329 778 54483

81476 08776 53858 68743 46847 51557 17963 21295 40370 76744 65204 01082 63431

63)1.3497 1)2.0663 7.5302 4.5457 3.3580

::: l/4 l/5

(63)1.5967 ( 1)2.0736 7.5478 4.5537 3.3627

823

24

l/2 l/3 l/4 l/5

72093 44135 42314 28292 43107

(63)2.1073 ( 1)2.0856 7.5769 4.5669 3.3705

l/2 l/3 l/4 l/5

24

l/2 l/3

16666 65361 84852 08540 27318

I 63)2.2267 c 1)2.0880 7.5827 4.5695 3.3720

440 1 93600 84000

(63)2.7724 ( 1)2.0976 7.6059 4.5799 3.3782 1 881 10 3.9213 13 11.7450 15 1.7653 18 13.4555 21 1.5377 23 6.8428 26 I 3.0450 63)3.6361 1)2.1095 7.6346 4.5929 3.3858

18906 53949 88825 73502 05720

(63)1.7848 ( 1)2.0784 7.5595 4.5590 3.3658

436 1 90096 828 81856

1 89225 12875

851

24

,63)1.6883 , 1)2.0760 7.5536 4.5563 3.3643

857

98685 97832 48212 64877 37758

71952 61302 86527 30941 75562 441 1 94481 66121

834 10 3.6469 13 1 1.5937 15 6.9644 18 1 3.0434 21 1.3299 23 5.8120 26 I 2.5398 (63)2.3526 ( 1)2.0904 7.5885 4.5721 3.3736

83700 60969 26299 14114 65436 437 1 90969 53453 15896 02247 78818 77243 99555 98057 86851 34640 54496 79338 48834 20969

863 10 3.8167 13 1.6869 15 I 1.4564 18 3.2957 21 1 1.4567 23 6.4387 26 1 2.8459 (63)3.0912 ( 1)2.1023 7.6174 4.5851 3.3813

442 1 95364 50888 09250 85488 75858 62329 26950 33117 20038 52385 79604 11603 71321 05834

53276 17696 04922 75651 40216

6312.9276 , 1)2.1000 7.6116 4.5825 3.3797

445 98025 21125 90063 18578 32671 73039 30002 98510 89837 37215 02311 06721 31864 83431

d4h

447

1 98916 887 16536 I 10 I 3.9567 57506 113 1.7647 13847 I 15 I 7.8706 23760 118 3.5102 98197 21 1.5655 92996 / 23 6.9825 44761 ,26 3.1142 14964 63)3.8373 95917 / 1)2.1118 71208 7.6403 21250 4.5955 09991 3.3874 03811,

1 99804 893 14623 10 3.9923 63648 13 1.7845 86551 15 7.9771 01882 18 I 3.5657 64541 21 1.5938 96750 23 1 7.1247 18472 26)3.1847 49157 (63)4.0493 05610 ( 1)2.1142 37451 7.6460 27242 4.5980 83787 3.3889 21465,

$

97132 00000 62611 75695 14445

._

1

[ 1 II3 (--6)4 4

(63)1.4277 ( 1)2.0688 7.5361 4.5484 3.3596

432 1 86624 806 21568

431

1 85761 800 62991

24

429

12523

._-

430

428 1 83184 784 02752

44370 16087 22043 23998 09138

63)1.5099 1)2.0712 7.5419 4.5510 3.3611

433 1 87489 811 82737

63)1.8867 1)2.0808 7.5653 4.5616 3.3674

28946 65205 54712

50145 22267

817

99040 44954 63318 62238 63549

1 88%6 46504

(63)1.9941 ( 1)2.0832 7.5711 4.5642 3.3689

438 1 91844 840 21672

(63)2.4852 ( 1)2.0928 7.5943 4.5747 3.3751

93273 31518 86732 78463 77583

846

30189 66666 74278 81614 76223 439 1 92721 04519

26)2.6585 (63)2.6251 ( 1)2.0952 7.6001 4.5773 3.3767

443._

1 96249 869 38307 10)3.8513 67000

875

52264 15920 32684 38502 71171 03314 444 1 97136 28384

26)2.9109 (63)3.2635

66867 43677

(63)3.4450

( y.;;;;

y;;;

( 1) ;. 16;;; ;;‘b;;

415877 3.3828

62546 34454

16313

4:5903 3;3843

448

449

2 ooioi 899 15392

3.2567 (63)4.2724 ( 1)2.1166 7.6517 4.6006 3.3904

17891 04226 01049 24131 53268 36406,

49388 60316

2 01601 905 18849

26)3.3301 (63)4.5072 ( 1)2.1189

47041 55570 62010

7.6514

13148

4.6032 3.3919

18450 48644

C-47)4 I

42

ELEMENTARY

Table

3.1

ANALYTICAL

POWERS

AND

METHODS

ROOTS

nk

k 450 2 02500 911 25000

: : 2 7 9” 10 24

l/2 l/3 l/4 l/5

(b3)4.7544 ( 1)2.1213 7.6630 4.6057 3.3934

1

50505 20344 94324 79352 58190

917

(63)5.0146 ( 1)2.1236 7.6687 4.6083 3.3949

455 2 07025 941 96375

: 4

451 2 03401 33851

946

08183 76058 66491 35988 b5055 456 2 07936 188lb

2 l 9 10 24 :g l/4 l/5

923

452 2 04304 45408

(b3)5.2883 ( 1):

;;ii 4:6108 3.3964

954 10 I 4.3617 13 1.9933 15 9.1095 1.9025

(63)6.1983 ( 1)2.1330 7.6913 4.6185 3.4009

13235 72901 71681 20218 65915

(b3)6.5336 ( 1)2.1354 7.6970 4. b210 3.4024

.__

973

55778

59532

b3)6.8863 1)2.1377 7.7026 4. b235 3.4039

461 2 12521 979 72181

46”

: 3

55383 15650 02263

2 11600 36000

w

10

(63)8.0572

70802

( 1)2.1447 4.6311

61059 42629 56507

3.4084

07924

7.7194

l/4 l/5

(63)8.4883 ( 1)2.1470 7.1250 4.6336 3.4098

465 2 16225

: 1005

: 5

44625

29103 91055 32380

(63)5.5764

;;y;

( 1’;. yo’

08317 b9249

(63)8.9414 ( 1)2.1494 7.7306

4.6361 3.4113

71390 88554

2 17156

1018

94696

453 2 05209 59677 73368 lb236 01548 00201 13891

4:b134 3.3979

457 2 08849 43993 90480 38249 55800

96;

21619 45396 55833 24618 87171 50532

(63)7.2572 ( 1)2.1400 7.7082 4.6261 3.4054

(63)5.8795

01000

( 1) ;. y5”;

y;

416159 3.3994

458 09764 71912 93570 42855 :'4::: 82342 57127 03764 39774 93456 38770 14413 38923

16664

967

00476 69669 459 2 lob81 02579

( b3)7.6472 1)2.1424 1.7130 4.6286 3.4069

35292 28529 44772 37519 24718

464 2 15296 998 97344

73356

38903

(63)9.4176

18526 14052 82186 66616

( 1)2.1517 1.7361 4.6386

467 2 18089 47563

37619 19665 85719 36534 70784

2 Oblib 935

463 14369 9922 52847 06816

462

466

1011

77338

13444 11128 34114 95360 54565 59409 b2447 98505 63510

: 7" 8 9 10 24

924

3.4128

1025

27630 72096 76381 24642 93909 76852 43479 87677 88909 42121

(63)9.9181 ( 1)2.1540 7.7417 4.6411 3.4143

468 2 19024 03232

69666 65923 53281 91574 15079 469 2 19961

1031

61709

7" 9" 10 24

l/2 :;:

(b4)1.0444

09634

( 1)2.1563 7.7473

85865

4.6436

l/5

3.4157

10895 90198 85500

64)1.0996 1)2.1587

60547 84795

3.4172

53393

470 10 I 13 16 18 21 I 24

1038 4.8796 2.2934 1.0779 5.0662 2.3811 1.1191

26) 5.2599

24 :::

69046 03314

7.7520 4. b461

24259 18278

7.7504

02264 75300

4.6486 3.4187

471

2 20900

23000 81000 50070 21533 31205

1044

2 21841

87111

1051

18768

472 2 22784 54048

(64) 1.2187 ( 1)2.1633 7.7639 4.6511 3.4201

10278

(b4)1.2827

30765

(

36077 61968 81635

1)2.1656

7.7694 4. b53b 3.4216

.._

2 23729

23817

44575

42003 474 2 24676

471

1058

68318 40783 62012

1064

96424

286b6

30473 13224

(64)1.3500

46075

(b4)1.4206

98007

( 1) ;. ;;;;

y;;

( 1)2.1702

53441 90361 97902 55283

4:b561 3.4230

1.1577 ( 1)2.1610

(64)

23215 99883

7.7804

4.6585 3.4245

(64)1.4948 ( 1)2.1725

85630 56098

7.7059

92032

3.4260

08213

4.6610

68652

(b4)1.5727 ( 1)2.1748 7.7914 4.6635 3.4274

77826 56317 87536

(64)1.6545 ( 1)2.1771 7.7969

51159 54106 74500

35480

4.6659

98399

50603

3.4289

06701

ELEMENTARY

ANALYTICAL

POWERS

AND

43

METHODS

ROOTS

rzk

Table

3.1

k

475 2 25625 71875

:

1071

: 2 .z 1:

24

l/2 l/3 l/4 l/5

II! 10 lb 13 18 21 24

5.0906 2.4180 1.1485 5.4557 2.5914 1.2309 26 5.8470 (64) 1.7403 ( 1)2.1794 7.8024

60126 86060

55878 40422 90207 49472 53153 57424

3.4303

2 30400 92000

;lO I 5.3084

l/2 l/3 l/4 l/5

lb000 39680

2.5480 1.2230 5.8706 2.8179 1.3526

59046 83423

28043 05461 ObZll

26 6.4925 (b4)2.2376 (

31322

1)2.1908 1.0291 4. b8Ob

90230 35282 94639

3.4375

43855

1140

64) 1.8304

1)2.1817 7.8079 4.6709

52278

1105

24

476 2 265% 50176

81079 65430 b4063

4.6604

13 :lb 18 '21 '24 I

1078

3.4317

1112 10)5.3527 13 I 2.5746 lb 1.2384 1.3 5.9568

87912 42423 25322 12569 95422

79188

6. b290

41895

3.4389

41094

1147

l/3 l/4 l/5

(b4)2.8694 ( 1)2.2022 7.8568

4.6928 3.4446

1176 10)5.7648 13 2.8247 lb 1.3841 '18 6.1822 21 I 3.3232

70250 71555 28008 36620 75750

b4)3.0148

( l';;?;; 4: b952 3.4460

(b4)2.0242

75033

32967

3.4332

36143

( 1)2.1863 7.8188 4.6758

21111 45511 11278

3.4346

74449

1119

7.8405 4.6855 3.4404

14973 2 3b196 91256

482 2 32324 80168

57971

(b4)2.4724

1155

94846 b2762

03713

487 2 37169 01303

( 1)2.2135

94362

( y.

2.9718

lb I 1.4710

18 7.2817 21 3.6044

24

l/2

l/3 l/4 l/5

4.6976 7.8676

3.4475

1126 10 I 5.4423 13 2. b28b lb 1.2696 18 6.1323 21 I 2.9619 24 1.4306

35163 85081

'8;;; 4:7072 3.4531

49066

6.9783

3.4333 1.6892 8.3109

4: 7096 3.4545

;;;"7;

3.4418

1162 10 1 5.6712

21 3.2163 24 1.5695 26 7.6594 b4)3.3271 1)2.2090 7.8729 4.7000 3.4489

10 13 lb 18 I 21 24

488 2 38144 14272

10) 13 lb I 18 21

75702 09424

14999 61719 61191 75643

b4)3.4947

21879

72203

1)2.2113

34439 68425

94366

7.8783 4.7024 3.4503

76812 26700

lZO!i

89849 58895 91354 97638

71635

b99bO

3.0629 1.5253

y.9;;

( 1)2.2315

98967

7.5963

3.7829

53784

81640

91695

6.1505

82790 39037 494 ;! 44036

2 43049 23157

4.7120 3.4559

417215 3.4615

.:[(,5]

489 2 39121 1169 30169

73159

7.8997

1235

24411 15760 54092

56474

60331

( 1’;. ;;y;

+p6)1]

30083

1)2.2203

62231

55329

7.8514 4. b904 3.4432

32162 84825

22124 61227

( 1’:

19620

26 I 8.4814 64)4.2493

8”;;;;

59546

45383

I_

1198 5.9072 2.9122 1.4357 7.0782 3.4895 1.7203

;;;;

1)2.2248

92097

00000

417192 3.4601

(b4)4.9154

67488 46397

92362

64)5.4138

56201 06649

79904

( 1)2.2000

78653

23936 87226 84064 84096

484 ; ! 34256

1133

(b4)2.7307

55098

26)8.8318

64)4.6830

4-;)3]

ii,":

54870 10350

26098 13365 91145

(b4)5.1588

24 I 1.7842

41683 64874

( 1';.

71692 18873 56854 72689

06863

94186

50361

1)2.1977 7.8460 4.6879

61115 39113 17000 08320 15513

71463 13374

59893

y;;

73526 97749

95451

92764

46016 122;

25063 43906 62733 60531

1.9104

(64)4.0472

2 42064 95488 98010 73021

(b4)2.1283 ( 1)2.1886 7.824;! 4. b78:! 3.4361.

b4)2.5985

.._

2 45025 87375

4.7168 3.4561

12045

03609 02524 50214 07962 17248

26 6.9098

.._

71639 91376 83697 56794

67133 12960

;! 29441 02239

/ 18 lb 1 2.5216 13 10 21 2.7713 1.2070 5.2643 5.7856

483 2 33289 78587 75752

497

1190 5.6594 2.8828 1.4183

68562

l/2

10 6.0037

53740 95065

40045 (b4)3.8543

13

07649

63943 14959

30728 93057

36822

1212

42798

( 1)2.2068

94384

(b4)3.6703

7.8837 4.7048 3.4517

(64)3.lb73

ii-;;

04856

24 l/3 l/4 l/5

82996

491 41081 70771

2 40100 49000 01000 52490 28720

99311 80708 55578 19966 01440

45935 89232 63849

419

1099

93826 96049

1.9250 ( Q2.1840 7.8133 4.6733 (64)

26 b.2270

13 2.1675 lb) 1.3505 18 I 6.5908

l/2

478 2 28484 15352

49036

486

2 352% 84125

1092 10 I 5.2204 13 2.4953 lb 1.1927 18 5.7015 21 I 2.7253 24 1.3027

6.0979

481 2 31361 84641 91232 92583 27132 34506

1.3781 b4)2.3522

477

2 27529 1085 31333 10 1 5.1769 44584 13 2.4694 02567 lb 1.1779 05024 18 5.6186 06966 21 I 2. b800 75523

b4)4.4611 1)2.222B

7.9051 4.7144

65384

3.457'3

498 2 48004 05992

1242

49467 11077

29393 57633 66263

499 :! 49001 51499

98402

10) 6.200'1

98004

13)3.0938

49800 74750

73006 57570 86070

18 7.1031 21 3.8441

79067 85754

24 1.9182 26 I 9.5720

48691

16)1.5438

25162 91360

7.9264 4.7239

12221

3.4629

47190

08444 1

43500

60970

(64)5.6808

47029

( 1)2.2338 7.9317 4.7263 3.464'3

30790 10391 4191b

q-37)3]

36816

.

44

ELEMENTARY Tahlc

24

l/2 l/3 l/4 l/5

3.1

POWERS

1250 10 6.2500 13 I 3.1250 16)1.5625 18)7.8125 21)3.9062 24)1.9531 26)9.7656 64)5.9604 1)2.2360 7.9370 4.7287 3.4657

500 2 50000 00000 00000 00000 00000 00000 50000 25000 25000 64478 67977 05260 08045 24216

505 2 55025 1287 87625 10)6.5037 75063 13)3.2844 06407 16 1.6586 25235 18 1 8.3760 57438 21)4.2299 09006 24

l/2 l/3 l/4 l/5

64)7.5682 1)2.2472 7.9633 4.7404 3.4726

08268

20505 74242 85740 28104

510 2 60100 1326 51000 (10)6.7652 01000 52510 13)3.4502

24

(64)9.5&?70

3.4794

1365

24

l/2

l/3 l/4 l/5

(65)1.2116 ( 1)2.2693 8.0155 4.7637 3.4862

1406

24

l/2 l/3 l/4 l/5

ANALYTICAL

(65)1.5278 ( 1)2.2803 8.0414 4.7753 3.4930

33090

77522 515 2 65225 90875

39706 61144 94581 81212 73428 520 2 70400 08000

48342 50850 51517 01928 16754

+;)a]

2

126;

1257

64)6.2532 1)2.2383 7.9422 4.7310 3.4671

44659 02929 93073 70628 09398

506 2 56036 1295 54216 (10)6.5554 43330 (13)3.3170 54325 (16 1.6784 29488 (18 1 8.4928 53211 (21)4.2973 83725 24)2.1744 76165 27)1.1002 84939 (64)7.9361 96349 ( 1)2.2494 44376 7.9606 27129 30775 4.7428 3.4740 02314 511 2 61121 32831 1334 6.8184 17664 3.4842 11426 32039 16)1.7804 9.0980 07719 81944 4.6490 (65)1.0048

50848

3.4808

1373

(65)1.2693 ( Q2.2715 8.0207 4.7660 3.4076

1414 10 I 7.3680 13 3.8387 16 I 1.9999 19 1.0419 21 5.4287

40954 516 2 66256 88096

83471 63338 79314 92045 26271 521 2 71441 20761 21648 39279 83164 91229 74301

1.4735 (65)1.5999 ( 1)2.2825

91925 46126 42442

8.0466

02993

4.7775 3.4943

AHD

96092 59190

.:[(-,P]

(27)1.0163 (64)6.5597 ( 1)2.2405 7.9475 4.7334 3.4684

METHODS

ROOTS

rth-

502 2 52004 06008

1272

35678 79050 35650 73855 29676 92370

507 2 57049 1303 23843 10)6.6074 18840

64)8.3212 1)2.2516 7.9738 4.7451 3.4753

1342

27)1.2379 65)1.0531 1)2.2627 8.0000 4.7568 3.4822

1381

65)1.3297 1)2.2737 8.0259 4.7683 3.4889

1422

(65)1.6752 ( 1)2.2847 8.0517 4.7790 3.4956

55408

(64)6.8806 ( 1)2.2427 7.9528 4.7357 3.4698

84448 66149 47628 85203 73139

(64)8.7242 ( 1)2.2538 7.9791 4.7475 3.4767

512 2 62144 17728

16 1.8226 18 9.3502 21 4.7966 24 2.4606 ii27 1.2623 (65)1.1036 ( 1)2.2649 8.0052 4.7591 3.4835

517 2 67289 88413

53822 14108 59837 06497 32173 12886 50331 04946 49431 61427

522 2 72484 36648

1430

(65)1.7540 ( 1)2.2869 8.0568 4.7821 3.4970

ni[(-;)S]

16064

(64)7.2166 ( 1)2.2449 7.9581 4.7381 3.4712

04000 94432 14416 37221 51715

509 2 59081 1318 72229 (10)6.7122 96456

523 2 73529 55667

91340 69982 57150 93433 04452 72252 524 __. 2 74576

1438

(65)1.8363 ( 1)2.2891

44200 19325

18034 56810 03133 67011 18483

519 2 69361 1397 98359 10 7.2555 34832 13 3.7656 22576 16 1.9543 58118 19 1.0143 11863 21 I 5.2642 78570 1.4179 (65)1.4588 ( 1)2.2781 8.0362 4.7730 3.4916

81704 61335 28718 03654 25675

06897 02835 44383 45086 11950

514 2 64196 1357 96744

65)1.1564 1)2.2671 8.0104 4.7614 3.4849

518 2 68324 1389 91832

(65)1.3928 ( 1)2.2759 8.0311 4.7707 3.4903

3.2520

(64)9.1459 ( 1)2.2561 7.9843 4.7498 3.4781

69942 85534 12176 10436 44229

513 2 63169 1350 05697 6.9257 92256

40039 22917 41700 00000 28460 02253

98008 31932 47881 86957 99566

63527

(10)6.4013

2 58064 1310 96512

97020 66050 73099 72336 74353

59294 63400 57353 99522 77017

503 2 53009

17824

30669 04628

86203

8.0620

17979

74532 37889

4.7844 3.4983

58829 74167

1

$[‘-;‘“I

ELEMENTARY

POWERS

ANALYTICAL

METHODS

AND

nk

ROOTS

45 Table

3.1

k :

3 4 5 6 7 8 9 10 24

1447 10)7.5969 13)3.9883 16 2.0938 19 11.0992 21)5.7713

525 2 75625 03125 14063 79883 99438 97205 10327

(65)1.9223 ( 1)2.2912 8.0671 4.7867 3.4997

l/2 l/3 l/4 l/5

09365 87847 43230 39859 08406

526 2 76676 1455 110 7.6549 113 1 4.0265 I16 2.1179 I19 1.1140 I 21 5.8598 124 I 3.0822 127)1.6212 , 65)2.0121 / 1)2.2934 8.0722 4.7890 3.5010

530 2 80900 1488 77000 10)7.8904 81000

: : 2 7 9" 10 24

l/2 l/3 l/4 l/5

27)1.7488 (65)2.4133 ( 1)2.3021 8.0926 4.1980 3.5063

74704 53110 72887 12335 96379

49267

535 2 86225 1531 30375 (10)8.1924 75063

24

l/2 l/3 l/4 l/5

(65)3.0233 ( 1)2.3130 8.1180 4.8093 3.5129

1574

66304 06701 41379 72829 40196

:65)2.5250 : 1y;; 4:8003 3.5076

31576

60898 09432 43961 385'24 42634 17226 77821 38448 68988 61977 17632 40614 531 81961 21291 00552 56493 46498 14290 68R82 22076 53022 41417 43724 58868 58033 71420

1463 10)7.7133

(65)2.1059 ( 1)2.2956 8.0113 4.7912 3.5023

94141

24056 37598 02002 (65)2.6416 ( 1)2.3065 8.1028 4.8026 ' 3.5089

!j36

540 2 91600 64000

1583

57'189 49669 67381 96201 18626 52463

82534 48057 74241 92160 70797 532 83024 68768 58458 57499 95390

13716

12519 39019 16494 91583

(65)3.3066 ( 1)2.3173 8.1281 4.8138 3.5155

541 2 92681 40421

1592

09101 26045 44739 61283 62774 542 2 93764 20088 28710

I 10 13 I 4.6773 16 2.5351 8.6297 19)1.3740

24

l/2 l/3

:/,:

65) 3.1796 1)2.3237 8.1432 4.8205 3.5194

: : 5

1618

38253

(65)3.9512

48669

II 21 24 1.4472 2.1877 4.0363 (65)4.1303

90008 52850 70514 82029

( 1';;:;;

y;

( 1y;;

545 2 97025 78625

4:8228 3.5207

00711 84516

546 2 98116 1621

71336

1471 10 7.7720 13 4.1036 16)2.1667 19)1.1440 21 6.0404 24 I 3.1893 27)1.6839 (65)2.2040 ( 1)2.2978 8.0824 4.7935 3.5036

4:8250 3.5220

1636 10)8.9526

12993 03642 26174 21864 94250 25684 12169 89345 93862 27819 85199

528 2 78784 97952 51866 43385 23707 30117 79020 72923 88903 12944 25059 80041 63454 98962

(10)7.8310

(65)2.7634 ( 1)2.3086 8.1079 4.8048 3.5103

1601 10 8.6935 13 4.7206 16 2.5632 19 1.3918 21 I 7.5578 2.2284 65)4.3171 1)2.3302 8.1583 4.8272 3.5233

547 2 99209 67323 02568

1645

83632 10416 50910 55831 07963 00000 79399 31523 25117 534

58943 79276 12808 71774 09762

98937 82701 87014 00810 71134

529 2 79841 35889 98528

2 85iG 1522 73304 10 8.1313 94434 13 14.3421 64628 16 2.3187 15911 19 j 1.2381 94297 21 6.6119 57543 24 1 3.5307 85328 27)1.8854 39365 65)2.8906 14446 1)2.3108 44002 8.1129 80255 4.8071 23882 3.5116 25964

538 2 89444 1557 20872

(65)3.4575 1)2.3194 8.1331 4.8161 3.5168

1480

19 1.1592 21 6.1326 I 24 I 3.2441 (2711.7161 (65)2.3064 ( 1)2.3000 8.0875 4.7958 3.5050

533 2 84089 1514 19437 10)8.0706 55992 13)4.3016 59644 16 2.2921 84590 19 1.2220 54187 21 I 6.5135 48814

537 2 88369 1548 54153

2 '37296 1539 90656 8.2538 99162

(27)1.9572 (65)3.1619 ( 1)2.3151 8.1230 4.8116 3.5142

527 2 77729 63183 39744

539 2 90521 1565' 90819 10)8.4402 45144

(65)3.6151 ( 1)2.3216 8.1382 4.8183 3.5181

543 2 94849 03007 93280 21151 97285 70426 56412 26405 37789 36040 05107 51847 83903

1609

(65)4.5120 ( 1)2.3323 8.1633 4.8294 3.5246

548 3 00304 66592

1654

94612 39982 69477 26138 48871

(65)5.6199 f 112.34311 ' -'8; i882 4.8405 3.531:l

83652 37353 23044 31217

77550

544 2' 95936 89184

46770 80758 10204 72806 80696 549 3 01401 69149

7" : 10 24

l/2 l/3

(65)4.7153 ( 1)2.3345 8.1683 4.8316 3.5259

73024 23506 09170 90704 75582

+;)3]

(65)4.9274 ( 1)2.3366 8.1733 4.8339 3.5272

63602 64289 02026 05553 68570

(65)5.1486 ( 1)2.3388 8.1782 4.8361 3.5285

;[(-47181

79188 03113 88788 17361 59664

(65)5.3793 ( 1)2.3409 8.1832 4.8383 3.5298

+37)41

$-?3]

99369 74903 44110 31895 36198

46

ELEMENTARY

Table

3.1

ANALYTICAL

POWERS

AND

METHODS

ROOTS

nk

k 550 3 02500

1663 (10)9.1506

75000 25000

13)5.0328

43750

16)2.7680

551 3 03601 1672 84151 10 19.2173 56720 13 5.0787 63553 16 2.7983 98718 19 1.5419 17693 21 I 8.4959 66491 24 1 4.6812 17536

64063

27

24

l/2 l/3 l/4 l/5

(65)5.8708 98173 ( 1)2.3452 07880 8.1932 12706 4.8427 34641 3.5324'21650

1709

24

l/2 l/3 l/4 l/5

(65)7.2951 ( 1)2.3558 8.2179 4.8537 3.5388

(10)9.8344

24

m l/3 l/4 l/5

1756

3.0330 (65)9.0471 ( 132.3664 8.2425 4.8645 3.5451

54891 67858 31913 70600 98558 74407

l/3 l/4 l/5

(66)1.1198 ( 1)2.3769 8.2670 4.8754 3.5514

1851 11)1.0556

24

w l/3 l/4 l/5

11516

( 1)2.3473 8.1981 4.8449 3.5337

38919 75283 34384 05234

(65)7.6171 ( 1)2.3579 8.2228 4.8558 3.5400

560 3 13600 16000 96000 %"6 94850

l);.:t%: 4:8667 3.5464

57461 72865 29409 20869 82586

44725

(66)1.3835 ( 1)2.3874 8.2913 4.8861 3.5577

55344 67277 44342 71586 46263 1

q-y]

556 3 09136 79616 06650 17697 60240 68693

93672 65225 98519 88409 95340 561 3 14721 58481

1861 11 l.Ob30 13 6.0698 16 3.4659 19 r 1.9790 22 1.1300 24 6.4524 27 3.6843 (66)1.4430 ( 1)2.3895 8.2961 4.8883 3.5589

1691 10 9.3519 13 1 5.1716 16 2.8598 19 1.5815 21 8.7458

24 27

24 4.8364 27)2.6745

44203 53644

(65)6.4052 ( 1)2.3494 8.2031 4.8471 3.5349

76258 68025 31859 31136 86956

(65)6,6896 1)2.3515 ( 8.2080 4.8493 3.5362

46227 95203 82453 24905 66821

1728 10)9.6254

(65)7.9528 ( 1)2.3600 8.2278 4.8580 3.5413

1775

557 3 10249 08693 44200

84664 84744 25361 70341 67840 562 3 15844 04328

39138 71525 53918

68801 39637

4.8689 3.5477

36145 03064

571 3 26041 69411 27337 86093 04959 31732 27119 54848 51718 00887 b0629 90248 13236 93720

.;[,,)8]

12377 14448 08690 99605 24482 30384

98682 80873

(65)9.8553 ( 1)2.3706 8.2523

75451 03838 76704 38859

1

553 3 05809

4.7582 2.6265

71309 7";;;;

566 3 20356 1813 21496 11 1.0262 79667 13 5.8087 42917 16 3.2877 48491 19 1.8608 65646 22 1.0532 49956 24 5.9613 94749 27 3.3741 49428 (66)1.1684 07534 ( 1)2.3790 8.2719 4.8775 3.5527

570 3 24900 93000 00100

3.4296

1765

( 65)9.4429

565 3 19225 62125

83922

(65)6.1325

1718 10 9.5565 13 ! 5.3134 16)2.9542 1.6425

05803 43798 65765 03532 21007

I ::I :: F% 1.7270

1803

l/2

555 3 08025 53875

2.5793

552 3 04704 1681 96608 10)9.2844 52762 13 I 5.1250 17924 16 2.8290 09894 19 1.5616 13462 21 I 8.6201 'I6308

1822

3.4342 (66)1.2189 ( 1)2.3811 8.2767 4.8797 3.5539

187:

I 22 27 I 1.1459 24 3.7493 6.5548 (66)1.5048 ( 1)2.3916 8.3010 4.8904 3.5602

567 3 21489 84263

39578 71112 76180 72529 29685 93358 572 27184 49248 93699 23956 84103 20907 56759 72660 87161 89774 52149 30501 52074 39430

;[‘-p”]

558 3 11364 1737 41112 10 9.6947 54050 13 1 5.4096 72760 16)3.0185 97400 19)1.6843 77349 21)9.3988 25608 65)8.3027 1)2.3622 8.2327 4.8602 3.5426

1784

27)3.1995 66)1.0284 1)2.3727 8.2572 4.8711 3.5489

1832

(6631.2716 ( 1)2.3832 8.2816 4.8818 3.5552

27311 02362 46311 49337 38514

1700

(65)6.9860 ( 1)2.3537 8.2130 4.8515 3.5375

1746 (10)9,7644 1.7056 9.5344 24)5.3297 27)2.9793 (65)8.6672 ( 1)2.3643 8.2376 4.8624 3.5439

563 3 16969 53547

13511 93323 62104 63270 00598 64695 568 3 22624 50432

27927 75058 35499 79820 46087

1794

53980 17896 41841 65115 88109 8340;

92851 20459 27082 15700 44836 559 3 12481 76879 37536 21474 24040 43038 26358 91224 18084 61384 25407 07368 564 3 18096 06144

66)1.0732 1)2.3748 8.2621 4.8732 3.5502

44065 68417 49226 62170 24533

1842 11)1.0482 13 5.9643 16 I 3.3937 19 1.9310 22 j 1.0987

569 3 23761 20009 11851 25433 01172 15967 48085

3.5573 (66)1.3264 ( 1)2.3853 8.2864 4.8840 3.5564

573 3 28329 1881 32517

27)3.8154 66)1.5693 ( 1)2.3937 8.3058 4.8925 3.5614

3 06916 31464

17788 60719 72088 92764 27117 97054

574 1 79476 1891 19224

(66)1.6363 ( 1)2.3958 8.3106 4.8947 3.5627

1$[‘-321

84728 29710 94107 21351 25633

ELEMENTARY

ANALYTICAL

POWERS

47

METHODS

AND ROOTS

nk

Table: 3.1

k :

3

1901

575 3 30625 09375

1911

576 3 31716 02976 11 13 16 I 19

5" 6 i 1: 24

l/2

l/3 l/4 l/5

(66)1.7061 ( 1)2.3979 8.3155 4.8968 3.5639

93459 15762 17494 51807 66137

l66)1.7788 I 1)2.4000 8.3203 4.8989 3.5652

:

51122 00000 35292 79486 04916

1961

192: 1.1084 6.3955 3.6902 2.1292

66)1.8544 1)2.4020 8.3251 4.9011 3.5664

577 32929 00033 17190 67189 42268 69789 88668 56614 27966 68735 82430 47517 04396 41976

582 3 30724 1971 37368

3 375'61 225141

:

193:

(66)1.9331 ( 1)2.4041 8.3299 4.9032 3.5676

570 34084 00552 21191 80481 82318 36180 26512 99239 72960 61432 63056 54185 26546 17321

583 3 39889 1981 55287 1.1552 45323

579 3 35241 04539 65281

1941 11)1.1238

;66;2.0150 ( 1)2.4062 8.3347 4.9053 3.5689

48620 41883 55313 45944 10958

584 3 41;56 1991 76704

6' ; 9 10 24

l/2 :g

l/5 : 3

(66)2.1002 ( 1)2.4083 8.3395 4.9074 3.5701

2002 1.1711

54121 18916 50915 62599 42892

(66)2.1889 ( 1)2.4103 8.3443 4.9095 3.5713

585 3 42225 01625 79506

2012

06331 94:159 41009 76!j18 73'L27

66)2.2811 1)2.4124 8.3491 4.9116 3.5726

586 3 43396 30'356

2022

38380 67616 25609 87710 01670 587 3 44569 62003

5" ; : 10 24

l/2 :;:

l/5

1 (66)2.5807 ( 1)2.4186 8.3634 4.9180 3.5762

2053

(66)3.1655 ( 1)2.4289 8.3672 4.9284 3.5823

2106 II (11 1.2533 13 7.4573 16 4.4371 19 2.6400

24 ::: l/4 l/5

I ::I k :'42 (27j5.5612 (66)3.8762 ( 1)2.4392 8.4108 4.9388 3.5884

19397 77324 46607 05007 77194 590 3 48100 79000

43453 91560 06527 80050 69695 595 3 54025 44875 37006 55187 26336 90170 :z 14639 08928 62184 32585 80725 21030

(66)2.6887 ( 1)2.4207 8.3682 4.9201 3.5774

2064

(66)3.2968 ( 1)2.4310 8.3919 4.9305 3.5835

2117

(66)4.0356 ( 1)2.4413 8.4155 4.9409 3.5896

02707 43687 09391 05372 99018

27)4.8571 (66)2.8010 ( 1)2.4228 8.3729 4.9222 3.5787

591 3 49281 25,071

52680 49156 42387 67063 83235 596 3 5'5216 089736

19703 11123 41899 62581 26411

2074

(66)3.4333 ( l)?;;;; 4:9326 3.5847

2121 11 1.2702 13 7.5835 '16 I 4.5273 '19 2.7028 ;22 I 1.6135 24 9.6331 27 5.7509 (66)4.2013 ( 1)2.4433 8.4202 4.9490 3.5908

44372 08521 08288 66760 03051 19175 592 3 50464 74688

72793 '3;;;; 51429 95134 3 56409 76113 73753 34304 69980 39878 95407 64580 99254 02448 58345 45948 33830 30176

(66)2.3770 ( 1)2.4145 8.3539 4.9137 3.5738

88299 39294 04732 96184 28526

2032 'ii i.1953 :13 7.0288 16 '4.1329 '19 I 2.4301 22 1.4289 '24 I 8.4022 27 4.9405 66)2.9178 1)2.4248 0.3777 4.9242 3.5799

588 3 45744 97472 89135 88116 86212 95893 55185 56487 26815 02055 71131 18728 98052 37670

2085 11 1.2365 13 7.3328 16 I 4.3483 19 2.5785 22 1.5291 24 I 9.0675 27)5.3770 (66)3.5753 ( 1)2.4351

593 3 51649 27857 70192 61239 06715 93322 05840 97630 85394 01250 '59132

8.4013 4.9347 3.5860

213'3

98104 33156 05396 598 3 57604 47192

4.6145 (66j2.4768 ( 1)2.4166 8.3506 4.9159 3.5750

40597 99188 09195 70393 01946 53698

589 3 46921 2043 36469

(66)3. 039:! ( 1)2.4269 6.3824 4. 926'3 3.5811

2095

54545 32220 65312 90382 54508

594 3 52836 84584

7.3948

98628

27)5.4684 (66)3.7228 ( 1)2.4372 8.4061 4.9368 3.5872

52572 42640 11521 17992 12252 14026

2149

599 3 58801 21799

.

(66)4.3734 ( 1)2.4454 0.4249 4.9451 3.5920

92798 03852 44747 02478 32329

(27)5.9465 (66)4.5524 ( 1)2.4474 8.42'?6 4.9471 3.5932

93118 34829 47650 38310 68534 32875

48

ELEMENTARY

Table

ANALYTICAL

POWERS

3.1

2160

(66)4.7383 ( 1)2.4494 8.4343 4.9492 3.5944

2214

(6635.7826 ( 1)2.4596 8.4576 4.9595 3.6004

2269

(66)7.0455 ( 1)2.4698 8.4809 4.9697 3.6063

3 60000 00000

81338 89743 26653 32004 31819

77757 74775 90558 10838 02669 610 3 72100 81000

68477 17807 26088 26156 34171

33286 19354 34993 70868 26906

620 3 84400 2385 28000

(67)1.0408 ( 1)2.4899 8.5270 4.9899 3.6180

66)4.9315 1)2.4515 8.4390 4.9512 3.5956

605 3 66025 45125

615 3 78225 2326 OLi375

(66)8.5704 ( 1)2.4799 8.5040 4.9798 3.6122

2170

79722 79920 18983 69859 81437

(66)6.0164 ( 1)2.4617 8.4623 4.9615 3.6015

2280

(66)7.3280 ( 1)2.4718 8.4855 4.9717 3.6075

601 3 61201 81801 61624 16360 50833 82950 41953 87314 53756 94142 '0% 92896 29165 606 67236 45016 22797 54150 28415 92819 83448 82770 27505 86963 06725 47078 58954 92098 611 3 73321 99131

60494 41419 57944 61679 15802

616 3 79456 2337 44896

(27)7.8669 (6638.9112 ( 1)2.4819 8.5086 4.9819 3.6134

63254 18488 34729 41730 01975 00850

621 3 85641 2394 83061

(67)1.0819 ( 1)2.4919 8.5316 4.9919 3.6192

28109 87159 00940 80728 47808

METHODS

AND ROOTS

218:

(66)5.1323 ( 1)2.4535 8.4436 4.9533 3.5968

223:

rzk

602 624 4 672 i! 8 66592 66885 93065 ::z: 08947 21860 44384 68829 87734 51218 24918

2192

(66)5.3409 ( 1)2.4556 8.4483 4.9554 3.5980

607 68449 48543 46656 08202 67079

--.

603 3 63609 56227

12849 05832 60500 06978 19083

2203

(66)5.5575 ( 1)2.4576 8.4530 4.9574 3.5992

608 69664 55712

2258

3 64816 48064

90288 41145 28104 60182 11665 609 3 70881 66529

'3:::: (66)6.2593 ( 1)2,4637 8.4670 4.9636 3.6027

%i 40623 36999 00076 04536 79959

2292 ;ll 1.4028 13 8.5853 '16 5.2542 '19 1 3.2155 '22 1.9679 '25 I 1.2043 '27 7.3707 (66)7.6213 ( "UN',;

612 3 74544 20928 32079 32326 23383 84711 37843 77960 93114 89047 0~;;

4;9737 3.6086

94704 95885

(66)6.5115

3.6039

2406

(67)1.1245 ( 1)2.4939 8.5361 4.9939 3.6204

68280 48470 43484 22621 73271 622 3 86884 41848

25305 92783 77980 89170 12677

(66)6.7735

66255

(66)7.9259 ( 1)2.4758 8.4948 4.9758 3.6098

z:::: 46261 45058 10121 05604 20352 51097 83681 06516 25239 74428

66)9.6321 1)2.4859 8.5178 4.9859 3.6157

618 81924 29032 59418 15202 70395 59704 87297 10750 48432 53659 60579 40269 40813 44173

2418 11 1.5064 13 1 9.3851 16 5.8469 19 1 3.6426 2.2693 8.8080 (67)1.1687 ( 1)2,4959 8.5407 4.9959 3.6215

231:

(66)8.2421 ( 1)2.4779 8.4994 4.9778 3.6110

614 76996 75544 59840 35419 92747 68947 79533 67433 42041 57465 02339 23260 53291 51433

I 22 19 I 2.1553 16 3.4820 5.6252

86668 76757 46313

(67)1.0013 ( 1)2.4879 8.5224 4.9879 3.6169

26192 71061 32097 56556 13560

242:

(67)1.2145

( y;‘: 1

50991

619 3 83161 2371 76659

623 3 88129 04367 41206 28716 35190 40623 65108 64101 27115 96795 50116 95191 76049

29447

3.6051

613 75769 46397

617 3 80689 2348 85113

66)9.2649 1)2.4839 8.5132 4.9839 3.6145

72833

4:9979 3.6227

624 89376 70624 36694 z; 66786 70474 70376 71145 91262 99199 17363 98799 37928

1ELEMENTARY

ANALYTICAL

POWERS 625

2441

24

l/2 l/3

$2

(67)1.2621 ( 1)2.5000 8.5498 5.0000 3.6238

3 90625 40625

77448 00000 79733 00000 98318

630 3 96900 2500 47000

m l/3 l/4 l/S

I 16 13 I 1.5752 11 6.2523 9.9243

50221 65430 96100

II 19 27 3.9389 25 22 2.4815 1.5633 9.6493 (67)1.5281 ( 1)2.5099

80639 02919 81416 57803 75339 80080 18882 70139

8.5726

5.0099 3.6296

2560

24

l/2 l/3 l/4 l/5

(67)1.8474 ( 1)2.5199 8.5952 5.0198 3.6354

78090

635 4 03225 47875

36020 20634 38034 81108 21280

14376 67994

81641 14307 14356 76187 80893 18391 47419 99201 37239 98801

3.6250

2512

57224

631 3 98161 39591

(67)1.5874 ( 1)2.5119 0.5771 5.0119 3.6308

66692 71337 52262 57040 29638 636

2572 11)1.6361

4 04496 59456 70140

(67)1.9185 ( 1)2.5219 8.5997 5.0218 3.6365

24

l/2 l/3 l/4 l/5

(67)2.2300 ( 1)2.5298 8.6177

5.0297 3.6411

2683

39634 04043 47604 56273 65574

2464

67)1.3627 1)2.5039 6.5589 5.0039 3.6262

2524 11 I 1.5953 14 1.0082 16 6.3723 19 I 4.0273 22 2.5452 25 1.6086 28 1.0166 67)1.6489 1)2.5139 8.5816 5.0139 3.6319

- .-

2633

74520 22128

(67)2.3152

22362

( 1) 2 ‘6;;;

‘2:;;;

4 10881 74721

2646

67)2.4034 1)2.5337

645 4 16025 36125

5:0316 3.6422

2695

97308 65548

8.6267 5.0336 3.6434

646 4 17316 86136

2708 (11)1.7523

24

w

l/3 l/4 l/5

(67)2.6880 ( 1)2.5396 8.6401 5.0395 3.6467

2476

65028 96805 89894 95209 14650

(67)1.4158 ( 1)2.5059 8.5635

5.0059 3.6273

632 3 99424 35960

2536

95318 89841 91794 51614 86220 20891 48403 59081 61018 80854 41581 79727

1.6316 (67)1.7127 ( 1)2.5159

80891 71892

24057 85020 22598 28767

99973

4--97

(67)2.7898 ( 1)2.5416 8.6445 5.0414 3. b479

47292 53005 85472 80939 30063

.;[(-;I”]

(67)2.8953 ( 1);

5:0434 3.6490

30845 58755

1

633 4 00689 36137

73988

23768 28361

2658

95163 94164 66188 52582 99626 49851

58638

II 19 22 3.8954 27 25 9.6940 21:5411 4502 (67)1.4710 ( 1)2.5079 8.5680 5.007') 3.6285

22049 20469 76765 06675 09545 87241 80703 80871 25079

2548

634 4 01956 40104

82992 17605 35581

2670

_ 648 .

2720

(67)3.0046 ( 1)2.5455 8.6534 5.0453 3.6501

“;[(-;)3]

4 19904 97792

93247 84412 97422 78492 86051

1

32334 44932 48015 67827 89842 644 4 14736 89984

67740 15508 55108 74325 68481

-..

2733

4’-37

82860 61719 35662 23720 03608 75544

639 4 08321 260') 17119

(67)2.5897 ( 1)2.537'7 8.6356 5.0375 3.6450

44467

629 3 95641 58189 50275 18009 39823

(67)2.1479 ( 1)2.5278 8.6132 5.0277 3.639')

643 4 13449 47707

3.1

I 11 13 I 1.565:) 16 6.1930 9.8458

(28)1.0492 (67)1.7788 ( 1)2.5179 8.5907 5.0179 3.6342

30535 49125 04672

(2Q1.1173 67)2.0686 1)2.5258 8.6087 5.0257 3.6388

647 4 18609 40023 34949

y;

70600

5.0159 3.6331

3.6445

61105 p9';

96309 92817 37711 89230

8.5062

(67)2.4949 ( 1)2.5357 8.6311 5.0356

06237 58602 01272

2488

73152

638 4 07044 2596 94072

642 4 12164 09288

30760

33719 28406

Table 620 3 94384

3 93129 91883

637 4 05769 2584 74853 11 1.6464 84814 14 1 1.0488 10826 16 6.6809 24963 19 4.2557 49202 22 2.7109 12241 25 1.7268 51098 28 I 1.1000 04149 67)1.9922 61654 1)2.5238 85893 8.6042 52449 5.0238 29110 3.6377 08430

64.1

4 09600 44000

nk

621

3 91876

_.

2621

AND ROOTS

626

2453 11 1.5356 13 9.6132 16 I 6.0179 19 3.7672 22 2.3582 25 1.4762 27 I 9.2415 67)1.3115 1)2.5019 8.5544 5.0019

49

METHODS

(67)3.1179 ( 1)2.5475 8.657') 5.0473 3.6513

4 21201 59449

75679

47841 46522 23886 11957

50

ELEMENTARY

Table

3.1

ANALYTICAL

POWERS

METHODS

AND ROOTS

nk

k 651 2146

4 23801

4 22500 25000

2758

94451

2771

10364 ;U;

1.4097 (67)3.6134 ( 1)2.5553

14868 66091 17457 19800 30929 16597 02582 86468

46611 81368

8.6756 5.0550 3.6558

97359 83054 01749

:lb

1.1531

19 5.0628 ‘22 I 3.3060 ;25 ,28

44710 09757 91053

5.0492 3.6524

67033 36416

(b7)3.3569

41134

( 1)2.5514

70164

8.6668

5.0512 3.6535

31029

07939 59612

(67)3.4829 ( 1);;::; 5:0531 3.6546

656

2810

(67)3.8885 ( 1)2.5592 8. b845 5.0589 3.6580 : : 5 : 8 1: 24

112 l/3 l/4 l/5

4 29025 11375

2823

81447 96778

93654

1)2.5612 8.6889 5.0608 3.6591

49695 62971

45603

49271 38399

---

2874 lQ1.8974 14 1.2523 lb 1 8.2653 19 5.4551 22 3.6004 25 2.3762 28 i 1.5683 (67)4.6671 ( 1)2.5690 8.7065 5.0685 3.6636

4 35600 96000 73600 32576 95002 60701

ObOb3

68001 36881 78950 46516 87691

11 '14 lb i 19 22 25 28

2888 1.9089 1.2618 8.3408 5.5132 3.6442 2.4088 1.5922

(67) 4.8398

( 1)2.5709

8.7109 5.0704 3.6647

76246

06215 665

:

4 42225 2940

2954

79625

79069 54676

2.1588

2835

3.6602

661 4 36921 04781

(b7)

69592

4.3393

17689

3.6613

83152

(b7)5.0187

05901

1);;;;; 5: 0124 3.6658

666 4 43556 08296

2967

(67)5.2038

;;;;t 11720

( 1)2.5748 8.7197 5.0743 3.6669

23896

b67 4 44889 40963

298;

664 40896

_-.4 54944 -

2927

34241 09058

48947

(67)

5.3955

27431 19745

59553 78638

( 1)2.5768 8.7241 5.0762 3.6680

26200 30727

41343 38514

3b224

--.

bb8 46224 77632

4 47561 2994 18309

58582 93933 27470 98350 12498 27948 49870

i

10

l/3 l/4 l/5

87920 99531 88202 55239 95358

18918 31842 65311 00901

7”

24

4 34281 91179

~24063025

4’ 5

10

(67)4.5003 ( 1)2.5670 8.7021 5.0666 3.6624

4 39569 2914

79482 68738 62236

15727

17214 20758

3.6569

663

4 38244 17528

48737 20153 82121

82739 95071

5.0570

2861

662

2901

99602

84834 92026

72888 42371 23736

4 31649 93393

80288

(67)4.1837

26264

(67)3.7485 ( 1)2.5573 8.6801

657

4 30336 0041b

67) 4.0335

2797

2784

11 '14 i 1.1873

(67)3.2353 ( 1)2.5495 8.6623

654 4 27iid

653 4 26409 45077 1.8182 46353

4 25104 67808

67) 5.5939

61683

(67) 5.7993

79113

1)2.5787 8.7285 5.0781 3.6691

59392 18735 48670 40389

( 1)2.5806 8.7328 5.0800 3.6702

97580 91741

11)2.0151 14)1.3501

lb) 9.0458

19 6.0607 22 4.0606 25 I 2.7206 28) 1.8228

43226

3021 11 2.0271 '14 I 1.3602

12100 25107 38217 11605 76776 53440

‘lb

'19 :22

9.1271 6.1243

4.1094

8.7503

40123

5.0895 3.6757

64588

1

“[C-36)2]

44934

3.6735

37627

46;)“I

673

4 51584 64448

( 1);:;;; 5:0914

87644 13208 '9;;;; 59790

3.6768

32575

(67)7.1922

03431 84552

3.6724

24470

27799

96605 66769 91362

15340

44740 672

3034

(b7)b.4599

66242 24639

25) 2.7946 28) 1.8719

1.8502

( 1)2.5845

09844 69597

5.0838 8,741b

67120 93105 67366

(67)6.9396 ( 1)2.5903 8.7546

2321

62528

lb I 9.2090 19 6.1884 22 4.1586

88867 35821

(67)b.

( 1)2.5865 8.7459 5.0857

11711 69581 30789 48592 lb705 lb509

37805

67266 41314

5:0819 3.6713

4 50241

4 48900 63000

67) 6.6956 1)2.5884 5.0876 3.6746

14426 ;U;;

671

_.I

3007

56673

(67)6.0120 ( 1'2%;;

3048

(67)7.4535 ( 1)2.5942

t

4637)3]

4 52929 21217

3061

b7819 43810 674 4 54276

82024

22063

(67)l.

24354

15552 50997 43858 19196 18565

7239

5.0933 8.7633

52878 80887

( 1)2.5961 5.0952 8.1677

3.6779

26219

3.6790

4’-37’21

ISLEMENTARY

ANALYTICAL

POWERS

METHODS

AND ROOTS

51

nk

Table

k 675 4 55625 3075 46875

24

l/2 l/3 l/4 l/5

(11)2.0759 (14)1.4012 (16)9.4585 (19)6.3844

41406 60449 08032 92922

(22)4.3095 (25)2.9089 (28) 1.9635

32722 34587

(67)8.0036

95322 76211 53215 32735 09614

( 1)2.5980 8.7720 5.0971 3.6801

3144 I 14 16 11 I 2.1381 9.8867 1.4539

24

676 4 56976 15776

22820

( 1)2.6076 8.7936

a0962 59344

5.1065 3.6855

45762 45546

30685

685 4 69225 3214 19125

72571

(67)8.5926

68325

00000 82955 19514 99371

( lJ2.6019

22366

(

(11 (14 116 119 I22 125

1.4646 9.9743 6.7925 4.6256 3.1500

55745 05627 02132 93952

2.1452

97581 16453

167)9.8976 I 1)2.6095 8.7979

128

21241 42651

17949 97670 67850

68) 1.0252

5.1084

22134

38701 12971 72141

3.6866

28893

5.1102 3.6871

( 1)2.6172 8.8151 5.1159 3.6909

50466 59819 07022 49595

1)2.6191 a.8194

28446 19551 60171 47349

5.1177 3.6920

73120 26615

690 4 76100

l/2 l/3 l/4 1/5

68) 1.3563 1)2.6267 8.8365 5.1252 3.6963

70007 a5107

3357

55922 17173 22179 695 4 83025 02375

lJ2.6115 8.8022

96441

10968

3299

3242

(2aJ2.3418

42816

a7886

8.8408 5.1270 3.6973

3371

22729 73128 92956

91886

68484 30714

5.1196 3.6931

37179

3313

24 10

68)1.6130

1)2.6362 a.8578 5.1344 3.7016

03502

a5265 48911 76863 63101

(68) 1.6696 ( 1)2.63al 8.8620 5.1363 3.1027

35809 al192 95243

22801 28321

50418

( lj2.6057

62844

11 2.1761 14 1.4862 17 I 1.0151 19 6.9333 22 4.7354 25 I 3.2343 28)2.2090

97694 44925 57584

(6aJ1.0619 ( 112.6134 a.8065 5.1121 3.6887

32441 26869 12225 68688 91774

a.7893 5.1046 3.6844

(68)1.2650 ( 1)2.6229 8.8280 5.1214 3.6941

692 4 78864 73888

4 67856 3200

13504

11)2.1888

92367

6a)l.o998 1)2.6153 a.8108 5.1140 3.6898

a2878 39366 68115 38880 71315

4 73344 60672 45423

3270

3328

93189 75410

(68)1.3099 ( i)2.6248

4 74721 82769

09925 99204 76894

58832 72050

1.4539 ( lj2.6305 a.8450

39271 a9288 a5422

5.1289 3.6984

27069 62494

69927

a.8322

a0950 a4991

5.1233 3.6952

59200 50159

693 4 a0249

12557

3342

07305 30255 a6536

3.6387

3386

46612 67319 60923

689 3256

694 4 81636 55384

41112 1.5052 ( 1)2.6324 a.8493

11857 a9316 44010

5.1307 3.6995

79001 30796

(68)

08873

(6a)1.7281 ( 1)2.6400 8.8663 5.1381

70846 75756 37511 66751

3.7037

91713

(6a)1.5582

14678

( 1)2.6343 a.8535 5.1326 3.7005

a7974 98503 28931 97866

698 4 a7204

4 a5809

2.3736 1.6568 1.1564 a.0721

68392 77376 26809 65112 26484

22 5.6343 3.9321

44286 12312

3400

11 14 17 19

l/2

(67)9.2230

43313 29644 86801 75023 --a

02381

I 19 7.5987 (22 5.2583 (68)

74277

94737

( lj2.6210 a.8237

696 4 84416

53536

(11)2.1255 (14) 1.4432

679 4 61041 46639 '38037

13744

11987 19871 89872 35983 78761

4 71969 42703

(66)1.2216

11 2.2931 14 1.5868 17 1.0980

'68)1.4043

3130

4 66489 3186

(11)2.2405

691 4 77481 39371

: 1)2.6286

678 4 59684 65752 93799 11595 54097 89678 65402 82142 65092

687

4 70596 28'356 05952 19683 84703 87060 79523 72953

168) 1.1797

24

14 1.4754

686

03562

a3744 08784

11 2.1634

4 65124 14568

14196

31118

22)5.1379 3.5452

682 3172

3.1919

(28)2.2746

09000 12100 31349 81631 53253

a7040

17J1.0062 19 6.8626 22 1 4.6803

(68)1.1391

1)2.6038 a.7850 5.1027 3.6833

08428 04200

03354 41087 50822 30603 14071

24

3285 11 2.2667 14 1 1.5640 17)1.0791 19)7.4463

8.7807 5.1009 3.6822

4 63761

3158 2.1507

10

l/3 l/4 l/5

3116

t 1)2.6000 8.7763 5.0990 3.6811

3228 111)2.2146 114 1.5192 117 1 1.0421 119 7.1493 I22 4.9044 I25 3.3644 28 I 2.3080

l/2

677 4 58329 88733 54722 43247 09783 94923 50263 31928 91415

t 67)8.2931

4 62400 32000

33568 37600 48262

3102 11)2.1006 14)1.4221 I (16 9.6279 (19 6.5180 (22 4.4127 (25 2.9874 (28 2.0224

(11)2.1130 (14 1.4326 (16 1 9.7135 (19 6.5057 (22 4.4651 (25 3.0273 (28 I 2.0525 (67)8.9025

30847

2.1139 (67)9.5546

: 3 4 5 6 7 8 9

3089

3.1

3415

(68)1.7886

69670

(68)

( lJ2.6419

68963

(

8.8705 5.1400 3.7048

15722 08719 53884

1.8511

699 4 a8601 32099

a.8748

95210 60813 09888

5.1418 3.7059

48708 14839

1)2.6438

52

ELEMENTARY

Table

3.1

ANALYTICAL

POWERS

AND

METHODS

ROOTS

nk

k :

3430

4 90000 00000

3444

701 4 91401 72101

3459

702 4 92804 48408

3414

703 4 94209 28927

3489

4 95616 13664

: i 9" 10 24

l/2 ;g

: 3 4

(6a)1.9158 ( 1)2.6457 a.8790 5.1436 3.7069

3504

12314 51311 40017

(68)1.9825 ( 1)2.6476 8.8832 5.1455 3.7080

86124

74581 705 4 97025 02625

3518

a7808 40459 66120 22771 33112

(68)2.0515

90555

2q2.9481 (68)2.1228

74939 91511

( 1’;;;;:

gg;

( 1) 2 "8:;;

'0;;;;

511473 3.7090

706 4 98436 95816

56056 90435

511491 3.7101

a8981 46554

(68)2.1965 ( 1)2.6532 8.8959 5.1510 3.7112

708 5 01264 3548 94912

707

4 99849 3533 93243

63787 99832 20362 19154 01473

709 5 02681 3564 00829

2 i 1'0 24

l/2 ::: l/5

(68)2.2726 ( 1)2.6551 8.9001 5.1520 3.7122

82709 a3609 30453 47377 55193

(68)2.3513 ( 1)2.6570 a.9043 5.1546 3.7133

710

3594

24

v2

l/3 l/4 l/5

(68)2.6927 1)2.6645 a.9211 5.1619 3.7115

3655

24

1 : 4

(bQ2.7852 ( 1)2.6664 a.9253 5.1637 3.7185

76016

a2519 21404 59433 05928

715 5 11225 25875

3670

25887 66051 36564 73657 07718

13215

( 1)2.6589 8.9085 5.1564 3.7143

47160 38706 97998 59051

711 5 05521 25431

3609

89985 58325 07760 76065 52523

68)2.8808 1)2.6683 8.9294 5.1655 3.7195

716 5 12656 61696

(68)3.1867 ( 1)2.6739 a.9420 5.1710

28051 48391 14037 23488

(6CQ3.2954 ( 1)2.6758 8.9461 5.1728

33372 17632 80866 30591

3.7221

27165

3.1231

67905

720 5 18400 3732 48000

(68)2.4325

3686

(68)3.4076 ( 1)2.6776 8.9503 5.1746 3.7248

721 5 19841 3140 05361

(68)2.5165 ( 1)2.6608 8.9127 5.1583 3.7154

712 5 06944 44128

44702 32813 90191 90782 97942

3624

(68)2.9795 ( 1)2.6702 6.9336 5.1674 3.7206

717 5 14089 61613

87302 85568 43817 35801 07483

3701

(68)3.5236 ( 1)2.6795 8.9545 5.1164 3.7258

722 5 21284 3763 67048

07242 26939 36887 20404 09195

(ba)2.6032 ( 1)2.6627 8.9169 5.1601 3.7164

713 5 08369 67097

36544 05985 68708 03588 42186

3639

:;zt

I 19 17)1.3249 22 I 9.4599 4.8226 6.7543

a3171 20408 18825 29584

(ba)3.0814 ( 1)2.6720 8.9376 5.1692 3.7216

63889 77843 43321 14489 85260

._

719

3716

(68)3.6432

3795

33067

5 16961 94959

86875

3.7260

723 5 22729 3719

714 5 09796 94344

\ 2) :: ZE

718 5 15524 46232

00491 52201. 02899 39125 45902

12640 05391 31117 40881 58153

a3164 724 5 24176 03424

2 7 a 1: 24

(68) 3.7668 ( 1)2.6832 a.9628 5.1800 3.1279

63712

a1573 09493 40128 19273

1

(68)3.8944 ( 1)2.6851 a.9669 5.1818 3.7289

R2[(-36)2]

51981 44316 57022 37817 54232 1

(68)4.0261 ( 1)2.6870 a.9711 5.1836 3.7299

qc-,7,5]

75870 05769 00718 33637 a8042

(68)4.1621 ( 1)2.6888 a.9152 5.1854 3.7310

4637)2]

63488 65932 40590 27593 20708

(ba)4.3025 t 112.6907 'a;4793 5.1872 3.7320

4’137’2]

46659 24809 76646 19688 52232

ELEMENTARY

ANALYTICAL

POWERS

3810 (11)2.7628

725 5 25625

70125 16406

3826

11)2.7780 14)2.0168 17)1.4642

AND ROOTS

726 5 27076 57176

91098 94137 65143

3842

(11)2.7934

53

METHODS

Table

nk

727 5 28529 40583

29038

728 3858

(11)2.8088 (14)2.0448 (17 1.4886

30403 28533 35172 26405 28230 76552 23730 09683 47513

24

(68)4.4474

61095

(68)4.5970

46501

(68)4.7514

46686

20 1 1.0837 22 7.8895 25 1 5.7435 I 28)4.1813 (68)4.9108

l/2

( 1)2.6925

82404

(

38717 37347 98317 11864

(

1)2 a'9917 6962

93753 62009

(

5:1925 3.7351

84860 39979

l/3

0.9835 5.1890 3.7330

l/4 l/5

: 3890

08896 09928 82616

1)2.6944 8.9876 5.1907 3.7341

730 5 32900 17000

3906

:

731 5 34361 17891

3922

5 29984 28352

1)2.6981 5.1943 8.9958 3.7361

732 5 35824 23168

3938

(11)2.8242 (14)2.0589

24

l/3

l/4 l/5

11)2.9025 1.563;' 20 1.1478

l/3 :;:

38047

(68)5.4202

21655

(68)5.6010

04807

(68)5.7875

58467

(68)5.9800

51217

( 1)2.7037 9.0082 5.1997 3.7392

01167 22937 12653 41158

( 1):;;;;

;;;'8;

( 1)2.7073

97274

( 1)2

735 5 40225 65375

86185

(

2";;;;

1) ;. ;;:; 5:2068 3.7433

(68)6.3836

1) $.-p;

11253 24423

512085 3.7443

740 5 47600 24000

(68)7.2704

49690

( 1)2.7202

94102

9.0450 5.2156 3.7484

3986

5:2014 3.7402

736 5 41696 88256

4003

l/2

:5: l/5

9.0164 5.2032 3.7412

737 5 43169 15553

41696 43874

03580

4068

(68)7.5099 ( 1)2.7221 9.0491 5.2174 3.7494

745 5 55025 93625

57129 68813 67701 31847 55355

( 1';.

;;;; 5:2261 3.7544

70867 81700

(22 I 8.7993

16095 24678 56123 24254

74542

(68)6.8132

( q.;;;;

'0;;;;

( l'y&

741 5 49081 69021

49065 31518 14206 05023 16115

5:2103 3.7453

4085

11)3.0312

49693 59393

01490 98844

l'$.;:f 5:2191 3.7504

4168

5:2121 3.7463

29268 39324 05277

(11)2.9824 (14)2.2040 (17)1.6287

4035

(68)7.0382

SW& 16213

5:2138 3.7473

75222

52049 72407 E: 14035 33628 49854 98142 93019

.5;5;;

(68)8.0118 ( 1)2.7258 9.0572

26396 48245 02634

64391 27557

5.2209 3.7514

21982 37909

747 5 58009 32723

748 5 59504 4185 08992

739 II 46121 a3419

81466 53804 95761

79698 y;

._ 410:

07181

28)5.0587

43437

9:02OI1 5.205(1 3.7423

( 1';.

742 5 50564 18488

( 68)7.7569

7092

730 5 44644 47272

(14)2.1891 (17 1.6156 (20 1.1923

(68)6.5950

61314 42461

30890 65584 85019

(11)2.9663

y7’;

5 56516 4151 60936

(68)8.8253

4019

27605

28)5.4101 (68)8.5457 ( 1)2.7294 9.0653 5.2244 3.7534

90029 63647

46422 09120 30506 36620 07703 24153 68916

(68)6.1786

4134

24

58576

(68)5.2450

13346 33452 17550

80275 93922 82539 16384 72255 29635 04352

( 1)2.7018 9.0041 5.1979 3.7382

734 Ia 38756 46904

t 22 8.4249

25 6.183(1 28 4.5390

4052

l/2

00000 00000 52423 92819

1)2.7OOll 9.000[1 5.1961 3.7371

(17

I 14 11 I 2.9184 17 2.1450 1.5766 II 20 28 4.6012 25 22 6.2601 8.5172 1.1588

24

87861

(

I 14)2.1304

3970

l/2

(68)5.0752

3954

9”

95365 11321 46353 89891 44308 73700 15828

5 37284 32637

10

729 5 31441 20489

(17)1.5009 (20)1.0941 (22)7.9766 (25 5.814(1 (28 1 4.2391

82891 69560 66963

2 1

://: l/4 l/5

3874

3.1

4118

2;;;; 80938 89950 744 II 53536 30784

(28)5.196

20453

(68)8.2746 ( 1)2.7270 9.061:) 5.2226, 3.7524

65623 36339 09792 77799 47174

(11)3.1472

749 ! j 61001 420: 89749

,21220

09038

48404

(68)9.1136

94019

(68)9.4110

55807

(68)9.7177

03069

2";;;;

( u;.;;;;

y;;

( l)$m~

;9"6";;

( 1)2.7367 9.081!5 5.2314 3.7574

a6437

84131 62453

512279 3.7554

34653

68472

1

5:2296 3.7564

a3419 7341;

63122 30432 77202

.

54

ELEMENTARY

Table

3.1

ANALYTICAL

POWERS

AND

METHODS

ROOTS

nk

k : 3 4 5 ?

4218 11)3.1640 14)2.3730 17 1.7797 ;20 1 1.3348 23 1.0011

10

;

25 7.5084 28 I 5.6313

24

69) 1.0033

l/2

1)2.7386

l/3 :::

: 3

9.0856 5.2331 3.7584

4303

75000 62500 46875 85156 38867 29150

11 3.1809 14 2.3889 17 1.7940 20 1 1.3473 23 1.0118

64751 71280 09431 70983 47308 57828

68628

25 1 1.5990 2Q5.7068

52291

51471 91278 12788 02964 75697 80079

755 5 70025 68875

11)3.2492 14 2.4532 17 1.8521 20 1.3983

85006 10180 73686 91133

10

23 i 1.0557 25 7.9711 28 I 6.0182

85305 79054

24

l/2

2 6 1 a 9

l/3 $2

9.0896 5.2349 3.7594

11 3.2665 14 I 2.4694 17 1.8669

1)2.7495

14)2.5355 17)l. 9269

20 1.4645 23 1 1.1130 8.4590

48491 75936 78075

09750

05271 33069 49662

9.1097 5.2436 3.7644

24

24

10 ::: l/5

20 1 1.5333 23 1.1729 25 1 8.9733 28)6.8645 69)1.6138 1)2.7658 9.1457 5.2591 3.7733

97125 83006 35500 27157

10275 82361 15059 86020

91907 63337 14274 47590 95151

770 5 92900 4565 33000 11)3.5153 04100 14 2.7067 84157 17 2.0842 23801 20 1.6048 52327 23 1.2357 36292 25 i 9.5151 69445 28)7.3266 80473 (69)1.8870 23915 ( 1)2.7748

9.1656 5.2677 3.7783

87385

56454 19986 14849 1

.i[,-,2]

16698

(69)

61840 71888 60997

( y.;;;;

1.1042 5:2384 3.7614

a2467

51214 45417 b6916 10795 74495

69) 1.2540

10313

1)2.7513 9.1137 5.2453 3.1654

63298

81798 43934 69862

69)1.4230 ( 1)2.7586 9.1298 5.2522 3.7694

4494 11)3.4428 14 2.6372 17 2.0200 20 1.5473 23 / 1.1853 6.9548 (69) 1.6652 ( 1)2.7676 9.1497 5.2b08

3.7743

69) 1.4686

22845 Ob063

1)2.7604

766 5 86756 55096 26035 04743 98833 95706 05111 48857 92289

70501 57625 65424

81144

._

771

5 94441 4583 14011

460:

27094 ;i;;; 29452

(69)2.0082 ( 1)2.7784 5.2711 9.1735

38127 88798 37257 85227

3.7792

95720

3.7802

75573

4-37151

.i[,,,2]

80599

4372

5 76681 45479

753568 b4176

(69)1.3359 ( 1)2.7549 9.1218 5.2488 3.1614

4459

88198 95463 00968 05067 57442

764 5 83696 43744

12784

5:2557 3.1714

(69)1.9467 ( l';.;:;; 5:2694

79980 93146

763 82169 ;;;;z 6528:l 91509 68822 72711

( uy;;:

772 95984 99648

39353

97112

(69)1.2943 ( 1)2.7531 9.1177 5.2470 3.7664

(69)1.5156

767 88289 45125 17663 11)3.4608 39475 63877 73794 91900 40987 73373 78477 69)1.7182 59425 1) '9. ;g;': 76485 37512 5:2625 81576 3.7753 66108

26517

17168 14132

50728 74547 63605 26467 11368 84062 32555 87807

9.1338 5.2539 3.7704

59366 40838

9.1017 5.2401 3.7624

72453 53520

762

88020

19555 06044

38329

80644

442: 11)3.3714

1.1400 ( 1)2.7459

(69)

:;;A:

75634 07855 28546 68109

41772

a4547 00985 01041 82064

14564

98093 51564

99699

38985 03656 80565

758

757 73049

81216 33993

761

165

5 85225 4476 11 3.4248 14 I 2.6200 17 2.0043

9.0936 5.2366 3.7604

81806

19457 34787 64385

1)2.7568

17395

35750

19217

61064

89022 29433 65863

06051

17600 25376 99286

79182

9.1258 5.2505 3.7684

56701

69) 1.0696

4286

92061

52239 56084 86975

( 1)2.7422

754 5 68516

753 5 67009 4269 57711

17667

760 5 77600 76000

69)1.3788

152 c. 65504 59008 47740

46588

26333

9.1057 5.2418 3.7634

88271 96977 37921 39217

425;

756 5 71536 4320

1)2.7477

2

::: l/4 l/5

1)2.7404

69) 1.2148

4

1: 24

69) 1.0359

69)1.1768

4389 11)3.3362

i

4235

40186 65520

1 :

.__

751 5 64001

750 5 62500

(11)3.4789

4529

16435 15056 45463 97144 06863 2006N8 768 5 a9az14 84832 23510 13255 52500 99582 z;t 86072

(b9)1.7728 (

y.;;;;

5: 2642 3.7763

(69)1.5640 ( 1)2.7640 9.1417 5.2574 3.7724

4547 11 3.4970 14 1 2.6892 17 2.0680 20 1.5903 23 1.2229 25 9.4045 28 i 7.2320 69)1.8290 1)2.7730 9.1616 5.2660 3.7773

(69)2.0716 ( 1)2.7802 9.1775 5.2728 3.7812

97529 a9917 ;;s:;; 22t54 35044 82084 65S13 92641 09310 87755 44479 43403 54412 .[C,,l]

769 5 91361 56609 78323 53231 35734 19480 55680 29178 82938 77701 84925 86919 08854 32958

774 5 99076 4636 84824

713

461;

13890 54992 a7449 28071 08126

(69)2.1368 ( 1)2.7820

9.1815 5.2745 3.7822

94378 85549

00317 47894 32239

ZLEMENTARY

ANALYTICAL

POWERS

55

METHODS

AND ROOTS

nk

Table

3.1

k

465i

715 00625 04375 03906

15527 57034

36701 08443

111 I 114 ( 17 I 20 I 23

4672 3.6261 2.8138 2.1835 1.6944 1.3149

91544 (69)2.2041

l/2 l/3 l/4 l/5

( 1';.

;;;i 5:2762 3.7832

82181 52750 50735 09055 780 08400 52000 05600 74368 96007 56885

14371 89209 24

(69)2.5719 ( 1)2.7928 9.2051 5.2847 3.7880

75831 97041 48009 64083 40305 78066

785 6 16225 4837 36625

24

l/2 l/3 l/4 l/5

(69)2.9982 ( 1)2.8017 9.2247 5.2931 3.7929

: : 5

4930

77060 85145 91357 89157 22172 790 6 24100 39000

86133 62839

03163 64854

84463 24

776 6 02176 88576 59350 99655

169)2.2734 I 1)2.7856 9.1894 5.2719 3.7841

476; I 11 3.7205 114 I 2.9057 I 17 2.2693 I 20 1.7723 123 I 1.3842 I 26 1.0810 f 28 8.4432 169)2.6523 I l'$;;W; 5:2864 3.7890

31271 28553 17655

01784 51928 84864 781 09961 79541 24215 29412 74671 81618 30044 83664 63416 13239 ;'6;;; 33318 48871

4855 11)3.8167 ?I4 2.9999 ,117 2.3579 (20 1.8533 123 I 1.4567 1,26)1.1449 '128)8.9996 / 69)3.0912 1)2.8035 9.2287 5.2948 3.1938

786 6 17796 87656 18976 41115 53117 51621 34374 93218 46695 99652 69154 06804 74081 88029

4949

791 6 25681 13671

: 8

777

4690 11 3.6448 14 2.832U 17 1 2.2005 20 1.7098 23 1.3285 26 I 1.0322 28 8.0206 (69)2.3447 ( 1)2.7874 9.1933 5.2796 3.7851

6 03729 97433 87054 77241 24016 07161 20164 60167 61501 92689 71973 47428 51478 59667

4782

6 11524 11768

(69)2.7350 ( 1)2.7964 9.2130 5.2881 3.7900

29868 26291 25029 24706 18681

787 6 19369 4874 43403

1.4716

27378

(69)3.1870 ( 1)2.8053 9.2326 5.2965 3.7948

84488 52028 18931 57399 52904

4709

11)3.6636

11)3.9346 14 3.1162 17 2.4680 20 1.9546 23 I 1.5481

93088

00846

( y.;;;;

i;;;$

5:2813 3.7861

(69)2.4940 ( 1)2.7910 9.2012 5.2830 3.7871

49388 33467

.__

4800 11 3.7587 14 1 2.9431 17 2.3044 20 I 1.8043 23 1.4128 26 1.1062 28 8.6619 69)2.8202 1)2.7982 9.2169 5.2898 3.7909

6 13089 48687 81219 25695 67419 97989 43625 56559 88854 15463 13716 50477 14473 87500

4893 11)3.8557

h __... ,“%I4 _

03872 14511

(69)3.2857 ( 1)2.8071 9.2365 5.2982 3.7958

09926 33770 21746 39113 16799

49%

__ 7775-l-.

(69)3.3872 ( 1)2.8089 9.2404 5.2999 3.1967

60118 49456 30018 50005 62619

(69)3.8243 ( 1)2.8160 9.2560 5.3066 3.8006

10648 26944 90066 24523 15447

24 l/3 l/4 l/5

(69)3.4918 ( 1)2.8106 9.2443 5.3015 3.7971

06676 93865 35465 91145 41656

69)3.5994 1)2.8124 9.2482 5.3032 3.7987

.,-

795

:

5024

6 32025 59875

5043

45514 72222 34384 74670 02623

69)3.7102 1)2.8142 9.2521 5.3049 3.7996

796 6 33616 58336

I

: 5 7” 8

5062

04737 07803 19075 17219 49232 52362 45624

24 l/3 l/4 l/5

(69)4.0626

65702

:69)4.1871

02820

( 1’;. y;

;‘i:;;

( 1’;. ;‘6;;

y&

(69)4.3151 ( 1)2.8231 9.2715 5.3133 3.8044

87922 18843 59160 02968 48104

5:3099 3.8025

66512 36800

5:3116 3.8034

35526 92932

56439 14381 33255 19227 79716

(11)3.9744 (14 3.1557 {17{2.5056 20 1.9894 (23 1.5796 26 1.2542 28 I 9.9587 (69)3.9417 ( 1)2.8178 9.2599 5.3082 3.8015

5100 11 4.0755 14 1 3.2563 17 2.6018 20 2.0788 23 1.66101 26 I 1.3271 29)1.0603 (69)4.5827 ( 1)2.8266# 9.279? 5.316b 3.8063

6' 38401 82399 58368 71136 40538 70590 17601 53063 95298 13463 58805 08064 33150 55574

7911

5081

6 36804 69592

1:

l/2

40422 00000 72584 02622 55329

794 6 30436 66184 95501 49428 65046 98046 61449 51190 54451 77065 00561 11460 95923 79705

5005

,-

797 6 35209 61573

I 11 17 4.0349 14 2.5630 3.2158 II 20 26 2.0427 23 29 1.2975 1.0341 1.6280

39997 25568 22375 23755 21646

6 14656 90304 19983

789 6 22521 4911 69069

793 6 28849

(11 3.9545 (14 3.1359 (17 2.4867 (20 1 1.9720 (23)1.5638

14558 57147 28569 45663 06266

._.

4818

(69)2.9079 ( 1)2.8000 9.2208 5.2915 3.7919

788

01257 04196 33723 82708 08705

779 6 06841 29139

784

(11)3.7780

lo’

l/2

4127

87207

(69)2.4183

792 6 21264 4967

778 6 05284 10952

(69)4.4470

23172

( 1) $ ;;;i

"3;;;;

5:3149 3.8054

68841 02317

56

ELEMENTARY

Table

3.1

ANALYTICAL

POWERS

AND

METHODS

ROOTS

nk

I;

_._

: 3

5120

801 41601 22401 18432

6 40000 00000

: 6

72400 52492 39346

1'0

l/2 l/3 l/4 l/5

(69) 4.7223 ( Q2.8284 9.2831 5.3182 3.8073

: 5216

4.1993

:

66483 27125 77667 95897 07877

(69)4.8660

92789

( 1)2.8301 9.2870 5.3199 3.8082

94340 44047 57086 59229

805 6 48025 60125

5236

64006

2 i 9

10 24

(69)5.4840

46503

l/2

( 1)2.8372

52192

9.3024 5.3265 3.8120

l/3 l/4 l/5

71468 86329 55159

(69)5.6499 ( ";.;;;; 513282 3.8130

l/4 l/5

(6936.3626 ( 1)2.8460 9.3216 5.3348 3.8167

5413

28962

1.2929

21104 33200

24

(69)7.3753 ( 1)2.8548 9.3408 5.3430 3.8214

49576 20485 38634 52016 79391

l/3 l/4 l/5

: : 5

I 23 29 I 1.3881 26 1.1146 1.7287 (69)5.1662

( 1)2.8319 9.2909 5.3216 3.8092

60452 07211 16720 09631

( 1)2.8337 9.2947 5.3232 3.8101

22264

(69)

25463 67164 74803 59085

( lj2.8354

(b9)5.8205

60843

:% 19524 78976 52346

(69)5.9961

74542 91690 75012 47468

9 3101 513298 3.8139

5:3315 3.8148

10420

(69)6.7506

( 1)2.8478 9.3255

06173 32030

5.3364 3.8177

84023 20859

( 1)2,8495 9.3293 5.3381 3.8186

61370 63391 28295 61880

6

5353

I

17 I 4.3473 14 11 2.8663 3.5300

816 65856

23915 31338

07593

__. 6 54481 75129

(69)7.5956

30157

(69)7.8222

( "yp;

;g;;

( ";A:%:

nm;

5:3463 3.8233

5554 11)4.5654 3.7528

13927

( lj2.8442

92531

(69)7.1611

98588

15486 9160H 71049 01974

( 1)2.8530 9.3370 5.3414 3.9205

68524 16687 12288 41144

(69)8.0552

81B 69124 43432 69274 06266 48326 03930 94015 57904 21966 54907

( 1y;g

m!;

h

5:3479 3.8242

822 6 75684 12248 88679 31694

59849 90912 36029

( 1)2.8513 9.3331 5.3397 3.8196

5473

26950 53125

9.3178 5.3331 3.8158

(69)6.9530

i286j 04529 14300 07941

(69)6.1768

:;o"l"i 42067 92216

RI7

:% 51976 67212 68445 18251 26093 85292

821 6 74041 5533 87661

89376

813 60969 67797 00190 34554 41493 52533 41510s 25547 52870 13847

67489 38513 15651 74587 40938 09746

543;

_-.

61548

809

5294

( "t%

(69)6.5539

6 72400 5513 68000 11)4.5212 17600

9.2986 5.3249 3.8111

14112 14025

812 59344 87328 45103 44224 95910 13479 40945 32047 21222 36166

90236 16717

5.3228

804 6 46416 18464

49732 11384

811 57721

5:3446 3.8224

5197

808 52864 525;

11731

815 6 64225 43375

(23 1.9465 (26 I 1.5864

(29

l/2

85441 49694 97518 38230 78910

05879

;;;;; 39778 01783

80748 22686 00899 06129 59670

l/3

803 44809 81627 86465 02531 78133 25440 18829 61219 93459

( 1)2.8407

%:f

24

(69)5.0140

806 6 49636 06616 69325 37076 38883 60940 67318 40258 45448 03151

810 6 56100

l/2

802 6 43204 49608

'6::f;: 71036

;

24

5158

5393 11)4.3903

5493

814 6 62566 53144 34592

819 6 70761 53259

I 11 23 20 17 14 1 4.4992 2.0242 2.4716 3.0178 3.684& 26 i 1.6578 29 1.3578 (69)8.2949

82936 51936 90032 47414 03191

( lj2.8618

17604

9.3560 5.3495 3.8252

62163 88616

'6 77% 5574 41767

5594

87724 10046 47511 95237 95877 23193 824 6 78976 76224

7" 9" 10 24

w l/3 l/4 l/5

(69)8.5414

66801

( lj2.8635

64213

9.3599 5.3512 3.8261

01623 28095 56858

(29)1.3913

34555

( 69) 8.7949 ( "29';;;; 5:3528

98523 ;i;;; 58822

3.8270

89612 1

(69) 9.0557 ( 1)2.8670 9.3675 5.3544 3.8280

33244 54237 05121 88059 21458

1

( 69) 9.3238 ( 1)2.8687 9.3713 5.3561 3.8289

66467 97658 02245 15810 5239;

(69)

9.5995

98755

( 1)2.8705 9.3750 5.3577 3.8298

40019 96295 42079 02432

ELEMENTARY

ANALYTICAL

POWERS k 1 2 3 4 2 i 1: 24

l/2

l/3 l/4 l/5

825 5615 11 4.6325 14 3.8218 17 3.1529 20 I 2.6012 23 2.1460 26 I 1.7704 29)1.4bOb 69)9.8831

1)2.8722 9.3788 5.3593 3.8308

15625 03906

: 5

828 _-. b a5564

827

3.8683

5676

l/2 l/3 l/4 l/5

09244 57626

27542 35853 81323 87277

(70)1.0174

68882

( 1)2.8740

21573

9.3826 5.3609 3.8317

bb869

11564

75196 90182 39795

23 2.1879 26 1.8094 i 29 I 1.4964 (70)1.0474 ( 1)2.8757 9.3864 5.3626 3.8326

831

6 90561

87000

5738

56191

(70)1.1425

47375

( 1)2.8809 9.3977

96375 72058

5.3674 3.8354

68731

5: 3690

84709

43756

3.8363

b7514

;70;1.1760 ; 1; g. m;

835 5821 III 11 17 4.8612 14 3.3893 4.0591

20 2.8301 23 2.3631 26 1.9732 29 1.6476 (70)1.3197 ( 1y;;

513755 3.8400 1 2

5759

46709 ;;"o';;

(70)1.2104 ( 1)2.8844 9.4053 5.3706 3.8372

829 b 87241

63552

22789

5697

83671 62496

25484 47415 b0769

60060 12021

67128

(70)1.0782 ( 1)2.8774 9.3902 5.3642 3.8335

832 6 92224 30368

b 98896

TA42 77056

24550 b8999 27006 23115 52801 32589

5863

(70)1.1099 ( 1)2.8792 9.3940 5.3658 3.8345

86167 41020 38751 99229 90383

76253

5084

63591 36010 20643 51293 19107 834 b 95556

5800 11)4.8379

1.6280 (70)1.2822 ( 1)2.8879 9.4128 5.3739 3.8391

838 7 02244 80472

93704 81491

22889 86929

05816 b9049 23907 33463 839 7 03921

I I

49211

00592

(70)1.3581

59133

(29)1.6'375 (70)1.3976

41959 90431

y;

(

;;;;;

( 1’;. i;:f

;:q2:;

5:3787 3.8418

50067 91464

34071 53677

71392 98914 41873 32391 93565

833 6 93889 5780 09537 11 4.8148 19443 14 1 4.0107 44596 17 3.3409 50249 20 2.7830 11557 23 I 2.3182 48627 26 1.9311 01106 29 I 1.6086 07222 70)1.2458 90957 1)2.8861 73938 9.4091 05407 5.3723 12294 3.8382 12366

837 7 00569

836

b 97225 82075

840 7 05600 5927 04000

:

3.1

b5913

! i 10 24

Table

23326

6 88900

5717

57

nk

b 83929 5656 09283

15723 97971

830

:

AND ROOTS

826 b 82276 5635 59976

6 80625

METHODS

1y;;;

5:3?71 3.8409

42790 73010

841 7 07281 5948 23321

70)1.4383 1)2.8948 9.4278 5.3803 3.8428

842 7 08964 5969

47688

( 70) 1. b124 ( 1)2.9017 9.4428 5.3867 3.8464

b4626

23072 22965 93606 55904 09040 I

(70) 1.4800' 1) 2.8965 9.4316, 5.3819 3.8437

86372 49672

42272 60304 25741

-., Rdd

.<

7 10649 5990 77107

1 12336

6012

11584

2 7 i 10 24

l/2 l/3 l/4 l/5

: 3

(70)1.5230

10388

( 1)2,8982 9.4353 5.3835 3.8446

75349 87961 b3271

6033

41568

045 7 14025 51125

(70)1.5671 ( 1)2.9000 9.4391 5.3851 3.0455

25939 00000 30677

64807 56523

7 15716 95736

23626

70428 64916 70609

-..

846

6054

29)1;812i 70)1.6590 1)2.9034 9.4466 5.3883 3.8473

Ad7

6076

7 17409 45423

6098

90531 58848 46228 07220 63600

70)1.7069 1)2.9051 9.4503 5.3899 3.8482

83826

848 7 19104 00192

6119

41821 b7809 41057 60862

96177 849 7 20801 60049

5" I

; 8 2.2436

1;: 24

10 l/3 l/4 l/5

70)1.7561 1)2.9068 9.4540 5.3915 3.8492

47601 88371 71946 56705

07664

?@l]

(70)1.8067 ( 1)2.9086 9.4577 5.3931 3.8501

11101 07914 99893 51133 18288

;[‘-;I”]

29)1.9003 (70)1.8586 ( 1)2.9103 9.4615 5.3947 3.8510

27182 52223 68111 26442 24903 44148 28051 I

I (70) 1.9120 ( 1)2.9120 9.4652 5.3963 3.8519

‘y-37)2]

55324 43956

46902 35753 36956

1

q-37)1]

(7O)l. 9669 ( 1)2.9137 9.468') 5.3979 3.8528

10351 60457

66137 25951 45003

ELEMENTARY

Table

: 5” F i

3.1

POWERS

3

10 24

l/2

1/q

6141

70883 05250

64710 57268 09335 28744 79034

(70)2.1406

90429

( 1)2.9189

(70)2.0232

71747

( 1)2.9154

75947

( 1)2.9171

1.9687

,

9.4726 5.3995 3.8537

I

15345 07421

(29)2.0876 (70)2.3290

66620 92589

( 1)2.9240 9.4912

38303 19958

5.4074 3.8582

31751 75391

15789 39031

( 1)2.9325

75660

l/3 l/4 l/5

67035 14574

06506 21628 67708 70568 15541 96443 98423 36164

( 1)2.9410

88234 79435 80095

57569

47768

1)2.9274

6382

38067

18797 18180 77490 861 7 41321 77381

68250

I :;I 44: '0%

i%;;

1:;j

%9"

;: '02":;

9.4986 5.4105 3.8600

6405

11 5.5211 14 4.7592 17 I 4.1024

29381 83597

(70)2.7547 ( 1)2.9342 9.5133 5.4168 3.8636

08410 80150 69910 99621 75378

866 7 49956 6494 61896

94533

METHODS

ROOTS 852 25904 70208 66172 99979 53982 45993 21986

6206

7 27609 50477

(11)5.2941 (14)4.5159 (17)3.8520

48569 08729 70146

(20 3.2858 (23 1 2.8028 (26)2.3907 (29)2.0393

15835 00907 89174 43165

72719

(70)2.2017

94325

03904 06107 88131

( 1)2.9206

16373

64021 857 7 34449 22793

53336 89409 30523

03059 89021 03491 18192 27165

56234 14756 97225 78746

9.4838 5.4042 3.8564

I 14)4.6498 (17)3.9895 (20)3.4230

23191 48298 32440

48329

( 1)2.9291

63703

( 1)2.9308

9.5023 5.4121 3.8609

6427 (11)5.5468

07842 74889 79161

(70)2.9124

54150

83651

( 1)2.9376

86164 30354 42587

9.5207 5.4200 3.8654

867 7 51689 6517 14363

63527 65178 16109

20 3.6824 23 I 3.1926

23067 60799

(70)3.3455

95291

( 1)2.9427

87794

1)2.9444

86373

( 1)2.9461

83973

6607

871 7 58641 76311

9.5354 5.4263 3.8690

17196 12167 45344 872

7 60384 6630 11)5.7818

54848 38275

62010

(70)3.5355

91351

(70)3.6344

25075

76241

( 1) ;. p;

;;g;:

5:4325 3.0726

60090 08827

1

9.5390 5.4270 3.8699

6653

81845 76171 37445

(70)2.9945

5.8084

06126 54753

I 20 23 3.8645 3.3137

58173 56899

27839 18234 55935

72440 859 7 37881 39779

28654 57114 58061 69474 17279 69182

70178 98059 51174 78737

37938

1)2.9393 9.5244

5.4216 3.8663

87691 06312

12022 64090

869 7 55161 6562 34909

29)2.4558 70)3.4393

(

1)2.9478 9.5427 5.4294 3.8708

873 7 62129 38617

(17 I 4.4267

86409

864

868 7 53424 6539 72032

05644

854 7 29%6 35864 18279 41610 45135 75345 95545

7 46496

68684

( 70)3.2543

49727 46809 52418

9.5059 5.4137 3.8618

863 7 AA769 .._. 35647 08634

1)2.9359

11 5.6503 14 14.6900 17 4.2473

4.0175

(70)2.5333

23218 29097

72447

(17

I

2.2650 70)2.8325

71787

6338

lZOj3.4510 23 2.9644 (26 2.5464 (29 2.1874 (70)2.6051

14329 02952

51555

5.4058 3.8573

1 36164 28712 74349

3.5363 3.0483

9.5170 5.4184 3.8645

70)2.2645 1)2.9223 9.4875

858 6316 11)5.4193

862 1 43044 05928

43859 26007 52818

6228 11)5.3190 14 4.5424 17 1 3.8792 20)3.3128 23)2.8291

13619 72729 68659

05907

870 7 56900 03000

1

857

7

(70)3.1654 9.5317 5.4247 3.8681

02709 00130 19185

nk

1

20 23

26)2.6003 I 29)2.2388

( 1)2.9495 9.5464 5.4310 3.8117

1 I

( 1)2.9257

3.6527

58668

11)5.3941 14)4.6227 17 3.9617

(70)2.3953

865 7 40225 14625

5.5984

6294

20 3.3952 23 2.9096 26 2.4936 29 2.1370 70)2.4634

(11)5.4955

85413 26084 77475

[14{4.8426 17 4.1888 20 3.6233 23 3.1342 26 2.7110 29 2.3450 (70)3.0788

17)4.3362

20 17 1 3.9340 3.3675 (23 2.8826

9.4949 5.4090 3.8591

81600 70176

9.4801 5.4026 3.8555

_-._.

856 7 327% 22016

20457 81511

1

I

95693 02137 58534

I 11)5.3690 14 4.5958

860 7 39600 56000

(70)2.6789

9.5280 5.4231 3.8672

6272

79694 25639

(29)2.2130

6585

24

7 31025 26315

I 20)3.3401 17)3.9065 23)2.8558 26)2.4417

(11

l/2

52195

99058

9.5096 5.4153 3.8627

9.4763 5.4011 3.8546

14744

75506

4.5690

6472

l/5

82372

(14

::i l/5

:;: 1/q

44043

(11 1 5.3439

(11)5.4700 14)4.7042

24

1

20 3.2322 23)2.7506 26)2.3408 I 29)1.9920 (70)2.0811

6360

l/2

I

70884 14922 95899

6184 11)5.2693 14 4.4094 17 3.8250 20 3.2589 23)2.7766

(20)3.2057 (23)2.7249

::z

24

_.___

7 7A?“l 95051

11)5.2446 14)4.4632 (17 3.7981

6250

l/3

6162

62500 53125 95156

l/5

24

850 7 22500 25000

AND

11)5.2200 14)4.4370 (17)3.7714

l/3

l/2

ANALYTICAL

6676

26970 36231 80595 43681 38824 20725 874 1 63816 27624

70)3.7359

03403

(70)3.8400

93943

(70)3.9470

65953

1)2.9529

64612

( 1) ;. ;;y;

;;;g

( 1)2.9563

49100 10846 31924 72857

9.5537 5.4341 3.8734

12362

18707 97651

514356 3.0743

75984 85661

9.5610 5.4372 3.8752

21,EMENTARY

ANALYTICAL

POWERS

59

METHODS

AND

ROOTS

6745 11)5.9155 14)5.1879 (17 4.5498 (20 13.9902 (23)3.4994

69129 26133 94186 76101 55041 22871 25458

nk

Tabile 3.1

k a75

6699

(70)4.0568 ( 1)2.9580 9.5646 5.4387 3.8761

6814 11)5.9969

24

l/2 l/3 :s:

70)4.6514 1)2.9664 9.5828 5.4465 3.8805

7 65625 21875

90376 39892 55914 86530 59242 7 74400 72000 53600

04745 79395 39714 39631 19041

6122

;11)5.8886 14)5.1584 '17)4.5188 '20 3.9584 23 3.4676 26 I 3.0376 29)2.6609 70)4.1696 1)2.9597 9.5682 5.4403 3.8770

a76 7 67376 21376 59254 65506 15784 82626 30781 44564 76638 39882 29717 98205 39803 44816

[:~~::E (70)4.2853

( l’p; 514418 3.8779

881 7 76161 6831

11 6.0242 14 1 5.3073 17 4.6151 20 1 4.1193 23 3.6291 26 1 3.1972 29)2.8168 70)4.7799 1)2.9681 9.5864 5.4480 3.8814

97841

58979 72161 94874 75284 69625 98440 19925 32920 64416 68204 86284 60596

879 7 72641 6791 51439 (11)5.6697 41149

E: 88904 18579 37725 91747 29583

(11)5.9426 (14 5.2176 (17 4.5810 (20 4.0221 (23 i 3.5314 (26 3.1006 (29 12.7223 (70)4.4042 ( 1)2.9631 9.5755 5.4434 3.8788

a78 7 70884 36152 21415 21602 71767 81011 74928 34987 57518 13682 06478 74480 42365 13542

a82 11924 28968 57498 61913 29607 17514 55847 09657 56718 60716 48481 93948 31621 41346

6884 11)6.0791 14)5.3678 17 4.7398 20 1 4.1852 23 3.6956 26 3.2632 29 I 2.8814 70)5.0472 ( 1)2.9715 9.5937 5.4511 3.8832

883 7 79689 65387 49367 88891 45891 83922 05703 19836 23115 74047 31592 16954 75645 21296

884 7 81456 07104 (11)6.1067 34799 14 5.3983 53563 17 4.1121 44549 75782 I 20 I 4.2185

7002 11 6.2180 14 5.5215 17 I 4.9031 20 4.3540 23 1 3.8663

7 88544 27072 16399 98563 19524 23417 72794

877

686:

(11)6.

0516

(29)2.8489 (70)4.9118

( l’y& 5:4496 3.8823

t : : 5

885 7 83225 6931 54125

t t 10 24

l/2 ::: l/5

(70)5.3289 ( 1)2.9748 9.6009 5.4542 3.8849

: :

7049

11365 94956 54166 59763 78808 890 7 92100 69000

886 7 84996 6955 06456 11)6.1621 87200 114)5.4596 97859 17 4.8372 92303 20 1 4.2858 40981 123 3.7912 55109 126 3.3643 68021 29 I 2.9808 30072 70)5.4753 17719 1)2.9765 75213 9.6045 69584 5.4551 99862 3.8858 56313

7073

697:

17587

98020 45144 (70)5.6255

14442

( 1’;. 2;;;

54522 81682 38658 33146

5:4573 3.8867

891 7 93881 47971

2 .l 1: 24

l/2 :,': l/5

(70)6.1004 ( 1)2.9832 9.6190 5.4619 3.8893

25945 86778 01716 47252 58728

70)6.2670 1)2.9849 9.6226 5.4634 3.8902

895

7169

24

l/2 l/3

(70)6.9783 ( 1)2.9916 9.6369 5.4696 3.8937

8 01025 17375

51604 55060 81200 02417 19006

.y-,$l]

7193 111 1 6.4451 14 5.7748 17 I 5.1742 120 4.6361 23 4.1539 26 3.7219 29 3.3348 70)7.1679 1)2.9933 9.6405 5.4711 3.8945

75070 62311 02990 80860 32348 896 8 02816 23136 35299 41228 57740 34935 76902 63304 79120 04854 25909 69057 29599 88722

887 86769 64103 54594 78425 43063 16896

(70)6.4380 ( u;.;;;; 5:4650 3.8911

7717

892 95664 32288 12009 84312 99206 81692 18069 62918 56123 82017 36905 01570 13179 05185 a97 8 04609 34213 56429 38917 03608 76231 E2”

86240 (70)7.3623

86846

( 11;. ;;i;

95826 54244 55504 57662

4637)3]

5:4726 3.8954

4-;~2]

676.3

(29j2.7535 (70)4.5261 ( 1)2.9647 9.5192 5.4449 3.8796

6908

29)2.9142 (70)5.1862 ( 1)2.9732 9.5973 5.4527 3.8841

888

3.0488 (70)5.7797 ( 1)2.9799 9.6117 5.4588 3.8876

05069 78281 32885 91067 76153 09128

23268 92303 93416 08475 91658 96696

7025 11116.2460

(70)5.9380 ( 1)2.9816 9.6153 5.4604 3.8884

22119 60897 13749 37224 18358 00450 889 7 90321 95369 72830

28303 10303 97744 12350 84321 R9d II

7121 11 6.3592 14 1 5.6788 17 5.0711 20 1 4.5285 23 4.0440 26 1 3.6112 29)3.2248 (70)6.6135

7 91449 21957 49076 09425 76816 60897 04881 96359 87648 55666

( 1) ;. ;;;;

;;:;;

514665 3.8919

7241 11)6.5028 14 5.8395 17 1 5.2439 20 4.7090 23 4.2287 26 I 3.1974 29)3.4100 (70)7.5619 ( 1)2.9966 9.6477 5.4741 3.8963

44210 71239 898 8 06404 50792 74112 80953 43696 61439 31112

7 99236 7145 16984

(70)6.7936 ( 1)2.9899 9.6333 5.4680 3.8928

7265

07487 83278 90671 73955 48512

-..

8 08201 72699

05980

70570 20026 64813 36769 a0133 25828

.:[,-,,l]

(70)7.7666 ( 1)2.9983 9.6513 5.4151 3.8971

29743 32870 16634 03489 93220

60

ELEMENTARY

POWERS

Table 3.1 k 1 2

900 8 10000 7290 00000

(70)8.1920

901 11801 32701 08636 77981 37961 94103 84987 19573 20735 95066

( 1) 2. pi

y”o;

4' 5 7" 8 lo' 24

10 l/3 l/4 l/5

24

l/2 v3 l/4 l/5

24

l/2 l/3 l/4 l/5

24

l/2 l/3 l/4 l/5

(70)7.9766 ( 1)3.0000 9.6548 5.4772 3.8980

44308 00000 93846 25575 59841

7412 11)6.7080 14)6.0707 17)5.4940 20 4.9721 23 4.4997 26 4.0722 29 I 3.6854 (70)9.1109 ( 1)3.0083 9.6727 5.4848 3.9023

905 8 19025 17625 19506 57653 35676 02287 52570 76076 09848 96943 21791 40271 17035 81426

7535

910 8 28100 71000

(71)1.0399 ( 1)3.0166 9.6905 5.4923 3.9066

7660 11 7.0094 14 6.4136 17 I 5.0684 20 5.3696 23 4.9132 26 I 4.4956 29)4.1134 71)1.1860 1)3.0248 9.7082 5.4999 3.9109

04400 20626 21083 77104 83951 915 8 37225 60875 57006 53161 92642 70767 48752 22608 94687 58902 96692 36884 06083 67606

97n .__ 8 46400 7786 88000

24

l/2 l/3 l/4 l/5

ANALYTICAL

29)4.3438 (71)1.3517 ( 1)3.0331 9.7258 5.5074 3.9152

84542 85726 50178 88262 04268 32576

+-;P]

514787 3.8989

7436

( 1';.

7338 (11)6.6195 14)5.9708 17)5.3856

5:4863 3.9032

7560

(71)1.0676 ( 1)3.0182 9.6940 5.4938 3.9075

768;

(71)1.2175 ( l);.;;;; 5:5014 3.9118

7812

7461

nk

7363 11 6.6489 6.0039 14 17 I 5.4215 20 4.8956 / 23 1 4.4208

40328 65946 90774

24265

3.7676 (70)9.6067

70117 14616

908

20)5.0886

(70)9.8641 ( "6

3.9041

58031

7585

04712 912 8 31744 50528

7610 11)6.9483

(71)1.0961 ( 1)3.0199 9.6976 5.4953 3.9083

916 39056 15296 49711 77136 79856 85148 70796 35649 72654 62793 ;;;;i 08174

917 40889 77180 95213 ii(ii ;. 07:; 43103 54826 17 519458 78275 20 5.4523 70378 23 4.9998 23637 26)4.5848 38275 (29)4.2042 96698 (71)1.2498 67732 ( 1);;;;; 00;;;; 5:5029 09036 3.9126

29961

94035 98181 10859 00236 83344

7837

1

23 5.2221 26 4.8148 29 I 4.4392 (71)1.4241 ( 1)3.0364 9.7329 5.5103 3.9169

q-37)3]

65476 33774 15172 92410 99668

(71)1.1253 ( 1)3.0215 9.7011 5.4968 3.9092

7736

y;;;; 56824 65216

48497 72778

918 8 42724 20632

(29)4.2503 (71)1.2829 ( 1)3.0298 5.5044 9.7188

70729 93183 51482 35404 08671

3.9135

28819

7063

(71)1.4616 ( 1)3.0380 9.7364 5.5118 3.9177

q-37)2]

923 8 51929 30467

41363 91506 48410 88520 82664

904 8 172ib 63264 19907 91596 11602 71289 29245 56837 88981 24888 59276 76254 01264 18640

909 8 26281 89429 (11)6.8274 02910 (14)6.2061 09245 17)5.6413 53304 20)5.1279 90153 43049 I 23)4.6613 26)4.2371 60832 79196 I 29)3.8515 (71)1.0128 22166 ( 1)3.0149 62686 9.6869 70141 5.4908 67587 3.9058 24962 7510

7635 11 1 6.9788 14 6.3786 17 I 5.8301 20 5.3287 23 4.8704

78622 88986 58327 98203 56397

09448

77448

1

30769

5.0436

75026

26266 00417 45985 05308 45290 30906 94986 33373

13312 08873 18909 47257

65825 ;'8;:

5:4893 3.9049

79852 77655 69425 85370 42186

22089

744h4 _ _.._.

R

7486 I 11 17 I 6.1720 14 5.6042 6.7974

y; 31551 43449

7387 11 6.6784 14 16.0372 17)5.4577 20 4.9337 23 1 4.4601 26)4.0319 29)3.6448 70)8.8724 1)3.0066 9.6691 5.4833 3.9015

46120 95840 09608 84235 55089

5.4817 9.6656 3.9006

907 8 22649 42643

4.5799

9n3 8 15409 14327 18373 73291 87881 93857 11553

(70)8.6398 ( 1)3.0049

16465 31484

9.6620 5.4802 3.8997

9"h .__ 8 20836 77416

y;

902 8 13604 70808 14688 02249 63628

(70)8.4131 ( 1)3.0033

09844

(71)1.3874 ( 1)3.0347 9.7294 5.5089 3.9160

AND ROOTS

46393 25692

(70)9.3557

MFlTIODS

1

“[‘-.;‘l]

914 8 35396 51944 64760 82398 15712 25761 55345

(71)1.1553 1)3.0232 9.7046 5.4984 3.9101

37042 43292 98896 02760 12376

7761 11 I 7.1328 14 6.5550 17 6.0241 20 I 5.5361 23 5.0877 26 4.6756 29 4.2968 (71)1.3169 ( 1)3.0315 9.7223 5.5059 3.9143

919 ._. 8 44561 51559 32827 73368 12425 59319 30414 24251 98666 59057 01278 63112 07081 81068

(11)7.2893 (14)6.7353

7888

(71)1.5001 ( 1)3.0397 9.7399 5.5133 3.9186

924 8 53776 89024 34582 45154

24518 36831 63373 80842 31220

:ELEMENTARY

ANALYTICAL

POWERS 925 55625 53125 41406 70801 80491 81954 18307 46934 23414 77607

11)7.3209 14)6.7718 17)6.2639 20 5.7941 23 1 5.3596

1) ;. y;;

%: 71952 79042

5:5148 3.9194

(11)7.4805 (14)6.9568

24

8043

930 8 64900 57000 20100 83693

(71)1.7522

l/2 l/3 l/4 l/5

( 1)3.0495 9.7610 5.5223 3.9237

28603 90136 00077 09423 07185

l/3 l/4 l/5

71)1.9928 1)3.0577 9.7784 5.5297 3.9279

8305

24 :;: l/4 l/5 :

29)5.3861 71)2.2650 1)3.0659 9.7958 5.5370 3.9321

68584 76970 61652 16964 17180

:71)2.0446 [ 1)3.0594 9.7819 5.5311 3.9287

940 8 83600 84000

51141 01461 41943 61087 94855 09204

(71)2.3235 ( 1)3.0675 9.7993 5.5385 3.9329

945 8 93025 8436 08625

l/2 :;: l/5

71)1.6214 1)3.0446 9.7504 5.5178 3.9211

8095 (11)7.5450 (14)7.0320

(71)2.5725 ( 1)3.0740 9.8131 5.5444 3.9362

932 8 68624 57568 76534 11329

(29)4.9449 (71)1.8449 ( 1';;;;;

10997 29260 97390 93317 50630

18334 38512 fJ;J;;

5:5252 3.9253

71)2.0977 1)3.0610 9.7854 5.5326 3.9295

56558 11708 46493 94905 57017

8 85481

8358 11 7.8741 14 I 7.4174 17)h.9872 20)6.5819 23 6.2002 26 15.8406 29)5.5018 (71)2.3835 ( 1';.;06;;

37621 66014 60819 49330 21220 61168 43259 25707 44328 72330 33566 66899 45467

515400 3.9337

04376 47511 85230 98931 43371 83427

( 1)3.0757 9.8166 5.5459 3.9371

1

q--y]

11300 59156 09574 16151

nk

5:5193 3.9220

(11)7.5775 (14)7.0698 (17)6.5961 (20)6.1541 (23)5.7418

I

q-3”3]

8492

6.1256

8017 (11)7.4483 (14 6.9195 (17 1 6.4282 (20 5.9718 (23 I 5.5478 (26)5.1539 (29)4.7880 (71)1.7075 ( 1)3.0479 9.7575 5.5208 3.9228

933 8 70489 66237 10991 17755 39965 98588 67282

5:5267 3.9262

8385 11 7.9076 14 1 7.4569 17)7.0318 20 6.6310 23 6.2530 26 I 5.8966 29)5.5605 (71)2.4450 ( 1)3.0708 9.8062 5.5415 3.9346

934 8 72356 80504 49907 86613 72697

(7131.9423 ( 1)3.0561 9.7749 5.5282 3.9270

38514 ;iJ;o" 57521 35348

3.1

929 8 63&l 65089 97677 61442 72579 65226 62795 64537 33055 64573 50131 00256 24332 63013

8147 11 I 7.6100 14 7.1077 17 6.6386

38996 41358 14326 37837 76625 019

938 8 79844 8252 93672 11 7.7412 54643 14 1 7.2612 96855 17 6.8110 96450 20 1 6.3888 08471 23 5.9927 02345 26 1 5.6211 54800 29)5.2726 43202 71)2.1521 28115 1)3.0626 78566 9.7889 08735 5.5341 47239 3.9304 34540

8 96809 78123

I/

8 81721 8279 36019 (11)7.7743 19218 (14)7.3000 85746 (17 6.8547 80516 (20 16.4366 38904 (23 6.0440 03931 (26 5.6753 19691 (29 I 5.3291 25190 71)2.2078 73640 1)3.0643 10689 9.7923 86145 5.5356 21636 3.9312 72229

943 8 89249 61807 37840 02483 59042 43076 73621 48424 39464 09921 30507 71149 07472 15863

8412 11)7.9412 14 7.4965 17 7.0767 20 I 6.6804 5.6197 (71)2.5080 ( 1)3.0724 9.8091 5.5429 3.9354.

948 8 98704 8519 71392 (11)8.0766 88.796

8546

8 91136 32384 33705 24617 19239 22962 88134 01911 58299 36263 76005 49998

$1 00601 70349

16600

(71)2.7064

46809

( I';.;;;;

:;;::

5:5473 3.9379

8121

(71)1.8930 ( 1y;:

32860 45573 28852 71663 96137

37771 81020

928 61184 78752 78819 99544 66776 12369 67478 48220 42348 92748 09242 97922 38042 18115

( "y;;

76015 93351

942 8 87364 9688.3 48685 48061 36074 76381 21751 08890 53574 35733 ;;;;;

Table

799; (11)7.4163 (14 1 6.8823 (17 6.3868 (20 I 5.9270 (23 5.5002 (26 5.1042 (2934.7367 (71)1.6639

937 8 77969 8226 56953 (11 7.7082 95650 jl417.2226 73024 17) 6.7616 44623

946 8 94916 8465 90536 11 8.0087 46471 14 7.5762 74161 17 7.1671 55356 20 1 6.7801 20967 23 6.4140 02003 26 6.0676 45895 29 I 5.7399 93016 (71)2.6386 83331

? 5.6796

927 59329 97983 63302 97481 83465 48572 29826 58649 46668 87554 61470 93072 50550 72488

941

8332

: 5

i 10 24

931 8 66761 54491

AND ROOTS

796: 11 i 7.3844 14 6.8453 17 6.3456 20 5.8824 23 5.4530

936 8 76096 8200 25856 11 7.6754 42012 '14 1 7.1842 13723 6.7244 24045 6.2940 60906 5.8912 41008

F

l/2

8069

71)1.7980 1)3.0512 9.7644 5.5237 3.9245

: 3 4 5

t 10 24

794: (11 I 7.3526 (14 6.8085 (17)6.3047 (20)5.83'31 (23 I 5.4061 (26 5.0060 (29)4.6356 (71)1.5800 ( 1)3.0430 9.7469 5.5163 3.9203

926 57476 22776 50906 54739 21688 72283 47534 92617 41763 23988 24811 85700 61854 26131

61

METHODS

74614 48170 I

q-37)2]

29)5.8625 (71)2.7758

06988 76218

( 1)3.0789 9.8235 5.5488 3.9387

60864 72299 38494 79481

I

q-y

(71)2.8470 ( 1)3.0805 9.8270 5.5503 3.9396

10693 84360 25224 01217 10103

I

62

ELEMENTARY

Table

3.1

ANALYTICAL

POWERS

AND

METHODS

ROOTS

I&

k : :

5

8573 111\8.1450

950 9 02500 75000 62500

8600

b fi 2b)b.3024

1'0

24

l/2 l/3 l/4 l/5

(29)5.9873 (71)2.9198 ( 1)3.0822 9.8304 5.5517 3.9404

94097

77556 53956

62784

93712 55775 28789 23805 23198

69392 90243 07001 75725 40019

955 9 12025 a709 83875

(li)a.3l7a (14)7.9435 (17 7.5861 (20 1 7.2447 (23) 6.9187

951 9 04401 a5351 11688 20515 ball0 92173

(71)2.9945 ( l'y& 5:5532 3.9412

69236

873;

96006 90686

29105 53295 39397

13936 22816 90121 b7356 15592 23306 10280

l/2 1/3 1/4 l/5

(7u3.3119 ( l)3.0903 9.8476 5.5590 3.9445

8847

l/2 l/3

:,/2

l/2

l/3 l/4 l/5

79145 960 9 21600 36000

74781 95790 39958 26360

(71)3.7541 ( 1)3.0983 9.8648 5.5663 3.9487

32467

(71)4.2526 ( 1)3.1064 9.8819 5.5735 3.9528

9126

24

53362

20)7.5144 7.2138 6.9253 (29) 6.6483

8986 11)8.6718

24

28238 07428 92005

(71)4.8141 ( 1)3.1144 9. a989 5.5807 3.9568

86677 48297 15367 00972 965 9 31225 32125 00006

09649 44913 45122 49061 05659 970 9 409on 73000

72219 82300 a2992

54698 93368

71)3.3961 1)3.0919 9.8511 5.5605 3.9454

a875

49474 69948 24967 28046 08040 04889

(71)3.0710 ( 1'9'. 8";;;

49109 49724

5:5546 3.9420

82461 97756

10374 a2870 27838 39352 31217

69469

957 9 15849 8764 (li)a.3877 (14)8.0271

(17)7,6819 20 7.3516 23 7.0355 26 I I 6.7329 (29)6.4434 (71)3.4824 ( 1)3.0935 9.8545 5.5619 3.9462

67493 93908 la770 52663

28698 08664

al792 63575 b2966

41660 61691 61578 29943

a902

f11)8.5644

__..

77128

65971

36200 05288

(7u3.8491 ( 1)3.1000 9.8682 5.5677 3.9495

9014 11 8.7078 14 1 a.4117 17)8.1257 20)7.8494 23 7.5825 26 7.3247 29 I 7.0757 (71)4.3596 ( 1)3.1080 9.8853 5.5749 3.9536

9154

(71)4.9347 ( 1)3.1160 9.9023 5.5821 3.9577

lab99

(71)3.9464

05693 12484 94135

00000 72403 64363 23275

12228 44894

966 9 33156 01203 35962 36940 61884

971 9 42841 98611

08664

87290 83537 92482 08886

(ll)a.2484 (14 7.8607

(17 7.4913 (20 7.1392 (23 I 6.8036 (71)3.1494 ( i)3.0870 9.8408 5.5561 3.9429

(71)4.4692 ( 1)3.1096 9.8887 5.5764 3.9544

9183

(71)5.0581 ( 1)3.1176 9.9057 5.5836 3.9585

2::: 57504 62361 67316 34668

42771 972 9 44784 30048

34323 91454 a1747 29155 23732

23177 35877 59391 03699

12425

954 9 1Olib 8682

50664

(11)8.2831 (14)7.9020 (17)7.5385 (20)7.1918

11335 88213 92155 lb916

(71)3.2296 ( i)3.0886 9.8442 5.5575 3.9437

91146 a9042 53565 97541 52709

b9441

12996 69808 12721 40574

25580

__-

959 9 19681 8819 74079 (11)8.4581 31418 (14)8.1113 48029

9 17764 8792 17912 (11 a.4229 07597 (14 1 a.0691 45478 (17)7.7302 41368 (20)7.4055 71230 (23)7.0945 37239

1 :g: ::z? (23)7.1539 6.5793

96686

(71)3.5708 ( 1)3.0951 9.8579 5.5634 3.9470

(71)3.6613 ( 1)3.0967 9.8614 5.5648 3.9478

94899 72513 71813

55021 57508 92945 13977 54307

.-_

9 27369 '3930 56347

(26)7.1225 (29)b.a590 (71)4.0460 ( 1)3.1032 9.8751 5.5706 3.9511

967

35089 31063 14379 65205 38153 18994 03867

28696

80180 72454 30190 44069 54054 57396 92425 24554

953 9 08209 8655

962

961 9 23521 03681

952 Ob304 01408 69404 38087 65059 20336 20160

956

26228

24

862:

9070

(29)7.2235 (71)4.5815 ( 1)3.1112 9.8921 5.5778 3.9552

9211 11)8.9629 14 8.7209 17 I a.4854 20)8.2563 23)8.0334 (71)5.1845 ( 1)3.1192 9.9091 5.5850 3.9593

67671

z: 98708

65240

77983

._.

9 29296 a958 41344 (11 8.6359 10556 (14 1 8.3250 17776 (17 a.0253 17136 (20 1 7.7364 05719 (23)7.4578 95113

32667 46699

24130 13495 58964

65831. 968 9 37024 39232

98067 09331 69837 74886 75794 60312

15371 94792 77627 64719

1

96142 34939 30490 04575 86085

,-_

9h9

9 38961 9098 53209 11)8.8164 77595 14 8.5431 66790 (17 a.2783 28619 (70 a.0217 00432 (23 I 7.7730 27719 (71)4.6964 ( 1)3.1128

971 .._ 9 467;!9 673:L7 57994 58129 922'59 a3968 61601

37908

(71)4.1480 ( i)3.1048 9.8785 5.5721 3.9519

60232

9.8955

76483 80110

5.5793 3.9560

15803 77177 914

9240

9 486% 10424

29)7.ba40

39148

(71)5.3139 f i)3.1208 ,~-9.9125 5.5864 3.9601

19427 97307

71181 99178 51415

ELEMENTARY

ANALYTICAL

POWERS k

63

METHODS

AND ROOTS

Table

nk

.._

975 : : 2 7

(11 (14 17 20 23

9268 I 9.0368 8.8109 8.5906 18.3759 8.1665

9 50625 59375 78906 56934 83010 15935 18037

9” 10

24

(71)5.4464 ( 1)3.1224 9.9159 5.5879 3.9609

m l/3 l/4 l/5

15584 98999 62413 32533 64254

(71)5.5820 ( 1)3.1240 9.9193 5.5893 3.9617

74443 99870 51328 64785 76427

(71)5.7209 ( 1)3.1256 9.9227 5.5907 3.9625

68141 99922 37928 95938 87934

9354 11)9.1486 14)8.9413 17)8.7505 20)8.5579 23 8.3697 26 8.1855 29 I 8.0055 71)5.8631 1)3.1272 9.9261 5.5922 3.9633

978 9 56484 41352 16423 46861 05230 94115 18245 84443 01586 70383 99154 22218 25992 98776

9498

983 9 66289 62087

.__

WI”

:

9 60400 92000 (11)9.2236 81600 (14)9.0392 07968 (17 8.8584 23809 20 1 8.6812 55332 23)8.5076 30226 9411

: 2 l3 lo' 24

l/2 l/3 l/4 l/5

I (71)6.1578 ( 1)3.1304 9.9328 5.5950 3.9650

03365 95168 83884 82813 18474

71)6.3103 1)3.1320 9.9362 5.5965 3.9658

89657 91953 61267 09584 27331

(71)6.4665 ( 1)3.1336 9.9396

95666 87923 36356

5.5919 3.9666

35265 35529

986 :

9615 11 9.4900 I4 9.3666 17 I 9.2449

: 2 i 9 10 24

l/2 l/3 l/4 l/5

: 3 : ! 8 9 10 24 ://: $2

(71)6.9577 ( 1)3.1384 9.9497 5.6022 3.9690

9702 (11 9.6059 14 9.5099 I 17 9.4148 20 9.3206 I 23 I 9.2274 26)9.1351 I 29)9.0438 (71)7.8567 ( 1)3.1464 9.9665 5.6093 3.9730

61406 70965 47896 05785 56179 990 9 80100 99000 60100 00499 01494 53479 46944 72475 20750 81408 26545 54934 01690 77521

(71)7.1292 ( 1)3.1400 9.9531 5.6036 3.9698

84708 63694 13846 27123 61152

( 71)7.3048 1)3.1416 9.9564 5.6050 3.9706

( 1)3.1480 9.9699 5.6107 3.9738

15248 09547 17644 79839

(71)8.2466 ( 1)3.1496 9.9732 5.6121 3.9746

991; (11)9.8805 (14)9.8508

l/2 :::

l/5

(71)8.8665 ( 1)3.1543 9.9833 5.6163 3.9770

35105 62059 05478 70767 82648

+y]

79843 12862 44315 26396 99386

(71)6.6265 ( l';.;:;;

03443 ;;3;;

5: 5993 3.9674

(71)9.0828 ( 1)3.1559 9.9866 5.6177 3.9778

91413 46768 48849 81384 81740

,:[(-231

(71)9.3043 ( 1)3.1575 9.9899 5.6191 3.9786

59857 43069

988 9 76144 9644 30272 [ ::I z: :::: (17 9.3012

EZ 57495

I 20 23 9.1896 9.0193

66691 42406

56083 55614 77521 47381 66671

I 29 26 8.9704 8.8621 (71)7.4845 ( 1)3.1432 9.9598 5.6064 3.9714

14296 69325 66822 46729 38925 66560 70939

98779 03150 61904 32527 81509

9791 11 i 9.7229 14 9.6548 17 9.5872 20 9.5201 23 9.4535 26 9.3873 29 9.3216 (71)8.4485 ( 1)3.1511 9.9766 5.6135 3.9754

I

996

24

987 9 74169 04803 52406 81724 14862

i 11 23 I19.1784 20 17 14 9.0223 9.3371. 8.7182 8.8689

997 94009 26973 38921 97304 44612 80578 04937 11422 17769 02025 30681 89983 90939 80191

“q-p]

(11 (14 (17

993 9 86049 46657 26304 65820 81759 70787 29591 54884 43400 45822 90251 12009 46340 82534

99490 9.9202 9.9003 9.8805

I

(29)9.8017 (71)9.5308 ( 1)3.1591 9.9933 5.6205 3.9794

(71)6.0087 ( 1)3.1288 9.9295 5.5936 3.9642

9527

(71)6.7901 ( 1)3.1368 9.9463 5.6007 3.9682

9673

8.9528 (71)7.6685 ( 1)3.1448 9.9631 5.6078 3.9122

3.1

56477 97569 04202 54950 08956 984 9 68256 63904

96812 77428 19667 83363 49952 9 78iil 61669

83144 10178 37039 98061 84662 74555 994 9 88036

(71)8.6551 ( 1)3.1527 9.9199 5.6149 3.9762

22630 76554 59866 59086 82913

(71)9.7627 ( 1)3.1606 9.9966 5.6220 3.9802

39866 96126 65555 06871 75173

998 96004 11992 39680 99201 98402 37206 15531 iEi 79767 13800 28884 99434 78001

.:[(-y]

.

4. Elementary Logarithmic,

Expwential,

Transcendental Circular

Functions

and Hyperbolic

Functions

RUTY ZUCKER l

Contents Page 67

Mathematical Prcbperties ..................... 4.1. Logarithmic Function ...................

4.2. 4.3. 4.4. 4.5. 4.6. Numerical

Methods

References 4.1.

log,, 5,

71

79 83 86

.......................

4.7. Use and Extension

Table

67 69

Exponential Function ................... Circular lknctions .................... Inverse C rcular Functions ................. Hyperbol:.c Functions ................... Inverse Hyperbolic Functions ............... of the Tables

89

..............

89

. . . . . . . . . . . . . . . . . . . . . . , , . . .

Commcln Logarithms (100&11350) 2=1(0(1)1350, 10D

. . . . . . . . . . .

93

95

Table

4.2. Natural Logarithms (0 5s<_2.1) In z, s=O(.O101)2.1, 16D

. , , . , . . . . . , , . 100

Table

Radix ‘Table of Natural Logarithms . . . . . . . . . . . . 114 -in (l-cc), r=lO-n(lO-n)lO-n+l, n=lO(-1)1, 2511 ln (l+z),

Table

4.4. Exponential ez, fz=o(.o31)1, x=5(.1)10, *tz=0(1)100,

Table

4.5. Radix Table of the Exponential Function . . . . . . . . . 140 ez, eez, x=1(1-n(lO-“)lO-“+l, n=lO(-l)l, 25D

Table

4.6. Circular Sines and Cosines for Radian Arguments sin 2, co9 5, ~=0(.001)1.6, 23D

Table

4.7. Radix Table of Circular Sines and Cosines . . . . . . . . . 174 sin 2, cos 2, ;c=lO-n(lO-“)lO-n+‘, n=lO(-1)4, 25D

4.3.

Function (0 5 1215 100) , , . . . . . . . . . . 116 18D, s=O(.1)5, ’ 15D 12D, --2=0(.1)10, 20D 19s

(0
4.8. Circular Sines and Cosines for Large Radian Arguments (0~z~1000). . . . . . . . . . . . . . . . . . . . . . . . . 175 sin 2, cos 2, ;e=O(l)lOO, 23D, ~=100(1)1000, 8D

Table

1 National

Bureau of Standards.

ELEMENTARY

Table

TRANSCENDENTAL

.FUNCTIONS

Tangents, Cotangents, Secants and Cosecants for Page (0
Radian 5

Arguments

-1 -cot

csc

x,

x=0(.01).5,

x-x-l,

8D

Table 4.10. Circular Sines and Cosines to Tenths of a Degree (0’ 5 8 5 90”) . 189

sin 8, cos 8, 8=O”(.lo)900,

15D

Table 4.11. Circular

Five Tenths tan e, cot set 8, csc

Tangents, Cotangents, Secants and Cosecants to of a Degree (0” 5 0<90’) . . . . . . . . . . . . . . 198 8, e=o”(.50)900, l5D 8, e=O”(.50)900, 8D

Table 4.12. Circular

Functions

for the Argument

f z

sin %, cos %, tan %, cot %c, set %, csc L, 2

2

2

Table 4.13. Harmonic

sin *‘T, cos z,

Analysis. r=l(l)[s/2],

2

s=O(.Ol)l,

2

20D

. . . . . . . . . . . . , . . . . . 202 s even

s odd

T&&2],

s=3(1)25,

2

(05 z < 1) . . . . 200

10D

Table 4.14. Inverse Circular

Sines and Tangents (O
. . . . , . 203

Table 4.15. Hyperbolic

Functions (0 5 az5 10) . . . . . . . . . . . . 213 sinh x, cash x, x=0(.01)2, 9D, 2=2(.1)10, 9D tanh z, coth 5, x=0(.01)2, 8D, 7D, 2=2(.1)10, 10D

Table 4.16. Exponential

and Hyperbolic Functions for the Argument KZ (O<x
Table 4.17. Inverse Hyperbolic

Functions (0 5~5~)) . . . . . . . . . 221 arcsinh x, arctanh 5, z=O(.Ol)l, 9D arcsinh x, (arccosh X)/(X?- l)‘, 2= 1(.01)2, 9D, 8D arcsinh z-ln z, arccosh z-ln 2, s-‘=.5(-.01)0, 10D

Table 4.18. Roots z, of cos x, cash z~= f 1, n= 1(1)5, Table 4.19. Roots 2, of tan zn=kzn -x=0(.05)1, n=1(1)9, 5D

x-*=-1(.05)1,

n=1(1)9,

(--00
7D . . . . . . 223

l<Xlm)

. . . . . 224

5D

Table 4.20. Roots 2, of cot x,,=kr,, (0 <Xla X=0(.05)1, n=l(1)9, 5D X-‘=1(-.05)0, n=1(1)9, 5D

) . . . . . . . . . . . 225

The author acknowledges the assistance of Lois K. Cherwinski io the preparation and checking of the tables.

and Elizabeth

F. Godefroy

.

4. Elementary Logarithmic,

Transcendental

Exponential,

Circular

Mathematical 4.1. Logarithmic Integral

4.1.1

and Hyperbolic

IFunctions

Properties

Function

Representat

In z=

Functions

Logarithmic

4.1.6

ion

Identities

Ln (z,z2)=Ln

zl+Ln

z2.

(i.e., every value of Ln (z1z2) is one of the values of Ln zr+Ln z2.)

z dt 17

S

4.1.7

In (z1z2)=ln q+ln

z2

iY

(--?r<arg

n

0

1

x

-I

4.1.8

Ln z=Ln

4.1.9

In z=ln

zl--Ln z,-ln

z=r+iy

FIGURE

4.1.

(a not an integer or zero.)

where the path of integration does not pass through the origin or cross the negative real axis. In z is a single-valued function, regular in the z-plane cut along the negative real axis, real when z is positive. z=x+iy=reie. 4.1.2 4.1.3

In z=ln r+iO r= (x2+y2)*,

(-*
Ln zn=n Ln 2

4.1.11

In P=n In 2 (n integer,

4.1.13

In 0=-

4.1.14

ln (-l)=*i

4.1.15

In (&i)=f& In e=l,

03

6 dt

1 t=’

“=2.71828 18284.. . (see 4.2.21) Logarithms

4.1.18

S

arg 25~)

S

The general logarithmic function is the manyvalued function Ln z defined by Lnz=

-s
e is the real number such that

e=arctan f.

4.1.4

(n integer)

In l=O

x=r cos 0, y=r sin 0:

z dt 1t

z2
(see chapter 1)

Values

4.1.12

4.1.16

zl-arg

;z2

4.1.10

Special

z217r)

z2

(-r<arg

Brunch cut for, In z and 9.

zl+arg

to General

log, z=ln

Base

z/in 1s

4.1.19

where the path does not pass through the origin. 4.1.20 4.1.5 Ln (re@)=ln (refe)+2kti=ln k being an arbitrary integer. principal branch of Ln z.

r+i(e+akr), In z is said to be the

4.1.21 4.1.22

log, z=ln 2 log,, z=ln z/in lO=log,, e In z = (.43429 44819 . . .)ln 2 67

68

ELEMENTARY

TRANSCENDENTAL

4.1.23 ln z=ln 10 log,, z= (2.30258 50929 . . .) loglo z (log, x=ln x, called natural, Napieriun, or hyperbolic logarithms; log,, x, called common or Briggs logarithms.) Series

4.1.24

Expansions

FUNCTIONS

4.1.35 4.1.36 4.1.37

In (1+2)=2-~2z++23-. (/z/11

Iln (l--x)\<

. . and z#-1)

4.1.38

, . .

4.1.39

111

In x
($J+

/In

4.1.26 In z=(z-1)-~(z-1)*+Q(z-l)3-.

(1+2)(:2-ln

(Iz-1111,

ZZO)

(sy+.

. .]

(sy+i

(922 In (%)=2

(k+&+&,+

a+2

(z in the cut plane of Figure Polynomial

zffl)

Approximations

log10 x=alt+u3t”+t(x),

(& rel="nofollow">

(a>O, Limiting

4.7.)

*

t=(x-1)/(x+1)

al= .86304

(&J5+

* . .]

a,=.36415

4.1.42

921--a#z) log10 x=a~t+a~t~+a~t~+a,t’+u~t”+~(x)

Values

lim x-0 In x=0 z+m (a constant,

4.1.31

..

le(x)l56XlO-’ [(&)+f +f

4.1.30

2 42 42 92

422 922 . . =---2;: 22 --. l-- 3- 5- 7-

4.1.29 In (z+a)=ln

Fractions

4.1.41

. . .) jl421,

W<1)

(l-120

4.1.40

z#O)

0,

n

(z in the plane cut from - 1 to - co)

. .

4.1.27 In 2=2 [(s)+i

(XX)

I’*-- 1) for any positive

In (1+z)=*&GG5+6+.

(92 13)

4.1.28

2_<2-1

Continued

(?J+a

(O
(XX)

4.1.25 In z=(G)+:

g

lim xa In x=0 z--10 (CYconstant,

t=(x-.1)/(x$1) je(x)/_
9&~>0) aI=.

1718

u3= .28933 5524

2&~>0)

a’= .09437 6476 f&=.19133

7714

as= .17752 2071

4.1.32 4.1.43 lim $I i-ln m-+m(

m =y (Euler’s >

=.57721 56649 . . . (see chapters 1, 6 and 23) Inequalities

4.1.33

ex


U+X)<J: (x>-1,

4.1.34

z<-In

(l--2)<

01:x<

1

constant)

x#O)

& (x
‘XZO)

In (1+x)=u1x+~x2+u3x3+u4x4+usx5+~(x) Is(x)/ 5 1 x 10-s UI= .99949 556

a4= - .13606 275

a2= - .49190 896

as= .03215 845

a3= .28947 478

* The approximations ings, Jr., Approsimations ,Univ. Press, Princeton,

4.1.41 to 4.1.44 are from C. Hastfor digital computers. Princeton N.J., 1955 (with permission).

ELEiMENTARY

4.1.44

TRANSCENDENTAL

O_
4.1.52

In (1+2)=a~z+u~22+~~23+uq~4S~u~25+ug~6

4.1.53 fw’

+ fws+f

(2)

Approximation

64239 41238 90258 38084

a$= . 16765 ue= --. 09532 u7= . 03608 us= --. 00645

in Terms

4.1.45

S

ln [z+(z2&1)(]dz=z

le(z)1<3XlO-* uq= . 99999 u2= -. 49987 a,= . 33179 u4= -. 24073

69

FUNCTIONS

of Chebyohev

S

zn In [z+(z2&l)i]dz=

2

S

-&+f)+

Definite

In (l+d=n$o

(see chapter 22)

In [z+(z2*l)+]

-- &

3

O
Z’,*(X)=COS n8, cos ~=!kc-l

[2+(22&1)+]-(22511))

4.1.54

40711 93897 84937 35442

Polynomials

In

dz

(n#--1)

Integrals

4.1.55

-4,T,,*@)

4.1.56 n

n

AZ 0 . 37645 2813 1 34314 5750 2 -: 02943 7252 3 .00336 7089 4 -. 00043 3276 5 . 00005 9471

A, 6 -.. 00000 00000 ; -: 00000 9 00000 10 -: 00000 11 . 00000

Differentiation

8503 1250 0188 0029 0004 0001

4.1.57

Trdt-ln t-Zz(x) 4.2. Exponential Beries

Function

Expansion

4.2.1

Formulas

z

22

e’=exp ~.=l+~+~+~+ 4.1.46

5.1.3)

(see

0

z3

. . .

(z=z+iy)

& In z=Jj where e is the real number defined in 4.1.16

4.1.47

&In

Fundamental

z=(-l)n-l(n-l)!z-fl 4.2.2 Integration

4.2.3

4.1.49 4.1.50

s

S

2

4.2.4

In z dz=z In z---z

p+l

4.2.6

n integer)

S

zn (ln z)“-‘dz

Clenshaw, functions, (with per-

z Powers

then z=Log,

N

a”= exp (2 In cc) If u=juj exp (i arg a) l&l = ((+-Y

arg (u’)=y

(--?r<arg

ul?r)

*rg =

In lul+z arg a

4.2.11

Ln u’=z In a 3 The approximation 4.1.45 is from C. W. Polynomial approximations to elementary Math. Tables Aids Comp. 8, 143-l‘L7 (1954) mission).

of General

If N=u*,

4.2.9 4.2.10

(n#-1)

(--?T<:J~IT)

d z exp z=exp

4.2.7 4.2.8

4.1.51

S

(k any integer)

exp (ln 2) = exp (Ln 2) = 2

Definition

(n#-1,

z?+l (In 2)” --- nt n-t-l n+l

In (exp z)=z

4.2.5

zn In zdz=- Zn+l ln z-n+l (n+02

z” (In 2)‘” dz=

Ln (exp 2)=2+2k7ri

Formulas

s dz T=ln

4.1.48

Properties

4.2.12

4.2.13’

for one of the values of Ln uz

In uz=z ln a

(a real and positive) le*[=ez

,

70

ELEMENTARY

TRANSCENDENTAL

arg (eZ)= y &l&Zz=&3

4.2.14 4.2.15 4.2.16 u%*= (cd)"

(--?r<arg

FUNCTIONS Special

a+arg

bS?r)

4.2.22 4.2.23 4.2.24

Values

(see

e=2.71828

18284 . .

eO=l

em= co

4.2.25 4.2.26

e-- =0 efd=-l

4.2.27

e*?= fi

4.2.28

e2*ki

=

(k any integer)

1

Exponential

Inequalities

If 2 is real and different -z 4.2.29 e l-r
4.2.30 4.2, Logarithmic

FIGURE

Periodic

4.2.17

and exponential junctions.

4.2.31

Property

4.2.32

&Wkl= ,f Exponential

4.2.18 4.2.19

(k any integer)

(ezl)Zs=~rlr2

Limiting

(- ?r<921< 7r)

Values

(Jarg 21_
6.5.11)

e'<j-$

@a

+x<(l-“-q<x

4.2.33

x<(e"--I)<&

4.2.34

l+x>t!G

4.2.35

(x>-1)

(x-1)

e">l+$ (

4.2.37

(n>O, z>O)

>

‘>,ezs

P
(O<xI

(~>O,

Y>O)

1.5936)

(Yconstant)

Continued

e,(z) see

ez>l+x

4.2.36 eZ> 1-t;

4.2.21

(For

@-a

5 r) can be removed

4.2.20 /iy zae-z=O * m

from zero

Identities

&z22=ez1+z2

The restriction (- n
1)

chapter

4.2.38

$4
4.2.39

(ez-l(<e~“-ll(z~e’r~

Fractions

@<14<1) (all z)

ELEMENTARY

TRANSCENDENTAL

4.2.42

Approximations

2a nrctnn

e

tzl

2a z--a+

I

a?+4 a2+9 32+ !iz+ 72+ * ”

a2+1

71

FUNCTIONS in

Terms

4.2.48

of

Chebyshev

4.2.43

Approximzltions

O<x
T:(x) =cos no,

cos 8=2x-- 1 (see chapter 22) e-=2

’

2=.693

1.75338 7654 .85039 1654 . 10520 8694 .00872 2105 * 00054 3437 .00002 7115 . 00000 1128 . 00000 0040

e-==1+u~x+u2x2$~E(x) le(x)J<3XlO-” a,=-.9664

u2= .3536

4.2.44

O<x
2

. 00000

le(x)/<3XlO-”

4.2.45

-- .15953 32 .0293641

u3= u4=

O<x
2

+u6x*+u,x’+E(x) le(x)112X10-10 u5= -- .00830 13598 u6= .00132 98820

aa=-.

a’=--.00014

d dz ez=e”

4.2.50

d” ear= anen’ dz”

4.2.51

d d=d dz

4.2.46 6

Integration

4.2.54

a2= .6774323 4.2.47

J

u3= .20800 30 a4= .12680 89

O_<x
10z= (1 +alx+u2x2+a3x3+u4x4t

a& +w6+~7~7)2+~(~)

a2=

.66273

088429

u3= .25439 357484 u4= .07295 173666

us=:. 01742

[(uz)n-n(az)n-1+n(n-1)(uz)n-2

+ . . . +(-1>m-in!+%!]

eaz 5 &=(n-1)2"-'+laY S s5 dz@>I>

(See chapters 5, 7 and 29 for other integrals involving exponential functions.) 4.3. Circular

4 The a proximations 4.2.43 to 4.S1.45 are from B. Carlson, M. E oldstein, Rational approximation of functions, Los Alamos Scientific Laboratory LIA-1943, Los Alamos, N. Mex., 1955 (with permission). 5 The approximations 4.2.46 to 4.2.47 are from C. Hastings, Jr., Approximations for digital computers. Princeton Univ. Press, Princeton, N.J., 1955 (with permission).

Functions

Definitions

111988

u6=. 00255 491796 a’=:. 00093 264267

(n20)

4.2.56

Ic(x)~<5xlO-~ 15129 277603

Formulas

eazdz= earla

4.2.55 f z”e”‘dz=s

JE(x)~~7X10-4

96

z)z’

s

10Z=(l+a,x+azx2+a3;c”+a4x4)2+r(x)

al=l.

$ Y=(l+ln

O<xil

a1=1.14991

In a:

-d y&(&-l dz

4.2.52

13161

.04165 73475

An

64503 5270 31284 1606 03870 4116 00320 8683 00019 9919 00000 9975 00000 0415 00000 0015

For!mulas

4.2.49

4.2.53

a, = - .99999 99995 u2= .49999 99206

A,T:(x) n=o

0001

e-z=1+a~x+a2x2+~3x3+~4x4+~~~5

u4=

ID ; -: -: 3 4 85 -: 16 7 -:

Differentiation

e-z=l+a,x+u2x2+u3x3+u~x4+e(x)

al = - .99986 84 u2= .49829 26

n

A,

..

6

O
(z in the cut plrtne of Figure 4.4.) Polynomial

Polynomials

ei3~e-iz 4.3.1

4.3.2

sin z=

2i

eiL+e--t” cos z=-2

6 The approximations 4.2.48 are from C. W. Clenshaw, Polynomial approximations to cxlementary functions, Math. Tables Aids Camp. 8, 143-147 (1954) (with permission).

72

ELEMENTARY

TRANSCENDENTAL

4.3.17

tan 2=%

4.3.3

FUNCTIONS

43.18 csc

4.3.5

SW z=-

4.3.6

1 cot z=- tan 2 Periodic

4.3.7

cos

(2

2

4.3.19

1 cos 2

z2

tan

2

2

(zl+z2,=“~~t~;“;i”,,“zz’

cot

2

Half-Angle

1

Formulas

4.3.20

sin $= &(qc)’

4.3.21

cos ;= *(

4.3.22

t,an i= &(~~>1=~=~2

Properties

2 +2/h)

(k any integer) = cos

2

tan (z+&r)=tan

4.3.9

tan z,+tan 1

(zl+z2)=cAtan

z1 sin z2

sin 2

sin (2+2h)=sin

4.3.8

tan

q cos z2-sin

(z~+~~)=c~s

1

4.3.4

2=-r--

cos

2

+=)”

The ambiguity in sign may be resolved with the aid of a diagram.

V

,

I Transforkation

of Trigonometric

Integrals

If t,an %=z then 4.3.23

22

sinx co3 x tonx

FIGURE 4.3. Relations

Circularfunctions.

Between

Circular

Functions

4.3.10

sin2 z+cos2 z=l

4.3.11

sec2 z-tan2

2=1

4.3.12

csc2 z-cot?

z=l

Angle

sin (--)=-sin

4.3.14

cos (- z)=cos tan (--)=-tan Addition

4.3.16

sin 22=2 sin ,a cos 2=,“,“t”,“,~2

4.3.25

cos

4.3.26

tan 22=--= 2 tan 2 l-tan22

sin (zl + z2)=sin

co9 z-1=1-2

sin2 z l-tan2 2 =cos2 z-sin2 z= 1+tarP 2

2~2

2 cot

2

cot22-1

2

=-cot z-tan

4.3.27

sin 3z=3 sin z-4 sin3 2

4.3.28

cos 3z=-3

4.3.29

sin 42=8 cos” 2 sin 2-4 cos 2 sin 2

4.3.30

cos 42=8 co& z-8 Products

cos z+4 cosa 2

of Sines

cos2 23-1 and

Cosines

2 sin z1 sin zz=cos (ZI-2&-co8

(21+22)

Formulas

4.3.13

4.3.15

Formulas

4.3.24

4.3.31 Negative

du

cos u=l+zz

Multiple-Angle

1 70

-------

l-z2

sin u=--, 1+z2

2 2 2

Formulas

z1 cos z2+cos z1 sin 22

4.3.32

2 cos

4.3.33

2 sin

Addition

and

z1 z1

cos Z:~=COS (a-z2)+cos

cos z:!=sin (zl-z2)+sin

Subtraction

of

Two

Circular

4.3.34

sin zl+sin

z2=2 sin(‘+)

cOS(y)

(21+22) (21+22)

Functions

2

ELEMENTARY

TRANSCENDENTAL

4.3.41 cos2 a-cos2

4.3.35

sin zl-sin

z2=2 cos(F)

sin(v)

73

FUNCTIONS

z2=-sin

(zl+z2) sin (~~-2~)

4.3.36

4.3.42 cos2 zl--sin2 z2=cos (z1+z2) cos (zl-z2) cos 21--cos z2=-2

sin(Tk)

sin(y)

4.3.43 Signs of the Circular Functions in the Four Quadrants

4.3.38

z2= sin (215 z2) cos 21 cos 22

tan z,i-tan

Quadrant

4.3.39

sin csc ___~-

(~2fzJ cot 21&cot z2= sin . sin z1 sin z2 Relations

Between

Squares

of ISines

and

Cosines

:I III IV

4.3.40

sin2 q-sin2

z2=sin (zl+zJ

sin (~~-4

cos set

tan cot

+ -

+

J

7 -

$ -

4.3.44 Functions

of Angles in Any Quadrant

(0 2 0 15,

in Terms of Angles in the First Quadrant.

k any integer)

2kufe

sin-------cos-- -.---tan--_---csc------set------cot

-sin

cos

e

Fsin

e e

+sec

8 e e e

FCSC

e

e

Ftan8

-tan0 --SC

set

_______

‘fc0t

--cot

Relations ==

4.3.45

8

cos

sin x=u

Between Circular =

cos x=u

Fsin

e

--OS

-set

e e e e

*cot

8

&tan FCSC

fsin ‘fc0t

( 1-u2)*

a(l+a2)-+

a-’

cos x------

(l--d)+

CC

(l+d)-+

u-‘(d-

tan x------

a(l--a2)-*

tc-‘(l--a2)*

a

csc x------

a-1

:1-a”)-*

set xv-----

(l-a2)-*

4

cot x------

a-‘(l-4)*

‘z(l--d)-*

(wx~;)

Illustration:

f00t

e

Functions

=

=

set x=a --

.a-‘(cc-

1)”

(1+a2)-4

(a”- 1)-’

(a2- 1)”

a-’

a-y1+a*y

a

a(a”- I)-”

(1 +a214

(lfa214

u(&-- 1)-t

a

a-‘(1+a2)*

a-l

(&-I)+

(a2- 1) -*

a

-

(a”- 1)-i

1)’

cot x=a

a(1 +a2)-i

cot ~=a-‘(l--a2)t

arcsec u=arccot

+sec

e 8 8 e

a-’

-

If sin x=u,

e

._

8

~CSC

csc x=a

a

-

itan

=

&sin +COS

ftan

~CSC

-set

sin x------

-1

8 e e 8 e

(or Inverse Circular)

tan x=a

-

--OS

-

-

74

ELEMENTARY

4.3.46 -

Circular -

a/12 15O

is -- - _sin

Functions

for Certain 7d6 3o”

TRANSCENDENTAL

Angles

*I4 45O

--

FUNCTIONS

*I3 60’ --

Euler’s

4.3.47

Formula

ez=ez+fu=ez

(cos y+i

De Moivre’s

4.3.48

(cos z+i

sin y)

Theorem

sin z)“=cos

(-*
vzfi

sin ~2


0

$ (J3-1)

cos

1

JZ z cd3+ 1)

4512

112

4.3.49

sin z= --i sinh iz

tan

0

2-J3

1

43

4.3.50

cos z=cosh

csc

co

tan z= --i tanh iz

1)

Jz

2&/3

4.3.51

llz<&+

4.3.52

csc z=:i csch iz

set

1

&qJ3-

1)

4

2

4.3.53

set z=:sech

cot

00

2+i3

4.3.54

cot z==i coth iz

&I2

l/2

&I2 Relation

1

= = 5~/12 75O

4313

=

=

30°

2~13 1200 .-

-

sin

1

4512

$(&i-l)

0

tan

w&

co

csc

j5<&-

1)

1

set

J%&+

1)

Q1

cot

a-& =

cos

A.-

0 = 37r/4 135O

sin cos

-Jz 4

(d3-

JR&-1, +m+ 1)

112

Functions

in

Tgazi

4.3.55

sin z=sin

x cash yfi

cos x sinh y

4.3.56

cos z=cos

x cash y--i

sin 2 sinh y

4 . 3 . 57

tan Z=sin 2x-j-i sinh 2y cos 2x f cash 2y

of

Real

4

Modulus

and

24313

4.3.59

Phase

(Argument)

=[a

-2

(cash

4.3.60

arg sin z=arctan

4.3.61

(cos z~=(co@ =[a

GO0

(Jc1)

4.3.62

Functions

y)+

2y-cos

2x)]+

(cot x tanh y) z+sinh2

(cash

y)f

2y+cos

arg cos z= --arctan

22)]+

(tan x tanh y)

4.3.63

‘cash 2y-cos Itan ‘l=(,cosh 2yfcos

4.3.64

arg tan z:=arctan

0

andImaginary

of Circular

/sin zj= (sin2 z+sinh2

2x 4 2x

(@S&$!)

-1 Series

tan

-1

-a/3

-(2-&i)

0

csc

liz

2

&m+1>

co

set

-42

- 2&/3

-&M-

cot

-1

4

-@+&I

-

iz

2x--i sinh cot z_sincash 2y- cos 2x 2y

-&/a

4%

to 4.5.12)

iz

4 . 3 . 58

’ lln/12 165’

4-$

(see 4.5.7

Circular

-l/2

-&3 ==

4243) =

5n/6 150°

lj5/2

1)

-(2f43)

=I

Functions

=

7~112 105O

n/2

to Hyperbolic

1)

-1 co

Expansions

4.3.65 sin z=z-$+&$+

...

(14-c-1

4.3.66 cos z=l -$+2-S+. . .

.

..

@4<4

ELEMENTARY

TRANSCENDENTAL

75

FUNCTIONS Inerlualities

4.3.67 23 225 tan z=z+~+~+~+

1727

...

4.3.79

22n-1$- . .

+(-l).-i22.(22n-l)Bzn

(2n)!

(

lH<;

>

4.3.80

sin 21 x_< tan 2

4.36% 12 csc

7

2=;+,+,,,

4.3.81

31

z5+

23+15120

. . .

(bl
*2n-l,+

+(-l)n-12(22n-‘-l)BZn

(2n) ! 4.3.69 z2 5z4 61z6 set ~=l+~+~+m+. +(-

..

l)n&

22n--

.

(2n) !

.*

4.3.70 1

23 2z5

2

cot z=;-j-~-py-

( )

(O<x
4.3.33

lsinh y/ 5 lsin ZJ5 cash y

4.3.84

lsinh y/ 5 Jcos 21 Icosh

4.3.85

lcsc z[ Icschlyl

4.3.86

/cos zl Scosh(z)

4.3.87

/sin zI Isinhlzl

y

lH<5

... 4.3.88 22n-1,- .

_ (-- w122n&n

(2n) !

lcos zl< 2,

/sin

(l2l
2[<$[2(

(l2l<7r) Infinite

Products

4.3.71

4.3.72 l*

4.3.89

sin

2=2

4.3.90

cos

2=,i1

cos 2=.& W-1

(-- w2n-1wn(2n) !

m,

p

(1-&)

ln tan T=gI 2

(l-

Expansion

(14<37r>

(2&$

in Partial

>

Fractions

4.3.91

4.3.73 (- l)‘-‘y..;;~l-

1) Bzn 22n

w
4.3.74

sin 2 lim ----xl z+o x

4.3.75

lim tan=1 z-10 x

Values

1

4.3.92

csc224

-

(z-kT)2

4.3.93 . . .>

(z#O,f?r,f%,

Continued

tan z=c

lim n sin E=x n+-

4.3.95

4.3.77

lim n tan 2=x ?a+-

a tan tan az=-----1+

4.3.78

lim cos z=l n-+-

k-i--m

and Euler

4.3.94 4.3.76

jiI

(I4

In

2

2

Fractions 22

--3-

(1-a’) 3f

22

22

5- 7-

...

(2

tan2 2 (4--a2) tan2 5+

(9-a”) tan2 2., . 2
#i*n*)

2

uz #+r

>

76

ELEMENTARY Polynomial

4.3.96

Approximations

TRANSCENDENTAL 1

FUNCTIONS

o<x-$

4.3.101

o<x<;

tan x -= 2

1 +a~x2+u4x4+agx6+agx8+~,ox10

la(s)] <2x10-5 a,=-.16605

u4= .00761

as= .33333

14036 a4= .13339 23995

a3= .02456 50893 alo= .00290 05250

a6= .05337 40603

a12= .00951 68091 05x<;

4.3.102

*

[E(x)(<2x10-9 a2=-. a4= as=-.

16666 66664 .00833 33315 00019 84090

4.3.98

as= .OOOOO27526 alo= - .OOOOO00239

x cot x= I $-a2x2+a4x4+ e (2) 16(2')1<3x10-5 a2= - .332867 02x<

4.3.103

o_<xg;

a2= - .33333 33410 a4= - .02222 20287 a6=-. 00211 77168

a4= .03705 O<X$

Approximations

cos x=1+a~xz+a4x4+a,xe+a8x8+a,0x10+t(x) la(x)ll2X

a2= - .49999 99963 a4=

.04166 66418 as=-.00138 88397

alo=

.00002 -. 00000

47609 02605

--l+a2x2+a4x4+t(x) 2

(a(x)(
7 The a proximations 4.3.96 to 4.3.103 are from B. Carlson, M. e oldstein, Rational approximation of functions, Los Alamos Scientific Laboratory LA-1943, Los Alamos, N. Mex., 1955 (with permission).

aa= - .00020 78504 alo= - .00002 62619

of Chebyshev

Polynomials

--11x51

T: (x) = cos ti, cos 8=2xsin ~~x=x *co A,T,*(,x*)

1 (see chapter 22) cos $rx=2

n-0

A,T,*(z*)

A, 1.27627 8962 1 -.28526 1569 2 .00911 8016 3 - .00013 6587

n A, 0 .47200 12 16 1 - .49940 3258 .02799 2080 2 3

- .00059 6695

4

4 5

.OOOOO6704 - .ooooo 0047

0

tan 2

a*= .31755

Terms

n

o<x$

4.3.100

in

4.3.104

10-Q aa=

c (x)

Ir(x)l<4Xlo-'O

16(X)1<9x 10-h

4.3.99

;

x cot x= 1 +a2x2+a4x4+a~x6+a8x8+aIox10+

cos x=1+a,x2+a,x4++(x)

a2= - .49670

a,=-.024369

5

.OOOOO1185 -.ooooo 0007

n The approximations 4.3.104 are from C. W. Clenshan, Polynomial approxinlations to elementary functions, Math. Tables Aids Camp. 8, 143-147 (1954) (with permission). *see page Ix.

8

ELEMENTARY Differentiation

TRANSCENDENTAL

Formulas

4.3.105

d 2; sin z=cos 2

4.3.106

z cos z=-sin

d

FUNCTIONS

77

4.3.122

S

z dz

-2

sin”=(n-

1 1) (n-2) sinnm2 2

cos 2

1) sinn-l z-(n-

z 4.3.123

-$ tan z=sec2 z

4.3.107

. 4.3.108

d & csc z=-csc

4.3.109

z set z=sec z tan 2

4.3.110

d -& cot 2=-csc2

S

d

2

S‘

sdz

S

tan z+ln

Ldz=z cos2

2

cos

2 dz cosn=+1)

S

2 cop-1

(La)

z-(%-l)

.I-

S

sinm

z

tan zdz=-ln

4.3.116 I-

J

csc zdz=ln

2 cos”

2

dz =

sirP+’

2 coC1

2

m+n

S

sin”’

cos z dz=sin ;:

S

4.3.115

cm>21

4.3.127

Formulas

s

4.3.114

cosn-2Z

S5

++-2) (n-1)

sin zdz=-cos

2

sin

2

4.3.112

4.3.113

(n>l)

4.3.126

4.3.111

Integration

zdz

zn-‘sin

4.3.124

2 cot 2

4.3.125

cosz==ln secz

=-

sinme

2 cosnfl

2

cosnm2 z dz

2

m+n

tan :=ln

(csc z-cot

S

sin”‘-*

z)=- 1 In- 1-cosz 2 lfcosz

4.3.117 s

S

sin z-n

zdz=zn

zn cos

2 COS”

2

dz

(m#-n)

4.3.128

S

seczdz=ln(secz+tanz)=lntan =Inverse

sinm z”Z,oP

Gudermannian

z=(n--1)

1 sinmS1

+

Function

2 cosnel

m+n-2

gd 2=2 arctan eZ-i

n-l

2

dz

S

sinm

2 coP2

2

(n>l)

S

cot zdz=ln

4.3.118 4.3.119

S

Psinzdz=-zz”

sin z=--1n

cos z+n

csc z

=(m-1)

sin:: +

z”-’ cos zdz

2 cosnml

m-b-2 m-l

S

2

sirF2

dz 2 cosn (m>l>

4.3.120

S 4.3.121

4.3.129

S

Adz=--z sin* 2

cot z-rln sin z

4.3.130

S S

tannzdz=ts-fian”-2zda

cot”zdz=---

COtn-’

n-l

(n#l> 2

S

Cotn-2ZdZ (n#l)

2

78

ELEMENTARY

TRANSCENDENTAL

4.3.131 a tan clz =------ 2 arctan (a”-- b2)t a+b sin z (a”--b”)*

a tan

1

0

In

(bz-a2)+

a tan

0

S

S

lsin2 nt dt== *cos2nt dt=%

4.3.141

S

=-

FUNCTIONS

0

(az>b2)

0

(n an integer,

1

z +b-(b2-a*))

S

msirltmt &=;

4.3.142

=o

(b2>a2) ---7-=Ttan lfsm 2

4.3.133 dz

S

a+b cos z

4.3.143

2 arctan (a2-b2)*

(a”-b2)i

(b-a)

=&h-l

2

tan c- (b2-a2)+ 2

0

(b2>a?)

S S

dz

-------=tan 1+cos 2

4.3.134

%

1-cos 2

,

0

s/2

In cos t dt=-;

In 2

.O

m cos mt o T+T dt=ae-”

S

4.3.146

Formulas

e”” sin bz dz=-

In sin t dt=

J

i

involv-

4.3.147

4.3.136

S

S f

-cos t2 dt=i

0

4.3.145

r/2

(b/a)

(See chapters 5 and. 7 for other integrals ing circular functions.) (See [5.3] for Fourier transforms.)

-2 2

dz=-cot

4.3.135

S

1 S

(m

2

mcos at--cos bt ___- L dt=ln 6

-sin t2 dt=

4.3.144

W>W

tan z+(b2-a2)+

(b-a)

Jo

tan i

(a-b)

=-----

r

(m=O) n

=--

dz

S

4.3.132

(m>O)

0

E +b+(b2-a2)t

n#O)

ea” (a sin bz-b

a2+b2

for

Solution

of Plane

Right

Triangles

cos bz)

4.3.137

S

e”” cos bz dz=-

em (a cos bz+b sin bz) a2fb2

4.3.138

C

eazsin“-I bz

S

earsinn bz dz=

a2+n2b2

(a sin bz-nb

+

eazsinne2 bz dz

4.3.139

S

e”’ cosn bz dz=

cos bz)

S

n(n-l)b2 a2+n2b2

eaLcos”-’ bz (a cos bz+nb sin bz) a2fn2b2 +

Definite

n(n-W2

S eaz

a2+n2b2

S S

If A, B and C are the vertices (C the right angle), and a, b and c the sides opposite respectively, sin A=:=c

1 CSCA

cos A=-= 6 - 1

c set A

COSn-2

bz dz

tan A=i=c+A

Integrals

versine A=vers

*

4.3.140

A

b

A=l-cos

A

sin mt sin nt dt=O

0

coversine A=covers

(mfn,

m and n integers)

*

cos mt cos nt dt=O

0

haversine A=hav

A=l-sin A=$

vers A

exsecant A=exsec A=sec A-l

A

ELEMENTARY

TRANSCENDENTAL

4.3.148

4.4. Formulaa

for

Solution

of Plane

79

FUNCTIONS

Inverse

Circular

Triangles

Functions

Definitions

4.4.1

* ~rcsm z= s o

s

A A/

‘1 b

cos A=

c*+ b*-a* 2bc

a=b cos C-l-c cos B a+b tan $(A+B) a--b=tan #(A-B) area

bc sin A =-=[s(s-a)

(s- b)(s,-c)]+

2

s=j(a+b+c) 4.3.149 Formulas

for

Solution

S

arctan z=

s

4.4.3

c

of Spherical

Triangles

l

arccos z= C

In a triangle with angles A, B and C and sides opposite a, b and c respectively, a b sin=sin=sin

(2=x+$

4.4.2

0

C

dt (l-t*)*

dt

----=9-arcsin z (l-t*)* 2

2

p dt

0 1+t*

The path of integration must not cross the real axis in the case of 4.4.1 and 4.4.2 and the imaginary axis in the case of 4.4.3 except possibly inside the unit circle. Each function is singlevalued and regular in the z-plane cut along the real axis from - 0~ to - 1 and + 1 to + 03in the case of 4.4.1 and 4.4.2 and along the imaginary axis from i to i 0~ and --i to --i 03in the case of 4.4.3. Inverse circular functions are also written arcsin 2= sin-’ 2, arccos z=cos-’ 2, arctan 2 =tan+ 2, . . . . When -1 _<x_<1, arcsin x and arccos x are real and 4.4.4

- $ir 5 arcsin 2 5 &r,

0 <;Irccos x,
4.4.5

arctan z+arccot

R220 RZ
z=&)s

4.4.6

arccsc 2= urcsiu l/z

4.4.7

arcsec z=arccos l/z

4.4.8

arccot z=arctan

l/z

arcsec 2 + arccsc 2= +a

4.4.9

(see 4.3.45)

s

C (I

A 0

b

m -I

C

sin B

sm b

+I

I -i

If A, B and C are the three argles and a, b and c the opposite sides, sin A y=,=-sin a

0

arcsin arccos

z and z

orccsc orcsec

2 and L

arctan

z

orccot

L

sin C sm c

cos a=cos b cm c+sin b sin c cos A =cos b cos (CH) CO8

e

where tan B=tan b cos A FIGURE

cos A=-cos

B cos C+sin 13 sin C cos a

4.4.

Branch cuts for jzbnctions.

inverse

circular

80

ELEMENTARY Fundamental

TRANSCENDENTAL

Property

The general solutions

of the equations

sin t=z

FUNCTIONS

4.4.23

Arccsc

z=i

4.4.24

Arcsec

z= f i Arcsech

4.4.25

Arccot

z=:i Arccoth

cos t=z

Logarithmic

tan t=2 are respectively 4.4.10

z= k arccos z+Zkr

t=Arccos

4.4.12 t=Arctan

z+ka

z=arctan

k is an arbitrary

where 4.4.13

iz z iz

Representations

4.4.26

Arcsin

x=-ii:Ln

4.4.27

Arccos

x=--i

4.4.28

Arctan

x=i

[(l-x”)*+ix]

(~“51)

Ln [x+i(l-x2)*]

(x251)

z= (- 1)” arcsin z+kn

t=Arcsin

4.4.11

Arccsch

Interval Y

Ln (x real)

(9#-1)

integer.

containing principal value x positive x negative or zero

arcsin x and arctan x

0 < y 5 a/2

- ?r/2 < y
arccos x and arcsec x

0 5 y 5~12

arccot x and arccsc x

0 i y 57~12

d2
4.4.29

Arccsc

x=-i

Ln [(x2-l)r+i]

4.4.30

Arcsec

x=--i

Ln

4.4.31

Arccot

(~“2

s ?I-

1)

(5 real)

- ?r/2 _
Addition

and

Subtraction of Functions

Two

Inverse

Circular

4.4.32 Arcsin

z1 f Arcsin

z2 =:Arcsin

[21(1-z3+~22(1-2T)t]

4.4.33 Arccos

21 f

Arccos z2 =Arccos{

L-20 -

FIGURE

4.5.

-

DlCILC OlECOl

I

Arctan

Inverse circular junctions. Arguments

4.4.35

4.4.14

arcsin (- 2) = - arcsin 2

Arcsin

4.4.15

arccos (- 2) = 7r- arccos 2

4.4.16

arctan

2

4.4.36

4.4.17

arccsc (- 2) = - arccsc 2

Arctan

4.4.18

arcsec (- 2) = 7r-arcsec

4.4.19

arccot

Functions

Relation

to Inverse

of Negative

(- 2) = - arctan

(-z)=-arccot Hyperbolic 4.6.19)

(l-z$]‘j

z1f Arccos

z,=Arctan

(e)

2:)

=Arccos

2

Functions

zlfArctan

=Arcsin{wdz[(l--zT)

z1 &Arccot

Inverse (see 4.6.14

to

Circular

(l-$)1’}

[zz(l-2~)~~zz1(l-z~)*]

z2

=Arctan

2

(*)=Arccot

Functions Imaginary

(--E?-E?)

in Terms Parts

Arcsin

z= -i

Arcsinh

iz

Arcsin

4.4.21

Arccos

z= f i Arccosh

z

4.4.38

4.4.22

Arctan

z---i

iz

~=kd-(-l)~

Real

arcsin/ f(-l)%In

(zz# -1)

of

4.4.37

4.4.20

Arctanh

zlz2F[(l-23

4.4.34

I

Arccos z=2knf

{arccosfi-i

[cYS(d--l)q

In [(~+(a~-l)~I}

and

ELElMENTARY

TRANSCENDENTAL

4.4.39 Arctan

FUNCTIONS

81

4.4.46

z=ks++

O<x
arctan ($&) arcsin ~=T-(l--z)~(a,+a~z+a,22+a3~3 2

+; lu [$H]

(zZ#-1) +u4x4+ujx5+u6x6+u,x’)

k is an integer

where

or zero

uild

le(x>l<2X 10-s

Cx=~[(2+1)~+?J~]~+~[(x--l)~+y~]~ P=~[(2+1)2+y2];-3[(2--1)2f?12]; Series

Expansions

4.4.40 23 1 a325 1 --*3.5;:’,7+. . . arcsIn 2=2+~~+~.~ .5+2e4e6

(l4<1)

uo= 1.57079 63050

CIQ= .03089 lSS10

aq= -.21459 88016

us= -.01708 81256

I&=

us=

.08897 89874

a3= -.05017 43046 4.4.41

4.4.41 arcsin (l-2)+(22)’

4.4.42

u1= 5-, “?+ . . . (/z/_
u3=

-

.99986 60

aI= -.08513 30

.33029 95

uQ=

.18014 10 -11&l

arctan x=Continued

Fractions

4.4.49 l1 arctan x

in the cut plane of Figure 4.4.)

(2

2

4.4.44

2 1.2.~~1.2.~~3.4~~ 3.4~~ ae22=1--r-cs-. .. in the cut plane of Figure 4.4.)

(2 Polynomial

ApproximaGons

4.4.45

8

O<x
arcsin 2=~-(1-~)*(a,+am-~a?rz+asr3)+~(r) /e(x)1

u2=:

aq= -.21211 44

u3=-

Q The approximations

Univ. Press, Princeton,

Ojril =l+k~u2&k+r(r) le(x)l12xlO-*

u2= -.33333 u4=

14528

.19993 55085

ulo= - .07528 96400 a12=

90429096138

u8= - .14208 89944

u14= -.01616

ua=

am=

.10656 26393

57367

.00286 62257

55x10-5

a,,= 1.57072 88

ings,Jr., Approximat.ions

1+:8x2+‘(“) Ir(x>l_<5x10-3

- z2 -4z2 -.92 -1622 1+ 3+ 5+ 7-t 9+. . .

z=z

.02083 51

. .((zl>landz2#-1) 4.4.48 lo

arctan

24911

le(x) 15 10-b

uj=

4.4.43

a,=-.00126

-l_<s
(121<2>

=i-i+&-&+.

.00667 00901

arctan x=ulx+u3x3+u~x6+u,x7+uaexQ+~(x)

1 +ks

arctan z=z 2,z”3

-tE(X)

.07426 10 -

.01872 93

4.4.45 to 4.4.47 are from C. Hastfor digital computers. Princeton N.J.. 1955 (with permission).

10The approximation 4.4.48 is from C. Hastings, Jr., Note 143, Math. Tables Aids Comp. 6, 68 (1958) (with permission). 11 The approximation 4.4.49 is from B. Carlson, M. Goldstein, Rational approximation of functions, Los Alamos Scientific Laboratory LA-1943, Los Alamos, N. Mex., 1955 (with permission).

82

ELEMENTARY

Approximations

in

Terms

4.4.50

of Chehyshev

TRANSCENDENTAL

Polynomials

I2

FUNCTIONS

d z arccsc 2=-2

4.4.57

-l<S
Tf(x)=cos ne,

cos 0=2x-

1

(see chapter

Integra-tion

22) 4.4.58

arctan x=x

s

g0 A,T~(zz)

:: 3 4 5 For

A,

i!t

00000 3821

-: 01113 10589 2925 5843 -: 00138 1195 .00018 5743 -. 00002 6215

7 ii 10

-: . 00000 0570 0086 -. 00000 0013 . 00000 0002

2 >l,

use arctan

s=$7r-arctan

s

arctan 2 dz==z arctan 2--3ln (1-j-z”)

4.4.61 arccsc 2 d2=2

arccsc 2fln

[~+(2~-1)*]

c

O<arccsc 2<5

4.4.51

-$Ji<X<$JZ

arcsin x=x

-i<arccsc

nTOA,Tt(2x2)

arccos x=&r-x n 0

A?& 1.05123 1959

;

: 05494 00408 6487 0631

i

.00004 6985 .00040 7890

For $~
5

ItaO

A,Tz(2x2)

O<arcsec 2
n :

4 00000 5881 : 00000 0777 0107

9

[

arc-

;<

arcsec 2
4.4.63

S

arccot 2 dz:=z arccot 2+$ In (l+z2)

:. 00000 0015 0002

use arcsin x=arccos(l-x2)+, (1--x2)+. Differentiation

2<0

4.4.62 . arcsec z dz=.z arcsec zTln [2+(22-l)+]

o<x<@

4.4.52

4.4.60

s

(l/z)

arcsin z dz ==z arcsin z + (1- ,z2)i

arccos z dz:=z arccos z-(3-z2)t

A,

88137 3587

Formulas

S

4.4.59

21

1 (e2-l)i

4.4.64 arcsin 2+$

(1-22)4

Formulas

& arcsin z=(l-z2)-+

4.4.65 9+1 arcsin n-j-1

S

2" arcsin 2 dz=--

4.4.53

(n#-1)

$ arccos z=-(I-z2)-f 4.4.66

4.4.54

d 1 arctan z=z 1+22

4.4.55

d -1 z arccot 2=1+22

S

z arccos z dz.7

d 1 z aJJcsec2=2 (2”- 1)t

la The approximations Clenshaw, Polynomial functions, Math. Tables (with permission).

4.4.50 to 4.4.51 are from C. W. approximations to elementary Aids Comp. 8, 143-147 (1954)

(l--z*)+

4.4.67

S

2n arccos 2 dz=Ez

4.4.56

arccos z-f

n,+l

arccos @Z-l)

,

4.4.68 1 2 arctan z dz=- (l+29 2

S

arctan 2-g

2

ELEMENTARY

TRANSCENDENTAL

4.4.69

1 arctan z---n+l

p+l

S

z” arctan z dz=-

n+l

S

gdz (n#-I)

4.4.70

S

z arccot z dz=i

nrccot z+:

(14-S)

83

FUNCTIONS

L.5.8

cash 2=cos

E.5.9

tanh z= --i’tan

L.5.10

csch z=i csc iz

E.5.11

sech z=sec iz

h.5.12

coth z=i cot iz Periodic

4.4.71

S

n+l

arcco t z J,-

Properties

z (k any integer)

(n#-1) 4.5. Hyperbolic

iz

sinh (2+2k7ri) =sinh

1.5.13

,$a+1

zn arccot z dz=-

iz

Functions

cash (zf2ksi)

a.5.14

tanh (z+k&)=tanh

1.5.15

Definitions

=cosh z

Relations

Between

z

Hyperbolic

Functions

4.5.1

ez-e-2 sinh z=2

4.5.2

e”+e-” cash z=2

4.5.3

tanh z=sinh

4.5.19

cash z+sinh

z=e’

4.5.4

csch z = l/sinh

4.5.20

cash z-sinh

z=e-”

4.5.5

sech z= l/cash z

4.5.6

coth z=l/tanh

(z=x+iy>

Z/COS:I z z

4.5.16

cosh2 z-sinh2

4.5.17

tanh2 z+sech2 z= 1

4.5.18

coth2 z-csch2 z= 1

Negative

Angle

z=l

Formulas

4.5.21

sinh (-.z)=-sinh

4.5.22

cash ( - z) = cash z

4.5.23

tanh (-z)=-tanh

z

Addition

4.5.24

z

z

Formulas

sinh (zl+z2) =sinh

z1 cash 22 +cosh z1 sinh 22

4.5.25

cash (zl+ 22)=cosh ZI cash

22

+sinh 4.5.26

tanh (q+zQ=(tanh

zI+tanh

z1 sinh zz &)/

(1 + tanh z1 tanh 22) FIGURE

4.6.

4.5.27

Hyperbolic jusctions.

coth (zl+zz)=(coth

zl coth %+l)/ (coth zz+coth

Relation

to Circular

Hyperbolic trigonometric

Functions

(see 4.3.49

to 4.3.34)

can be derived from formulas identities by replacing z by iz

Half-Angle

4.5.28 sinh E=(,,sh;-l)i

4.5.7

sinh z=--i

sin i;r

Formulas

ZI)

ELEMENTARY

84

TRANSCENDENTAL

FUNCTIONS

4.5.44 cash z,-cash

z2=2 s:inh (p)

sinh (v)

4.5.45 zl+tanh

sinh (21+z2) %=E;h L 2 cosh 2 1 2

coth z,+coth

sinh (2, +2,) zz=- smh 2, sinh zz

tad 4.5.46 Multiple-Angle

Formulas

4.5.31

sinh 22=2 sinh z cash ~=~~‘a~~f,

4.5.32

cash 22=2

cosh2 z--l=2

Relations

Between

Squares of COosines

4.5.47 sinh2 zL-sinh2 zz=sinh =cosh2 4.5.48 sinh2 zl+cosh2 z2=cosh =cosh2

sinh2 z-j-1

= cosh2 z + sinh2 z

Hyperbolic

Sines

(zl+zz) sinh (zl--q-cosh2 zz (q+zJ cash zl+sinh2 z2

z2)

(2,-z2)

4.5.33

tanh 2~=12+tt&annhhez,

4.5.34

sinh 32=3

4.5.35

cash 32=-3

4.5.36

sinh 42=4 sinh3 z cash 2+4 cosh3 z sinh z

4.5.49

sinh z=sinh

x cos y+i

cash x sin y

4.5.37

cash 4z=cosh4 z-j-6 sinh2 z cosh2 z+sinh’

4.5.50

cash z=cosh x cos yfi

sinh x sin y

Products

4.5.38

Hyperbolic

sinh 2+4 sinh3 z

of Hyperbolic

Sines

and

(Q-.zz>

z

and

Subtraction

of Two

De Moivre’s

(q-q)

(~~-2~)

4.5.55 z2=2 sinh (‘9)

cash (‘9)

4.5.42 sinh z,-sinh

4.5.54

4.5.56

4.5.57 z2=2 cash (9)

(cash z+sinh and

Phase

Theorem

z)“=cosh

nz+sinh

(Argument) Functions

of

m

Hyperbolic

lsinh z/ = (sinh2 x+sin2 y)* =[+(cosh 2x-cos 2y)]”

Functions

4.5.41 sinh z,+sinh

4.5.53 Modulus

(zI+zz)

Hyperbolic

Imaginary

4 . 5 .52 coth 2=sinh 2x---i sin 2y cash 2x-cos 2y

2 cash 21 cash zz=cosh (zl+zJ

+sinh Addition

and

4 . 5 .51 tanh 2=sinh 2x-j-i sin 2y cash 2x+cos 2y

Cosines

2 sinh z1 sinh zz=cosh (zl+z2)

2 sinh z1 cash z2=sinh

of Real

(z:=zfiy)

+cosh 4.5.40

in z;;;s

cash 2+4 cosh3 z

-cash 4.5.39

Functions

and

sinh (y) 4.5.58

arg sinh z=a.rctan

(coth x tan y)

(cash zl= (sinh2 x+cos2 y)” =[+(cosh 2x+cos 2y)]h arg cash z=arctan

(tanh x tan y)

cash 2x-cos 2y * ltanh Z1=(cosh 2x+cos 2y

4.5.43 cash zl+cosh z2=2 cash (‘+)

cash (y)

4.5.59

arg tanh z=arctan

~ (f:2,“,) Y

ELEMENTARY

Relations

4.5.60

TRANSCENDENTAL

Bet ween Hyperbolic

sinh x=a

CO~ll x=u

85

FUNCTIONS

(or Inverse Hyperbolic)

tanh z=a

Functions

csch x=u

sech x=,z

= --

coth x=a

sinh x----?.

a

(a’-. l)+

a(l-u2)-1

CL-’

a-‘(1-a2y

(a’- 1) -1

cash x----o

(l+u*)+

a

(l-a”)+

a-‘(l+a2)t

a-l

a(a2- 1)-i

tanh x-----

a(l+a’)-4

u-‘(a2- 1)’

a

(l+ay-+

(l-a’)+

a-’

csch x-----

am1

(a’- 1)-’

a-‘(l-a2)+

a

a(l-a2)+

(a2- 1)’

soch x-----

(l+u’)-+

a-*

(l-4)+

a(l+a2)-*

a

a-‘(a2- 1)’

coth x-----

a-‘(a2+l)f

u(a2-- 1)-i

a-l

u+a2y

(1-a2)-4

a

-

Illustration:

If sinh x=a, coth x=aB1(a2+ arcsech a=arccoth

Special Values of the Hypl>rbolic

4.5.61

Functions z

* . -a 2

0

2

1)’

(l--a2)-)

4.5.66 sech z=l

-f+&

24-go

zs+

. . . +a*

22*+

(

0

i

0

--i

co

4.5.67

z-------

1

0

-1

0

cu

coth

tanh

z------

0

-i

0

-coi

1

csch

2 _______

0~

4

OD

i

0

sech

2 _______

1

OD

-1

co

0

coth

2 _______

0~

0

co

0

1

sinh

2 ____

cash

_ __

. . .

co

2=;+;-$+$f-

IM<;

. . . +2~22n-1+

>

. . .

(l4<*)

where B, and E,, are the nth Bernoulli and Euler numbers, see chapter 23. Inhite

4.5.68

Products

sinh z=z ii

(l+&)

-. Series

Expansions

4.5.69

cash z=fi

l+ k=l

23

ZfJ

4.5.62

smh ’

4.5.63

cash z=l+$+$+;+ . .

1

(2Zzf)%2

C

2

~=2+9+3+~+

.

. . .

(14-c

...

(l4< m)

-1

Continued

4.5.70

tanh 2=&

Fraction 22

22

3+ 5f

24

7+ . . . (

Differentiation

*

+

...

+22W2”-1)B,,

22n-l+

(h)!

’’*

4.5.65 csch z=I

’

z--T;+360

’

23

page

II.

Formulas

4.5.71

d z sinh z=cosh

2

4.5.72

d cash z=sinh z

2

4.5.73

& tanh z=sech2 z

4.5.74

zd csch z=-csch

-j&o25+...

-2(22”-‘-l)B2, (2n)!

*See

2 #i

2”-1 +... 2 (l4<3d

z coth z

ifnni

>

86

ELEMENTARY

d z sech z=-sech

4.5.75

TRANSCENDENTAL

FUNCTIONS,

4.5.87

z tanh z

S

tanh”

4.5.76

-$ coth z=-csch2 Integration

z

s

cash zdz=sinh

tanh z dz=ln

4.6. Inverse

(sinh z)

coth zdz=ln

sinh z

4.5.83 P cash z-n

en sinh z dz=z”

4.5.84

S S

(See chapters 5 and 7 for other volving hyperbolic functions.)

z

tanh :

sech z dz=arctan

4.5.82

z” cash 2 dz=z”

r J

9-i

arcsinh

z=

4.6.2

arccosh

z=

4.6.3

arctanh

z=

integrals

in-

Functions

z cash” z dz=-

1

sinh”‘+l

=--&

S

dz sinh” z coshn z=s - m+n-2 m--l

z cosh’+l

to --i and i

Inverse hvnerbolic functions are also written ” -. sinh-’ z, arsmh z, JS’T sinh z, etc.

z

sinh” z coshnm2 z dz

S

4.6.4

arccsch z = arcsinh 1/z q

4.6.5

arcsech z ==arccosh 1/z

4.6.6

arccoth z==arctanh 9

l/z

iy

ti

-1

1

sinh”-’

z coshn-l

dz sinhmm2 z cash” z 1 n-l

sinh”

X

(m+n#O>

S S

z dt -

0 1-P

sinhm+ z cash” z dz

=-

m+n-2 + n-l

iz (t2fl)t

4.6.3 real axis from - ~0 to - 1 and + 1 to +a

sinhmel z coshn+l z

m-l -m+-

(z=x+iy)

4.6.2 real axis fro.m - 03 to + 1

+- S m+n

Oz(lTt2)1

4.6.1 imaginary axis from --im to iw

zn-l sinh z dz

m+n

S S S

The paths of integration must not cross the following cuts.

cash z dz

z-n S

sinh

n-l

4.5.86

Hyperbolic

4.6.1

4.5.85

sinh”

z dz

Definitions

csch zdz=ln

4.5.81

J

coth” z de= -%!@.?+SCoth”-2 n-l

cash z

S S S

4.5.80

Wl)

Wl>

s

4.5.79

z dz

s

z

s

4.5.78

tanh”-‘z+Stanh”-” n-l

4.5.88

Formulas

sinh z dz=cosh

4.5.77

z dz=--

z

3 -i arcsinh

z

arccsch

z

--CD

4-I

X

t-

X

-I

0 arccosh

.z

1 arcsech

z

iy I0 +I

arctanh

(m#l> 1

sinh”-’ dz z cosh”-2

5 cash”-’ z

Wl>

z FIGURE

4.7.

Branch cuts for junctions.

1 arccothz

inverse

hyperbolic

z

ELESMENTARY

arctanh z=arccoth

4.6.7

TRANSCENDENTAL

z& $ri

(see 4.5.60) Fundamental

arccoth x=i

4.6.25

(according

87

FUNCTIONS

as J&SO)

x+1 In x-l

(x2>1)

Property

The general solutions of the equations z=sinh t z=cosh t z=tanh

t

are respectively 4.6.8

t=Arcsinh

~=(-l)~

arcsinh z+kri

4.6.9

t=Arccosh

2= harccosh

4.6.10

t=Arctanh

z=arctanh

z+2k& z.+kti (k, integer)

Functions

of Negative

Arguments

FIGURE

arcsinh (- z) = - arcsinh z

4.6.11

arccosh (- z) = ai- arccosh z

*4.6.12

Addition

4.6.13 Relation

Inverse hyperbolic junctiolas.

4.8.

arctanh to

Inverse

(-2) = -arctanh Circular

and

Subtraction

z

Functions

(see

of

Two

Inverse

Hyperbolic

[2,(1

+z:)*d~

~~(1

+z;>+]

[ (zi-

1) (z$-

l)]‘}

Functions 4.4.20

to

4.6.26

4.4.25)

Hyperbolic identities can be derived from trigonometric identities by replacing 2 by i2. 4.6.14

Arcsinh

4.6.15

Arccosh z= k i Arccos

4.6.16

Arctanh

Arcsin iz

2=-i

2=

4.6.17

Arccsch z=i

4.6.18

Arcsech

4.6.19

Arccoth z=i

-i

Arcsinh

z1

&Arcsinh

z2

=Arcsinh 4.6.27

2

Arccosh

2,

Arctrm iz

f Arccosh

z2

=Arccosh

{ 2,z2k

Arccsc iz 4.6.28

2=

5 i Arcwc z

Arctanh Logarithmic

z,&Arctanh

z2=Arctanh

Arccot iz 4.6.29

Representrltions

4.6.20

arcsinh z=ln

[x+(x*-+l)t]

4.6.21

arccosh x=ln

[z+ (x2-- l)i]

4.6.22

arctanh x=4 In -1+x l-x

4.6.23

arccsch x=ln [:+($-I-1)1]

4.6.24

arcsech s=ln

[i+($-

Arcsinh z,fArccosh (x21)

=Arcsinh

/[za(l

+z:)‘&

2,(2[--

4.6.30

Arctanh

(x+0) (O<X<

{ 2,22~:[~1+2:)(2~-1l)]‘j

=Arccosh

Wx2<1)

1>‘1

z2

1)

2,fArccoth

z2=Arctanh

=Arccoth ‘see pageII.

(S> ( 2ri:i;)

l)‘]

ELEMENTARY

88 Series

TRANSCENDENTAL

Expansions

FUNCTIONS

4.6.42

4.6.31 arcsinh z=z --.A2 . 3 z3++-&

-$ arccoth. 2=(1-S)+

25

Integration

1.3.5 -2.4.6.7

“+

’ ’ *

4.6.44

(lzl
1.3 =In 2z+1 2 ’ 222 - 2 * 4 * 424 +

4.6.43

4.6.45

l-3.5

2.4.6.&e-

'*'

4.6.46

s s

Formulas

arcsinh z dz=z

arcsinh z-(1+,9)+

arccosh zdz=z

arccosh z-(z2-1)f

Sarctanhzdz=z s

arctanh z++ln

arccsch z dz=z

arccsch

arccosh z=ln 22---

4.6.33

1 1.3 2 * 222 2 * 4 * 4z4 1.3.5 -2.4 . 6. 69--

arctanh z=z+$+f+g+

...

4.6.47 4.6.48 * **

Sarcsech z dz = z arcsech Sarccoth z dz=z arccoth

arccoth z=$-&+&~+$+

z arcsinh z dz=--

(14>1)

s

(121<1)

4.6.50

(I 01) Continued

4.6.35

(z in the cut plane of Figure 4.7.)

4.6.36 arcsinh z 2 1 * 222 1 ’ 222 3 * 422 3 * 422 =1+3+5+-7+ 9+ -** 1/1+z2

s

Differentiation

4.6.37

S2 arccosh

2 arccosh z=(z2-1)-f

arcsinhz

z dz:= 2221 arccosh z-$94

4.6.52 2”arccosh zdz=*‘-

S

,,n+1 arccoshz--

7t+1

$1

1)”

p+1

(z”- l,td2 (nit-l)

S

4.6.53 z arctsnh a dz--2

S

22-l

arctanh z+f

4.6.54

s 2”

4.6.38

n-i-1

(n#-1)

Formulas

$ arcsinh z=(l +z2)-i

.z++ In (~~-1)

4.6.51

Fractions

22 422 922 arctanh z=$- - 3- 5-- 7-. . .

*

2 f arcsin z

z!z2+ 1 arcsinh z-z 4 (z’+lY 4

p+1

S

...

*

4.6.49

zR arcsinh zdz=--

4.6.34

z f alcsinh z

(according as 9~~0)

(14>1) 4.6.32

(l-9)

pa+1

arctanh

2

dz:=-

n+l

arctanh z -&Ssdz (n#-

1)

4.6.55 4.6.39

2 arctanhz=(l-,9)-I

4.6.40

z arccsch z=T

d

Sz arccsch ’ z(l+S)) (according as 9?z=O)

4.6.41 *See

pageII.

1 2 arcsech z= 7 dz z(l-.zy

z de=<

arccsh z&-f (l+zz)* (according as &z~O)

4.6.66

p+1arccsch

.z”arccsch z dz- -

S

n+l

2f

(n# -1)

*

ELEMENTARY

4.6.57 s

*

z arcsech z dz=$

TRANSCENDENTAL

arcsech tri

4.6.59

(l-z*)1

S

z arccoth z dz- -2

(acc.ording as %ZO) 4.6.58

89

FUNCTIONS

I 4.6.60 1 -n+1

p+1

s

z” arcsech z dz =-

nfl

arcsech z + -

S

zdz (l--z*)’

z” arcc0t.h z dz=-

(n#-1)

of the Tables

1. Computation

of Common

arccoth z+-

1

n+l

2.

Compute xw314for x=9.19826 to 10D usjng t,he Table of Common Logarithms. From Table 4.1, four-point Lagrangian intierpolation gives log,, (9.19826) = .96370 56812. Then, log,, (x) = - .72277 92609=9.27722 073!$1- 10.

Linear inverse interpolation in Table 4.1 yields antilog (‘i.27722) =. 18933. For 10 place ndcurncy subtnbulation with 4- oint Lngrnnginn intcrpolnnt’s produces the tn Ele N .18933 .18934

log,, 5 * 10’

A:’

A

log,, N .27721 94350

2 29379 2 29366

.2’7724 23729 .2’7726 53095

.18935 x. 10’

n+l

Methods

-;

Logarithms.

To comput,e common lognrithms, the number must be expressed in the form x. log, (1 Iz
*ll+1

1

(hrE-1)

Example

NOTE: In the examples given it is assumed t,hat the arguments are esact. Example

arccoth z+

I

Numerical 4.7. Use and Extension

z*- 1

-13

By linear inverse interpolation

so09836 9.836. 1O-3 5.99281:35=(--200718

15)

.09836 9.836. lo-’

-i.99281,85=(-1.00718

15)

Example 3.

.9836

9.836.10~’

i.99281;35=(-0.00718

15)

9.836. loo

0.99281 85

9.836.10!

1.99281:35

Convert log,, x t,o In x for x=.009836. Using 4.1.23 nnd Table 4.1, In (.009836)= In 10 log,, (.009836)=2.30258 5093 (-2.00718 15) = -4.62170 62:.

9,836.10*

2.99281:35

Example 4.

9.836 98.36 983.6

x-314= .18933 05685.

Interpolation in Table 4.1 between 983 nnd 984 gives -99281 85 as t,he mantissa of 9836. Note t.hat 5.99281 85=-3+.!39281 85. When p is negative the common logarithm can be expressed in the alternative forms

Compute In x for x=.00278 t*o 6D. Using 4.1.7, 4.1.11 nndTable 4.2, ln (.00278)= In (.278.10-*)==ln (.278)-2 ln lo=-5.886304. between x=.002 nnd Linenr lntrrpolntion ivc ln(.00278)=-5.808. TO x= .003 would obtain 5 decimn f plncc accurncy with linenr interpolatio’n it, is neccssnry t’hnt x>.175.

log,, (.009836)=?k99281

Example 5.#

85=7.99281

= -2.00718

85-10

15.

The last form is convenient for conversion from common logarit.hms to natural lognrithms. The inverse of log,,x is called l;he ant,ilogarithm of x, and is written antilog x or lo -I x. The logarithm of the reciprocal of a num %er is called the cologarithm, written colog.

Compute In x for x=1131.718 t,o SD. Using 4J.7, 4.1.11 and Table 4.2 1131.718 1131

In 1131.718:=hl 1131

:=ln 1’~:~‘8+1n :=ln(1.00063

1.131 +ln

lo3

4836)+1x1 1.131+3 lu 10.

90

ELEMENTARY

Then

from

4836)-$(.00063

4836)2

1.131+3 In 10=.00063 4836-.OOOOO 0202 +.12310 2197+6.90775

Example

6.

Example

.

68.

13489 24685 12693.

Let a=x-.867. Using 4.2.1, compute successively 1.00000 00000 00000 00000

376199

a4 ---cm* 4

8e.W728

Compute ez to 18D for x=.86725

1167 693059

aa -= 3

‘3B==e4.

10.

.00048 32583 282384

a2 --=-. 2

Example

e4.80728

e’eg72* 6g=(134.28978)(1.10217 67)=148.0111.

4.2 we compute successively a=

Compute e4.gg72a 6gto 7s. Using 4.2.18 and Table 4.4,

Linear interpolation gives e.og72* “= 1.10217 67 with an error of 1X 10m7,

Since &g- - 1.00048 32583 282384= 1+a, using and Table

9.

5279=7.03149 211.

Compute In x working with 16D for x= 1.38967 12458 179231.

4.1.24

FWNCTIONS

Example

4.1.24

In 1131.718=(.00063 +ln

TRANSCENDENTAL

a= .00025 13489 24685 12693

136

a2 g=

-

315 88140 97019

In (l-ta)=

.00048 31415 965388

aa fg=

*

2646 54842

In 1.389 = ln x=

.32858 40637 722067 .32906 72053 687455.

a4 ig=

.

16630

e”=1.00025

7.

Compute the principal value of In (f2 f 3i). 4.2 and 4.14. From 4.1.2, 4.1.3 and Tables In (2+3i)=i

e.867=2.37976 08513 29496 863 from Table eae.a67=ez=2.38035 Example

In (22+32)+i arctan i

13805 15472 81184 90:768

39006

4.4

089.

11.

Compute eelsto 7s. =L282475+i(.982794) Let n=&. In (-2+3i)=i

In 13+i

7-arctan

i

>

=1.282475fi(2.158799) In (-2--3i)=i

In 13-i-i (--?r+arctan

Then exp x=exp

g >

13+i

-arctan

( =1.282475-G(.982794).

Example

Compute (.227).6gto 7D. Using 4.2.7 and Tables 4.2 and 4.4,

=e--l.o~la

ln lO)=exp

[(n+d) In lo]

=lO” exp (d In 10) 3 2>

From Table

4.4

es48=exp (E

In 10) ==exp (281.42282 42 In 10)

=10281exp (.42282 42 In 10)=lOzsl exp .97358 87

8.

(.227).8g=e. 68

(&

=exp (In 10”) exp (d In 10)

=1.282475-i(2.158799) In (2-33i)=iln

and d=the decimal part of Go*

In C.227) ,e.68(-1.48%0 b@l=

.35g46

=10281(2.647428)=(!281)2.647428. Example

12.

5262)

60.

Compute em2for x= .75 using the expansion in Chebyshev polynomials.

. ELE:MENTARY

Following 4.2.48

the procedure

TRANSCENDENTAL

in [4.3] we have from

Example

15.

Compute sin x and cos x for z=2.317 From 4.344 and Table 4.6

e-z=&m(Z) I *

91

FUNCTIONS

where a(r) are the Chebyshev polynomials defined in chapter 22. Assuming be=kO we F;WE&IZ ba, k=7,6,5,. . . 0 from the recurrence

sin (2.317)=sin

(r-2.317)=sin

(.82459 2654) =.73427

cos (2.317)=cos

(r-2.317)=

-cos

be

Linear interpolation error of 9X lo-*.

- .ooooo 00 15

6

.ooooo 0400 - .OOOOO9560

5 4

Example

.03550 4993 - .27432 74.49 .33520 2828

Example

7449)

32925 19943 !Z9577 r 08882 08665 ‘72159 62 r 48481 36811 139535 9936 r 38O=. 66322 51 r 42’=.01221 73 r

32”=.00015 51 r -38’42’32”= .675598 r. 14.

Express X= 1.6789 radians in degrees, minutes and seconds to the nearest tenth of a second. From Table 1.1 giving the mathematical constants we have 1 r-l8oo --=57.29577 r 1.6789 r=96.19388’ .19388oX6O=11.633’ .633’X60=38.0” 1.6789 r=96°11‘38.0”. *See page II.

142.

951300 . . .

17.

Compute sin x to 19D for x=.86725 13489 24685 12693. Let

in radians to 6D.

Therefore

Example

12 sin 367

The method of reduction to an angle in the first quadrant which was given in Example 15 may also be used.

13.

lo=.01745 1’=.00029 l”=.OOOOO

12 cos .867+cos

=.29612

e-.‘6=.33520 2828-(.5)(--.27432 = .47236 6553.

Express 38’42’32”

2654 gives an

16.

sin (12.867)=sin

sincef(z)=b,,-((2x-l)bl,

Example

for x=.82459

Compute sin x for x=12.867 to 8D. From 4.3.16 and Tables 4.6 and 4.8

.00018 9959 -.00300 9164

3 2 1 0

12

(.82459 2654) =-. 67885 60.

b,=(4s-2)b,+l-bc+~+AI k 7

to 71).

Table

a=.867,

/3=x-a.

From

4.3.16

and

4.6

sin (a+@)=sin a co9 @+cos a sin 0 sin a=.76239 10208 07866 22598 cos a=.64711

66288 94312 75010

With the series expansions for sin B and cos fl we compute successively 1.00000 d -2=-

00000 315

*

8’ a=

00000 OOOOO 88140 97019 16630

*

co9 /3=

.99999 8=

11859

196li

.00025 13489 24685 12693

Ba T!=-

*

!c 51-

*

sin /3=

99684

BOO25 sin a cos /3=.76239 ma a sin fl=.OOOlS sin x=.76255

2646 54842 1 13489 09967 26520 36487

22038 25351 67105 92457

57352 31308 82436 13’74

.

92

ELEMENTARY

This procedure is equivalent Taylor’s formula 3.6.4.

to interpolation

Example

TRANSCENDENTAL

with

ABC,

a=123,

B=29”16’,

cos B=(123)2+(321)2

bs=az+d--2ac

-2(123)(321) b=221.99934

cos 29’16’

00

sin A-a--= sin B b

(123)(-48887 50196)=.27086 221.99934 00

3gg18

ABC,

c2+b2-a2=81+49-16 2bc

sin A=.42591 sin B=7(*425g1

l/20= 1.52083 7931

arccot 20=;-arctan

2O=arctan

a=4,

b=7,

114

2-7 -9

===.90476

c=9,

Express z=3+9i

5954=25’12’31.6”

770g’,

in polar form.

k is an

f+2rk,

~=(3~+9~)f=

9/3=arctan

4

770g),

For k=O,

3= 1.24904 58.

exp (1.24964 58i).

24.

Compute arctan x for. z= l/3 to l2D. From 4.4.34 and 4.4.42 we have Bz.84106

8670

arctan s=arctan

(x0+@

=48Y1’22.9”

sin C=g(‘425g1

integer.

qt%=9.486833

Thus 3+93=9.486833 Example

7709

4

8396.

where r==($+y2)+,

B=arctan 1905

.05=.04995

23.

0=arctan

19.

A=.43997

arctan 20=;-arctan

z=x+iy=refe,

In the plane triangle find A, B, and C. cos A=

22.

Example

A=15’42’56.469”. Example

Example

Compute arctan 20 and arccot 20 to 9D. * Using 4.4.5,4.4.8, and Table 4.14

18.

In the plane triangle c=321; find A, b.

FUNCTIONS

Ccl.86054

h

=arctan

xo+arctan

: .L+xoh+g

=arcm

xo+(i-$&&&+~+xJ+.

803 =106’36’5.6”

.. .

where the supplementary an le must be chosen for (7. As a check we get A+ 27+C=180”00’.1”.

We have

Example

x= f = .33333 33333 33 so that h= .00033 33333 33

20.

Compute cot x for x=.4589 to 6D. Since 2<.5, using Table 4.9 with interpolation 1 in (z-l-cot x), we find --cot(.4589)= .4589 .155159. Therefore cot (.4589)=2.179124.155159=2.023965. Example

and,

from

= .32145

Table

4.1,4, arctan

05244

03.

zO=arctan

Since

.00030 00300 03 we get arctan x= .32145 05244 03+.00030

00300 03

--.ooooo

21.

Compute arcsin x for 5= .99511. For z>.95, using Table 4.14 with interpolation in the auxiliary functionf(z) we find arcsin 2=:-[2(1-x)I?f(x)

=.32175

00000 09

05543 97.

If x is given in the form b/a it is convenient use 4.4.34 in the form b arctan ,=arctan

xO+arctan b-axe -* a+b

arcsin (.99511)=;-[2(.00489)]*j(.99511) In the present example we get =1.57079

6327-(.09889

=1.47186

2100.

388252)

(1.00040 7951)

.333

h 1 +x&+x;=

arctan jj=arctan *see page II.

.333+arctan

1 -. 3333

to

ELEMENTARY

Example

TRANSCENDENTAL

arctanh .96035=+ In 1+ .96035=$ln- 1.96035 1- .96035 .03965

25.

Compute arcsec 2.8 to 5D. Using 4.3.45 and Table 4.14

=a In 49.44136 191 =$(3.90078 7359) = 1.950394.

(z”-- I)+ arcsec z=arcsin --y .Example 27.

. [(2.8)2-l]+ arcsec 2.8=arcsm 2.8 =arcsin

93

FUNCTIONS

Compute arccosh x for x=1.5368 Using Table 4.17

.93404 97735

to 6D.

arccosh x=arccosh 1.5368=.852346 (x2-- 1)’ [(1.5368)2-l]*

= 1.20559 or using 4.3.45 and Table 4.14

arccosh 1.5368=(.852346)(1.361754)* arcsec z=arctan arcsec 2.8=arctan

(z2-l)*

=(.852346)(1.166942)

2.61533 9366

= .994638. =z-arctan

.38235 951564, f:rom 4.4.3 and 4.4.8

=1.570796-.365207

Example 28. Compute arccosh x for x=31.2 to 5D. Using Tables 4.2 and 4.17 with l/x=1/31.2 = .03205 128205

= 1.20559. Example 26.

arccosh 31.2-m

Compute arctanh 2 for x=.96035 to 6D. From 4.6.22 and Table 4.2

31.2=.692886

arccosh 31.2=.692886+3.440418=4.13330.

References Texts

14.11 B. Carlson! M. Goldstein, R,ttional approximation of functions, Los Alamos Scientific Laboratory LA-1943 (Los Alamos, N. Mex., 1955). [4.2] C. W. Clenshaw, Polynomial approximations to elementary functions, Mat!l. Tables Aids Comp. 8, 143-147 (1954). of [4.3] C. W. Clenshaw, A note c’n the summation Chebyshev series, Math. ‘l’ables Aids Comp. 9, 118-120 (1955). 14.41 G. H. Hardy, A course of pure mathematics, 9th ed. (Cambridge Univ. Press, Cambridge, England, and The Macmillan Co., New York, N.Y., 1947). [4.5] C. Hastings, Jr., Approximations for digital computers (Princeton Univ. F’ress, Princeton, N.J., 1955). [4.6] C. Hastings, Jr., Note 143, Math. Tables Aids Comp. 6, 68 (1953). [4.7] E. W. Hobson, A treatise on plane tri~metry, 4th ed. (Cambridge Univ. Press, ambndge, England, 1918). [4.8] H. S. Wall, Analytic theory of continued fractions (D. Van Nostrand Co., Iuc., New York, N.Y., 1948). Tables

[4.9] E. P. Adams, Smithsonian mathematical formulae and tables of elliptic functions, 3d reprint (The l$~$onian Institution, Washington, D.C., [4.10] H.

Andoyer, Nouvelles tables tri onometrlques fondamentales (Hermann et fils, !baris, France, 1916).

[4.11] British Association for the Advancement of Science, Mathematical Tables, vol. I. Circular and hyperbolic functions, exponential, sine and cosine integrals, factorial function and allied functions, Hermitian probabilit functions, 3d ed. (Cambridge Univ. Press, 8 ambridge, England, 1951). [4.12] Chemical Rubber Corn any! Standard mathematical tables, 12th ed. ( B hemmal Rubber Publ. Co., Cleveland! Ohio, 1959). [4.13] L. J. Comne, Chambers’ six-figure mathematical tables, vol. 2 (W. R. Chambers, Ltd., London, [4.14] [4.15] Laboratory., Tables of the [4.16] Harvard. Computation function arcsin z (Harvard Univ. Press, Camz=x+iy, O<x<475, bridge, Mass., 1956). 0 5~ 1475, 6D, varying intervals. [4.17] Harvard Computation Laboratory, Tables of inverse hyperbolic functions (Harvard Univ. Press, Cambridge, Mass., 1949). arctanh x, O
ELEMENTARY

94

TRANSCENDENTAL

14.191 National Bureau of Standards, Table of natural logarithms for arguments between zero and five to sixteen decimal places, 2d ed., Applied Math. Series 31 (U.S. Government Printing office, Washington, D.C., 1953). 2=0(.0001)5, 16 D. [4.20] National Bureau of Standards, Tables of the exponential function eZ, 3d ed., Applied Math. Series 14 (U.S. Government Printing Office, Washing.9999, ton, D.C., 1951). Z= -2.4999(.0001) 18D, s=l(.OOOl) 2.4999, 15D, z=2.5(.001)4.999, 15D, z=5(.01)9.99, 12D, z== -.000099(.000001) .000099, 18D, z= - 100(1)100, lQS, z= -9X 10-n(10-n)9X, lo-*, n= 10, 9, 8, 7, 18D; values of [4.21] N~t~?al%ur~~~2,5u56oD. Standards, Table of the descending exponential, 2=2.5 to z=lO, Applied Math. Series 46 (U.S. Government Printing Office, Washington, D.C., 1955). s=2.5(.001)10, ZOU.

[4.22] National Bureau of Standards, Tables of sines and cosines for radian arguments, 2d ed., Applied Math. Series 43 (U.S. Government Printing Office, Washingtor, D.C., 1955). sin 5, cos z, z=O(.OO1)25.2, O(l)lOO, 8D, x=10-“(lo-“)9X lo-n, n=5,4,3, 2, 1, 15D, ~=0(.00001) 91, 12D. [4.23] xational Bureau of Standards, Tables of circular and hyperbolic sines and cosines for radian arguments, 2d ed., Applied Math. Series 36 (U.S. Government Printing O&e, Washington, D.C., 1953). sin x, cos z, sinh Z, cash 2, z=O(.OOOl) 1.9999, O(.l)lO, QD. [4,24] National Bureau of Standards, Table of circular and hyperbolic tangents and cot,angents for radian arguments, 2d printing (Columbia Univ. Press, Fey;; prk, N.Y., 1947). tan 5, cot Z, tanh z, 8D or 8S, z=O(.l)lO, ) 2=0(.0001)2, 10D. [4.25] National Bureau of Standards, Table of sines and cosines to fifteen decimal places at hundredths of a degree, Applied Math. Series 5 (U.S. Government Printing Office, Washington, D.C., 1949). sin 5, cos z, z=O”(.O1o)QOo, 15D; supplementary table 30 D. of sin Z, cos z, z=l”(lo)8Qo, [4.26] National Bureau of Standards, Table of secants and cosecants to nine significant figures at hundredths of a degree! Applied Math. Series 40 (U.S. Government Printing Office, Washington, D.C., 1954). [4.27] National Bureau of Standards, Tables of functions and of zeros of functions, Collected short tables of the Computation Laboratory, Applied Math. Series 37 (U.S. Government Printing Office, Washington, D.C., 1954).

FUNCTIONS

[4.28] National Bureau of Standards, Table of arcsin z (Columbia Univ. Fress, New York, N.Y., 1945). arcsin z, s=0(.0001).989(.00001) 1, 12D; auxiliary table of f(u) =[$a-arcsin (l--)]/(2v) %, v=O(.OOOOl).OOO5, 13D. [4.29] National Bureau of Standards, Tables of arctan z, 2d ed., Applied Math. Series 26 (U.S. Government Printing Office, Washington D.C., 1953). s=0(.001,7(.01)50(.1)300(1)2000(10)10000, 12D. [4.30] National Bureau of 8tandards, Table of hyperbolic sines and cosines, x=2 to x=10, Applied Math. Series 45 (U.S. Government Printing Office, Washington, D.C., 1955). 2=2(.001)10, 9s. [4.31] B. 0. Peirce, A short table of integrals, 4th ed. (Ginn and Co., Boston, Mass., 1956). [4.32] J. Peters, Ten-place logarithm table, ~01s. 1, 2 (together with an appendix of mathematical tables) (Berlin, 1~922; rev. ed., Frederick Ungar Publ. Co., New York, N.Y., 1957). [4.33] J. Peters, Seven-place values of trigonometric functions for every thousandth of a de ree (Berlin-Friedenau., 1918; D. Van Nostrand 5 o., Inc., New York, N.Y., 1942). [4.34] L. W. Pollak, R.echentafeln zur harmonischen Analyse (Johann Ambrosius Barth, Leipzig, Germany, 1926). [4.35] A. J. Thompson, Standard table of logarithms to twenty decimal places, Tracts for Computers, No. 22 (Cambridge Univ. Press, Cambridge, England, and New York, :N.Y., 1952). [4.36] J. Todd, Table of arctangents of rational numbers, NBS Applied Math. Series 11 (U.S. Government Printing Office, ‘Washington, D.C., I D51). arctan m/n and arccot m/n, O<m
ELEMENTARY

TRANSCENDENTAL COMMON

x

log10

1

x

log10 T

100 00000 00000 12591 150 176119 101 00432 13738 151 1789769473 102 0086001718 152 181r3435879 103 0128372247 153 184159 14308 104 0170333393 154 187152 07208

95

FUNCTIONS

LOGARITHMS

Table 4.1

log10 z

2

200 201 202 203 204

3010299957 3031960574 3053513694 3074960379 3096301674

250 251 252 253 254

3979400087 3996737215 4014005408 4031205212 4048337166

E

log10

2:

x

log10x 300 47712 12547 301 47856 64956 % 4814426285 48000 69430 304 48287 35836

105 106 107 108 109

0211892991 0253058653 0293837777 03342 37555 0374264979

155 156 157 158 159

1903316982 205 193112 45984 206 1950996524 207 198b570870 208 2013971243 209

3117538611 3138672204 3159703455 3180633350 3201462861

255 256 257 258 259

4065401804 4082399653 40993 31233 4116197060 4132997641

305 306 307 308 309

4842998393 48572 14265 48713 83755 48855 07165 48995 84794

110 112 113 114

0413926852 0453229788 0492180227 0530784435 0569048513

160 161 162 163 164

204:.1 99827 210 3222192947 2061i258760 211 3242824553 2091il 50145 212 3263358609 212:.8 76044 213 3283796034 2140438480 214 3304137733

260 261 262 263 264

4149733480 4166405073 4183012913 41995 57485 4216039269

310 311 312 313 314

49136 16938 4927603890 4941545940 4955443375 4969296481

115 116 117 118 119

0606978404 0644579892 0681858617 0718820073 0755469614

165 166 167 168 169

217118 39442 220:.0 80880 222:'l 64711 225:bO92817 227118 67046

215 216 217 218 219

3324384599 3344537512 3364597338 3384564936 3404441148

265 266 267 268 269

4232458739 4248816366 4265112614 4281347940 4297522800

315 316 317 318 319

49831 05538 4996870826 5010592622 5024271200 5037906831

120 121 122 123 124

0791812460 0827853703 0863598307 0899051114 0934216852

170 171 172 173 174

2304489214 232'1961104 235112 84469 2380461031 240114 92483

220 221 222 223 224

3424226808 3443922737 3463529745 3483048630 3502480183

270 271 272 273 274

4313637642 4329692909 4345689040 4361626470 4377505628

320 321 322 323 324

5051499783 5065050324 5078558717 5092025223 5105450102

125 126 127 128 129

0969100130 1003705451 1038037210 1072099696 1105897103

175 176 177 178 179

243t1380487 225 3521825181 245f1126678 226 3541084391 247517 32664 227 3560258572 250l.2 00023 228 3579348470 252fl5 30310 229 3598354823

275 276 277 278 279

4393326938 4409090821 4424397691 4440447959 4456042033

325 326 327 328 329

5118833610 5132176001 5145477527 5158738437 5171958979

130 131 132 133 134

1139433523 1172712957 1205739312 1238516410 1271047984

180 181 182 183 184

2552'725051 257617 85749 26Ot1713880 2621.510897 264tll 78230

230 231 232 233 234

3617278360 3636119799 3654879849 3673559210 3692158574

280 281 282 283 284

4471580313 4487063199 4502491083 4517864355 4533183400

330 331 332 333 334

5185139399 5198279938 5211380837 5224442335 5237464668

135 136 137 138 139

1303337685 1335389084 1367205672 1398790864 1430148003

185 186 187 188 189

2671717284 269fsl 29442 271tb416065 2741578493 2761618042

235 236 237 238 239

3710678623 3729120030 3747483460 3765769571 3783979009

285 286 287 288 289

4548448600 4563660331 4578818967 4593924878 4608978428

335 336 337 338 339

5250448070 5263392774 5276299009 5289167003 5301996982

140 141 142 143 144

1461280357 1492191127 1522883444 1553360375 1583624921

190 191 192 193 194

27875

36010 281tl3 33672 283:sO12287 2855573090 287t~O17299

240 241 242 243 244

3802112417 3820170426 3838153660 3856062736 3873898263

290 291 292 293 294

4623979979 4638929890 4653828514 4668676204 4683473304

340 341 342 343 344

5314789170 5327543790 53402 61061 5352941200 5365584426

145 146 147 148 149

1613680022 1643528558 1673173347 1702617154 1731862684

195 196 197 198 199

29OC346114 2922'560714 2941.662262 2966651903 298E15 30764

245 246 247 248 249

3891660844 3909351071 3926969533 3944516808 3961993471

295 296 297 298 299

4698220160 4712917111 4727564493 4742162641 4756711883

345 346 347 348 349

5378190951 53907 60988 5403294748 5415792439 5428254270

150

1760912591 200 3OlCl299957 250 3979400087 300 (-;I6

111

[ 1

[(-F2 1

4771212547 350 54406 80444 C-l)6

[ 1

For use of common logarithms see Examples 1-3. For 100<~,<135 interpolate in the range 1000<:~~<1350.Compiled from A. J. Thompson, Standard table of logarithms to twenty decimal places, Tracts for Computers, No. 22. Cambridge Univ. Press, Cambridge, England, 1952 (with permission).

96

ELEMENTARY

TRANSCENDENTAL

Table 4.1 2’

COMMON

log10

2

.c

log10

z

z

FUNCTIONS

LOGARITHMS log10

2

2

log10

z

z

log10

2

350 351 352 353 354

54406 54530 54654 54777 54900

80444 71165 26635 47054 32620

400 401 402 403 404

60205 60314 60422 60530 60638

99913 43726 60531 50461 13651

450 451 452 453 454

65321 65417 65513 65609 65705

25138 65419 84348 82020 58529

500 501 502 503 504

69897 69983 70070 70156 70243

00043 77259 37171 79851 05364

550 551 552 553 554

74036 74115 74193 74272 74350

26895 15989 90777 51313 97647

355 356 357 358 359

55022 55144 55266 55388 55509

83531 99980 82161 30266 44486

405 406 407 408 409

60745 60852 60959 61066 61172

50232 60336 44092 01631 33080

455 456 457 458 459

65801 65896 65991 66086 66181

13967 48427 62001 54780 26855

505 506 507 508 509

70329 70415 70500 70586 70671

13781 05168 79593 37123 77823

555 556 557 558 559

74429 74507 74585 74663 74741

29831 47916 51952 41989 18079

360 361 362 363 364

55630 55750 55870 55990 56110

25008 72019 85705 66250 13836

410 411 412 413 414

61278 61384 61489 61595 61700

38567 18219 72160 00517 03411

460 461 462 463 464

66275 66370 66464 66558 66651

78317 09254 19756 09910 79806

510 511 512 513 514

10757 70842 70926 71011 71096

01761 09001 99610 73651 31190

560 561 562 563 564

74818 74896 74973 75050 75127

80270 28613 63156 83949 91040

365 366 367 368 369

56229 56348 56466 56504 56702

28645 10854 60643 78187 63662

415 61804, 416 61909 417 62013 418 62117 419 -62221

80967 33306 60550 62818 40230

465 ‘77; 468 469

66745 66838 66931 67024 67117

29529 59167 68806 58531 28427

515 516 517 518 519

71180 71264 71349 71432 71516

72290 97016 05431 97597 73578

565 566 567 568 569

75204 75281 75350 75434 75511

84478 64312 30589 83357 22664

370 371 372 373 374

56820 56937 57054 57170 57287

17241 39096 29399 88318 16022

420 421 422 423 424

62324 62428 62531 62634 62736

92904 20958 24510 03674 58566

470 471 472 473 474

67209 67302 67394 67486 67577

78579 09071 19986 11407 83417

520 521 522 523 524

71600 71683 71767 71850 71933

33436 77233 05030 16889 12870

570 571 572 573 574

75587 75663 75739 75815 75891

48557 61082 60288 46220 18924

375 376 377 378 379

57403 57518 57634 57749 57863

12677 78449 13502 17998 92100

425 426 427 428 429

62838 62940 63042 63144 63245

89301 95991 78750 37690 72922

475 476 477 478 479

67669 67760 67851 67942 68033

36096 69527 83790 78966 55134

525 526 527 528 529

72015 72098 72181 72263 72345

93034 57442 06152 39225 56720

575 576 577 578 579

75966 76042 76117 76192 76267

78447 24834 58132 78384 85637

380 381 382 383 384

57978 58092 58206 58319 58433

35966 49757 33629 87740 12244

430 431 432 433 434

63346 63447 63548 63648 63748

84556 72702 37468 i'8964 97295

480 481 482 483 484

68124 68214 68304 68394 68484

12374 50764 70382 71308 53616

530 531 532 533 534

72427 72509 72591 72672 72754

58696 45211 16323 72090 12570

580 581 582 583 584

76342 76417 76492 76566 76641

79936 61324 29846 85548 28471

385 386 387 388 389

58546 58658 58771 58883 58994

07295 73047 09650 17256 96013

435 436 437 438 439

63848 63948 64048 64147 64246

92570 64893 14370 41105 45202

485 486 487 488 489

68574 68663 68752 68841 68930

17386 62693 89612 98220 88591

535 536 537 538 539

72835 72916 72997 73078 73158

37820 47897 42857 22757 87652

585 586 587 588 589

76715 76789 76863 76937 77011

58661 76160 81012 73261 52948

390 391 392 393 394

59106 59217 59328 59439 59549

46070 67574 60670 25504 62218

440 441 442 443 444

64345 64443 64542 64640 64738

26765 85895 22693 37262 29701

490 491 492 493 494

69019 69108 69196 69284 69372

60800 14921 51028 69193 69489

540 541 542 543 544

73239 73319 73399 73479 73559

37598 72651 92865 98296 88997

590 591 592 593 594

77085 77158 77232 77305 77378

20116 74809 17067 46934 64450

395 396 397 398 399

59659 59769 59879 59988 60097

70956 51859 05068 30721 28957

445 446 447 448 449

64836 64933 65030 65127 65224

00110 48587 75231 80140 63410

495 496 497 t;;

69460 69548 69635 69722 69810

51989 16765 63887 93428 05456

545 546 547 548 549

73639 73719 73798 73878 73957

65023 26427 73263 05585 23445

595 596 597 598 599

77451 77524 77597 77610 77742

69657 62597 43311 11840 68224

400

60205 99913

450

65321 25138

500

69897 00043

550

74036 26895

600

77815 12504

[ 1 C-l)4

[93 1

[ 1 (73

[c-y1

ELEMENTARY

TRANSCENDENTAL

COMMON X

log10 x

97

FUNCTIONS

LOGARITHMS

Table 4.1

600 601 602 603 604

77815 77887 77959 78031 78103

12504 44720 64913 73121 69386

650 651 652 653 654

81291 81358 al424 81491 81557

33566 39886 75957 31813 77483

700 701 702 703 704

log10 2 84509 80400 84571 80180 84633 71121 84695 53250 84757 26591

605 606 607 608 609

78175 78247 78318 78390 78461

53747 26242 86911 35793 72926

655 656 657 658 659

81624 81690 81756 81822 81888

13000 38394 53696 58936 54146

705 706 707 708 709

84818 84880 84941 85003 85064

91170 47011 94138 32577 62352

755 756 757 758 759

87794 87852 87909 87966 88024

69516 17955 58795 92056 17759

805 806 807 808 809

90579 90633 90687 90741 90794

58804 50418 35347 13608 85216

610 611 612 613 614

78532 78604 78675 78746 78816

98350 12102 14221 04745 83711

660 661 662 663 664

81954 82020 82085 82151 82216

39355 14595 79894 35284 80794

710 711 712 713 714

85125 85186 85247 85308 85369

83487 96007 99936 95299 82118

760 761 762 763 764

88081 88138 88195 88252 88309

35923 46568 49713 45380 33586

810 811 812 813 814

90848 90902 90955 91009 91062

50189 085412 6029:2 05456 44049

615 616 617 618 619

78887 78958 79028 79098 79169

51158 07122 51640 a4751 06490

665 666 667 668 669

82282 82347 82412 82477 82542

16453 42292 58339 64625 61178

715 716 7L7 718 719

85430 85491 85551 85612 85672

60418 30223 91557 44442 88904

765 766 767 768 769

88366 88422 88479 88536 88592

14352 87696 53639 12200 63398

815 816 817 818 819

91115 91169 91222 91275 91328

76087 01588 20565 33037 39018

620 621 622 623 624

79239 79309 79379 79448 79518

16895 16002 03847 80467 45897

670 671 672 673 674

82607 82672 82736 82801 82865

48027 25202 92731 50642 98965

720 721 722 723 724

85733 a5793 85853 85913 a5973

24964 52647 71976 82973 85662

770 771 772 773 774

88649 88705 88761 88817 88874

07252 43781 73003 94939 09607

820 821 822 823 824

91381 91434 91487 91539 91592

38524 31571 18175 98352 72117

625 626 627 628 629

79588 79657 79726 79795 79865

00173 43332 75408 96437 06454

675 676 677 678 679

82930 82994 83058 83122 83186

37728 66959 86687 96939 97743

725 726 727 728 729

86033 86093 86153 86213 86272

80066 66207 44109 13793 75283

775 776 777 778 779

88930 88986 89042 89097 a9153

17025 17213 10188 95970 74577

825 826 827 828 829

91645 91698 91750 91803 91855

39485 00473 55096 03368 45306

630 631 632 633 634

79934 80002 80071 80140 80208

05495 93592 70783 37100 92579

680 681 682 683 684

83250 83314 83378 83442 83505

89127 71119 43747 07037 61017

730 731 732 733 734

86332 86391 86451 86510 86569

28601 73770 10811 39746 60599

780 781 782 783 784

89209 89265 89320 89376 89431

46027 10339 67531 17621 60627

:zi 832 833 834

91907 91960 92012 92064 92116

80924 10238 33263 50014 60506

635 80277 37253 636 80345 711% 637 80413 94323 638 80482 06787 639 80550 08582

685 686 687 688 689

83569 83632 83695 a3758 83821

05715 41157 67371 84382 92219

735 736 737 738 739

86628 86687 86746 86805 86864

73391 78143 74879 63618 44384

785 786 787 788 789

89486 89542 89597 89652 89707

96567 25460 47324 62175 70032

835 836 837 838 839

92168 92220 92272 92324 92376

64755 62774 54580 40186 19608

640 641 642 643 644

80617 80685 80753 80821 80888

99740 80295 50281 09729 58674

690 691 692 693 694

83884 83947 84010 84073 84135

90907 80474 60945 32346 94705

740 741 742 743 744

86923 86981 87040 87098 87157

17197 82080 39053 88138 29355

790 791 792 793 794

89762 89817 89872 89927 89982

70913 64835 51816 31873 05024

840 841 842 843 844

92427 92479 92531 92582 92634

92861 59958 20915 75746 24466

645 646 647 648 649

80955 81023 81090 81157 81224

97146 25180 42807 50059 46968

695 696 697 698 699

84198 84260 84323 84385 84447

48046 92396 27781 54226 71757

745 746 747 748 749

87215 a7273 87332 87390 87448

62727 88275 06018 15979 18177

795 796 797 T2

90036 90091 90145 90200 90254

71287 30677 83214 28914 67793

845 846 847 848 849

92685 92737 92788 92839 92890

67089 036:30 34103 58523 76902

650 81291 33566 c-y

700

84509 80400

750

87506 12634

800

90308 99870

850

92941 89257 C-t,8

[ 1

X

loglcl

2

[c-p11

X

750 751 752 753 754

log10 x 87506 12634 87563 99370 87621 78406 87679 49762 a7737 13459

800 801 802 803 804

log10 2 90308 99870 90363 25161 90417 43683 90471 55453 90525 60487

2

C-47) 1 [ I

X

[ 1

98

ELEMENTARY

Table 4.1

F’UNCTIONS

COM MON LOGARITHMS log10

X

TRANSCENDENTAL

%

X

log10

x

X

log10

x

X

log10

x

2

log10

2

00000

00000

1002 1003 1004

00043 00086 00130 00173

40775 77215 09330 37128

1050 1051 1052 1053 1054

02118 02160 02201 02242 02284

92991 27160 57398 83712 06109

33716 78923 19378 55091 86072

1005 1006 1007 1008 1009

00216 00259 00302 00346 00389

60618 79807 94706 05321 11662

1055 1056 1057 1058 1059

02325 02366 02407 02448 02489

24596 39182 49873 56677 59601

98227 98272 98317 98362 98407

12330 33877 50720 62871 70339

1010 1011 1012 1013 1014

00432 00475 00518 00560 0060:3

13738 11556 05125 94454 79550

1060 1061 1062 1063 1064

02530 02571 02612 02653 02694

58653 53839 45167 32645 16280

965 966 967 968 969

98452 98497 98542 98587 98632

73133 71264 64741 53573 37771

1015 1016 1017 1018 1019

00646 00689 00732 00774 00817

60422 37079 09529 77780 41840

1065 1066 1067 1068 1069

02734 02775 02816 02857 02897

96078 72047 44194 12527 77052

78273 96302 09211 17010 19712

970 971 972 973 974

98677 98721 98766 98811 98855

17343 92299 62649 28403 89569

1020 1021 1022 1023 1024

00860 00902 00945 00987 01029

01718 57421 08958 56337 99566

1070 1071 1072 1073 1074

02938 02978 03019 03059 03100

37777 94708 47854 97220 42814

96614 96661 96707 96754 96801

17327 09867 97341 79762 57140

975 976 977 978 979

98900 98944 98989 99033 99078

46157 98177 45637 88548 26918

1025 1026 1027 1028 1029

01072 01114 01157 01199 01241

38654 73608 04436 31147 53748

1075 1076 1077 1078 1079

03140 03181 03221 03261 03302

84643 22713 57033 87609 14447

930 931 932 933 934

96848 96894 96941 96988 97034

29486 96810 59124 16437 68762

980 981 982 983 984

99122 99166 99211 99255 99299

60757 90074 14878 35178 50984

1030 1031 1032 1033 1034

01283 72247 01325 86653 01367 96973 014:LO 03215 01452 05388

1080 1081 1082 1083 1084

03342 03382 03422 03462 03502

37555 56940 72608 84566 92822

32707 37219 36198 29658 17610

935 936 937 938 939

97081 97127 97173 97220 97266

16109 58487 95909 28384 55923

985 986 987 988 989

99343 99387 99431 99475 99519

62305 69149 71527 69446 62916

1035 1036 1037 1038 1039

01494 01535 01577 01619 01661

03498 97554 87564 73535 55476

1085 1086 1087 1088 1089

03542 03582 03622 03662 03702

97382 98253 95441 88954 78798

94939 94987 95036 95085 95133

00066 77040 48544 14589 75188

940 941 942 943 944

97312 97358 97405 97451 97497

78536 96234 09028 16927 19943

990 991 992 993 994

99563 99607 99651 99694 99738

51946 36545 16722 92485 63844

1040 1041 1042 1043 1044

01703 01745 01786 01828 01870

33393 07295 77190 43084 04987

1090 1091 1092 1093 1094

03742 03782 03822 03862 03901

64979 47506 26384 01619 73220

895 896 897 898 899

95182 95230 95279 95327 95375

30353 80097 24430 63367 96917

945 946 947 948 949

97543 97589 97634 97680 97726

18085 11364 99790 83373 62124

995 996 997 998 999

99782 99825 99869 99913 99956

30807 93384 51583 05413 54882

1045 1046 1047 1048 1049

01911 01953 01994 02036 02077

62904 16845 66817 12826 54882

1095 1096 1097 1098 1099

03941 03981 04020 04060 04099

41192 05541 66276 23401 76924

900

95424 25094

950

97772 36053

1000

00000 00000 ‘-iI 6

1050

02X18 92991 c-3815

1100

04139 26852 C-38)5

850 851 852 853 854

92941 92992 93043 93094 93145

89257 95601 95948 90312 78707

900 901 902 903 904

95424 95472 95520 95568 95616

25094 47910 65375 77503 84305

950 951 952 953 954

97772 97818 97863 97909 97954

36053 05169 69484 29006 83747

1000 ii01

855 856 857 858 859

93196 93247 93298 93348 93399

61147 37647 08219 72878 31638

905 906 907 908 909

95664 95712 95260 95808 95856

85792 81977 72871 58485 38832

955 956 957 958 959

98000 98045 98091 98136 98181

860 861 562 363 164

93449 93500 93550 93601 93651

84512 31515 72658 07957 37425

910 911 912 913 914

95904 95951 95999 96047 96094

13923 83770 48383 07775 61957

960 961 962 963 964

865 866 867 868 869

93701 93751 93801 93851 93901

61075 78920 90975 97252 97764

915 916 917 918 919

96142 96189 96236 96284 96331

10941 54737 93357 26812 55114

870 871 872 873 874

93951 94001 94051 94101 94151

92526 81550 64849 42437 14326

920 921 922 923 924

96378 96425 96473 96520 96567

875 876 877 878 879

94200 94250 94299 94349 94398

80530 41062 95934 45159 88751

925 926 927 928 929

880 881 882 a83 884

94448 94497 94546 94596 94645

26722 59084 85851 07036 22650

885 886 887 888 889

94694 94743 94792 94841 94890

890 891 892 893 894

[

c-y

I

[ 1

[ 1

[1 I

ELEMENTARY

TRANSCENDENTAL

COMMON n

log10

1’

log10

2

FUNCTIt

99

INS

LOGARITHMS 1ogu.l T

4.1

log10

1:

r

x

07918 07954 07990 08026 08062

12460 30074 44677 56273 64869

1250 1251 1252 1253 1254

09691 09725 09760 09795 09829

00130 73097 43289 10710 75365

1300 1301 1302 1303 1304

11394 11427 11461 11494 11527

33523 72966 09842 44157 75914

1205 1206 1207 1208 1209

08098 08134 08170 08206 08242

70469 73078 72701 69343 63009

1255 1256 1257 1258 1259

09864 09898 09933 09968 10002

37258 96394 52777 06411 57301

1305 1306 1307 1308 1309

11561 11594 11627 11660 11693

05117 31769 55876 77440 96466

791392 22:197 61:!81 971.47 29803

1210 1211 1212 1213 1214

08278 08314 08350 08386 08421

53703 41431 26198 08009 86867

1260 1261. 1262 1263 1264

10037 10071 10105 10140 10174

05451 50866 93549 33506 70739

1310 1311 1312 1313 1314

11727 11760 11793 11826 11859

12957 26917 38350 47261 53652

06632 06669 06707 06744 06781

59;!54 851jO4 08fj60 28428 45:112

1215 1216 1217 1218 1219

08457 08493 08529 08564 08600

62779 35749 05782 72883 37056

1265 1266 1267 1268 1269

10209 10243 10277 10311 10346

05255 37057 66149 92535 16221

1315 1316 1317 1318 1319

11892 11925 11958 11991 12024

57528 58893 57750 54103 47955

1170 1171 1172 1173 1174

06818 06855 06892 06929 06966

58017 68951 76:117 80:121 80969

1220 1221 1222 1223 1224

08635 08671 08707 08742 08778

98307 56639 12059 64570 14178

1270 1271 1272 1273 1274

10380 10414 10448 10482 10516

37210 55506 71113 84037 94280

1320 1321 1322 1323 1324

12057 12090 12123 12155 12188

39312 28176 14551 98442 79851

25224 83905 39160 90996 39419

1175 1176 1177 1178 1179

07003 07040 07077 07114 07151

781i66 73?17 64h28 52905 38051

1225 1226 1227 1228 1229

08813 08849 08884 08919 08955

60887 04702 45627 83668 18829

1275 1276 1277 1278 1279

10551 10585 10619 10653 10687

01848 06744 08973 08538 05445

1325 1326 1327 1328 1329

12221 12254 12287 12319 12352

58783 35241 09229 80750 49809

05307 05346 05384 05422 05461

84435 26049 64269 99099 30546

1180 1181 1182 1183 1184

07188 07224 07261 07298 07335

201173 98976 74'165 47446 17024

1230 1231 1232 1233 1234

08990 09025 09061 09096 09131

51114 80529 07078 30766 51597

1280 1281 1282 1283 1284

10720 10754 10788 10822 10856

99696 91297 80252 66564 50237

1330 1331 1332 1333 1334

12385 12417 12450 12483 12515

16410 80555 42248 01494 58296

1135 1136 1137 1138 1139

05499 05537 05576 05614 05652

58615 83314 04647 22621 37241

1185 1186 1187 1188 1189

07371 07408 07445 07481 07518

83503 46;390 07190 64,106 18546

1235 1236 1237 1238 1239

09166 09201 09236 09272 09307

69576 84708 96996 06447 13064

1285 1286 1287 1288 1289

10890 10924 10957 10991 11025

31277 09686 85469 58630 29174

1335 1336 1337 1338 1339

12548 12580 12613 12645 12678

12657 64581 14073 61134 05770

1140 1141 1142 1143 1144

05690 05728 05766 05804 05842

48513 56444 61039 62304 60245

1190 1191 1192 1193 1194

07554 07591 07627 07664 07700

69514 171515 62354 04137 43,268

1240 1241 1242 1243 1244

09342 09377 09412 09447 09482

16852 17815 15958 11286 03804

1290 1291 1292 1293 1294

11058 11092 11126 11159 11193

97103 62423 25137 85249 42763

1340 1341 1342 1343 1344

12710 12742 12775 12807 12839

47984 87779 25158 60127 92687

1145 1146 1147 1148 1149

05880 05918 05956 05994 06032

54867 46176 34179 18881 00287

1195 1196 1197 1198 1199

07736 07773 07809 07845 07881

79,353 11797 41504 68181 91331

1245 1246 1247 1248 1249

09516 09551 09586 09621 09656

93514 80423 64535 45853 24384

1295 1296 1297 1298 1299

11226 11260 11293 11327 11360

97684 50015 99761 46925 91511

1345 1346 1347 1348 1349

12872 12904 12936 12968 13001

22843 50599 75957 98922 19497

1150

06069

78404

1200

07918

12160

1250

09691

00130

1300

11394

33523

1350

13033

1100 1101 1102 1103 1104

04139 04178 04218 04257 04296

26852 73190 15945 55124 90734

1105 1106 1107 1108 1109

04336 04375 04414 04453 04493

1110 1111 1112 1113 1114

2

1150 1151 1152 1153 1154

06069 06107 06145 06182 06220

781104 53:!36 24.791 93073 58088

1200 1201 1202 1203 1204

22780 51270 76209 97604 15461

1155 1156 1157 1158 1159

06258 06295 06333 06370 06408

19842 78341 331590 851594 34360

04532 04571 04610 04649 04688

29788 40589 47872 51643 51908

1160 1161 1162 1163 1164

06445 06483 06520 06557 06595

1115 1116 1117 1118 1119

04727 04766 04805 04844 04883

48674 41946 31731 18036 00865

1165 1166 1167 1168 1169

1120 1121 1122 1123 1124

04921 04960 04999 05037 05076

80227 56126 28569 97563 63112

1125 1126 1127 1128 1129

05115 05153 05192 05230 05269

1130 1131 1132 1133 1134

c 1 c-3815

[ 1 c-3814

log10

Table

[ I c-3814

[ I c-3813

2

37685

[ I C-38)3

100

ELEMENTARY

NATURAL

Table 4.2 3

0.000 0: 001 -6.90775

TRANSCENDENTAL

In x -m

FUNCTIONS

LOGARITHMS

In 2

X

In x

X

821371 221917 140274 622464

0.050 0.051 0.052 0.053 0.054

-2.99573 -2.97592 -2.95651 -2.93746 -2.91877

22735 96462 15604 33654 12324

539910 578113 007097 300152 178627

0.100 0.101 0.102 0.103 0.104

-2.30258 -2.29263 -2.28278 -2.27302 -2.26336

50929 47621 24656 62907 43798

940457 408776 978660 525013 407644

0.002 0.003 0.004

-6.21460 -5.80914 -5.52146

52789 80984 29903 09178

0.005 3.006 0.007 0.008 0.009

-5.29831 -5.11599 -4.96184 -4.82831 -4.71053

73665 58097 51299 37373 07016

480367 540821 268237 023011 459177

0.055 0.056 0.057 0.058 oIo59

-2.90042 -2.88240 -2.86470 -2.84731 -2.83021

20937 35882 40111 22684 78350

496661 469878 475869 357177 764176

0.105 0.106 0.107 0.108 0.109

-2.25379 -2.24431 -2.23492 -2.22562 -2.21640

49288 61848 64445 40518 73967

246137 700699 202309. 579174 529934

0.010 0.011 0.012 0.013 0.014

-4.60517 -4.50986 -4.42284 -4.34280 -4.26869

01859 00061 86291 59215 79493

880914 837665 941367 206003 668784

0.060 0,061 0.062 0.063 0.064

-2.81341 -2.79688 -2.78062 -2.76462 -2.74887

07167 14148 08939 05525 21956

600364 088258 370455 906044 224652

0.110 0.111 0.112 0.113 0.114

-2.20727 -2.19822 -2.18925 -2.18036 -2.17155

49131 50776 64076 74602 68305

897208 698029 870425 697965 876416

0.015 0.016 OiO17 0.018 0.019

-4.19970 -4.13516 -4.07454 -4.01738 -3.96331

50778 65567 19349 35210 62998

799270 423558 259210 859724 156966

0.065 0.066 0.067 0.068 0.069

-2.73336 -2.71810 -2.70306 -2.68824 -2.67364

80090 05369 26595 75738 87743

864999 557115 911710 060304 848777

0.115 0.116 0.117 0.118 0.119

-2.16282 -2.15416 -2.14558 -2.13707 -2.12863

31506 50878 13441 06545 17858

188870 757724 843809 164723 706077

0.020 0.021 0.022 0.023 0.024

-3.91202 -3.86323 -3.81671 -3.77226 -3.72970

30054 28412 28256 10630 14486

281461 587141 238212 529874 341914

0.070 0.071 0.072 0.073 0.074

-2.65926 -2.64507 -2.63108 -2.61729 -2.60369

00369 54019 91599 58378 01857

327781 408216 660817 337459 779673

0.120 0.121 0.122 0.123 0.124

-2.12026 -2.11196 -2.10373 -2.09557 -2.08747

35362 47333 42342 09236 37133

000911 853960 488805 097196 771002

0.025 0.026 0.027 0.028 0.029

-3.68887 -3.64965 -3.61191 -3.57555 -3.54045

94541 87409 84129 07688 94489

139363 606550 778080 069331 956630

0.075 0.076 0.077 0.078 0.079

-2.59026 -2.57702 -2.56394 -2.55104 -2.53830

71654 19386 98571 64522 74265

458266 958060 284532 925453 151156

0.125 0.126 0.127 0.128 0.129

-2.07944 -2.07147 -2.06356 -2.05572 -2.04794

15416 33720 81925 50150 28746

798359 306591 235458 625199 204649

0.030 0.031 0.032 0.033 0.034

-3.50655 -3.47376 -3.44201 -3.41124 -3.38139

78973 80744 93761 77175 47543

199817 969908 824105 156568 659757

0.080 0.081 0.082 0.083 0.084

-2.52572 -2.51330 -2.50103 -2.48891 -2.47693

86443 61243 60317 46711 84801

082554 096983 178839 855391 388234

0.130 0.131 0.132 0.133 0.134

-2.04022 -2.03255 -2.02495 -2.01740 -2.00991

08285 79557 33563 61507 54790

265546 809855 957662 603833 312257

0.035 0.036 0.037 0.038 0.039

-3.35240 -3.32423 -3.29683 -3.27016 -3.24419

72174 63405 73663 91192 36328

927234 260271 379126 557513 524906

0.085 0.086 0.087 0.088 0.089

-2.46510 -2.45340 -2.44184 -2.43041 -2.41911

40224 79827 71603 84645 89092

918206 286293 275533 039306 499972

0.135 0.136 0.137 0.138 0.139

-2.00248 -1.99510 -1.98777 -1.98050 -1.97328

05005 03932 43531 15938 13458

437076 460850 540121 249324 514453

0.040 0.041 0.042 0.043 0.044

-3.21887 -3.19418 -3.17008 -3.14655 -3.12356

58248 32122 56606 51632 56450

682007 778292 987687 885746 638759

0.090 0.091 0.092 0.093 0.094

-2.40794 -2.39689 -2.38596 -2.37515 -2.36446

56086 57724 67019 57858 04967

518720 652870 330967 288811 121332

0.140 0.141 0.142 0.143 0.144

-1.96611 -1.95899 -1.95192 -1.94491 -1.93794

28563 53886 82213 06487 19794

728328 039688 808763 222298 061364

0.045 0.046 0.047 0.048 0.049

-3.10109 -3.07911 -3.05760 -3.03655 -3.01593

27892 38824 76772 42680 49808

118173 930421 720785 742461 715104

0.095 0.096 0.097 0.098 0.099

-2.35387 -2.34340 -2.33304 -2.32278 -2.31263

83873 70875 43004 78003 54288

815962 143008 787542 115651 475471

0.145 0.146 0.147 0.148 0.149

-1.93102 -1.92414 -1.91732 -1.91054 -1.90380

15365 86572 26922 30052 89730

615627 738006 034008 180220 366779

0.050

-2.99573

22735 539910

0.100

-2.30258

50929 940457

0.150

-1.89711

99848 858813

For use of natural logarithms

see Examples 4-7.

In 10 = 2.30258 50929 940457

ELEMJCNTARY TRANSCENDENTAL NATURAL x 11 47 87 31 80

X

101

FUNCTIONS

LOGARITHMS

Table 4.2 X

In x

In 2 99848 54421 47581 73575 26765

858813 672127 358607 897016 685079

0.200 0.201 0.202 0.203 0.204

-1.60943 -1.60445 -1.59948 -1.59454 -1.58963

79124 03709 75815 92999 52851

341004 230613 809323 403497 379207

0.250 0.251 0.252 0.253 0.254

-1.38629 -1.38230 -1.37832 -1.37436 -1.37042

43611 23398 61914 57902 10119

In 2 198906 503532 707137 546168 636005

0.150 0.151 0.152 0.153 0.154

-1.897 -1.890 -1.883 -1.877 -1.870

0.155 0.156 0.157 0.158 0.159

-1.86433 -1.85789 -1.85150 -1.84516 -1.83885

01620 92717 94736 02459 10767

628904 326000 338290 551702 619055

0.205 0.206 0.207 0.208 0.209

-1.58474 -1.57987 -1.57503 -1.57021 -1.56542

52998 91101 64857 71992 10270

437289 925560 167680 808191 173260

0.255 0.256 0.257 0.258 0.259

-1.36649 -1.36257 -1.35867 -1.35479 -1.35092

17338 78345 91940 56940 72172

237109 025746 869173 605196 825993

0.160 0.161 0.162 0.163 0.164

-1.83258 -1.82635 -1.82015 -1.81400 -1.80788

14637 09139 89437 50781 88511

483101 976741 497530 753747 579386

0.210 0.211 0.212 0.213 0.214

-1.56064 -1.55589 -1.55116 -1.54646 -1.54177

77482 71455 90043 31132 92639

646684 060706 101246 727119 602856

0.260 0.261 0.262 0.263 0.264

-1.34707 -1.34323 -1.33941 -1.33560 -1.33180

36479 48716 07752 12468 61758

666093 594436 210402 043725 358209

0.165 0.166 0.167 0.168 0.169

-1.80180 -1.79576 -1.78976 -1.78379 -1.77785

98050 74906 14665 12995 65640

815564 255938 653819 788781 590636

0.215 0.216 0.217 0.218 0.219

-1.53711 -1.53247 -1.52785 -1.52326 -1.51868

72508 68712 79254 02161 35491

544743 979720 416775 930480 656362

0.265 0.266 0.267 0.268 0.269

-1.32802 -1.32425 -1.32050 -1.31676 -1.31304

54529 89702 66205 82984 38993

959148 004380 818875 712804 802979

0.170 0.171 0.172 0.173 0.174

-1.77195 -1.76609 -1.76026 -1.75446 -1.74869

68419 17224 08021 36844 99797

318753 794772 686840 843581 676080

0.220 0.221 0.222 0.223 0.224

-1.51412 -1.50959 -1.50507 -1.50058 -1.49610

77326 25774 78971 35075 92271

297755 643842 098576 220183 270972

0.270 0.271 0.272 0.273 0.274

-1.30933 -1.30563 -1.30195 -1.29828 -1.29462

33199 64581 32126 34837 71725

837623 024362 861397 971773 940668

0.175 0.176 0.177 0.178 0.179

-1.74296 -1.73727 -1.73160 -1.72597 -1.72036

93050 12839 55464 17286 94731

586230 439853 083079 900519 413821

0.225 0.226 0.227 0.228 0.229

-1.49165 -1.48722 -1.48280 -1.47840 -1.47403

48767 02797 52615 96500 32754

777169 098512 007344 276963 278974

0.275 0.276 0.277 0.278 0.279

-1.29098 -1.28735 -1.28373 -1.28013 -1.27654

41813 44132 77727 41652 34971

155658 649871 947986 915000 607714

0.180 0.181 0.182 0.183 0.184

-1.71479 -1.70925 -1.70374 -1.69826 -1.69281

84280 82477 85919 91261 95213

919267 163113 053417 407161 731514

0.230 Oi231 0.232 Oi2j3 0.234

-1.46967 -1.46533 -1.46101 -1i45671 -1.45243

59700 75684 79073 68254 41636

589417 603435 158271 164365 244356

0.280 0.281 0.282 0.283 0.284

-1.27296 -1.26940 -1.26584 -1.26230 -1.25878

56758 06096 82080 83813 10408

128874 483913 440235 388994 209310

0.185 0.186 0.187 0.188 0.189

-1.68739 -1.68200 -1.67664 -1.67131 -1.66600

94539 86052 66621 33161 82639

038122 689358 275504 521878 224947

3-22;: 0:237 0.238 0.239

-1.44816 -1.44392 -1.43969 -1.43548 -1.43129

97648 34739 51378 46053 17270

379781 565270 470059 106624 506264

0.285 0.286 0.287 0.288 0.289

-1.25526 -1.25176 -1.24827 -1.24479 -1.24132

60987 34681 30632 47988 85908

134865 622845 225159 461911 697049

0.190 0.191 0.192 0.193 0.194

-1.66073 -1.65548 -1.65025 -1.64506 -1.63989

12068 18509 99069 50900 71199

216509 355072 543555 772515 188089

0.240 0.241 0.242 0.243 0.244

-1.42711 -1.42295 -1.41881 -1.41469 -1.41058

63556 83454 75528 38356 70536

401457 914821 254507 415886 889352

0.290 0.291 0.292 0.293 0.294

-1.23787 -1.23443 -1.23100 -1.22758 -1.22417

43560 20118 14767 26699 55116

016173 106445 138553 650697 434554

0.195 0.196 0.197 0.198 0.199

-1.63475 -1.62964 -1.62455 -1.61948 -1.61445

57204 06197 15502 82482 04542

183903 516198 441485 876018 576447

0.245 0.246 0.247 0.248 0.249

-1.40649 -1.40242 -1.39836 -1.39432 -1.39030

70684 37430 69423 65328 23825

374101 497742 541599 171549 174294

0.295 0.296 0.297 0.298 0.299

-1.22077 -1.21739 -1.21402 -1.21066 -1.20731

99226 58246 31401 17924 17055

423172 580767 794374 767326 914506

0.200

-1.60943

79124 341004

0.250

-1.38629

43611 198906

0.300

-1.20397

28043 259360 C-7612

[ 1 In 10 = 2.30258 50929 940457

102

ELEMENTARY

Table 4.2

TRANSCENDENTAL

NATURAL .I’

In $7’

2’

FUNCTIONS

LOGARITHMS :c

In .f’

In .I’

;.;N; 0:302 0.303 0.304

-1.20397 -1.20064 -1.19732 -1.19402 -1.19072

28043 50142 82616 24734 75775

259360 332613 072674 727679 759154

0.350 0.351 0.352 0.353 0.354

-1.04982 -1.04696 -1.04412 -1.04128 -1.03845

21244 90555 41033 72220 83658

986777 162712 840400 488403 483626

0.400 0.401 0.402 0.403 0.404

-0.91629 -0.91379 -0.91130 -0.90881 -0.90634

07318 38516 31903 87170 04010

741551 755679 631160 354541 209870

0.305 0.306 0.307 0.308 0.309

-1.18744 -1.18417 -1.18090 -1.17765 -1.17441

35023 01770 75313 54960 40020

741254 297563 949399 085626 843916

0.355 0.356 0.357 0.358 0.359

-1.03563 -1.03282 -1:OSOOl -1.02722 -1.02443

74895 45481 94972 22925 28904

067213 301066 024980 814367 938582

0.405 0.406 0.407 0.408 0.409

-0.90386 -0.90140 -0.89894 -0.89648 -0.89404

82118 21193 20935 81045 01229

755979 804044 395421 779754 393353

0.310 0.311 0.312 0.313 0.314

-1.17118 -1.16796 -1.16475 -1.16155 -1i15836

29815 23668 20911 20884 22930

029451 029029 726547 419838 738837

0.360 0.361 0.362 0.363 0.364

-1.02165 -1.01887 -1.01611 -1.01335 -1.01060

12475 73206 10671 24447 14113

319814 492561 563660 172863 453964

0.410 0.411 0.412 0.413 0.414

-0.89159 -0.88916 -0.88673 -0.88430 -0.88188

81192 20644 19296 76860 93051

837836 859024 326107 211043 568227

0.315 0.316 Oi317 0.318 0.319

-1.15518 -1.15201 -1i14885 -1.14570 -1.14256

26401 30653 35051 38962 41761

565040 952249 048564 019602 972925

0.365 0.366 0.367 0.368 0.369

-1.00785 -1.00512 -1.00239 -0.99967 -0.99695

79253 19455 34309 23408 86349

996455 807708 275668 132061 416099

0.415 0.416 0.417 0.418 0.419

-0.87947 -0.87707 -0.87466 -0.87227 -0.86988

67587 00187 90571 38464 43590

514388 208738 833356 573807 599993

0.320 0.321 0.322 0.323 0.324

-1.13943 -1.13631 -1.13320 -1.13010 -1.12701

42831 41558 37334 29557 17631

883648 521212 377287 594805 898077

0.370 0.371 0.372 0.373 0.374

-0.99425 -0.99155 -0.98886 -0.98617 -0.98349

22733 32163 14247 68593 94815

438669 747019 089905 383215 676051

0.420 0.421 0.422 0.423 0.424

-0.86750 -0.86512 -0.86274 -0.86038 -0.85802

05677 24452 99649 30999 18237

047231 997556 461252 358591 501793

0.325 0.326 0.327 0.328 0.329

-1.12393 -1.12085 -1.11779 -1.11474 -1.11169

00966 78976 51080 16705 75282

523996 154294 848837 979933 167652

0.375 0.376 0.377 0.378 0.379

-0.98082 -0.97816 -0.97551 -0.97286 -0.97021

92530 61355 00915 10833 90738

117262 922425 341263 625494 997107

0.425 0.426 0.427 0.428 0.429

-0.85566 -0.85331 -0.85097 -0.84863 -0.84629

61100 59327 12657 20834 83600

577202 127666 535125 003403 541201

0.330 0.331 0.332 0.333 0.334

-1.10866 -1.10563 -1.10262 -1.09961 -1.09661

26245 69036 03100 27890 42860

216111 050742 656485 016932 054366

0.380 0.381 0.382 0.383 0.384

-0.96758 -0.96495 -0.96233 -0.95972 -0.95711

40262 59038 46703 02898 27263

617056 554361 755619 014911 944102

0.430 0.431 0.432 0.433 0.434

-0.84397 -0.84164 -0.83932 -0.83701 -0.83471

00702 71888 96907 75509 07448

945289 783893 380267 796472 817322

0.335 0.336

0: s>i

0.338 0.339

-1.09362 -1.09064 -ii08767 -1.08470 -1.08175

47471 41190 23486 93834 51716

570706 189328 297753 991183 016868

0.385 0.386 0.387 0.388 0.389

-0.95451 -0.95191 -0.94933 -0.94674 -0.94417

19446 79095 05859 99393 59353

943528 173062 523552 588636 636908

0.435 OI4% 0.431 0.438 0.439

-0.83240 -0.83011 -0.82782 -0i82553 -0.82325

92478 30356 20838 63686 58659

934530 331027 865469 056909 069657

0.340 0.341 0.342 0.343 0.344

-1.07880 -1.07587 -1.07294 -1.07002 -1.06711

96613 28016 45419 48318 36216

719300 986203 195319 161971 087387

0.390 0.391 0.392 0.393 0.394

-0.94160 -0.93904 -0.93649 -0.93394 -0.93140

85398 77189 34391 56671 43696

584449 967713 916745 128758 842032

0.440 0.441 0.442 0.443 0.444

-0.82098 -0.81871 -0.81644 -0.81418 -0.81193

05520 04035 53969 55089 07165

698302 352911 044389 370014 499123

0.345 Oi346 0.347 0.348 0.349

-1.06421 -1.06131 -1.05843 -1.05555 -1.05268

08619 65039 04990 27992 33567

507773 244128 352779 076627 797099

0.395 0.396 0.397 0.398 0.399

-0.92886 -0.92634 -0.92381 -0.92130 -0.91879

95140 10677 89982 32736 38620

810152 276565 949466 976993 922736

0.445 0.446 0.447 0.448 0.449

-0.80968 -0.80743 -0.80519 -0.80296 -0.80073

09968 63269 66843 20465 23912

158968 620730 685682 671519 398828

0.350

-1.04982

21244 986777

0.400

-0.91629

07318 741551

0.450

-0.79850

[‘-,6’1] L

’

J

In 10 = 2.30258 50929 940457 *see page n.

76962 177716 * (-77)8 [ I

ELEMEiNTARY

TRANSCENDENTAL

NATURAL In

X

LOGARITHMS In

X

2

103

FUNCTIONS

x

Table

In

X

4.2

x

0.450 0.451 0.452 0.453 0.454

-0.79850 -0.79628 -0.79407 -0.79186 -0.78965

76962 79394 30991 31534 80809

177716 794587 499059 991030 407891

0.500 0.501 0.502 0.503 0.504

-0.69314 -0.69114 -0.68915 -0.68716 -0.68517

71805 91778 51592 51088 90109

599453 972723 904079 823978 107684

0.550 0.551 0.552 0.553 0.554

-0.59783 -0.59602 -0.59420 -0.59239 -0.59059

70007 04698 72327 72774 05922

556204 292226 050417 598023 348532

0.455 0.456 0.457 0.458 0.459

-0.78745 -0.78526 -0.78307 -0.78088 -0.77870

78600 24694 18880 60948 50689

311866 677510 879324 679521 215919

0.505 0.506 0.507 0.508 0.509

-0.68319 -0.68121 -0.67924 -0.67727 -0.67530

68497 86096 42753 38314 72624

067772 946715 909539 036552 316143

0.555 0.556 0.557 0.558 0.559

-0.58878 -0.58698 -0.58519 -0.58339 -0.58160

71652 69847 00390 63166 58058

357025 315547 548530 008261 270379

0.460 0.461 0.462 0.463 0.464

-0.77652 -0.77435 -0.77219 -0.77002 -0.76787

87894 72359 03879 82248 07267

989964 854885 003982 959030 558818

0.510 0.511 0.512 0.513 0.514

-0.67334 -0.67138 -0.66943 -0.66747 -0.66553

45532 56887 06539 94338 20135

637656 784326 426293 113675 269719

0.560 0.561 0.562 0.563 0.564

-0.57981 -0.57803 -0.57625 -0.57447 -0.57270

84952 43734 34290 56508 10274

529421 594407 884460 424467 840782

0.465 0.466 0.467 0.468 0.469

-0.76571 -0.76356 -0.76142 -0.75928 -0.75715

78733 96448 60213 69830 25105

947807 564912 132397 644903 358577

0.515 0.516 0.517 0.518 0.519

-0.66358 -0.66164 -0.65971 -0.65778 -0.65585

83783 85135 24044 00367 13958

184009 005743 737079 226540 162484

0.565 0.566 0.567 0.568 0.569

-0.57092 -0.56916 -0.56739 -0.56563 -0.56387

95478 12007 59752 38602 48448

356961 789541 543850 609857 558061

0.470 0.471 0.472 0.473 0.474

-0.75502 -0.75289 -0.75077 -0.74865 -0.74654

25842 71849 62933 98904 79572

780328 657193 965817 902041 870606

0.520 0.521 0.522 0.523 0.524

-0.65392 -0.65200 -0.65008 -0.64817 -0.64626

64674 52372 76910 38149 35946

066640 287701 994983 172142 610949

0.570 0.571 0.572 0.573 0.574

-0.56211 -0.56036 -0.55861 -0.55686 -0.55512

89181. 60693 62876 95622 58826

535412 261268 023392 673975 625706

0.475 0.476 0.477 0.478 0.479

-0.74444 -0.74233 -0.74023 -0.73814 -0.73605

04749 74247 87880 45464 46815

474958 507170 937958 906811 712218

0.525 0.526 0.527 0.528 0.529

-0.64435 -0.64245 -0.64055 -0.63865 -0.63676

70163 40662 47304 89952 68471

905133 444272 407747 758756 238377

0.575 0.576 0.577 0.578 0.579

-0.55338 -0.55164 -0.54991 -0.54818 -0.54645

52381 76182 30124 14103 28014

847866 862458 740375 097596 091418

0.480 0.481 0.482 0.483 0.484

-0.73396 -0.73188 -0.72981 -0.72773 -0.72567

91750 80088 11649 86253 03722

802004 763759 315367 295644 655053

0.530 0.531 0.532 0.533 0.534

-0.63487 -0.63299 -0.63111 -0.62923 -0.62735

82724 32577 17896 38548 94400

359695 401982 404927 162925 219422

0.580 O;iiSl 0.582 0.583 0.584

-0.54472 -0i54300 -0.54128 -0.53956 -0.53785

71754 45221 48312 80926 42961

416720 302258 506992 316447 539100

0.485 0.486 0.487 0.488 0.489

-0.72360 -0.72154 -0.71949 -0.71743 -0.71539

63880 66550 11558 98731 27895

446539 816433 995473 289899 072650

0.535 0.536 0.537 0.538 0.539

-0.62548 -0.62362 -0.62175 -0.61989 -0.61803

85320 11179 71844 67188 97080

861305 113351 732724 203526 731399

0.585 0.586 0.587 0.588 0.589

-0.53614 -0.53443 -0.53273 -0.53102 -0.52932

34317 54894 04591 83310 90953

502806 051244 540406 835101 305503

0.490 0.491 0.492 0.493 0.494

-0.71334 -0.71131 -0.70927 -0.70724 -0.70521

98878 11511 65624 61049 97617

774648 876165 898289 394469 942145

0.540 0.541 0.542 0.543 0.544

-0.61618 -0.61433 -0.61248 -0.61064 -0.60880

61394 60001 92775 59590 60321

238170 356555 424908 482016 261944

0.590 ;A;; 0:593 0.594

-0.52763 -0.52593 -0.52424 -0.52256 -0.52087

27420 92615 86440 08799 59596

823719 760389 981314 844116 194921

0.495 0.496 0.497 0.498 0.499

-0.70319 -0.70117 -0.69916 -0.69715 -0.69514

75164 93522 52528 52019 91832

134468 572096 855083 574841 306184

0.545 0.546 0.547 0.548 0.549

-0.60696 -0.60513 -0.60330 -0.60147 -0.59965

94843 63032 64765 99920 68374

188930 372320 601558 341215 726064

0.595 0.596 0.597 0.598 0.599

-0.51919 -0.51751 -0.51583 -0.51416 -0.51249

38734 46119 81655 45250 36808

365073 167873 895350 315053 666877

0.500

-0.69314

71805

599453

0.550

-0.59783

70007

556204

0.600

-0.51082

56237

659907

[ I C-76

[ 1 95

ln lo= 2.30258 50929 940457

104

ELEMENTARY

TRANSCENDENTAL

Table 4.2

NATURAL

In z

2

56231

659907 469295 733160 549516 473221

0.650 0.651 0.652 0.653 0.654

-0.43078 -0.42924 -0.42771 -0.42617 -0.42464

-0.50252 -0.50087 -0.49922 -0.49758 -0.49593

68209 52929 64879 70112

512956 128226 226388 159700 722400

0.655 0.656 0.657 0.658 0.659

0.610 0.611 0.612 0.613 0.614

-0.49429 -0.49265 -0.49102 -0.48939 -0.48776

63218 83198 29964 03430 03508

147801 105417 698110 459257 349946

0.660 0.661 0.662 0.663 0.664

-0.41400 -0.41248 -0.41098 -0.40947

0.615 0.616 0.617 0.618 0.619

-0.48613 -0.48450

30111 83154

756192 486173

-0.48126 -0.47965

68215 00062

244463

0.665 0.666 0.667 0.668 0.669

0.620 0.621 0.622 0.623 0.624

-0.47803 -0.47642 -0.47481 -0.47320 -0.47160

58009 41970 51862 87601 49106

429998 486583 429576 946839 127094

0.670 0.671

0.625

-0.47000

36292

457356

0.626

-0.46840 -0.46680 -0.46521

49078 87383

820385

492164

51125 40222

139384 816965

0.675 0.676 0.677 0.678 0.679

54595 94164 58848

965587 409239 352796

0.680

-0.51082 -0.50916 -0.50749 -0.50583 -0.50418

0.605 0.606 0.607 0.608 0.609

:* 2 0: 629 0.630

03970

-0.48288 62550 767492

-0.46362 -0.46203 -0.46044 -0.45886 -0.45728

975409

0.672 0.673 0.674

0.681 0.682

924543

56367

735678

07170 81497 79275

554841 057060 249384

-0.42312 -0.42159

00433 44900

468851 380480

-0.42007 -0.41855 -0.41703

12604

975265

03476 17444

568199 796298

54439 14391 97230 02887 31295

616658 304508 451288 962745 057032

-0.40796 -0.40646 -0.40496 -0.40346 -0.40197

82383 56084 52330 71054

262829 417479 665133 454913

x

0.700 0.701 0.702 0.703 0.704

-0.35667 -0.35524 -0.35382 -0.35239 -0.35097

49439 73919 18749 83871 69228

387324 475470 563259 714721

0.705 0.706 0.707 0.708 0.709

-0.34955 -0.34814 -0.34672 -0.34531 -0.34389

74761 00414 46130 11852 97524

698684 888950 855643 884173 500096

0.710 0.711 0.712

-0.34249 -0.34108

03089 28491

788962

-0.33967

73675

0.713

701613

-0.33827 -0.33687

38585

678411

-0.33547 -0.33407

27362

0.716

51120

881294 214914

12188 539086

ao@ . 5:i 0.719

-0.33128 -0.33267 -Or32989

94383 57099 39212

825167 339129 610904

-0.40047 -0.39898 -0.39749 -0.39600 -0.39452

75665 61420 69384 99493 51680

971253 104553 589875 374092 698300

0.720 0.721 0.722 0.723 0.724

-0.32850 -0.32711 -0.32573 -0.32434 -0.32296

40669 61416 01400 60568 38865

720361 971880 893108 233724 964207

-0.39304 -0i39i56 -0.39008

25881 22029 40060

096072 391730 698621

0.7i!5 0.726 0.727

-0.32158 -0.32020 -0.31882 -0.31745

36241 52641 88014 42307

274623 573410 486177 854511

-0.41551

-0Ij8860 79910 417415

0.714

0.71!i

0.728

240947

467759

23166 425527

-0.38713

41514

234409

0.729

-0.31608 15469 734789

-0.38566 -0.38419 -0.38272

24808 29728 56211 04194 73613

119847 326247

0.7'30

-0.31471

07448

-0.31334 -0.31197

18192

323585

386750

a. 731 0.732

113470 595866

0.733

-0.31060

47650 95770

0.134

-0.30924

62503

208255 954856 676215

-0.45570

48568 63245

379609

0.683

-0.38126

449111

0.684

-0.37979

0.635 0.636 0.637 0.638 0.639

-0.45413 -0.45255 -0.45098 -0.44941 -0.44785

02800 67156 56234 69956 08246

894454 420149 099737 373472 046022

0.685 0.686

-0.37833 -0.37687

0.681 0.688

-0.37542 -0.37396

0.689

0.640 0.641 0.642 0.643 0.644

-0.44628 -0.44472 -0.44316 -0.44161 -0.44005

71026 58220 05547 65528

284195 614670 921759 445177 777834

0.645 0.646 0.647

0.649

-0.43850 -0.43695 -0.43540 -0.43386 -0.43232

49621 57751 89844 45826 25622

0.650

-0.43078

29160

0.648

In

2

29160

i* t;: 0:633 0.634

69752

LOGARITHMS

In 2

X

03444 78336 80822 10810

0.600 OibOl 0.602 0.603 0.604

FUNCTIONS

64407

199118

76512 09867

562518 597877

0.735 0.736 0.137

-0.30788 -0.30652 -0.30516

-0.37251

64410 40079

487934 684785

0.738 0.739

0.690 0.691 0.692 0.693 0.694

-0.37106 -0.36961 -0.36816 -0.36672 -0.36528

36813 54552 93233 52797 33184

908320 144672 644675 922330 753326

863646 995352 812365 298624 780471

0.695 0.696 0.697 0.698 0.699

-0.36384 -0.36240 -0.36096 -0.35953 -0.35810

34334 56186 98682 61762 45367

216132 197646 483268

924543

0.700

-0.35667

49439

381324

173449

477174

In lO= 2.30258 50929 940457

397002

47797

693004

-0.30381

51602 73867 14543

532608 928004 816646

-0: 30245

73580

339353

0.740 0.741

-0.30110 -0.29975

50927

839216

0.742

-0.29840

46536 60358

0.743 0.744

-0.29705 -0.29571

92342 42441

0.745 0.746 0.747 0.748

-0.29437 -0.29302 -0i29169 -0.29035

10606 96787 00938 23010

0.749

-0.28901

62954

076598 649176

0.750

-0.28768

20724

517809

860502 147566 643779 490452 025775 783762 493197

ELEMENTARY

TRANSCENDENTAL

NATURAL

In

X

x

LOGARITHMS

In T

X

105

FUNCTIONS

Table 4.2

In

X

x

0.750 0.751 0.752 0.753 0.754

-0.28768 -0.28634 -0.28501 -0.28369 -0.28236

20724 96272 89550 00511 29109

517809 180023 322973 822435 741810

0.800 0.801 0.802 0.803 0.804

-0.22314 -0.22189 -0.22064 -0.21940 -0.21815

35513 43319 66711 05650 60098

142098 137778 156226 353754 031707

0.850 0.851 0.852 0.853 0.854

-0.16251 -0.16134 -0.16016 -0.15899 -0.15782

89294 31504 87521 57314 40851

977749 087629 528213 904579 935672

0.755 0.756 0.757 0.758 0.759

-0.28103 -0.27971 -0.27839 -0.27707 -0.27575

75297 39028 20255 18933 35015

331123 026041 446883 397654 865071

0.805 0.806 0.807 0.808 0.809

-0.21691 -0.21567 -0.21443 -0.21319 -0.21195

30015 15364 16107 32204 63619

635737 755088 121883 610417 236454

0.855 0.856 0.857 0.858 0.859

-0.15665 -0.15548 -0.15431 -0.15315 -0.15198

38100 49028 73603 11794 63569

453768 403950 843573 941748 978817

0.760 0.761 0.762 0.763 0.764

-0.27443 -0i27312 -0.27180 -0.27049 -0.26918

68457 19211 87232 72476 74898

017603 204512 954908 976800 156166

0.810 0.811 0.812 0.813 0.814

-0.21072 -0.20948 -0.20825 -0.20702 -0.20579

10313 72248 49388 41694 49129

156526 667241 204591 343265 795968

0.860 0.861 0.862 0.863 0.864

-0.15082 -0.14966 -0.14850 -0.14734 -0.14618

28897 07745 00083 05878 25101

345836 544063 184440 987091 780814

0.765 0.766 0.767 0.768 0.769

-0.26787 -0.26657 -0.26526 -0.26396 -0.26266

94451 31092 84776 55458 43094

556012 415458 148809 344649 764931

0.815 0.816 0.817 0.818 0.819

-0.20456 -0.20334 -0.20211 -0.20089 -0.19967

71657 09240 61841 29423 11951

412743 180300 221342 793900 290676

0.865 0.866 0.867 0.868 0.869

-0.14502 -0.14387 -0.14271 -0.14156 -0.14041

57720 03704 63022 35643 21537

502577 197019 015952 217869 167450

0.770 0.771 0.772 0.773 0.774

-0.26136 -0.26006 -0.25877 -0.25747 -0.25618

47641 69054 07289 62303 34053

344075 188076 573609 947151 924099

0.820 0.821 0.822 0.823 0.824

-0.19845 -0.19723 -0.19601 -0.19479 -0.19358

09387 21695 48839 90783 47490

238383 297088 259571 050672 726654

0.870 0.871 0.872 0.873 0.874

-0.13926 -0.13811 -0.13696 -0.13581 -0.13467

20673 33021 58550 97231 49033

335076 296343 731574 425348 266016

0.775 0.776 0.777 0.778 0.779

-0.25489 -0.25360 -0.25231 -0.25102 -0.24974

22496 27587 49286 87548 42331

287901 989183 144896 037454 113888

0.825 0.826 0.827 0.828 0.829

-0.19237 -0.19116 -0.18995 -0.18874 -0.18753

18926 05054 05839 21245 51238

474561 611590 584457 968774 468421

0.875 0.876 0.877 0.878 0.879

-0.13353 -0.13238 -0.13124 -0.13010 -0.12897

13926 91880 82866 86853 03812

245226 457456 099540 470204 969601

0.780 0.781 0.782 0.783 0.784

-0.24846 -0.24718 -0.24590 -0.24462 -0.24334

13592 01291 05384 25829 62586

984996 424511 368260 913340 317292

0.830 0.831 0.832 0.833 0.834

-0.18632 -0.18512 -0.18392 -0.18272 -0.18152

95781 54841 28381 16368 18766

914934 266889 609285 152944 233903

0.880 0.881 0.882 0.883 0.884

-0.12783 -0.12669 -0.12556 -0.12443 -0.12329

33715 76530 32229 00783 82163

098849, 459575 753457 781770 444936

0.785 0.786 0.787 0.788 0.789

-0.24207 -0.24079 -0.23952 -0.23825 -0.23698

15611 84865 70305 71891 89581

997286 529305 647338 242579 362628

0.835 0.836 0.837 0.838 0.839

-0.18032 -0.17912 -0.17793 -0.17673 -0.17554

35541 66658 12084 71785 45725

312816 974354 926617 000540 149309

0.885 0.886 0.887 0.888 0.889

-0.12216 -0.12103 -0.11991 -0.11878 -0.11765

76339 83283 02966 35359 80434

742075 770561 725576 899670 682325

0.790 Oi791 0.792 0.793 0.794

-0.23572 -0i23445 -0.23319 -0.23193 -0.23067

23335 73112 38871 20573 18177

210699 144832 677112 472891 350013

0.840 0.841 0.842 0.843 0.844

-0.17435 -0.17316 -0.17197 -0.17078 -0.16960

33871 36190 52647 83209 27843

447778 091890 398103 802816 861799

0.890 0.891 0.892 0.893 0.894

-0.11653 -0.11541 -0.11428 -0.11316 -0.11204

38162 08515 91464 86981 95038

559515 113277 021277 056380 086229

0.795 0.796 0.797 0.798 0.799

-0.22941 -0.22815 -0.22690 -0.22564 -0.22439

31643 60931 06001 66815 43332

278052 377540 919220 323283 158624

0.845 0.846 0.847 ;.;&I; .

-0.16841 -0.16723 -0.16605 -0.16487 -0.16369

86516 59193 45843 46431 60926

249632 759138 300827 902340 707897

0.895 0.896 0.897 ;.;N: .

-0.11093 -0.10981 -0.10869 -0.10758 -0.10647

15607 48660 94169 52106 22445

072817 072066 233409 799374 105168

0.800

-0.22314

35513

142098

0.850

-0.16251

89294

977749

0.900

-0.10536

05156

578263

[ 1 (72

[ 1 C-67)2

In lO= 2.30258 50929 940457

[ 1 (72

106

ELEMENTARY

Table 4.2

NATURAL

In z

x

TRANSCENDENTAL

FUNCTIONS

LOGARITHMS

In

X

In x

X

2

0.900 0.901 0.902 0.903 0.904

-0.10536 -0.10425 -0.10314 -0.10203 -0.10092

05156 00213 07589 27255 59185

578263 737991 195134 651516 899606

0.950 0.951 0.952 0.953 0.954

-0.05129 -0.05024 -0.04919 -0.04814 -0.04709

32943 12164 02441 03753 16075

875505 367467 907717 279349 338505

1.000 0.00000 00000 000000 1.0011 0.00099 95003 330835 1 OOZ! 0.00199 80026 626731 1:003 0.00299 55089 797985 1.004 0.00399 20212 695375

0.905 0.906 0.907 0.908 0.909

-0.09982 -0.09871 -0.09761 -0.09651 -0.09541

03352 59729 28288 09003 01848

822109 391577 670004 808438 046582

0.955 0.956 0.957 0.958 0.959

-0.04604 -0.04499 -0.04395 -0.04290 -0.04186

39385 73659 18875 75010 42040

014068 307358 291828 112765 986988

1.005 1.006 1.007 1.008 1.009

0.00498 0.00598 0.00697 0.00796 0.00895

75415 20716 56137 81696 97413

110391 775475 364252 491769 714719

0.910 0.911 0.912 0.913 0.914

-0.09431 -0.09321 -0.09211 -0.09101 -0.08992

06794 23817 52889 93983 47075

712413 221787 078057 871686 279870

0.960 0.961 0.962 0.963 0.964

-0.04082 -0.03978 -0.03874 -0.03770 -0.03666

19945 08700 08283 18671 39843

202551 118446 164306 840115 715914

1.010 1.011. 1 OK! 1:013 1.014

0.00995 0.01093 0.01192 0.01291 0.01390

03308 99400 85708 62252 29051

531681 383344 652738 665463 689914

0.915 0.916 0.917 0.918 0.919

-0.08883 12137 -0iO8773 89143 -0.08664 78067 -0iO8555 78883 -0.08446 91566

066157 080068 256722 616466 264500

0.965 Oi966 0.967 0.968 0.969

-0.03562 -0iO3459 -0.03355 -0.03252 -0.03149

71776 14447 67835 31917 06670

431511 696191 288427 055600 913708

1.015 1.016 1.017 l.OlEl 1.019

0.01488 OiOi587 0.01685 0.01783 0.01882

86124 33491 71170 99181 17542

937507 562901 664229 283310 405878

0.920 Oi921 0.922 Oi923 0.924

-0.08338 -0.08229 -0.08121 -0.08012 -0.07904

16089 52427 00554 60444 32073

390511 268302 255432 792849 404529

0.970 0.971 0.972 0.973 0.974

-0.03045 92074 847085 -0iO2942 88106 908121 -0.02839 94745 216980 -0.02737 11967 961320 -0.02634 39753 396020

1.020 1.021. l.O2i! 1.023 1.024

0.01980 OiO2078 0.02176 OiO2273 0.02371

26272 25391 14917 94869 65266

961797 825285 815127 694894 173160

0.925 0.926 0.927 0.928 0.929

-0.07796 -0.07688 -0.07580 -0.07472 -0.07364

15414 10443 17134 35461 65401

697119 359577 162819 959365 682985

0.975 0.976 0.977 0.978 0.979

-0.02531 -0.02429 -0.02326 -0.02224 -0.02122

78079 26925 86269 56089 36364

842899 690446 393543 473197 516267

:- i:: 1: 02i: 1.028 1.029

0.02469 0.02566 0.02664 0.02761 0.02858

26125 77467 19309 51670 74568

903715 485778 464212 329734 519126

'0.930 0.931 0.932 0.933 0.934

-II.07257 -6.07149 -0.07042 -0.06935 -0.06827

06928 60017 24642 00781 88407

348354 050700 965459 347932 532944

0.980 0.981 0.982 0.983 0.984

-0.02020 -0.01918 -0.01816 -0.01714 -0.01612

27073 28194 39706 61588 93819

175194 167740 276712 349705 298836

1.030 1.031. 1 03i! 11033 1.034

0.02955 0.03052 0.03149 0.03246 0.03343

88022 92050 86670 71901 47760

415444 348229 593710 375015 862374

0.935 0.936 0.937 0.938 0.939

-0.06720 -0.06613 -0.06507 -0.06400 -0.06293

87496 98025 19967 53299 97997

934501 045450 437149 759124 738741

0.985 0.986 0.987 0.988 0.989

-0.01511 -0.01409 -0.01308 -0.01207 -0.01106

36378 89243 52395 25812 09473

100482 795016 486555 342692 594249

1.035 1.036 1.037 1.038 1.039

0.03440 0.03536 0.03633 0.03729 0.03825

14267 71438 19292 57847 87121

173324 372913 473903 436969 170903

0.940 0.941 0.942 0.943 0.944

-0.06187 -0.06081 -0.05975 -0.05868 -0.05762

54037 21393 00044 89963 91128

180875 967574 057740 486796 366364

0.990 o1991 0.992 0.993 0.994

-0.01005 -0~00904 -0.00803 -0.00702 -0.00601

03358 07446 21716 46149 80723

535014 521491 972643 369645 255630

1.040 1.041. 1 042 1:043 1.044

0.03922 OiO4018 0.04114 0.04210 0.04305

07131 17896 19433 11760 94894

532813 328318 311752 186354 604470

0.945 0.946 0.947 0.948 0.949

-0.05657 -0.05551 -0.05445 -0iO5340 -0.05234

03514 27099 61857 07767 64803

883943 302588 960588 271152 722092

0.995 0.996 0.997 0.998 0.999

-0.00501 -0.00400 -0.00300 -0.00200 -0.00100

25418 80213 45090 20026 05003

235443 975388 202987 706731 335835

1.045 1.046 1.047 1.048 1.049

0.04401 0.04497 0.04592 0.04688 0.04783

68854 33656 89318 35858 73294

167743 427312 883998 988504 141601

0.950

-0.05129

32943 875505

1.000

0.00000 00000 000000

1.050

0.04879 01641 694320

[ 1 C-67)2

[ In 10=2.30258

(-iI 1 I 50929 940457

11 [C-67)

ELEMENTARY

TRANSCENDENTAL

NATURAL

In 2

X

107

FUNCTIONS

LOGARITHMS

In

X

Table 4.2

In

X

x

x

1.050 1.051 1.052 1.053 1.054

0.04879 OiO4974 0.05069 0.05164 0.05259

01641 20918 31143 32331 24501

694320 948141 155181 518384 191706

1.100 1.101 1.102 1.103 1.104

0.09531 0.09621 0.09712 0.09803 0.09893

01798 88577 67107 37402 99478

043249 405429 307227 713654 549036

1.150 1.151 1.152 1.153 1.154

0.13976 0.14063 0.14149 0.14236 0.14323

19423 11297 95622 72412 41680

751587 397456 736995 869220 859078

1.055 1.056 1.057 1.058 1.059

0.05354 0.05448 0.05543 0.05638 0.05732

07669 81852 47068 03334 50666

280298 840697 881006 361076 192694

1.105 1.106 1.107 1.108 1.109

0.09984 0.10074 0.10165 0.10255 0.10345

53349 99031 36537 65883 87083

697161 001431 264998 250921 682300

1.155 1.156 1.157 1.158 1.159

0.14410 0.14496 0.14583 0.14669 0.14755

03439 57702 04482 43791 75643

737569 501857 115395 508035 576147

1.060 1.061 1.062 1.063 1.064

0.05826 0.05921 0.06015 0.06109 0.06203

89081 18596 39228 50993 53909

239758 318461 197471 598109 194526

1.110 1.111 1.112 1.113 1.114

0.10436 0.10526 0.10616 0.10705 0.10795

00153 05106 01958 90722 71415

242428 574929 283906 934078 050923

1.160 1.161 1.162 1.163 1.164

0.14842 0.14928 0.15014 0.15100 0.15186

00051 17027 26584 28735 23493

182733 157544 297195 365274 092461

1.065 1.066 1.067 1.068 1.069

0.06297 0.06391 0.06485 0.06578 0.06672

47991 33257 09723 77405 36320

613884 436528 196163 380031 429082

1.115 1.116 1.117 1.118 1.119

0.10885 0.10975 0.11064 0.11154 0.11243

44049 08639 65200 13747 54293

120821 591192 870637 329074 297882

1.165 1.166 1.167 1.168 1.169

0.15272 0.15357 0.15443 0.15529 0.15614

10870 90879 63533 28844 86824

176639 283006 044189 060353 899314

1.070 1.071 1.072 1.073 1.074

0.06765 0.06859 0.06952 OiO7045 0.07138

86484 27914 60626 84636 99960

738148 656117 486102 485614 866729

1.120 1.121 1.122 1.123 1.124

0.11332 0.11422 0.11511 0.11600 0.11689

86853 11440 28071 36757 37514

070032 900229 005046 563061 714993

1.170 1.171 1.172 1.173 1.174

0.15700 0.15785 0.15871 0.15956 0.16041

37488 80846 16911 45696 67214

096648 155803 548209 713384 059047

1.075 1.076 1.077 1.078 1.079

0.07232 0.07325 0.07417 0.07510 0.07603

06615 04617 93981 74724 46862

796261 395927 742515 868054 759976

1.125 1.126 1.127 1.128 1.129

0.11778 0.11867 0.11955 0.12044 0.12133

30356 15297 92350 61530 22851

563835 174986 576392 758672 675250

1.175 1.176 1.177 1.178 1.179

0.16126 0.16211 0.16296 0.16381 0.16466

81475 88494 88282 80852 66215

961223 764352 781397 293950 552339

1.080 1.081 1.082 1.083 1.084

0.07696 0.07788 0.07881 0.07973 0.08065

10411 65386 11804 49680 79030

361283 570712 242898 188536 174545

1.130 1.131 1.132 1.133 1.134

0.12221 0.12310 0.12398 0.12486 0.12575

76327 21971 59797 89820 12053

242492 339834 809912 458693 055603

1.180 1.181 1.182 1.183 1.184

0.16551 0.16636 0.16720 0.16805 0.16889

44384 15372 79189 35849 85364

775734 152253 839065 962497 618139

1.085 1.086 1.087 1.088 1.089

0.08157 0.08250 0.08342 0.08434 0.08525

99869 12215 16081 11484 98439

924229 117437 390724 337509 508234

1.135 1.136 1.137 1.138 1.139

0.12663 0.12751 0.12839 0.12927 0.13015

26509 33202 32147 23357 06844

333660 989596 683990 041392 650451

1.185 1.186 1.187 1.188 1.189

0.16974 0.17058 0.17142 0.17227 0.17311

27745 63005 91156 12209 26177

870945 755337 275310 404532 086448

1.090 1.091 1.092 1.093 1.094

0.08617 0.08709 0.08801 0.08892 0.08984

76962 47068 08773 62091 07039

410523 509338 227133 944015 997895

1.140 1.141 1.142 1.143 1.144

0.13102 0.13190 0.13278 0.13365 0.13453

82624 50708 11112 63848 08929

064041 799386 338185 126736 576062

1.190 1.191 1.192 1.193 1.194

0.17395 0.17479 0.17563 0.17647 0.17730

33071 32903 25686 11431 90149

234380 731631 431580 157791 704103

1.095 1.096 1.097 1.098 1.099

0.09075 0.09166 0.09257 0.09349 0.09440

43632 71885 91812 03430 06754

684641 258238 930932 873389 214843

1.145 1.146 1.147 1.148 1.149

0.13540 0.13627 0.13714 0.13802 0.13889

46370 76182 98381 12978 19988

062030 925478 472336 973747 666186

1.195 1.196 1.197 1.198 1.199

0.17814 0.17898 0.17981 0.18065 Oil8148

61853 26555 84265 34996 78760

834740 284400 758361 932576 453772

1.100

0.09531 01798 043249 6;) 1 [ I

1.150

0.13976 19423 751587

1.200

0.18232 15567 939546

In 10 = 2.30258 50929 940457

[c-y1

108

ELEMENTARY

TRANSCENDENTAL

NATURAL

Table 4.2

In 2

X

FUNCTIONS

LOGARITHMS

In

X

x

In

X

x

1.200 1.201 1.202 1.203 1.204

0.18232 0.18315 0.18398 0.18481 0.18564

15567 45430 68361 84369 93468

939546 978465 130158 925418 866293

1.250 1.251 1.252 1.253 1.254

0.22314 0.22394 0.22474 0.22554 0.22633

35513 32314 22726 06759 84422

142098 847741 779068 139312 107290

1.300 1.301 1.302 1.303 1.304

0.26236 0.26313 0.26390 0.26466 0.26543

42644 31995 15437 92981 64635

674911 303682 863775 427081 044612

1.205 1.206 1.207 1.208 1.209

0.18647 0.18730 0.18813 0.18896 0.18979

95669 90983 79421 60995 35716

426183 049937 153944 126232 326556

1.255 1.256 1.257 1.258 1.259

0.22713 0.22793 0.22872 0.22952 0.23031

55725 20680 79296 31582 77550

837472 460069 081104 782488 622101

1.305 1.306' 1.307 1.308 1.309

0.26620 0.26696 0.26773 0.26849 0.26926

30407 90308 44346 92530 34869

746567 542393 420849 350070 277629

1.210 1.211 1.212 1.213 1.214

0.19062 0.19144 0.19227 0.19309 0.19392

03596 64645 18876 66299 06926

086497 709552 471227 619131 373065

1.260 1.261 1.262 1.263 1.264

0.23111 0.23190 0.23269 0.23348 0.23428

17209 50569 77641 98433 12957

633866 827825 190214 683541 246657

1.310 1.311 1.312 1.313 1. 314

0.27002 0.27079 0.27155 0.27231 0.27307

71372 02047 26905 45953 59200

130602 815628 218973 206591 624188

1.215 1.216 1.217 1.218 1.219

0.19474 0.19556 0.19638 0.19721 0.19803

40767 67835 88140 01692 08504

925118 439753 053901 877053 991345

1.265 1.266 1.267 1.268 1.269

0.23507 0.23586 0.23665 0.23744 0.23822

21221 23237 19013 08560 91887

794836 219844 390020 150342 322506

1. 31'5 1.316 1.317 1.318 1.319

0.27383 Oi27459 0.27535 0.27611 0.27687

66656 68329 64227 54360 38737

297279 031255 611440 803155 351775

1.220 1.221 1.222 1.223 1.224

0.19885 0.19967 0.20048 0.20130 0.20212

08587 01951 88607 68567 41840

451652 285676 494036 050353 901343

1.270 1.271 1.272 1.273 1.274

0.23901 0.23980 0.24059 0.24137 0.24216

69004 39922 04649 63195 15571

704999 073170 179304 752695 499716

0.27763 :3322: 0.27838 1:322 0.27914

1.32:3 1.3214

17365 90255 57414 0.27990 18851 0.28065 74575

982795 401883 294945 328186 148165

1.225 1.226 1.227 1.228 1.229

0.20294 0.20375 0.20457 0.20538 0.20620

08439 68375 21657 68297 08305

966903 140197 287744 249507 838978

1.275 1.276 1.277 1.278 1.279

0.24294 0.24373 0.24451 0.24529 0.24607

61786 01849 35770 63559 85225

103895 225981 504022 553431 967056

1.325 1.326 1.3i!7 1.328 1.329

0.28141 0.28216 0.28292 0.28367 0.28442

24594 68917 07553 40510 67797

381855 636708 500705 542421 311083

1.230 1.231 1.232 1.233 1.234

0.20701 0.20782 0.20863 0.20945 0.21026

41693 68472 88651 02241 09254

843261 023165 113280 822072 831961

1.280 1.281 1.282 1.283 1.284

0.24686 0.24764 0.24842 0.24920 0.24998

00779 10229 13584 10856 02052

315258 145972 984783 334994 677694

1.330 1.331 1.332 1.333 1.334

0.28517 0.28593 0.28668 0.28743 0.28818

89422 05394 15721 20411 19474

336624 129746 181974 965716 934320

1.235 1.236 1.237 1.238 1.239

0.21107 0.21188 0.21268 0.21349 0.21430

09700 03590 90934 71742 46026

799405 354990 103508 624044 470054

1.285 1.286 1.287 1.288 1.289

0.25075 0.25153 0.25231 0.25309 0.25386

87183 66258 39286 06276 67239

471831 154276 139896 821619 570503

1.335 1.336 1. 337 1.338 1.339

0.28893 0.28968 0.29042 0.29117 Oi29192

12918 00751 82981 59617 30667

522129 144540 198061 060367 090355

1.240 1.241 1.242 1.243 1.244

0.21511 0.21591 0.21672 0.21752 0.21833

13796 75062 29835 78125 19943

169455 224702 112870 285741 169877

1.290 1.291 1.292 1.293 1.294

0.25464 0.25541 0.25619 0.25696 0.25773

22183 71118 14053 50997 81960

735807 645054 604101 897204 787088

1.340 1.341 1.342 1.343 1.344

0.29266 0.29341 0.29416 0.29490 0.29565

96139 56042 10385 59175 02421

628200 995415 494901 411005 009578

1.245 1.246 1.247 1.248 1.249

6.21913 0.21993 0.22074 0.22154 0.22234

55299 84203 06666 22699 32311

166709 652614 978994 472359 434406

1.295 1.296 1.297 1.298 1.299

0.25851 0.25928 0.26005 0.26082 0.26159

06951 25979 39053 46182 47376

515011 300830 343068 818983 884625

1.1345 1.346 1.347 1.348 1.349

0.29639 0.29713 0.29787 0.29862 0.29936

40130 72312 98974 20124 35772

538024 225361 282269 901153 256188

1.250

0.22314 35513 142098

1.300

0.26236 42644 674911

1.350

0.30010 45924 503381

cc-pI

In 10 = 2.30258 50929 940457

[

c-y

I

ELEMENTARY

TRANSCENDENTAL

NATURAL

In

x

x

109

FUNCTIONS

LOGARITHMS

In

X

Table 4.2

In

X

x

x

1.350 1.351 1.352 1.353 1.354

0.30010 0.30084 0.30158 0.30232 0.30306

45924 50589 49776 43491 31744

503381 780618 207723 886510 900833

1.400 1.401 1.402 1.403 1.404

0.33647 0.33718 0.33789 0.33861 0.33932

22366 62673 97886 28011 53056

212129 548700 123983 203239 036194

1.450 1.451 1.452 1.453 1.454

0.37156 0.37225 0.37294 P.37363 0.37431

35564 29739 19164 03845 83791

324830 020508 026043 881459 113276

1.355 1.356 1.357 1.358 1.359

0.30380 0.30453 Oi30527 0.30601 0.30674

14543 91895 63808 30291 91351

316642 182038 527321 365044 690067

1.405 1.406 1.407 1.408 1.409

0.34003 0.34074 0.34145 0.34217 0.34288

73027 87933 97781 02577 02329

857091 884732 322520 358507 165432

1.455 1.456 1.457 1.458 1.459

0.37500 0.37569 0.37637 0.37706 0.37775

59006 29497 95272 56335 12695

234558 744942 130678 864664 406486

1.360 1.361 1.362 1.363 1.364

0.30748 0.30821 0.30895 0.30968 0.31042

46997 97236 42077 81527 15594

479606 693290 273206 143956 212704

1.410 1.411 1.412 1.413 1.414

0.34358 0.34429 0.34500 0.34571 0.34642

97043 86728 71390 51037 25674

900769 706770 710503 023904 743810

1.460 1.461 1.462 1.463 1.464

0.37843 0.37912 0.37980 0.38048 0.38117

64357 11327 53613 91220 24155

202451 685624 275868 379873 391198

1.365 1.366 1.367 1.368 1.369

0.31115 0.31188 0.31261 0.31334 0.31408

44286 67611 85577 98192 05463

369231 485983 418125 003587 063118

1.415 1.416 1.417 1.418 1.419

0.34712 0.34783 0.34854 0.34924 0.34995

95310 59952 19607 74281 23981

952009 715280 085434 099358 779056

1.465 1.466 1.467 1.468 1.469

0.38185 Or38253 0.38321 0.38390 0.38458

52424 76034 94991 09301 18971

690306 644597 608447 923238 917403

1.370 :* z: 1:373 1.374

0.31481 0.31554 0.31626 0.31699 0.31772

07398 04005 95293 81267 61938

400335 801773 036935 858340 001576

1.420 1.421 1.422 1.423 1.424

0.35065 0.35136 0.35206 0.35276 0.35346

68716 08491 43313 73191 98129

131694 149636 810491 077153 897840

1.470 1.471 1.472 1.473 1.474

0.38526 0.38594 0.38662 0.38730 0.38797

24007 24416 20203 11374 97937

906449 193005 066845 804932 671449

1.375 1.376 1.377 1.378 1.379

0.31845 0.31918 0.31990 0.32063 0.32135

37311 07395 72197 31725 85988

185346 111519 465178 914668 111648

1.425 1.426 1.427 1.428 1.429

0.35417 0.35487 0.35557 0.35627 0.35697

18137 33219 43384 48639 48989

206138 921042 946994 173926 477304

1.475 1.476 1.477 1.478 1.479

0.38865 0.38933 0.39001 0.39068 0.39136

79897 57261 30035 98225 61837

917831 782808 492427 260100 286627

1.380 1.381 1.382 1.383 1.384

0.32208 0.32280 0.32353 0.32425 0.32497

34991 78744 17253 50526 78571

691133 271551 454782 826212 954778

1.430 1.431 1.432 1.433 1.434

0.35767 0.35837 0.35907 0.35977 0.36046

44442 35005 20685 01488 77421

718159 743139 384539 460348 774286

1.480 1.481 1.482 1.483 1.484

0.39204 0.39271 0.39339 0.39406 0.39474

20877 75352 25268 70631 11447

760237 856617 738951 557950 451887

1.385 1.386 1.387 1.388 1.389

0.32570 0.32642 0.32714 0.32786 0.32858

01396 19007 31413 38620 40637

393018 677115 326945 846128 722067

1.435 1.436 1.437 1.438 1.439

0.36116 0.36186 0.36255 0.36325 0.36394

48492 14706 76070 32592 84279

115844 260324 968879 988549 052308

1.485 1.486 1.487 1.488 1.489

0.39541 0.39608 0.39676 0.39743 0.39810

47722 79462 06674 29364 47537

546629 955674 780180 109001 018719

1.390 1.391 1.392 1.393 1.394

0.32930 0.33002 0.33074 0.33145 0.33217

37471 29129 15619 96947 73123

426004 413059 122279 976686 383321

1.440 1.441 1.442 1.443 1.444

0.36464 0.36533 0.36603 0.36672 0.36741

31135 73170 10388 42797 70404

879093 173850 627573 917338 706345

1.490 1.491 1.492 1.493 1.494

0.39877 0.39944 0.40011 0.40078 0.40145

61199 70357 75017 75185 70867

573678 826014 815691 570533 106256

1.395 11.z: 1:398 1.399

0.33289 0.33361 0.33432 0.33504 0.33575

44152 10043 70802 26438 76956

733290 401807 748248 116185 833441

1.445 1.446 1.447 1.448 1.449

0.36810 0.36880 0.36949 0.37018 0.37087

93215 11237 24476 32939 36633

643955 365729 493468 635246 385453

1.495 1.496 1.497 1.498 1.499

0.40212 0.40279 0.40346 0.40413 0.40479

62068 48795 31054 08850 82191

426497 522855 374913 950277 204607

1.400

0.33647 22366 212129

1.450

0.37156 35564 324830 (-fV

1.500

0.40546 51081 081644

[

c-y

I

[ 1

In lo= 2.30258 50929 940457

110

ELEMENTARY

TRANSCENDENTAL

NATURAL

Table 4.2

In x

X

FUNCTIONS

LOGARITHMS

In z

5

In

X

1.500 1.501 1.502 1.503 1.504

0.40546 0.40613 0.40679 0.40746 0.40812

51081 15526 75533 31107 82255

081644 513249 419430 708374 276481

1.550 1.551 1.552 1.553 1.554

0.43825 0.43889 0.43954 0.44018 0.44083

49309 98841 44217 85441 22519

311553 944018 610270 665500 454557

1.600

1.505 1.506 1.507 1.508 1.509

0.40879 0.40945 0.41012 0.41078 0.41144

28982 71293 09196 42695 71797

008391 777018 443584 857643 857118

1.555 1.556 1.557 1.558 1.559

0.44147 0.44211 0.44276 0.44340 0.44404

55456 84257 08928 29474 45900

1.510 1: 511 1.512 1.513 1.514

0.41210 0141277 0.41343 Oi41409 0.41475

96508 16832 32777 44348 51550

268330 906025 573413 062189 152570

1.560 1.561 1.562 1.563 1.564

0.44468 0.44532 0.44596 0.44660 0.44724

1.515 1.516 1.517 1.518 1.519

0.41541 0.41607 0.41673 0.41739 0.41805

54389 52872 47003 36789 22236

613325 201799 663952 734382 136358

1.565 1.566 1.567 1.568 1.569

1.520 1.521 1.522 1.523 1.524

0.41871 0.41936 0.42002 0.42068 0.42133

03348 80132 52594 20739 84572

581850 771558 394941 130248 644545

1.525 1.526 1.527 1.528 1.529

0.42199 0.42264 0.42330 0.42395 0.42461

44100 99328 50262 96907 39269

1.530 1.531 1.532 1.533 1.534

0.42526 0.42592 0.42657 0.42722 0.42787

1.535 1.536 1.537 1.538 1.539

x

:- El: 1:603 1.604

0.47000 0.47062 0.47125 0.47187 0.47250

36292 84340 28486 68736 05094

457356 145776 461675 274159 443228

311975 561999 518613 485565 756395

1.605 1.606 1.607 1.608 1.609

0.47312 0.47374 0.47436 0.47499 0.47561

37565 66155 90867 11707 28680

819792 245699 553755 567746 102462

58212 66415 70514 70514 66421

614457 332950 174942 393396 231193

1.610 1.611 1.612 1.613 1.614

0.47623 Oi47685 0.47747 Oi47809 0.47871

41789 51041 56440 57991 55698

963716 948373 844365 430718 477571

0.44788 0.44852 0.44916 0.44980 0.45043

58239 45975 29633 09219 84737

921165 686114 738838 282161 508955

1.615 1.616 1.617 1.618 1.619

0.47933 0.47995 0.48057 0.48119 0.48180

49566 39600 25805 08186 86746

746199 989036 949698 362999 954981

1.570 1.571 1.572 1.573 1.574

0.45107 0.45171 0.45234 0.45298 0.45362

56193 23592 86940 46240 01499

602167 734841 070148 761408 952115

1.620 1.621 1.622 1.623 1.624

0.48242 0.48304 0.48365 0.48427 0.48489

61492 32427 99556 62885 22417

442927 535391 932212 324542 394862

593749 622653 364954 443287 469252

1.575 1.576 1.577 1.578 1.579

0.45425 0.45488 0.45552 0.45615 0.45679

52722 99914 43079 82224 17352

775964 356874 809013 236825 735050

1.625 1.626 1.627 1.628 1.629

0.48550 0.48612 0.48673 0.48735 0.48796

78157 30111 78282 22675 63296

817008 256188 369007 803486 199081

77354 11166 40713 65998 87029

043441 755467 183996 896771 450644

1.580 1.581 1.582 1.583 1.584

0.45742 0.45805 0.45868 0.45932 0.45995

48470 75582 98693 17808 32933

388754 273350 454621 988751 922341

1.630 1.631 1.632 1.633 1.634

0.48858 0.48919 0.48980 0.49041 0.49103

00148 33236 62565 88139 09964

186710 388768 419153 883281 378111

0.42853 0.42918 0.42983 0.43048 0.43113

03810 16347 24645 28710 28548

391605 254804 564588 834522 567422

1.585 1.586 1.587 1.588 1.589

0.46058 0.46121 0.46184 0.46247 0.46310

44073 51232 54415 53628 48875

292439 126562 442720 249440 545789

1.635 1.636 1.637 1.638 1.639

0.49164 0.49225 0.49286 0.49347 0.49408

28043 42381 52983 59854 62997

492167 805553 889979 308777 616926

1.540 1.541 1.542 1.543 1.544

0.43178 0.43243 0.43308 0.43372 0.43437

24164 15563 02751 85733 64516

255378 379787 411377 810238 025844

1.590 1.591 1.592 1.593 1.594

0.46373 0.46436 0.46499 0.46561 0.46624

40162 27493 10874 90309 65803

321402 556498 221913 279115 680233

1.640 1.641 1.642 1.643 1.644

0.49469 0.49530 0.49591 0.49652 0.49713

62418 58121 50110 38390 22966

361071 079538 302365 551310 339882

1.545 1.546 1.547 1.548 1.549

0.43502 0.43567 0.43631 0.43696 0.43760

39103 09501 75715 37751 95614

497088 652302 909291 675354 347316

1.595 1.596 1.597 1.598 1.599

0.46687 0.46750 0.46812 0.46875 0.46937

37362 04990 68692 28473 84338

368079 276170 328754 440829 518172

1.645 1.646 1.647 1.648 1.649

0.49774 0.49834 0.49895 0.49956 0.50016

03842 81022 54511 24314 90435

173352 548781 955033 872800 774619

1.550

0.43825 49309 311553 C-58)6

1.600

0.47000 36292 457356

1.650

0.50077 52879 124892

E 1

[(-;I51

In 10 = 2.30258 50929 940457

c-p I: 3

ELEMXNTARY

TRANSCENDENTAL

NATURAL

X

LOGARITHMS

2

In x

111

FUNCTIONS

In 2

Table 4.2

In x

X

1.650 1.651 1.652 1.653 1.654

0.50077 0.50138 0.50198 0.50259 0.50319

52879 11649 66750 18188 65966

124892 379910 987863 388871 014996

1.700 1.701 1.702 1.703 1.704

0.53062 0.53121 0.53180 0.53239 0.53297

82510 63134 40301 14016 84284

621704 137247 511824 805512 071240

1.750 1.751 1.752 1.753 1.754

0.55961 0.56018 0.56075 {Or56132 0.56189

57879 70533 79925 86059 88939

354227 037148 141997 390974 499913

1.655 1.656 1.657 1.658 1.659

0.50380 0.50440 0.50500 0.50561 0.50621

10088 50559 87384 20567 50112

290262 630679 444259 131032 083074

1.705 1.706 1.707 1.708 1.709

0.53356 0.53415 0.53473 0.53532 0.53590

51107 14490 74438 30953 84041

354801 694874 123036 663781 334538

1.755 1.756 1.757 1.758 1.759

0.56246 0.56303 0.56360 0.56417 0.56474

88569 84952 78092 67992 54657

178291 129249 049601 629853 554211

1.660 1.661 1.662 1.663 1.664

0.50681 0.50741 0.50802 0.50862 0.50922

76023 98306 16964 32002 43423

684519 311578 332564 107906 990168

1.710 1.711 1.712 1.713 1.714

0.53649 0.53707 0.53766 0.53824 0.53882

33705 79949 22777 62193 98201

145685 100564 195504 419829 755880

1.760 1.761 1.762 1.763 1.764

0.56531 0.56588 0.56644 0.56701 0.56758

38090 18295 95275 69034 39575

500604 140691 139878 157332 845996

1.665 1.666 1.667 1.668 1.669

0.50982 0.51042 0.51102 0.51162 0.51222

51234 55437 56037 53039 46446

324071 446509 686569 365550 796980

1.715 1.716 1.717 1.718 1.719

0.53941 0.53999 0.54057 0.54116 0.54174

30806 60010 85819 08235 27264

179032 657705 153385 620636 007122

1.765 1.766 1.767 1.768 1.769

0.56815 0.56871 0.56928 0.56984 0.57041

06903 71021 31933 89642 44151

852601 817683 375593 154517 776482

1.670 1.671 1.672 1.673 1.674

0.51282 0.51342 0.51402 0.51461 0.51521

36264 22496 05146 84220 59720

286637 132567 625099 046869 672836

1.720 1.721 1.722 1.723 1.724

0.54232 0.54290 0.54348 0.54406 0.54464

42908 55172 64060 69575 71722

253617 294024 055391 457926 415014

1.770 1.771 1.772 1.773 1.774

0.57097 0.57154 0.57210 0.57267 0.57323

95465 43588 88521 30270 68838

857378 006965 828892 920708 873877

1.675 1.676 1.677 1.678 1.679

0.51581 0.51641 0.51700 0.51760 0.51819

31652 00020 64828 26080 83780

770298 598913 410718 450144 954038

1.725 1.726 1.727 1.728 1.729

0.54522 0.54580 0.54638 0.54696 0.54754

70504 65926 57991 46703 32067

833231 612362 645415 818639 011534

1.775 1.776 1.777 1.778 1.779

0.57380 0.57436 0.57492 0.57548 0.57605

04229 36445 65491 91370 14086

273791 699783 725143 917128 836981

1.680 1.681 1.682 1.683 1.684

0.51879 0.51938 0.51998 0.52057 0.52117

37934 88544 35615 79152 19158

151676 264786 507563 086690 201350

1.730 1.731 1.732 1.733 1.734

0.54812 0.54869 0.54927 0.54985 0.55043

14085 92761 68101 40107 08783

096876 940722 402434 334690 583501

1.780 1.781 1.782 1.783 1.784

0.57661 0.57717 0.57773 0.57829 0.57885

33643 50043 63290 73388 80341

039938 075246 486176 810034 578176

1.685 1.686 1.687 1.688 1.689

0.52176 0.52235 0.52295 0.52354 0.52413

55638 88595 18035 43961 66378

043250 796637 638312 737654 256630

1.735 1.736 1.737 1.738 1.739

0.55100 0.55158 0.55215 0.55273 0.55331

74133 36162 94872 50268 02353

988225 381584 589679 432003 721460

1.785 1.786 1.787 1.788 1.789

0.57941 0.57997 0.58053 0.58109 0.58165

84152 84824 82361 76767 68045

316024 543073 772910 513224 265821

1.690 1.691 1.692 1.693 1.694

0.52472 0.52532 0.52591 0.52650 0.52709

85289 00699 12611 21031 25962

349821 164432 840315 509983 298627

1.740 1.741 1.742 1.743 1.744

0.55388 0.55445 0.55503 0.55560 0.55618

51132 96607 38784 77665 13254

264377 860520 303111 378839 867879

1.790 1.791 1.792 1.793 1.794

0.58221 0.58277 0.58333 0.58389 0.58444

56198 41230 23145 01946 77636

526636 785747 527387 229958 366044

1.695 1.696 1.697 1.698 1.699

0.52768 0.52827 0.52886 0.52945 0.53003

27408 25373 19862 10878 98426

324136 697113 520893 891556 897950

1.745 1.746 1.747 1.748 1.749

0.55675 0.55732 0.55790 0.55847 0.55904

45556 74574 00311 22772 41960

543905 174105 519195 333437 364650

1.795 1.796 1.797 1.798 1.799

0.58500 Oi58556 0.58611 Oi58667 0.58723

50219 19698 86078 49360 09549

402422 800079 014220 494285 683961

1.700

0.53062 82510 621704

1.750

0.55961 57879 354227

1.800

0.58778 66649 021190

C-58)5

[ 1

[ I C-5814

In 10 = 2.30258 50929 940457

[C-5814I

112

ELEMENTARY

NATURAL

Table 4.2

FUNCTIONS

LOGARITHMS

x

In x

X

TRANSCENDENTAL

In x

In x

2

1.800 1.801 1.802 1.803 1.804

0.58778 0.58834 0.58889 0.58945 0.59000

66649 20661 71591 19442 64216

021190 938190 861462 211802 404319

1.850 1.851 1.852 1.853 1.854

0.61518 0.61572 0.61626 0.61680 0.61734

56390 60335 61362 59473 54671

902335 913605 239876 032227 436634

1.900 1.901 1.902 1.903 1.904

0.64185 0.64238 0.64290 0.64343 0.64395

38861 00635 59641 15883 69363

723948 062921 231986 140124 691736

1.805 1.806 1.807 1.808 1.809

0.59056 0.59111 0.59166 0.59222 0.59277

05917 44549 80116 12619 42064

848442 947937 100914 699848 131581

1.855 1.856 1.857 1.858 1.859

0.61788 0.61842 0.61896 0.61950 0.62003

46960 36343 22823 06403 87087

593985 640088 705687 916468 393070

1.905 1.906 1.907 1.908' 1.909

0.64448 0.64500 0.64553 0.64605 0.64657

20085 68052 13266 55730 95437

786643 320104 182820 260948 436106

1.810 1.811 1.812 1.813 1.814

0.59332 0.59387 0.59443 0.59498 0.59553

68452 91789 12076 29317 43516

777344 012763 207876 727140 929449

1.860 1.861 1.862 1.863 1.864

0.62057 0.62111 0.62165 0.62218 0.62272

64877 39776 11788 80916 47162

251099 601137 548753 194514 633994

1.910 1.911. 1.912 1.913 1.914

0.64710 0.64762 0.64814 0.64867 0.64919

32420 66652 98146 26904 52930

585385 581360 292095 581158 307625

1.815 1.816 1.817 1.818 1.819

0.59608 0.59663 0.59718 0.59773 0.59828

54677 62801 67894 69957 68995

168141 791016 140341 552871 359852

1.865 1.866 1.867 1.868 1.869

0.62326 0.62379 0.62433 0.62486 0.62540

10530 71024 28645 83398 35284

957789 251521 595856 066509 734258

1.915 1.916 1.917 1.918 1.919

0.64971 0.65023 0.65076 0.65128 0.65180

76226 96795 14640 29764 42170

326093 486688 635074 612465 255629

1.820

0.59883 65010 887040 0.59938 58007 454709

;.;;;' 1:823 1.824

0.60048 34956 0.59993 47988 377666 965260 0.60103 18916 521396

1.870 1.871 1.872 1.873 1.874

0.62593 0.62647 0.62700 0.62754 0.62807

84308 30472 73780 14234 51838

664953 919526 554003 619515 162304

1.92:O 1. 9i!l 1.9;!2 1.9:!3 1.924

0.65232 0.65284 0.65336 0.65388 0.65440

51860 58837 63105 64666 63522

396902 864196 481007 066427 435147

1.825 1.826 1.827 1.828 1.829

0.60157 0.60212 0.60267 0.60322 Oi60376

99870 77821 52773 24730 93694

344548 727767 958697 319583 087286

1.875 1.876 1.877 1.878 1.879

0.62860 0.62914 0.62967 0.63020 0.63073

86594 18505 47576 73807 97204

223741 840329 043718 860712 313283

1.925 1.926 1.927 1.928 1,929

0.65492 0.65544 0.65596 0.65648 0.65700

59677 53133 43894 31961 17339

397475 759338 322293 883539 235920

1.830 1.831 1.832 1.833 1.834

0.60431 0.60486 0.60540 0.60595 0.60649

59668 22656 82662 39688 93738

533296 923737 519385 575680 342731

1.880 1.881 1.882 1.883 1.884

0.63127 0.63180 0.63233 0.63286 0.63339

17768 35503 50411 62496 71761

418578 188933 631879 750154 541713

1.930 1.931 1.932 1.933 1.934

0.65752 0.65803 0.65855 0.65907 0.65959

00029 80034 57357 32002 03970

167942 463774 903263 261938 311026

1.835 1.836 1.837 1.838 1.839

0.60704 0.60758 0.60813 0.60867 0.60922

44815 92921 38062 80239 19456

065336 982987 329886 334953 221840

1.885 1.886 1.887 1.888 1.889

0.63392 0.63445 0.63498 0.63551 0.63604

78208 81842 82663 80677 75885

999741 112658 864132 233089 193725

1.935 1.936 1.937 1.938 1.'939

0.66010 0.66062 0.66114 0.66165 0.66217

73264 39888 03844 65134 23762

817451 543853 248588 685745 605148

1.840 1.841 1.842 1.843 1.844

0.60976 0.61030 0.61085 0.61139 0.61193

55716 89022 19378 46786 71251

208943 509408 331151 876862 344021

1.890 1.891 1.892 1.893 1.894

0.63657 0.63710 0.63763 0.63816 0.63869

68290 57896 44706 28722 09947

715510 763204 296865 271858 638865

1.940 1.941 :* zt; 1:944

0.66268 0.66320 0.66371 0.66423 0.66474

79730 33041 83698 31703 77060

752368 868732 691332 953030 382473

1.845 1.846 1.847 1.848 1.849

0.61247 0.61302 0.61356 0.61410 Or61464

92774 11360 27012 39732 49524

924905 806604 171029 194924 049878

1.895 1.896 1.897 1.898 1.899

0.63921 0.63974 0.64027 0.64080 0.64132

88385 64038 36909 07001 74318

343897 328301 528772 877361 301488

1.945 1.946 1.947 1.948 1.949

0.66526 0.66577 0.66628 0.66680 0.66731

19770 59837 97263 32052 64205

704096 638133 900626 203434 254238

1.850

0.61518 56390 902335 C-58)4

1.900

0.64185 38861 723948 ‘-;8’4

L950

0.66782 93725 756554

[ 1

[ I

In 10 = 2.30258 50929 940457

[ 3 C-i)3

ELEIKENTARY

TRANSCENDENTAL

NATURAL

In x

X

LOGARITHMS

In

X

113

FUNCTIONS

2

Table 4.2

In x

X

1.950 1.951 1.952 1.953 1.954

0.66782 Oi66834 0.66885 Oi66936 0.66987

93725 20616 44879 66518 85536

756554 409742 909007 945419 205910

2.000 2.001 2.002 2.003 2.004

0.69314 0.69364 0.69414 0.69464 0.69514

71805 70556 66808 60566 51832

599453 015964 930288 836812 226184

2.050 2.051 2.052 2.053 2.054

0.71783 0.71832 0.71881 0.71930 0.71978

97931 74790 49273 21380 91115

503168 902436 085231 367965 063665

1.955 1.956 1.957 1.958 1.959

0.67039 0.67090 0.67141 0.67192 0.67243

01934 15716 26884 35441 41389

373291 126256 139392 083186 624037

2.005 2.006 %*00:: 2:009

0.69564 0.69614 0.69664 0.69713 0.69763

40607 26895 10698 92018 70858

585325 397438 142011 294828 327974

2.055 2.056 2.057 2.058 2.059

0.72027 0.72076 0.72124 0.72173 0.72222

58479 23475 86106 46374 04280

481979 929187 708201 118579 456524

1.960 1.961 1.962 1.963 1.964

0.67294 Oi67345 0.67396 0.67447 0.67498

44732 45472 43611 39152 32099

424259 142092 431713 943240 322741

2.010 2.011 2.012 2.013 2.014

0.69813 0.69863 0.69912 0.69962 0.70012

47220 21107 92522 61466 27942

709844 905150 374928 576544 963706

2.060 2.061 2.062 2.063 2.064

0.72270 0.72319 0.72367 0.72416 0.72464

59828 13019 63855 12340 58476

014897 083220 947682 891148 193163

1.965 1.966 1.967 1.968 1.969

0.67549 0.67600 0.67650 0.67701 0.67752

22453 10217 95394 77986 57996

212246 249748 069220 300617 569885

I* 00:: 2:017 2.018 2.019

0.70061 0.70111 0.70161 0.70210 0.70260

91953 53502 12589 69219 23393

986463 091222 720747 314172 307004

2.065 2.066 2.067 2.068 2.069

0.72513 0.72561 0.72609 0.72658 0.72706

02264 43706 82806 19566 53987

129961 974468 996312 461827 634060

1.970 11971 1.972 11973 1.974

0.67803 Oi678ii 0.67904 Oi67955 0.68006

35427 ib281 82561 52270 19410

498971 7Oii83i 804437 404783 112898

2.020 :- 8;: 2:023 2.024

0.70309 0.70359 0.70408 0.70458 0.70507

75114 24384 71205 15581 57514

131134 214840 982797 856084 252191

2.070 2.071 2.072 2.073 2.074

0.72754 0.72803 0.72851 0.72899 0.72947

86072 15824 43243 68334 91098

772777 134471 972366 536425 073356

1.975 1.976 1.977 1.978 1.979

0.68056 0.68107 0.68158 0.68208 0.68259

83983 45993 05441 62332 16666

530852 256761 884799 005204 204287

2.025 27026 2.027 z*. 82298

0.70556 0.70606 0.70655 0.70705 0.70754

97005 34058 68674 30608 00857

585025 264916 698630 436777 289367

2.075 2.076 2.077 2.078 2.079

0.72996 0.73044 0.73092 0.73140 0.73188

11536 29653 45448 58926 70088

826616 036422 939753 770357 758759

1.980 1.981 1.982 1.983 1.984

0.68309 68447 0.6836$'?7677 0.68410 64359 0.68461 08495 0.68511 50088

064439 164139 077962 376589 626811

2.030 2.031 2.032 2.033 2.034

0.70803 0.70852 0.70902 0.70951 0.71000

57930 82825 05297 25346 42976

536960 982476 162355 462096 263682

2.080 2.081 2.082 2.083 2.084

0.73236 0.73284 0.73332 0.73380 0.73428

78937 85474 89701 91622 91238

132266 114974 927771 788349 911205

1.985 1.986 1.987 1.988 1.989

0.68561 0.68612 0.68662 0.68712 0.68763

89141 25656 59635 91082 19998

391537 229808 696798 343823 718351

2.035 2:036 2.037 2.038 2.039

0.71049 0.71098 0.71147 0.71196 0.71245

58188 70986 81372 89348 94915

945583 882763 446688 005331 923181

2.085 2.086 2.087 2.088 2.089

0.73476 0.73524 0.73572 0.73620 0.73668

88552 83565 76280 66700 54825

507648 785807 950637 203923 744287

1.990 11991 1.992 1.993 1.994

0.68813 Or68863 0.68913 0.68964 0.69014

46387 70250 91591 10412 26715

364010 820592 624065 306577 396466

0.71294 %FA! 0.71343 2:042 0.71392

2.043 2.044

98078 98838 97197 0.71441 93158 0.71490 86723

561250 277077 424738 354850 414580

2.090 2.091 2.092 2.093 2.094

0.73716 0.73764 0.73812 0.73859 0.73907

40659 24204 05462 84434 61124

767196 464965 026765 638627 483451

1.995 i;99i 1.997 1.998 1.999

0.69064 Oi69114 0.69164 0.69214 0.69264

40503 51778 60544 66802 70555

418268 892722 336782 263618 182630

2.045 2iO46 2.047 2.048 2.049

0.71539 0.71588 0.71637 0.71686 0.71735

947651 294347 791525 772614 567627

2.095 2.096 2.097 2.098 2.099

0.73955 0.74003 0.74050 0.74098 0.74146

35533 07664 77519 45099 10408

741011 587957 197829 741054 384959

2.000

0.69314 71805 599453

2.050

0.71783 97931 503168

2.100

0.74193 73447 293773

r (91

L 5 J For x>2.1 see Example 5.

77894 66675 53066 37071 18692

[ 1 In 10 = 2.30258 50929 940457 C-i)3

[C-i)3 1

114 Table

ELEMENTARY

4.3

RADIX

TRANSCENDENTAL

TABLE

OF

10 10 10 10 10 10 10 10 10

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

; 9 ;

0.00000 0.00000 0.00000

00009 99999 99950 00019 99800 00000 00029 99999 99550 00000 00049 99999 98750 00039 99200 00000

9

0.00000

00059 99999 98200 00001

; 9

0.00000 0.00000

00069 99999 97550 00079 96800 00001 00002 00089 99999 95950 00002

8"

0.00000

00199 99999 95000 00099 80000 00027 00003

8 8

0.00000 0.00000

00299 99999 55000 00090 00399 99999 20000 00213

ii 8"

0.00000 0.00000

00499 99998 75000 00599 20000 00417 00720 00799 00699 99996 99997 55000 80000 01707 01143

8

0.00000

00899 99995 95000 02430

7

0.00000

00999 99995 00000 03333

s :

0.00000 0.00000

01999 02999 99955 99980 00000 26667 90000 03999 99875 04999 99920 00004 00002 16667 13333

7

0.00000

05999 99820 00007 20000

; 7

0.00000 0.00000

6 6

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

6" i 6 6" For n>lO,ln

(1+x10-n, 99999 99999 99999 99998 99999 99995 99999 99992 99999 99987 99999 99982 99999 99975 99999 99968 99999 99959

NATURAL

In 00000 00001 00002 00003 00004 00005 00006 00007 00008

n

FUNCTIONS

LOGARITHMS

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

-In (I-210Wn) 0000~1 00000 00000 OOOCl200000 00002 oooCl3 00000 00004 00004 00000 00008 oooCl5 00000 00012 00006 00000 00018 00007 00000 00024 00008 00000 00032 00009 00000 00040

50000 00000 50000 00000 50000 00000 50000 00000 50000

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

00010 ooo;!o 00030 00040 00050 00060 00070 00080 00090

00000 00000 00000 00000 00000 00000 00000 00000 00000

00050 00200 00450 00800 01250 01800 02450 03200 04050

00000 00000 00000 00000 00000 00001 00001 00002 00002

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

00100 00200 00300 00400 00500 00600 007OO 00800 00900

00000 00000 00000 00000 00001 00001 00002 00003 00004

05000 20000 45000 80000 25000 80000 45000 20000 05000

00003 00027 00090 00213 00417 00720 01143 01707 02430

07999 99680 06999 99755 00011 00017 43333 06666 08999 99595 00024 29998

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

OlOOO 00005 02000 00020 03000 00045 040'00 00080 05000 00125 06000 00180 07000 00245 08000 00320 09000 00405

00000 00000 00000 00002 00004 00007 00011 00017 00024

03333 26667 90000 13333 16667 20000 43334 06668 30002

09999 19999 29999 39999 49999 59999 69999 79999 89999

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

10000 20000 30000 40000 50000 60000 70000 80000 90000

00033 00266 00900 02133 04166 07200 11433 17066 24300

33336 66707 00203 33973 68229 03240 39336 76907 16403

99500 98000 95500 92000 87500 82000 75500 68000 59500

00033 00266 00899 02133 04166 07199 11433 17066 24299

(l~~10-~)=f~10-~-~~210-2~to

50000 00000 50000 00000 50000 00000 50000 00000 50000

33331 66627 99798 32693 65104 96760 27331 56427 83598

25D.

00500 02000 04500 08000 12500 18000 24500 32000 40500

ELEMENTARY

RADIX

OF

NATURAL

FUNCTIONS

Table

-In

21

:

0.00001 0.00000

99998 !jOOOO 99999 00002 33333 66662 08334 66673

; 5

5 s

0.00003 0.00002 0.00004

99992 99995 130021 150008 99979 33269 75049 33538 99987 50041 66510 42292

; 9"

z 5

0.00005 0.00006 0.00008 0.00007

99975 130071 99982 50114 99676 32733 01555 11695 99968 130170 99959 65642 86809 50242 98359 73220

: 3 4 2

1 4 4 4

0.00009 0.00019 0.00029 0.00039 0.00059 0.00049

99950 99800 99550 99200 98200 98750

; 9

: t

0.00079 0.00069 0.00089

96801 70564 97551 14273 34192 33215 77369 90059 95952 42836 09300 94948

13 2 ;

;

0.00099 0.00199 0.00299 0.00399

95003 33083 53316 68094 80026 62673 05601 82538 20212 79798 55089 47881 90751 69537 45299 16106

5

;

0.00498

75415 11039 07361 21022

; 8 9

3 ; 3

0.00697 0.00598 0.00796 0.00895

20716 56137 77547 36425 46378 24209 20189 95222 81696 49176 87351 07973 97413 71471 90444 31465

;

2

0.01980 0.00995

26272 53168 03308 96179 71302 08284 82154 60291

; 5

2; 2

0.02955 0.03922 0.04879

88022 53281 07131 41544 40273 29626 26194 92009 01641 69432 00306 53744

;

;

0.06765 0.05826

86484 23975 89081 73814 77552 80526 57184 84159

9"

2

0.08617 0.07696

76962 10411 41052 36128 32498 33234 42170 13335

;

:

0.18232 0.09531

15567 04324 01798 62621 17180 93954 86004 39521

3 4

i

0.26236 0.33647

42644 67491 05203 54960 22366 21212 93050 45934

2

1

0.47000 0.40546

36292 45735 51081 08164 55365 38197 80131 09370

ii 9

1' 11

0.58778 0.53062 0.64185

82510 02119 66649 62170 39623 00818 15432 97311 38861 72394 77599 10360

1

0

0.69314

71805 59945 30941 72321

00333 32666 D8997 21326 71967 41651

115

LOGARITHMS

In (1+x10-")

n

X

‘TABLE

TRANSCENDENTAL

26673 30833 97548 93538 61554 04791

53332 06560 58785 06509 40636 42280

4.3

(l-x10-n)

0.00001 0.00002 0.00003 0.00004 0.00005 0.00006 0.00007 0.00008 0.00009

00000 00002 00004 00008 00012 00018 00024 00032 00040

50000 00002 50009 00021 50041 00072 50114 00170 50243

33333 66670 00020 33397 66822 00324 33933 67690 01640

58334 66673 25049 33538 92292 01555 61695 73221 36811

0.00010 0.00020 0.00030 0.00040 0.00050 0.00060 0.00070 0.00080 0.00090

00050 00200 00450 00800 01250 01800 02451 03201 04052

00333 02667 09002 21339 41682 72032 14393 70769 43164

35833 06673 02548 73538 29791 41555 39196 13224 14318

53335 06773 61215 20162 92719 97800 69533 63873 66419

0.00100 0.00200 0.00300 0.00400 0.00501 0.00601 0.00702 0.00803 0.00904

05003 20026 45090 80213 25418 80723 46149 21716 07446

33583 70673 20298 97538 23544 25563 36964 97264 52149

53350 07735 72181 81834 28204 01620 45987 25903 06220

01430 16511 32509 87927 30937 19350 41123 86494 55241

0.01005 03358 53501 44118 35489 0.02020 27073 17519 44840 80453 0.03045 92074 84708 54591 92613 0.04082 19945 20255 12955 45771 0.05129 32943 87550 53342 61961 0.06187 54037 18087 47179 78001 0.07257 06928 34835 43071 15733 0.08338 16089 39051 05839 47658 0.09431 06794 71241 32687 71427 0.10536 0.22314 0.35667 0.51082 0.69314 0.91629 1.20397 1.60943 2.30258

05156 35513 49439 56237 71805 07318 28043 79124 50929

57826 14209 38732 65990 59945 74155 25935 34100 94045

30122 75576 37891 68320 30941 06518 99262 37460 68401

75010 62951 26387 55141 72321 35272 27462 07593 79915

ELEMENTARY Table 4.4

TRANSCENDENTAL

EXPONENTIAL

FTJNCTIONS

FUNCTION

e-z

N

0.001 0.002 0.003 0.004

0.000

1.00000 1.00100 1.00200 iI 1.00400

00000 05001 20013 45045 80106

00000 66708 34000 03377 77341

000 342 267 026 872

1.00000 0.99900 0.99800 0.99700 0.99600

00000 04998 19986 44955 79893

00000 33374 67333 03372 43991

000 992 067 976 472

0. 003 0.006 0.007 0.008 0.009

1.00501 1.00601 1.00702 1.00803 1.00904

25208 80360 45572 20855 06217

59401 54064 66848 04273 73867

063 865 555 431 814

0.99501 0.99401 0.99302 0.99203 0.99104

24791 79640 44429 19148 03787

92682 53935 33235 37060 72883

313 265 105 630 662

0.010 0.011 0.012 0.013 0.014

1.01005 1.01106 1.01207 1.01308 1.01409

01670 07224 22888 48673 84589

84168 44719 66077 59809 38492

058 556 754 158 345

0.99004 0.98906 0.98807 0.98708 0.98609

,d337 02787 17128 41350 75442

49168 75368 61930 20287 62861

054 698 540 583 903

0.015 0.016 0.017 0.018 0.019

1.01511 1.01612 1.01714 1.01816 1.01918

30646 86854 53223 29763 16486

15718 06094 25240 89793 17408

979 822 748 761 011

0.98511 0.98412 0.98314 0.98216 0.9811'7

19396 73200 36846 10323 93622

03062 55285 34909 58300 42806

661 115 635 718 006

0.020 0.021 0.022 0.023 0.024

1.02020 1.02122 1.02224 1.02326 1.02429

13400 20516 37844 65395 03178

26755 37528 70438 47217 90621

810 653 235 475 534

0.98019 0.97921 0.97824 0.97726 0.97628

86733 89645 02350 24837 57097

06755 69459 51210 73277 57909

302 588 045 073 314

0.025 0.026 0.027 0.028 0.029

1.02531 1.02634 1.02736 1.02839 1.02942

51205 09484 78027 56844 45944

24428 73442 63489 21425 75130

841 115 392 045 820

0.97530 0.97433 0.97336 0.97238 0.97141

99120 50896 12415 83668 64644

28332 08749 24336 01246 66604

669 328 791 891 825

0.030 0.031 0.032 0.033 0.034

1.03045 1.03148 1.03251 1.03355 1.03458

45339 55038 75053 05392 46067

53516 86522 05118 41305 28117

856 716 420 472 894

0.97044 0.96947 0.96850 0.96753 0.96657

55335 55730 65820 85595 15046

48508 76025 79197 89032 37506

177 948 585 009 651

0.035 0.036 0.037 0.038 0.039

1.03561 1.03665 1.03769 1.03873 1.03977

97087 58464 30208 12328 04836

99623 90923 38157 78497 50157

260 727 074 733 831

0.96560 0.96464 0.96367 0.96271 0.96175

54162 02934 61353 29408 07091

57566 83123 49053 91199 46366

478 030 452 529 723

0.040 0.041 0.042 0.043 0.044

1.04081 1.04185 1.04289 1.04393 1.04498

07741 21055 44787 78948 23548

92388 45479 50763 50612 88443

227 549 238 586 779

0.96078 0.95982 0.95886 0.95791 0.95695

94391 91299 97805 13900 39574

52323 47798 72484 67030 73046

209 914 552 669 678

0.045 0.046 0.047 0.048 0.049

1.04602 1.04707 1.04812 1.04917 liO5022

78599 44109 20090 06553 03507

08716 56937 79655 24470 40028

943 184 638 516 148

0.95599 0.95504 0.95408 0.95313 0.95218

74818 19621 73975 37870 11296

33099 90714 90371 77504 98504

907 635 141 745 853

0.050

1.05127 10963 76024 040

For use and extension of the table see Examples 8-11. 2 -23 SW Table 7.1 for values of -e and Table 26.1 for & &

0. 95122 94245 00714 009

-- 23 e 2.

ELEMENTARY

TRANSCENDENTAL

EXPONENTIAL

Table 4.4

FUNCTION

ez

X

117

F’UNCTIONS

e-z

0.050 0.051 0.052 0.053 0.054

1.05127 1.03232 1.05337 1.05442 1.05548

10963 28932 57425 96451 46021

76024 83203 13364 19355 55080

040 913 763 907 041

0.95122 0.95027 0.94932 0.94838 0.94743

94245 86705 88668 00124 21065

00714 32426 42889 82298 01798

009 935 583 184 300

0.055 0.056 0.057 0.058 0.059

1.05654 1.05759 1.05865 1.05971 1.015077

06146 76837 58103 49957 52407

75494 36611 95500 10287 40159

286 252 087 540 012

0.94648 0.94553 0.94459 0.94364 0.94270

51479 91358 40693 99474 67691

53483 90396 66523 36798 57099

869 267 349 514 754

0.060 0.061 0.062 0.063 0.064

l.Otj183 1.015289 1.015396 1.015502 1.015609

65465 89141 23447 68392 23987

45359 87195 28033 31305 61505

622 264 669 464 244

0.94176 0.94082 0.93988 0.93894 0.93800

45335 32397 28867 34736 49995

84248 76009 91088 89133 30729

710 730 928 241 488

0.065 0.066 0.067 0.068 0.069

l.Oli715 1.015822 1.0~5929 1.07036 1.07143

90243 67171 54781 53084 62091

84192 65993 74600 78774 48346

625 321 202 366 205

0.93706 Oi93613 0.93519 0.93426 0.93332

74633 08642 52013 04735 66800

77403 91618 36776 77213 78201

433 844 558 542 958

0.070 0.071 0.072 0.073 0.074

1.0'7250 1.0'7358 1.0'7465 1.0'7573 1.0'7680

81812 12258 53440 05369 68054

54216 68357 63813 14703 96219

479 383 620 476 891

0.93239 0.93146 0.93053 0.92960 0.92867

38199 18921 08958 08300 16938

05948 27592 11205 25792 41287

229 106 732 713 187

0.075 0.076 0.077 0.078 0.079

1.0'7788 1.0'7896 1.013004 1.0:3112 1.0;3220

41508 25741 20763 26586 43220

84631 57281, 92600 70083 70314

536 889 313 133 717

0.92774 0.92681 0.92588 0.92496 0.92403

34863 62065 98536 44265 99244

28552 59382 06495 43539 45086

892 237 377 280 807

0.080 0.081 0.082 0.083 0.084

1.013328 1.013437 1.0~3545 1.013654 1.013762

70676 08965 58098 18085 88938

74958 66760 29549 48238 08826

554 341 059 061 156

0.92311 0.92219 0.92127 0.92035 0.91943

63463 36914 19586 11472 12560

86635 44608 96348 20124 95124

783 072 654 706 674

0.085 0.086 0.087 0.088 0.089

1.0;3871 1.013980 1.0'3089 1.0'3198 1.0'3308

70666 63283 66797 81220 06563

98398 05128 18277 28197 26330

696 660 747 460 201

0.91851 Oi91759 0.91667 0.91576 0.91484

22844 42312 70956 08767 55735

01457 20150 33152 23325 74452

356 982 295 631 003

0.090 0.091 0.092 0.093 0.094

1.0'3417 1.0'3526 1.0'3636 1.0'3746 1.0'3855

42837 90052 48220 17352 97459

05210 58465 80816 68081 17173

358 401 975 994 736

0.91393 0:91301 0.91210 0.91119 0.91028

11852 77108 51495 35002 27622

71228 99265 45090 96140 40766

187 803 403 557 940

0.095 0.096 0.097 0.098 0.099

1.0'3965 1.10075 1.113186 1.11296 1.10406

88551 90639 03736 27851 62995

26102 93978 21010 08507 58881

942 912 606 743 902

0.90937 0.90846 0.90755 0.90664 0.90574

29344 40160 60061 89037 27080

68231 68706 33272 53920 23548

420 150 654 921 496

0.100

1.13517 09180 75647 625

[c-J)1 1

0.90483 74180 35959 573

[(-;I1 1

118

ELEMENTARY

Table

TRANSCENDENTAL

4.4

EXPONENTIAL

FUNCTIONS

FUNCTION e-z

ez

X

0.101 0.102 0.103 0.104

0.100

1.10517 1.10627 1.10738 1.10849 1.10960

09180 75647 625 66417 63423 521 34717 27933 371 14090 76007 230 04549 15582 540

0.90483 0.90393 0.90302 0.90212 0.90122

74180 35959 573 30328 85864 089 95516 68876 819 69734 81516 470 52974 21204 780

0.105 0.106 0.107 0.108 0.109

1.11071 1.11182 1.11293 1.11404 1.11516

06103 55705 232 18765 06530 839 42544 79325 605 77453 86467 594 23503 41447 807

0.90032 0.89942 0.89852 0.89762 0.89673

45225 86265 613 46480 75924 059 56729 90305 534 75964 30434 876 04174 98235 450

0.110 0.111 0.112 0.113 0.114

1.11627 1.11739 1.11851 1.11963 1.12075

80704 58871 292 49068 54458 258 28606 45045 196 19329 48585 987 21248 84153 031

0.89583 0.89493 0.89404 0.89315 0.89225

41352 96528 251 87489 29031 000 42575 00357 257 06601 16015 519 79558 82408 325

0.115 0.116 0.117 0.118 0.119

1.12187 1.12299 1.12411 1.12524 1.12636

34375 71938 354 58721 33254 738 94296 90536 839 41113 67342 307 99182 88352 913

0.89136 0.89047 0.88958 0.88869 0.88780

61439 06831 368 52232 97472 599 51931 63411 334 60526 14617 364 78007 61950 067

0.120 0.121 0.122 0.123 0.124

1.12749 1.12862 1.12975 1.13088 1.13201

68515 79375 671 49123 67343 967 41017 80318 682 44209 47489 324 58709 99175 153

0.88692 0.88603 0.88514 0.88426 0.88337

04367 17157 516 39595 92875 591 83685 02627 096 36625 60820 866 98408 82750 886

0.125 0.126 0.127 0.128 0.129

1.13314 1.13428 1.13541 1.13655 1.13769

84530 66826 317 21682 83024 976 70177 81486 442 30026 97060 307 01241 65731 582

0.88249 0.88161 0.88073 0.87985 0.87897

69025 84595 403 48467 83416 046 36725 97156 940 33791 44643 827 39655 45583 178

0.130 0.131 0.132 0.133 0.134

1.13882 1.13996 1.14110 1.14224 1.14339

83833 24621 831 77813 11990 306 83192 67235 091 99983 30894 235 28196 44646 898

0.87809 0.87721 0.87634 0.87546 0.87459

54309 20561 324 77143 91043 564 09950 79373 297 50921 08771 138 00646 03334 043

0.135 0.136 0.137 0.138 0.139

1.14453 1.14568 1.14682 1.14797 1.14912

67843 51314 488 18935 94861 807 81485 20398 195 55502 74178 672 41000 03605 088

0.87371 0.87284 0.87197 0.87109 0.87022

59116 88034 434 26324 88719 322 02261 32109 436 86917 45798 347 80284 58251 595

0.140 0.141 0.142 0.143 0.144

1.15027 1.15142 1.15257 1.15372 1.15488

37988 57227 268 46479 84744 161 66485 37004 992 98016 66010 407 41085 24913 632

0.86935 0.86848 0.86762 0.86675 0.86588

82353 98805 820 93116 97667 890 12564 85914 032 40688 95488 962 77480 59205 017

0.145 0.146 0.147 0.148 0.149

1.15603 1.15719 1.15835 1.15951 1.16067

95702 68021 623 61880 50796 218 39630 29855 297 28963 62973 936 29892 09085 563

0.86502 22931 10741 288 0.86415 77031 84642 755 0.86329 39774 16319 421 0.862431 11149 42045 443 0.86156 91148 98958 277

0.150

1.16183 42427 28283 123

0.86070 79764 25057 807

[c-p1 1

[(-l’l1

ELEMENTARY

TRANSCENDENTAL EXPONENTIAL

X

FUNCTIONS

119

FUNCTION

Table

4.4

e-1

ez

0.150 0.151 0.152 0.153 0.154

1.16183 1.16299 1.16416 1.16532 1.16649

42427 66580 02364 49789 08867

28283 81820 32112 42737 78439

123 230 335 886 490

0.86070 0.85984 0.85898 0.85812 0.85727

79764 76986 82807 97218 20210

25057 59205 41123 11393 11457

807 488 482 800 440

0.155 0.156 0.157 0.158 0.159

1.16765 1.16882 1.16999 1.17116 1.17233

79611 62030 56139 61947 79466

05125 89869 00913 07669 80717

080 080 572 465 662

0.85641 0.85555 0.85470 0.85384 0.85299

51774 91903 40588 97819 63589

83613 71018 17685 68481 69131

531 473 083 735 511

0.160 0.161 0.162 0.163 0.164

1.17351 1.17468 1.17586 1.17703 1.17821

08709 49688 02413 66896 43150

91810 13871 20999 88467 92722

235 592 654 025 171

0.85214 0.85129 0.85044 0.84959 0.84874

37889 20711 12045 11884 20218

66211 07151 40232 14590 80206

338 144 998 263 741

0.165 0.166 0.167 0.168 0.169

1.17939 iIi8Oi7 1.18175 ii18293 1.18412

31187 31017 42653 66106 01389

11390 23276 08361 47810 23969

594 011 533 843 378

0.84789 0.84704 0.84619 0.84535 Oi84450

37040 62341 96113 38346 89033

87915 89399 37188 84658 86034

828 660 270 733 326

0.170 0.171 0.172 0.173 0.174

1.18530 1.18649 1.18767 1.18886 1.19005

48513 07490 78332 61050 55658

20365 21711 13905 84032 20362

514 746 874 188 660

0.84366 0.84282 0.84197 0.84113 0.84029

48165 15734 91731 76148 68976

96383 71619 68499 44623 58431

682 939 904 201 438

0.175 0.176 0.177 0.178 0.179

1.19124 1.19243 1.19363 1.19482 1.19602

62166 80586 10931 53212 07441

12358 50669 27138 34800 67883

122 468 834 796 563

0.83945 0.83861 0.83777 0.83694 Oi83610

70207 79833 97845 24234 58993

69207 37074 22993 88768 97035

358 003 869 073 511

0.180 0.181 0.182 0.183 0.184

1.19721 ii19841 1.19961 1.20081 1.20201

73631 51792 41938 44080 58230

21810 93199 79868 80830 96301

165 657 311 812 462

0.83527 0.83443 0.83360 0.83276 0.83193

02114 53586 13404 81557 58038

11272 95789 15735 37090 26671

021 549 309 951 728

0.185 0.186 0.187 0.188 0.189

1.20321 1.20442 1.20562 1.20683 1.20804

84401 22603 72850 35153 09524

27695 77629 49924 49605 82901

376 686 742 317 811

0.83110 0.83027 0.82944 0.82861 0.82778

42838 35949 37363 47072 65066

52125 81932 85403 32680 94733

659 701 915 634 637

0.190 0.191 0.192 0.193 0.194

1.2'0924 iIi1045 1.21167 1.2'1288 1.2'1409

95976 94520 05169 27935 62829

57251 81299 64900 19119 56233

458 533 562 527 085

0.82695 0.82613 0.82530 0.82448 0.82365

91339 25881 68684 19741 79042

43362 51193 91682 39108 68576

318 854 387 186 832

0.195 0.196 0.197 0.198 0.199

1.2'1531 1.211652 1.211774 1.2'1896 1.212018

09864 69053 40407 23938 19658

89730 34316 05908 21642 99872

774 229 396 747 499

0.82283 ii82201 0.82119 0.82036 0.81954

46580 22346 06333 98531 98933

56018 78186 12657 37831 32925

384 562 919 021 626

0.200

1.;!2140 27581 60169 834

[ (-Y1

0.81873 07530 77981 859

[c-y1

120

ELEMENTARY

TRANSCENDENTAL EXPONENTIAL

Table 4.4

FUNCTIONS

FUNCTION e-z

X

0.200 0.201 0.202 0.203 0.204

1.22140 1.22262 1.22384 1.22507 1.22629

27581 47718 80081 24682 81534

60169 23327 11358 47499 56210

834 112 099 185 607

0.81873 0.81791 0.81709 0.81627 Oi81546

07530 24315 49279 82414 23711

77981 53859 42236 25609 87292

859 397 649 934 668

0.205 0.206 0.207 0.208 0.209

1.22752 1.22875 1.22998 1.23121 1.23244

50649 32039 25717 31695 49985

63177 95312 80752 48867 30254

678 005 723 721 869

0.81464 0.81383 0.81301 0.81220 0.81139

73164 30762 96499 70367 52356

11414 82920 87570 11939 43411

545 720 998 015 427

0.210 0.211 0.212 0.213 0.214

1.23367 1.23491 1.23614 1.23738 1.23862

80599 23550 78850 46512 26547

56743 61394 78503 43600 93452

251 396 512 719 285

0.81058 0.80977 0.80896 0.80815 0.80734

42459 40668 46975 61372 83850

70187 81276 66499 16488 22681

100 291 845 379 475

0.215 0.216 0.217 0.218 0.219

1.23986 1.24110 1.24234 1.24358 1.24483

18969 23790 41021 70676 12766

66061 00671 37764 19061 87531

862 728 020 978 187

0.80654 0.80573 0.80492 0.80412 0.80332

14401 53018 99693 54416 17181

77326 73479 05001 66559 53626

874 662 467 655 521

0.220 0.221 0.222 0.223 0.224

1.24607 1.24732 1.24857 1.24982 1.25107

67305 34305 13778 05737 10194

87380 64064 64283 35983 28362

820 879 447 926 294

0.80251 0.80171 0.80091 0.80011 0.79931

87979 66802 53643 48492 51343

62478 90195 34659 94554 69365

483 284 186 165 114

0.225 0.226 0.227 0.228 0.229

1.25232 l.25357 1.25482 1.25608 1.25734

27161 56652 98679 53254 20390

91864 78186 40279 32344 09839

345 948 295 151 113

0.79851 0.79771 0.79692 0.79612 0.79532

62187 81016 07822 42598 85335

59377 65674 90139 35453 05093

043 '274 647 721 973

0.230 0.231 0.232 0.233 0.234

1.25860 1.25985 1.26111 1.26238 1.26364

00099 92394 97288 14793 44922

29477 4v31 28329 27261 07777

863 426 426 349 797

0.79453 0.79373 0.79294 0.79215 0.79136

36025 94660 61233 35735 18158

03334 35242 06683 24314 95583

008 758 687 003 855

0.235 0.236 0.237 0.238 0.239

1.26490 1.26617 1.26744 1.26870 1.26997

87687 43101 11177 91928 85365

32891 66879 75283 24910 83836

756 857 640 818 547

0.79057 0.78978 0.78899 0.78820 0.78741

08496 06739 12880 26910 48823

28735 32802 17609 93770 72687

550 754 706 426 922

0.240 0.241 0.242 0.243 0.244

1.27124 1.27252 1.27379 1.27506 1.27634

91503 10353 41928 86241 43304

21404 08229 16194 18459 89454

692 095 849 570 665

0.78662 0.78584 0.78505 0.78427 0.78348

78610 16263 61775 15137 76342

66553 88345 51829 71556 62862

409 515 496 451 532

0.245 0.246 0.247 0.248 0.249

1.27762 1.27889 1.28017 1.28145 1.28274

13132 95735 91127 99321 20330

04886 41738 78269 94021 69811

611 230 966 162 341

0.78270 0.78192 0.78114 0.78035 0.77957

45382 22249 06935 99432 99733

41868 25477 31376 78034 84700

168 270 458 273 396

0.250

1.28402 54166 87741 484 C-i)2

[

1

0.77880 07830 71404 868

[c-y1

ELEMENTARY

TRANSCENDENTAL

EXPONENTIAL

FUNCTION

ez

X

121

FUNCTIONS Table

4.4

e-2

0.250 0.251 0.252 0.253 0.254

1.28402 1.26531 1.28659 1.28788 ii28917

54166 00843 60372 32768 18042

87741 31195 84840 34630 67804

484 317 591 366 299

0.77880 0.77802 0.77724 0.77646 0.77569

07830 23715 47380 78818 18020

71404 58957 68946 23737 46476

868 312 150 828 034

0.255 0.256 0.257 0.258 0.259

1.29046 1.29175 1.23304 1.29433 1.29563

16208 27279 51267 88186 38048

72889 39703 59353 24237 28048

931 974 603 745 373

0.77491 0.77414 0.77336 0.77259 0.77182

64979 19687 82137 52321 30230

61080 92248 65449 06928 43703

928 360 096 045 483

0.260 0.261 0.262 0.263 0.264

1.29693 1.29822 1.29952 1.30082 1.30212

00866 76654 65424 67189 81963

65771 33689 29381 51724 00894

798 967 755 266 131

0.77105 0.77028 0.76951 0.76874 0.76797

15858 09196 10237 18973 35396

03566 15079 07575 11160 56706

284 142 806 303 173

0.265 0.266 0.267 0.268 0.269

1.30343 1.30473 1.30604 1.30734 1.30865

09757 50586 04463 71400 51410

78368 86927 30654 14936 46466

808 883 372 028 646

0.76720 0.76643 0.76567 0.76490 0.76414

59499 91275 30714 77811 32556

75855 01019 65373 02864 48198

698 133 938 015 937

0.270 0.271 0.272 0.273 0.274

1.30996 1.31127 1.31258 1.31390 1.31521

44507 50703 70013 02448 48022

33247 84587 11108 24739 38724

364 979 252 218 500

0.76337 0.76261 0.76185 0.76109 0.76033

94943 64964 42610 27876 20752

36853 05065 89837 28934 60882

186 386 543 278 066

0.275 0.276 0.277 0.278 0.279

1.31653 ii31784 1.31916 1.32048 1.32180

06748 78640 63710 61972 73438

67621 27303 34958 09095 69539

623 324 873 387 151

0.75957 0.75881 0.75805 0.75729 0.75653

21232 29307 44971 68215 99032

24968 61241 10508 14335 15047

476 409 337 547 380

0.280 0.281 0.282 0.283 0.284

1.32312 li32445 1.32577 1.32710 1.32843

98123 36039 87199 51618 29307

37436 35257 86792 17157 52794

936 318 007 164 731

0.75578 0.75502 0.75427 0.75351 0.75276

37414 83354 36845 97878 66447

55725 80208 33088 59717 06196

472 002 932 250 222

0.285 0.286 0.287 0.288 0.289

1.32976 ii33109 1.33242 1.33375 1.33509

20281 24552 42134 73041 17285

21473 52291 75675 23384 28508

753 710 843 488 403

0.75201 0.75126 0.75051 0.74976 0.74901

42543 26159 17288 15922 22054

19382 46886 37067 39041 02669

630 026 974 301 348

0.290 0.291 0.292 0.293 0.294

1.33642 1.33776 1.33910 1.34044 1.34178

74880 45839 30176 27904 39036

25472 50035 39293 31681 66971

103 199 724 481 373

0.74826 0.74751 0.74676 0.74602 0.74527

35675 56780 85359 21406 64914

78565 18091 73357 97221 43288

215 016 128 444 626

0.295 0.296 0.297 0.298 0.299

1.34312 1.34447 1.34581 1.34716 1.34850

63586 01568 52994 17878 96234

86276 32052 48097 79554 72910

747 735 594 052 654

0.74453 0.74378 0.74304 0.74230 0.74155

15874 74280 40123 13397 94094

65909 20179 61939 47774 35010

357 599 843 369 502

0.300

1.34985

88075

76003

104

0.74081

82206

[c-p1

866

[1 1 C-l)2

81717

122

ELEMENTARY Table

TRANSCENDENTAL EXPONENTIAL

4.4

FUNCTIONS

FUNCTION e-2

ez

X

0.300 0.301 0.302 0.303 0.304

1.34985 1.35120 1.35256 1.35391 1.35526

88075 76003 104 93415 38015 618 12267 09482 272 44644 42288 348 90560 89671 692

0.74081 0.74007 0.73933 0.73859 0.73786

82206 81717 866 77727 46707 647 80648 89531 848 90963 70482 549 08664 50591 171

0.305 0.306 0.307 0.308 0.309

1.35662 1.35798 1.35934 1.36070 1.36206

50030 06224 066 23065 47892 497 09680 71980 642 09889 37150 137 23705 03421 961

0.73712 0.73638 0.73565 0.73491 0.73418

33743 91627 732 66194 56100 112 06009 07253 313 53180 09068 726 07700 26263 391

0.310 0.311 0.312 0.313 0.314

1.36342 1.36478 1.36615 1.36752 1.36888

51141 32177 794 92211 86161 378 46930 29479 880 15310 27605 258 97365 47375 624

0.73344 0.73271 0.73198 0.73124 0.73051

69562 24289 264 38758 69332 482 15282 28312 628 99125 68882 001 90281 59424 881

0.315 0.316 0.317 0.318 0.319

1.37025 1.37163 1.37300 1.37437 1.37575

93109 56996 611 02556 26042 743 25719 25458 804 62612 27561 208 13249 06039 370

0.72978 0.72905 0.72833 0.72760 0.72687

88742 69056 797 94501 67623 797 07551 25701 720 27884 14595 463 55493 06338 254

0.320 0.321 0.322 0.323 0.324

1.37712 1.37850 1.37988 1.38126 1.38264

77643 35957 085 55808 93753 895 47759 57246 476 53509 05630 003 73071 19479 542

0.72614 0.72542 0.72469 0.72397 0.72325

90370 73690 925 32509 90141 181 81903 29902 880 38543 67915 300 02423 79842 419

0.325 0.326 0.327 0.328 0.329

1.38403 1.38541 1.38680 1.38818 1.38957

06459 80751 421 53688 72784 617 14771 80302 136 89722 89412 403 78555 87610 642

0.72252 73536 42072 189 Oi72180 51874 31715 812 0.72108 37430 26607 016 0.72036 30197 05301 338 0.71964 30167 47075 395

0.330 0.331 0.332 0.333 0.334

1.39096 1.39235 1.39375 1.39514 1.39654

81284 63780 266 97923 08194 268 28485 12516 609 72984 69803 608 31435 74505 339

0.71892 0.71820 0.71748 0.71677 0.71605

37334 31926 170 51690 40570 286 73228 54443 294 01941 55698 947 37822 27208 486

0.335 0.336 0.337 0.338 0.339

1.39794 1.39933 1.40073 1.40214 1.40354

03852 22467 023 90248 10930 424 90637 38535 249 05034 05320 540 33452 12726 081

0.71533 0.71462 0.71390 0.71319 0.71248

80863 52559 924 31058 16057 326 88399 02720 095 52878 98282 260 24490 89191 756

0.340 0.341 0.342 0.343 0.344

1.40494 1.40635 1.40776 1.40916 1.41057

75905 63593 797 32408 62169 155 02975 14102 572 87619 26450 817 86355 07678 418

0.71177 0.71105 0.71034 0.70963 0.70892

03227 62609 715 89082 06409 751 82047 09177 248 82115 60208 649 89280 49510 748

0.345 0.346 0.347 0.348 0.349

1.41198 1.41340 1.41481 1.41623 1.41764

99196 67659 075 26158 17677 066 67253 70428 658 22497 40023 522 91903 41986 146

0.70822 0.70751 0.70680 0.70609 0.70539

03534 67799 973 24871 06501 685 53282 57749 463 88762 14384 398 31302 69954 390

0.350

1.41906 75485 93257 248

[(-;I21

0.70468 80897 18713 434

[

C-29

1

ELEMENTARY

TRANSCENDENTAL EXPONENTIAL

FUNCTIONS

FUNCTION

e=

X

1.23 Table

4.4

e-=

0.350 0.351 0.352 0.353 0.354

1.41906 1.42048 1.42190 1.42333 1.42475

75485 73259 85237 11434 51864

93257 12195 18577 33601 79888

248 200 438 886 380

0.70468 0.70398 0.70328 0.70257 0.70187

80897 37538 01219 71933 49673

18713 55620 76340 77241 55394

434 921 929 521 037

0.355 0.356 0.357 0.358 0.359

1.42618 1.42760 1.42903 1.43046 1.43189

06542 75482 58698 56204 68015

81480 63844 53876 79897 71658

082 915 979 983 672

0.70117 0.70047 0.69977 0.69907 0.69837

34432 26202 24977 30750 43513

08572 35252 34611 06525 51573

398 !399 008 666 587

0.360 0.361 0.362 0.363 0.364

1.43332 1.43476 1.43619 1.43763 1.43907

94145 34608 89419 58592 42141

60340 78555 60351 41209 58046

258 848 880 556 276

0.69767 0.69697 0.69628 0.69558 0.69489

63260 89984 23678 64334 11947

71031 66872 41770 99095 42910

057 738 967 062 621

0.365 0.366 0.367 0.368 0.369

1.44051 1.44195 1.44339 1.44484 1.44628

40081 52426 79191 20389 76036

49217 54516 15177 73879 74739

078 071 881 090 677

0.69419 0.69350 0.69280 0.69211 0.69142

66508 28012 96450 71816 54104

77978 09755 44391 88730 50308

831 768 707 425 508

0.370 0.371 0.372 0.373 0.374

1.44773 1.44918 1.45063 1.45208 1.45353

46146 30733 29812 43398 71504

63324 86644 93158 32775 56852

462 554 799 223 487

0.69073 0.69004 0.68935 0.68866 0.68797

43306 39415 42425 52328 69118

37354 58789 24222 43955 28979

660 010 423 806 422

0.375 0.376 0.377 0.378 0.379

1.45499 1.45644 1.45790 1.45936 1.46082

14146 71337 43093 29428 30357

18201 71086 71225 75796 43431

336 052 910 632 842

0.68728 0.68660 0.68591 0.68523 0.68454

92787 23330 60738 05006 56126

90972 42301 96020 65870 66278

199 040 141 297 222

0.380 0.381 0.382 0.383 0.384

1.46228 1.46374 1.46521 1.46667 1.46814

45894 76054 20851 80300 54416

34224 09728 32959 68398 81989

532 512 881 485 380

0.68386 0.68317 0.68249 0.68181 0.68113

14092 78896 50532 28992 14271

12355 19899 05390 85990 79547

858 696 084 553 125

0.385 0.386 0.387 0.388 0.389

1.46361 1.47108 1.47255 1.47402 1.47550

43214 46708 64912 97842 45513

41144 14743 73135 88141 33054

302 133 370 592 939

0.68045 0.67977 0.67909 0.67841 0.67773

06362 05256 10949 23432 42700

04587 80321 26636 64104 13971

638 060 810 077 142

0.390 0.391 0.392 0.393 0.394

1.47698 1.47845 1.47993 1.48141 1.48290

07938 85134 77114 83893 05486

82642 13147 02288 29264 74753

577 180 401 352 084

0.67705 0.67638 0.67570 0.67502 0.67435

68744 01560 41139 87475 40562

98164 39289 60626 86133 40444

700 177 058 209 198

0.395 0.396 0.397 0.398 0.399

1.48438 1.48586 1.48735 1.48884 1.49033

41909 93175 59300 40299 36186

20914 51389 51306 07277 07402

066 667 642 615 565

0.67368 0.67300 0.67233 0.67166 0.67099

00392 66959 40256 20276 07013

48867 37386 32657 62009 53445

624 438 274 771 901

0.400

1.49182

46976

41270

318

0.67032

00460.35639 C-l)9

[ 1 c-y

1 1

301

124

ELEMENTARY

TRANSCENDENTAL

Table 4.4

EXPONENTIAL

FUNCTIONS

FUNCTION

ez

X

e+

0.400 0.401 0.402 0.403 0.404

1.49182 1.49331 1.49481 1.49630 1.49780

46976 72684 13326 68916 39469

41270 99960 76042 63582 58138

318 030 686 585 840

0.67032 0.66965 0.66898 0.66831 0.66764

00460 00610 07456 20993 41212

35639 37934 90346 23560 68928

301 596 733 309 902

0.405 0.406 0.407 0.408 0.409

1.49930 1.50080 1.50230 1.50380 1.50531

25000 25524 41056 71611 17204

56766 58019 61950 70111 85559

870 898 452 860 754

0.66697 0.66631 0.66564 0.66497 0.66431

68108 01674 41903 88788 42323

58474 24886 01521 22401 22216

400 338 227 888 786

0.410 0.411 0.412 0.413 0.414

1.50681 1.50832 1.50983 1.51134 1.51285

77851 53565 44363 50259 71268

12853 58058 28744 33993 84394

578 082 838 742 526

0.66365 0.66298 0.66232 0.66166 0.66100

02501 69316 42760 22828 09512

36319 00727 52122 27848 65912

366 386 256 372 454

0.415 0.416 0.417 0.418 0.419

1.51437 1.51588 1.51740 1.51892 1.52044

07406 58688 25129 06744 03547

92048 70568 35084 02239 90196

265 894 718 927 115

0.66034 0.65968 0.65902 0.65836 0.65770

02807 02704 09199 22284 41952

04982 84389 44120 24827 67816

886 050 673 158 932

0,420 0,421 0.422 0.423 0.424

1.52196 1.52348 1.52500 1.52653 1.52806

15556 42784 85246 42959 15937

18633 08753 83279 66456 84057

796 926 422 685 126

0.65'704 0.65639 0.65573 0.65507 0.65442

68198 01014 40393 86331 38819

15056 09171 93441 11806 08858

782 201 728 293 560

0.425 0.426 0.427 0.428 0.429

1.52959 1.53112 1.53265 1.53418 1.53572

04196 07751 26617 60809 10343

63378 33247 24018 67579 97349

690 382 802 666 347

0.65376 0.65311 0.65246 0.65181 0.65115

97851 63421 35522 14147 99291

29847 20675 27900 98731 81032

271 593 462 930 515

0.430 0.431 0.432 0.433 0.434

1.53725 1.53879 1.54033 1.54187 l.54341

75235 55499 51151 62207 88681

48281 56865 61127 00632 16487

402 110 008 428 038

0.65050 0.64985 0.64920 0.64856 0.64791

90947 89107 93766 04918 22554

23316 74749 85147 04976 85350

545 506 398 075 604

0.435 0.436 0.437 0.438 0.439

1.54496 1.54650 1.54805 1.54960 1.55115

30589 87947 60770 49074 52873

51338 49377 56340 19508 87714

384 427 096 826 108

0.64726 0.64661 0.64597 0.64532 0.64468

46670 77259 14314 57828 07796

78034 35439 10624 57294 29801

611 635 479 565 285

0.440 0.441 0.442 0.443 0.444

1.55270 1.55426 1.55581 1.55737 1.55893

72185-11336 07023 42305 57404 34107 23343 41779 04856 21915

042 879 580 367 277

0.64403 0.64339 0.64274 0.64210 0.64146

64210 27065 96354 72070 54208

83141 72956 55531 87795 27319

359 185 200 233 863

0.445 0.446 0.447 0.448 0.449

1.56049 1.56205 1.56361 1.56517 1.56674

01958 14665 42992 86956 46571

719 035 055 663 356

0.64082 0.64018 0.63954 0.63890 0.63826

42760 37720 39082 46840 60986

32318 61647 74800 31916 93768

776 123 880 208 809

0.450

1.56831

21854 90168 811

0.63762

81516 21773 293

32666 33744 86418 53521 99451

1 1 t-p

IIc-y1

ELEMENTARY

TRANSCENDENTAL

EXPONENTIAL 5

125

FUNCX’IONS

FUNCTION

Table

ez

4.4

ecz

0.450 0.451 0.452 0.453 0.454

1.56831 1.56988 1.57145 1.57302 1.57459

21854 90168 811 12820 93202 449 19485 77649 003 41865 14175 089 79974 75018 775

0.63762 0.63699 0.63635 0.63571 0.63508

81516 21773 293 08421 77982 535 41697 25087 037 81336 26414 293 27332 45928 153

0.455 0.456 0.457 0.458 0.459

1.57617 1.57775 1.57932 1.53090 1.53249

33830 33991 152 03447 66477 911 88842 49440 916 90030 61419 781 07027 82533 449

0.63444 0.63381 0.63318 0.63254 0.63191

79679 48228 182 38370 98549 030 03400 62759 794 74762 07363 387 52448 99495 898

0.460 0.461 0.462 0.463 0.464

1.53407 39849 94481 775 1.55565 88512 80547 101 1.5~372453032 25595 846 1.5'3883 33424 16080 087 1.5'3042 29704 40039 147

0.63128 0.63065 0.63002 0.62939 0.62876

36455 06925 969 26773 98054 154 23399 41912 291 26325 08162 872 35544 67098 411

0.465 0.466 0.467 0.468 0.469

1.5'3201 41888 87101 182 1.5'3360 69993 48484 772 1.5'3520 14034 17000 511 1.5'3679 74026 87052 601 1.59839 49987 54640 444

0.62813 0.62750 0.62688 0.62625 0.62562

51051 89640 814 72840 47340 750 00904 12377 027 35236 57555 956 75831 56310 730

0.470 0.471 0.472 0.473 0.474

1.5'3999 41932 17360 241 1.61)159 49876 74406 589 1.60319 73837 26574 077 1.61148013829 76258 891 1.60640 69870 27460 416

0.62500 0.62437 0.62375 0.62313 0.62250

22682 82700 796 75784 11411 229 35129 17752 104 00711 77657 876 72525 67686 754

0.475 0.476 0.477 0.478 0.479

1.60801 41974 85782 835 1.60962 30159 58436 741 1.6X123 34440 54240 740 1.6:!284 54833 83623 064 1.6:1445 91355 58623 174

0.62188 0.62126 0.62064 0.62002 0.61940

50564 65020 075 34822 47461 685 25292 93437 314 21969 81993 957 24846 92799 250

0.480 0.481 0.482 0.483 0.484

l-6:.607 1.6:.769 1.6:.930 1.6:!092 1.6:!255

44021 92893 382 12849 01700 456 97853 01927 238 99050 12074 265 16456 52261 382

0.61878 0.61816 0.61754 0.61692 0.61631

33918 06140 853 49177 02925 827 70617 64680 018 98233 73547 436 32019 12289 639

0.485 0.486 0.487 0.488 0.489

1.6:!417 1.6;!579 1.6;!742 1.6;!905 1.6:,068

50088 44229 364 99962 11341 538 66093 78585 406 48499 72574 272 47196 21548 865

0.61569 0.61508 0.61446 0.61385 0.61323

71967 64285 113 18073 13528 659 70329 44630 776 28730 42817 043 93269 93927 508

0.490 0.491 0.492 0.493 0.494

1.6:1231 62199 55378 970 1.63394 93526 05565 057 1.63558 41192 05239 912 1.6:,722 05213 89170 270 1.6:)885 85607 93758 453

0.61262 0.61201 0.61140 0.61079 0.61018

63941 84416 069 40740 01349 867 23658 32408 668 12690 65884 251 07830 90679 799

0.495 0.496 0.497 0.498 0.499

1.64049 1.64213 1.64378 1.64542 1.64707

82390 57044 002 95578 18705 315 25187 20061 292 71234 04072 971 33735 15345 173

0.60957 09072 96309 287 Oi60896 16410 72896 868 0.60825 29838 11176 269 Oi60774 49349 02490 178 0.60713 74937 38789 634

0.500

1.61187212707 00128 147

0.60653 06597 12633 424

1(72 1

[ c-y1

ELEMENTARY

Table

TRANSCENDENTAL

4.4

EXPONENTIAL

FUNCTIONS

FUNCTION

e2

X

e-x

0.500 0.501 0.502 0.503 0.504

1.64872 1.65037 1.65202 1.65367 1.65532

12707 08166 20128 48611 93631

00128 06319 83464 82760 57054

147 214 418 175 920

0.6065:3 0.60592 0.60531 0.60471 0.60410

06597 44322 88106 37943 93828

12633 17187 46224 94122 55864

424 470 228 075 709

0.505 0.506 0.507 0.508 0.509

1.65698 1.65864 1.66030 1.66196 li66362

55204 33347 28076 39409 67361

60850 50305 83232 19105 19058

766 156 516 918 736

0.603510 0.602910 0.60229 0.6016'3 0.60109

55754 23715 97704 77717 63747

27040 03842 83065 62109 38975

541 093 390 362 237

0.510 0.511 0.512 0.513 0.514

1.66529 1.66695 1.66862 1.67029 1.67196

11949 73190 51101 45698 56998

45886 64047 39666 40535 36112

308 601 871 333 826

0.60049 0.59989 0.5992'3 0.5986'3 0.5980'3

55788 53833 57878 67916 83940

12265 81185 45538 05729 62761

943 502 434 153 369

0.515 0.516 0.517 0.518 0.519

1.67363 1.67531 1.67698 1.67866 1.68034

85017 29773 91283 69562 64627

97529 97587 10762 13205 82744

486 414 348 342 439

0.59750 0.596910 0.596310 0.59571 0.59511

05946 33926 67876 07789 53658

18237 74358 33920 00321 77550

489 019 965 238 053

0.520 0.521 0.522 0.523 0.524

1.68202 1.68371 1.68539 1.68708 1.68876

76496 05186 50712 13093 92344

98886 42818 97408 47211 78463

347 123 851 326 738

0.59452 0.59392 0.59333 0.5927.3 0.59214

05479 63245 26951 96589 72156

70194 83436 23052 95412 07481

339 138 015 460 294

0.525 0.526 0.527 0.528 0.529

1.69045 1.69215 1.69384 1.69553 1.69723

88483 01527 31492 78396 42254

79091 38708 48618 01819 93001

359 232 855 881 803

0.5915s 0.59096 0.5903'7 0.58978 0.58919

53643 41046 34359 33576 38690

66815 81562 60463 12850 48643

082 533 912 450 749

0.530 0.531 0.532 0.533 oI534

1.69893 1.70063 1.70233 1.70403 1.70574

23086 20906 35733 67583 16474

18550 76549 66781 90727 51574

654 702 146 817 883

0.58860 0.5880:1 0.58742 0.58684 0.5862!5

49696 66589 89361 18008 52523

78355 13085 64523 44946 67219

196 372 463 670 626

0.535 0.536 0.537 0.538 0.539

1.70744 1.70915 1.71086 1.71257 1.71429

82422 65445 65559 82781 17130

54211 05232 12940 87347 40175

545 748 887 510 036

0.58566 0.58508 0.5844'3 0.58391 0.58333

92901 39135 91221 49151 12920

44793 91706 22582 52628 97638

803 932 409 716 836

0.540 0.541 0.542 0.543 0.544

1.71600 1.71772 1.71944 1.72116 1.72288

68621 37273 23102 26125 46360

84858 36547 12106 30118 10887

460 069 159 747 296

0.58274 0.58216 0.581513 0.581013 0.58042

82523 57953 39205 26273 19151

73989 98641 89137 63601 40742

665 430 107 839 351

0.545 0.546 0.547 0.548 0.549

1.72460 1.72633 1.72806 1.72978 1.73152

83823 38533 10506 99760 06311

76435 50509 58581 27847 87233

429 656 095 197 477

0.57984 0.57926 0.57868 0.57810 0.57752

17833 22313 32586 48646 70487

39846 80782 83997 70519 61954

373 055 389 631 718

0.550

1.73325 30178 67395 237 c-y

1 1

0.57694 98103 80486 695 C-i)8

[

1

ELEMENTARY

TRANSCENDENTAL EXPONENTIAL

FUNCTIONS

FUNCTION

ez

X

127 Table

4.4

e-2

0.550 0.551 0.552 0.553 0.554

1.73325 1.73498 1.73672 1.73846 1.74019

30178 71378 29927 05843 99144

67395 00719 21325 65069 69542

237 302 750 647 780

0.57694 0.57637 0.57579 0.57522 0.57464

98103 31489 70638 15546 66205

80486 48877 90464 29163 89465

695 132 548 839 693

0.555 0.556 0.557 0.558 0.559

1.74194 1.74368 1.74542 1.74717 1.74892

09847 37970 83529 46543 27028

74075 19737 49342 07446 40349

399 955 837 121 310

0.57407 0.57349 0.57292 0.57235 0.57178

22611 84758 52640 26251 05586

96436 75715 53518 56633 12420

024 391 425 257 941

0.560 0.561 0.562 0.563 0.564

1.75067 1.75242 1.75417 1.75593 1.75768

25002 40484 73489 24037 92143

96101 24499 77091 07179 69816

083 041 459 036 648

0.57120 0.57063 0.57006 0.56949 0.56892

90638 81402 77873 80045 87911

48814 94320 78013 29541 79121

886 280 522 648 561

0.565 0.566 0.567 0.568 0.569

1.75944 1.76120 1.76297 1.76473 1.76649

77827 81105 01995 40515 96682

21815 21742 29927 08459 21189

104 902 989 520 621

0.56836 0.56779 0.56722 0.56665 0.56609

01467 20706 45624 76213 12469

57540 96153 26884 82224 95233

464 288 125 657 792

0.570 0.571 0.572 0.573 0.574

1.76826 1.77003 1.77180 1.77357 1.77535

70514 62029 71244 98177 42846

33735 13479 29574 52941 56273

152 471 208 024 392

0.56552 0.56496 0.56439 0.56383 0.56326

54386 01959 55181 14047 78551

99537 29326 19358 04955 22004

097 229 370 664 648

0.575 0.576 0.577 0.578 0.579

1.77713 1.77890 1.78068 1.78246 1.78425

05269 85463 83445 99235 32850

14038 02478 99612 85240 40940

362 341 864 377 016

0.56270 0.56214 0.56158 0.56101 0.56045

48688 24451 05837 92838 85449

06955 96822 29181 42170 74490

693 437 224 538 445

0.580 0.581 0.582 0.583 0.584

1.78603 1.78782 1.78961 1.79140 1.79319

84307 53624 40820 45912 68918

50073 97786 71010 58466 50662

382 336 772 414 599

0.55989 0.55933 0.55877 0.55822 0.55766

83665 87480 96888 11884 32463

65402 54726 82846 90701 19791

033 843 320 245 179

0.585 0.586 0.587 0.588 0.589

1.79499 1.79678 1.79858 1.80038 1.80218

09856 68744 45599 40441 53286

39900 20272 87669 39776 76077

067 757 600 313 198

0.55710 0.55654 0.55599 0.55543 0.55488

58618 90344 27635 70486 18892

12173 10464 57836 98018 75294

905 868 621 264 892

0.590 0.591 0.592 0.593 0.594

1.80398 1.80579 1.80760 1.80940 1.81121

84153 33061 00026 85067 88202

97856 08202 12004 15959 28572

940 413 477 787 596

0.55432 0.55377 0.55321 0.55266 0.55211

72847 32345 97380 67948 44043

34507 21050 80873 60481 06930

035 107 848 771 610

0.595 0.596 0.597 0.598 0.599

1.81303 1.81484 1.81666 1.81847 1.82029

09449 48827 06353 82045 75923

60156 22836 30550 99051 45908

569 588 566 264 101

0.55156 0.55101 0.55046 0.54991 0.54936

25658 12789 05431 03577 07222

67829 91340 26176 21601 27429

766 753 649 542 984

0.600

1.82211 88003 90508 975 c-y

[ 1

0.54881 16360 94026 433 C-i)7

[

1

128

ELEMENTARY

TRANSCENDENTAL

EXPONENTIAL

Table 4.4

FUNCTIONS

FUNCTION e-a

ez

5

0.600 0.601 0.602 0.603 0.604

1.82211 ii82394 1.82576 1.82759 1.82942

88003 18305 66846 33645 18719

90508 54062 59597 31970 97859

975 083 740 203 499

0.54881 0.54826 0.54771 0.54716 0.54662

16360 30987 51097 76683 07741

94026 72304 13727 70305 94597

433 710 448 543 605

0.605 0.606 0.607 0.608 0.609

1.83125 1.83308 1.83491 1.83675 1.83859

22088 43770 83782 42143 18872

85773 26048 50853 94189 91893

244 479 497 676 312

0.54607 0.54552 0.54498 0.54443 0.54389

44266 86251 33692 86582 44917

39709 59293 07547 39217 09589

413 368 943 140 946

0.610 0.611 0.612 0.613 0.614

1.84043 1.84227 1.84411 1.84596 1.84780

13987 27507 59448 09832 78674

81637 02933 97134 07433 78869

455 750 270 364 496

0.54335 0.54280 0.54226 0.54172 0.54118

08690 77897 52533 32591 18066

74499 90323 13983 02940 15202

787 981 200 922 890

0.615 0.616 0.617 0.618 0.619

1.84965 1.85150 1.85335 1.85521 1.85707

65995 71812 96145 39011 00429

58327 94538 38085 41401 58772

090 381 258 120 725

0.54064 0.54010 0.53956 0.53902 0.53848

08953 05246 06940 14030 26510

09316 44370 79994 76357 94167

571 616 313 053 789

0.620 0.621 0.622 0.623 0.624

1.85892 1.86078 1.86264 1.86451 1.86637

80418 78996 96182 31995 86452

46342 62108 65928 19523 86472

044 121 925 215 402

0.53794 0.53740 0.53686 0.53633 0.53579

44375 67620 96238 30226 69576

94674 39663 91459 12924 67456

492 618 568 149 037

0.625 0.626 0.627 0.628 0.629

1.86824 1.87011 1.87198 1.87385 1.87573

59574 51378 61683 91108 39071

32222 24085 31242 24743 77511

407 530 321 442 543

0.53526 0.53472 0.53419 0.53365 0.53312

14285 64346 19754 80505 46591

18990 31997 71484 02990 92592

242 571 093 602 086

0.630 0.631 0.632 0.633 0.634

1.87761 1.87948 1.88136 1.88325 1.88513

05792 91289 95581 18687 60625

64343 61910 48763 05331 13924

132 454 361 198 678

0.53259 0.53205 0.53152 0.53099 0.53046

18010 94754 76818 64198 56888

06897 13047 78717 72114 61974

190 683 927 344 883

0.635 0.636 0.637 00. .E

1.88702 1.88891 1.89079 1.89269 1.89458

21414 01074 99623 17079 53463

58737 25849 03226 80722 50084

766 565 199 703 912

0.52993 0.52940 0.52887 0.52834 0.52781

54883 58177 66765 80641 99802

17568 08694 05682 79390 01207

489 574 485 975 673

0.640 0.641 0.642 0.643 0.644

1.89648 1.89837 1.90027 1.90217 1.90408

08793 83087 76365 88646 19949

04951 40855 55225 47391 18580

353 140 865 502 301

0.52729 0.52676 0.52623 0.52571 0.52518

24240 53951 88930 29172 74670

43048 77357 77105 15790 67436

557 426 369 242 140

0.645 0.646 0.647 0.648 0.649

1.90598 1.90789 1.90980 1.91171 1.91362

70292 39696 28178 35758 62456

71922 12453 47112 84748 36119

692 188 287 384 674

0.52466 0.52413 0.52361 0.52309 0.52256

25421 81418 42656 09131 80836

06592 08335 48263 02500 47694

872 432 478 807 830

0.650

1.91554 08290 13896 070

0.52204 57767 61016 048 C-i)7

c 1

ELEMENTARY

TRANSCENDENTAL

EXPONENTIAL

129

FUNCTIONS

FUNCTION

Table

4.4

e-z

X

0.650 0.651 0.652 0,653 0.654

1.91!554 08290 13896 070 1.91'745 73279 32661 108 1.91'337 57443 08913 867 1.92129 60800 61070 883 1.92321 83371 09468 067

0.52204 0.52152 0.52100 0.52048 0.51996

57767 61016 048 39919 20157 530 27286 03334 394 19862 89283 277 17644 57261 823

0.655 0.656 0.657 0.658 0.659

1.92514 25173 76362 630 1.92706 86227 85934 997 1.92899 66552 64290 740 1.92'092 66167 39462 496 1.92128585091 41411 902

0.51944 0.51892 0.51840 0.51788 0.51736

20625 87048 156 28801 58940 364 42166 53755 974 60715 52831 438 84443 38021 612

0.660 0.661 0.662 0.663 0.664

1.9:!479 1.93672 1.93866 1.94060 1.94254

23344 02031 522 80944 55146 776 57912 36517 879 54266 83841 774 70027 36754 070

0.51685 0.51633 0.51581 0.51530 0.51478

13344 91699 238 47414 96754 426 86648 36594 140 31039 95141 674 80584 56836 146

0.665 0.666 0.667 0.668 0.669

1.94449 1.94643 1.94838 1.95033 1.95228

05213 36830 982 59844 27591 272 33939 54498 192 27518 64961 432 40601 08339 065

0.51427 0.51375 0.51324 0.51273 0.51222

35277 06631 974 95112 29998 365 60085 12918 798 30190 41890 516 05423 03924 002

0.670 0.671 0.672 0.673 0.674

1 =1542373206 35939 496 1:(:,5619 25354 01023 417 1.05814 97063 58805 754 1.')6010 88354 66457 630 1.96206 99246 83108 314

0.51170 0.51119 0.51068 0.51017 0.50966

85777 86542 478 71249 77781 383 61833 66187 865 57524 40820 271 58316 91247 632

0.675 0.676 0.677 0.678 0.679

l/36403 29759 69847 187 ii 96599 79912 89725 700 1.96796 49726 07759 335 1.96993 39218 90929 575 1.97190 48411 08185 868

0.50915 0.50864 0.50813 0.50763 0.50712

64206 07549 157 75186 80313 718 91254 00639 348 12402 60132 723 38627 50908 661

0.680 0.681 0.682 0.683 0.684

1.97387 1.97585 1.97782 1.97980 1.98178

77322 30447 594 25972 30606 040 94380 83526 371 82567 66049 605 90552 56994 589

0.50661 0.50611 0.50560 0.50509 0.50459

69923 65589 610 06285 97305 142 47709 39691 448 94188 86890 827 45719 33551 185

0.685 0.686 0.687 0.688 0.689

1.98377 18355 37159 979 1: 98575 65995 89326 220 1,98774 33493 98?57 531 1,98973 20869 50703 885 1.99172 28142 35403 001

0.50409 0.50358 0.50308 0.50258 0.50207

02295 74825 526 63913 06371 449 30566 24350 644 "2250 25428 387 .3960 06773 037

0.690 0.691 0.692 0.693 0.694

1.99371 1.99571 1.99770 1.99970 2.00170

55332 43082 329 02459 66461 043 69544 00252 033 56605 41163 899 63663 87902 948

0.50157 0.50107 0.50057 0.50007 0.49957

60690 66055 534 47437 01448 895 39194 11627 713 35956 95767 658 37720 53544 971

0.695 0.696 0.697 0.698 0.699

Z!.OO37090739 41175 193 :!.00571 37852 03688 356 :!.00772 05021 80153 865 2.00972 92268 77288 865 2.01173 99613 03818 219

0.49907 0.49857 0.49807 0.49757 0.49708

44479 85i.35 969 56229 91216 541 72965 72961 653 94682 32044 844 21374 70637 732

0.700

2.01375 27074 70476 522

0.49658 53037 91409 515 (-;I6

!Lc-y 1

[I 1

ELEMENTARY

Table

TRANSCENDENTAL

EXPONENTIAL

4.4

FVNCTIONS

FUNCTION e-z

ez

X

0.700 0.701 0.702 0.703 0.704

2.01375 2.01576 2.01778 2.01980 2.02182

27074 74673 42430 30365 38498

70476 90010 77179 48759 23544

522 108 065 247 296

0.49658 0.49608 0.49559 0.49509 0.49460

53037 89666 31256 77802 29299

91409 97526 92651 80943 67056

515 471 465 451 976

0.705 0.706 0.707 0.708 0.709

2.02384 2.02587 2.02789 2.02992 2.03195

66849 15438 84286 73414 82840

22347 68004 85374 01341 44819

653 586 210 511 374

0.49410 0.49361 0.49312 0.49262 0.49213

85742 47126 13446 84698 60875

56141 53841 66295 00135 62485

685 826 756 445 987

0.710 0.711 0.712 0.713 0.714

2.03399 2.03602 2.03806 2.04010 2.04214

12586 62672 33118 23945 35173

46750 40109 59906 43184 29026

612 996 288 280 822

0.49164 0.49115 0.49066 0.49017 0.48968

41974 27990 18916 14750 15485

60965 03682 99240 56730 85736

102 649 129 197 169

0.715 0.716 0.717 0.718 0.719

2.04418 2.04623 2.04827 2.05032 2.05237

66822 18913 91467 84503 98043

58556 74939 23384 51146 07529

873 531 083 049 226

0.48919 0.48870 0.48821 0.48772 0.48723

21117 31641 47053 67346 92516

96331 99079 05032 25731 73205

534 460 312 153 263

0.720 0.721 0.722 0.723 0.724

2.05443 2.05648 2.05854 2.06060 2.06266

32106 86714 61886 57644 74008

43887 13628 72211 77154 88034

743 106 257 626 189

0.48675 0.48626 0.48577 0.48529 0.48480

22559 57469 97243 41873 91357

59971 99034 03884 88500 67343

650 560 990 207 253

0.725 0.726 0.727 0.728 0.729

2.06473 2.06679 2.06886 2.07093 2.07300

10999 68637 46943 45938 65642

66486 76210 82971 54598 60992

529 896 273 438 036

0.48432 0.48384 0.48335 0.48287 0.48239

45689 04864 68878 37725 11401

55362 67990 21146 31229 15125

467 997 315 734 923

0.730 0.731 0.732 0.733 0.734

2.07508 2.07715 2.07923 2.08131 2.08339

06076 67261 49218 51967 75529

74122 68033 18844 04749 06025

645 852 323 882 589

0.48190 0.48142 0.48094 0.48046 0.47998

89900 73219 61352 54295 52042

90202 74309 85778 43422 66536

427 180 027 238 031

0.735 0.736 0.737 0.738 0.739

2.08548 2108756 2.08965 2109174 2.09384

19925 85175 71302 78325 06265

05027 86196 36056 43220 98392

819 344 419 868 173

0.47950 0.47902 0.47854 0.47806 0.47759

54589 61931 74064 90982 12680

74894 88751 28841 16377 73052

090 082 182 589 052

0.740 0.741 0.742 0.743 0.744

2.09593 2.09803 2.10013 2.10223 2.10433

55144 24983 15801 27621 60464

94364 26026 90360 86450 15478

563 109 816 725 007

0.47711 0.47663 0.47616 0.47568 0.47520

39155 70400 06412 47186 92716

21034 82972 81989 41687 86144

388 004 423 803 466

0.745 0.746 0.747 0.748 0.749

2.10644 2.10854 2.11065 2.11277 2.11488

14349 89299 85335 02477 40747

80727 87586 43551 58226 43325

065 641 917 625 155

0.47473 0.47425 0.47378 0.47331 0.47283

42999 98029 57801 22312 91555

39912 28019 75969 09739 55779

416 867 767 326 537

0.750

2.11700

00166 12674 C-l)3

669

0.47236

65527

41014

707

[

1

[

(96

1

ELEMENTARY

TRANSCENDENTAL

EXPONENTIAL

131

FVNCTIONS

FUNCTION

Table

ez

X

4.4

e-z

0.750 0.751 0.752 0.753 0.754

2.11700 2.11911 2.12123 2.12336 2.12548

00166 12674 669 80754 a2217 212 a2534 70011 a30 05526 96236 688 49752 a3191 190

0.47236 0.47189 0.47142 0.47095 0.47048

65527 41014 707 44222 92841 982 27637 39130 a75 15766 08222 791 08604 28930 562

0.755 0.756 0.757 0.758 0.759

2.12761 2.12974 2.13187 2.13400 2.13613

15233 55298 098 0.1990 39105 663 10044 63289 745 39417 58655 946 90130 58141 739

0.47001 0.46954 0.46907 0.46860 0.46813

06147 30537 969 08390 42799 274 15328 95938 749 26958 20650 211 43273 48096 543

0.760 0.761 0.762 0.763 0.764

2.13827 2.14041 2.14255 2.14470 2.14684

62204 96818 602 55662 11894 152 70523 42714 282 06810 30765 301 64544 19676 075

0.46766 0.46719 0.46673 0.46626 0.46579

64270 09909 234 89943 38187 907 20288 65499 852 55301 24879 557 94976 49828 242

0.765 0.766 0.767 0.768 0.769

2.14899 2.15114 2.15329 2.15545 2.15760

43746 55220 173 44438 a5318 010 66642 60038 993 10379 31603 678 75670 54385 916

0.46533 0.46486 0.46440 0.46394 0.46347

39309 74313 393 88296 32768 297 41931 60091 573 00210 91646 708 63129 63261 598

0.770 0.771 0.772 0.773 0.774

2.15976 2.16192 2.16409 2.16625 2.16842

62537 a4915 008 71002 ala77 a66 01087 06121 167 52812 20653 514 26199 90647 604

0.46301 0.46255 0.46208 0.46162 0.46116

30683 11228 073 02866 72301 444 79675 83700 034 61105 83104 714 47152 08658 446

0.775 0.77 0.77 t 0.778 0.779

2.17059 2.17276 2.17493 2.17711 2.17929

21271 a3442 386 38049 68545 234 76555 17634 114 36810 04559 757 18836 05347 a30

0.46070 0.46024 0.45978 0.45932 0.45886

37809 98965 ala 33074 93092 580 32942 30565 la9 37407 51370 344 46465 95954 527

0.780 0.781 0.782 0.783 0.784

2.18147 2.18365 2.18583 2.18802 2.19021

22654 98201 117 48288 63501 691 95758 a3813 099 65087 43882 545 56296 30643 070

0.45840 60113 05223 545 Oi45794 78344 20542 069 0.45749 01154 a3733 175 0.45703 28540 37077 a90 0.45657 60496 23314 727

0.785 0.786 0.787 0.788 0.789

2.19240 2.19460 2.19679 2.19899 2.20119

69407 33215 744 04442 42911 a52 61423 53235 086 40372 59883 740 41311 60752 903

0.45611 0.45566 0.45520 0.45475 0.45429

97017 85639 236 38100 67703 540 83740 13615 885 33931 67940 176 88670 75695 532

0.790 0.791 0.792 0.793 0.794

2.20339 2.20560 2.20780 2.21001 2.21222

64262 55936 659 09247 47730 288 76288 40632 465 65407 41347 466 76626 58787 377

0.45384 0.45339 0.45293 0.45248 0.45203

47952 82355 822 11773 33849 215 80127 76557 724 53011 57316 754 30420 23414 649

0.795 0.796 0.797 0.798 0.799

2.21444 2.21665 2.21887 2.22109 2.22331

09968 04074 299 65453 90542 561 43106 33740 936 42947 51434 850 64999 63608 607

0.45158 0.45112 0.45067 0.45022 0.44977

12349 22592 237 98794 03042 379 89750 13409 518 a5213 02789 227 a5178 20727 758

0.800

2.22554 09284 92467 605 C-l)3

[

1

0.44932 a9641 17221 591 (96

[1 1

132

ELEMENTARY

Table

TRANSCENDENTAL

EXPONENTIAL

4.4

FUNCTIONS

FUNCTION

@

X

ecz

0.800 0.801 0.802 0.803 0.804

2.22554 2.22776 2.22999 2.23222 2.23446

09284 75825 64644 75762 09202

92467 62440 00181 34513 96726

605 556 717 111 759

0.44932 0.44887 0.44843 0.44798 0.44753

89641 98597 12042 29971 52381

17221 42716 48109 84743 04412

591 986 530 691 369

0.805 0.806 0.807 0.808 0.809

2.23669 2.23893 2.24117 2.24341 2.24566

64988 43140 43681 66635 12022

19986 39932 94378 23379 69230

909 270 249 186 599

0.44708 0.44664 0.44619 0.44574 0.44530

79265 10621 46442 86726 31467

59356 02264 86271 64960 92358

447 340 556 242 738

0.810 0.811 0.812 0.813 0.814

2.24790 2.25015 2.25240 2.25466 2.25691

79866 70189 83014 18363 76258

76471 91886 64507 45618 88752

419 242 569 061 788

0.44485 0.44441 0.44396 0.44352 0.44308

80662 34305 92392 54918 21880

22941 11626 13780 85209 82167

134 826 063 512 806

0.815 0.816 0.817 0.818 0.819

2.25917 2.26143 2.26369 2.26596 2.26823

56723 59779 85450 33758 04725

49701 86510 59486 31195 66470

480 786 532 979 087

0.44263 0.44219 0.44175 0.44131 0.4408'7

93273 69092 49333 33992 23064

61351 79898 95392 65856 49756

106 654 332 218 146

0:822 0.823 0.824

2.27049 2.27277 2.27504 2.27732 2.27960

98375 14729 53812 15645 00251

32405 98368 35993 19188 24138

781 215 046 700 650

0.44043 0.43999 0.43955 0.43911 0.43867

16545 14429 16714 23395 34466

05999 93933 73347 04469 47967

263 588 574 662 847

0.825 0.826 0.827 0.828 0.829

2.28188 2.28416 2.28644 2.28873 2.29102

07653 37874 90936 66863 65678

29303 15424 65522 64904 01164

690 217 506 998 583

0.43823 0.43779 0.43735 0.43692 0.43648

49924 69765 93983 22575 55537

64949 16959 65982 74441 05193

237 611 985 171 342

0.830 0.831 0.832 0.833 0.834

2.29331 2.29561 2.29790 2.30020 2.30251

87402 32060 99674 90267 03862

64182 46132 41479 46985 61709

888 567 593 553 945

0.43604 0.43561 0.43517 0.43474 0.43430

92863 34549 80592 30987 85729

21535 87200 66356 23608 23995

593 502 699 428 109

0.835 0:838 0.839

2.30481 2.30712 2.30942 2.31173 2.31405

40482 00151 82890 88724 17676

87012 26555 86305 74537 01834

474 358 628 437 366

0.43387 0.43344 0.43300 0.43257 0.43214

44814 08238 75996 48084 24498

32990 16504 40877 72886 79740

906 293 616 664 233

0.840 0.841 0.842 0.843 0.844

2.31636 2.31868 2.32100 2.32332 2.32565

69767 45023 43465 65117 10003

81091 27518 58641 94304 56672

734 913 644 351 462

0.43171 0.43127 0.43084 0.43041 0.42998

05234 90286 79652 73326 71304

29079 88978 27942 14906 19239

693 558 052 679 788

0.845 0.846 0.847 0.848 0.849

2.32797 2.33030 2.33263 2.33497 2.33730

78145 69567 84292 22343 83745

70234 61805 60527 97872 07647

734 575 370 812 233

0.42955 0.42912 0.42869 0.42827 0.42784

73582 80155 91020 06172 25606

10739 59632 36577 12659 59395

148 516 204 654 005

0.850

2.33964

68519 25990 C-i)3

937

0.42741

49319

48726

670

8*Ei

8. $76

[

1

[ 1 (536

ELEMENTARY

TRANSCENDENTAL

EXPONENTIAL

133

FUNCTIONS

FUNCTION

ez

X

Table

4.4

e-2

0.850 0.851 0.852 0.853 0.854

2.33964 2.34198 2.34433 2.34667 2.34902

68519 76689 08280 63314 41814

25990 91381 44636 28914 89719

937 538 295 459 607

0.42741 0.42698 0.42656 0.42613 0.42570

49319 77306 09563 46085 86869

48726 53025 45091 98148 85850

67n 901

0.855 0.856 0.857 0.858 0.859

2.35137 2.35372 2.35608 2.35843 2.36079

43805 69310 18352 90954 87141

74901 34660 21547 90464 98674

997 911 002 656 336

0.42528 0.42485 0.42443 0.42400 0.42358

31910 81204 34746 92533 54560

82274 61924 99730 71047 51652

123 574 893 281 373

0.860 0.861 0.862 0.863 0.864

2.36316 2.36552 2.36789 2.37026 2.37263

06937 50363 17445 08206 22669

05794 73806 67050 52237 98442

948 196 946 586 400

0.42316 0.42273 0.42231 0.42189 0.42147

20823 91317 66039 44983 28147

17748 45962 13343 97363 75917

817 841 840 945 606

0.865 0.866 0.867 0.868 0.869

2.37500 2.37738 2.37976 2.38214 2.38452

60859 22799 08513 18024 51357

77111 62065 29496 57978 28460

933 359 863 010 126

0.42105 0.42063 0.42021 0.41979 0.41937

15526 07115 02910 02908 07103

27321 30312 64050 08114 42503

165 439 296 234 963

0.870 0.871 0.872 0.873 0.874

2.38691 2.38929 2.39168 2.39408 2.39647

08535 89582 94522 23379 76177

24276 31145 37171 32849 11065

682 671 999 872 184

0.41895 0.41853 0.41811 0.41769 0.41727

15492 28071 44834 65779 90901

47638 04358 93919 97998 98691

983 162 324 822 126

0.875 0.876 0.877 0.878 0.879

2.39887 2.40127 2.40367 2.40608 2.40849

52939 53690 78455 27255 00117

67097 98624 05720 90861 58929

915 518 327 947 666

0.41686 0.41644 0.41602 0.41561 0.41519

20196 53660 91288 33076 79020

78508 20380 07652 24088 53866

403 096 513 408 560

0.880 0.881 0.882 0.883 0.884

2.41089 2.41331 2.41572 2.41814 2.42056

97064 18119 63308 32654 26181

17209 75397 45597 42330 82530

851 361 956 708 413

0.41478 0.41436 0.41395 0.41354 0.41312

29116 83360 41748 04276 70938

81581 92242 71274 04515 78218

367 420 097 140 250

0.885 0.886 0.887 0.888 0.889

2.42298 2.42540 2.42783 2.43026 2.43269

43914 85877 52094 42590 57387

85550 73163 69565 01380 97656

015 018 911 593 799

0.41271 0.41230 0.41188 0.41147 0.41106

41732 16653 95698 78861 66139

79049 94088 10827 17170 01433

666 753 593 568 949

0.890 0.891 0.892 0.893 0.894

2.43512 2.43756 2.44000 2.44244 2.44488

96512 59989 47841 60092 96769

89874 11946 00220 93481 32956

527 472 460 882 134

0.41065 0.41024 0.40983 0.40942 0.40901

57527 53022 52620 56315 64106

52345 59044 11078 98410 11406

488 001 959 082 922

0.895 0.896 0.897 0.898 0.899

2.44733 2.44978 2.45223 2.45468 2.45714

57894 43493 53589 88208 47374

62311 27659 77561 63026 37516

060 394 203 343 904

0.40860 0.40819 0.40779 0.40738 0.40697

75986 91952 12001 36127 64327

40848 77922 14226 41763 52948

458 685 207 826 135

0.900

2.45960 31111 56949 664 C-l)3

[ 1

d,

-

?A7 I”.

726 193

0.40656 96597 40599 112 (-;I5

[

1

'

ELEMENTARY TRANSCENDENTAL FUNCTIONS Table

EXPONENTIAL

4.4

FUNCTION

eI

X

e+

0.900 0.901 0.902 0.903 0.904

2.45960 2.46206 2.46452 2.46699 2.46946

31111 56949 664 39444 79698 548 72398 66597 083 29997 80940 863 12266 88490 006

0.40656 0.40616 0.40575 0.40535 0.40494

96597 40599 112 32932 97943 710 73330 18615 453 17784 96654 028 66293 26504 879

0.905 0.906 0.907 0.908 0.909

2.47193 2.47440 2.47688 2.47935 2.48183

19230 57471 626 50913 58582 298 07340 64990 529 88536 52339 232 94525 98748 200

0.40454 0.40413 0.40373 0.40333 0.40292

18851 03018 802 75454 21451 540 36098 77463 377 00780 67118 736 69495 86885 773

0.910 0.911 0.912 0.913 0.914

2.48432 25333 84816 587 2.48680 80984 93625 386 2.48929'61504 10739 912 2.49178 66916 24212 291 2.49427 97246 24583 942

0.40252 0.40212 0.40171 0.40131 0.40091

42240 33635 975 19010 04643 753 99800 97586 047 84609 10541 915 73430 41992 136

0.915 0.916 0.917 0.918 0.919

2.49677 2.49927 2.50177 2.50427 2.50678

52519 04888 075 32759 60652 177 37992 89900 513 68243 93156 620 23537 73445 810

0.40051 0.40011 0.39971 0.39931 0.39891

66260 90818 809 63096 56304 950 63933 38134 089 68767 36389 877 77594 51555 677

0.920 0.921 0.922 0.923 0.924

2.50929 2.51180 2.51431 2.51682 2.51934

03899 36297 671 09353 89748 577 39926 44344 189 95642 13141 971 76526 11713 703

0.39851 0.39812 0.39772 0.39732 0.39692

90410 84514 173 07212 36546 962 27995 09334 165 52755 04954 021 81488 25882 492

0.925 0.926 0.927 0.928 0.929

2.52186 2.52439 2.52691 2.52944 2.53197

82603 58147 991 13899 73052 794 70439 79557 936 52249 03317 633 59352 72513 022

0.39653 0.39613 0.39573 0.39534 0.39494

14190 74992 866 50858 55555 360 91487 71236 720 36074 26099 830 84614 24603 311

0.930 0.931 0.932 0.933 0.934

2.53450 2.53704 2.53958 2.54212 2.54466

91776 17854 680 49544 72585 166 32683 72481 544 41218 55857 927 75174 63568 010

0.39455 0.39415 0.39376 0.39337 0.39297

37103 71601 130 93538 72342 199 53915 32469 987 18229 58022 122 86477 55429 996

0.935 0.936 0.937 0.938 0.939

2.54721 2.54976 2.55231 2.55486 2.55742

34577 39007 611 19452 28117 220 29824 79384 537 65720 43847 026 27164 75094 464

0.39258 0.39219 0.39180 0.39140 0.39101

58655 31518 373 34758 93504 997 14784 49000 198 98728 06006 497 86585 72918 221

0.940 0.941 0.942 0.943 0.944

2.55998 2.56254 2.56510 2.56767 2.57024

14183 29271 496 26801 65080 189 65045 43782 593 28940 29203 299 18511 87732 007

0.39062 0.39023 0.38984 0.38945 0.38906

78353 58521 102 74027 71991 894 73604 22897 977 77079 21196 971 84448 77236 34i

0.945 0.946 0.947 0.948 0.949

2.57281 2.57538 2.57796 2.58054 2.58312

33785 88326 089 74788 02513 161 41544 04393 651 34079 70643 376 52420 80516 117

0.38867 0.38829 0.38790 0.38751 0.38712

95709 01753 010 10856 05872 971 29886 01110 896 52794 99369 747 79579 12940 390

0.950

2.58570 96593 15846 199 C-l)3

[ 1

0.38674 10234 54501 207

[C-l)5 1

'

ELEMENTARY

TRANSCENDENTAL

EXPONENTIAL x

135

FUNCTIONS

FUNCTION

Table

er

4.4

e-=

0.950 0.951 0.952 0.953 0.954

2.58570 2.58829 2.59088 2.59347 2.59607

96593 66622 62535 84356 32112

15846 61051 03133 31686 38890

199 072 898 135 126

0.38674 0.38635 0.38596 0.38558 0.38519

10234 44757 83143 25389 71491

54501 37117 74242 79713 67755

207 707 140 111 194

0.955 0.956 0.957 0.958 0.959

2.59867 2.60127 2.60387 2.60647 2.60908

05829 05532 31248 83003 60823

19521 70952 93153 88696 62757

695 740 828 799 366

0.38481 0.38442 0.38404 0.38365 0.38327

21445 75247 32893 94380 59703

52978 50378 75335 43613 71361

545 516 273 409 560

0.960 0.961 0.962 0.963 0.964

2.61169 2.61430 2.61692 2.61954 2.62216

64734 94761 50932 33272 41807

23117 80169 46914 38971 74573

718 136 592 373 688

0.38289 0.38251 0.38212 0.38174 0.38136

28859 01844 78654 59286 43734

75112 71780 78665 13447 94189

023 368 061 076 517

0.965 0.966 0.967 0.968 0.969

2.62478 2.62741 2.63004 2.63267 2.63530

76564 37569 24848 38428 78334

74575 62452 64304 08861 27480

291 101 825 583 539

0.38098 0.38060 0.38022 0.37984 0.37946

31997 24069 19947 19628 23107

39337 67716 98534 51378 46217

233 437 325 697 574

0.970 0.971 0.972 0.973 0.974

2.63794 2.64058 2.64322 2.64587 2.64851

44593 37232 56276 01753 73689

54152 25503 80798 61940 13478

532 708 158 558 808

0.37908 0.37870 0.37832 0.37794 0.37756

30381 41445 56296 74931 97345

03398 43649 88076 58165 75777

818 757 798 054 964

0.975 0.976 0.977 0.978 0.979

2.65116 2.65381 2.65647 2.65913 2.66179

72109 97042 48512 26548 31174

82606 19166 75651 07209 71643

682 470 628 434 642

0.37719 0.37681 0.37643 0.37606 0.37568

23535 53497 87227 24721 65976

63156 42920 38065 71965 68367

913 859 949 147 855

0.980 0.981 0.982 0.983 0.984

2.66445 2.66712 2.66979 2.67246 2.67513

62419 20308 04868 16127 54109

29417 43654 80145 07345 96380

138 602 169 099 441

0.37531 0.37493 0.37456 0.37418 0.37381

10988 59753 12267 68527 28529

51399 45561 75729 67156 45466

539 350 751 142 482

0.985 0.986 0.987 0.988 0.989

2.67781 2.68049 2.68317 2.68585 2.68854

18844 10356 28673 73822 45830

21049 57826 85862 86989 45722

708 547 418 272 235

0.37343 0.37306 0.37269 0.37232 0.37194

92269 59743 30948 05880 84535

36660 67113 63571 53155 63358

918 412 361 231 181

0.990 0.991 0.992 0.993 0.994

2.69123 2.69392 2.69662 2.69932 2.70202

44723 70528 23273 02984 09688

49262 87498 53013 41079 49668

289 962 016 142 652

0.37157 0.37120 0.37083 0.37046 0.37009

66910 53000 42802 36313 33529

22045 57455 98195 73247 11961

691 187 674 362 296

0.995 0.996 0.997 0.998 0.999

2.70472 2.70743 2.71013 2.71285 2.71556

43412 04184 92030 06977 49053

79452 33802 18796 43219 18566

181 382 637 755 687

0.36972 0.36935 0.36898 0.36861 0.36824

34445 39058 47366 59363 75046

44058 99632 09141 03418 13662

983 024 744 822 921

1.000

2.71828

18284 59045 C-l)3

235

0.36787

94411 (-!I5

71442

322

[

1

[

1

ELEMENTARY TRANSCENDENTAL FUrVCTIONS Table

EXPONENTIAL

4.4

FUNCTION e-l

e2

2

1.00000 1.10517 1.22140 1.34985 1.49182

00000 00000 09180 75648 27581 60170 88075 76003 46976 41270

1.00000

00000

0.90483 0.81873 0.74081 0.67032

74180 35959 57316 07530 77981 85867 82206 81717 86607 00460 35639 30074

1.64872 1.82211 2.01375 2.22554 2.45960

12707 00128 88003 90509 27074 70477 09284 92468 31111 56950

0.60653 0.54881 0.49658 0.44932 0.40656

06597 12633 42360 16360 94026 43263 53037 91409 51470 89641 17221 59143 96597 40599 11188

2.71828 3.00416 3.32011 3.66929 4.05519

18284 59045 60239 46433 69227 36547 66676 19244 99668 44675

0.36787 0.33287 0.30119 0.27253 0.24659

94411 71442 32160 10836 98079 55329 42119 12202 09664 17930 34012 60312 69639 41606 47694

4.48168 4.95303 5.47394 6.04964 6.68589

90703 38065 24243 95115 73917 27200 74644 12946 44422 79269

0.22313 0.20189 0.18268 0.16529 0.14956

01601 48429 82893 65179 94655 40849 35240 52734 65022 88882 21586 53830 86192 22635 05264

22:10 s-23 2:4

7.38905 8.16616 9.02501 9.97418 11.02317

60989 30650 99125 67650 34994 34121 24548 14721 63806 41602

0.13533 0.12245 0.11080 0.10025 0.09071

52832 36612 69189 64282 52981 91022 31583 62333 88333 88437 22803 73373 79532 89412 50338

2.5

12.18249 13.46373 14.87973 16.44464 18.17414

39607 03473 80350 01690 17248 72834 67710 97050 53694 43061

0.08208 0.07427 0.06720 0.06081 0.05502

49986 23898 79517 35782 14333 88043 55127 39749 76513 00626 25217 96500 32200 56407 22903

20.08553 22.19795 24.53253 27.11263 29.96410

69231 87668 12814 41631 01971 09349 89206 57887 00473 97013

0.04978 0.04504 0.04076 0.03688 0.03337

70683 67863 94298 92023 93557 80607 22039 78366 21517 31674 01240 00545 32699 60326 07948

33.11545 36.59823 40.44730 44.70118 49.40244

19586 92314 44436 77988 43600 67391 44933 00823 91055 30174

0.03019 0.02732 0.02472 0.02237 0.02024

73834 22318 50074 37224 47292 56080 35264 70339 39120 07718 56165 59578 19114 45804 38847

t-23 414

54.59815 60.34028 66.68633 73.69979 81.45086

00331 44239 75973 61969 10409 25142 36995 95797 86649 68117

0.01831 0.01657 0.01499 0.01356 0.01227

56388 88734 18029 26754 01761 24754 55768 20477 70621 85590 12200 93176 73399 03068 44118

i-56 4: 7 4.8 4.9

90.01713 99.48431 109.94717 121.51041 134.28977

13005 21814 56419 33809 24521 23499 75187 34881 96849 35485

0.01110 0.01005 0.00909 0.00822 0.00744

89965 38242 30650 18357 44633 58164 52771 01695 81709 97470 49020 02884 65830 70924 34052

E t:

0: 4 0. 5 i-76 0: 8 0.9 1. 0 1.1 1'3 1:4 1. 5 1.6 11-i 1:9

3:; ;:t 3. 0 z 3:s 3.4 3.5 3:: z 210

00000

00000

148.41315 91025 76603 0.00673 79469 99085 46710 5.0 From C. E. Van Orstrand, Tablesof the exponentialfunction and of the circular sine and cosine to radian arguments, Memoirsof the National Academy of Sciences, vol. 14, Fifth Memoir. U.S. GovernmentPrinting Office, Washington,D.C., 1921(with permission) for e-",x12.4.

ELEMENTARY

TRANSCENDENTAL

EXPONENTIAL

137

FUNCTIONS

FUNCTION

Table

ez

4.4

e-2

148.41315 164.02190 181.27224 200.33680 221.40641

91025 77 72999 02 18751 51 99747 92 62041 87

0.00673 0.00609 0.00551 0.00499 0.00451

79469 99085 46710 67465 65515 63611 65644 20760 77242 15939 06910 21621 65809 42612 66798

244.69193 270.42640 298.86740 330.29955 365.03746

22642 20 74261 53 09670 60 99096 49 78653 29

0.00408 0.00369 0.00334 0.00302 0.00273

67714 38464 06699 78637 16482 93082 59654 57471 27277 75547 45375 81475 94448 18768 36923

403.42879 34927 35 445.85777 00825 17 492.74904 10932 56 544.57191 01259 29 601.84503 78720 82

0.00247 0.00224 0.00202 0.00183 0.00166

87521 76666 35842 28677 19485 80247 94306 36295 73436 63047 77028 90683 15572 73173 93450

665.14163 735.09518 812.40582 897.84729 992.27471

30443 62 92419 73 51675 43 16504 18 56050 26

0.00150 0.00136 0.00123 0.00111 0.00100

34391 92977 57245 03680 37547 89342 09119 02673 48118 37751 47844 80308 77854 29048 51076

1096.63315 84284 59 1211.96707 44925 77 1339.43076 43944 18 1480.29992 75845 45 1635.98442 99959 27

0.00091 0.00082 0.00074 0.00067 0.00061

18819 65554 51621 51049 23265 40427 65858 08376 67937 55387 75193 84424 12527 61129 57256

1808.04241 44560 63 1998.19589 51041 18 2208.34799 18872 09 2440.60197 76244 99 2697.28232 82685 09

0.00055 0.00050 0.00045 0.00040 0.00037

30843 70147 83358 04514 33440 61070 28271 82886 79706 97349 78979 78671 07435 40459 08837

2980.95798 70417 28 3294.46807 52838 41 3640.95030 73323 55 4023.87239 38223 10 4447.06674 76998 56

0.00033 0.00030 0.00027 0.00024 0.00022

54626 27902 51184 35391 38078 86666 46535 69972 14233 85168 27107 95202 48673 24178 84827

4914.76884 02991 34 5431.65959 13629 80 6002.91221 72610 22 6634.24400 62778 85 7331.97353 91559 93

0.00020 0.00018 0.00016 0.00015 0.00013

34683 69010 64417 41057 93667 57912 65858 10987 63341 07330 75095 47660 63889 26482 01145

8103.08392 75753 84 8955.29270 34825 12 9897.12905 87439 16 10938.01920 81651 84 12088.38073 02169 84

0.00012 0.00011 0.00010 0.00009 0.00008

34098 04086 67955 16658 08490 11474 10394 01837 09335 14242 31478 17334 27240 65556 63226

13359.72682 96618 72 14764.78156 55772 73 16317.60719 80154 32 18033.74492 78285 11 19930.37043 82302 89

0.00007 0.00006 0.00006 0.00005 0.00005

48518 29887 70059 77287 36490 85387 12834 95053 22210 54515 99432 17698 01746 82056 17530

22026.46579 48067 17

0.00004 53999 29762 48485

ELEMENTARY TRANSCENDENTAL FUNCTIONS Table

EXPONENTIAL

4.4

FUNCTION

e*

0)1.00000 00000 00000 000 0)2.71828 18284 59045 235 I 112.00855 0)7.38905 60989 36923 30650 18766 774 227 e( lj5.45981 50033 14423 908

:: 12 13 14

1.48413 4.03428 1.09663 2.98095 3)8.10308

15910 25766 034 79349 27351 226,-31584 28458 599 79870 41728 275 39275 75384 008 --

4)2.20264 5.98741 1.62754 4.42413 1.20260

65794 80671 65241715 19781 846 -79141 90039 208 39200 89205 03342841 64776 778--c

( 6)3.26901 6)8.88611 7)2.41549 6.56599 1.78482

73724 72110 639 05205 07872 637 52.75357529 821 69137 33051 114~ 30036 31872 608

4.85165 1.31881 3.58491 9.74480 2.64891

19540 97902 78057344 83214 69728461 31591 562-p 34462 48902 60022129 84347 229

25 26 27 28 29

7.20048 1.95729 5.32048 1.44625 3.93133

30

e-=

1.00000 3.67879 1.35335 4.97870 1.83156

00000 00000 000 44117 14423 216 28323 66126 919 68367 86394 298 38888 73418 029

Ii

f- 3j6.73794 69990 85467 097 - 3 4 2.47875 3.35462 21766 9.11881 1.23409 62790 66358 80408 96555 25118 45162 66795 423 388 495 080 4.53999 1.67017 6.14421 2.26032 8.31528

29762 48485 154 00790 24565 931 23533 28209 759 94069 81054 326 71910 35678 841

!- 713.05902 32050 18257 884

Ii

- 7 9 1.12535 8 1.52299 17471 5.60279 4.13993 64375 85166 79744 77187 37267 844 71262 92591 540 660 145 2.06115 7.58256 2.78946 1.02618 3.77513

36224 38557 828 04279 11906 728 80928 68924 808 79631 70189 030 45442 79097 752

99337 38587 252--. 60942 88387 643 24060 17986 167 70642 91475 17442971 44042 074-

-11 1.38879 -12 5.10908 -12 1.87952 -13 6.91440 I -13 I 2.54366

43864 96402 059 90280 63324 720 88165 39083 295 01069 40203 009 56473 76922 910

1.06864 2.90488 7.89629 2.14643 5.83461

74581 52446 215, 49665 24742 5232 60182 68069 516 57978 59160 646~ 74252 74548 814

-14 9.35762 -14 3.44247 -14 1.26641 -15 4.65888 I -15 I 1.71390

29688 40174 605 71084 69976 458 65549 09417 572 61451 03397 364 84315 42012 966

(15)1.58601 4.31123 1.17191 3.18559 (16)8.65934

34523 13430 728 15471 15195 227, 42372 80261 1311 31757 11375 622~ 00423 99374 695

6.30511 2.31952 8.53304 3.13913 1.15482

67601 46989 386 28302 43569 388 76257 44065 794 27920 48029 629 24173 01578 599

(17)2.35385 6.39843 1.73927 4.72783 (19)1.28516

26683 70199 854 49353 00549 492 49415 20501 047 94682 29346 561 00114 35930 828

4.24835 1.56288 5.74952 2.11513 7.78113

42552 91588 995 21893 34988 768 22642 93559 807 10375 91080 487 22411 33796 516

45 46 47 48 49

I 19 20I 9.49611 3.49342 28861 2.58131 7.01673 59120 48509 94206 71057 97631 90067 02448 535 396 739 875

2.86251 1.05306 3.87399 1.42516 5.24288

85805 49393 644 17357 55381 238 76286 87187 113 40827 40935 106 56633 63463 937

50

(21)5.18470 55285 87072 464

:: ;34

40 2 43 44

(21)1.90734 65724 95099 691

(-22)1.92874 98479 63917 783

ELEMENTARY

TRANSCENDENTAL

EXPONENTIAL

FUNCTIONS

FUNCTION

ez

Table

4.4

ec2

5.18470 1.40934 3.83100 1.04137 2.83075

55285 90824 80007 59433 33032

87072 26938 16576 02908 74693

464 796 849 780 900

1.92874 7.09547 2.61027 9.60268 3.53262

98479 41622 90696 00545 85722

63917 84704 67704 08676 00807

783 139 805 030 030

7.69478 2.09165 5.68571 1.54553 4.20121

52651 94960 99993 89355 04037

42017 12996 35932 90103 90514

138 154 223 930 255

1.29958 4.78089 1.75879 6.47023 2.38026

14250 28838 22024 49256 64086

07503 85469 24311 45460 94400

074 081 649 326 606

1.14200 3.10429 8.43835 2.29378 6.23514

73898 79357 66687 31594 90808

15684 01919 41454 69609 11616

284 909 489 879 883

8.75651 3.22134 Iv.18506 4.35961 1.60381

07626 02859 48642 00000 08905

96520 92516 33981 63080 48637

338 089 006 974 853

(28)1.69488 4.60718 1.25236 3.40427 9.25378

92444 66343 31708 60499 17255

10333 31291 42213 31740 87787

714 543 781 521 600

2.51543 6.83767 1.85867 5.05239 1.37338

86709 12297 17452 36302 29795

19167 62743 84127 76104 40176

006 867 980 195 188

3.97544 1.46248 5.38018 1.97925 7.28129

97359 62272 61600 98779 01783

08646 51230 21138 46904 21643

808 947 414 554 834

3.73324 1.01480 2.75851 7.49841 2.03828

19967 03881 34545 69969 10665

99001 13888 23170 90120 12668

640 728 206 435 767

2.67863 9.85415 3.62514 1.33361 4.90609

69618 46861 09191 48155 47306

08077 11258 43559 02261 49280

944 029 224 341 566

(34j5.54062 1.50609 4.09399 1.11286 3.02507

23843 73145 69621 37547 73222

93510 85030 27454 91759 01142

053 548 697 412 338

1.80485 6.63967 2.44260 8.98582 3.30570

13878 71995 07377 59440 06267

45415 80734 40527 49380 60734

172 401 679 670 298

8.22301 2.23524 6.07603 1.65163 4.48961

27146 66037 02250 62549 28191

22913 34715 56872 94001 74345

510 047 150 856 246

1.21609 4.47377 1.64581 6.05460 2.22736

92992 93061 14310 18954 35617

52825 81120 82273 01185 95743

564 735 651 885 739

1.22040 3.31740 9.01762 2.45124 6.66317

32943 00983 84050 55429 62164

17840 35742 34298 20085 10895

802 626 931 786 834

8.19401 3.01440 1.10893 4.07955 1.50078

26239 87850 90193 86671 57627

90515 65374 12136 77560 07394

430 553 379 158 888

1.81123 4.92345 1.33833 3.63797 9.88903

90828 82860 47192 09476 03193

89023 12058 04269 08804 46946

282 400 500 579 771

5.52108 2.03109 7.47197 2.74878 1.01122

22770 26627 23373 50079 14926

28532 34810 42990 10214 10448

732 926 161 930 530

(43)2.68811 For 1zj>lOO see

139

Example

11.

71418 16135 448

II

f-29)5.90009

05415 97061 391

-30 -29 2.93748 7.98490 20113 2.17052 1.08063 21117 03639 42456 92777 10802 412 07278 86978 808 495 947

(-44)3.72007

59760 20835 963

140 Table

ELEMENTARY

4.5

TRANSCENDENTAL

RADIX @O-

TABLE

FUNCTIONS

OF THE

EXPONENTIAL

fl

FUNCTION e-2210-n

1.00000 1.00000 1I00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

00001 00002 00003 00004 00005 00006 00007 00008 00009

00000 00000 00000 00000 00000 00000 00000 00000 00000

00000 00002 00004 00008 00012 00018 00024 00032 00040

50000 00000 50000 00000 50000 00000 50000 00000 50000

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

99999 99998 99997 99996 99995 99994 99993 99992 99991

00000 00000 00000 00000 00000 00000 00000 00000 00000

00000 00002 00004 00008 00012 00018 00024 00032 00040

50000 00000 50000 00000 50000 00000 50000 00000 50000

1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

00010 00020 00030 00040 00050 00060 00070 00080 00090

00000 00000 00000 00000 00000 00000 00000 00000 00000

00050 00200 00450 00800 01250 01800 02450 03200 04050

00000 00000 00000 00000 00000 00000 00001 00001 00001

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

99990 99980 99970 99960 99950 99940 99930 99920 99910

00000 00000 00000 00000 00000 00000 00000 00000 00000

00050 00200 00450 00800 01250 01800 02449 03199 04049

00000 00000 00000 00000 00000 00000 99999 99999 99999

1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

00100 00200 00300 00400 00500 00600 00700 00800 00900

00000 00000 00000 00000 00001 00001 00002 00003 00004

05000 00002 20000 00013 45000 00045 80000 00107 25000 00208 80000 00360 45000 00572 20000 00853 05000 -01215

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

99900 99800 99700 99600 99500 99400 99300 99200 99100

00000 00000 00000 00000 00001 00001 00002 00003 00004

04999 19999 44999 79999 24999 79999 44999 19999 04999

99998 99987 99955 99893 99792 99640 99428 99147 98785

1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

01000 02000 03000 04000 05000 06000 07000 08000 09000

00005 00020 00045 00080 00125 00180 00245 00320 00405

00000 00000 00000 00001 00002 00003 00005 00008 00012

01667 13333 45000 06667 08333 60000 71667 53334 15000

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.94999

99000 98000 97000 96000 95000 94000 93000 92000 91000

00004 00019 00044 00079 00124 00179 00244 00319 00404

99999 99999 99999 99998 99997 99996 99994 99991 99987

98333 86667 55000 93333 91667 40000 28333 46667 85000

1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

10000 20000 30000 40000 50000 60000 70000 80000 90000

00500 02000 04500 08000 12500 18000 24500 32000 40500

00016 00133 00450 01066 02083 03600 05716 08533 12150

66667 33340 00034 66773 33594 00540 67667 35040 02734

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

90000 80000 70000 60000 50000 40000 30000 20000 10000

00499 01999 04499 07999 12499 17999 24499 31999 40499

99983 99866 99550 98933 97916 96400 94283 91466 87850

33334 66673 00034 33440 66927 00540 34334 68373 02734

Compiled from C. E. Van Orstrand, Tables of the exponential function and of the circular sine and cosine to radian arguments, Memoirs of the National Academy of Sciences,vol. 14, Fifth Memoir. U.S. Government Printing Office, Washington, D.C., 1921 (with permission),

ELEMENTARY

RADIX

13

Table

4.5

16666 33334 50003 66677 83359 00054 16766 33504 50273

70833 00000 37502 33342 37526 00065 70973 00273 37992

0.99999 0.99998 0.99997 0.99996 0.99995 0.99994 0.99993 0.99992 0.99991

00000 00001 00004 00007 00012 00017 00024 00031 00040

49999 99998 49995 99989 49979 99964 49942 99914 49878

83333 66667 50003 33343 16692 00053 83433 66837 50273

1.00010 1.00020 1.00030 1.00040 1.00050 1.00060 1.00070 1.00080 1.00090

00050 00200 00450 00800 01250 01800 02450 03200 04051

001% 01333 04500 10667 20835 36005 57176 85350 21527

67083 40000 33752 73341 93776 40064 67223 40273 34242

34167 26668 02510 86724 04384 80648 40801 10308 14882

0.99990 0.99986 0.99970 0.99960 0.99950 0.99940 0.99930 0.99920 0.99910

00049 30199 00449 00799 01249 01799 02449 03199 04048

99833 98666 95500 89334 79169 64005 42843 14683 78527

33749 99167 73333 06668 33747

97512

39991 27057 39935 33609 73060 33257

46724 29384 20648 95801 30307 99880

1.00100 1.00200 1.00300 1.00400 1.00501 1.0060.1 1.00702 1.00803 1.00904

05001 20013 45045 80106 25208 80360 45572 20855 06217

66708 34000 03377 77341 59401 54064 66848 04273 73867

34166 26675 02601 87235 06338 86485 55523 43117 81406

80558 55810 29341 88080 35662 55845 16000 20736 25705

0.99900 0.99800 0.99700 0.99600 0.99501 0.99401 0.99302 0.99203 0.99104

04998 19986 44955 79893 24791 79640 44429 19148 03787

33374 67333 03372 43991 92682 53935 33235 37060 72883

99166 06675 97601 47235 31335 26474 10490 63033 66216

80554 55302 20662 23064 25642 44988 47970 98697 45648

1.01005 1.02020 1.03045 1.04081 1.05127 1.06183 1.07250 1.08328 1.09417

01670 13400 45339 07741 10963 65465 81812 70676 42837

84168 26755 53516 92388 76024 45359 54216 74958 05210

05754 81016 85561 22675 03969 62222 47905 55443 35787

21655 01439 24400 70448 75176 46849 31039 59878 28976

0.99004 0.98019 0.97044 0.96078 0.95122 0.94176 0.93239 0.92311 0.91393

98337 86733 55335 94391 94245 45335 38199 63463 11852

49168 06755 48508 52323 00714 84248 05948 86635 7122‘8

05357 30222 17693 20943 00909 70953 22885 78291 18674

39060 08141 25284 92107 14253 71528 79726 07598 73535

09180 27581 88075 46976 12707 88003 27074 09284 31111

75647 60169 76003 41270 00128 90508 70476 92467 56949

62481 83392 10398 31782 14684 97487 52162 60457 66380

17078 10720 37443 48530 86508 53677 45494 95375 01266

0.90483 0.81873 0.74081 0.67032 0.60653 0.54881 0.49658 0.44932 0.40656

74180 07530 82206 00460 06597 16360 53037 89641 96597

35959 77981 81717 35639 12633 94026 91409 17221 40599

57316 85866 86606 30074 42360 43262 51470 59143 11188

42491 99355 68738 44329 37995 84589 48001 01024 34542

18284 59045 23536 02875

0.36787

94411 71442 32159 55238

1

0

2.71828

;

FUNCTION

50000 00001 50004 00010 50020 00036 50057 00085 50121

1 1

8'

EXPONENTIAL

00000 00002 00004 00008 00012 00018 00024 00032 00040

9

6

OF THE

141

FTJNCTIONS

1.00001 1.00002 1.00003 1.00004 1.00005 1.00006 1.00007 1.00008 1.00009

1.10517 1.22140 1.34985 1.49182 1.64872 1.82211 2.01375 2.22554 2.45960

54 ;

TABLE

TRANSCENDENTAL

37500 33333 37498 99991 70807 99935 37360 33060 37008

142

ELEMENTARY Table

4.6

CIRCULAR

TRANSCENDENTAL SINES

AND

COSINES

0.000

sin z 0.00000 00000 00000 00000 000

0.001 0.002 0.003 0.004

0.00099 0.00199 0.00299 0.00399

99998 99986 99955 99893

33333 66666 00002 33341

34166 93333 02499 86666

0.005 0.006 0.007 0.008 0.009

0.00499 0.00599 0.00699 0.00799 0.00899

99791 99640 99428 99146 98785

66692 00064 33473 66939 00492

0.010 0.011 00. “0:: 0: 014

0.00999 0.01099 0.01199 0.01299 0.01399

98333 97781 97120 96338 95426

0.015 0.016 0.017 0.018 0.019

0.01499 0.01599 0.01699 0.01799 0.01899

0.020 0.021 0.022 0.023 0.024

I

FUNCTIONS FOR

RADIAN

ARGUMENTS

cos x

667 331 957 342

1.00000 0.99999 0.99999 0.99999 0.99999

00000 95000 80000 55000 20000

00000 00041 00666 03374 10666

00000 66666 66657 99898 66097

000 528 778 750 778

70831 79994 39150 73291 07405

783 446 327 723 100

0.99998 0.99998 0.99997 0.99996 0.99995

75000 20000 55001 80001 95002

26041 53999 00041 70666 73374

64496 93520 50326 30257 26188

529 004 542 El’9 857

34166 68008 02073 36427 71148

66468 75446 59289 42921 51241

254 684 053 659 801

0.99995 0.99993 0.99992 0.99991 0.99990

00004 95006 80008 55011 20016

16665 10039 63995 90034 00656

27778 20617 85281 96278 20901

026 059 066 551 438

94375 93173 91811 90280 88568

06328 42071 78498 15746 53967

09109 41340 72691 27852 31431

944 585 726 832 205

0.99988 0.99987 0.99985 0.99983 0.99981

75021 20027 55034 80043 95054

09359 30643 80008 73952 29976

17975 36508 14243 76107 32558

106 430 829 331 650

0.01999 0.02099 0.02199 0.02299 0.02399

86666 84565 82253 79722 76960

93333 34033 76279 20302 66354

07936 81764 77175 18277 28999

649 335 771 769 311

0.99980 0.99977 0.99975 0.99973 0.99971

00066 95081 80097 55116 20138

66577 03255 60509 59836 23734

77841 88132 19593 06320 58193

270 556 878 750 002

0.025 0.026 0.027 0.028 0.029

0.02499 0.02599 0.02699 0.02799 0.02899

73959 70707 67196 63414 59353

14712 65676 19572 76750 37589

33066 53973 14955 38952 48577

217 517 411 746 881

0.99968 0.99966 0.99963 0.99960 0.99957

75162 20190 55221 80256 95294

75702 40237 42836 09997 69215

58624 62215 92299 38394 53557

967 698 214 779 207

0.030 0.031 0.032 0.033 0.034

0.02999 0.03099 0.03199 0.03299 0.03399

55002 50350 45389 40108 34497

02495 71904 46280 26119 11951

66076 13288 11602 81908 44553

853 752 188 762 435

0.99955 0.99951 0.99948 0.99945 0.99942

00337 95384 80436 55494 20556

48987 78809 89175 11581 78521

51627 04381 38584 32936 14926

216 810 710 824 773

0.035 0.036 0.037 0.038 0.039

0.03499 0.03599 0.03699 0.03799 0.03899

28546 22245 15584 08553 01142

04336 03869 11180 26937 51841

19281 25183 80633 03228 09720

702 461 489 414 085

0.99938 0.99935 0.99931 0.99927 0.99923

75625 20699 55780 80868 95963

23488 80976 86478 76484 88487

57581 76116 24487 91840 98862

460 700 902 819 358

0.040 0.041 0.042 0.043 0.044

0.03998 0.04098 0.04198 0.04298 0.04398

93341 85141 76530 67500 58040

86634 32096 89047 58349 40905

15945 36751 85918 76078 18626

255 449 946 755 492

0.99920 0.99915 0.99911 0.99907 0.99903

01066 96177 81296 56424 21561

60977 33444 46376 41262 60588

94031 49770 58494 28564 80138

457 040 043 524 853

0.045 0.046 0.047 0.048 0.049

0.04498 0.04598 0.04698 0.04798 0.04898

48140 37790 26980 15701 03941

37660 49604 77774 23249 87159

23632 99745 54095 92191 17808

066 054 689 340 403

0.99898 0.99894 0.99889 ii99884 0.99879

76708 21865 57033 82211 97401

47842 47508 05071 67013 80818

40921 41817 12480 76767 48087

992 869 849 299 272

0.050

0.04997

91692

70678

32879

487

0.99875

02603

94966

24656

287

C-79)6 For conversion from degrees to radians see Example 13. For use and extension of the table see Examples 15-17. From C. E. Van Orstrand, Tables of the exponential functionand of thecircular sine and cosine to radian arguments,Memoirs of the National Academy of Sciences, vol. 14, Fifth Memoir. U.S. Government Printing Office, Washington, D.C., 1921 (with permission). Known errors have been corrected.

ELEMENTARY CIRCULAR

SINES

AND

COSINES

TRANSCENDENTAL FOR

RADIAN

ARGUMENTS

sin 2

2

143

FUNCTIONS Table co9

4.6

5

0.050 0.051 0.052 0.053 0.054

0.04997 0.05097 0.05197 0.05297 0.05397

91692 70678 32879 487 78943 75032 37375 800 65685 01496 29184 649 51906 51396 03981 925 37598 26109 55099 505

0.99875 0.99869 0.99864 0.99859 0.99854

02603 94966 24656 287 97818 58936 84647 237 83046 23208 81242 407 58287 39259 37585 623 23542 59564 41634 531

0.055 0.056 0.057 0.058 0.059

0.05497 0.05597 0.05696 0.05796 0.05896

22750 27067 73387 446 07352 55755 47070 891 91395 13712 61601 567 74868 02534 99503 794 57761 23875 40214 896

0.99848 0.99843 0.99837 0.99831 0.99826

78812 37598 40913 005 24097 27834 37163 704 59397 85743 80900 770 84714 67796 65862 676 00048 31461 23365 235

0.060 0.061 0.062 0.063 0.064

0.05996 0.06096 0.06196 0.06295 0.06395

40064 79444 59919 909 21768 71012 31380 500 02863 00408 23757 982 83337 69523 02430 343 63182 80309 28803 166

0.99820 0.99814 0.99807 0.99801 0.99795

05399 35204 16554 766 00768 38490 34561 437 86156 01782 86552 769 61562 86542 95687 334 26989 55229 92968 628

0.065 0.066 0.067 0.068 0.069

0.06495 0.06595 0.06694 0.06794 0.06894

42388 34782 60114 361 20944 35022 49232 601 98840 83173 44449 361 76067 81445 89264 458 52615 32117 22165 004

0.99788 0.99782 0.99775 0.99768 0.99762

82436 71301 10999 144 27904 99211 77634 635 63395 04415 09538 592 88907 53362 05636 926 04443 13501 40472 866

0.070 0.071 0.072 0.073 0.074

0.06994 0.07094 0.07193 0.07293 0.07393

28473 37532 76397 655 03632 00106 79734 071 78081 22323 54229 480 51811 06738 15974 250 24811 55977 74838 360

0.99755 0.99748 0.99740 0.99733 0.99726

10002 53279 57462 091 05586 42140 62048 084 91195 50526 14757 726 66830 49875 24157 139 32492 12624 39707 777

0.075 0.076 0.077 0.078 0.079

0.07492 0.07592 0.07692 0.07792 0.07891

97072 72742 34208 684 68584 59805 90718 980 39337 20017 33972 485 09320 56301 46257 015 78524 71660 02252 478

0.99718 0.99711 0.99703 0.99695 0.99688

88181 12207 44522 774 33898 23055 48023 568 69644 20596 78496 785 95419 81256 75551 417 11225 82457 82476 279

0.080 0.081 0.082 0.083 0.084

0.07991 0.08091 0.08190 0.08290 0.08390

46939 69172 68730 688 14555 51998 04247 389 81362 23374 58826 394 47349 86621 73635 718 12508 45140 80655 638

0.99680 0.99672 0.99663 0.99655 0.99647

17063 02619 38497 771 12932 21157 70937 933 98834 18485 87272 823 74769 76013 67091 212 40739 76147 53953 598

0.085 0.086 0.087 0.088 0.089

0.08489 0.08589 0.08689 0.08788 0.08888

76828 02416 02338 544 40298 62015 51260 514 02910 27592 29764 492 64653 02885 29594 973 25516 91720 31524 112

0.99638 0.99630 0.99621 0.99613 0.99604

96745 02290 47151 570 42786 38841 93367 506 78864 71197 78234 626 04980 85750 17797 412 21135 69887 49872 388

0.090 0.091 0.092 0.093 0.094

0.08987 0.09087 0.09187 0.09286 0.09386

85491 98011 04969 125 44568 25760 07600 919 02735 79059 84943 819 59984 62093 69966 323 16304 79136 82662 751

0.99595 0.99586 0.99577 0.99567 0.99558

27330 11994 25309 284 23565 01450 99152 586 09841 28634 21703 483 86159 84916 29482 217 52521 62665 36090 844

0.095 0.096 0.097 0.098 0.099

0.09485 0.09585 0.09684 0.09784 0.09883

71686 34557 29625 724 26119 32817 03609 347 79593 78472 83083 006 32099 76177 31775 683 83627 30679 98210 683

0.99549 0.99539 0.99529 0.99520 0.99510

08927 55245 22976 426 55378 57015 30094 649 91875 63330 46473 881 18419 70541 00679 686 35011 75992 51179 796

0.100

0.09983 34166 46828 15230 681

0.99500 41652 78025 76609 556

144

ELEMENTARY TRANSCENDENTAL FUNCTIONS Table

4.6

CIRCULAR

SINES

AND COSINES

FOR RADIAN

cos x

sin x

X

ARGUMENTS

0.100 0.101 0.102 0.103 0.104

0.09983 0:10082 0.10182 0;10281 0.10381

34166 46828 15230 681 83707 29567 99512 975 32239 83945 51074 864 79754 15107 52769 040 26240 28302 69768 897

0.99500 0.99490 0.99480 0.99470 0.99459

41652 78025 76609 556 38343 75976 65937 840 25085 70176 08533 469 01879 61949 84132 117 68726 53618 52703 737

0.105 0.106 0.107 0.108 0.109

0.10480 0.10580 0.10679 0.10779 0.10878

71688 28882 49043 655 16088 22302 18823 209 59430 14121 88052 588 01704 10007 45835 941 42900 15731 60869 939

0.99449 0.99438 0.99428 0.99417 0.99406

25627 48497 44220 501 72583 50896 48325 268 09595 66120 03900 596 36665 00466 88538 307 53792 61230 07909 607

0.110 0.111 0.112 0.113 0.114

0.10977 0.11077 0.11176 0.11275 0.11375

83008 37174 80866 495 22018 80326 31964 714 59921 51285 18131 952 96706 56261 20553 909 32364 01575 97013 636

0.99395 0.99384 0.99373 0.99362 0.99350

60979 56696 85035 784 58226 96148 49459 483 45535 89860 26316 578 22907 49101 25308 652 90342 86134 29576 080

0.115 0.116 0.117 0.118 0.119

0.11474 0.11574 0.11673 0.11772 0.11871

66883 93663 81259 372 00256 39072 82361 097 32471 44465 84055 722 63519 16621 44080 790 93389 62434 93496 613

0.99339 0.99327 0.99316 0.99304 0.99292

47843 14215 84471 755 95409 47595 86235 439 33043 01517 70568 768 60744 92218 01110 921 78516 36926 57814 950

0.120 0.121 0.122 0.123 0.124

0.11971 0.12070 0.12169 0.12269 0.12368

22072 88919 35996 735 49559 03206 47206 615 75838 12547 73970 447 00900 24315 33626 003 24735 46003 13267 407

0.99280 0.99268 0.99256 0.99244 0.99232

86358 53866 25224 810 84272 62252 80653 067 72259 82294 82259 329 50321 35193 57029 382 18458 43142 88655 070

0.125 0.126 0.127 0.128 0.129

0.12467 0.12566 0.12665 0.12765 0.12864

47333 85227 68995 744 68685 49729 25157 389 88780 47372 73569 978 07608 86148 72735 909 25160 74174 47043 273

0.99219 76672 29329 05314 910 Oi99207 24964 17930 67355 462 0.99194 63335 34118 54873 474 0.99181 91787 04055 55198 803 0.99169 10320 54896 50278 123

0.130 0.131 0.132 0.133 0.134

0.12963 0.13062 0.13161 0.13260 0.13359

41426 19694 85954 121 56395 31083 43179 968 70058 16843 35844 433 82404 85608 43632 907 93425 46144 07929 171

0.99156 0.99143 0.99130 0.99116 0.99103

18937 14788 03959 451 17638 12868 49177 481 06424 79267 75039 751 85298 45107 13813 659 54260 42499 27814 325

0.135 0.136 0.137 0.138 0.139

0.13459 0.13558 0.13657 0.13756 0.13855

03110 07348 30938 844 11448 78252 74799 575 18431 68023 60677 867 24048 85962 67852 453 28290 41508 32784 107

0.99090 0.99076 0.99063 0.99049 0.99035

13312 04547 96193 339 62454 65348 01628 375 01689 59985 16913 714 31018 24535 91451 667 50441 96067 37644 937

0.140 0.141 0.142 0.143 0.144

0.13954 0.14053 0.14152 0.14251 0.14350

31146 44236 48171 799 32607 03861 61995 092 32662 30237 76542 691 31302 33359 47427 025 28517 23362 82584 791

0.99021 0.99007 0.98993 0.98979 0.98964

59962 12637 17189 895 59580 13293 27270 829 49297 38073 86655 145 29115 28007 21689 546 99035 25111 52197 214

0.145 0.146 0.147 0.148 0.149

0.14449 0.14548 0.14647 0.14746 0.14844

24297 10526 41263 332 18632 05272 32992 773 11512 18167 16543 800 02927 59922 98870 997 92868 41398 34041 627

0.98950 0.98936 0.98921 0.98906 0.98892

59058 72394 77275 984 09187 13854 60997 551 49421 94478 18007 704 79764 60241 99027 617 00216 58111 76256 193

0.150

0.14943 81324 73599 22149 773 p-p1 L

’

J

0.98877 10779 36042 28673 498

cc-y1

ELEMENTARY

~IR~UIAR SINES

TRANSCENDENTAL

FUNCTIONS

AND COSINES FOR RADIAN

ARGUMENTS

Table 4.6 cos x

sin z

X

145

0.150 0.151 0.152 0.153 0.154

0.14943 0.15042 Oil5141 0.15240 0.15339

81324 68286 53744 37687 20107

73599 67680 34944 86847 34994

22149 08215 81070 72225 54727

773 725 532 604 267

0.98877 0.98862 0.98847 0.98831 0.98816

10779 36042 11454.42977 02243 28849 83147 44579 54168 42076

28673 27245 20028 17178 75856

498 283 611 614 382

0.155 0.156 0.157 0.158 0.159

0.15438 0.15536 0.15635 0.15734 0.15833

00992 80334 58122 34347 08998

91143 67205 75247 27490 36311

41996 86651 79319 47428 53983

190 555 902 529 354

0.98801 0.98785 0.98770 0.98754 0.98738

15307 66566 07947 39451 61079

74239 94954 59094 22522 42087

85038 50224 78054 60814 60855

006 794 663 736 150

0.160 0.161 0.162 0.163 0.164

0.15931 0.16030 0.16129 0.16227 0.16326

82066 53540 23412 91670 58306

14245 73987 28387 90460 73379

96331 04906 41960 00278 01876

146 020 095 226 705

0.98722 0.98706 0.98690 0.98674 0.98658

72833 74715 66727 48869 21144

75626 81965 20914 53272 40826

94904 18284 09029 51905 22328

095 099 574 638 234

0.165 0.166 0.167 0.168 0.169

0.16425 0.16523 0.16622 0.16721 0.16819

23309 86670 48378 08424 66798

90480 55265 81396 82704 73183

96685 61216 97208 30268 08481

825 228 916 843 981

0.98641 0.98625 0.98608 0.98592 0.98575

83553 36098 78780 11602 34564

46347 33596 67316 13241 38088

70185 03560 72356 51818 25966

554 791 233 712 434

0.170 0.171 0.172 0.173 0.174

0.16918 0.17016 0.17115 0.17213 0.17312

23490 78490 31789 83376 33241

66996 78473 22117 12595 64750

01015 96702 02607 42577 55773

762 805 812 560 865

0.98558 0.98541 0.98524 0.98507 0.98490

47669 50917 44312 27854 01546

09560 96348 68126 95555 50280

70917 38117 37476 20391 62691

193 998 124 598 158

0.175 0.176 0.177 0.178 0.179

0.17410 0.17509 0.17607 0.17706 0.17804

81375 27769 72411 15292 56403

93595 14318 42278 93011 82230

95189 26146 24778 76492 74417

433 505 176 317 975

0.98472 0.98455 0.98437 0.98419 0.98402

65389 19384 63534 97840 22304

04933 33129 09469 09537 09903

47463 47797 09416 33225 57745

670 052 699 443 046

0.180 0.181 0.182 0.183 0.184

0.17902~95734 0.180Dl 33274 0.18099 69014 0.18198 02944 0.18296 35054

25824 39859 40581 44417 67974

17834 10581 59452 72574 57756

180 029 980 233 116

0.98384 0.98366 0.98348 0.98330 0.98311

36927 41713 36661 21775 97056

88121 22728 93246 80179 65017

41459 45058 13586 58485 39552

272 522 083 974 448

0.185 0.186 0.187 0.188 0.;89

0.18394 0.18492 0,18591 0.18689 0.18787

65335 93776 20368 45100 67964

28041 41589 25775 97940 75611

20836 64000 84083 70855 05288

370 231 224 554 013

0.98293 0.98275 0.98256 0.98237 0.98219

62506 18126 63919 99886 26029

30231 59276 36591 47595 78693

46781 82121 41132 94537 69683

122 799 959 971 022

0.190 0.191 0.192 0.193 0.194

0.18885 0.18984 0.19082 0.19180 0.19278

88949 08046 25244 40533 53905

76500 18510 19732 98445 73120

57799 86484 35325 32380 87958

285 571 424 691 485

0.98200 0.98181 0.98162 0.98143 0.98124

42351 48852 45535 32402 09455

17270 51693 71313 66461 28451

31896 65751 56228 69777 35290

788 875 034 178 214

0.195 0.196 0.197 0.198 0.199

0.19376 0.19474 0.19572 0.19670 Oil9768

65349 74855 82414 88016 91650

62421 85204 60517 07604 45907

92769 16058 03723 76404 27565

058 510 204 820 917

0.98104 0.98085 0.98065 0.98046 0.98026

76695 34125 81746 19561 47571

49577 23115 43322 05437 05677

24965 35080 66661 06062 05434

723 479 867 170 796

0.200

0.19866 93307 95061 21545 941 c-y

c 1

0.98006 65778 41241 63112 420 c-y

1

146

ELEMENTARY

Table

4.6

CIRCULAR

TRANSCENDENTAL

SINES

AND COSINES

FUNCTIONS

FOR RADIAN

ARGUMENTS

sin x

cos x

0.200 0.201 0.202 0.203 0.204

0.19866 93307 95061 21545 941 0.19964 92978 74900 91597 545 OI20062 90653 05459 37903 151 0.20160 86321 06969 25571 640 0.20258 79972 99863 82615 083

0.98006 65778 41241 63112 420 0.97986 74185 10310 03887 090 0.97966 72793 12041 59192 306 0.97946 61604 46575 47187 084 OI97926 40621 15030 52742 047

0.205 0.206 0.207 0.208 0.209

0.20356 0.20454 0.20552 0.20650 0.20748

71599 04777 97905 397 61189 42549 19110 856 48734 34218 50612 330 34224 01031 51399 175 17648 64439 32944 665

0.97906 0.97885 0.97865 0.97844 0.97823

09845 19505 07327 536 69278 63076 68803 784 18923 49802 01113 156 58781 84716 53874 491 88855 73834 41879 553

0.210 0.211 0.212 0.213 0.214

0.20845 0.20943 0.21041 0.21139 0.21237

98998 46099 57060 871 78263 67877 33732 895 55434 51846 18932 346 30501 20289 12409 982 03453 95699 55467 398

0.97803 0.97782 0.97761 0.97740 0.97718

09147 24148 24491 614 19658 43628 84946 201 20391 41225 09554 014 11348 26863 66806 039 92531 11448 86380 882

0.215 0.216 0.217 0.218 0.219

0.21334 0.21432 0.21530 0.21627 0.21725

74283 00782 28707 677 42978 58454 49764 905 09530 91846 71012 439 73930 24303 77249 851 36166 79385 83368 434

0.97697 0.97676 0.97654 0.97633 0.97611

63942 06862 38054 344 25583 25963 10511 247 77456 82586 90059 555 19564 91546 39246 782 51909 68630 75378 736

0.220 0.221 0.222 0.223 0.224

0.21822 0.21920 0.22018 0.22115 0.22213

96230 80869 31995 179 54112 52747 91115 124 09802 19233 51671 977 63290 04757 25146 920 i4566 33970 41115 484

0.97589 0.97567 0.97545 0.97523 0.97501

74493 30605 48940 602 87317 95212 21920 392 90385 81168 46034 788 83699 08167 40857 388 67259 96877 71849 392

0.225 0.226 0.227 0.228 0.229

0.22310 0.22408 0.22505 0.22602 0.22700

63621 31745 44782 417 10445 23176 94494 428 55028 33582 59230 720 97360 88504 16071 214 37433 13708 47642 363

0.97479 0.97457 0.97434 0.97412 0.97389

41070 68943 28292 737 05133 46983 01125 708 59450 54590 60681 052 04024 16334 34326 607 38856 57756 84008 477

0.230 0.231 0.232 0.233 0.234

0.22797 0.22895 0.22992 0.23089 0.23187

75235 35188 39540 462 10757 79163 77732 354 43990 72082 45933 437 74924 40621 22962 869 03549 11686 80075 884

0.97366 0.97343 0.97320 0.97297 0.97274

63950 0.537483696 773 79306 86678 96733 940 84929 30133 53085 695 80819 65176 26494 602 66980 22218 11536 294

0.235 0.236 0.237 0.238 0.239

0.23284 0.23381 0.23478 0.23575 0.23673

29855 12416 78273 112 53832 70180 65586 809 75472 12580 74343 904 94763 67453 18405 752 11697 62868 90384 520

0.97251 0.97228 0.97204 0.97181 0.97157

43413 32643 00578 389 10121 28807 60642 091 67106 44041 10166 529 14371 12644 95675 843 51917 69892 68349 034

0.240 0.241 0.242 0.243 0.244

0.23770 0.23867 0.23964 0.24061 0.24158

26264 27134 58836 079 38453 88793 65429 334 48256 76627 22091 869 55663 19655 08131 828 60663 47136 67335 933

0.97133 0.97109 0.97086 0.97062 0.97037

79748 52029 60492 618 97865 96272 61916 095 06272 40809 96210 262 04970 24800 96928 391 93961 88375 83670 294

0.245 0.246 0.247 0.248 0.249

0.24255 0.24352 0.24449 0.24546 0.24643

63247 88572 05043 522 63406 73702 85196 546 61130 32513 27365 389 56408 95231 03750 445 49232 92328 36159 337

0.97013 0.96989 0.96965 0.96940 0.96915

73249 72635 38069 313 42836 19650 79682 233 02723 72463 41782 166 52914 75084 47054 425 93411 72494 83195 397

0.250

0.24740 39592 54522 92959 685 c-:)3

2

[ 1

0.96891 24217 10644 78414 459

1c-y1

ELEMENTARY CIRCULAR

SINES

TRANSCENDENTAL

.4ND COSINES

FOR RADIAN

ARGUMENTS

Table

4.6

cos x

sin x

X

147

FUNCTIONS

0.250 0.251 0.252 0.253 0.254

0.24740 0.24837 0.24934 0.25030 0.25127

39592 54522 92959 685 27478 12778 86007 332 12879 98307 67549 922 95788 42569 27105 742 76193 77272 88317 722

0.96891 0.96866 0.96841 0.96816 0.96791

24217 10644 78414 459 45333 36453 76838 955 56762 97810 13822 250 58508 43570 91154 897 50572 23561 52178 941

0.255 0.256 0.257 0.258 0.259

0.25224 0.25321 0.25418 0.25514 0.25611

54086 34378 05782 506 29456 46095 61854 486 02294 44888 63424 714 72590 63473 38674 587 40335 34820 33804 209

0.96766 0.96741 0.96715 0.96690 0.96664

32956 88575 56805 375 05664 90374 56434 780 68698 81687 68781 180 22061 16211 52599 126 65754 48609 82314 035

0.260 0.261 0.262 0.263 0.264

0.25708 0.25804 0.25901 0.25997 0.26094

05518 92155 09735 339 68131 68959 38788 820 28163 98972 01336 401 85606 16189 82426 844 40448 54868 68386 239

0.96638 0.96613 0.96587 0.96561 0.96535

99781 34513 22555 822 24144 30519 02595 835 38845 94190 90687 131 43888 84058 68308 107 39275 59618 04309 520

0.265 0.266 0.267 0.268 0.269

0.26190 0.26287 0.26383 0.26480 0.26576

92681 49524 43392 399 42295 34933 86023 278 89280 46135 65779 278 33627 18431 39579 372 75325 87386 48230 942

0.96509 0.96483 0.96456 0.96430 0.96403

25008 81330 28964 923 01091 10622 07924 537 67525 09885 16072 584 24313 42476 11288 118 71458 72716 08109 368

0.270 0.271 0.272 0.273 0.274

0.26673 0.26769 0.26865 0.26962 0.27058

14366 88831 12873 229 50740 58861 31394 301 84437 33839 74821 451 15447 50396 83684 915 43761 45431 64354 828

0.96377 0.96350 0.96323 0.96296 0.96269

08963 65890 51301 623 36830 88248 89328 696 55063 07004 47727 972 63662 90334 02389 084 62633 07377 52736 246

0.275 0.276 0.277 0.278 0.279

0.27154 0.27250 0.27347 0.27443 0.27539

69369 56112 85351 302 92262 19879 73627 557 12429 74443 10825 981 29862 57786 29507 043 44551 08166 09350 952

0.96242 0.96215 0.96188 0.96160 0.96133

51976 28237 94814 248 31695 23980 94278 169 01792 66634 59286 807 62271 29189 13299 879 13133 85596 67778 997

0.280 0.281 0.282 0.283 0.284

0.27635 0.27731 0.27827 0.27923 0.28019

56485 64113 73331 967 65656 64435 83865 270 72054 48215 38926 293 75669 54812 68142 411 76492 23866 28856 909

0.96105 54383 10770 94792 459 0.96077 86021 80586 99523 878 0.96050 08052 71880 92684 682 0.96022 20478 62449 62830 504 Oi95994 23302 31050 48581 495

0.285 0.286 0.287 0.288 0.289

0.28115 0.28211 0.28307 0.28403 0.28499

74512 95294 02165 110 69722 09293 88922 591 62110 06345 05725 374 51667 27208 80861 997 38384 12929 50237 384

0.95966 0.95938 0.95909 0.95881 0.95852

16526 57401 10746 590 00154 22179 04351 746 74188 07021 50572 193 38630 94525 08568 713 93485 68245 47227 984

0.290 0.291 0.292 0.293 0.294

0.28595 0.28691 0.28786 0.28882 0.28978

22251 04835 53268 394 03258 44540 28750 981 81396 73943 10698 841 56656 35230 24153 475 29027 70875 80965 551

0.95824 0.95795 0.95767 0.95738 0.95709

38755 12697 16807 013 74442 13353 20481 688 00549 56644 85799 478 17080 29961 36036 308 24037 21649 61457 636

0.295 0.296 0.297 0.298 0.299

0.29073 Oi29169 0.29265 Oi29360 0.29456

98501 23642 75547 489 65067 36583 80597 155 28716 53042 42792 582 89439 16653 78457 616 47225 71345 69198 389

0.95680 21423 21013 90483 768 Oi95651 09241 18315 60759 429 0.95621 87494 04772 90127 632 Oi95592 56184 72560 47507 858 0.95563 15316 14809 23678 590

0.300

0.29552 02066 61339 57510 532

0.95533 64891 25606 01964 231

[C-f)4 1

[ (-;)I1

148

ELEMENTARY Table

4.6

CIRCULAR

TRANSCENDENTAL

SINES

AND COSINES

FUNCTIONS FOR RADIAN

cos x

sin x

X

ARGUMENTS

0.300 0.301 0.302 0.303 0.304

0.29552 0.29647 0.29743 0.29838 0.29933

02066 61339 57510 532 53952 31151 42357 025 02873 25592 74716 586 48819 89771 53102 518 91782 69093 19051 897

0.95533 0.95504 0.95474 0.95444 0.95414

64891 25606 01964 231 04912 99993 28826 414 35384 33968 84359 763 56308 24485 52692 116 67687 69450 92289 242

0.305 0.306 0.307 0.308 0.309

0.30029 0.30124 0.30220 0.30315 0.30410

31752 09261 52585 026 68718 56279 67635 045 02672 56451 07447 613 33604 56380 39950 549 61505 02974 53093 365

0.95384 0.95354 0.95324 0.95294 0.95263

69525 67727 06164 084 61825 19130 11990 559 44589 24430 12121 945 17820 85350 63513 878 81523 04568 47552 001

0.310 0.311 0.312 0.313 0.314

0.30505 0.30601 0.30696 0.30791 0.30886

86364 43443 50156 564 08173 25301 45030 632 26921 96367 57464 615 42601 04767 08284 189 55200 98932 14579 138

0.95233 0.95202 0.95172 0.95141 0.95110

35698 85713 39784 281 80351 33367 79558 038 15483 53066 39561 711 41098 51295 95271 383 57199 35494 94302 111

0.315 0.316 0.317 0.318 0.319

0.30981 i:j1076 0.31171 0.31266 0.31361

64712 27602 84860 120 71125 39828 14184 658 74430 84966 79252 234 74619 12688 33468 402 71680 72974 01977 833

0.95079 0.95048 0.95017 0.94986 0.94954

63789 14053 25664 080 60870 96311 88923 617 48447 92562 63269 094 26523 14047 76481 749 95099 72959 73811 467

0.320 0.321 0.322 0.323 0.324

0.31456 0.31551 0.31646 0.31741 0.31836

65606 16117 76666 176 56385 92727 11130 659 44010 53724 15619 332 28470 50346 51938 844 09756 34148 28330 674

0.94923 0.94892 0.94860 0.94828 0.94796

54180 82440 86757 531 03769 56583 01754 395 43869 10427 28762 501 74482 59963 69764 173 95613 22130 87164 613

0.325 0.326 0.327 0.328 0.329

0.31930 0.32025 0.32120 0.32215 0.32309

87858 57000 94315 718 62767 71094 35507 128 34474 28937 68391 319 02968 83360 35077 048 68241 87512 98012 460

0.94765 0.94733 0.94701 0.94668 0.94636

07264 14815 72098 048 09438 56853 12639 034 02139 68025 61918 976 85370 69063 06147 877 59134 81642 32541 351

0.330 0.331 0.332 0.333 0.334

0.32404 0.32498 0.32593 0.32687 Oij2782

30283 94868 34670 020 89085 59222 32199 224 44637 34694 82047 011 96929 75730 74545 756 45953 37100 93468 777

0.94604 0.94571 0.94539 0.94506 0.94473

23435 28386 97152 941 78275 32866 92611 768 23658 19598 15765 535 59587 14042 35228 939 86065 42606 58837 502

0.335 0.336 0.337 0.338 0.339

0.32876 0.32971 0.33065 0.33160 0.33254

91698 73903 10553 241 34156 41562 79990 386 73316 95834 32882 957 09170 92801 71669 766 41708 88879 64517 288

0.94441 0.94408 0.94375 0.94341 0.94308

03096 32643 01006 864 10683 12448 49997 577 08829 11264 35085 413 97537 59275 93637 243 76811 87612 38092 499

0.340 0.341 0.342 0.343 0.344

0.33348 0.33442 0.33537 0.33631 0.33725

70921 40814 39678 177 96799 05684 79816 635 19332 40903 16300 519 38512 04216 23460 104 54328 53706 12813 399

0.94275 0.94242 0.94208 0.94174 0.94141

46655 28346 22850 264 07071 14493 11062 025 58062 80011 41330 105 99633 59801 94311 834 31786 89707 59229 468

0.345 0.346 0.347 0.348 0.349

0.33819 0.33913 0.34007 0.34101 0.34195

66772 47791 27257 928 75834 45227 35228 880 81505 05108 24823 531 83774 86866 97891 850 82634 50276 64093 188

0.94107 0.94073 0.94039 0.94005 0.93971

54526 06513 00285 905 67854 47944 22986 218 71775 52668 40365 059 66292 60293 39119 944 51409 11367 45650 473

0.350

0.34289 78074 55451 34918 963

0.93937 27128 47378 92003 503

ELEMENTARY CIRCULAR

SINES

TRANSCENDENTAL

AND COSINES

FUNCTIONS

FOR RADIAN

149

ARGUMENTS

Table

4.6

cos x

sin z

0.34289 78074 55451 34918 963 0.34383 70085 62847 17681 237

0.93937 0.93902

93454

0.34477 0.34571 0.34665

58658

0.93799

36103

03447

99266

461

0.355 0.356 0.357 0.358 0.359

0.34759 0.34852 0.34946 0.35040

03652 78377 49617 17363

0.93764 0.93729 0.93694 0.93659 0.93624

64888 84296 94332

19349 88839 59978

27409 87337 89202

412 915 418

86299

04124

73578

312

0.360 0.361 0.362 0.363 0.364

0.35227 0.35320 0.35414 0.35508 0.35601

42332 99538 53211 03343 49923

75089 05683 26351 01729

0.365 0.366 0.367 0.368 0.369

0.35694 0.35788

92944 32396

76911 07756

39203

863

0.35975 0.36068

00552 29239

86229 67042

93504 91160

0.36161 0.36254 0.36347

54319 75783 93621

64961 47479 82448

25290 28534

86047 16625

0.350 0.351 0.352 0.353 0.354

33263

09467

102

43783 27841 91058 778 25451 08071 20819 319 35784 73161 82729 27364

28543 09276 17091

852 237 064

58840 a91 0.35133 al604 70292 87868 632 97684 15610 96384 15734

991 866 608 065

96801 63913 294

27128 47378 92003 503 10755 al724 321 0.93868 50389 44865 55613 a41 0.93833 97937 94014 57391 a69

94998 al762 72980 716

0.93589 68236 77934 85835 091 40815 54999 29438 322 0.93519 04038 88060 13742 042 0.93483 57910 30795 02492 a55 0.93448 02433 37816 78462 165 0.93554

37611 63448 79948

354 721

0.93412 0.93376 0.93340 0.93304

a7113 33740 a7371 606 0.93268 84948 14096 19348 a71

97803 21412 31502

729 373

al3 0.36441 07825 38085 52343 006 0.36534 la384 a2970 56131 067

0.93232 0.93196 0.93160

73456 52640 22505

06034 70704

42320 74082

381 737

70188 65151 560 0.93123 a3054 67499 62553 347 0.93087 34291 26582 73524 125

0.375 0.376 0.377 0.378 0.379

0.36627 0.36720

56137 99809

291 733

0.93050 0.93014 0.92977 0.92940 0.92903

76219

12314

29114

948

0.380

0.37092

04694 12982 67184 549 0.37184 89484 33562 49909 aal

0.92866

46355

76510

24949

253

i* 33:: 0: 383

0.37277 0.37370 0.37463

70556

0.92792 0.92754

09378

03592

76777

471

0.385 0.386

0.37555 0.37648 0.37741

91367 57472

0.390 0.391 0.392 0.393 0.394

0.38018 0.38111 0.38203 0.38296

a4151 23161 42823 118 31339 34675 51860 671 74716 33087 43373 349 14272 94059 55222 774

0.38388

49999

0.395 0.396

0.370 0.371 0.372 0.373 0.374

0.384

41380 647 0.35881 68268 55391 65142 021

0.36813 28105 44381 61843 251 0.36906 0.36999

23995 16194

39357 71964

05224

37211 34164

88020

926 758

096

47899 99862 72083 la4 21506 a9741 70366 479 21610 27762

089 945

07984 05404 97425

897 739 922

08841 90501 47704 265 32163 27881 98417 211 46186 92123 64451 a36 50916 51824 06312 328

0.92829 32508 36638 24806 a58 76968 49686 81063 030 0.92717 35283 48161 29943 792 84326 73184 70454 235

0.92679 0.92642 0.92604 0.92566

24101 54613

99852

0.92490 0.92452 0.92414 0.92376

90598 84090 68337 43342

57313 51063

04145 22192

068 776

35142

86457

070

0.92299 0.92261 0.92222

65643 12947 51025

63117 40199 06064

25225 97879 84939

693 040 589

it ;z; 0:399

0.38480 ala88 08245 02477 888 0.38573 09928 14697 01854 707 0.38665 34110 90188 34186 658 0.38757 54427 12300 79611 426 0.38849 70867 59002 a3601 363

0.400

0.38941 a3423 08650 49166 631

0.92106

09940

02885 08279 a53

2 ;:i 01389

47501 46155

64673 67846 04751

19812 59093 46681 397 0.37833 78378 60081 a4790 240 0.37926 33161 23263 89706 110

93636

r(-_sbq L

'_I

29011

366

66966

223

04187 63679 438 75863 63138 47019 143 oI9252a 87857 54580 07941 297

16481 39537 314

0.92338 09109 a9547 07898 401

0.92183 79880 46904 06602 584 0.92144 99517 49832 05558 150

cC-f)’ 1

150

ELEMENTARY

Table

4.6

CIRCULAR

TRANSCENDENTAL

FUNCTIONS

SIR’ES AND COSINES

FOR RADIAN

cos x

sin x

X

ARGUIMENTS

0.400 0.401 0.402 0.403 0.404

0.38941 0.39033 0.39125 0.39217 0.39309

83423 92084 96842 97687 94611

08650 39988 32150 64660 17434

49166 29019 17700 43663

631 595 358 363

61324 955

0.92028 03157 16118 16919 248 0.91988 85959 56976 45007 979 0.91949 59563 09315 43137 110

0.405 0.406 0.407 0.408 0.409

0.39401 0;j9493 0.39585 0.39677 0.39769

87603 76656 61759 42903 20080

70780 05398 02384 43226 09812

43071

820

0.91910

29995 97324 36782

816 356 508

0.91870 0.91831 0.91791 0.91751

0.410 0.411 0.412 0.413 0.414

0.39860 0.39952

93279 62493

84422 49738

89359 65238

380 251

71230 202

0.40044 27711 88838 35528 558 0.40135 88925 85200 23958 010 0.40227

46126

22702

98524

766

0.40318 99303 85626 63109 550 0.40410 48449 58653 49047 645 0.40501 93554 26869 06660 654

0.415 0.416 0.417 0.418 0.419

0.40593 0.40684

34608 71603

75762 91229

96747 82037

0.420 0.421 0.422 0.423 0.424

0.40776 0.40867 0.40958 0.41049 0.41140

04530 33379 58142 78808 95370

59570 67491 02108 50946 01936

0.425 0.426 0.421 0.428 0.429

0.41232 0.41323

07817

43424

16141 64165 31825 593 0.41414 20333 53326 15081 889 0.41505 20384 00488 14189 067 0.41596

16283

95646

32014

0.430 0.431 0.432 0.433 0.434

0.41687 0.41777 0.41868 0.41959 0.42050

08024 95595 78989 58196 33207

29210 92007 75279 70687 70310

0.435 0.436

0.42141 0,42231 0.42322

04013 70605 32974

ii* t;; 0:439

0.42412

0.92106 0.92067

0.91712 0.91672 0.91632 0.91592 0.91551

09940 11151

02885 95020

08279 86221

853 075

23971 79189 25219 62067 89735

65774 19913 66209 00060 17781

72800 45073 81253 73416 44815

745 295 568 956 737

08228 17549

16605 94682

10547 37232 03796 61266 92832

564 150 202 649 194

17704 51081 08695 90527

85785 99696

0.91511 63204 94631 73753 232

939 655

0.91471 0.91430 0.91390 0.91349

26730 81109 26344 62441

73322 39416 97475 52975

31180 03880 01872 65972

180 251 722 725

18597 47203 84671 15143 81337

279 546 703 980 201

0.91308 0.91268 0.91227 0.91186 0.91145

89403 07233 15937 15518 05981

12308 82776 12591 90901 47728

27243 66357 72866 04379 45647

609 915 996 332 576

75749

435

301

0.91103 0[91062 0.91021 Oi90979 0.90938

87329 59567 22698 76727 21659

54033 21681 63449 93022 24998

67564 86066 20950 54591 90577

373 990 808 701 360

76621 52231 50136 39579 58584

692 243 257 028 774

0.90896 0.90854 0.90813 0.90771 0.90729

57496 84244 01906 10488 09991

74885 59097 94960 00709 95484

12247 41143 95366 47844 84510

591 638 563 729 435

0.42503

66648 52619 21565 91110 67248 45005 83856

04753 26011 11315 81323 79016

684 018 146 456 027

0.90687 Oi90644 0.90602 0.90560 0.90517

00422 81785 54083 17320 71502

99336 33221 19003 79452 38245

62385 67577 73181 97096 59741

731 465 601 848 647

0.440 0.441 0.442 0.443 0.444

0.42593 0.42684 0.42774 0.42865 0.42955

94650 40036 81153 17992 50545

65999 08712 07458 58123 57025

60276 84433 04751

972 381 750

59317

145

0.90475 0.90432 0.90389 0.90346 0.90304

16632 52714 79753 97753 06718

19963 50093 55027 62061 99394

41716 41286 31889 19473 99766

554 061 904 892 305

0.445 0.446 0.447 0.448 0.449

0.43045 0.43136 0.43226 0.43316 0.43406

78803 02755 22395

00908 86947 12746

83666 65073 82453

443 141 917

48696

76203

11100

244

0.90261 0.90217 0.90174 0.90131 0.90088

06653 97562 79449 52319 16175

96132 82279 88745 47341 90780

15457 13291 01061 04523 24210

899 573 718 319 832

0.450

0.43496

55341

11230

21042

084

0.90044

71023

58891 823

37711 76342 50745 219

[1(-y 1 52676

92166

884

ELEMENTARY

CIRCULAR

SINES

TRANSCENDENTAL

AND COSINES

x

151

FUNCTIONS

FOR RADIAN

ARGCMENTS

Table

4.6

cos x

sin x

0.450 0.451 0.452 0.453 0.454

0.43496 0.43586 0.43676 0;43766 0.43856

55341 57635 55571 49140 38331

11230 80759 84561 22842 96246

21042 44573 42243 61170 25020

084 567 681 507 568

0.90044 0.90001 0.89957 Oi89913 0.89870

71023 16866 53709 81556 00412

52676 67546 70803 98765 88646

92166 28580 98337 67474 59552

884 847 319 569 965

0.455 0.456 0.457 0.458 0.459

0.43946 0.44036 0.44125 0.44215 0.44305

23138 03549 79557 51152 18326

05853 53183 40194 69287 43301

23944 04468 59344 17350 33053

492 918 542 215 008

0.89826 0.89782 0.89738 0.89693 0.89649

10281 11168 03076 86010 59975

78561 07522 15441 43127 32287

11933 31966 53089 90836 98759

463 167 030 721 714

0.460 0.461 0.462 0.463 0.464

0.44394 0.44484 0.44573 0.44663 0.44752

81069 39373 93228 42626 87558

65519 39668 69916 60878 17615

76524 23010 42563 89618 92537

151 752 218 275 506

0.89605 0.89560 0.89516 0.89471 0.89426

24975 81014 28097 66229 95414

25525 66339 99127 69179 22683

24253 64298 21110 57699 53342

639 937 867 908 602

0.465 0.466 0.467 0.468 0.469

0.44842 0.44931 0.45020 0745110 0.45199

28014 63986 95465 22442 44907

45634 50888 39782 19166 96343

43101 85958 08029 27868 84976

319 244 479 603 342

0.89382 Oi89337 0.89292 Oi89247 0.89202

15656 26959 29329 22770 07286

06720 69266 59190 26256 21120

58962 52423 93730 80142 01196

873 883 459 134 857

0.470 0.471 0.472 0.473 0.474

0.45288 0.45377 0.45466 0.45555 0.45644

62853 76270 85149 89482 89259

79068 75545 94432 44843 36343

29070 09309 63474 07100 22566

327 736 735 635 671

0.89156 0.89111 0.89066 0.89020 0.88974

82881 49562 07330 56193 96153

95328 01323 92437 22891 47800

93645 96296 04773 26178 33674

402 541 005 292 367

0.475 0.476 0.477 0.478 0.479

0.45733 0.45822 0.45911 0.46000 0.46089

84471 75110 61167 42633 19498

78955 83158 59888 20540 76967

48139 66969 96047 75103 55473

307 994 279 180 739

0.88929 0.88883 0.88837 0.88791 0.88745

27216 49386 62667 67065 62583

23168 05888 53744 25407 80438

20970 56721 38842 48723 05369

288 822 074 197 212

0.480 0.481 0.482 0.483 0.484

0.46177 0.46266 0.46355 0.46443 0.46532

91755 59394 22406 80783 34515

41482 26861 46338 13613 42849

88913 16364 56679 95295 72867

664 968 522 430 132

0.88699 0.88653 0.88606 0.88560 0.88514

49227 27001 95910 55958 07150

79284 83281 54652 56506 52837

19439 47206 44417 20075 90129

995 469 051 401 517

0.485 0.486 0.487 0.488 0.489

0.46620 0.46709 0.46797 0.46886 Oi46974

83594 28011 67757 02823 33201

48672 46175 50915 78918 46678

73849 15033 34040 77761 90760

162 451 104 558 024

0.88467 0.88420 0.88374 0.88327 0.88280

49491 82984 07636 23450 30432

08528 89343 61933 93833 53462

31072 33453 55301 75463 46844

223 094 874 416 214

0.490 0.491 0.492 0.493 0.494

0.47062 0.47150 0.47238 0.47327 Oi47415

58881 79855 96114 07649 14451

71158 69788 60472 61583 91970

03618 21242 11121 91533 19709

136 715 556 149 261

0.88233 0.88186 0.88138 0.88091 0.88044

28586 17916 98427 70125 33014

10121 33995 96151 68537 23984

49570 44058 23994 69230 98588

547 307 541 763 075

0.49'5 0.496 0.497 0.498 0.499

0.47503 0.47591 0.47679 0.47766 0.47854

16512 13823 06374 94157 77164

70950 18319 54345 99774 75827

79950 71693 97532 51191 05452

264 150 118 668 099

0.87996 0.87949 0.87901 0.87853 0.87806

87098 32382 68872 96571 15485

36204 79786 30204 63808 57828

22574 96012 70581 47270 28743

157 154 529 917 023

0.500

0.47942

55386

04203

00027

329

0.87758

25618

r, 90372-\.-,

71611

628

c(6;W 1

l’-:“l

152

ELEMENTARY

Table

4.6

CIRCULAR

TRANSCENDENTAL

SINES

AND COSINES

FUNCTIONS

FOR RADIAN

sin x

X

ARGUMENTS cos 2

0.500 0.501 0.502 0.503 0.504

0.47942 0.48030 0.48117 0.48205 0.48293

55386 04203 00027 329 28813 07080 29394 947 97437 07116 30578 414 61249 27448 70881 314 20240 91696 35573 583

0.87758 0.87710 0.87662 0.87614 0.87565

25618 90372 71611 628 26976 40428 38630 733 19562 87859 50795 903 03383 13407 39357 847 78441 98689 97748 295

0.505 0.506 0.507 0.508 0.509

0.48380 0.48468 0.48555 0.48643 0.48730

74403 23960 15529 617 23727 48823 94818 170 68204 91355 38243 967 07826 77106 78840 928 42584 32116 05316 931

0.87517 0.87469 0.87420 0.87371 0.87323

44744 26201 33418 203 02294 79311 19588 355 51098 42264 46912 391 91160 00180 75052 318 22484 39053 84166 561

0.510 0.511 0.512 0.513 0.514

0.48817 0.48904 0.48992 0.49079 0.49166

72468 82907 49450 013 97471 56492 73435 934 17583 80371 57187 006 32796 82532 85582 104 43101 91455 35667 778

0.87274 0.87225 0.87176 0.87127 0.87078

45076 45751 26310 581 58941 08013 76750 129 64083 14454 85187 176 60507 54560 26898 565 48219 18687 53787 441

0.515 0.516 0.517 0.518 0.519

0.49253 0.49340 0.49427 0.49514 0.49601

48490 36108 63810 364 48953 45953 92799 025 44482 50944 98899 617 35068 81528 98859 309 20703 68647 36861 855

0.87029 0.86979 0.86930 0.86881 0.86831

27222 98065 45347 504 97523 84793 59540 132 59126 71841 83584 429 12036 53049 84660 240 56258 23126 60524 189

0.520 0.521 0.522 0.523 0.524

0.49688 0.49774 0.49861 0.49948 0.50034

01378 43736 71433 446 77084 38729 62299 043 47812 86055 57189 109 13555 18641 78596 658 74302 69914 10484 518

0.86781 0.86732 0.86682 0.86632 0.86582

91796 77649 90038 785 18657 13065 83614 647 36844 26688 33565 898 46363 16698 64378 779 47218 82144 82893 524

0.525 0.526 0.527 0.528 0.529

0.50121 0.50207 0.50294 0.50380 0.50467

30046 73797 84942 748 80778 64718 68796 092 26489 77603 50161 411 67171 47881 24954 981 02815 11483 83349 596

0.86532 0.86482 0.86431 0.86381 0.86331

39416 22941 28399 561 22960 39868 22644 077 97856 34571 19753 996 64109 09560 56071 436 21723 68210 99902 671

0.530 0.531 0.532 0.533 0.534

0.50553 0.50639 0.50725 0.50811 0.50898

33412 04846 96181 366 58953 64911 01306 143 79431 29121 89905 473 94836 35431 92741 999 05160 22300 66364 220

0.86280 0.86230 0.86179 0.86128 0.86077

70705 14761 01380 670 11058 54312 41041 248 42788 92829 81312 894 65901 37140 13920 311 80400 94932 10201 726

0.535 0.536 0.537 0.538 0.539

0.50984 0.51070 0.51156 0.51241 0.51327

10394 28695 79260 534 10529 94093 97962 456 05558 58481 73096 946 95471 62356 25387 754 80260 46726 31605 686

0.86026 0.85975 0.85924 0.85873 0.85822

86292 74755 70140 025 83581 86021 71507 760 72273 39001 18926 068 52372 44824 92837 581 23884 15482 98393 339

0.540 0.541 0.542 0.543 0.544

0.51413 0.51499 0.51585 0.51670 0.51756

59916 53113 10467 728 34431 23551 08484 914 03796 00588 85758 874 68002 27290 01726 969 27041 47234 00855 920

0.85770 0.85719 0.85667 0.85616 0.85564

86813 63824 14253 797 41166 03555 41303 947 86946 49241 51282 623 24160 16304 35326 032 52812 21022 52425 567

0.545 0.546 0.547 0.548 0.549

0.51841 0.51927 0.52012 0.52098 0.52183

80905 04516 98283 861 29584 43752 65410 714 73071 10073 15436 812 11356 49129 88849 675 44432 07094 38858 868

0.85512 72907 80530 77799 957 0.85460 84452 12819 51181 787 0.85408 87450 36734 25018 472 0.85356 81907 71975 12587 703 Oi85304 67829 39096 36027 442

0.550

0.52268 72289 30659 16778 838 C-f)7

0.85252 45220 59505 74280 498 C-J)1

1 1

[ 1

ELEMENTARY CIRCULAR

SINES

AND

X

TRANSCENDENTAL COSINES

FOR

153

FUNCTIONS RADIAN

ARGUMENTS

sin 2

Table

4.6

cos x

0.550 0.551 0.552 0.553 0.554

0.52268 0.52353 0.52439 0.52524 0.52609

72289 94919 12314 24465 31364

30659 67038 63969 69712 33053

16778 57359 64065 94301 44585

838 653 565 297 976

0.85252 0.85200 0.85147 0.85095 0.85042

45220 14086 74432 26263 69585

59505 55464 50084 67333 32026

74280 10953 82092 23867 20180

498 761 114 il0 431

0.555 0.556 0.557 0.558 0.559

0.52694 0.52779 0.52864 0.52949 0.53033

33002 29370 20460 06264 86773

03301 30292 64391 56488 58002

35674 97627 54824 10933 33815

635 180 757 415 002

0.84990 0.84937 0.84884 0.84831 0.84778

04402 30721 48545 57881 58734

69831 07267 71701 91352 95285

50182 35704 88608 58049 77652

218 287 318 047 517

0.560 0.561 0.562 0.563 0.564

0.53118 0.53203 0.53287 0.53372 0.53457

61979 31872 96446 55691 09598

20883 97610 41195 05179 43639

40385 81418 26300 47726 06347

187 533 543 585 607

0.84725 0.84672 0.84619 0.84565 0.84512

51110 35012 10448 77421 35938

13416 76506 16165 64850 55863

12609 06683 29136 21564 44654

452 799 481 438 991

0.565 0.566 0.567 0.568 0.569

0.53541 0.53626 0.53710 0.53794 0.53878

58160 01367 39212 71686 98780

11183 62956 54637 42441 83121

35362 25057 07291 39926 91211

572 521 168 969 553

0.84458 0.84405 0.84351 0.84297 0.84244

86004 27624 60803 85547 01861

23353 02313 28580 38838 70611

24855 00958 70603 36691 53715

579 945 796 011 445

0.570 0.571 0.572 0.573 0.574

0.53963 0.54047 0.54131 0.54215 0.54299

20487 36797 47702 53195 53266

33969 52812 98021 28505 03714

24099 80524 65614 31859 63213

446 005 465 028 905

0.84190 0.84136 0.84082 0.84027 0.83973

09751 09222 00279 82928 57175

62268 53020 82920 92863 24582

74013 93925 99876 14368 41893

376 658 632 839 605

0.575 0.576 0.577 0.578 0.579

0.54383 0.54467 0;54551 0.54634 0.54718

47906 37109 20865 99165 72002

83642 28825 00342 59818 69423

59158 18694 24296 25797 24232

222 718 136 231 321

0.83919 0.83864 0.83810 0.83755 0.83701

23024 80481 29551 70241 02555

20654 24493 80354 33330 29351

14757 38825 39176 05683 38499

543 019 658 918 807

0.580 0.581 0.582 0.583 0.584

0.54802 0.54886 0.54969 0.55053 0.55136

39367 01252 57649 08548 53942

91873 90432 28912 71672 83624

55618 74682 38538 90300 42652

270 851 382 563 424

0.83646 0.83591 0.83536 0.83481 0.83426

26499 42078 49298 48164 38683

15186 38442 47559 91816 21326

93465 27434 43511 36205 36508

789 927 337 988 907

0.585 0.586 0.587 0.588 0.589

0.55219 0.55303 0.55386 0.55469 0.55552

93823 28181 57009 80299 98041

30227 77494 91989 40829 91685

61353 48692 26889 21434 44380

309 799 504 637 278

0.83371 0.83315 0.83260 0.83205 0.83149

20858 94697 60204 17385 66245

87037 40732 35026 23370 60044

56877 36143 84331 27399 51895

861 543 337 720 332

0.590 0.591 0.592 0.593 0.594

0.55636 0.55719 0.55802 0.55885 0.55968

10229 16852 17904 13375 03258

12783 72905 41388 88127 83575

77572 55827 50056 50327 48880

254 556 192 409 201

0.83094 0.83038 Oi82982 0.82926 Oi82870

06791 39026 62959 78593 85934

00163 99672 15348 04797 26455

49524 61643 23660 09361 75147

800 346 255 243 786

0.595 0.596 0.597 0.598 0.599

0.56050 0.56133 0.56216 0.56299 0.56381

87544 66226 39293 06739 68555

98744 05205 75090 81092 96468

23078 18307 30821 90525 43709

004 516 541 792 545

0.82814 0.82758 0.82702 0.82646 Oi82589

84988 75761 58257 32484 98446

39590 04294 81491 32932 21193

04193 50517 82974 29164 19254

468 407 799 660 799

0.600

0.56464 24733 95035 35720 095 C-f)7

[ 1

0.82533 56149 09678 29724 095

154

ELEMENTARY

Table

4.6

CIRCULAR

TRANSCENDENTAL

SINES

AND

COSINES

FUNCTIONS

FOR

RADIAN

sin x

X

ARGUMENTS cos x

0.601 0.602 0.603 0.604

0.56464 O;i6546 0.56629 0.56711 0.56793

24733 95035 35720 095 75265 51175 93580 897 20142 39837 08553 336 59356 36531 18642 028 92899 17336 91043 574

0.82533 0.82477 0.82420 0.82363 0:82307

56149 09678 29724 095 05598 62617 27022 123 46800 45065 11146 193 79760 22901 59135 858 04483 62830 68484 934

0.605 0.606 0.607 0.608 0.609

0.56876 20762 58900 04538 687 Oi56958 42938 38434 31827 607 0.57040 59418 33722 21808 719 0.57122 70194 23115 81800 299 0.57204 75257 85537 59705 300

0.82250 0.82193 0.82136 0.82079 0.82022

20976 32380 00471 116 29243 99900 23403 216 29292 34564 55786 102 21127 06368 09403 380 04753 86127 32317 893

0.610 0.611 0.612 0.613 0.614

0.57286 0:$7368 0.57450 O:i7532 0.57614

74601 00481 26119 098 68215 48012 56380 111 56093 08770 12563 221 38225 63966 25415 904 14604 95387 76236 989

0.81964 0.81907 0.81850 0.81792 0.81734

80178 45479 51790 075 47406 56882 17114 225 06443 93612 42372 770 57296 29766 49108 549 99969 40259 08915 198

0.615 0.616 0.617 0.618 0.619

0.57695 0.57777 0.57859 0.57940 0.58022

85222 85396 78697 975 50071 16931 60606 809 09141 73507 45614 047 62426 39217 34861 330 09916 98732 88572 073

0.81677 0.81619 0.81561 0.81503 0.81445

34469 00822 85945 685 60800 88007 79339 051 78970 79180 65565 411 88984 52524 40689 288 90847 87037 62551 318

0.620 0.621 0.622 0.623 0.624

0.58103 0.58184 0.58266 0.58347 0.58428

51605 37305 07584 296 87483 40765 14825 522 17542 95525 36729 641 41775 88579 84595 681 60174 07505 35888 387

0.81387 0.81329 0.81271 0.81213 0.81154

84566 62533 92868 400 70146 59641 39252 335 47593 59801 97147 027 16913 45270 91684 290 78111 99116 19458 331

0.625 0.626 0.627 0.628 0.629

0.58509 0.58590 0.58671 0.58752 0.58833

72729 40462 15480 540 79433 76194 76836 923 80279 04032 83139 861 75257 13891 88356 252 64359 96274 18246 006

0.81096 0.81037 0.80979 0.80920 0.80861

31195 05217 90218 953 76168 48267 68483 556 13038 13768 15067 973 41809 88032 28536 214 62489 58182 86569 178

0.630 0.631 0.632 0.633 0.634

0.58914 0.58995 0.59075 0.59156 0.59237

47579 42269 51311 811 24907 43555 99690 151 96335 92400 89983 484 61856 81661 44033 509 21462 04785 59635 440

0.80802 0.80743 0.80684 0.80625 0.80566

75083 12151 87252 371 79596 38679 90282 722 76035 27315 58094 522 64405 68414 96904 569 44713 53140 97676 566

0.635 0.636 0.637 0.638 0.639

0.59317 0.59398 0.59478 0.59559 0.59639

75143 55812 91193 198 22893 29375 30315 454 64703 20697 86352 425 00565 25599 66873 364 30471 40494 58084 641

0.80507 0.80447 0.80388 0.80328 0.80269

16964 73462 77004 837 81165 22155 17917 411 37320 92798 10598 548 85437 79775 93030 752 25521 78276 91556 338

0.640 0.641 0.642 0.643 0.644

0.59719 0.59799 0.59879 0.59959 0.60039

54413 62392 05188 355 72383 88897 92681 375 84374 18215 24594 757 90376 49145 04673 426 90382 81087 16496 070

0.80209 0.80149 0.80089 0.80030 0.79970

57578 84292 61358 611 81614 94617 26862 715 97636 06847 22056 216 05648 19380 30729 469 05657 31415 26635 842

0.645 0.646 0.647 0.648 0.649

0.60119 0.60199 0.60279 0.60359 0.60439

84385 14041 03535 151 72375 48606 49156 949 54345 85984 56561 576 30288 27978 28662 868 00194 76993 47908 070

0.79909 0.79849 0.79789 0.79729 0.79668

97669 42951 13571 848 81690 54786 65377 243 57726 68519 65855 159 25783 86546 48612 327 85868 12061 36819 444

0.650

0.60518 64057 36039 56037 252

0.600

0.79608 37985 49055 82891 760

ELEMENTARY CIRCULAR

SINES

AND

TRANSCENDENTAL COSINES

FOR

RADIAN

ARGKMENTS

sin x

X

155

FUNCTIONS Table

4.6

cos x

0.650 0.651 0.652 0.653 0.654

0.60518 0.60598 0.60677 0.60757 0.60836

64057 36039 56037 252 21868 08730 33782 358 73618 99284 80505 ala 19302 12527 93778 646 58909 53891 48897 929

0.79608 0.79547 0.79487 0.79426 0.79365

37985 49055 a2891 760 a2142 02318 08089 927 la343 77432 42041 la3 46596 80778 62180 929 66907 19531 33114 757

0.655 0.656 0.657 0.658 0.659

0.60915 0.60995 0.61074 0.61153 0.61232

92433 29414 78343 652 19865 45745 51174 755 41198 10140 52364 359 56423 30466 62074 073 65533 15201 34867 307

0.79304 0.79243 0.79182 0.79121 0.79060

79281 01659 45900 987 83724 35925 57253 785 80243 31885 28666 909 68843 99886 65458 154 49532 51069 55734 550

0.660 0.661 0.662 0.663 0.664

0.61311 0.61390 0.61469 0.61548 0.61627

68519 73433 78861 515 65375 14865 34819 272 56091 49810 55178 137 40660 a9197 a3019 la6 19075 44570 30974 165

0.78999 0.78937 0.78876 0.78814 0.78753

22314 97365 09278 382 a7197 51494 96354 080 44186 26970 a6436 061 93287 38093 86857 558 34506 99953 81380 523

0.665 0.666 0.667 0.668 0.669

0.61705 0.61784 0.61863 0.61941 0.62020

91327 28086 60071 171 57408 52521 58518 785 17311 31267 20428 576 71027 78333 24475 901 la550 08348 12498 919

0.78691 0.78629 0.78568 0.78506 0.78444

67851 28428 68686 643 93326 40184 00789 551 10938 52672 21368 279 20693 a4132 04022 017 22598 53587 90446 244

0.670 0.671 0.672 0.673 0.674

0.62098 0.62176 0.62255 0.62333 0.62411

59870 36559 68035 744 94980 78835 94799 654 23873 51665 95092 281 46540 72160 48154 700 62974 58052 a8456 349

0.78382 0.78320 0.78257 0.78195 0.78133

16658 80849 28530 294 02880 86510 10376 414 81270 91948 10240 374 51835 19324 22393 698 14579 91581 98907 578

0.675 0.676 0.677 0.678 0.679

0.62489 0.62567 0.62645 0.62723 0.62801

73167 27699 a3921 682 77111 00082 14094 496 74797 94805 48239 849 66220 32101 23383 477 51370 32827 22288 658

0.78070 0.78008 0.77945 0.77882 0.77820

69511 32446 a7358 526 16635 66425 68455 a30 55959 18805 93590 a77 a7488 15655 22308 414 11228 a3820 59699 786

0.680 0.681 0.682 0.683 0.684

0.62879 0.62957 0.63034 0.63112 0.63189

30240 18468 51370 418 02822 11138 la547 018 69108 33578 11028 644 29091 09159 73043 207 a2762 61884 a3499 197

0.77757 0.77694 0.77631 0.77568 0.77505

27187 50927 93718 239 35370 45381 32416 339 35783 96362 41105 566 28434 33829 79438 156 13327 88518 38411 247

0.685 0.686 0.687 0.688 0.689

0.63267 0.63344 0.63422 0.63499 0.63576

30115 16386 33585 498 71140 97929 04308 084 05832 32410 43963 542 34181 46361 45549 306 56180 66947 24110 566

0.77441 0.77378 0.77315 0.77251 0.77188

90470 91938 77293 390 59869 76376 60473 500 21530 74891 94232 293 75460 21318 63436 286 21664 50263 68154 418

0.690 0.691 0.692 0.693 0.694

0.63653 0.63730 0.63807 0.63884 0.63961

71822 21967 94023 743 al098 39859 46216 467 84001 49694 25323 984 80523 81182 06781 a99 70657 64670 73855 200

0.77124 0.77060 0.76997 0.76933 0.76869

60149 97106 60197 354 90922 97998 79579 541 13989 89862 90904 069 29357 10392 19670 418 37030 98049 @SO5 132

0.695 0.696 0.697 0.698 0.699

0.64038 0.64115 0.64192 0.64268 0.64345

54395 31146 94603 464 31729 12236 98782 Ii5 02651 40207 54600 136 67154 47966 i5a92 698 25230 69063 48031 063

0.76805 0.76741 0.76677 0.76612 0.76548

37017 92068 53315 502 29324 32449 39366 321 13956 59961 77279 757 90921 16142 38958 434 60224 43294 73431 759

0.700

0.64421 76872 37691 05367 261

[ c-y1

0.76484 21872 84488 42625 586

[c-y 1

156

ELEMENTARY Table

4.6

CIRCULAR

TRANSCENDENTAL

SINES

AND COSINES

FUNCTIONS FOR RADIAN

sin 5

X

ARGUMENTS cos x

0.700 0.701 0.702 0.703 0.704

0.64421 0.64498 0.64574 0.64650 0.64727

76872 37691 05367 261 22071 88685 07414 902 60821 57525 65445 583 93113 80337 88940 870 18940 93892 61979 783

0.76484 21872 84488 42625 586 0.76419 75872 83558 57055 252 Oi76355 22230 85105 i1442 075 0.76290 60953 34492 20253 368 0.76225 92046 77847 53166 023

0.705 0.706 0.707 0.708 0.709

0.64803 0.64879 0.64955 0.65031 0.65107

38295 35607 19561 705 51169 43546 23864 641 57555 56422 40438 747 57446 13597 14335 062 50833 55081 46169 354

0.76161 0.76096 0.76031 0.75966 0.75901

15517 62061 70453 752 31372 34787 58298 030 39617 44439 64022 815 40259 40193 31253 107 33304 71984 34997 406

0.710 0.711 0.712 0.713 0.714

0.65183 0.65259 0.65334 0.65410 0.65486

37710 21536 68121 013 18068 54275 19866 915 91900 95261 24450 173 59199 87111 64083 709 19957 73096 55888 565

0.75836 0.75770 0.75705 0.75640 0.75574

18759 90508 16654 146 96631 47219 18942 159 66925 94330 20755 235 29649 84811 71940 852 84809 72391 28003 128

0.715 0.716 0.717 0.718 0.719

0.65561 0.65637 0.65712 0.65787 0.65863

74166 97140 27566 883 21820 03821 93009 463 62909 38376 27837 851 97427 46694 44880 853 25366 75324 69585 417

0.75509 0.75443 0.75378 0.75312 0.75246

32412 11552 84730 074 72463 57536 12745 203 04970 66335 91983 563 29939 94701 46092 263 47378 00135 76755 558

0.720 0.721 0.722 0.723 0.724

0.65938 0.66013 0.66088 0.66163 0.66238

46719 71473 15361 800 61478 83004 58862 952 69636 58443 15198 027 71185 46973 13079 967 66117 98439 69907 065

0.75180 0.75114 0.75048 0.74982 0.74916

57291 40894 97944 549 59686 75987 70091 576 54570 65174 34189 363 41949 68966 45814 983 21830 48626 09078 707

0.725 0.726 0.727 0.728 0.729

0.66313 0.66388 0.66463 0.66537 0.66612

54426 63349 66778 441 36103 92872 23443 354 11142 38839 73184 280 79534 53748 37633 666 41272 90759 01524 309

0.74849 0.74783 0.74717 0.74650 0.74584

94219 66165 10497 806 59123 84344 52795 369 16549 66673 88624 209 66503 77410 54215 910 08992 81559 02955 103

0.730 0.731 0.732 0.733 0.734

0.66686 0.66761 0.66835 0.66910 0.66984

96350 03697 87373 259 44758 47057 30099 195 86490 75996 51573 181 21539 46342 35102 739 49897 14589 99849 159

0.74517 0.74450 0.74383 0.74317 0.74250

44023 44870 38879 013 71602 33841 50102 364 91736 15714 42167 693 04431 58475 71321 153 09695 30855 77713 862

0.735 0.736 0.737 0.738 0.739

0.67058 0.67132 0.67206 0.67280 0.67354

71556 37903 75177 973 86509 74117 74942 523 94749 81736 71700 537 96269 19936 70863 650 91060 48565 84779 796

0.74183 0.74115 0.74048 0.73981 0.73914

07534 02328 18528 866 97954 43109 01033 791 80963 24156 15559 237 56567 17168 68402 998 24772 94586 14660 158

0.740 0.741 0.742 0.743 0.744

0.67428 0.67502 0.67576 0.67650 0.67723

79116 28145 06748 388 60429 19868 84968 216 34991 85605 96417 996 02796 87900 20669 485 63836 89971 13633 096

0.73846 0.73779 0.73711 0.73644 0.73576

85587 29587 90979 142 39016 96092 48243 787 85068 68756 84181 492 23749 22975 75897 532 55065 34881 12335 582

0.745 0.746 0.747 0.748 0.749

0.67797 0.67870 0.67944 0.68017 0.68090

18104 55714 81235 936 65592 49704 53032 193 06293 37191 55745 803 40199 84105 86745 313 67304 57056 87450 880

0.73508 0.73440 0.73373 0.73305 0.73237

79023 81341 26664 537 95631 39960 28591 681 04894 89077 36602 285 06821 07766 10125 695 01416 75833 81627 975

0.750

0.68163 87600 23334 16673 324

[(-;GJ 1

0.73168 88688 73820 88631 184

[(-;)I1

ELEMENTARY

CIRCULAR

SINES

TRANSCENDENTAL

AND COSINES

FOR RADIAN

.4RGUMENTS

sin x

X

157

FUNCTIONS

Table

4.6

cos x

0.750 0.751 0.752 0.753 0.754

0.68163 0.68237 0.68310 0.68383 0.68456

87600 23334 16673 324 01079 50908 23885 163 07735 08431 22423 554 07559 65237 62625 080 00545 91345 04892 285

0.73168 0.73100 0.73032 0.72964 0.72895

88688 73820 88631 184 68643 83000 05659 342 41288 85375 76111 160 06630 63683 44059 608 64676 01388 85978 367

0.755 0.756 0.757 0.758 0.759

0.68528 0.68601 0.68674 0.68747 0.68819

86686 57454 92691 917 65974 34953 25484 772 38401 95911 31587 089 03962 13086 40963 419 62647 59922 57950 885

0.72827 0.72758 0.72689 0.72621 0.72552

15431 82687 42395 268 58904 92503 49472 750 95102 16489 70515 436 24030 41026 27404 8k7 45696 53220 31961 494

0.760 0.761 0.762 0.763 0.764

0.68892 0.68964 0.69036 0.69109 0.69181

14451 10551 33914 776 59365 39792 39835 383 97383 23154 38826 030 28497 36835 58582 200 52700 57724 63761 700

0.72483 0.72414 0.72345 0.72276 0.72207

60107 40905 17233 969 67269 92639 68715 814 67190 97707 55489 548 59877 46116 61298 318 45336 28598 15545 123

0.765 0.766 0.767 0.768 0.769

0.69253 0.69325 0.69397 0.69469 0.69541

69985 63401 28295 794 80345 32137 07631 223 83772 42896 10903 039 80259 75335 73038 195 69800 09807 26789 802

0.72138 0.72068 0.71999 0.71930 0.71860

23574 36606 24219 693 94598 62317 00753 084 58415 98627 96800 072 15033 39157 32949 410 64457 78243 29362 010

0.770 0.771 0.772 0.773 0.774

0.69613 0.69685 0.69756 0.69828 0.69900

52386 27356 74701 988 28011 09725 61005 296 96667 39351 43442 524 58347 99368 65024 972 13045 73609 25718 983

0.71791 0.71721 0.71651 0.71581 0.71512

06696 10943 36337 129 41755 33033 64806 626 69642 41008 lb757 355 90364 32078 15581 770 03928 04171 36356 807

0.775 0.776 0.777 0.778 0.779

0.69971 0.70043 0.70114 0.70185 0.70256

60753 46603 54062 747 01464 03580 78713 256 35170 30469 99923 379 61865 13900 60948 949 81541 41203 19385 818

0.71442 0.71372 0.71302 0.71231 0.71161

10340 55931 36051 117 09608 86716 83660 709 01739 96600 90273 093 86740 86370 39059 972 64618 57525 15198 564

0.780 0.781 0.782 0.783 0.784

0.70327 0.70398 0.70469 0.70540 0.70611

94192 00410 18436 790 99809 80256 58108 374 98387 70180 66337 280 89918 60324 70046 581 74395 41535 66131 480

0.71091 0.71020 0.70950 0.70880 0.70809

35380 12277 35721 626 99032 53550 79296 239 55582 84980 15931 435 05038 10910 36614 737 47405 36395 82877 671

0.785 0.786 0.787 0.788 0.789

0.70682 0.70753 0.70823 0.70894 0.70964

51811 05365 92374 614 22158 44073 98290 801 85430 50625 15901 193 41620 18692 30436 730 90720 42656 50970 857

0.70738 0.70668 0.70597 0.70526 0.70455

82691 67199 76290 330 10904 09793 47885 059 32049 71355 67509 330 46135 59771 73107 880 53168 83632 99934 173

0.790 0.791 0.792 0.793 0.794

0.71035 0.71105 0.71175 0.71246 0.71316

32724 17607 80981 403 67624 39345 88841 574 95414 04380 78239 979 16086 09933 58529 620 29633 53937 15005 776

0.70384 0.70313 0.70242 0.70171 0.70099

53156 52236 09691 278 46105 75582 19602 208 32023 64376 31409 812 10917 30026 60306 275 82793 84643 63792 314

0.795 0.796 0.797 0.798 0.799

0.71386 0.71456 0.71526 0.71596 0.71665

36049 35036 79112 713 35326 52590 98579 148 27458 06672 07482 391 12436 98066 96241 109 90256 28277 81536 630

0.70028 0.69957 0.69885 0.69814 0.69742

47660 41039 70466 123 05524 12728 08742 151 56392 13922 35499 779 00271 59535 64661 971 37169 65179 95703 964

0.800

0.71735 60908 99522 76162 718

[c-y1

0.69670 67093 47165 42092 075

[

C-f,9

1

158

ELEMENTARY

Table

4.6

CIRCULAR

TRANSCENDENTAL

SINES

AND COSINES

FUNCTIONS

FOR RADIAN

co5 x

sin x

X

ARGUMENTS

0.800 0.801 0.802 0.803 0.804

0.71735 0.71805 0.71874 0.71944 0.72013

60908 24388 80686 29797 71714

99522 14736 77571 92397 64303

76162 58803 43741 50488 73354

718 753 255 651 263

0.69670 0.69598 0.69527 0.69455 0.69383

67093 90050 06047 15091 17189

47165 22499 08886 24727 89116

42092 59652 74871 13123 26831

075 695 538 218 236

0.805 0.806 0.807 0.808 0.809

0.72083 0.72152 0.72221 0.72290 0.72359

06429 33937 54228 67298 73139

99098 03310 84188 49704 08550

50932 35522 62476 19472 15721

396 503 322 935 677

0.69311 0.69239 0.69166 0.69094 0.69022

12350 00579 81884 56273 23752

21844 43394 74945 38365 56214

23558 94027 40074 02528 89026

425 956 951 784 151

0.810 0.811 0.812 0.813 0.814

0.72428 Oi72497 0.72566 0.72635 0.72703

71743 63105 47217 24072 93664

70142 44620 42849 76416 57636

51092 85175 06266 00277 19583

818 959 069 085 027

0.68949 0.68877 0.68804 0.68732 0.68659

84329 38011 84805 24719 57759

51747 48903 72316 47306 99881

01754 65129 53394 18165 15892

964 158 472 280 545

0.815 0.816 0.817 0.818 0.819

0.72772 Oi72841 0.72909 Oi72977 0.73046

55985 11030 58790 99259 32430

99550 15926 21260 30776 60427

51786 884L4 93542 72343 39565

534 775 651 223 302

0.68586 0.68514 0.68441 0.68368 0.68295

83934 03250 15714 21335 20118

56737 45257 93509 30246 84907

35262 24529 18772 67094 59742

969 414 652 544 692

0.820 0.821 0.822 0.823 0.824

0.73114 0.73182 0.73250 0.73318 0.73386

58297 76852 88089 92001 88581

26895 47595 40670 24998 20187

87938 56503 98872 51414 01366

131 084 320 329 283

0.68222 0.68148 0.68075 0.68002 0.67929

12072 97204 75521 47030 11740

87613 69169 61060 95457 05207

55166 07005 91008 31885 30088

656 802 857 232 213

0.825 0.826 0.827 0.828 0.829

0.73454 0.73522 0.73590 0.73658 0.73725

77822 59718 34261 01446 61264

46578 25249 78008 27404 96715

54873 04953 99391 08557 93150

150 477 793 557 579

0.67855 0.67782 0.67708 0.67635 0.67561

69656 20786 65139 02720 33538

23839 85563 25264 78508 81536

88530 39229 69888 50409 59331

058 106 949 750 781

0.830 0.831 0.832 0.833 0.834

0.73793 0.73860 0.73927 0.73995 0.74062

13711 58777 96458 26746 49635

09962 91899 68020 64557 08481

71872 89026 82039 48913 15603

858 752 434 544 989

0.67487 0.67413 0.67339 0.67265 0.67191

57600 74913 85485 89323 86434

71267 85293 61885 39984 59207

10211 77928 24928 27394 01352

246 481 580 537 983

0.835 0.836 0.837 0.838 0.839

0.74129 0.74196 0.74263 0.74330 0.74397

65117 73186 73836 67059 52849

27503 50074 05390 23383 34732

03320 95758 06248 44844 85324

808 049 576 755 932

0.67117 0.67043 0.66969 0.66895 0.66820

76826 60506 37482 07761 71351

59842 82850 69864 63185 05786

28712 83235 56439 83438 68708

570 098 445 385 357

0.840 0.841 0.842 0.843 0.844

0.74464 0.74531 0.74597 0.74664 0.74730

31199 02103 65554 21545 70070

70859 63927 46848 53275 17609

32125 87199 16798 18184 86260

657 577 923 539 385

0.66746 0.66671 0.66597 0.66522 0.66447

28258 78491 22056 58962 89216

41308 14059 69016 51824 08791

11792 32935 98654 47240 14192

267 396 482 065 152

0.845 0.846 0.847 0.848 0.849

0.74797 0.74863 0.74929 0.74995 0.75062

11121 44693 70779 89371 00464

74999 61339 13273 68190 64233

80133 89598 01550 66317 63922

429 886 724 368 547

0.66373 0.66298 0.66223 0.66148 0.66073

12824 29796 40137 43857 40961

86891 33764 97713 27703 73363

57589 83391 70674 96802 62530

286 100 409 946 783

0.850

0.75128

04051

40292 i-8)9

70271

207

0.65998

31458

[ 71

[c-w1 84982 7

17039

542

ELEMENTARY

CIRCULAR

TRANSCENDENTAL

SINES AND COSINES

FOR RADIAN

ARGUMENTS

sin z

X

159

FUNCTIONS

Table 4.6

cos x

0.850 0.851 0.852 0.853 0.854

0.75128 0.75194 0.75259 0.75325 0.75391

04051 00125 88679 69708 43204

40292 36009 91775 48737 48790

70271 23260 88815 26849 57151

207 432 295 594 380

0.65998 0.65923 0.65847 0.65772 0.65697

31458 15356 92661 63381 27524

84982 13509 10556 28392 19944

17039 82909 81024 55410 98010

542 449 321 547 152

0.855 0.856 0.857 0.858 0.859

0.75457 0.75522 0.75588 0.75653 0.75718

09161 67572 18431 61731 97466

34586 49528 37776 44244 14602

25193 67867 79144 75659 62217

237 227 450 143 260

0.65621 0.65546 0.65470 0.65395 0.65319

85097 36108 80564 18474 49843

38799 39199 76042 04884 81933

73388 43373 91635 48193 13861

013 300 218 134 148

0.860 0.861 0.862 0.863 0.864

0.75784 0.75849 0.75914 0.75979 0.76044

25628 46213 59212 64620 62430

95276 33451 77068 74826 76186

97229 58068 06350 53141 24087

459 441 566 684 122

0.65243 0.65167 0.65092 0.65016 0.64940

74681 92995 04791 10079 08865

64051 08756 74216 19251 03332

84627 75966 47091 25131 29254

203 794 357 418 574

0.865 0.866 0.867 0.868 0.869

0.76109 0.76174 0.76239 0.76303 0.76368

52636 35230 10208 77561 37284

31366 91346 07866 33429 21299

24465 04168 22598 13500 49706

750 673 272 144 858

0.64864 0.64787 0.64711 0.64635 0.64559

01156 86962 66288 39144 05536

86580 29767 94312 42282 36391

94718 96858 75010 56369 79782

373 196 176 276 561

0.870 0.871 0.872 0.873 0.874

0.76432 0.76497 0.76561 0.76625 0.76690

89370 33813 70606 99742 21217

25505 00837 02852 87869 12977

07814 32779 02438 91953 38182

480 191 134 834 114

0.64482 0.64406 0.64329 0.64253 0.64176

65472 18960 66007 06621 40810

40001 17117 32390 51117 39234

19477 08727 63447 05733 87329

766 234 280 091 202

0.875 0.876 0.877 0.878 0.879

0.76754 0.76818 0.76882 0.76946 0.77010

35022 41152 39600 30359 13424

36027 15638 11198 82862 91555

03963 42337 60682 84773 22769

458 736 252 027 271

0.64099 0.64022 0.63946 0.63869 0.63792

68581 89942 04901 13466 15643

63325 90610 88955 26862 73477

13035 64049 21244 88380 15258

656 903 528 872 639

0.880 0.881 0.882 0.883 0.884

0.77073 0.77137 0.77201 0.77264 0.77328

88788 56445 16388 68611 13107

98969 67568 60587 42032 76680

29120 68399 79051 37074 19618

965 506 337 497 049

0.63715 0.63638 0.63560 0.63483 0.63406

11441 00868 83931 60638 30997

98580 72592 66570 52208 01835

20801 16079 27264 18529 14874

550 131 710 695 218

0.885 0.886 0.887 0.888 0.889

0.77391 0.77454 0.77518 0.77581 0.77644

49871 78895 00174 13701 19470

30081 68560 59214 69915 69310

68504 53673 36552 33343 78237

290 706 600 321 045

0.63328 0.63251 0.63174 0.63096 0.63018

95014 52699 04059 49102 87834

88415 85546 67460 09021 85724

24894 63485 74481 53235 69127

213 020 571 256 530

0.890 0.891 0.892 0.893 0.894

0.77707 0.77770 0.77832 0.77895 0.77958

17475 07709 90165 64839 31723

26823 12654 97778 53950 53704

86549 17776 38577 85676 28683

033 316 722 211 432

0.62941 0.62863 0.62785 0.62707 0.62629

20265 46402 66252 79824 87125

73696 49694 91105 75942 82849

88020 94643 14919 38222 39581

355 540 057 428 242

0.895 0.896 0.897 0.898 0.899

0.78020 0.78083 0.78145 0.78208 0.78270

90811 42097 85575 21238 49080

70350 77980 51465 66458 99392

32846 21716 39740 14771 20508

443 548 163 667 171

0.62551 0.62473 0.62395 0.62317 0.62239

88163 82946 71482 53778 29842

91096 80578 31818 25961 44779

01810 37587 11458 61790 22654

880 545 656 683 524

0.900

0.78332 69096 27483 38846 138

[c-p1 1

0.62160 99682 70664 45648 472

[ (-f)S 1

160

ELEMENTARY

Table

4.6

CIRCULAR

TRANSCENDENTAL

SINES sin

X

AND COSINES

FUNCTIONS

FOR RADIAN

ARGUMENTS cos x

x

0.900 0.901 0.902 0.903 0.904

0.78332 0.78394 0.78456 0.78518 0.78580

69096 27483 38846 138 81278 28730 22159 796 85620 81914 55501 279 82117 66602 18722 439 70762 63143 48518 260

0.62160 99682 70664 45648 472 0.62082 63306 86633 21658 870 0.62004 20722 76323 02558 530 0.61925 71938 23992 22842 983 Oib1847 16961 14519 21204 658

0.905 0.906 0.907 0.908 0.909

0.78642 0.78704 0.78765 0.78827 0.78888

51549 52674 00391 817 24472 17115 10540 713 89524 39174 57664 940 46700 02347 24696 094 95992 90915 60447 888

0.61768 0.61689 0.61611 0.61532 0.61453

55799 33401 62045 040 88460 66755 56924 921 14953 01314 85952 792 35284 24430 19111 466 49462 24068 37523 020

0.910 0.911 0.912 0.913 0.914

0.78950 0.79011 0.79072 0.79134 0.79195

37396 89950 41187 896 70905 85311 32130 474 96513 63647 48850 789 14214 12398 18619 897 24001 19793 41660 812

0.61374 0.61295 0.61216 0.61137 0.61058

57494 88811 54652 118 59390 07856 37447 803 55155 71013 27423 839 44799 68705 61677 674 28329 91968 93848 110

0.915 0.916 0.917 0.918 0,919

0.79256 0.79317 0.79378 0.79438 0.79499

25868 74854 52325 499 19810 67394 80192 738 05820 88020 11086 785 83893 28129 48016 785 54021 79915 72036 860

0.60979 0.60899 0.60820 0.60741 0.60661

05754 32450 15011 758 77080 82406 74518 350 42317 34706 00764 999 01471 82824 21909 476 54552 20845 86522 589

0.920 0.921 0.922 0.923 0.924

0.79560 0.79620 0.79681 0.79741 0.79801

16200 36366 03026 828 70422 91262 60393 471 16683 39183 23692 319 54975 75501 93169 858 85293 96389 50226 129

0.60582 0.60502 0.60422 0.60343 0.60263

01566 43462 84179 741 42522 45973 65991 745 77428 24282 65074 984 06291 74899 16960 980 29120 94936 79945 468

0.925 0.926 0.927 0.928 0.929

0.79862 0.79922 0.79982 0.80042 0.80102

07631 98814 17797 639 21983 80542 20660 537 28343 40138 45653 978 26704 76967 01823 638 17061 91191 80485 294

0.60183 0.60103 0.60023 0.59943 0.59863

45923 82112 55377 043 56708 34746 07885 466 61482 51758 85549 703 60254 32673 40005 791 53031 77612 46494 584

0.930 0.931 0.932 0.933 0.934

0.80161 0.80221 0.80281 0.80340 0.80400

99408 83777 15208 432 73739 56488 41719 806 40048 11892 57726 899 98328 53358 82661 218 48574 85059 17341 371

0.59783 0.59703 0.59622 0.59542 0.59462

39822 87298 23849 491 20635 63051 54424 260 95478 06791 03960 905 64358 21032 41397 846 27284 08887 58618 345

0.935 0.936 0.937 0.938 0.939

0.80459 0.80519 0.80578 0.80637 0.80696

90781 11969 03555 863 24941 39867 83565 545 51049 75339 59525 671 69100 25773 52827 488 79086 99364 63359 313

0.59381 0.59301 0.59220 0.59140 0.59059

84263 74063 90139 324 35305 20863 32740 634 80416 54181 65034 867 19605 79507 66977 785 52881 02922 39319 443

0.940 0.941 0.942 0.943 0.944

0.80755 0.80814 0.80873 0.80932 0.80991

81004 05114 28687 022 74845 52830 83153 915 60605 53130 16899 872 38278 17436 34799 758 07857 57982 15321 017

0.58978 0.58898 0.58817 0.58736 0.58655

80250 31098 22996 099 01721 71298 18462 976 17303 31375 04967 973 27003 19770 59766 388 30829 45514 77276 748

0.945 0.946 0.947 0.948 0.949

0.81049 0.81108 0.81166 0.81225 0.81283

69337 87809 69300 383 22713 20770 98639 669 67977 71528 54920 560 05125 55555 97938 351 34150 89138 54154 591

0.58574 0.58493 0.58412 0.58330 0.58249

28790 18224 88177 827 20893 48104 78446 913 07147 45944 08339 436 87560 23117 31310 012 62139 91583 12874 994

0.950

0.81341 55047 89?73 75068 542

1 1 c-y

0.58168 30894 63883 49416 618

[ !-;I81

ELEMENTARY

CIRCULAR

SINES

TRANSCENDENTAL

AND COSINES

FOR RADIAN

ARGUMENTS

sin x

X

161

FUNCTIONS

Table

4.6

cos x

0:950 0.951 0.952 0.953 0.954

0.81341 0.81399 0.81457 0.81515 0.81573

55047 89373 75068 542 67810 74171 95507 433 72433 62256 91835 411 68910 73166 40081 165 57236 27252 73984 145

0.58168 30894 63883 49416 618 0.58086 93832 53142 86928 810 0.58005 50961 73067 39704 748 0.57924 02290 37944 08966 251 Oi57842 47826 62640 01435 096

0.955 0.956 0.957 0.958 0.959

0.81631 0.81689 0.81746 0.81804 0.81861

37404 45683 42959 322 09409 50441 69980 433 73245 64327 09381 654 28907 10956 04577 644 76388 14762 45701 891

0.57760 0.57679 0.57597 0.57515 0.57433

87578 62601 47846 300 21554 53853 21403 511 49762 52997 56176 536 72210 77213 65441 113 88907 44256 59961 007

0.960 0.961 0.962 0.963 0.964

0.81919 0.81976 0.82033 0.82090 0.82147

15683 00998 27163 322 46785 95734 05121 101 69691 25859 54877 569 84393 19084 28189 263 90886 03938 10495 962

0.57351 0.57270 0.57188 0.57105 0.57023

99860 72456 66212 505 05078 80718 44551 395 04569 88520 07322 513 98342 15912 36911 940 86403 83518 03741 923

0.965 0.966 0.967 0.968 0.969

0.82204 0.82261 0.82318 0.82375 0.82432

89164 09771 78067 694 79221 66757 55069 656 61053 05889 70544 986 34652 58985 15315 328 00014 58683 98799 136

0.56941 0.56859 0.56777 0.56694 0.56612

68763 12530 84208 614 45428 24714 78562 699 16407 42403 28733 004 81708 88498 36093 162 41340 86469 79171 417

0.970 0.971 0.972 0.973 0.974

0.82488 0.82545 0.82601 0.82657 0.82714

57133 38450 05747 662 06003 32571 52898 564 46618 76161 45547 087 78974 05158 34034 750 03063 56326 70155 495

0.56529 0.56447 0.56364 0.56282 0.56199

95311 60354 31303 653 43629 34754 78229 727 86302 34839 35633 190 23338 86340 66624 480 54747 15554 99167 663

0.975 0.976 0.977 0.978 0.979

0.82770 0.82826 0.82882 0.82938 0.82993

18881 67257 63479 226 26422 76369 37592 699 25681 22907 86257 689 16651 46947 29486 397 99327 89390 69534 022

0.56116 0.56034 0.55951 0.55868 0.55785

80535 49341 43450 813 00712 15121 09200 110 15285 40876 22937 736 24263 55149 45183 654 27654 87042 87601 358

0.980 0.981 0.982 0.983 0.984

0.83049 0.83105 0.83160 0.83216 0.83271

73704 91970 46808 453 39776 97248 95697 028 97538 48619 00310 290 46983 90304 50142 703 88107 67360 95650 254

0.55702 0.55619 0.55536 0.55452 0.55369

25467 66217 30087 666 17710 22891 37806 645 04390 87840 78167 757 85517 92397 37748 295 61099 68448 39160 207

0.985 0.986 0.987 0.988 0.989

0.83327 0.83382 0.83437 0.83492 0.83547

20904 25676 03744 902 45368 11970 13205 801 61493 73796 90007 262 69275 59543 82563 379 68708 18432 76889 279

0.55286 0.55202 0.55119 0.55036 0.54952

31144 48435 57861 376 95660 65354 38911 453 54656 52753 13672 322 08140 44732 16453 272 56120 75943 01100 969

0.990 0.991 0.992 0.993 0.994

0.83602 0.83657 0.83712 0.83766 0.83821

59786 00520 51678 926 42503 56699 33299 444 16855 38697 50701 883 82835 99079 90248 385 40439 91248 50455 694

0.54868 0.54785 0.54701 0.54617 0.54534

98605 81587 57534 313 35603 97417 28224 252 67123 59732 24618 647 93173 05380 43512 268 13760 71756 83362 006

0.995 0.996 0.997 0.998 0.999

0.83875 0.83930 0.83984 0.84038 0.84093

89661 69442 96654 953 30495 88741 15567 733 62937 05059 69798 245 86979 75154 52241 668 02618 56621 40408 555

0.54450 0.54366 0.54282 0.54198 0.54114

28894 96802 60547 375 38584 19004 25576 412 42836 77392 79237 026 41661 11542 88693 907 35065 61572 03531 067

1.000

0.84147 09848 07896 50665 250

IIC-f)1 1

0.54030 23058 68139 71740 094

[c-p7 1

162

ELEMENTARY Table

4.6

CIRCULAR

TRANSCENDENTAL

SINES

AND COSINES

FUNCTIONS FOR RADIAN

cos x

sin x

X

ARGUMENTS

1.000 1. 001 1.002 1.003 1.004

0.84147 0.84201 0.84254 0.84308 0.84362

09848 07896 50665 250 08662 88256 92390 268 99057 57821 22046 578 81026 77549 97169 747 54565 09246 30271 873

0.54030 0.53946 0.53861 0.53777 0.53693

23058 68139 71740 094 05648 72446 55654 214 82844 16233 47828 237 54653 41780 86864 465 21084 91907 73184 669

1.005 1.006 1.007 1.008 1.009

0.84416 0.84469 0.84523 0.84576 0.84629

19667 15556 42661 273 76327 59970 18177 851 24541 06821 56844 116 64302 21289 28431 774 95605 69397 25943 853

0.53608 0.53524 0.53439 0.53355 0.53270

82147 09970 84748 188 37848 39863 92716 262 88197 26016 77062 668 33202 13394 42130 747 72871 47496 32136 904

1.010 1.011 1.012 1.013 1.014

0.84683 0.84736 0.84789 0.84842 0.84895

18446 18015 19012 310 32818 34859 07211 051 38716 88491 73284 331 36136 48323 36290 466 25071 84612 04660 810

0.53186 0.53101 0.53016 0.52931 0.52846

07213 74355 46620 673 36237 40537 55841 426 59950 93140 16121 808 78362 79791 85137 984 91481 48651 37156 798

1.015 1.016 1.017 1.018 1.019

0.84948 0.85000 0.85053 0.85105 0.85158

05517 68464 29173 940 77468 71835 55845 003 40919 67530 78730 164 95865 29204 92646 111 42300 31363 45804 549

0.52761 0.52677 0.52591 0.52506 0.52421

99315 48406 78219 896 01873 28274 61274 932 99163 37999 01253 921 91194 27850 90098 832 77974 48627 11734 503

1.020 1.021 1.022 1.023 1.024

0.85210 0.85263 0.85315 0.85367 0.85419

80219 49362 92361 655 09617 59411 44882 415 30489 38569 26719 808 42829 64749 24308 778 46633 16717 39374 945

0.52336 0.52251 0.52166 0.52080 0.51995

59512 51649 56988 961 35816 88764 38461 245 06896 12341 05336 792 72758 75271 58150 502 33413 30969 63497 542

1.025 1.026 1.027 1.028 1.029

0.85471 0.85523 0.85575 0.85626 0.85678

41894 74093 41057 997 28609 17351 17949 715 06771 27819 30046 586 76375 87681 60616 931 37417 79977 67982 525

0.51909 0.51824 0.51738 0.51653 0.51567

88868 33369 68691 985 39132 36926 16373 373 84213 96612 59061 276 24121 67920 73657 956 58864 06859 75899 186

1.030 1.031 1.032 1.033 1.034

0.85729 0.85781 0.85832 0.85883 0.85935

89891 88603 37214 627 33792 98311 31744 398 69115 94711 44887 626 95855 64271 51283 734 14006 94317 58248 998

0.51481 0.51396 0.51310 0.51224 0.51138

88449 69955 34753 350 12887 14248 86768 878 32184 97296 50370 116 46351 77168 40101 715 55396 12447 80821 625

1.035 1.036 1.037 1.038 1.039

0.85986 0.86037 0.86088 0.86139 0.86189

23564 73034 57043 938 24523 89466 7.4054 819 16879 33518 21889 224 00625 95953 50385 634 75758 68397 97536 975

0.51052 0.50966 0.50880 0.50794 0.50708

59326 62230 21842 776 58151 86122 51023 535 51880 44242 08807 028 40520 97216 02209 404 24082 06180 18757 138

1.040 1.041 1.042 1.043 1.044

0.86240 0.86291 0.86341 0.86391 0.86442

42272 43338 40328 079 00162 14123 45486 997 49422 74964 20150 131 90049 20934 62441 124 22036 47972 11963 456

0.50622 0.50535 0.50449 0.50363 0.50276

02572 32778 40373 447 76000 39161 57213 919 44374 87986 81451 427 07704 42416 61010 426 65997 66117 93250 711

1.045 1.046 1.047 1.048 1.049

0.86492 0.86542 0.86592 0.86642 0.86692

45379 52878 00206 699 60073 33318 00866 385 66112 87822 80077 424 63493 15788 46561 037 52209 17477 01685 140

0.50190 0.50103 0.50017 0.49930 0.49843

19263 23261 38600 728 67509 78520 34140 520 10745 97070 07134 396 48980 44586 88513 415 82221 87247 26307 756

1.050

0.86742 32255 94016 89438 141

[(-;)I1

0.49757 10478 91726 99029 085 c-8)7 7

II 1

ELEMENTARY CIRCULAR

SINES

TRANSCENDENTAL

AND

COSINES

FOR

RADIAN

ARGUMENTS

sin z

5

163

FUNCTIONS Table

4.6

cos x

1.050 1.051 1.052 1.053 1.054

0.86742 0.86792 0.86841 0.86891 0.86940

32255 03628 66321 20330 65651

94016 47403 80499 97035 01611

89438 46316 51123 74685 29477

141 092 146 276 198

0.49757 0.49670 0.49583 0.49496 0.49409

10478 33760 52074 65430 73836

91726 25200 55338 50311 78782

99029 29002 95651 48726 21490

085 975 499 051 510

1.055 1.056 1.057 1.058 1.059

0.86990 0.87039 0.87088 0.87137 0.87186

02276 30203 49427 59941 61741

99694 97621 02601 22711 66899

19162 88046 70443 39954 58660

460 624 529 543 794

0.49322 0.49235 0.49148 0.49061 0.48974

77302 75835 69444 58139 41927

09910 13349 59246 18239 61459

43854 55459 18707 31756 41446

806 008 979 732 534

1.060 1.061 1.062 1.063 1.064

0.87235 0.87284 0.87333 0.87381 0.87430

54823 39181 14811 81707 39866

44986 67663 46494 93918 23243

26228 28925 88556 11299 36468

295 947 345 356 402

0.48887 0.48799 0.48712 0.48625 0.48537

20818 94820 63943 28194 87582

60527 87554 15140 16372 64825

56191 58818 19351 07757 06632

864 317 528 202 362

1.065 1.066 1.067 1.068 1.069

0.87478 0.87527 0.87575 0.87623 0.87671

89281 29948 61863 85020 99415

48654 85211 48845 56366 25458

85179 08932 38105 30362 18969

424 453 753 492 874

0.48450 0.48362 0.48275 0.48187 0.48100

42117 91807 36660 76686 11893

34560 00124 36547 19345 24514

23847 05142 46667 07484 22014

867 311 387 800 811

1.070 1.071 1.072 1.073 1.074

0.87720 0.87768 0.87815 0.87863 0.87911

05042 01898 89976 69274 39784

74681 23471 92149 01900 74797

61030 85627 41877 46904 33716

706 336 919 963 111

0.48012 0.47924 0.47836 0.47749 0.47661

42290 67886 88689 04709 15953

28534 08365 41447 05700 79522

12436 01039 22529 36282 38551

509 904 904 289 762

1.075 1.076 1.077 1.078 1.079

0.87959 0.88006 0.88053 0.88101 0.88148

01504 54428 98551 33868 60376

33788 02703 06248 70011 20461

98997 50816 56244 88879 76291

101 869 731 619 297

0.47573 0.47485 0.47397 0.47309 0.47221

22432 24153 21126 13359 00861

41788 71851 49538 55152 69469

74632 50968 47223 28292 56273

160 911 840 396 392

1.080 1.081 1.082 1.083 1.084

0.88195 0.88242 0.88289 0.88336 0.88383

78068 86941 86990 78210 60596

84947 91699 69831 49337 61096

47373 79609 46247 63390 36998

533 169 031 660 790

0.47132 0.47044 0.46956 0.46868 0.46779

83641 61708 35070 03737 67717

73740 49685 79499 45845 31856

02391 58871 50767 47743 75803

353 547 810 217 727

1.085 1.086 1.087 1.088 1.089

0.88430 Oi88476 0.88523 0.88570 0.88616

34144 98849 54706 01710 39858

36869 09301 11921 79145 46272

09797 08104 88562 84791 53940

534 243 972 522 000

0.46691 0.46602 0.46514 0.46425 0.46337

27019 81651 31624 76945 17623

21135 97750 46239 51605 99315

28984 80991 96791 44159 05181

862 522 014 401 235

1.090 1.091 1.092 1.093 1.094

0.88662 0.88708 0.88755 0.88801 0.88846

69144 89564 01113 03786 97579

49487 25861 13352 50807 77956

23160 35990 98641 26207 88779

860 371 470 951 948

0.46248 0.46159 0.46071 0145982 0.45893

53668 85088 11892 34089 51687

75300 65958 58145 39181 96847

87702 36738 45833 68372 28855

790 852 190 764 783

1.095 1.096 1.097 1.098 1.099

0.88892 0.88938 0.88984 0.89029 0.89075

82488 58507 25633 83860 33184

35422 64713 08227 09252 11966

57470 50354 78315 90807 21527

660 274 047 488 609

0.45804 0.45715 0.45626 0.45537 0.45448

64697 73125 76983 76277 71018

19382 95485 14314 65483 39062

34113 84487 84956 56224 45757

686 142 158 382 688

1.100

0.89120 73600 61435 33995 180

c(-;I1 1

0.45359 61214 25577 38777 137 C-78)6

c 1

164

ELEMENTARY Table

4.6

CIRCULAR

SINES

TRANSCENDENTAL AND

COSINES

FUNCTIONS FOR

RADIAN

cos x

sin x

2

ARGUMENTS

1.100 1.101 1.102 1.103 1.104

0.89120 0.89166 0.89211 0.89256 0.89301

73600 61435 33995 180 05105 03618 67046 971 27692 85365 80240 901 41359 54417 99171 080 46100 59408 60693 678

0.45359 0.45270 0.45181 0.45092 0.45002

61214 25577 38777 137 46874 16008 69206 400 28007 01790 30573 730 04621 74808 86868 576 76727 27402 83352 928

1.105 1.106 1.107 1.108 1.109

0.89346 0.89391 0.89436 0.89480 0.89525

41911 49863 58063 585 28787 76201 85981 812 06724 89735 85553 594 75718 42671 89157 146 35763 88110 65223 027

0.44913 0.44824 0.44734 0.44645 0.44555

44332 52361 57327 478 07446 42924 48852 689 66077 92780 11424 866 20235 96065 22607 305 69929 47363 94616 628

1.110 1.111 1.112 1.113 1.114

0.89569 0.89614 0.89658 0.89702 0.89747

86856 80047 62924 063 28992 73373 56775 801 62167 23874 91147 427 86375 88234 24683 120 01614 24030 74633 785

0.44466 0.44376 0.44286 0.44197 0.44107

15167 41706 84864 374 55958 74570 06453 951 92312 41874 38633 030 24237 39984 37201 474 51742 65707 44874 890

1.115 1.116 1.117 1.118 1.119

0.89791 0.89835 0.89878 0.89922 0.89966

07877 89740 61099 138 05162 44737 51180 079 93463 49293 03041 321 72776 64577 09884 230 43097 52658 43829 826

0.44017 0.43927 0.43838 0.43748 0.43658

74837 16293 01603 891 93529 89431 54849 166 07829 83253 69812 438 17745 96329 39623 410 23287 27666 95482 777

1.120 1.121 1.122 1.123 1.124

0.90010 0.90053 0.90097 0.90140 0.90183

04421 76504 99711 910 56744 99984 38780 263 00062 87864 32313 880 34371 05813 05144 201 59665 20399 79088 276

0.43568 0.43478 0.43388 0.43298 0.43207

24462 76712 16761 399 21281 43347 41055 736 13752 27890 74199 612 01884 31095 00232 420 85686 54146 91323 845

1.125 1.126 1.127 1.128 1.129

0.90226 0.90269 0.90312 0.90355 0.90398

75940 99095 16291 842 83194 10271 62482 258 81420 23203 90131 256 70615 08069 41527 464 50774 35948 71758 658

0.43117 65167 98666 17655 197 0.43027 40337 66704 57257 452 0.42937 11204 60745 05806 078 0.42846 77777 83700 86372 749 0142756 40066 38914 59134 030

1.130 1.131 1.132 1.133 1.134

0.90441 0.90483 0.90526 0.90568 0.90611

21893 78825 91603 708 83969 09589 10334 160 36996 02030 78425 425 80970 30848 30177 523 15887 71644 26245 348

0.42665 0.42575 0.42485 0.42394 0.42303

98079 30157 31037 122 51825 61627 65422 763 01314 37950 91605 376 46554 64178 14410 540 87555 45785 23669 902

1.135 1.136 1.137 1.138 1.139

0.90653 0.90695 0.90737 0.90779 0.90821

41744 00926 96078 401 58534 96110 80269 960 66256 35516 72815 632 64903 98372 63281 260 54473 64813 78880 126

0.42213 0.42122 0.42031 0.41941 0.41850

24325 88672 03673 585 56874 99161 42580 219 85211 83998 41784 656 09345 50349 25243 478 29285 05800 48758 379

1.140 1.141 1.142 1.143 1.144

0.90863 0.90905 0.90946 0.90988 0.91029

34961 15883 26459 422 06362 33532 34395 940 68673 00620 94400 939 21889 00918 03234 153 66006 19102 04326 885

0.41759 0.41668 0.41577 0.41486 0.41395

45039 58358 09217 519 56618 16446 53794 933 64029 88907 89108 094 67283 85000 90333 707 66389 14400 10281 852

1.145 1.146 1.147 1.148 1.149

0.91071 0.91112 0.91153 0.91194 0.91235

01020 40761 29314 164 26927 52394 39475 912 43723 41410 67087 073 51403 96130 56676 684 49965 05786 06195 821

0.41304 0.41213 0.41122 0.41031 0.40940

61354 87194 88428 529 52190 13888 59906 732 38904 05397 64456 120 21505 73050 55331 381 00004 28587 08169 395

1.150

0.91276 39402 60521 08094 403

0.40848 74408 84157 29815 258

ELEMENTARY CIRCULAR

SINES

AND

TRANSCENDENTAL COSINES

FOR

RADIAN

ARGUMENTS

sin x

5

165

FUNCTIONS Table

4.6

cos x

1.150 1.151 1.152 1.153 1.154

0.91276 0.91317 0.91357 Oi91398 0.91439

39402 19712 90890 52933 05835

60521 51391 70367 10330 65075

08094 90306 57146 30107 88579

403 792 165 602 865

0.40848 0.40757 0.40666 0.40574 0.40483

74408 44728 10972 73149 31269

84157 52320 46045 78706 64086

29815 67107 15621 28372 24481

258 284 071 706 224

1.155 1.156 1.157 1.158 1.159

0.91479 0.91519 0.91560 0.91600 0.91640

49594 84204 09663 25966 33109

29314 98669 69679 39800 07401

10465 12711 91743 63815 05261

816 431 383 143 556

0.40391 0.40300 0.40208 0.40117 0.40025

85341 35373 81375 23357 61326

16372 50159 80441 22620 92496

97790 25449 76456 20152 34689

397 945 266 779 958

1.160 1.161 1.162 1.163 1.164

0.91680 0.91720 0.91759 0.91799 0.91839

31087 19898 99536 69999 31282

71766 33100 92520 52063 14682

92661 42911 53200 40902 83374

866 136 023 883 147

0.39933 0.39842 0.39750 0.39658 0.39566

95294 25267 51257 73271 91320

06273 80553 32340 79035 38435

15445 83402 93491 42889 79278

164 355 775 706 377

1.165 1.166 1.167 1.168 1.169

0.91878 0.91918 0.91957 0.91996 0.92035

83380 26291 60010 84533 99857

84250 65556 64310 87139 41592

57652 80075 45798 68222 18336

941 906 178 492 360

0.39475 0.39383 0.39291 0.39199 0.39107

05412 15556 21762 24039 22396

28737 68530 76800 72926 76682

09066 05567 17146 75312 02789

125 898 187 486 366

1.170 1.171 1.172 1.173 1.174

0.92075 0.92114 0.92152 0.92191 0.92230

05977 02889 90590 69076 38343

36135 80158 83968 58796 16793

63957 08886 31967 26061 36915

301 071 851 369 902

0.39015 0.38923 0.38830 0.38738 0.38646

16843 07387 94040 76809 55705

08230 88126 37316 77135 29304

21533 60718 64679 00821 67479

266 072 599 054 575

1.175 1.176 1.177 1.178 1.179

0.92268 0.92307 0.92345 0.92384 0.92422

98386 49203 90789 23140 46253

71033 35513 25145 55777 44173

01956 88974 34733 83468 25312

127 783 097 944 701

0.38554 0.38462 0.38369 0.38277 0.38184

30736 01911 69240 32733 92397

15936 59525 82956 09495 62792

01753 87293 62048 25982 48743

942 547 718 487 902

1.180 1.181 1.182 1.183 1.184

0.92460 0.92498 0.92536 0.92574 0.92612

60124 64748 60123 46244 23108

08020 65932 37446 43024 04056

34610 08156 03329 76141 19188

754 619 642 242 645

0.38092 0.38000 0.37907 0.37814 0.37722

48243 00280 48517 92963 33627

66881 46178 25478 29958 85174

77302 43547 71840 86542 19493

960 271 534 917 444

1.185 1.186 1.187 1.188 1.189

0.92649 0.92687 0.92724 0.92762 0.92799

90710 49047 98116 37912 68432

42853 82657 47634 62876 54404

99516 96383 38942 43819 52606

095 480 352 290 588

0.37629 0.37537 0.37444 0.37351 0.37258

70520 03649 33025 58656 80552

17058 51921 16451 37709 43133

17454 49518 14476 48149 30684

471 342 334 962 752

1.190 1.191 1.192 1.193 1.194

0.92836 0.92874 0.92911 0.92947 0.92984

89672 01628 04297 97675 81758

49166 75038 60825 36260 32004

69260 97404 77546 24192 62877

202 950 899 928 403

0.37165 0.37073 0.36980 0.36887 0.36794

98722 13176 23922 30970 34330

60532 18091 44362 68273 19116

93806 28040 89893 08995 95213

568 589 026 672 382

1.195 1.196 1.197 1.198 1.199

0.93021 0.93058 0.93094 0.93131 0.93167

56542 22025 78201 25068 62622

79650 11719 61664 63866 53638

67095 95146 26876 00337 48349

956 303 083 679 974

0.36701 0.36608 0.36515 0.36422 0.36328

34010 30020 22369 11066 96122

26558 20629 31729 90622 28438

45714 52004 06923 11604 82399

570 819 698 876 631

1.200

0.93203 90859 67226 34967 013

[C-f)1 1

0.36235 77544 76673 57763 837 p-y51 L

’

J

166

ELEMENTARY Table

4.6

CIRCULAR

TRANSCENDENTAL

SINES

AND COSINES

FUNCTIONS FOR RADIAN

sin x

X

ARGUMENTS cos x

1.200 1.201 1.202 1.203 1.204

0.93203 0.93240 0.93276 0.93312 0.93348

90859 67226 34967 013 09776 41805 91853 542 19369 15485 54567 367 19634 27305 98748 519 10568 17240 76215 175

0.36235 0.36142 0.36049 0.35956 0.35862

77544 76673 57763 837 55343 67184 05108 539 29528 32190 27614 189 00108 04273 71008 651 67092 16376 30309 065

1.205 1.206 1.207 1.208 1.209

0.93383 92167 26196 50966 302 0.93419 64427 96013 35090 992 Oi93455 27346 69465 24584 444 0.93490 80919 90260 35070 567 0.93526 25144 03041 37431 162

0.35769 0.35675 0.35582 0.35488 0.35395

30490 01799 56527 660 90310 94203 63341 607 46564 27606 33727 018 99259 36382 26557 166 48405 55261 83165 039

1.210 1.211 1.212 1.213 1.214

0.93561 0.93596 0.93632 0.93667 0.93702

60015 53385 93341 646 85530 87806 90713 291 01686 53752 79041 926 08478 99608 04663 095 05904 74693 45913 598

0.35301 0.35208 0.35114 0.35021 0.34927

94012 19330 33870 301 36088 64027 04470 775 74644 25144 22698 521 09688 38826 24640 616 41230 41568 61124 730

1.215 1.216 1.217 1.218 1.219

0.93736 0.93771 0.93806 0.93841 0.93875

93960 29266 48199 416 72642 14521 58969 959 41946 82590 62598 617 01870 86543 15169 574 52410 80386 79170 848

0.34833 0.34739 0.34646 0.34552 0.34458

69279 70217 04069 578 93845 61966 52800 358 14937 54360 40329 260 32564 85289 39601 140 46736 92990 69704 455

1.220 1.221 1.222 1.223 1.224

0.93909 0.93944 0.93978 0.94012 0.94046

93563 19067 58093 524 25324 58470 30937 151 47691 55418 86621 257 60660 67676 58302 957 64228 53946 57600 622

0.34364 0.34270 0.34176 0.34082 0.33988

57463 16047 02047 552 64752 93385 66500 405 68615 64277 57501 890 69060 68336 40132 702 66097 45517 56153 996

1.225 1.226 1.227 1.228 1.229

0.94080 0.94114 0.94148 0.94181 0.94215

58391 73872 08723 559 43146 88036 82507 685 18490 57965 30357 157 84419 46123 18091 912 40930 15917 59701 104

0.33894 0.33800 0.33706 0.33612 0.33518

59735 36117 30011 855 49983 80771 74807 668 36852 20455 98234 533 20349 96483 08479 750 00486 50503 20093 523

1.230 1.231 1.232 1.233 1.234

0.94248 0.94282 0.94315 0.94348 0.94381

88019 31697 51002 382 25683 58754 03206 998 53919 63320 76390 684 72724 12574 12870 299 82093 74633 70486 175

0.33423 0.33329 0.33235 0.33140 0.33046

77271 24502 59823 955 50713 60802 72418 427 20823 02059 26391 462 87608 91261 19759 164 51080 71729 85740 328

1.235 1.236 1.237 1.238 1.239

0.94414 0.94447 0.94480 0.94513 0.94545

82025 18562 55790 164 72515 14367 57139 322 53560 32999 77695 223 25157 46354 68328 851 87303 27272 60431 046

0.32952 0.32857 0.32763 0.32668 0.32574

11247 87117 98424 316 68119 81408 78405 786 21705 98914 98386 387 72015 84277 88743 487 19058 82466 43066 054

1.240 1.241 1.242 1.243 1.244

0.94578 0.94610 0.94643 0.94675 0.94707

39994 49538 98628 471 83227 87884 73405 063 17000 17986 53628 942 41308 16467 18984 738 56148 60895 92311 309

0.32479 0.32385 0.32290 0.32195 0.32101

62844 38776 23657 769 03381 98828 67007 475 40681 08569 89227 042 74751 14269 91456 764 05601 62521 65238 364

1.245 1.246 1.247 1.248 1.249

0.94739 0.94771 0.94803 0.94835 0.94866

61518 29788 71844 815 57414 02608 63367 118 43832 59766 12259 472 20779 82619 35461 479 88225 53474 53335 262

0.32006 0.31911 0.31816 0.31721 0.31627

33242 00239 97855 712 57681 74660 77643 341 78930 33339 99262 871 96997 24152 68947 423 11891 95292 09714 116

1.250

0.94898 46193 55586 21434 849

0.31532 23623 95268 66544 754

[c-y1

ELEMENTARY CIRCULAR

SINES

AND

X

TRANSCENDENTAL COSINES

FOR

167

FUNCTIONS

RADIAN

ARGUMENTS

sin x

Table

4.6

cos x

1.250 1.251 1.252 1.253 1.254

0.94898 0.94929 0.94961 0.94992 0.95023

46193 94671 33656 63145 83135

55586 73157 91340 96237 74899

21434 62180 96439 75008 10006

849 713 444 528 196

0.31532 0.31437 0.31342 0.31247 0.31152

23623 32202 37637 39938 39114

95268 72909 77355 58064 64805

66544 11534 49010 20615 10363

754 791 665 601 979

1.255 1.256 1. 257 1.258 1.259

0.95054 0.95085 0.95116 0.95147 0.95178

93623 94605 86078 68040 40486

15326 06469 38232 01466 87975

06166 92038 51091 52726 83188

303 225 729 783 287

0.31057 0.30962 0.30867 0.30772 0.30676

35175 28130 17989 04761 88456

47660 57024 43600 58403 52756

49664 22311 69445 94485 68021

355 242 729 052 196

1.260 1.261 1.262 1.263 1.264

0.95209 0.95239 0.95270 0.95300 0.95330

03415 56824 00708 35065 59892

90515 02793 19468 36151 49407

76385 44617 09200 31003 40886

682 416 227 222 709

0.30581 0.30486 0.30391 0.30295 0.30200

69083 46652 21173 92654 61106

78289 86939 30948 62866 35544

32688 08001 95158 81822 46859

634 291 833 373 693

1.265 1.266 1.267 1.268 1.269

0.95360 0.95390 0.95420 0.95450 0.95480

75186 80944 77163 63840 40972

56753 56660 48552 32808 10759

70045 80258 94039 24694 06289

767 512 032 963 671

0.30105 0.30009 0.29914 0.29819 0.29723

26538 88959 48379 04808 58254

02136 16100 31192 01472 81295

65060 11818 67791 23518 84019

070 814 595 675 121

1.270 1.271 1.272 1.273 1.274

0.95510 0.95539 0.95569 0.95598 0.95627

08555 66588 15067 53989 83351

84692 57849 34427 19578 19409

23509 41432 35209 19640 78657

018 673 944 104 170

0.29628 0.29532 0.29437 0.29341 0.29245

08729 56240 00799 42413 81094

25318 88493 26068 93588 46891

73355 39166 57175 35661 19906

114 425 182 000 579

1.275 1.276 1.277 1.278 1.279

0.95657 0.95686 0.95715 0.95744 0.95772

03150 13383 14048 05142 86661

40985 92326 82408 21166 19488

94719 78101 96095 02109 64678

118 497 419 886 437

0.29150 0.29054 0.28958 0.28863 0.28767

16850 49691 79626 06666 30819

42108 35665 84278 44951 74982

96613 98290 07609 61732 56616

869 890 308 860 726

1.280 1.281 1.282 1.283 1.284

0.95801 0.95830 0.95858 0.95887 0.95915

58602 20964 73742 16935 50539

89224 43180 95120 59764 52796

96370 82604 10371 96853 17957

075 453 286 962 320

0.28671 0.28575 0.28479 0.28383 0.28288

52096 70505 86057 98761 08626

31955 73742 58503 44681 91007

51277 72036 16730 58895 51923

939 934 332 050 831

1.285 1.286 1.287 1.288 1.289

0.95943 0.95971 0.95999 0.96027 0.96055

74551 88969 93790 89011 74629

90853 91535 73400 55966 59710

36739 31748 25260 11427 84322

577 357 814 805 094

0.28192 0.28096 0.28000 0.27904 Oi27808

15663 19881 21288 19896 15714

56494 00438 82417 62291 00198

33192 28157 54428 25809 56310

303 651 993 577 871

1.290 1.291 1.292 1.293 1.294

0.96083 0.96111 0.96138 0.96166 0.96193

50642 17046 73839 21018 58580

06072 17450 17203 29652 80080

65890 33810 49249 84528 50693

556 354 056 675 590

0.27712 0.27615 0.27519 3.27423 0.27327

08750 99015 86519 71271 53280

56557 92064 67693 44692 84588

64138 75651 29289 79480 00512

661 234 769 997 263

1.295 1.296 1.297 1.298 1.299

0.96220 0.96248 0.96275 0.96302 0.96329

86523 04845 13541 12610 02048

94730 00807 26481 00880 54098

24982 78203 02013 36103 95282

339 231 782 915 920

0.27231 0.27135 0.27038 0.26942 0.26846

32557 09111 82951 54087 22529

49177 00534 01003 13198 00008

90379 74605 10035 88600 41057

053 108 206 711 992

1.300 /

0.96355 81854 17192 96470 135

[c-y1

0.26749 88286 24587 40699 798 C-7834

L 1

168

ELEMENTARY

Table

4.6

CIRCULAR

x

TRANSCENDENTAL

SINES

AND COSINES

FUNCTIONS

FOR RADIAN

sin 2:

ARGUMENTS CO8 2

1.300 1.301 1.302 1.303 1.304

0.96355 81854 17192 96470 135 0.96382 52024 22181 85589 331 0.96409 12556 02048 64366 761 0.96435 63446 90740 17032 855 0796462 04694 23167 36927 537

0.26749 0.26653 0.26557 0.26460 0.26364

88286 24587 40699 798 51368 50360 07039 695 11785 41018 09469 650 69546 60519 70890 877 24661 73088 71318 016

1.305 1.306 1.307 1.308 1.309

0.96488 0.96514 0.96540 0.96566 0.96592

36295 35205 53009 126 58247 63694 56266 806 70548 46439 26036 635 73195 22209 56221 061 66185 30740 81411 924

0.26267 0.26171 0.26074 0.25978 0.25881

77140 43213 51456 761 26992 35646 16255 031 74227 15401 38427 774 18854 47755 61955 494 60883 98246 05556 626

1.310 1.311 1.312 1.313 1.314

0.96618 0.96644 0.96669 0.96695 0.96720

49516 12734 02916 926 23185 09856 14689 520 87189 64740 29162 218 41527 20986 02983 276 86195 23159 62656 736

0.25785 0.25688 0.25591 0.25495 0.25398

00325 32669 66133 818 37188 17082 22194 242 71482 17797 37244 030 03217 01385 63156 911 32402 34673 43517 173

1.315 1.316 1.317 1.318 1.319

0.96746 0.96771 0.96796 0.96821 0.96846

21191 16794 3gO85 794 46512 48390 48019 478 62156 65416 05402 607 68121 16306 62628 991 64403 50465 76697 879

0.25301 0.25204 0.25108 0.25011 0.24914

59047 84742 16937 022 83163 18927 20348 457 04758 04816 92269 738 23842 10251 76046 556 40425 03323 23067 996

1.320 1.321 :* 2: 1:324

0.96871 0.96896 0.96920 0.96945 0.96970

51001 18265 26273 590 27911 71045 36648 340 95132 61115 04608 211 52661 41752 23202 252 00495 67204 06414 685

0.24817 0.24720 0.24623 0.24526 0.24429

54516 52372 95957 398 66126 25991 71738 199 75263 93018 44974 865 81939 22539 30889 004 86161 83886 68450 760

1.325 1.326 1.327 1.328 1.329

0.96994 0.97018 0.97042 0.97066 0.97090

38632 92687 13740 188 67070 74387 74662 236 85806 69462 13034 465 94838 36036 71365 051 94163 33208 35004 060

0.24332 0.24235 0.24138 0.24041 0.23944

87941 46638 23445 582 87287 80615 91516 463 84210 55885 01181 759 78719 42753 16828 662 70824 11769 41682 448

1.330 1.331 1.332 1.333 1.334

0.97114 0.97138 0.97162 0.97185 0.97209

83779 21044 56233 768 63683 60583 78261 900 33874 13835 59117 786 94348 43780 95451 405 45104 14372 46235 282

0.23847 0.23750 0.23653 0.23556 0.23458

60534 33723 20751 578 47859 79643 43748 768 32810 20797 47988 097 15395 28690 21258 288 95624 75063 04672 221

1.335 1.336 1.337 1.338 1.339

0.97232 0.97256 0.97279 0.97302 0.97325

86138 90534 56369 230 17450 38163 80187 900 39036 24129 04871 129 50894 16271 73757 0146 53021 83406 09557 931

0.23361 0.23264 0.23167 0.23069 0.22972

73508 31892 95492 805 49055 71391 49935 286 22276 66003 85946 099 93180 88407 85958 358 61778 11512 99624 085

1.340 1.341 1.342 1.343 1.344

0.97348 0.97371 0.97394 0.97416 0.97439

45416 95319 37478 787 28077 22772 08238 616 01000 37498 N994 365 64184 12205 46167 522 17626 20575 48173 349

0.22875 0.22777 0.22680 0.22583 0.22485

28078 08459 46523 264 92090 52617 18849 831 53825 17584 84074 691 13291 77188 87585 859 70500 05482 55305 819

1.345 1.346 1.347 1.348 1.349

0.97461 0.97483 0.97506 0.97528 0.97550

61324 37264 08052 713 95276 37901 46006 501 19479 99092 43832 603 33932 98416 67265 423 38633 14428 88217 916

0.22388 0.22290 0.22193 0.22095 0.21998

25459 76744 96286 212 78180 65480 05279 929 28672 46415 65290 729 76944 94502 50100 463 23007 84913 26774 007

1.350

0.97572 33578 26659 06926 111 (-;I1

[ 1

0.21900 66870 93041 58142 002 c-p3

II 1

ELEMENTARY CIRCULAR

SINES

AND

x

TRANSCENDENTAL COSINES

FOR

169

FUNCTIONS RADIAN

ARGUMENTS

sin x

Table

4.6

cos x

1.350 1.351 1.352 1.353 1.354

0.97572 0.97594 0.97615 0.97637 0.97659

33578 18766 94194 59861 15764

26659 15612 62771 50591 62506

06926 73996 12353 39095 87244

111 110 536 407 418

0.21900 0.21803 0.21705 0.21607 0.21510

66870 08543 48036 85358 20520

93041 94501 65124 80961 18281

58142 05261 29854 96725 76154

002 504 627 291 163

1.355 1.356 1.357 1.358 1.359

0.97680 0.97701 0.97723 0.97744 0.97765

61901 98270 24869 41696 48748

82927 97238 91804 53965 72037

27405 89325 83352 21803 40225

609 386 894 706 805

0.21412 0.21314 0.21217 0.21119 0.21021

53530 84399 13137 39753 64257

53567 63517 25046 15278 11553

46271 95410 24434 49048 02083

899 772 790 406 908

1.360 1.361 1.362 1.363 1.364

0.97786 0.97807 0.97828 0.97848 0.97869

46024 33521 11237 79171 37319

35316 34074 59561 04006 60615

18567 02249 23135 20406 61343

849 690 125 864 685

0.20923 0.20826 0.20728 0.20630 0.20532

86658 06968 25195 41349 55440

91419 32637 13175 11211 05130

35767 23964 64404 80880 25435

598 842 112 089 952

1.365 1.366 1.367 1.368 1.369

0.97889 0.97910 0.97930 0.97950 0.97970

85681 24253 53035 72024 81217

23574 88047 50175 07082 56868

61999 07786 73954 45982 39862

774 196 516 521 027

0.20434 0.20336 0.20238 0.20140 0.20042

67477 77471 85432 91369 95291

73521 95182 49113 14517 70801

80524 61151 16990 34489 38946

932 240 457 495 217

1.370 1.371 1.372 1.373 1.374

0.97990 0.98010 0.98030 0.98050 0.98069

80613 70211 50007 20000 80189

98614 32380 59206 81114 01103

22288 30754 93540 49613 68424

769 328 094 233 652

0.19944 0.19846 0.19748 0.19650 0.19552

97209 97133 95072 91036 85036

97572 74640 82010 99890 08682

96568 16515 52911 06852 28380

820 079 545 798 853

1.375 1.376 1.377 1.378 1.379

0.98089 0.98108 0.98128 0.98147 0.98166

30570 71142 01903 22852 33986

23155 52232 94276 56212 45944

69608 42586 66065 27452 42153

920 155 826 479 343

0.19454 0.19356 0.19258 0.19160 0.19062

77079 67178 55340 41577 25898

88987 21600 87511 67905 44156

18444 30840 74132 13553 72884

822 918 912 129 094

1.380 1.381 1.382 1.383 1.384

0.98185 0.98204 0.98223 0.98241 0.98260

35303 26802 08480 80336 42368

72359 45326 75694 75296 56947

72787 48298 82965 95320 26961

813 791 850 221 571

0.18964 0.18865 0.18767 0.18669 0.18571

08312 88831 67462 44217 19105

97834 10696 64691 41955 24813

36320 50314 25395 37980 32156

915 508 757 715 930

1.385 1.386 1,387 1.388 1.389

0.98278 0.98297 0.98315 0.98333 0.98352

94574 36952 69500 92216 05100

34442 22562 37068 94707 13205

61276 42059 92032 31273 95537

561 162 708 673 148

0.18472 0.18374 0.18276 0.18178 0.18079

92135 63119 32665 00183 65884

95776 37540 32988 65185 17379

21451 90577 97169 73489 28124

016 542 360 451 404

1.390 1.391 1.392 1.393 1.394

0.98370 0.98388 0.98405 0.98423 0.98441

08148 01359 84731 58262 21952

11276 08614 25898 84790 07939

54484 29809 13274 84637 29485

004 722 870 207 405

0.17981 0.17882 0.17784 0.17686 0.17587

29776 91871 52177 10704 67464

72999 15656 29142 97424 04651

47659 98336 27690 66173 28756

616 311 484 860 976

1. 39-5 1.396 1.397 1.398 1.399

0.98458 0.98476 0.98493 0.98510 0.98527

75797 19796 53948 78250 92701

18974 42512 04152 30479 49063

56974 17462 20048 50013 86162

360 083 145 670 846

0.17489 0.17390 0.17292 0.17193 0.17095

22464 75715 27228 77011 25074

35146 73409 04115 12112 82423

16514 18192 11759 65937 41718

467 681 690 830 833

1.400

0.98544

97299

947

0.16996

71429

00240

93861

675

[ (-/I11 88460

18065

II(-;j31

170

ELEMENTARY

Table

4.6

CIRCULAR

TRANSCENDENTAL

SINES

AND COSINES

FUNCTIONS

FOR RADIAN

cos x

sin z

1:

ARGUMENTS

1.400 1.401 1.402 1.403 1.404

0.98544 0.98561 0.98578 0.98595 0.98612

97299 88460 18065 947 92043 78208 63203 840 76931 48834 84013 966 51961 31850 04837 776 17131 59751 28769 609

0.16996 0.16898 0.16799 0.16701 0.16602

71429 00240 93861 675 16083 50929 72373 233 59048 20024 23971 842 00332 93227 93533 854 39947 56412 25523 303

1.405 1.406 1.407 1.408 1.409

0.98628 0.98645 0.98661 0.98677 0.98693

72440 66021 54406 982 17886 85129 92502 294 53468 52531 82515 912 79184 04669 09070 631 95031 78970 18307 486

0.16503 0.16405 0.16306 0.16207 0.16109

77901 95615 65404 770 14205 97042 61039 544 48869 47062 64065 184 81902 32209 31258 571 13314 39179 25882 568

1.410 1.411 1.412 1.413 1.414

0.98710 0.98725 0.98741 0.98757 0.98773

01010 13850 34142 909 97117 48711 74427 198 83352 23943 67004 304 59712 80922 65672 895 26197 62012 66048 706

0.16010 0.15911 0.15812 0.15714 0.15615

43115 54831 19016 356 71315 66184 90869 577 97924 60420 32080 359 22952 24876 44997 336 46408 47050 44945 751

1.415 1.416 1.417 1.418 1.419

0.98788 0.98804 0.98819 0.98834 0.98850

82805 10565 21328 142 29533 70919 57953 120 66381 88402 91177 144 93348 09330 40532 586 10430 81005 45199 170

0.15516 0.15417 0.15319 0.15220 0.15121

68303 14596 61477 752 88646 15325 39606 967 07447 37202 41027 471 24716 68347 45317 231 40463 97033 51126 135

1.420 1.421 1.422 1.423 1.424

0.98865 0.98880 0.98895 0.98909 0.98924

17628 51719 79273 627 14939 70753 66940 521 02362 88375 97544 222 79896 55844 40562 021 47539 25405 60478 351

0.15022 0.14923 Oil4824 0.14725 0.14626

54699 11685 77348 698 67432 00880 64281 559 78672 53344 74765 840 88430 57953 95314 499 96716 03732 37224 747

1.425 1.426 1.427 1.428 1.429

0.98939 0.98953 0.98967 0.98982 0.98996

05289 50295 31560 129 53145 84738 52533 174 91106 83949 61159 714 19171 04132 48716 941 37337 02480 74376 619

0.14528 0.14429 0.14330 0.14231 0.14132

03538 79851 37675 648 08908 75628 60810 986 12835 80526 98807 514 15329 84153 72928 666 16400 76259 34563 848

1.430 1.431 1.432 1.433 1.434

0.99010 0.99024 0.99038 0.99052 0.99065

45603 37177 79485 729 43968 67397 01748 121 32431 53301 89307 176 10990 56046 14729 460 79644 37773 88889 346

0.14033 0.13934 0.13835 0.13736 0.13637

16058 46736 66253 390 14312 85619 82699 275 11173 83083 31761 733 06651 29440 95441 799 00755 15144 90849 940

1.435 1.436 1.437 1.438 1.439

0.99079 0.99092 0.99106 0.99119 0.99132

38391 61619 74754 605 87230 91709 01072 941 26160 93157 75959 459 55180 32073 00385 060 74287 75552 81565 735

0.13537 0.13438 0.13339 0.13240 0.13141

93495 30784 71160 849 84881 67086 26554 495 74924 14910 85143 546 63632 65254 13887 244 51017 09245 19491 852

1.440 1.441 1.442 1.443 1.444

0.99145 0.99158 0.99171 0.99184 0.99197

83481 91686 46252 760 82761 49554 53923 766 72125 19229 09874 676 51571 71773 78212 505 21099 79243 94748 990

0.13042 0.12943 0.12844 0.12744 0.12645

37087 38145 49297 752 21853 43347 92153 306 05325 16375 79275 576 87512 48881 85098 002 68425 32647 28105 135

1.445 1.446 1.447 1.448 1.449

0.99209 0.99222 0.99234 0.99247 0.99259

80708 14686 79795 055 30395 52141 50856 088 70160 66639 35228 024 00002 34203 82494 216 19919 31850 76923 086

0.12546 0.12447 0.12348 0.12248 0.12149

48073 59580 71654 525 26467 21717 24785 871 03616 11217 43017 513 79530 20366 29130 391 54219 41572 33939 548

1.450

0.99271 29910 37588 49766 535

1c-y1

0.12050 27693 67366 57053 287 (-92

[ 1

ELEMENTARY CIRCULAR

SINES

.4ND

TRANSCENDENTAL

COSINES

FOR

RADIAN

ARGUMENTS

sin x

X

171

FTJNCTIONS Table

4.6

cos x

1.450 1.451 1.452 1.453 1.454

0.99271 0.99283 0.99295 0.99307 0.99318

29910 37588 49766 535 29974 30417 91459 118 20109 90332 63717 946 00315 98319 11543 325 70591 36356 75120 114

0.12050 0.11950 0.11851 0.11752 0.11653

27693 67366 57053 287 99962 90401 47620 080 71037 03450 05063 327 40925 99404 79804 068 09639 71276 73971 735

1.455 1.456 1.457 1.458 1.459

0.99330 0.99341 0.99353 0.99364 0.99375

30934 87418 01619 777 81345 35468 56903 143 21821 65467 37123 830 52362 63366 80232 355 72967 16112 77380 893

0.11553 0.11454 0.11355 0.11255 0.11156

77188 12194 42103 061 43581 15402 91829 237 08828 74262 84551 407 72940 82249 36104 618 35927 32951 17410 313

1.460 1.461 1.462 1.463 1.464

0.99386 0.99397 0.99408 0.99419 0.99430

83634 11644 84228 683 84362 38896 32148 075 75150 87794 39331 194 55998 49260 21797 223 26904 15209 04300 286

0.11056 0.10957 0.10858 0.10758 0.10659

97798 20069 55117 465 58563 37417 32232 463 18232 78917 88737 835 76816 38604 22199 915 34324 10617 88365 556

1.465 1.466 1.467 : . tb6:

0.99440 0.99451 0.99461 0.99472 0.99482

87866 78550 31137 923 38885 33187 76860 141 79958 74019 56879 043 11085 96938 37979 012 32265 98831 48727 437

0.10559 0.10460 0.10361 0.10261 0.10162

90765 89208 01747 983 46151 68730 36201 884 00491 43646 25487 846 53795 08521 63826 230 06072 58026 06440 584

1.470 1.471 1.472 1.473 1.474

0.99492 0.99502 0.99512 0.99522 0.99531

43,49777580 89785 993 44780 32063 44122 430 36112 62150 87122 898 17493 68709 96604 762 88922 53602 62729 932

0.10062 57333 86931 70090 698 0.09963 07588 90112 33595 391 0.09863 56847 62542 38345 147 0.09764 05119 99295 88804 678 OiO9664 52415 95545 53005 525

1.475 1.476 1.477 1.478 1.479

0.99541 0.99551 0.99560 0.99569 0.99578

50398 19685 97818 664 01919 70812 46063 854 43486 11829 93145 787 75096 48581 75747 356 96749 87906 90969 720

0.09564 0.09465 0.09365 0.09266 0.09166

98745 46561 63028 806 44118 47711 15478 186 88544 94456 71943 189 32034 82355 59452 948 74598 07058 70920 484

1.480 1.481 1.482 1.483 1.484

0.99588 0.99597 0.99606 0.99614 0.99623

08445 37640 05648 408 10182 06611 65569 851 01959 04648 04588 337 83775 42571 53643 374 55630 32200 49677 461

0.09067 0.08967 0.08867 0.08768 0.08668

16244 64309 65577 623 56984 49943 69400 641 96827 59886 75526 752 35783 90154 44661 519 73863 36851 05477 303

1.485 1.486 1.487 1.488 1.489

0.99632 0.99640 0.99649 0.99657 0.99665

17522 86349 44454 246 69452 18829 13277 079 11417 44446 63607 933 43417 79005 43586 693 65452 39305 50450 815

0.08569 0.08469 0.08369 0.08270 0.08170

11075 96168 55002 845 47431 64385 59004 070 82940 37866 52356 240 17612 13060 39407 518 51456 86499 94334 076

1.490 1.491 1.492 1.493 1.494

0.99673 0.99681 0.99689 0.99697 0.99705

77520 43143 38855 320 79621 09312 29093 143 71753 57602 15215 811 53917 08799 73054 448 26110 84688 68141 099

0.08070 0:0797i 0.07871 0.07771 0.07672

84484 54800 61486 832 i6705 i4659 55729 907 48128 62854 62770 926 78764 96243 39483 234 08624 11762 14220 152

1.495 1.496 1.497 :*. t;:

0.99712 ii99720 0.99727 0.99735 0.99742

88334 08049 63530 364 40586 02660 27521 334 62865 93295 41279 821 15173 05727 06360 877 37506 66724 52131 595

0.07572 0.07472 0.07372 0.07273 0.07173

37716 06424 87121 354 66050 77322 30411 478 93638 21620 88691 060 20488 36561 79219 898 46611 19459 92192 943

1.500

0.99749 49866 04054 43094 172

r’-,7”1 L

’

-I

0.07073 72016 67702 91008 819 C-7812

[ 1

172

ELEMENTARY Table

4.6

CIRCULAR

TRANSCENDENTAL

SINES

AND

COSINES

FUNCTIONS FOR

RADIAN

sin x

X

ARGUMENTS cos x

1.500 1.501 1.502 1.503 1.504

0.99749 0.99756 0799763 0.99770 0.99776

49866 52250 44659 27091 99547

04054 46480 23765 66667 06942

43094 86109 37519 10173 80349

172 251 509 501 750

0.07073 0.06973 0.06874 0.06774 0.06674

72016 96714 20715 44028 66664

67702 78750 50131 79447 64365

91008 12531 67342 39990 89231

819 065 208 761 245

1.505 1.506 1.507 1.508 1.509

0.99783 0.99790 0.99796 0.99802 0.99809

62024 14524 57044 89585 12145

77346 11631 44547 11841 50260

94581 76379 32859 61264 55394

063 092 104 976 397

0.06574 0.06475 0.06375 0.06275 0.06175

88633 09943 30607 50633 70031

02623 92023 30434 15789 46086

48257 24928 01988 37280 63952

343 268 470 758 953

1.510 1.511 1.512 2.513 1.514

0.99815 0.99821 0.99827 0.99833 0.99838

24724 27322 19938 02571 75221

97548 92446 74695 85033 65198

11924 36636 50542 95912 42198

274 332 912 947 118

0.06075 0.05976 0.05876 0.05776 0.05676

88812 06985 24560 41548 57959

19385 33809 87538 78816 05945

90658 01748 57464 94113 24248

160 769 281 053 072

1.515 1.516 1.517 1.518 1.519

0.99844 0.99849 0.99855 0.99860 0.99865

37887 90569 33265 65976 88701

57923 06943 56990 53793 44081

91859 86092 10456 00399 46683

188 495 612 163 784

0.05576 0.05476 0.05377 0.05277 0.05177

73801 89086 03823 18023 31695

67282 61243 86301 40981 23862

36836 97425 48297 08625 74620

851 545 399 609 716

1.520 1.521 1.522 1.523 1.524

0.99871 0.99876 0.99880 0.99885 0.99890

01439 04190 96954 79730 52517

75583 97023 58128 09621 03224

00717 79776 72136 42098 34913

231 634 872 089 328

0.05077 0.04977 0.04877 0.04777 0.04677

44849 57495 69644 81305 92488

33579 68814 28305 10835 15238

19672 94487 27218 23593 67036

613 284 360 598 388

1.525 1.526 1.527 1.528 1.529

0.99895 0.99899 0.99904 0.99908 0.99912

15314 68123 10941 43769 66606

91658 28645 68902 68148 83100

81616 03749 17990 40684 92265

285 180 729 234 762

0.04578 0.04478 0.04378 0.04278 0.04178

03203 13460 23270 32642 41586

40397 85239 48738 29915 27830

18782 17991 81854 05695 63073

371 291 166 871 262

i.530 1.531 1.532 1.533 1.534

0.99916 0.99920 0.99924 0.99928 0.99932

79452 82306 75169 58038 30915

71476 91989 04354 69286 48498

01592 10170 76285 79026 22220

427 755 152 436 463

0.04078 0.03978 0.03878 0.03778 0.03678

50112 58230 65951 73283 80238

41591 70343 13276 69617 38633

05868 64380 47406 42326 15178

899 513 277 008 390

1.535 1.536 1.537 1.538 1.539

0;99935 0.99939 0.99942 0.99946 0.99949

93799 46689 89585 22486 45393

04701 01607 03928 77374 88654

38256 91817 83506 53376 84360

819 592 202 306 752

0.03578 0.03478 0.03378 0.03279 0.03179

86825 93054 98935 04478 09693

19628 11943 14956 28078 50755

10734 52566 43115 63750 74831

312 435 073 505 796

1.540 1.541 1.542 1.543 1.544

0.99952 0:99955 0.99958 0.99961 0.99964

58306 61222 54144 37069 09999

05479 96555 31593 81300 17383

05600 95674 85726 62497 71251

596 180 242 095 832

0.03079 0.02979 0.02879 0.02779 0.02679

14590 19180 23471 27475 31200

82466 22720 71058 27051 90300

15762 05041 40314 98418 35423

248 568 858 526 217

1.545 1.546 1.547 1.548 1.549

0.99966 0.99969 0.99971 0.99974 0.99976

72932 25868 68807 01749 24694

12550 40506 75959 94615 73179

18609 75272 78656 35418 23886

586 821 660 249 150

0.02579 0.02479 0.02379 0.02279 0.02179

34658 37858 40810 43524 46010

60430 37097 19980 08784 03238

86673 66826 69885 69229 17647

867 971 184 328 934

1.550

0.99978 37641 89356 96389 761

[(-;)I1

0.02079 48278 03092 47364 391

[(-;I91

ELEMENTARY CIRCULAR

SINES AND

TRANSCENDENTAL

COSINES

FOR RADIAN

FUNCTIONS

173

ARGUMENTS

Table 4.6

.c 1.550 1.551 1.552 1.553 1.554

0.99978 0.99980 0.99982 0.99984 0.99985

37641 40591 33542 16495 89450

sin x 89356 21853 50374 55624 19308

96389 81488 86102 97539 85428

761 767 606 966 298

0.02079 0.01979 0.01879 0.01779 0.01679

48278 50338 52200 53874 55370

03092 08120 18116 32894 52286

47364 70061 76905 38564 05229

391 827 802 929 507

1.555 1,556 1.557 1.558 1.559

0.99987 0.99989 0.99990 0.99991 0.99993

52406 05363 48321 81281 04241

24131 53795 93 07 279470 43888

03543 91538 76575 74851 93030

342 676 277 093 623

0.01579 56698 0.01479 57869 0.01379 58891 0.01279 59775 0.01179,60532

76142 04329 36731 73245 13782

06628 52043 30323 09896 38778

284 433 849 874 533

1.560 1.561 1.562 1.563 1.564

0.99994 0.99995 0.99996 0.99996 0.99997

17202 20163 13125 96087 69050

29966 74406 66914 98192 59945

29574 75969 17856 36062 07529

517 172 344 758 731

0.01079 0.00979 0.00879 0.00779 0.00679

61170 61701 62133 62478 62744

58267 06636 58835 14822 74562

44582 34527 95443 93777 75597

392 146 014 062 546

1.565 1.566 1.567 1.568 1.569

0.99998 0.99998 0.99999 0.99999 0.99999

32013 84976 27939 60902 83866

44876 46689 60087 80775 05456

06142 03461 69348 72499 80873

794 318 142 201 162

0.00579 0.00479 0.00379 0.00279 0.00179

62943 63084 63176 63231 63258

38028 05200 76064 50611 28835

66597 72096 77045 46023 23243

372 784 359 436 059

1.570 1.571 1.572 1.573 1.574

0.99999 0.99999 0.99999 0.99999 0.99999

96829 99792 92755 75719 48682

31834 58612 85495 13185 43386

62021 83315 12082 15626 61164

053 895 337 285 539

+0.00079 -0.00020 -0.00120 -0.00220 -0.00320

63267 36732 36729 36714 36677

10733 03695 14450 21533 24944

32548 22583 59042 14087 45343

541 254 804 901 613

1.575 1.576 1.577 1.578 1.579

0.99999 0.99998 0.99998 0.99997 0.99996

11645 64609 07572 40536 63500

78803 23138 81096 58379 61693

15654 45523 16298 92137 35254

423 419 798 261 568

-0.00420 -0.00520 -0.00620 -0.00720 -0.00820

36608 36497 36334 36109 35811

24688 20771 13205 02006 87197

30802 68822 78129 97812 87324

109 280 029 142 647

1.580 1.581 1.582 1.583 1.584

0.99995 0.99994 0.99993 0.99992 0.99991

76464 79429 72395 55361 28327

98740 78223 09847 04315 73330

05255 58361 46545 16554 08844

179 895 499 408 324

-0.00920 35432 68808 26480 539 -0.01020. 34961 46876 15451 796 -0.01120 34388 21448 74764 568 -0.01220 33702 92583 45294 454 -0.01320 32895 60348 88260 743

1.585 1.586 1.587 1.588 1.589

0.99989 0.99988 0.99986 0.99985 0.99983

91295 44263 87233 20204 43177

29595 86814 59691 63927 16226

56407 83504 04289 21344 24106

893 374 313 232 322

-0.01420 -0.01520 -0.01620 -0.01720 -0.01820

31956 30874 29641 28245 26678

24825 86108 44304 99538 51948

85219 38055 68973 20485 55400

553 737 475 440 452

1.590 1.591 1.592 1.593 1.594

0.99981 0.99979 0.99977 0.99975 0.99973

56151 59127 52105 35085 08068

34290 36823 43527 75103 53254

87198 68657 08066 24582 14867

158 422 646 972 933

-0.01920 -0.02020 -0.02120 -0.02220 -0.02320

24929 22987 20843 18488 15910

01692 48945 93900 36773 77799

56809 28070 92788 94801 98151

503 065 583 039 502

1.595 1.596 1.597 1.598 1.599

0.99970 0.99968 0.99965 0.99963 0.99960

71054 24042 67033 00029 23027

00681 41086 99171 00635 72179

50917 77790 11241 35248 99440

259 702 891 219 759

-0.02420 -0.02520 -0.02620 -0.02720 -0.02819

13101 10049 06745 03180 99342

17236 55365 92491 28945 65082

87068 65939 59282 11714 87922

552 492 234 764 093

1.600

0.99957 36030 41505 16434 211

-0.02919

95223 01288 72620 577 C-79)3

For 01.6

cos 2

c(-;I11

see Example 16.

;=I.57079

63267 94896 61923 132

[ 1

u=3.14159 26535 89793 23846 264

174

ELEMENTARY Table

TRANSCENDENTAL

RADIX

4.7

TABLE

OF

FUNCTIONS

CIRCULAR

SINES

AND

sin .~10-a 0.00000

00001

00000

00000

cos ,110-n 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0;99999 0.99999 0.99999

99999 99999 99999 99999 99999 99999 99999 99999 99999

99999 99999 99999 99999 99999 99999 99999 99999 99999

99999 99998 99995 99992 99987 99982 99975 99959

50000 00000 50000 00000 50000 00000 50000

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

99999 99999 99999 99999 99999 99999 99999 99999 99999

99999 99999 99999 99999 99999 99999 99999 99999 99999

99950 99800 99550 99200 98750 98200 97550

00000 00000 00000 00000 00000 00000 00000

96800

00000

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

99999 99999 99999 99999 99999 99999 99999 99999 99999

99999 99999 99999 99999 99998 99998 99997

95000 80000 55000 20000 75000 20000 55000

00000 00000 00000 00000 00000 00000 00000

99996

80000

00000

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

99999 99999 99999 99999 99999 99999 99999 99999 99999

99995 99980 99955 99920 99875 99820 99755

00000 00000 00000 00000 00000 00000 00000

00000 00000 00000 00000 00000 00000 00000

99680

00000

00000

99999 99999 99999 99999 99999 99999 99999 99999 99999

99500 98000 95500 92000 87500 82000 75500

00000 00000 00000

00000 00007 00034

00000 00000

00107 00260

68000

00000

00000

0:99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

59500

00000

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

00002 00000 00000 00003 00000 00000 00004 00000 00000 00005 00000 00000 00006 00000 00000 00007 00000 00000 00008 00000 00000 00009 00000 00000

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

00010 00020 00030 00040 00050 00060

00000 00000 00000 00000 00000 00000

00000 00000 00000 00000 00000 00000

00000 00000 00000 00000 00000 00000

0.00000

00069

99999

99999

99999

0.00000 0.00000

00079 00089

99999 99999

99999 99999

00000 00000. 00000 00000 00000 00000 00000 00000

99999 99999

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

00099 00199 00299 00399 00499 00599

99999 99999 99999 99999 99999 99999

99999 99999 99999 99999 99999 99999

0.00000

00699

99999

99999

99640 99428

99999 99999

99147 98785

o;ooooo 0.00000

00799 00899

99999 99999

99998 99987 99955 99893 99792

0.00000

00999

99999

99999

98333

0.00000

01999

99999

99999

86667

0.00000 0.00000 0.n0n00 0.00000 0.00000 0.00000 0.00000

02999 03999 114999 05999 06999

07999 08999

99999 99999 99999 99999 99999 99999 99999

99999 99998 99997 99996 99994 99991 99987

COSINES

55000 93333 91667 40000 28333 46667 85000

99595

95950

95000

00000

00000

00000

00000

00000

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

09999 99999 19999 99999 29999 99999 39999 99999 49999 99999 59999 99999

99963

0.00000

69999 79999

99999

89999

99999 99999

94283

00000

00540 01000 01707 02734

0.00000 0.00001 0.00002 0.00003 0.00004 0.00005

99999

99999

83333

33333

0.99999

99999

50000

00000

041bl

99999

99998

bbbbb

bbbbl

0.99999

99996

00000

00000

bbbb7

99999 99999 99999 99999

99995 99989 99979

50000 33333 16666

00002 33342 66693

99987

00000 50000

00026

66667 04167

O.OOOOb 0.00007

99999 99999

99964 99942

00000 83333 bbbbb

00065 33473 bb940

0.99999

0.99999

99968

00492

99959

0.00009 0.00019 0.00029 0.00039 0.00049 0.00059

99833 98666 95500 89333 79166

33333 bbbbb 00002 33341 bbb92

34167 93333 02500 86667 70833

0.99999 0.99999

64000

00064

80000

0.00069 0.00079

99999 99999 99999 99999 99999 99999 99999

39167

0.99999 0.99999 0.99999 0.99999 0.99999

99999

bb939 00492

73333 07499

0.99999

96800

99998

14666 78500

0.99999

0.00099

99998

33333

34166

bbbb5

0.99999

0.00000 0.00000

0.00008

0.00089

99999

99914 99878

42833

99866

99550 98933 97916 96400 91466 87850

50000

33473

33333 bbbb7 00000 33333 66667 00000

99995

99968

50000

33333 66667

0.99999 0.99999 0.99999 0.99999 0.99999

99995 99992 99982 99975

50000 00000 50000

00000 00000

00003 00010

37500

50000

00054 00100 00170 00273

04167 bbbb7 37500

99950

00000

0041b

66667

99800

00000

06666

66666

99550 99200 98750 98200 97550

33749

99990 66610

95950

00000 00001 00002 00005 OOOlD 00017 00027

95000

00041

00000

06666

60416 39999 00416

00000

66450

99352 65033

Obbbb

63026

33749

92619

bb6bb

52118

For )I >lO, sin ,110-n = .rlO-n; cos .rlO-n = 1 -i .c210-2n; to 25~. From C. E. Van Orstrand, Tables of ttie exponential function and of the circular sine and cosine to radian arguments, Memoirs of the National Academy of Sciences, vol. 14, Fifth Memoir. U.S. Government Printing Office, Washington, D.C., 1921 (with permission).

ELEMENTARY CIRCULAR

TRANSCENDENTAL

SINES AND COSINES

X

FOR LARGE

175

FUNCTIONS

RADIAN

ARGUMENTS

Table 4.8

00000 23058 68365 24966 36208

cos x 00000 68139 47142 00445 63611

00000 71740 38699 45727 91463

000 094 757 157 917

0.00000 +0.84147 +0.90929 +0.14112 -0.75680

00000 09848 74268 00080 24953

sin 5 00000 07896 25681 59867 07928

-0.95892 -0.27941 +0.65698 +0.98935 +0.41211

42746 54981 65987 82466 84852

63138 98925 18789 23381 41756

46889 87281 09039 77780 56975

315 156 700 812 627

+0.28366 +0.96017 +0.75390 -0.14550 -0.91113

21854 02866 22543 00338 02618

63226 50366 43304 08613 84676

26446 02054 63814 52586 98836

664 565 120 884 829

-0.54402 -0I99999 -0.53657 +0:42016 +0.99060

11108 02065 29180 70368 73556

89369 50703 00434 26640 94870

81340 45705 97166 92186 30787

475 156 537 896 535

-0.83907 +0.00442 +0.84385 +0.90744 +0.13673

15290 56979 39587 67814 72182

76452 88050 32492 50196 07833

45225 78574 10465 21385 59424

886 836 396 269 893

+0.65028 -0.28790 -0.96139 -0.75098 +0.14987

78401 33166 74918 72467 72096

57116 65065 79556 71676 62952

86582 29478 85726 10375 32975

974 446 164 016 424

-0.75968 -0.95765 -0.27516 +0.66031 +0.98870

79128 94803 33380 67082 46181

58821 23384 51596 44080 86669

27384 64189 92222 14481 25289

815 964 034 610 835

+0.91294 52507 27627 65437 610 +0.83665 56385 36056 03186 648 -0.00885 13092 90403 87592 169 -0.84622 04041 75170 63524 133 -0.90557 83620 06623 84513 579

+0.40808 -0.54772 -0I99996 -0.53283 +0.42417

20618 92602 08263 30203 90073

13391 24268 94637 33397 36996

98606 42138 12645 55521 97593

227 427 417 576 705

-0.13235 17500 97773 +0.76255 84504 79602 +0.95637 59284 04503 +0.27090 57883 07869 -0.66363 38842 12967

02890 73751 01343 01998 50215

201 582 234 634 117

+0.99120 +0.64691 -0:29213 -0.96260 -0.74805

28118 93223 88087 58663 75296

63473 28640 33836 13566 89000

59808 34272 19337 60197 35176

329 138 140 545 519

-0.98803 -0.40403 +0.55142 +0.99991 +0.52908

16240 76453 66812 18601 26861

92861 23065 41690 07267 20023

78998 00604 55066 14572 82083

775 877 156 808 249

+0.15425 +0.91474 +0.83422 -0.01327 -0.84857

14498 23578 33605 67472 02747

87584 04531 06510 23059 84605

05071 27896 27221 47891 18659

866 244 553 522 997

-0.42818 -0.99177 -0.64353 +0.29636 +0.96379

26694 88534 81333 85787 53862

96151 43115 56999 09385 84087

00440 73683 46068 31739 75326

675 529 567 230 066

-0.90369 -0.12796 +0.76541 +0.95507 +0.26664

22050 36896 40519 36440 29323

91506 27404 45343 47294 59937

75984 68102 35649 85758 25152

730 833 108 654 683

ti

+0.74511 31604 79348 78698 771 -0.15862 26688 04708 98710 332 -0.91652 15479 15633 78589 899 -0.83177 47426 28598 28820 958 +0.01770 19251 05413 57780 795

-0.66693 -0.98733 -0.39998 +0.55511 +0.99984

80616 92775 53149 33015 33086

52261 23826 88351 20625 47691

84438 45822 29395 67704 22006

409 883 471 483 901

442 47 48 49

+0.85090 +0.90178 +0.12357 -0.76825 -0.95375

+0.52532 -0.43217 -0.99233 -0.64014 +0.30059

19888 79448 54691 43394 25437

17729 84778 50928 69199 43637

69604 29495 71827 73131 08368

746 278 975 294 703

:: 12 13 14

20 t: 23 24

310 ;23 34 ;56 ;78 39 40 t:

35245 83476 31227 46613 26527

34118 48809 45224 23666 59471

00000 50665 69539 22210 25137

000 250 602 074 264

1.00000 +0.54030 -0.41614 -0.98999 -0.65364

42486 18503 00406 79904 81836

238 329 153 497 042

50 +0.96496 60284 92113 27406 896 -0.26237 48537 03928 78591 439 From C. E. Van Orstrand, Tables of the exponential function and of the circular sine and cosine to radian arguments, Memoirs of the National Academy of Sciences, vol. 14, Fifth Memoir. U.S. Government Printing Office, Washington, D.C., 1921 (with permission) for x_
176

ELEMENTARY

Table 4.8

CIRCULAR

TRANSCENDENTAL

SINES

AND COSINES

FUNCTIONS

FOR LARGE

RADIAN

sin x

ARGUMENTS co5 x

-0.26237 +0.67022 +0.98662 +0,39592 -0.55878

48537 91758 75920 51501 90488

03928 43374 40485 81834 51616

78591 73449 29658 18150 24581

439 435 757 339 787

+0.96496 +0.74215 -0.16299 -0.91828 -0.82930

60284 41968 07807 27862 98328

92113 13782 95705 12.118 63150

27406 53946 48100 89119 14772

896 738 333 973 785

-0.99975 -0.52155 +0.43616 +0.99287 +0.63673

51733 10020 47552 26480 80071

58619 86911 47824 84537 39137

83659 88018 95908 11816 88077

863 741 053 509 123

+0.02212 +0.85322 +0.89986 +0.11918 -0.77108

67562 01077 68269 01354 02229

61955 22584 69193 48819 75845

73456 11396 78650 28543 22938

356 968 300 584 744

-0.30481 -0.96611 -0.73918 +0.16735 +0.92002

06211 77700 06966 57003 60381

02216 08392 49222 02806 96790

70562 94701 86727 92152 68335

565 829 602 784 154

-0.95241 -0.25810 +0.67350 +0.98589 +0.39185

29804 lb359 71623 65815 72304

15156 38267 23586 82549 29550

29269 44570 25288 69743 00516

382 121 783 864 171

+0.82682 -0.02655 -0.85551 -0.89792 -0.11478

86794 11540 99789 76806 48137

90103 23966 75322 89291 83187

46771 79446 25899 26040 22054

021 384 683 073 507

-0.56245 -0.99964 -0.51776 +O, 44014 +0.99339

38512 74559 97997 30224 03797

38172 66349 89505 96040 22271

03106 96483 06565 70593 63756

212 045 339 105 155

+0.77389 +0.95105 +0.25382 -0.67677 -0.98514

06815 46532 33627 19568 62604

57889 54374 62036 87307 68247

09778 63665 27306 62215 37085

733 657 903 498 189

+0.63331 -0.30902 -0.96725 -0.73619 +0.17171

92030 27281 05882 27182 73418

86299 66070 73882 27315 30777

83233 70291 48729 96016 55609

201 749 171 815 845

-0.38778 +0.56610 +0.99952 +0.51397 -0.44411

lb354 76368 01585 84559 26687

09430 98180 80731 87535 07508

43773 32361 24386 21169 36850

094 028 610 609 760

+0.92175 +0.82433 -0.03097 -0.85780 -0.89597

12697 13311 50317 30932 09467

24749 07557 31216 44987 90963

31639 75991 45752 85540 14833

230 501 196 835 703

-0.99388 -0.62988 +0.31322 +0.96836 +0.73319

86539 79942 87824 44611 03200

23375 74453 33085 00185 73292

18973 87856 15263 40435 lb636

081 521 353 015 321

-0.11038 +0.77668 +0.94967 +0.24954 -0.68002

72438 59820 76978 01179 34955

39047 21631 82543 73338 87338

55811 15768 20471 12437 79542

787 342 326 735 720

-0.17607 -0.92345 -0.82181 +0.03539 +0.86006

56199 84470 78366 83027 94058

48587 04059 30822 33660 12453

07696 80260 54487 68362 22683

212 163 211 543 685

-0.98437 -0.38369 +0.56975 +0.99937 +0.51017

66433 84449 03342 32836 70449

94041 49741 65311 95124 41668

89491 84477 92000 65698 89902

821 893 851 442 379

+0.89399 +0.10598 -0.77946 -0.94828 -0.24525

66636 75117 60696 21412 19854

00557 51156 15804 69947 67654

89051 85002 68855 23213 32522

827 021 400 104 044

-0.44807 -ii99436 -0.62644 +0.31742 +0.96945

36161 74609 44479 87015 93666

29170 28201 10339 19701 69987

15236 52610 06880 64974 60380

548 672 027 551 439

+0.68326 +0.98358 +0.37960 -0.57338 -0.99920

17147 77454 77390 18719 68341

36120 34344 27521 90422 86353

98369 85760 69648 88494 69443

958 773 192 922 272

+0.73017 -Oil8043 -0.92514 -Or81928 +0.03982

35609 04492 75365 82452 08803

94819 91083 96413 91459 93138

66479 95011 89170 25267 89816

352 850 475 566 180

-0.50636

56411

09758

79365

656

+0.86231

88722

87683

93410

194

ELEMENTARY CIRCULAR X

SINES

TRANSCENDENTAL

AND COSINES

sin 5

FOR LARGE

CO8x

RADIAN X

177

FUNCTIONS ARGUMENTS

Table 4.8

sin z

co9 2

102 103 104

-0.50636 +0.45202 +0.99482 -0.62298 -0.32162

564 579 679 863 240

+0.86231 +0.89200 +0.10158 -0.78223 -0.94686

887 487 570 089 801

150 151 152 153 154

-0.71487 +0.20214 +0.93332 +0.80640 -0.06192

643 988 052 058 034

+0.69925 +0.97935 +0.35904 -0.59136 -0.99808

081 460 429 968 109

105 106 107 108 109

-0.97053 -0.72714 +0.18478 +0.92681 +0.81674

528 250 174 851 261

-0.24095 +0.68648 +0.98277 +0.37550 -0.57700

905 655 958 960 218

155 156 157 158 159

-0.87331 -0.88178 -0.07954 +0.79582 +0.93951

198 462 854 410 973

-0.48716 +0.47165 +0.99683 +0.60552 -0.34249

135 229 099 787 478

110 111 112 113 114

-0.04424 -0.86455 -0.88999 -0.09718 +0.78498

268 145 560 191 039

-0.99902 -0.50254 +0.45596 +0.99526 +0.61952

081 432 910 664 061

160 161 162 163 164

+0.21942 -0.70240 -0.97845 -0.35491 +0.59493

526 779 035 018 278

-0.97562 -0.71177 +0.20648 +0.93490 +0.80377

931 476 223 040 546

115 116 117 118 119

+0.94543 +0.23666 -0.68969 -0.98195 -0.37140

533 139 794 217 410

-0.32580 -0.97159 -0.72409 +0.18912 +0.92847

981 219 720 942 132

165 166 167 168 169

+0.99779 +0.48329 -0.47555 -0.99717 -0.60199

728 156 019 329 987

-0.06633 -0.87545 -0.87968 -0.07513 +0.79849

694 946 859 609 619

120 121 122 123 124

+0.58061 +0.99881 +0.49871 -0.45990 -0.99568

118 522 315 349 699

+0.81418 -0.04866 -0.86676 -0.88796 -0.09277

097 361 709 891 620

170 171 172 173 174

+0.34664 +0.97659 +0.70865 -0.21081 -0.93646

946 087 914 053 197

+0.93799 +0.21510 -0.70555 -0.97752 -0.35076

475 527 101 694 911

125 126 127 128 129

-0.61604 +0.32999 +0.97263 +0.72103 -0.19347

046 083 007 771 339

+0.78771 +0.94398 +0.23235 -0.69289 -0.98110

451 414 910 582 552

175 176 177 178 179

-0.80113 +0.07075 +0.87758 +0.87757 +0.07072

460 224 979 534 217

+0.59848 +0.99749 +0.47941 -0.47943 -0.99749

422 392 231 877 605

130 131 132 133 134

-0.93010 -0.81160 +0.05308 +0.86896 +0.88592

595 339 359 576 482

-0.36729 +0.58420 +0.99859 +0.49487 -0.46382

133 882 007 222 887

180 181 182 183 184

-0.80115 -0.93645 -0.21078 +0.70868 +0.97658

264 140 107 041 438

-0.59846 +0.35079 +0.97753 +0.70552 -0.21513

007 734 329 964 471

135 136 137 138 139

+0.08836 -0.79043 -0.94251 -0.22805 +0.69608

869 321 445 226 013

-0.99608 -0.61254 +0.33416 +0.97364 +0.71796

784 824 538 889 410

185 186 187 188 189

+0.34662 -0.60202 -0.99717 -0.47552 +0.48331

118 394 102 367 795

-0.93800 -0.79847 +0.07516 +0.87970 +0.87544

520 804 615 293 489

140 141 142 143 144

+0.98023 +0.36317 -0.58779 -0.9.9834 -0.49102

966 137 501 536 159

-0.19781 -0.93172 -0.80900 +0.05750 +0.87114

357 236 991 253 740

190 191 192 193 194

+0.99779 +0.59490 -0.35493 -0.97845 -0.70238

928 855 836 657 633

+0.06630 -0.80379 -0.93488 -0.20645 +0.71179

686 339 971 273 593

145 146 147 148 149

+0.46774 +0.99646 +0.60904 -0.33833 -0.97464

516 917 402 339 865

+0.88386 +0.08395 -0.79313 -0.94102 -0.22374

337 944 642 631 095

195 196 197 198 199

+0.21945 +0.93953 +0.79580 -0.07957 -0.88179

467 006 584 859 884

+0.97562 +0.34246 -0.60555 -0.99682 -0.47162

270 646 186 859 571

150

-0.71487 643

+0.69925 081

200

-0.87329 730

11ooY

+0.48718 768

178

ELEMENTARY

Table 4.8 X

CIRCULAR

SINES

TRANSCENDENTAL

AND COSINES co9 x

sin x

FUNCTIONS

FOR LARGE 5

RADIAN sin x

ARGUMENTS cos x

200 201 202 203 204

-0.87329 -0.06189 +0.80641 +0.93330 +0.20212

730 025 841 970 036

+0.48718 +0.99808 +0.59134 -0.35907 -0.97936

768 296 538 242 069

250 251 252 253 254

-0.97052 -0.32159 +0.62301 +0.99482 +0.45199

802 386 221 373 890

+0.24098 +0.94687 +0.78221 -0.10161 -0.89201

831 771 211 569 850

205 206 207 208 209

-0.71489 -0.97464 -0.33830 +0.60906 +0.99646

751 190 503 793 664

-0.69922 +0.22377 +0.94103 +0.79311 -0.08398

926 033 651 806 947

255 256 257 258 259

-0.50639 -0.99920 -0.57335 +0.37963 +0.98359

163 803 717 563 318

-0.86230 -0.03979 +0.81930 +0.92513 +0.18040

361 076 553 609 080

210 211 212 213 214

+0.46771 -0.49104 -0.99834 -0.58777 +0.36319

852 785 709 062 945

-0.88387 -0.87113 -0.05747 +0.80902 +0.93171

747 260 243 763 141

260 261 262 263 264

+0.68323 -0.24528 -0.94829 -0.77944 +0.10601

970 121 171 719 749

-0.73019 +X96945 -0.31740 +0.62646 +0.99436

416 197 012 794 427

215 216 217 218 219

+0.98024 +0.69605 -0.22808 -0.94252 -0.79041

562 849 161 453 474

+0.19778 -0.71798 -0.97364 -0.33413 +0.61257

403 508 202 697 207

265 266 267 268 269

+0.89401 +0.86005 +0.03536 -0.82183 -0.92344

017 403 818 501 688

+0.44804 -0.51020 -0.99937 -0.56972 +0.38372

667 297 435 556 628

220 221 222 223 224

+0.08839 +0.88593 +0.86895 +0.05305 -0.81162

871 880 084 349 100

+0.99608 517 +Q.46380 216 -0.49489 841 -0.99859 167 -0.58418 435

270 271 272 273 274

-0.17604 +0.73321 +0.96835 +0.31320 -0.62991

595 082 694 015 141

+0.98438 195 +0168000 139 -0.24956 931 -0194968 714 -0.77666 699

225 226 227 228 229

-0.93009 -0.19344 +0.72105 +0.97262 +0.32996

488 382 860 306 237

+0.36731 +0.98111 +0.69287 -0.23238 -0.94399

937 135 409 a42 409

275 276 277 278 279

-0.99388 -0.44408 +0.51400 +0.99952 +0.56608

533 566 431 109 279

+0.11041 +0.89598 +0.85778 +0.03094 -0.82434

720 433 760 490 840

230 231 232 233 234

-0.61606 -0.99568 -0.45987 +0.49873 +0.99881

420 419 672 928 669

-0.78769 +0.09280 +0.88798 +0.86675 +0.04863

594 622 277 206 350

280 281 282 283 284

-0.38780 -0.98515 -0.67674 +0.25385 +0.95106

942 144 976 252 397

-0.92173 -Oil7168 +0.73621 +0:96724 +0.30899

958 765 312 294 406

235 236 237 238 239

+0.58058 -0.37143 -0.98195 -0.68967 +0.23669

664 209 787 611 068

-0.81419 -0.92846 -0.18909 +0.72411 +0.97158

847 012 982 799 506

285 286 287 288 289

+0.77387 -0.11481 -0.89794 -0.85550 -0.02652

159 476 095 437 102

-0.63334 -0.99338 -0.44011 +0.51779 +0.99964

253 692 595 559 826

240 241 242 243 244

+0.94544 +0.78496 -0.09721 -0.89000 -0.86453

515 171 191 935 630

+0.32578 -0.61954 -0.99526 -0.45594 +0.50257

131 428 371 228 038

290 291 292 293 294

+0.82684 +0.92001 +0.16732 -0.73920 -0.96610

563 423 598 100 999

+0.56242 -0.39188 -0.98590 -0.67348 +0.25813

893 496 163 488 076

245 246 247 248 249

-0.04421 +0.81676 +0.92680 +0.18475 -0.72716

256 000 719 212 319

+0.99902 +0.57697 -0.37553 -0.98278 -0.68646

215 756 754 515 463

295 296 297 298 299

-0.30478 +0.63676 +0.99286 +0.43613 -0.52157

191 125 906 763 672

+0.95242 +0.77106 -0.11921 -0.89987 -0.85320

217 103 006 997 439

250

-0.97052 802

+0.24098 831

300

-0.99975 584

-0.02209 662

ELEMENTARY

CIRCULAR

SINES

x

sin x

TRANSCENDENTAL

AND COSINES

FUNCTIONS

FOR LARGE

cos x

RADIAN x

179

ARGUMENTS sin x

Table 4.8 cos x

300 301 302 303 304

-0.99975 -0.55876 +O. 39595 +O. 98663 +O. 67020

584 405 283 250 680

-0.02209 +O. 82932 +O. 91827 +O. 16296 -0.74217

662 668 085 104 440

350 351 352 353 354

+O. 14114

985

+o. 90930 +O. 84145

997 470

305 306 307 308 309

-0.26240 -0.95376 -0.76823 +O. 12360 +O. 90180

394 171 536 304 137

-0.96495 -0.30056 +O. 64016 +O. 99233 +O. 43215

812 379 750 174 076

355 356 357 358 359

-0.00003 -0.84148 -0.90928 -0.14109 +O. 75682

014 727 488 017 220

-1.00000

000

-0.54027 +O. 41617 +O. 98999 +O. 65362

694 425 675 081

310 311 312 313 314

+O. 85088 +O. 01767 -0.83179 -0.91650 -0.15859

769 179 148 949 291

-0.52534 -0.99984 -0.55508 +O. 40001 +O. 98734

764 384 823 294 406

360 361 362 363 364

+O. 95891 +O. 27938 -0.65700 -0.98935 -0.41209

572 655 932 386 102

-0.28369 -0.96017 -0.75388 +O. 14552 +O. 91114

109 871 245 986 268

315 316 317 318 319

+O. 74513 +O. 96378 +O. 29633 -0.64356 -0.99177

326 735 979 121 500

+O. 66691 -0.26667 -0.95508 -0.76539 +O. 12799

560 199 258 465 359

365 366 367 368 369

+O. 54404 +o. 99999 +O. 53654 -0.42019 -0.99061

640 007 748 439 148

+O. 83905 -0.00445 -0.84387 -0.90743 -0.13670

513 584 013 412 736

320 ;s: 323 324

-0.42815 +O. 52910 +O. 99991 +o. 55140 -0.40406

543 827 226 153 522

+o. 90370 +O. 84855 +O. 01324 -0.83423 -0.91473

511 433 661 998 018

370 371 372 373 374

-0.65026 +O. 28793 +O. 96140 +O. 75096 -0.14990

494 218 579 734 701

+O. 75970 +O. 95765 +O. 27513 -0.66033 -0.98870

752 080 436 935

325 326 327 328 329

-0.98803 -0.66361 +O. 27093 +O. 95638 +O. 76253

627 133 481 473 895

-0.15422 +O. 74807 +O. 96259 +O. 29210 -0.64694

167 753 770 998 231

375 376 377 378 379

-0.91295 755 -0.83663 913 +o. 00888 145 +O. 84623 647 +O. 90556 557

-0.40805 +O. 54775 +O. 99996 +O. 53280 -0.42420

454 448 056 751 631

330 331 332 333 334

-0.13238 -0.90559 -0.84620 -0.00882 +O. 83667

163 115 434 117 215

-0.99119 -0.42415 +O. 53285 +O. 99996 +o. 54770

882 171 853 109 404

380 381 382 383 384

+O. 13232 -0.76257 -0.95636 -0.27087 +O. 66365

187 795 712 677 643

-0.99120 -0.64689 +O. 29216 +O. 96261 +O. 74803

680 634 764 403 752

335 336 337 338 339

+O. 91293 +O. 14984 -0.75100 -0.96138 -0.28787

295 741 715 920 445

-0.40810 -0.98870 -0.66029 +O. 27519 +O. 95766

958 914 407 232 816

385 386 387 388 389

+O. 98802 +o. 40401 -0.55145 -0.99991 -0.52905

697 007 183 146 711

-0.15428 -0.91475 -0.83420 +O. 01330 +O. 84858

123 454 674 689 622

340 341 342 343 344

+O. 65031 +O. 99060 +O. 42013 -0.53659 -0.99999

074 323 968 836 034

+O. 75966 -0.13676 -0.90745 -0.84383 -0.00439

831 708 945 778 555

390 391 392 393 394

+O. 42820 +O. 99178 +O. 64351 -0.29639 -0.96380

991 271 506 737 342

+O. 90367 +O. 12793 -0.76543 -0.95506 -0.26661

930 379 345 471 388

345 346 347 348 349

-0.54399 +O. 41214 +O. 98936 +O. 65696 -0.27944

582 595 263 387 444

+O. 83908 +O. 91111 +o. 14547 -0.75392 -0.96016

793 021 206 186

395 396 397 398 399

-0.74509 +O. 15865 +O. 91653 +O. 83175 -0.01773

306 243 361 801 206

+O. 66696 +O. 98733 +O. 39995 -0.55513 -0.99984

052 450 769 837 277

350

-0.95893

283

-0.28363

328

400

-0.85091

936

-0.52529

634

784

-0.95893 -0.75678

283 279

-0.28363 +O. 65366 +O. 98998 +O. 41611 -0.54032

328 643 824 943 767

010

180

ELEMENTARY

Table 4.8 5

.

CIRCULAR sin 2

SINES

TRANSCENDENTAL

AND COSINES co9

FUNCTIONS

FOR LARGE X

x

RADIAN sin x

ARGUMENTS co9

x

400 401 402 403 404

-0.85091 -0.90177 -0.12354 +0.76827 +0.95374

936 532 321 396 359

-0.52529 +0.43220 +0.99233 +0.64012 -0.30062

634 513 919 118 129

450 451 452 453 454

-0.68328 -0.98358 -0.37957 +0.57340 +0.99920

373 231 985 657 563

-0.73015 +0.18046 +0.92515 +0.81927 -0.03985

296 010 898 096 100

405 406 407 408 409

+0.26234 -0.67025 -0.98662 -0.39589 +0.55881

577 155 268 747 405

-0.96497 -0.74213 +0.16302 +0.91829 +0.82929

394 399 052 472 299

455 456 457 458 459

+0.50633 -0.45205 -0.99482 -0.62296 +0.32165

965 268 985 505 095

-0.86233 -0.89199 -0.10155 +0.78224 +0.94685

414 124 572 967 832

410 411 412 413 414

+0.99975 +0.52152 -0.43619 -0.99287 -0.63671

451 528 188 624 476

-0.02215 -0.85323 -0.89985 -0Ii1915 +0.77109

689 583 368 021 942

460 461 462 463 464

+0.97054 +0.72712 -0.18481 -0.92682 -0.81672

255 181 137 982 521

+0.24092 -0.68650 -0.98277 -0.37548 +0.57702

979 847 401 166 680

415 416 417 418 419

+0.30483 +0.96612 +0.73916 -0.16738 -0.92003

933 555 039 542 785

+0.95240 +0.25807 -0.67352 -0.98589 -0.39182

379 251 944 154 950

465 466 467 468 469

+0.04427 +0.86456 +0.88998 +0.09715 -0.78499

279 660 186 190 906

+0.99901 +0.50251 -0.45599 -0.99526 -0.61949

948 826 593 957 695

420 421 422 423 424

-0.82681 +0.02658 +0.85553 +0.89791 +0.11475

172 129 559 441 487

+0.56247 +0.99964 +0.51774 -0.44017 -0.99339

878 666 401 009 384

470 471 472 473 474

-0.94542 -0.23663 +0.68971 +0.98194 +0.37137

551 211 977 647 611

+0.32583 +0.97159 +0.72407 -0.18915 -0.92848

830 932 641 902 252

425 426 427 428 429

-0.77390 -0.95104 -0.25379 +0.67679 +0.98514

977 534 421 415 108

-0.63329 +0.30905 +0.96725 +0.73617 -0.17174

587 140 824 232 704

475 476 477 478 479

-0.58063 -0.99881 -0.49868 +0.45993 +0.99568

573 376 703 026 978

-0.81416 +0.04869 +0.86678 +0.88795 +0.09274

347 372 212 504 619

430 431 432 433 434

+0.38775 -0.56613 -0.99951 -0.51395 +0.44413

385 249 922 260 968

-0.92176 -0.82431 +0.03100 +0.85781 +0.89595

296 427 516 859 756

480 481 482 483 484

+0.61601 -0.33001 -0.97263 -0.72101 +0.19350

671 928 707 682 297

-0.78773 -0.94397 -0.23232 +0:69291 +0.98109

308 419 978 756 969

435 436 437 438 439

+0.99389 +0.62986 -0.31325 -0.96837 -0.73316

198 458 741 198 982

+0.11035 -0.77670 -0.94966 -0.24951 +0.68004

728 497 826 093 560

485 486 487 488 489

+0.93dll +0.81158 -0.05311 -0.86898 -0.88591

702 578 369 067 083

+0.36726 -0.58423 -0.99858 -0.49484 +0.46385

329 328 847 603 557

440 441 442 443 444

+0.17610 +0.92347 +0.82180 -0.03542 -0.86008

529 001 066 843 478

+0.98437 +0.38367 -0.56977 -0.99937 -0.51015

134 061 511 222 112

490 491 492 493 494

-0.08833 +0.79045 +0.94250 +0.22802 -0.69610

866 167 438 291 177

+0.99609 +0.61252 -0.33419 -0.97365 -0.71794

050 441 379 577 312

445 446 447 448 449

-0.89398 -0.10595 +0.77948 +0.94827 +0.24522

316 754 495 257 276

+0.44810 +0.99437 +0.62642 -0.31745 -0.96946

056 066 095 729 676

495 496 497 498 499

-0.98023 -0.36314 +0.58781 +0.99834 +0.49099

370 328 939 363 533

+0.19784 +0.93173 +0.80899 -0.05753 -0.87116

312 331 219 262 220

450

-0.68328 373

-0.73015 296

500

-0.46777 181

-0.88384 927

ELEMENTARY CIRCULAR x

SINES

TRANSCENDENTAL

AND COSINES

FOR LARGE

co9 x

sin z

RADIAN x

181

FUNCTIONS ARGUMENTS

Table 4.8

sin x

co9 x

500 501 502 503 504

-0.46777 -0.99647 -0.60902 +0.33836 +0.97465

181 170 011 176 539

-0.88384 -0.08392 +0.79315 +0.94101 +0.22371

927 940 478 611 157

550 551 552 553 554

-0.21948 -0.93954 -0.79578 +0.07960 +0.88181

408 038 759 864 305

-0.97561 -0.34243 +0.60557 +0.99682 +0.47159

608 814 585 620 913

505 506 507 508 509

+0.71485 -0.20217 -0.93333 -0.80638 +0.06195

535 940 135 275 042

-0.69927 -0.97934 -0.35901 +0.59139 +0.99807

236 850 615 399 923

555 556 557 ::98

+0.87328 +0.06186 -0.80643 -0.93329 -0.20209

261 016 623 888 084

-0.48721 -0.99808 -0.59132 +0.35910 +0.97936

400 483 107 055 678

510 511 512 513 514

+0.87332 +0.88177 +0.07951 -0.79584 -0.93950

667 040 849 235 941

+0.48713 -0.47167 -0.99683 -0.60550 +0.34252

502 887 339 389 310

560 561 562 563 564

+0.71491 +0.97463 +0.33827 -0.60909 -0.99646

859 516 666 184 411

+0.69920 -0.22379 -0.94104 -0.79309 +0.08401

771 971 671 970 951

515 516 517 518 519

-0.21939 +0.70242 +0.97844 +0.35488 -0.59495

585 924 413 199 701

+0.97563 +0.71175 -0.20651 -0.93491 -0.80375

593 358 172 110 753

565 566 567 568 569

-0.46769 +0.49107 +0.99834 +0.58774 -0.36322

187 411 883 623 754

+0.88389 +0.87111 +0.05744 -0.80904 -0.93170

157 780 234 534 046

520 521 522 523 524

-0.99779 -0.48326 +0.47557 +0.99717 +0.60197

528 517 670 555 580

+0.06636 +0.87547 +0.87967 +0.07510 -0.79851

701 403 426 603 433

570 571 572 573 574

-0.98025 -0.69603 +0.22811 +0.94253 +0.79039

158 684 096 460 628

-0.19775 +0.71800 +0.97363 +0.33410 -0.61259

448 607 514 856 589

525 526 527 528 529

-0.34667 -0.97659 -0.70863 +0.21084 +0.93647

773 735 787 000 255

-0.93798 -0.21507 +0.70557 +0.97752 +0.35074

430 583 237 059 088

575 576 577 578 579

-0.08842 -0.88595 -0.86893 -0.05302 +0.81163

874 278 592 338 861

-0.99608 -0.46377 +0.49492 +0.99859 +0.58415

251 546 461 327 989

530 531 532 533 534

+0.80111 -0.07078 -0.87760 -0.87756 -0.07069

655 230 424 088 210

-0.59850 -0.99749 -0.47938 +0.47946 +0.99749

837 179 586 522 818

580 581 582 583 584

+0.93008 +0.19341 -0.72107 -0.97261 -0.32993

380 424 948 606 391

-0.36734 -0.98111 -0.69285 +0.23241 +0.94400

740 719 235 774 403

535 536 537 538 539

+0.80117 +0.93644 +0.21075 -0.70870 -0.97657

068 083 160 168 790

+0.59843 -0.35082 -0.97753 -0.70550 +0.21516

592 557 965 828 415

585 586 587 588 589

+0.61608 +0.99568 +0.45984 -0.49876 -0.99881

795 139 996 541 816

+0.78767 -0.09283 -0.88799 -0.86673 -0.04860

737 623 663 702 339

540 541 542 543 544

-0.34659 +0.60204 +0.99716 +0.47549 -0.48334

290 801 876 715 434

+0.93801 +0.79845 -0.07519 -0.87971 -0.87543

565 989 621 726 032

590 591 592 593 594

-0.58056 +0.37146 +0.98196 +0.68965 -0.23671

210 008 357 428 997

+0.81421 +0.92844 +0.18907 -0.72413 -0.97157

597 893 022 878 792

545 546 547 548 549

-0.99780 -0.59488 +0.35496 +0.97846 +0.70236

128 432 654 280 487

-0.06627 +0.80381 +0.93487 +0.20642 -0.71181

678 133 901 324 710

595 596 597 598 599

-0.94545 -0.78494 +0.09724 +0.89002 +0.86452

497 304 191 309 115

-0.32575 +0.61956 +0.99526 +0.45591 -0.50259

281 794 078 545 644

550

-0.21948 408

-0.97561 608

600

+0.04418 245

-0.99902 348

182

ELEMENTARY

Table x

4.8

CIRCULAR sin x

SINES

TRANSCENDENTAL

AND COSINES cos x

FUNCTIONS

FOR LARGE X

RADIAN

ARGUMENTS cos x

sin x

600 601 602 603 604

+0.04418 -0.81677 -0.92679 -0.18472 +0.72718

245 739 586 249 389

-0.99902 -0.57695 +0.37556 +0.98279 +0.68644

348 294 547 072 271

650 651 652 653 654

+0.30475 -0.63678 -0.99286 -0.43611 +0.52160

320 449 546 050 244

-0.95243 -0;77104 +0.11923 +0:89989 +0.85318

136 183 999 312 866

605 606 607 608 609

+0.97052 iO:32156 -0.62303 -0i99482 -0.45197

075 532 579 067 201

-0.24101 -0.94688 -0.78219 +0.10164 +0.89203

756 740 333 568 212

655 656 657 658 659

+0.99975 +0.55873 -0.39598 -0.98663 -0.67018

651 905 051 742 443

+0.02206 -0.82934 -0.91825 -0.16293 +0.74219

648 352 891 130 460

610 611 612 613 614

+0.50641 +0.99920 +0.57333 -0.37966 -0.98359

763 923 248 351 862

+0.86228 +0.03976 -0.81932 -0.92512 -0.18037

834 064 281 465 115

660 661 662 663 664

+0.26243 +0.95377 +0.76821 -0.12363 -0.90181

303 077 607 295 440

+0.96495 +0.30053 -0.64019 -0.99232 -0.43212

021 504 066 802 358

615 616 617 618 619

-0.68321 +0.24531 +0.94830 +0.77942 -0.10604

769 043 128 830 746

+0.73021 +0.96944 +0.31737 -0.62649 -0.99436

475 458 153 144 107

665 666 667 668 669

-0.85087 -0.01764 +0.83180 +0.91649 +0.15856

185 165 821 743 314

+0.52537 +0:99984 +0.55506 -0.40004 -0.98734

329 437 315 057 884

620 621 622 623 624

-0.89402 -0.86003 -0.03533 +0.82185 +0.92343

368 865 805 218 531

-0.44801 +0.51022 +0.99937 +0.56970 -0.38375

972 890 542 079 412

670 671 672 673 674

-0.74515 -0.96377 -0.29631 +0.64358 +0.99177

337 931 100 428 114

-0.66689 +0.26670 +0.95509 +0.76537 -0.12802

314 104 151 525 348

625 626 627 628 629

+0.17601 -0.73323 -0.96834 -0.31317 +0.62993

627 132 941 153 482

-0.98438 -0.67997 +0.24959 +0.94969 +0.77664

726 929 850 658 801

675 676 677 678 679

+0.42812 -0.52913 -0.99991 -0.55137 +0.40409

819 384 266 639 279

-0.90371 -0.84853 -0.01321 +0.83425 +0.91471

802 838 646 660 800

630 631 632 633 634

+0.99388 +0.44405 -0.51403 -0.99952 -0.56605

200 865 017 202 794

-0.11044 -0.89599 -0.85777 -0.03091 +0.82436

716 772 210 477 546

680 681 682 683 684

+0.98804 +0.66358 -0.27096 -0.95639 -0.76251

092 878 382 354 945

+0.15419 -0.74809 -0.96258 -0.29208 +0.64696

188 754 953 115 529

635 636 637 638 639

+0.38783 +0.98515 +0.67672 -0.25388 -0.95107

721 661 757 168 328

+0.92172 +0.17165 -0.73623 -0.96723 -0.30896

789 795 352 528 539

685 686 687 688 689

+0.13241 +0.90560 +0.84618 +0.00879 -0.83668

151 393 828 102 866

+0.99119 +0.42412 -0.53288 -0.99996 -0.54767

483 441 404 136 882

640 641 642 643 644

-0.77385 +0.11484 +0.89795 +0.85548 +0.02649

250 470 421 876 089

+0.63336 +0.99338 +0:44008 -0.51782 -0.99964

586 346 889 138 905

690 691 692 693 694

-0.91292 -0.14981 +0.75102 +0.96138 +0.28784

065 760 706 090 558

+0.40813 +0.98871 +0.66027 -0.27522 -0.95767

710 365 143 130 684

645 646 647 648 649

-0.82686 -0.92000 -0.16729 +0.73922 +0.96610

259 241 626 130 221

-0.56240 +0.39191 +0.98590 +0.67346 -0.25815

400 270 667 260 988

695 696 697 698 699

-0.65033 -0I99059 -0.42011 +0.53662 +0.99999

364 911 233 379 047

-0.75964 +0.13679 +0.90747 +0.84382 +0.00436

871 694 211 161 541

650

+0.30475 320

-0.95243 136

700

+0.54397 052

-0.83910 433

ELEMENTARY CIRCULAR x

SINES

TRANSCENDENTAL

AND COSINES

sin x

FOR LARGE

cos x

FUNCTIONS

RADIAN X

183

ARGUMENTS

Table

sin z

cos x

4.8

700 701 702 703 704

+0.54397 -0.41217 -0.98936 -0.65694 +0.27947

052 342 702 115 339

-0.83910 -0.91110 -0.14544 +0.75394 +0.96015

433 541 039 186 344

750 751 752 753 754

+0.74507 -0.15868 -0.91654 -0.83174 +0.01776

295 219 566 127 220

-0.66698 -0.98732 -0.39993 +0.55516 +0.99984

298 971 006 345 224

705 706 707 708 709

+0.95894 +0.75676 -0.14117 -0.90932 -0.84143

137 309 969 251 841

+0.28360 -0.65368 -0.98998 -0.41609 +0.54035

437 925 399 202 304

755 756 757 758 759

+0.85093 +0.90176 +0.12351 -0.76829 -0.95373

519 229 330 325 453

+0.52527 -0.43223 -0.99234 -0.64009 +0.30065

069 231 292 802 004

710 711 712 713 714

+O.OOOOb029 +0.84150 356 +0.90927 234 +0.14106 032 -0.75684 190

+1.00000 +0.54025 -0.41620 -0.99000 -0.65359

000 157 166 100 799

760 761 762 763 764

-0.26231 +0:67027 +0.98661 +0.39586 -0.55883

668 392 776 979 905

+0.96498 +0.74211 -0.16305 -0.91830 -0.82927

184 379 026 665 614

715 716 717 718 719

-0.95890 -0.27935 +0.65703 +0.98934 +0.41206

717 761 205 947 355

+0.28372 +0.96018 +0.75386 -0.14555 -0.91115

000 713 264 968 511

765 766 767 768 769

-0.99975 -0.52149 +0.43621 +0.99287 +0.63669

384 956 901 983 152

+0.02218 +0.85325 +0.89984 +0.11912 -0.77111

703 155 053 028 861

720 721 722 723 724

-0.54407 -0.99998 -0.53652 +0.42022 +0.99061

170 994 204 174 560

-0.83903 +0.00448 +0.84388 +0.90742 +0.13667

873 599 631 145 750

770 771 772 773 774

-0.30486 -0.96613 -0.73914 +0.16741 +0.92004

804 333 009 514 966

-0.95239 -0.25804 +0.67355 +0.98588 +0.39180

460 339 173 649 176

725 726 727 728 729

+0.65024 -0.28796 -0.96141 -0.75094 +0.14993

204 105 408 744 682

-0.75972 -0.95764 -0.27510 +0.66036 +0.98869

712 212 538 198 558

775 776 777 778 779

+0.82679 -0.02661 -0.85555 -0.89790 -Oil1472

477 142 119 114 492

-0.56250 -0.99964 -0.51771 +0.44019 +0.99339

370 585 822 716 730

730 731 732 733 734

+0.91296 +0.83662 -0.00891 -0.84625 -0.90555

985 262 160 253 279

+0.40802 -0.54777 -0.99996 -0.53278 +0.42423

702 970 029 200 360

780 781 782 783 784

+0.77392 +0.95103 +0.25376 -0.67681 -0.98513

886 602 505 634 591

+0.63327 -0.30908 -0.96726 -0.73615 +0.17177

255 007 589 192 673

735 736 737 738 739

-0.13229 +0.76259 +0.95635 +0.27084 -0.66367

199 745 831 775 898

+0.99121 +0:64687 -0.29219 -0.96262 -0.74801

079 335 647 220 752

785 786 787 788 789

-0.38772 +0.56615 +0.99951 +0.51392 -0.44416

606 733 829 674 668

+0.92177 +0.82429 -0.03103 -0.85783 -0.89594

465 720 529 408 417

740 741 742 743 744

-0.98802 -0.40398 +0.55147 +0.99991 +0.52903

232 250 697 106 153

+0.15431 +0.91476 +0.83419 -0.01333 -0.84860

102 672 011 703 217

790 791 792 793 794

-0.99389 -0.62984 +0.31328 +0.96837 +0.73314

531 117 604 950 932

-0.11032 +0.77672 +0.94965 +0.24948 -0.68006

732 396 881 174 770

745 746 747 748 749

-0.42823 -0.99178 -0.64349 +0.29642 +0.96381

715 657 199 616 146

-0.90366 -0.12790 +0.76545 +0.95505 +0.26658

639 390 285 577 483

795 796 797 798 799

-0.17613 -0.92348 -0.82178 +0.03545 +0.86010

497 158 349 855 016

-0.98436 -0.38364 +0.56979 +0.99937 +0.51012

603 277 988 115 519

750

+0.74507 295

-0.66698 298

800

+0.89396 965

-0.44812 751

184

ELEMENTARY Table X

4.8

CIRCULAR sin 5

SINES

TRANSCENDENTAL AND COSINES

FOR LARGE X

cos 5

FUNCTIONS RADIAN sin 2

ARGUMENTS co5 x

800 801 802 803 804

+0.89396 +0.10592 -0.77950 -0.94826 -0.24519

965 756 384 300 354

-0.44812 -0.99437 -0.62639 +0.31748 +0.96947

751 385 745 587 415

850 851 852 853 854

+0.98022 +0.36311 -0.58784 -0.99834 -0.49096

773 519 378 189 907

-0.19787 -0.93174 -0.80897 +0.05756 +0.87117

267 426 447 271 700

805 806 807 808 809

+0.68330 .tO.98357 +0.37955 -0.57343 -0.99920

573 687 196 126 443

+0.73013 -0.18048 -0.92517 -0.81925 +0.03988

237 975 042 368 112

855 856 857 858 859

+0.46779 +0.99647 +0.60899 -0.33839 -0.97466

845 423 620 013 214

+0.88383 +0.08389 -0.79317 -0.94100 -0.22368

517 936 314 591 219

810 811 812 813 814

-0.50631 +0.45207 +0.99483 +0.62294 -0.32167

365 956 291 147 949

+0.86234 +0.89197 +0.10152 -0.78226 -0.94684

940 762 573 845 862

860 861 862 863 864

-0.71483 +0.20220 +0.93334 +0.80636 -0.06198

427 893 217 493 051

+0.69929 +0.97934 +0.35898 -0.59141 -0.99807

390 241 802 830 736

815 816 817 818 819

-0.97054 -0.72710 +0.18484 +0.92684 +0.81670

981 111 099 114 782

-0.24090 +0.68653 +0.98276 +0.37545 -0.57705

054 039 844 372 142

865 866 867 868 869

-0.87334 -0.88175 -0.07948 +0.79586 +0.93949

135 618 845 060 908

-0.48710 $0.47170 +0.99683 +0.60547 -0.34255

870 545 579 989 142

820 821 822 823 824

-0.04430 -0.86458 -0.88996 -0.09712 +0.78501

291 174 811 190 774

-0.99901 -0.50249 +0.45602 +0:99527 +0.61947

814 220 276 249 329

870 871 872 873 874

+0.21936 -0.70245 -0.97843 -0.35485 +0.59498

644 070 790 381 124

-0.97564 -0.71173 +0.20654 +0.93492 +0.80373

254 241 122 180 959

825 826 827 828 829

+0.94541 +0.23660 -0.68974 -0.98194 -0.37134

569 282 159 076 812

-0.32586 -0.97160 -0.72405 +0.18918 +0.92849

680 646 561 862 371

875 876 877 878 879

+0.99779 +0.48323 -0.47560 -0.99717 -0.60195

328 878 322 782 173

-0.06639 -0.87548 -0.87965 -0.07507 +0.79853

709 859 992 597 248

830 831 832 833 834

+0.58066 +0.99881 +0.49866 -0.45995 -0.99569

027 229 090 702 258

+0.81414 -0.04872 -0.86679 -0.88794 -0.09271

596 383 716 118 618

880 881 882 883 884

+0.34670 +0.97660 +0.70861 -0.21086 -0.93648

601 383 660 947 312

+0.93797 +0.21504 -0.70559 -0.97751 -0.35071

385 639 373 423 265

835 836 837 838 839

-0.61599 +0.33004 +0.97264 +0.72099 -0.19353

297 774 407 594 254

+0.78775 +0.94396 +0.23230 -0.69293 -0.98109

165 424 046 929 386

885 886 887 888 889

-0.80109 +0.07081 +0.87761 +0.87754 +0.07066

851 237 869 643 203

+0.59853 +0.99748 +0.47935 -0.47949 -0.99750

252 965 940 167 031

840 841 842 843 844

-0.93012 -0.81156 +0.05314 +0.86899 +0.88589

809 816 379 559 685

-0.36723 +0.58425 +0.99858 +0.49481 -0.46388

525 775 687 983 228

890 891 892 893 894

-0.80118 -0.93643 -0.21072 +0.70872 +0.97657

871 025 213 294 141

-0.59841 +0.35085 +0.97754 +0.70548 -0.21519

177 380 600 692 358

845 846 847 848 849

+0.08830 -0.79047 -0.94249 -0.22799 +0.69612

863 014 431 356 342

-0.99609 -0.61250 +0.33422 +0.97366 +0.71792

316 058 221 264 213

895 896 897 898 899

+0.34656 -0.60207 -0.99716 -0.47547 +0.48337

463 208 649 063 073

-0.93802 -0.79844 +0.07522 +0.87973 +0.87541

610 174 627 159 575

850

+0.98022 773

-0.19787 267

900

+0.99780 327

+0.06624 670

ELEMENTARY CIRCULAR

SINES sin 5

TRANSCENDENTAL

AND COSINES

185

FUNCTIONS

FOR LARGE RADIAN 5 cos 2

ARGUMENTS sin x

Table 4.8 cos x

900 901 902 903 904

+0.99780 +0.59486 -0.35499 -0.97846 -0.70234

327 009 472 902 341

+0.06624 -0.80382 -0.93486 -0.20639 +0.71183

670 926 831 374 827

950 951 952 953 954

+0.94546 +0.78492 -0.09727 -0.89003 -0.86450

479 436 191 684 600

+0.32572 -0.61959 -0.99525 -0.45588 +0.50262

431 160 784 862 250

905 906 907 908 909

+0.21951 io.93955 +0.79576 -0.07963 -0.88182

349 070 933 869 727

+0.97560 +0.34240 -0.60559 -0.99682 -0.47157

947 981 984 380 255

955 956 957 958 959

-0.04415 +0.83679 +0.92678 +0.18469 -0.72720

233 478 454 287 458

+0.99902 +0.57692 -0.37559 -0.98279 -0.68642

481 832 341 629 079

910 911 912 913 914

-0.87326 -0.06183 +0.80645 +0.93328 +0.20206

792 008 406 805 131

+0.48724 +0.99808 +0.59129 -0.35912 -0.97937

032 669 676 869 287

960 961 962 963 964

-0.97051 -0.32153 +a.62305 +0.99481 +0.45194

349 677 937 760 512

+0.24104 +0.94689 +0.78217 -0.10167 -0.89204

682 709 455 567 574

915 916 917 918 919

-0.71493 -0.97462 -0.33824 +0.60911 +0.99646

966 841 829 575 158

-0.69918 +0.22382 +0.94105 +0.79308 -0.08404

616 909 690 134 955

965 966 967 968 969

-0.50644 -0.99921 -0.57330 +0.37969 +0.98360

362 043 778 140 406

-0.86227 -0.03973 +0.81934 +0.92511 +0.18034

308 052 009 320 150

920 921 922 923 924

+0.46766 -0.49110 -0.99835 -0.58772 +0.36325

523 037 056 184 562

-0.88390 -0.87110 -0.05741 +0.80906 +0.93168

567 299 224 306 952

970 971 972 973 974

+0.68319 -0.24533 -0.94831 -0.77940 +0.10607

568 966 084 942 744

-0.73023 -0.96943 -0.31734 +0.62651 +0.99435

535 718 294 493 787

925 926 927 928 929

+0.98025 +0.69601 -0.22814 -0.94254 -0.79037

754 520 031 467 781

+0.19772 -0.71802 -0.97362 -0.33408 +0.61261

493 705 827 015 972

975 976 977 978 979

+0.89403 +0.86002 +0.03530 -0.82186 -0.92342

718 327 793 936 374

+0.44799 -0.51025 -0.99937 -0.56967 +0.38378

277 482 648 601 195

930 931 932 933 934

+0.08845 +0.88596 +0.86892 +0.05299 -0.81165

877 676 100 328 622

+0.99607 +0.46374 -0.49495 -0.99859 -0.58413

984 875 080 487 542

980 981 982 983 984

-0.17598 +0.73325 +0.96834 +0.31314 -0.62995

660 181 189 290 823

+0.98439 +0.67995 -0.24962 -0.94970 -0.77662

256 719 769 602 902

935 936 937 938 939

-0.93007 -0.19338 +0.72110 +0.97260 +0.32990

273 467 037 905 546

+0.36737 +0.98112 +0.69283 -0.23244 -0.94401

544 302 061 706 398

985 986 987 988 989

-0.99387 -0.44403 +0.51405 +0.99952 +0.56603

867 164 603 296 309

+0.11047 +0.89601 +0.85775 +0.03088 -0.82438

712 111 661 464 252

940 941 942 943 944

-0.61611 -0.99567 -0.45982 +0.49879 +0.99881

169 859 319 154 962

-0.78765 +0.09286 +0.88801 +0.86672 +0.04857

880 625 049 199 328

990 991 992 993 994

-0.38786 -0.98516 -0.67670 +0.25391 +0.95108

499 179 538 083 260

-0.92171 -0.17162 +0.73625 +0.96722 +0.30893

620 825 392 763 672

945 946 947 948 949

+0.58053 -0.37148 -0.98196 -0.68963 +0.23674

755 806 927 246 926

-0.81423 -0.92843 -0.18904 +0.72415 +0.97157

347 773 062 957 078

995 996 997 998 999

+0.77383 -0.11487 -0.89796 -0.85547 -0.02646

341 465 748 315 075

-0.63338 -0.99338 -0.44006 +0.51784 +0.99964

919 000 182 716 985

+0.94546 479 +0.32572 431 For x>lOOOsee Example 16.

1000

+0.82687 954

+0.56237 908

186

ELEMENTARY Table

TRANSCENDENTAL

FUNCTIONS

4.9

CIRCULAR

TANGENTS,

COTANGENTS,

SECANTS

AND

COSECANTS

FOR

RADIAN

ARGUMENTS

67 35 39 79

x-l-cot 0.00000 0.00333 0.00666 0.01000 0.01333

x 000 335 684 060 476

0.00166 0.00333 0.00500 0.00666

668 349 053 791

20.00833 16.67667 14.29738 12.51334 11.12612

58 09 76 32 53

0.01666 0.02000 0.02334 0.02667 0.03001

944 480 096 805 621

0.00833 0.01000 O.Oll.67 0.01334 0.01501

576 420 334 330 419

09 07 35 99 07

10.01668 9.10926 8.35336 7.71401 7.16624

61 83 70 72 39

0.03335 0.03669 0.04003 0.04338 0.04672

558 628 845 223 776

0.01668 0.01835 0.02003 0.02170 0.02338

614 925 365 946 680

1.01135 1.01293 1.01462 1.01642 1.01832

64 80 61 16 55

6.69173 6.27674 5.91078 5.58566 5.29495

24 65 21 93 84

0.05007 0.05342 0.05677 0.06013 0.06348

516 458 615 000 628

0.02506 0.02674 0.02842 0.03011 0.03180

578 653 915 379 054

49 81 35 77 78

1.02033 1.02246 1.02469 1.02704 1.02950

88 26 78 58 78

5.03348 95 4.79708 57 4.58232 93 4.38639 73 4.20693.71

0.06684 0.07020 0.07357 0.07693 0.08030

512 667 105 841 889

0.03348 0.03518 0.03687 0.03857 0.04027

955 092 477 124 044

3.91631 3.75909 3.61326 3.47760 3.35106

74 41 32 37 28

1.03208 1.03477 1.03759 1.04052 1.04357

50 89 10 27 57

4.04197 3.88983 3.74908 3.61852 3.49708

25 14 94 56 77

0.08368 0.08705 0.09044 0.09382 0.09721

264 978 046 483 302

0.04197 0.04367 0.04538 0.04709 0.04881

250 754 569 707 181

625 751 941 487 688

3.23272 3.12180 3.01759 2.91949 2.82696

81 50 80 61 00

1.04675 1.05005 1.05347 1.05703 1.06072

16 22 94 51 13

3.38386 3.27805 3.17897 3.08600 2.99861

34 83 74 99 68

0.10060 0.10400 0.10740 0.11080 0.11421

519 147 202 697 648

0.05053 0.05225 0.05397 0.05570 0.05744

003 186 744 689 034

876 0:38 0.39

0.36502 0.37640 0.38786 0.39941 0.41105

849 285 316 272 492

2.73951 2.65672 2.57822 2.50367 2.43276

22 80 89 59 50

1.06454 1.06849 1.07258 1.07681 1.08118

02 38 47 50 74

2.91632 2.83869 2.76536 2.69599 2.63027

08 75 87 57 48

0.11763 0.12104 0.12447 0.12790 0.13133

070 976 383 306 759

0.05917 0.06091 0.06266 0.06441 0.06617

792 976 601 678 222

0.40 0.41 0.42 0.43 0.44

0.42279 0.43463 0.44657 0.45862 0.47078

322 120 255 102 053

2.36522 2.30080 2.23927 2.18044 2.12413

24 12 78 95 20

1.08570 1.09036 1.09518 1.10015 1.10527

44 89 36 15 57

2.56793 2.50872 2.45242 2.39882 2.34775

25 20 03 48 15

0.13477 0.13822 0.14167 0.14513 0.14859

758 318 456 185 524

0.06793 0.06969 0.07146 0.07324 0.07502

246 763 789 336 418

0.45 0.46 0.47 0.48 0.49

0.48305 0.49544 0.50796 0.52061 0.53338

507 877 590 084 815

2.07015 2.01837 1.96863 1.92082 1.87480

74 22 61 05 73

1.11055 1.11600 1.12161 1.12740 1.13335

94 60 91 22 91

2.29903 2.25251 2.20805 2.16553 2.12483

27 55 98 72 00

0.15206 0.15554 0.15902 0.16251 0.16600

486 089 348 280 901

0.07681 0.07860 0.08040 0.08220 0.08401

051 247 022 390 366

0.50

0.54630 249 c-y

0.01 0.02 0.03 0.04

tanx 0.00000 0000 0.01000 0333 0.02000 2667 0.03000 9003 0.04002 1347

0.05 0.06 0.07 0.08 0.09

0.05004 0.06007 0.07011 0.08017 0.09024

0.10 0.11 0.12 0.13 0.14

x 0.00

cot x

set x

csc x

99.990066666 49.99333 32 33.32333 27 24.98666 52

1.00000 1.00005 1.00020 1.00045 1.00080

00 00 00 02 05

100.0:166 50.00333 33.33833 25.00666

1708 2104 4558 1105 3790

19.98333 16.64666 14.26237 12.47332 11.08109

06 19 33 19 49

1.00125 1.00180 1.00245 1.00320 1.00406

13 27 50 86 37

0.10033 0.11044 0.12057 0.13073 0.14092

467 582 934 732 189

9.96664 9.05421 8.29329 7.64892 7.09612

44 28 49 55 94

1.00502 1.00608 1.00724 1.00850 1.00988

0.15 0.16 0.17 0.18 0.19

0.15113 0.16137 0.17165 0.18196 0.19231

522 946 682 953 984

6.61659 6.19657 5.82557 5.49542 5.19967

15 54 68 56 16

0.20 0.21 0.22 0.23 0.24

0.20271 0.21314 0.22361 0.23414 0.24471

004 244 942 336 670

4.93315 4.69169 4.47188 4.27088 4.08635

0.25 0.26 0.27 0.28 0.29

0.25534 0.26602 0.27675 0.28755 0.29841

192 154 814 433 279

0.30 0.31 0.32 0.33 0.34

0.30933 0.32032 0.33138 0.34252 0.35373

0.35

[ 1

1.83048 77

1.13949 39 C-j)2

II 1

2.08582 96

csc x-x-l 0.00000

0.16951 228

000

0.08582 964 c-p9 [hyperbolictangentsand 1 [.C-l)8cotan-1

Compilation of tansand cots from National Bureau of Standards, Table ofcircularand gents for radian arguments, 2d printing. Columbia Univ. Press, New York, N.Y., 1947 (with permission).

ELEMENTARY CIRCULAR

x

TANGENTS,

TRANSCENDENTAL COTANGENTS, FOR

0.50 0.51 0.52 0.53 0.54

tanx 0.54630 0.55935 0.57256 0.58591 0.59942

0.55 0.56 0.57 0.58 0.59

187

FUNCTIONS SECANTS

RADIAN

AND

COSECANTS

Table

4.9

ARGUMENTS

249 872 183 701 962

cot x 1.83048 1.78776 1.74653 1.70672 1.66825

772 154 626 634 255

set x 1.13949 1.14581 1.15231 1.15900 1.16589

39 07 38 77 70

csc x 2.08582 2.04843 2.01255 1.97810 1.94501

96 63 78 89 07

0.61310 0.62694 0.64096 0.65516 0.66955

521 954 855 845 565

1.63104 1.59502 1.56013 1.52632 1.49352

142 471 894 503 784

1.17298 1.18028 1.18778 1.19551 1.20345

68 21 81 06 53

1.91319 1.88257 1.85311 1.82473 1.79739

00 90 45 78 41

0.60 0.61 0.62 0.63 0.64

0.68413 0.69891 0.71390 0.72911 0.74454

681 886 901 473 382

1.46169 1.43078 1.40073 1.37152 1.34310

595 125 873 626 429

1.21162 1.22003 1.22868 1.23758 1.24673

83 59 47 16 39

1.77103 1.74560 1.72106 1.69737 1.67449

22 45 62 57 37

0.65 0.66 0.67 0.68 0.69

0.76020 0.77610 0.79225 0.80866 0.82533

440 491 417 138 611

1.31543 1.28848 1.26222 1.23661 1.21162

569 559 118 155 759

1.25614 1.26583 1.27580 1.28605 1.29660

92 52 04 34 31

1.65238 1.63101 1.61034 1.59034 1.57100

34 05 23 84 01

0.70 0.71 0.72 0.73 0.74

0.84228 0.85952 0.87706 0.89491 0.91308

838 867 790 753 953

1.18724 1.16342 1.14016 1.11742 1.09518

183 833 258 140 285

1.30745 1.31863 1.33013 1.34196 1.35415

93 17 09 77 38

1.55227 1.53413 1.51656 1.49954 1.48304

03 35 54 35 60

0.75 0.76 0.77 0.78 0.79

0.93159 0.95045 0.96966 0.98926 1.00924

646 146 833 154 629

1.07342 1.05213 1.03128 1.01085 0.99083

615 158 046 503 842

1.36670 1.37962 1.39293 1.40664 1.42076

11 24 10 08 67

1.46705 1.45154 1.43650 1.42190 1.40775

27 43 25 99 03

0.80 0.81 0.82 0.83 0.84

1.02963 1.05045 1.07171 1.09343 1.11563

857 514 372 292 235

0.97121 0.95196 0.93308 0.91455 0.89635

460 830 500 085 264

1.43532 1.45032 1.46580 1.48175 1.49821

42 96 02 42 08

1.39400 1.38066 1.36771 1.35513 1.34292

78 78 62 96 52

0.85 0.86 0.87 0.88 0.89

1.13833 1116155 1.18532 1.20966 1.23459

271 586 486 412 946

0.87847 0.86091 0.84365 0.82667 0.80997

778 426 058 575 930

1.51519 1.53271 1.55080 1.56948 1.58878

02 39 46 63 44

1.33106 1.31953 1.30833 1.29745 1.28688

09 53 72 63 25

0.90 0.91 0.92 0.93 0.94

1.26015 1.28636 1.31326 1.34087 1.36923

822 938 370 383 448

0.79355 0.77738 0.76146 0.74578 0.73033

115 169 169 232 510

1.60872 1.62933 1.65065 1.67270 1.69552

58 92 49 52 44

1.27660 1.26661 1.25691 1.24747 1.23830

62 84 05 40 10

0.95 0.96 0.97 0.98 0.99

1.39838 1.42835 1.45920 1.49095 1.52367

259 749 113 827 674

0.71511 0.70010 0.68530 0.67070 0.65630

188 485 649 959 719

1.71914 1.74361 1.76897 1.79525 1.82252

92 84 37 95 32

1.22938 1.22071 1.21228 1.20409 1.19613

40 57 91 77 51

1.00

1.55740 772

0.64209 262

[ Y21

1.85081 57

[(-$11

1.18839 51

1 1 c-412 5

ELEMENTARY Table

CIRCULAR

4.9

TRANSCENDENTAL

TANGENTS, FOR

FUNCTIONS

COTANGENTS, SECANTS RADIAN ARGUMENTS

AND

COSECANTS

262 942 141 260 722

set x 1.85081 1.88019 1.91070 1.94243 1.97542

57 15 a9 08 47

1.18839 1.18087 1.17356 1.16645 1.15954

51 20 01 42 90

0.57361 0.56040 0.54733 0.53441 0.52162

970 467 693 147 342

2.00976 _ 2.04552 2.08279 2.12166 2.16223

32 49 43 31 06

1.15283 1.14632 1.13999 1.13384 1.12787

98 17 02 11 01

97 82 53 01 51

0.50896 0.49644 0.48403 0.47175 0.45958

811 096 759 371 520

2.20460 2.24890 2.29524 2.34378 2.39466

44 16 97 77 75

1.12207 1.11644 1.11098 1.10569 1.10055

33 69 71 05 37

2.23449 2.29579 2.35998 2.42726 2.49789

69 85 11 64 94

0.44752 0.43557 0.42373 0.41198 0.40033

802 829 221 610 638

2.44805 2.50413 2.56310 2.62518 2.69063

57 48 57 99 21

1.09557 1.09074 1.08607 1.08154 1.07715

35 67 04 17 79

1.20 1.21 1.22 1.23 1.24

2.57215 2.65032 2.73275 2.81981 2.91192

16 46 42 57 99

0.38877 0.37731 0.36593 0.35463 0.34341

957 227 119 310 486

2.75970 2.83270 2.90997 2.99188 3.07885

36 55 35 25 30

1.07291 1.06881 1.06485 1.06102 1.05732

64 46 01 06 39

1.25 1.26 1.27 1.28 1.29

3.00956 3.11326 3.22363 3.34135 3.46720

97 91 32 00 57

0.33227 0.32120 0.31020 0.29928 0.28841

342 577 899 023 670

3.17135 3.26993 3.37517 3.48778 3.60853

77 04 57 15 36

1.05375 1.05032 1.04700 1.04382 1.04076

79 05 98 41 14

1.30 1.31 1.32 1.33 1.34

3.60210 3.74708 3.90334 4.07230 4.25561

24 10 78 98 79

0.27761 0.26687 0.25619 0.24556 0.23498

565 440 034 088 350

3.73833 3.87822 4.02940 4.19329 4.37153

41 33 74 31 10

1.03782 1.03499 1.03229 1.02970 1.02723

00 a5 53 88 77

1.35 1.36 1.37 1.38 1.39

4.45522 4.67344 4.91305 5.17743 5.47068

18 12 a1 74 86

0.22445 0.21397 0.20353 0.19314 0.18279

572 509 922 574 234

4.56607 4.77923 5.01379 5.27312 5.56133

06 14 49 60 39

1.02488 1.02263 1.02050 1.01848 1.01656

07 65 39 18 93

1.40 i.41 1.42 1.43 1.44

5.79788 6.16535 6.58111 7.05546 7.60182

37 61 95 38 61

0.17247 0.16219 0.15194 0.14173 0.13154

673 663 983 413 734

5.88349 6.24592 6.65666 7.12597 7.66731

01 80 08 85 76

1.01476 1.01306 1.01147 1.00999 1.00861

51 85 85 43 52

1.45 1.46 1.47 1.48 1.49

8.23809 8.98860 9.88737 10.98337 12.34985

28 76 49 93 64

0.12138 0.11125 0.10113 0.09104 0.08097

732 194 908 6660 2601

8.29856 9.04406 9.93781 11.02880 12.39027

45 25 58 87 66

1.00734 1.00616 1.00510 1.00413 1.00327

05 95 15 62 29

1.50 1.51 1.52 1.53 1.54

14.10141 16.42809 19.66952 24.49841 32.46113

99 17 78 04 89

0.07091 0.06087 0.05084 0.04081 0.03080

4844 1343 0061 8975 6066

14.13683 16.45849 19.69493 24.51881 32.47653

29 92 14 14 83

1.00251 1.00185 1.00129 1.00083 1.00047

13 09 15 27 44

1.55 1.56 1.57 1.58 1.59

48.07848 92.62049 +1255.76559 - 108.64920 - 52.06696

25 63 15 36 96

0.02079 0.01079 + 0.00079 - 0.00920 - 0.01920

9325 6746 6327 3933 6034

48.08888 92.62589 +1255.76598 - 108.65380 - 52.07657

10 45 97 55 18

1.00021 1.00005 1.00000 1.00004 1.00018

63 83 03 24 44

1.60

-

tan

2

cot

.I

1.00

1.01 1.02 1103 1.04

1.55740 1.59220 1.62813 1.66524 1.70361

77 60 04 40 46

0.64209 0.62805 0.61420 0.60051 0.58698

1.05 1.06 1.07 1.08 1.09

1.74331 1.78442 1.82702 1.87121 1.91709

53 48 82 73 18

1.10 1.11 1.12 1.13 1.14

1.96475 2.01433 2.06595 2.11975 2.17587

1.15 1.16 1.17 1.18 1.19

34.23253 27

For 01.6, use 4.3.44.

- 0.02921 1978 (-55’2 [ 3

-

csc .x

34.24713 56

1.00042 66 t--p3

[

1

ELEMENTARY CIRCULAR

SINES

AND

TRANSCENDENTAL COSINES

TO

TENTHS

A DEGREE

Table

4.10

sin 0 00000 53283 06514 59638 12602

00000 65898 15224 31420 97962

1.00000 0.99999 0.99999 0.99998 0.99997

cos e 00000 84769 39076 62922 56307

00000 13288 57790 47427 05395

90.0° 89.9 89.8 89. 7 89.6

0.00872 0.01047 0.01221 0.01396 0.01570

65354 17841 70008 21803 73173

98374 16246 35247 39145 11821

0.99996 0.99994 0.99992 0.99990 0.99987

19230 51693 53696 25240 66324

64171 65512 60452 09304 81661

89.5 89.4 89.3 89.2 89.1

0.01745 0.01919 0.02094 0.02268 0.02443

24064 74423 24198 73335 21781

37284 99690 83357 72781 52653

0.99984 0.99981 0.99978 0.99974 0.99970

76951 57121 06834 26093 14897

56391 21644 74845 22698 81183

E 88:8 88.7 88.6

0.02617 0.02792 0.02966 0.03141 0.03315

69483 16387 62440 07590 51783

07873 23569 85111 78128 88526

0.99965 0.99961 0.99955 0.99950 C.99945

73249 01150 98601 65603 02159

75557 40354 19384 65732 41757

88.5 88.4 88.3 88.2 88.1

0.03489 0.03664 0.03838 0.04013 0.04187

94967 37087 78090 17925 56537

02501 06556 87520 32560 29200

0.99939 0.99932 0.99926 0.99919 0.99912

08270 83937 29164 43951 28300

19096 78656 10621 14446 98858

88.0 87.9 87.8 87. 7 87.6

0.04361 0.04536 0.04710 0.04884 0.05059

93873 29881 64507 97697 29400

65336 29254 09643 95613 76713

0.99904 0.99897 0.99888 0.99880 0.99871

82215 05697 98749 61373 93571

81858 90715 61970 41434 84186

87.5 87.4 87.3 87.2 87.1

0.05233 0.05407 0.05582 0.05756 0.05930

59562 88129 15049 40269 63735

42944 84775 93164 59567 75962

0.99862 0.99853 0.99844 0.99834 0.99823

95347 66703 07641 18166 98279

54574 26212 81981 14028 23765

87.0 86.9 86.8 86. 7 86.6

0.06104 0.06279 0.06453 0.06627 0.06801

85395 05195 23082 39004 52906

34857 29313 52958 00000 65248

0.99813 0.99802 0.99791 0.99780 0.99768

47984 67284 56182 14682 42788

21867 28272 72179 92050 35605

86.5 86.4 86.3 86.2 86.1

0.06975 0.07149 0.07323 0.07497 0.07671

64737 74443 81971 87268 90281

44125 32686 27632 26328 26819

0.99756 0.99744 0.99731 0.99718 0.99705

40502 07829 44772 51335 27522

59824 30944 24458 25116 26920

86.0 85.9 85.8 85.7 85.6

0.07845 0.08019 0.08193 0.08367 0.08541

90957 89243 85086 78433 69231

27845 28859 30041 32315 37367

0.99691 0.99677 0.99663 0.99649 0.99634

73337 88784 73868 28592 52961

33128 56247 18037 49504 90906

85.5 85.4 85.3 85.2 85.1

0.99619 46980 91746 sin e

85.0 e

For conversionfrom radiansto degreesseeExample

page II.

OF

0.00000 0.00174 0.00349 0.00523 0.00698

0.08715 57427 47658 co.36

‘See

189

FUNCTIONS

14.

9o”-e

190

ELEMENTARY Table

4.10

CIRCULAR

SINES

.4ND

COSINES

FUNCTIONS TO

TENT.HS

sin 0

e 0 ::1” z-23 5: 4 5.5 5. 6 55'87 5:9 6.0 2: 6:3 6.4 6. 5 66.76 6:8 6.9 ::: ;*: 714 77.56 717 ::i 8. 0 88:; 2: $56 8:l 29"

T:Z x 9:9 10. 0 90°-e

page Il.

OF

A DEGREE

cos e

90°-e

0.08115 0.08889 0.09063 0.09237 0.09410

57427 42968 25801 05874 83133

47658 66442 97780 46562 18514

0.99619 0.99604 0.99588 0.99572 0.99556

46980 91746 10654 10770 43986 15970 46981 84582 19646 03080

85.0' 84.9 84.8 84.7 84.6

0.09584 0.09758 0.09931 0.10105 0.10279

57525 28997 97497 62971 25367

20224 59149 43639 82946 87247

0.99539 0.99522 0.99505 0.99488 0.99470

b1983

73999 81831 55699 61226 07088 28788 28171 17174

84.5 84.4 84.3 84.2 84.1

67653 36233 06023 91045

0.11146 89322 06325

0.99452 18953 68273 0.99433 79441 33205 0.99415 09639 72315 0.99396 09554 55180 0.99376 79191 60596

84.0 83.9 83.8 83.7 83. b

0.11320 0.11493 0.11667 0.11840 0.12013

32137 71504 07370 39683

67907 92867 99333 06501

68388

34647

0.99357 18556 76587 0.99337 27656 00396 0.99317 06495 38486 0.99296 55081 06537 0.99275 73419 29446

83.5 83.4 83.3 83.2 83.1

0.12186 0.12360 0.12533 0.12706 0.12879

93434 14767 32335 46086 55965

05147 40493 64304 01350 77563

0.99254 0.99233 0.99211 0.99189 0.99167

61516 19378 47013 44425 11623

41322 85489 14478 90030 83090

83.0 82.9 82.8 82.7 82.6

0.13052 0.13225 0.13398 0.13571 0.13744

61922 63902 61854 55724 45460

20052 57122 18292 34304 37147

0.99144 0.99121 0.99098 0.99074

48613 55402 31997 78404

73810 51542 14836 71444

Oi99050

94632

38309

82.5 82.4 82.3 82.2 82.1

0.13917 0.14090 0.14262 0.14435 0.14608

31009 12319 89337 62010 30285

bOO65 37583 05512 00973 62412

0.99026 0.99002 0.98977 0.98952 0.98927

80687 36577 62309 57890 23329

41570 lb558 07789

0.14780 0.14953 0.15126 0.15298 0.15471

94111 53434 08202 58362 03862

29611 43710 47219 84038 99468

0.98901 0.98875 0.98849 0.98822 0.98795

58633 61917 63810 47006 38868 08684 83814 46553

0.15643 0.15815

44650

40231

0.98768 0.98741 0.98713 0.98685

0.10452 84632 0.10626 0.10799 0.10973

40713 93557 43110

80672 0.15988 11876 0.16160 38211 0.16332 59622

54484 91835 03361 41622

0.16504 0.16676 0.16848 0.17020 0.17192

60678 16102 65003 66033 79410

76058 87467 93795 94991 91002

0.17364 81776 66930 cos e *

*see

TRANSCENDENTAL

[t--8)75 1

0.98657

67179

68969

62988

98657

69389

83405 38067 62650 57164 21616

95138 50911 72988 06807 06969

82.0 81.9 81.8 81.7 81.6 81.5 81.4 81.3 81.2 81.1 81.0 80.9 80.8 80.7 80.6

0.98628

56015 37231 0.98599 60370 70505 0.98570 34690 88854 0.98540 78984 83490 0.98510 93261 54774

80.5 80.4 80.3 80.2 80.1

C.98480 77530 12208 sin 8 t-l)4

80.0 a

c 1

191

ELEMENTARY TRANSCENDENTAL FUNCTIONS CIRCULAR

SINES

AND COSINES

TO TENTHS

OF A DEGREE

sin 0

e

Table

4.10

cos e

900-e

lO.OO 10.1 10.2 10. 3 10. 4

0.17364 81776 66930 0.17536 67260 91987 0.17708 47403 19583 0.17880 22151 16350 i1;18051 91452 50560

0.98480 0.98450 0.98419 0.98388 0.98357

77530 12208 31799 74437 56079 69242 50379 33542 14708 13386

80.0° 79.9 79.8 79.7 79.6

10.5 10.6 10.7 10. 8 10.9

0.18223 0.18395 0.18566 0.18738 0.18909

55254 92147 13506 12720 66153 85577 13145 85725 54429 89891

0.98325 0.98293 0.98261 0.98228 0.98195

49075 63955 53491 49554 27965 43615 72507 28689 87126 96444

79.5 79.4 79.3 79.2 79Il

11.0 11.1 11.2 11.3 11.4

0.19080 0.19252 0.19423 0.19594 0.19765

89953 76545 19665 25907 43512 19972 61442 42518 73403 79126

0.98162 0.98129 0.98095 0.98061 0.98027

71834 47664 26639 92245 51553 49192 46585 46613 11746 21722

79.0 78.9 78.8 78.7 78.6

11. 5 11. 6 11. 7 11.8 11.9

0.19936 0.20107 0.20278 0.20449 0.20620

79344 17197 79211 45965 72953 56512 60518 41790 41853 96630

0.97992 0.97957 0.97922 0.97886 0.97850

47046 20830 52495 99344 28106 21766 73887 61685 89851 01778

78.5 78.4 78.3 78.2 78.1

12. 0 12.1 12.2 12.3 12.4

0.20791 0.20961 0.21132 0.21303 0.21473

16908 17759 85629 03822 47964 55389 03862 74977 53271 67063

0.97814 0.97778 0.97741 0.97704 0.97667

76007 33806 32367 58606 58942 86096 55744 35264 22783 34168

78.0 77.9 77.8 77.7 77.6

12.5 12.6 12.7 12.8 12.9

0.21643 0.21814 0.21984 0.22154 0.22325

96139 38103 32413 96543 62043 52838 84976 19467 01160 10951

0.97629 0.97591 0.97553 0.97514 0.97476

60071 19933 67619 38747 45439 45857 93543 05563 11941 91222

77.5 77.4 77.3 77.2 77.1

13. 0 13.1 13.2 13.3 13.4

0.22495 0.22665 0.22835 0.23004 0.23174

10543 43865 13074 36855 08701 10656 97371 88104 79034 94157

0.97437 0.97397 0.97357 0.97317 0.97277

00647 85235 59672 79052 89028 73160 88727 77088 58782 09397

77.0 76.9 76.8 76.7 76.6

13. 5 13. 6 13.7 13.8 13.9

0.23344 0.23514 0.23683 0.23853 0.24022

53638 55905 21131 02590 81460 65619 34515 78581 80424 71264

0.97236 0.97196 0.97154 0.97113 0.97071

99203 97677 10005 78546 91199 97646 42799 09636 64815 78191

76.5 76.4 76.3 76.2 76.1

14. 0 14.1 14.2 14. 3 14.4

0.24192 0.24361 0.24530 0.24699 0.24868

18955 99668 50117 86023 73858 78803 90127 22743 98871 64855

0.97029 0.96987 0.96944 0.96901 0.96858

57262 75996 20152 84747 53498 95139 57314 06870 31611 28631

76.0 75.9 75.8 75.7 75.6

14. 5 14. 6 14.7 14.8 14.9

0.25038 0.25206 0.25375 0.25544 0.25713

00040 54441 93582 43114 79445 84806 57579 35791 27931 54696

0.96814 0.96770 0.96726 0.96682 0.96637

76403 78108 91704 81971 77527 75877 33886 04459 60793 21329

75.5 75.4 7513 75.2 75.1

15.0 90°-e

0.25881 90451 02521 cose * (--7)l 5

*see

c 1

page Il.

0.96592 58262 89068 sin e c-y4

II 1

15.0 e

ELEMENTARY

Table 4.10

CIRCULAR

TRANSCENDENTAL

SINES

AND COSINES

e

sin 0

15.0° 15.1 15.2 15. 3 15.4

0.25881 90451 02521 0.2605ti 45086 42648 0.26218 91786 40865 Oi26387 30499 65373 0.26555 61174 86809

15.5 15. 6 15.7 15.8 15.9

0.26723 0:2689i 0.27060 0:27228 0.27395

16.0 16.1 16.2 16.3 16.4

0.21563 0.27731 0.27899 0.28066 0.28234

16.5 16.6 16.1 16.8 16.9

FUNCTIONS

TO TENTHS

OF A DEGREE

cose

90°-e

0.96592 0.96547 0.96501 0.96455 0.96409

58262 89068 26308 79225 64944 72311 14184 5719i 54042 34110

74.9 74.8 74.7 14.6

83760 78257 98206 15266 04459 75864 02470 40574 92186 9'2432

0.96363 0.96316 0.96269 0.96221 0.96174

04532 08623 25667 97658 17464 26479 79935 29285 13095 49211

74.5 74.4 74.3 74.2 74.1

73558

16999 46533 02378 11060 39229 67089 20788 14568 42816

0.96126 0.96011 0.96029 0.95980 0.95931

16959 38319 91541 57594 36856 16943 52919 75187 39745 40058

74.0 73.9 73.8 73.7 73.6

0.28401 0.28568 0.28736 0.28903 0.29070

53447 03923 83674 04974 05198 49712 17969 44472 21935 98252

0.95881 0.95832 0.95782 0.95731 0.95681

97348 68193 25744 65133 24948 45315 94975 32067 35840 57607

73.5 73.4 13.3 73.2 73.1

17.0 17.1 11.2 17.3 17.4

0.29237 0.29404 0.29570 0.29737 0.29904

17041

03252 32304 80500 44047 48740 77786 07922 56087

0.95630 0.95579 0.95527 0.95476 0.95424

47559 63035 30147 98330 83621 22344 07995 02797 03285 16277

73.0 72.9 72.8 72.7 72.6

17.5 17.6 17.7 17.8 17.9

0.30070 0.30236 0.30403 0.30569 0.30735

57995 04273 98907 50445 30609 25490 53049 63106 66177 99807

0.95371 0.95319 0.95266 0.95212 0.95159

69507 48227 06677 92947 14812 53586 93927 42139 44038 79438

72.5 72.4 12.3 72.2 72.1

18.0 18.1 18.2 18.3 18.4

0.30901 0.31067 0.31233 0.31399 0.31564

69943 74947 64296 30732 49185 12233 24559 67405 90369 47102

0.95105 0.95051 0.94997 0.94942 0.94887

65162 95154 57316 27784 20515 24653 54776 41904 60116 44491

72.0 71.9 71.8 71.7 71.6

18. 5 18.6 18.1 18.8 18.9

0.31730 0.31895 0.32061 0.32226 0.32391

46564 05092 93092 98070 29905 85676 56952 30511 74181 98149

0.94832 0.94776 0.94721 0.94664 0.94608

36552 06199 84100 09586 02777 46029 92601 15696 53588 27545

71.5 71.4 71.3 71.2 71.1

19.0 19.1 19.2 19.3 19.4

0.32556 0.32721 0.32886 0.33051 0.33216

81544 57157 78989 79104 66467 38583 43927 13223 11318 83703

0.94551 0.94494 0.94437 0.94380 0.94322

85755 99317 89121 57531 63702 37481 09515 83229 26579 47601

71.0 70.9 70.8 70.7 70.6

19. 5 19.6 19.7 19.8 19.9

0.33380 0.33545 0.33709 0.33813 0.34037

68592 33771 15697 50255 52584 23082 79202 45291 95502 13050

0.94264 0.94205 0.94147 0.94088 0.94028

14910 92178 74521 87297 05448 12038 07689 54225 81270 10419

70.5 70.4 70.3 70.2 70.1

0.93969 26207 85908

70.0 e

20.0 90"-e

22737

0.34202 01433 25669 cos e *

[51 (-7)l

sin 0 c-:,4

c 1

15. o”

193

ELEMENTARY TRANSCENDENTAL FUNCTIONS CIRCULAR

SINES

AND COSINES

TO TENTHS

OF A DEGREE

sin t9

0

Table

4.10

co5e

90°-e

20.0° 20.1 20.2 20.3 20.4

0.34202 0.34365 0.34529 0.34693 0.34857

01433 25669 96945 85616 81989 98535 56515 73256 20473 21815

0.93969 0.93909 0.93849 0.93788 0.93728

26207 85908 42520 94709 30227 59556 89346 11898 19894 91892

70.0° 69.9 69.8 69.7 69.6

20.5 20.6 20.7 20.8 20.9

0.35020 0.35184 0.35347 0.35510 0.35673

73812 59467 16484 04702 48437 79257 69624 08137 79993 19625

0.93667 0.93605 0.93544 0.93482 0.93420

21892 48398 95357 38973 40308 29867 56763 96014 44743 21030

69.5 69.4 69.3 69.2 69.1

21.0 21.1 21.2 21.3 21.4

0.35836 0.35999 0.36162 0.36325 0.36487

79495 45300 68081 20051 45700 82092 12304 72978 67843 37620

0.93358 0.93295 0.93232 0.93169 0.93105

04264 97202 35348 25489 38012 15512 12275 85549 58158 62528

69.0 68.9 68.8 68.7 68.6

21.5 21.6 21.7 21.8 21.9

0.36650 0.36812 0.36974 0.37136 0.37298

12267 24297 45526 84678 67572 73829 78355 50235 77825 75809

0.93041 0.92977 0.92913 0.92848 0.92783

75679 82025 64858 88251 25715 34056 58268 80914 62538 98920

68.5 68.4 68.3 68.2 68.1

22.0 22.1 22.2 22.3 22.4

0.37460 0.37622 0.37784 0.37945 0.38107

65934 15912 42631 39366 07868 18467 61595 29005 03763 50274

0.92718 0.92652 0.92587 0.92520 0.92454

38545 66787 86308 71837 05848 09995 97183 85782 60336 12313

68.0 67.9 67.8 67.7 67.6

22.5 22.6 22.7 22.8 22.9

0.38268 0.38429 0.38590 0.38751 0.38912

34323 65090 53226 59804 60423 24319 55864 52103 39501 40206

0.92387 0.92321 0.92253 0.92186 0.92118

95325 11287 02171 12981 80894 56246 31515 88501 54055 65721

67.5 67.4 67. 3 67.2 67.1

23.0 23.1 23.2 23.3 23.4

0.39073 0.39233 0.39394 0.39554 0.39714

11284 89274 71166 03561 19095 90951 55025 62965 78906 34781

0.92050 0.91982 0.91913 0.91844 0.91775

48534 52440 14973 21738 53392 55234 63813 43087 46256 83981

67.0 66.9 66.8 66.7 66.6

23.5 23.6 23.7 23.8 23.9

0.39874 90689 25246 0.40034 90325 56895 0.40194 77766 55960 0.40354 52963 52390 Oi40514 15867 79863

0.91706 0.91636 0.91566 0.91495 0.91425

00743 85124 27295 62240 25933 39561 96678 49825 39552 34264

66.5 66.4 66.3 66.2 66.1

24.0 24.1 24.2 24.3 24.4

0.40673 66430 75800 0.40833 04603 81385 Oi40992 30338 41573 0.41151 43586 05109 0.41310 44298 24542

0.91354 0.91283 0.91212 0.91140 0.91068

54576 42601 41772 33043 01161 72273 32766 35445 36608 06177

66.0 65.9 65.8 65.7 65.6

24.5 24.6 24.7 24.8 24.9

0.41469 0.41628 0.41786 0.41945 0;42103

0.90996 0.90923 0.90850 0.90777 0.90704

12708 76543 61090 47069 81775 26722 74785 32909 40142 91465

65.5 65.4

0.90630 77870 36650

65.0 e

25.0 90°-e

0.42261 82617 40699 cos0 *

‘See page n.

32426 56239 07922 60401 70738 01077 20824 46177 58133 67491

5 1 [(-712

sin 0

cC-i)4 1

E-23 . 65.1

ELEMENTARY TRANSCENDENTAL FUNCTIONS Table

4.10

CIRCULAR

e

SINES

AND

COSINES

TO

TENTHS

sin 0

OF

A

DEGREE

cos e

9o”-e

25.0' 25.1 25.2 25.3 25.4

0.42261 82617 40699 0.42419 94227 45390 Oi42577 92915 65073 0.42735 78633 87192 0.42893 51334 03146

0.90630 0.90556 0.90482 i;90408 0.90333

77870 36650 87990 11140 70524 66020 25496 60778 52928 63301

65.0' 64.9 64.8 64.7 64.6

25.5 25.6 25.7 25.8 25.9

0.43051 0.43208 0.43365 0.43523 0.43680

10968 08295 57488 01982 90845 87544 10993 72328 17883 67702

0.90258 0.90183 0.90107 0.90031 0.89955

52843 49861 25264 05114 70213 22092 87714 02194 77789 55180

64.5 64.4 64.3 64.2 64.1

26.0 26.1 26.2 26.3 26.4

0.43837 0.43993 0.44150 0.44307 0.44463

11467 89077 91698 55915 58527 91745 11908 24180 51791 84927

0.89879 0.89802 0.89725 0.89648 0.89571

40462 99167 75757 60616 83696 74328 64303 83441 17602 39413

64.0 63.9 63.8 63.7 63.6

26.5 26.6 26.7 26.8 26.9

0.44619 0.44775 0.44931 0.45087 0.45243

78131 09809 90878 38770 89986 15897 75406 89431 47093 11783

0.89493 0.89415 0.89337 0.89258 0.89179

43616 02025 42368 39368 13883 27838 58184 52125 75296 05214

63.5 63.4 63.3 63.2 63.1

27.0 27.1 27.2 21.3 21.4

0.45399 0.45554 0.45709 0.45864 0.46019

04997 39547 49072 33516 79270 58694 95544 84315 97847 83852

0.89100 0.89021 0.88941 0.88861 0.88781

65241 88368 28046 11127 63732 91298 72326 54949 53851 36401

63.0 62.9 62.8 62.7 62.6

27.5 27.6 21.7 27.8 27.9

0.46174 0.46329 0.46484 0.46638 0.46792

86132 35034 60351 19862 20457 24620 66403 39891 98142 60573

0.88701 0.88620 0.88539 0.88458 0.88376

08331 78222 35792 31215 36257 54416 09752 15084 56300 88693

62.5 62.4 62.3 62.2 62.1

28.0 28.1 28.2 28.3 28.4

0.46947 0.47101 0.47255 0.47408 0.47562

15627 85891 18812 19410 07648 69054 82090 47116 42090 70275

0.88294 0.88212 0.88130 0.88047 0.87964

75928 58927 68660 17668 34520 64992 13535 09162 85728 66617

62.0 61.9 61.8 61.7 61.6

28.5 28.6 28.7 28.8 28.9

0.47715 0.47869 0.48022 0.48175 0.48328

87602 59608 18579 40607 34914 43189 36741 01715 23832 55002

0.87881 0.87798 0.87714 0.87630 0.87546

71126 61965 29754 27981 61637 05589 66800 43864 45270 00018

61.5 61.4 61. 3 61.2 61.1

29.0 29.1 29.2 29.3 29.4

0.48480 0.48633 0.48785 0.48938 0.49090

96202 46337 53804 23490 96591 38733 24517 48846 37536 15141

0.87461 0.87377 0.87292 0.87206 0.87121

97071 39396 22230 35465 20772 69810 92724 32121 38111 20189

61.0 60.9 60.8 60.7 60.6

29.5 29.6 29.7 29.8 29.9

0.49242 0.49394 0.49545 0.49697 0.49848

35601 03467 18665 84231 86684 32408 39610 27555 17397 53830

0.87035 56959 39900 0.86949 49295 05219 0.86863 15144 38191 0.86776 54533 68928 Oi86689 67489 35603

60.5 60.4 60.3 60.2 60.1

0.50000 00000 00000 cose * (-7)2 [ 5 I

0.86602 54037 84439

60.0 e

30.0 9o"-s

*See page n.

sin

e

5 1 [C-7)4

ELEMENTARY

CIRCULAR

SINES

ZRANSCENDENTAL

AND COSINES

195

FUNCTIONS

TO TENTHS

OF A DEGREE

sin 0

cos

0

Table

4.10 9o”-s

30.0° 30.1 30.2 30.3 30.4

0.50000 0.50151 0.50301 0.50452 0.50603

00000 00000 07371 59457 99466 30235 76238 15019 37641 21164

0.86602 54037 84439 0.86515 14205 69704 Oi86427 48019 53705 0.86339 55506 06772 0.86251 36692 07257

60.0' 59.9 59.8 59.7 59.6

30.5 30.6 30.7 30.8 30.9

0.50753 0.50904 0.51054 0.51204 0.51354

83629 60704 14157 50371 29179 11606 28648 70572 12520 58170

0.86162 0.86074 0.85985 0.85895 0.85806

91604 41526 20270 03944 22715 96873 98969 30664 49057 23645

59.5 59.4 59.3 59.2 59.1

31. 0 31.1 31.2 31.3 31.4

0.51503 0.51653 0.51802 0.51951 0.52100

80749 10054 33288 66642 70093 73130 91118 79509 96318 40576

0.85716 0.85626 0.85536 0.85445 0.85355

73007 02112 70846 00328 42601 60507 88301 32807 07972 75327

59.0 58.9 58.8 58.7 58.6

31.5 31.6 31.7 31.8 31.9

0.52249 0.52398 0.52547 0.52695 0.52843

85647 15949 59059 70079 16510 72268 57954 96678 83347 22347

0.85264 0.85172 0.85081 0.84989 0.84897

01643 54092 69341 43048 11094 24051 26929 86864 16876 29141

58.5 58.4 58.3 58.2 58.1

32.0 32.1 32.2 32.3 32.4

0.52991 0.53139 0.53287 0.53435 0.53582

92642 33205 85795 18083 62760 70730 23493 89826 67949 78997

0.84804 0.84712 0.84619 0.84526 0.84432

80961 56426 19213 82137 31661 27564 18332 21856 79255 02015

58.0 57.9 57.8 51.7 57.6

32.5 32.6 32.7 32.8 32.9

0.53729 0.53877 0.54024 0.54170 0.54317

96083 46824 07850 06863 03204 77655 82102 82740 44499 50671

0.84339 14458 12886 0.84245 23970 07148 0.84151 07819 45306 0.84056 66034 95684 Oi83961 98645 34413

57.5 57.4 57.3 57.2 57.1

33.0 33.1 33.2 33. 3 33.4

0.54463 0.54610 0.54756 0.54902 0.55048

90350 15027 19610 14429 32234 92550 28179 98132 07400 84996

0.83867 0.83771 0.83676 0.83580 0.83484

05679 45424 87166 20439 43134 58962 73613 68270 78632 63407

57.0 56.9 56.8 56.7 56.6

33.5 33.6 33.7 33.8 33.9

0.55193 0.55339 0.55484 0.55629 0.55774

69853 12058 15492 43344 44274 47999 56155 00305 51089 79690

0.83388 58220 67168 0.83292 12407 10099 0.83195 41221 30483 0.83098 44692 74328 0:SSOOl 22850 95367

56.5 56.4 56.3 56.2 56.1

34.0 34.1 34.2 34.3 34.4

0.55919 0.56063 0.56208 0.56352 0.56496

29034 70747 89945 63242 33778 52131 60489 37571 70034 24938

0.82903 0.82806 0.82708 0.82609 0.82511

75725 55042 03346 22494 05742 74562 82944 95764 34982 78295

56.0 55.9 55.8 55.7 55.6

34.5 34.6 34.7 34.8 34.9

0.56640 0.56784 0.56927 0.57071 0.57214

62369 24833 37450 53101 95234 30844 35676 84432 58734 45516

0.82412 0.82313 Oi82214 0.82114 0.82015

61886 22016 63685 34442 40410 30737 92091 33704 18758 73772

55.5 55.4 55.3 55.2 55.1

35.0 9o"- e

0.57357 64363 51046 cos e

0.81915 20442 88992

55. 0 e

sin 0

196

ELEMENTARY

Table

4.10

CIRCULAR

TRANSCENDENTAL

SINES

AND COSINES

FUNCTIONS

TO TENTHS

sin e

I9

OF A DEGREE

cos e

900-e

35.0° 35.1 35.2 35.3 35.4

0.57357 0.57500 0.57643 0.57785 0.57928

64363 51046 52520 43279 23161 69793 76243 83505 11723 42679

0.81915 0.81814 0.81714 0.81613 0.81512

20442 88992 97174 25023 48983 35129 75900 80160 77957 28554

55.0° 54.9 54.8 54.7 54.6

35.5 35.6 35.7 35.8 35.9

0.58070 29557 10940 0.58212 29701 57289 0.58354 12113 56118 0.58495 76749 87215 Oi58637 23567 35789

0.81411 0.81310 0.81208 0.81106 0.81004

55183 56319 07610 47028 35268 91806 38189 89327 16404 45796

54.5 54.4 54.3 54.2 54.1

36.0 36.1 36.2 36.3 36.4

0.58778 0.58919 0.59060 0.59201 0.59341

52522 92473 63573 53342 56676 19925 31787 99220 88866 03701

0.80901 0.80798 0.80696 0.80592 0.80489

69943 74947 98838 98031 03121 43802 82822 48516 37973 55914

54.0 53.9 53.8 53.7 53.6

36.5 36.6 36.7 36.8 36.9

0.59482 0.59622 0.59762 0.59902 0.60042

27867 51341 48749 65616 51469 75521 35985 15586 02253 25884

0.80385 0.80281 0.80177 0.80073 0.79968

68606 17217 74751 91115 56442 43754 13709 48733 46584 87091

E3.5 53.4 53.3 53.2 53.1

37.0 37.1 37.2 37.3 37.4

0.60181 0.60320 0.60459 0.60598 0.60737

50231 52048 79877 45282 91148 62375 84002 65711 58397 23287

0.79863 0.79758 0.79652 0.79547 0.79441

55100 47293 39288 25229 99180 24196 34808 54896 46205 35418

53.0 52.9 52.8 52.7 52.6

37.5 37.6 37.7 37.8 37.9

0.60876 0.61014 0.61152 0.61290 0.61428

14290 08721 51639 01268 70401 85831 70536 52976 52000 98943

0.79335 0.79228 0.79122 0.79015 0.78908

33402 91235 96433 55191 35329 67490 50123 75690 40848 34691

52.5 52.4

38.0 38.1 38.2 38.3 38.4

0.61566 0.61703 0.61840 0.61977 0.62114

14753 25658 58751 40749 83953 57554 90317 95140 77802 78310

0.78801 0.78693 0.78585 0.78477 0.78369

07536 06722 50219 61337 68931 75402 63705 33083 34573 25840

52.0 51.9 51.8 51.7 51.6

38.5 38.6 38. 7 38.8 38.9

0.62251 0.62387 0.62524 0.62660 0.62796

46366 37620 95967 09386 26563 35705 38113 64461 30576 49338

0.78260 0.78152 0.78043 0.77933 0.77824

81568 52414 04724 18819 04073 38330 79649 31474 31485 26021

51.5 51.4 51.3 51.2 51.1

39.0 39.1 39.2 39.3 39.4

0.62932 03910 49837 Oi63067 58074 31286 0.63202 93026 64851 Oi63338 08726 27550 0.63473 05132 02268

0.77714 0.77604 0.77494 0.77384 0.77273

59614 56971 64070 66546 44887 04180 02097 26506 35734 97351

51. 0 50.9 50.8 50.7 50.6

39.5 39.6 39.7 39.8 39.9

0.63607 0.63742 0.63876 0.64010 0.64144

0.77162 0.77051 0.76939 0.76828 0.76716

45833 87720 32427 75789 95550 46895 35235 93523 51518 15300

50.5 50.4 50.3 50.2 50.1

0.76604 44431 18978

50.0 e

40.0 go"- e

82202 77764 39897 48690 78175 15598 96994 84955 96315 69158

0.64278 76096 86539 cose *

[(-7)25 1

sin 9

[IC-57131

z 52:l

\

ELEMENTARY TRANSCENDENTAL FUNCTIONS CIRCULAR

SINES

AND

e

COSINES

TO

TENTHS

OF

A DEGREE

sin 0

cos e

197 Table

4.10 90”-e

40.0° 40.1 40.2 4.0.3 40.4

0.64278 0.64412 0.64545 0.64678 0.64811

76096 86539 36297 61387 76877 23951 97795 10460 99010 63131

0.76604 0.76492 0.76379 0.76266 0.76153

44431 18978 14009 18432 60286 34642 83296 95688 83075 36737

50.0° 49.9 49.8 49.7 49.6

40.5 40.6 40.7 40.8 40.9

0.64944 0.65077 0.65209 0.65342 0.65474

80483 30184 42172 65851 84038 30392 06039 90105 08137 17340

0.76040 0.75927 0.75813 0.75699 0.75585

59656 00031 13073 34881 43361 97652 50556 51756 34691 67640

49.5 49.4 49. 3 49.2 49.1

41.0 41.1 41.2 41.3 41.4

0.65605 0.65737 0.65868 0.66000 0.66131

90289 90507 52457 94096 94601 18680 16679 60937 18653 23652

0.75470 0.75356 0.75241 0.75126 0.75011

95802 22772 33923 01638 49088 95724 41335 03511 10696 30460

49.0 48.9 48.8 48.7 48.6

41.5 41.6 41.7 41.8 41.9

0.66262 00482 15737 0.66392 62126 52242 0.66523 03546 54361 0.66653 24702 49452 Oi66783 25554 71047

0.74895 0.74779 0.74663 0.74547 0.74431

57207 89002 80904 98532 81822 85391 59996 82862 15462 31154

48.5 48.4 48.3 48.2 48.1

42.0 42.1 42.2 42.3 42.4

0.66913 0.67042 0.67172 0.67301 0.67430

06063 58858 66189 58799 05893 22990 25135 09773 23875 83723

0.74314 0.74197 0.74080 0.73963 0.73845

48254 77394 58409 75616 45962 86750 10949 78610 53406 25884

48.0 47.9 47.8 47.7 47.6

42.5 42.6 42.7 42.8 42.9

0.67559 0.67687 0.67815 0.67944 0.68072

02076 15660 59696 82661 96698 68071 13042 61517 08689 58918

0.73727 0.73609 0.73491 0.73372 0.73254

73368 10124 70871 19734 45951 49960 98645 02876 28987 87379

47.5 47.4 47.3 47.2 47.1

43.0 43.1 43.2 43.3 43.4

0.68199 0.68327 0.68454 0.68581 0.68708

83600 62499 37736 80799 71059 28689 83529 27376 75108 04423

0.73135 0.73016 0.72896 0.72777 0.72657

37016 19170 22766 20752 86274 21412 27576 57210 46709 70976

47.0 46.9 46.8 46.7 46.6

43.5 43.6 43.7 43.8 43.9

0.68835 0.68961 0.69088 0.69214 0.69340

45756 93754 95437 35670 24110 76858 31738 70407 18282 75813

0.72537 0.72417 0.72296 0.72176 0.72055

43710 12288 18614 37468 71459 09568 02280 98362 11116 80330

46.5 46.4 46.3 46.2 46.1

44.0 44.1 44.2 44.3 44.4

0.69465 0.69591 0.69716 0.69841 0.69966

83704 58997 27965 92314 51028 54565 52854 31006 33405 13365

0.71933 0.71812 0.71691 0.71569 0.71447

98003 38651 62977 63189 06076 50483 27337 03736 26796 32803

46.0 45.9 45.8 45.7 45.6

44.5 44.6 44.7 44.8 44.9

0.70090 0.70215 0.70339 0.70463 0.70587

92642 99851 30529 95162 47028 10504 42099 63595 15706 78681

0.71325 0.71202 0.71079 0.70957 0.70833

04491 54182 60459 90996 94738 72992 07365 36521 98377 24529

45.5 45.4 45.3 45.2 45.1

45. 0 9o"-s

0.70710 67811 86548 cose * (-7)3 5

0.70710 67811 86548

45.0 e

[ 1

‘See

page

Il.

sin 0

[(-yj7)31

198

ELEMENTARY Table

4.11

CIRCULAR

TRANSCENDENTAL

FUNCTIONS

TANGENTS, COTANGENTS, SECANTS AND COSECANTS TO FIVE TENTHS OF A DEGREE

90”-e

0.00000 0.00872 0.01745 0.02618 0.03492

tan 0 cot e 00000 00000 68677 90759 114.58865 07293 09608 50649 28217 57.28996 16307 59424 59215 69187 38.18845 92970 25609 07694 91747 28.63625 32829 15603

set 9 csc9 1.00000 000 1.00003 808 114.5;301 348 1.00015 233 57.29868 850 1.00034 279 38.20155 001 1.00060 954 28.65370 835

0.04366 0.05240 0.06116 0.06992 0.07870

09429 08512 77792 83041 26201 50484 68119 43510 17068,24618

22.90376 19.08113 16.34985 14.30066 12.70620

55484 31198 66877 28211 54760 99672 62567 11928 47361 74704

1.00095 1.00137 1.00186 1.00244 1.00309

269 235 869 190 220

22.92558 19.10732 16.38040 14.33558 12.74549

563 261 824 703 484

87.5 87.0 86.5 86.0 85.5

0.08748 0.09628 0.10510 0.11393 0.12278

86635 25924 90481 97538 42352 65676 56083 01645 45609 02904

11.43005 10.38539 9.51436 8.77688 8.14434

23027 61343 70801 38159 44542 22585 73568 69956 64279 74594

1.00381 1.00462 1.00550 1.00646 1.00750

984 509 828 973 983

11.47371 10.43343 9.56677 8.83367 8.20550

325 052 223 147 905

85.0 84.5 84.0 83.5 83.0

0.13165 0.14054 0.14945 0.15838 0.16734

24975 87396 08347 02391 10013 49128 44403 24536 26090 81419

7.59575 7.11536 6.69115 6.31375 5.97576

41127 25150 97223 84209 62383 17409 15146 75043 43644 33065

1.00862 1.00982 1.01110 1.01246 1.01390

896 757 613 513 510

7.66129 7.18529 6.76546 6.39245 6.05885

758 653 908 322 796

82.5 82.0 81.5 81. 0 80.5

10. 0 10.5 11. 0 11.5 12.0

0.17632 69807 08465 0.18533 90449 31534 0.19438 03091 37718 0.20345 22994 23699 0121255 65616 70022

5.67128 5.39551 5.14455 4.91515 4.70463

18196 17709 71743 19137 40159 70310 70310 71205 01094 78454

1.01542 1.01703 1.01871 1.02048 1.02234

661 027 670 657 059

5.75877 5.48740 5.24084 5.01585 4.80973

049 427 307 174 435

80.0 79.5 79.0 78.5 78.0

12.5 13.0 13.5 14.0 14.5

0.22169 46626 42940 0.23086 81911 25563 0.24007 87590 80116 0.24932 80028 43180 0125861 75843 55890

4.51070 4.33147 4.16529 4.01078 3.86671

85036 62057 58742 84155 97700 90417 09335 35844 30948 98738

1.02427 1.02630 1.02841 1.03061 1.03290

951 411 519 363 031

4.62022 4.44541 4.28365 4.13356 3.99392

632 148 757 550 916

77.5 77.0 76.5 76.0 75.5

15.0 15.5 16. 0 16.5 17.0

0.26794 0.27732 0.28674 0.29621 0.30573

91924 31122 45440 59838 53857 58808 34949 62080 06814 58660

3.73205 3.60588 5.48741 3.37594 3.27085

08075 68877 35087 60874 44438 40408 34225 91246 26184 84141

1.03527 1.03774 1.04029 1.04294 1.04569

618 221 944 891 176

3.86370 3.74197 3.62795 3.52093 3.42030

331 754 528 652 362

75.0 74.5 74.0

17.5 18.0 18.5 19.0 19.5

0.31529 0.32491 0.33459 0.34432 0.35411

87888 78983 96962 32906 53195 02073 76132 89665 85725 30698

3.17159 3.07768 2.98868 2.90421 2.82391

48023 63212 35371 75253 49627 42893 08776 75823 28856 00801

1.04852 1.05146 1.05449 1.05762 1.06084

913 222 231 068 870

3.32550 3.23606 3.15154 3.07155 2.99574

952 798 530 349 431

72.5 72.0 71.5 71. 0 70.5

20.0 20.5 21.0 21.5 22.0

0.36397 0.37388 0.38386 0.39391 0.40402

02342 66202 46794 84804 40350 35416 04756 14942 62258 35157

2.74747 2.67462 2.60508 2.53864 2.47508

74194 54622 14939 26824 90646 93801 78956 64307 68534 16296

1.06417 1.06760 1.07114 1.07478 1.07853

777 936 499 624 474

2.92380 2.85545 2.79042 2.72850 2.66946

440 095 811 383 716

70.0 69.5 69.0 68.5 68.0

2.61312 593 set 0

67.5 e

3-Z 3:5 ::: E t-50 7:o i:'o E 9:5

22.5 90°-0

0.41421 35623 73095 cot e

[c-y1

2.41421 35623 73095 tan 0

1.08W0220

[(-;I11

90.0° 89.5 89.0 88.5 88.0

:33-z .

ELEMENTARY TRANSCENDENTAL FUNCTIONS

199

CIRCULAR

TANGENTS, COTANGENTS, SECANTS AND COSECANTS TO FIVE TENTHS OF A DEGREE tan 8 set H csc e cot e

Table

4.11

e 22.5O 23.0 23.5 24. 0 24.5

0.41421 0.42447 0.43481 0.44522 0.45572

35623 73095 48162 09604 23749 60933 86853 08536 62555 32584

2.41421 2.35585 2.29984 2.24603 2.19429

35623 73095 23658 23753 25472 36257 67739 04216 97311 65038

1.08239 1.08636 1.09044 1.09463 1.09894

220 038 110 628 787

2.61312 2.55930 2.50784 2.45859 2.41142

593 467 285 334 102

9o”-s 67.5' 67.0 66.5 66.0 65.5

25.0 25.5 26.0 26.5 27.0

0.46630 0.47697 0.48773 0.49858 0.50952

76581 54998 55326 98160 25885 65861 16080 53431 54494 94429

2.14450 69205 09558 2109654 35990 88174 2.05030 38415 79296 2.00568 97082 59020 1.96261 05055 05150

1.10337 1.10792 1.11260 1.11740 1.12232

792 854 194 038 624

2.36620 2.32282 2.28117 2.24115 2.20268

158 050 203 845 926

65.0 64.5 64.0 63.5 63.0

27.5 28.0 28.5 29. 0 29.5

0.52056 0.53170 0.54295 0.55430 0.56577

70505 51746 94316 61479 56996 38437 90514 52769 27781 87770

1.92098 1.88072 1.84177 1.80404 1.76749

21269 71166 64653 46332 08860 33458 77552 71424 40162 42891

1.12738 1.13257 1.13789 1.14335 1.14895

195 005 318 407 554

2.16568 2.13005 2.09573 2.06266 2.03077

057 447 853 534 204

62.5 62.0 61.5 61.0 60.5

30.0 30.5 31.0 31.5 32.0

0.57735 0.58904 0.60086 0.61280 0.62486

02691 89626 50164 20551 06190 27560 07881 39932 93519 09327

1.73205 1.69766 1.66427 1.63185 1.60033

08075 68877 31193 26089 94823 50518 16871 28789 45290 41050

1.15470 1.16059 1.16663 1.17282 1.17917

054 210 340 770 840

2.00000 1.97029 1.94160 1.91388 1.88707

000 441 403 086 991

60.0 59.5 59.0 58.5 58.0

32.5 33.0 33.5 34.0 34.5

0.63707 0.64940 0.66188 0.67450 0.68728

02608 07493 75931 97510 55611 95691 85168 42426 09586 01613

1.56968 1.53986 1.51083 1.48256 1.45500

55771 17490 49638 14583 51936 14901 09685 12740 90286 72445

1.18568 1.19236 1.19920 1.20621 1.21340

905 329 494 795 641

1.86115 1.83607 1.81180 1.78829 1.76551

900 846 103 165 728

57.5 57. 0 56.5 56.0 55.5

35.0 35.5 36.0 36.5 37.0

0.70020 0.71329 0.72654 0.73996 0.75355

75382 09710 30678 97005 25280 05361 10750 ,28487 40501 02794

1.42814 1.40194 1.37638 1.35142 1.32704

80067 42114 82944 76336 19204 71173 24379 45808 48216 20410

1.22077 1.22832 1.23606 1.24400 1.25213

459 691 798 257 566

1.74344 1.72205 1.70130 1.68117 1.66164

680 082 162 299 014

55.0 54.5 54.0 53.5 53.0

37.5 38.0 38.5 39.0 39.5

0.76732 0.78128 0.79543 0.80978 0.82433

69879 78960 56265 06717 59166 67828 40331 95007 63858 17495

1.30322 1.27994 1.25717 1.23489 1.21309

53728 41206 16321 93079 22989 18954 71565 35051 70040 92932

1.26047 1.26901 1.27777 1.28675 1.29596

241 822 866 957 700

1.64267 1.62426 1.60638 1.58901 1.57213

963 925 793 573 369

52.5 52.0 51.5 51. 0 50.5

40.0 40.5 41.0 41.5 42.0

0.83909 0.85408 0.86928 0.88472 0.90040

96311 77280 06854 63466 67378 16226 52645 55944 40442 97840

1.19175 1.17084 1.15036 1.13029 1.11061

35925 94210 95661 12539 84072 21009 43863 61753 25148 29193

1.30540 1.31508 1.32501 1.33519 1.34563

729 700 299 242 273

1.55572 1.53976 1.52425 1.50916 1.49447

383 904 309 050 655

50.0 49.5 49.0 48.5 48.0

42.5 43. 0 43.5 44.0 44.5

0.91633 0.93251 0.94896 0.96568 0.98269

11740 17423 50861 37661 45667 14880 87748 07074 72631 15690

1.09130 1.07236 1.05378 1.03553 1.01760

85010 69271 87100 24682 01252 80962 03137 90569 73929 72125

1.35634 1.36732 1.37859 1.39016 1.40203

170 746 847 359 206

1.48018 1.46627 1.45273 1.43955 1.42671

723 919 967 654 819

47.5 47.0 46.5 46. 0 45.5

45.0 90°-e

1.00000 00000 00000 cot tJ

[ 1 C-i)4

1.00000 o~ooy 00000

[ 1 C-f!4

[‘-4’31 L

1.41Q&356

-1

1.414&356

[ 1 c-:)3

45.0 e

ELEMENTARY

200 Table

4.12

CIRCULAR sin

1:

TRANSCENDENTAL FUNCTIONS

FOR

FUNCTIONS THE

ARGUMENT

co9 T

T r, 2

; x

l-x

taniz

0.00 0.01 0.02 0.03 0.04

0.00000 0.01570 0.03141 0.04710 0.06279

00000 73173 07590 64507 05195

00000 11820 78128 09642 29313

00000 1.00000 67575 0.99987 29384 0.99950 66090 0.99888 37607 0.99802

2" 00000 00000 66324 81660 65603 65731 98749 61969 67284 28271

0.05 0.06 0.07 0.08 0.09

0.07845 0.09410 0.10973 0.12533 0.14090

90957 83133 43110 32335 12319

27844 18514 91045 64304 37582

94503 31847 26802 24537 66116

0.99691 0.99556 0.99396 0.99211 0.99002

73337 19646 09554 47013 36577

33127 03080 55179 14477 16557

97620 01290 68775 83105 56725

0.07870 0.09452 0.11040 0.12632 0.14232

17068 78311 10278 93784 10757

24618 79282 15818 46108 02942

44806 04901 94497 17478 94229

0.95 0.94 0.93 0.92 0.91

0.10 0.11 0.12 0.13 0.14

0.15643 0.17192 0.18738 0.20278 0.21814

44650 91002 13145 72953 32413

40230 79409 85724 56512 96542

86901 54661 63054 48344 55202

0.98768 0.98510 0.98228 0.97922 0.97591

83405 93261 72507 28106 67619

95137 54773 28688 21765 38747

72619 91802 68108 78086 39896

0.15838 0.17452 0.19076 0.20709 0.22352

44403 79388 02022 00444 64828

24536 94365 18566 27938 97149

29384 08461 74856 70402 10184

0.90 0.89 0.88

0.15 0.16 0.17 0.18 0.19

0.23344 0.24868 0.26387 0.27899 0.29404

53638 98871 30499 11060 03252

55905 64854 65372 39229 32303

41177 78824 89696 25185 95777

0.97236 0.96858 0.96455 0.96029 0.95579

99203 31611 74184 36856 30147

97676 28631 57798 76943 98330

60183 11949 09366 07175 12664

0.24007 0.25675 0.27356 0.29052 0.30764

87590 63603 90430 68567 01696

80116 67726 82237 31916 59898

03926 78332 23655 45432 29067

0.85 0. 84 0.83 0. 82 0.81

0.20 0.21 0.22 0.23 0.24

0.30901 0.32391 0.33873 0.35347 0.36812

69943 74181 79202 48437 45526

74947 98149 45291 79257 84677

42410 41440 38122 12472 95915

0.95105 0.94608 0.94088 0.93544 0.92977

65162 53588 07689 40308 64858

95153 27545 54225 29867 88251

57211 31853 47232 32518 40366

0.32491 0.34237 0.36002 0.37786 0.39592

96962 65257 21530 85117 80087

32906 28683 95756 75820 97721

32615 05965 62634 93670 26049

0.80 0.79 0.78 0.77 0.76

0.25 0.26 E; 0:29

0.38268 0.39714 0.41151 0.42577 0.43993

34323 78906 43586 92915 91698

65089 34780 05108 65072 55915

77173 61375 77405 64886 14083

0.92387 0.91775 0.91140 0.90482 0.89802

95325 46256 32766 70524 75757

11286 83981 35445 66019 60615

75613 14114 24821 52771 63093

0.41421 0.43273 0.45151 0.47056 0.48989

35623 86422 73130 42812 49450

73095 47425 86983 12251 22477

04880 93197 28945 49308 05270

0.75 0.74 0.73 0.72 0.71

0.30 0.31 0.32 0.33 0.34

0.45399 0.46792 0.48175 0.49545 0.50904

04997 98142 36741 86684 14157

39546 60573 01715 32407 50371

79156 37723 27498 53805 30028

0.89100 0.88376 0.87630 0.86863 0.86074

65241 56300 66800 15144 20270

88367 88693 43863 38191 03943

86236 42432 58731 24777 63716

0.50952 0.52947 0.54975 0.57038 0.59139

54494 27451 46521 99296 83513

94428 82014 92770 73294 99471

81051 63252 07429 88698 09817

0.70 0.69 0.68 0.67 0.66

0.35 0.36 0.37 00% .

0.52249 0.53582 0.54902 0.56208 0.57500

85647 67949 28179 33778 52520

15948 78996 98131 52130 43278

86499 61827 74352 60010 56590

0.85264 0.84432 0.83580 0.82708 0.81814

01643 79255 73613 05742 97174

54092 02015 68270 74561 25023

22152 07855 25847 82492 43213

0.61280 0.63461 0.65687 0.67959 0.70281

07881 92975 72224 92982 17712

39931 44148 01279 24526 40357

99664 10071 37691 52184 33761

0. 65 0. 64 0.63 0.62 0.61

0.40 0.41 0.42 0.43 0.44

0.58778 0.60042 0.61290 0.62524 0.63742

52522 02253 70536 26563 39897

92473 25884 52976 35705 48689

12917 04976 49336 17290 71017

0.80901 0.79968 0.79015 0.78043 0.77051

69943 46584 50123 04073 32427

74947 87090 75690 38329 75789

42410 53868 36516 73585 23080

0.72654 0.75082 0.77567 0.80115 0.82727

25280 12380 95110 10705 19459

05360 38764 49613 58751 72475

88589 68575 10378 23382 63403

0.60 0.59 0.58 0.57 0.56

0.45 0.46 0.47 0.48 0.49

0.64944 0.66131 0.67301 0.68454 0.69591

80483 18653 25135 71059 27965

30183 23651 09773 28688 92314

65572 87657 33872 67373 32549

0.76040 0.75011 0.73963 0.72896 0.71812

59656 10696 10949 86274 62977

00030 30459 78609 21411 63188

93817 54151 69747 52314 83037

0.85408 0.88161 0.90992 0.93906 0.96906

06854 85923 99881 25058 74171

63466 63189 77737 17492 93793

63752 11465 46579 35255 27618

0.55 0.54 0.53 0.52 0.51

0.50

0.70710 67811 86547 52440

1.00000 00000 00000 00000

0.50

l-x

cos ;x

(-62

[ ii3 1

00000 59864 55700 97264 56195

0.00000 0.01570 0.03142 0.04715 0.06291

00000 92553 62660 88028 46672

00000 23664 43351 77480 53649

00000 91632 14782 47448 75722

1.00 0.99 0.98 0.97 0.96

0.70710 67811 86547 52440 sin

g2

[(-i$3 1

cot

;x

[(J4jl 1

00% .

2

ELEMENTARY CIRCULAR

TRANSCENDENTAL

FUNCTIONS

FOR

201

FUNCTIONS

THE

ARGUMENT

Table

f x

csc;x

X

4.12 1-X

1 00000 1'00012 1:00049 1.00111 1.00197

00000 33827 36832 13587 71730

00000 39761 37144 85243 71142

00000 81169 42400 76109 10978

63 66459 31:83622 21.22851 15.92597

53060:0564 52090 97622 50958 16816 11099 08654

58546 95566 17580 59358

1.00 0.99 0.98 0.97 0.96

0.05 12.70620 47361 74704 64602 0.06 10.57889 49934 05635 52417 0.07 9.05788 66862 38928 19329 0.08 7.91581 50883 05826 84427 0.09 7.02636 62290 41380 19848

1.00309 1.00445 1.00607 1.00794 1.01007

21984 78193 57361 79708 68726

82825 57019 86291 09297 13784

50283 51480 90575 28943 19104

12.74549 10.62605 9.11292 7.97872 7.09717

48431 37962 00161 97555 00264

82374 83115 49841 59476 69225

28619 99865 72675 60149 38129

0.95 0.94 0.93 0.92 0.91

0.10 0.11 0.12 0.13 0.14

6.31375 5.72974 5.24218 4.82881 4.47374

15146 16467 35811 73521 28292

75043 24314 13176 92759 11554

09898 86192 73758 97818 62415

1.01246 1.01511 1.01803 1.02121 1.02467

51257 57576 21481 80406 75534

88002 62501 91042 26567 55900

93136 87437 38259 47910 33566

6.39245 5.81635 5.33671 4.93127 4.58414

32214 10329 14122 53949 38570

99661 24944 92458 49859 27373

54704 03199 78659 96253 56913

0.90 0.89 0.88 0.87 0.86

0.15 0.16 0.17 0.18 0.19

4.16529 3789474 3.65538 3:44202 3.25055

97700 28549 43546 25766 08012

90417 29859 52259 69218 99836

20387 33474 73004 62809 37634

1.02841 1.03243 1.03674 1.04134 1.04625

51936 58734 49162 80947 16303

65208 17339 32016 70681 39647

54585 88710 53065 14007 78848

4.28365 4.02107 3.78970 3.58434 3.40089

75697 22333 11465 36523 40753

31185 75967 59780 72161 61802

03924 50952 81919 57038 31848

0. 85 0.84 0. 83 0.82 0. 81

0.20 0.21 0.22 0.23 0.24

3.07768 2.92076 2.77760 2.64642 2.52571

35371 09892 68539 32102 16894

75253 98816 14974 86631 47304

40257 40048 88865 86514 99451

1.05146 1.05698 1.06283 1.06901 1.07552

22242 70790 39243 10439 73070

38267 93232 36113 98926 22247

21205 61183 96396 01199 78234

3.23606 3.08720 2.95213 2.82905 2.71647

79774 66268 47928 56388 18916

99789 08416 09339 91501 65871

69641 38088 97327 64260 74307

0. 80 0.79 0.78 0.77 0.76

0.25 2.41421 35623 73095 04880 0.26 2.31086 36538 82410 63708 0.27 2.21475 44978 13361 51875 0.28 2.12510 81731 57202 76115 0..29 2.04125 39671 21703 26026

1.08239 1.08961 1.09720 1.10518 1.11355

22002 58646 91341 35787 15511

92393 48705 29537 56399 90413

96880 30888 26252 59380 37268

2.61312 2.51795 2.43004 2.34863 2.27304

59297 36983 88648 46560 15214

52753 10349 55296 54351 61957

05571 34110 52041 86300 72361

0.75 0.74 0.73 0.72 0.71

0.30 0.31 0.32 0.33 0.34

1.96261 1.88867 1.81899 1.75318 1.69090

05055 13416 32472 66324 76557

05150 31067 81066 72237 85011

58230 67620 27571 08332 24674

1.12232 1.13152 1.14115 1.15123 1.16178

62376 17133 30035 61494 82810

34360 97749 92241 81376 72765

80715 42882 17245 51287 98515

2.20268 2.13707 2.07574 2.01833 1.96447

92645 26325 96076 18280 66988

85266 27611 48793 89559 67248

62156 85837 05903 43676 48330

0.70 0.69 0.68 0. 67 0.66

0.35 0.36 0.37 0.38 0.39

1.63185 1.57574 1.52235 1.47145 1.42285

16871 78599 45068 53158 60774

28789 68651 96131 19969 31870

61767 08688 24085 04283 59031

1.17282 1.18437 1.19644 1.20907 1.22227

76966 39497 79450 20434 01770

14008 36918 89806 06541 86068

94955 17500 17366 15436 14117

1.91388 1.86627 1.82141 1.77909 1.73911

08554 47167 79214 54854 45497

30942 00567 74081 79867 30640

72280 54120 38479 33350 74960

0.65 0.64 0.63 0.62 0.61

0.40 0.41 0.42 0.43 0.44

1.37638 1.33187 1.28919 1.24820 1.20879

19204 49515 22317 40363 23504

71173 02597 85066 53049 09609

53820 59439 67042 43751 13115

1.23606 1.25049 1.26557 1.28134 1.29783

79774 29154 44560 42308 62271

99789 09784 72090 20677 84727

69641 85573 15648 31999 12712

1.70130 1.66550 1.63156 1.59937 1.56881

16167 01910 87575 90408 45035

04079 65749 13749 68062 05365

86436 08074 73007 88301 75750

0.60 0.59 0.58 0.57 0.56

0.45 0.46 0.47 0.48 0.49

1.17084 1.13427 1.09898 1.06489 1.03191

95661 73492 56505 18403 99492

12539 55405 36301 24791 80495

22520 46422 56382 86700 57182

1.31508 1.33313 1.35202 1.37180 1.39251

69998 59054 53634 11480 27141

90784 50172 40027 64918 49012

80424 40410 12805 28453 49662

1.53976 1.51214 1.48585 1.46081 1.43696

90432 58610 64735 98491 16493

22366 31226 81717 22513 57094

30748 40092 76608 12750 20394

0.55 0.54 0.53 0.52 0.51

0.50 l-x

1.00000 00000 00000 00000 tan;x

1.41421 35623 73095 04880

0.50

0.00

0.01 0.02 0.03 0.04

63.65674 31.82051 21.20494 15.89454

1162; 59537 87896 48438

71580 73958 88751 65303

99500 03934 52283 44576

1.41421 35623 73095 04880 cscTx 2

SW;X

X

202

ELEMENTARY Table

4.13

TRANSCENDENTAL HARMONIC

2r7 sin -;

0.86602

54038

0.86602 0.86602 0.00000

54038 54038 00000

0.64278 0.98480 0.86602 0.34202

76097 77530 54038 01433

0.50000 0.86602 1.00000 0.86602 0.50000 0.00000

00000 54038 00000 54038 00000 00000

0.40673 0.74314 0.95105 0.99452 0.86602 0.58778 0.20791

66431 48255 65163 18954 54038 52523 16908

0.34202 0.64278 0.86602 0.98480 0.98480 0.86602 0.64278 0.34202 0.00000

01433 76097 54038 77530 77530 54038 76097 01433 00000

0.29475 0.56332 0.78183 0.93087 0.99720 0.91492 0.86602 0.68017 0.43388 0.14904

51744 00580 14825 37486 37972 79122 54038 27378 37391 22662

0.25881 0.50000 0.70710 0.86602 0.96592 1.00000 0.96592 0.86602 0.70710 0.50000 0.25881 0.00000

9045; 00000 67812 54038 58263 00000 58263 54038 67812 00000 90451 00000

ANALYSIS

(30s 2s

s=3 -0.50000

84

00000

1.00000 0.00000

00000 00000

+o.ooooo -1.00000

00000 00000

0.95105 0.58778

65163 52523

00000 00000 00000

0.78183 0.97492 0.43388

14824 79122 37391

s=7 +0.62348 -0.22252 -0.90096

98019 09340 88679

0.70710 1.00000 0.70710 0.00000

67812 00000 67812 00000

44431 81777 00000 26208

0.58778 0.95105 0.95105 0.58778 0.00000

52523 65163 65163 52523 00000

s=lO 0.80901 to.30901 -0.30901 -0.80901 -1.00000

69944 69944 69944 69944 00000

0.54064 0.90963 0.98982 0.75574 0.28173

s=6 +0.50000 -0.50000 -1.00000

s=g 0.76604 +0.17364 -0.50000 -0.93969

FUNCTIONS

0.70710 +o.ooooo -0.70710 -1.00000

67012 00000 67812 00000

08174 19953 14419 95743 25568

0.84125 +0.41541 -0.14231 -0.65486 -0.95949

35328 50130 48383 07340 29736

0.43388 0.78183 0.97492 0.97492 0.78183 0.43388 0.00000

37391 14825 79122 79122 14825 37391 00000

s=14 0.90096 0;62348 +0.22252 -0.22252 -0.62348 -0.90096 -1.00000

88679 98019 09340 09340 98019 88679 00000

95325 67812 34324 00000 34324 67812 95325 00000

0 36124 0:67369 0.89516 0.99573 0.96182 0.79801 0.52643 0.18374

16662s= 56436 32913 41763 56432 72273 21629 95178

=17 0.93247 0.73900 0.44573 +0.09226 -0.27366 -0.60263 -0.85021 -0.98297

22294 89172 03558 a3595 29901 46364 71357 30997

72417 05094 81581 54872 93455 54247 15716 37512 13034

0.30901 0.58778 0.80901 0.95105 1.00000 0.95105 0.80901 0.58778 0.30901 0.00000

69944 52523 69944 65163 00000 65163 69944 52523 69944 00000

29736 35328 07340 50130 48383 48383 50130 07340 35328 29736 00000

0.26979 0.51958 0.73083 oiaa788 0.97908

67711 39500 59643 52184 40877 87692 09221 98930

s=ll

54038 00000 00000 00000 54038 00000

s=15 0.38268 0.70710 0.92387 1.00000 0.92387 0.70710 0.38268 0.00000

34324 67812 95325 00000 95325 67812 34324 00000

0.93969 0.76604 0.50000 +0.17364 -0.17364 -0.50000 -0.76604 -0.93969 -1.00000

26208 44431 00000 81777 81777 00000 44431 26208 00000

0 32469 0:61421 0.83716 0.96940 0.99658 0.91577 0.73572 0.47594 0.16459

94692? 27127 64782 02659 44930 33266 39107 73930 45903

-21 0.95557 0.82623 0.62348 0.36534 +0.07473 -0.22252 -0.50000 -0.73305 -0.90096 -0.98883

28058 a7743 98019 10244 00936 09340 00000 18718 88679 08262

0.28173 0.54064 0.75574 0.90963 0.98982 0.98982 0.90963 0.75574 0.54064 0.28173 0.00000

25568 08174 95743 19953 14419 14419 19953 95743 08174 25568 00000

58263 54038 67812 00000 90451 00000 90451 00000 67812 54038 58263 00000

0.24868 0.48175 0.68454 0.84432 0.95105 0.99802 ii98228 0.90482 0.77051

98872 36741 71059 79255 65163 67284 72507 70525 32428 52523 45527 32336

s=18

SC

s=16 -6.92307 0.70710 0.38268 +o.ooooo -0.38268 -0.70710 -0.92387 -1.00000

54576 06064 69944 a4633 00000 69944 76007

0.91354 0.66913 +0.30901 -0.10452 -0.50000 -0.80901 -0.97814

8 0.96592 0.86602 0.70710 0.50000 0.25881 + 0.00000 -0.25881 -0.50000 -0.70710 -0.86602 -0.96592 -1.00000

94581 0:78914 0.54694 +0.24548 -0.08257 -0.40169 -0.67728 -0.87947 -0.98636

s=22 0.95949 0.84125 0.65486 0.41541 +0.14231 -0.14231 -0.41541 -0.65486 -0.84125 -0.95949 -1.00000 b-=25

=24

0.58178

0.36812 0.12533

0.96858 0.87630 0.72896 0.53582 0.30901 +0.06279 -0.18738 -0.42577 -0.63742 -0.80901 -0.92977 -0.99211

69944 69944

~43

s=12 0.86602 0.50000 +o.ooooo -0.50000 -0.86602 -1.00000

s=5 +0.30901 -0.80901

31611 66801 86274 67950 69944 05196 13146 92916 39898 69944 64859 47013

0.99166

0.94226 0.81696 0.63108 0.39840 0.13616

s=20

0.95105 0.80901 0.50778 0.30901 +o.ooooo -0.30901 -0.58778 -0.80901 -0.95105 -1.00000

65163 69944 52523 69944 00000 69944 52523 69944 65163 00000

s==23 0. 96291 72874 0. a5441 94046 0. 68255 31432 0. 46006 50378 + 0. 20345 60131 -0. 06824 24134 33487 96122 10": 57668 03221 19443 77571 12907 10898 1; 91721 13015 66491 -0: 99068 59460

ELEMENTARY

INVERSE

TRANSCENDENTAL

CIRCULAR

SINES

0.o”o0 0.00000 00000 00 0.001 0.002 0.003 0.004

0.00100 0.00200 0.00300 0.00400

00001 67 00013 33 00045 00 00106 67

arctan z 0.00000 00000 00 0.00099 0.00199 0.00299 0.00399

0.005 0.006 0.007 0.008 0.009

0.00500 0.00600 0.00700 0.00800 0.00900

00208 34 00360 01 00571 68 00853 36 01215 04

0.010 0.011 0.012 0.013 0.014

0.01000 0.01100 0.01200 0.01300 0.01400

0.015 0.016 0.017 0.018 0.019

arcsin 1:

203

FWCTIONS

AND

Table

TANGENTS

4.14

arctan L

99996 67 99973 33 99910 00 99786 67

0.;50 0.051 0.052 0.053 0.054

0 OSO:i%?5:8 06 0:05102 21344 17 0.05202 34632 28 0.05302 48442 51 0.05402 62784 97

0.04995 0.05095 0.05195 0.05295 0.05394

83957 22 58518 77 32065 61 04578 05 76036 42

0.00499 0.00599 0.00699 0.00799 0.00899

99583 34 99280 02 98856 70 98293 40 97570 12

0.055 0.056 0.057 0.058 0.059

0.05502 0.05602 0.05703 0.05803 0.05903

77669 81 93107 15 09107 14 25679 92 42835 64

0.05494 0.05594 0.05693 0.05793 0.05893

46421 07 15712 34 83890 60 50936 23 16829 64

01666 74 02218 45 02880 19 03661 95 04573 74

0.00999 0.01099 0.01199 0.01299 0.01399

96666 87 95563 66 94240 50 92577 41 90'954 41

0.060 0.061 0.062 0.063 0.064

0.06003 0.06103 0.06203 0.06304 0.06404

605'84 45 789.36 52 97902 01 17491 09 37713 94

0.05992 0.06092 0.06192 0.06291 0.06391

81551 21 45081 38 07400 58 68489 26 28327 89

0.01500 0.01600 0.01700 0.01800 0.01900

05625 57 06827 45 08189 40 09721 42 11433 52

0.01499 0.01599 0.01699 0.01799 0.01899

88751 52 86348 76 83626 17 80!563 78 77141 62

0.065 0.066 0.067 0.068 0.069

0.06504 58580 75 0.06604 80101 69 0.06705 02286 97 0.06805 25146 79 0~06905 48691 36

0.06490 0.06590 0.06690 0.06789 0.06889

86896 93 44176 90 00148 29 54791 63 08087 46

0.020 0.021 0.022 0.023 0.024

0.02000 0.02100 0.02200 0.02300 0.02400

13335 73 15438 06 17750 53 20283 16 23045 97

0.01999 73339 73 0.02099 69113817 0.02199 64516 97 0.02299 59456 20 OiO2399 53935 92

0.070 0.071 0.072 0.073 0.074

0.07005 72930 88 0.07105 97875 58 0.07206 23535 68 0.07306 49921 42 OiO7406 77043 03

0.06988 0.07088 0.07187 0.07287 0.07386

60016 35 10558 85 59695 56 07407 09 53674 06

0.025 0.026 0.027 0.028 0.029

0.02500 0.02600 0.02700 0.02800 0.02900

26048 99 29302 25 32815 77 36599 58 40663 72

0.02499 0.02599 0.02699 0.02799 0.02899

47936 19 41437 08 34418 68 26861 07 18744 33

0.075 0.076 or 077 0.078 0.079

0.07507 04910 77 0.07607 33534 87 OiO7707 62925 62 0.07807 93093 26 0.07908 24048 07

0.07485 98477 11 0.07585 41796 89 OiO7684 83614 08 0.07784 23909 37 0.07883 62663 48

0.030 0.031 0.032 0.033 0.034

0.03000 0.03100 0.03200 0.03300 0.03400

45018 23 49673 15 54638 51 59924 37 65540 77

0.02999 0.03099 0.03198 0.03298 0.03398

10048 57 00753 89 30840 39 80288 21 69077 46

0.080 0.081 0.082 0.083 0.084

0.08008 0.08108 0.08209 0.08309 0.08409

55800 34 88360 35 21738 40 55944 79 90989 83

0.07982 0.08082 0.08181 0.08281 0.08380

99857 12 35471 05 69486 04 01882 86 32642 31

0.035 0.036 0.037 0.038 0.039

0.03500 0.03600 0.03700 0.03800 0.03900

71497 75 77805 38 84473 72 91512 81 98932 73

0.03498 0.03598 0.03698 0.03798 0.03898

57188 29 44600 82 31295 22 17251 64 02450 25

0.085 ii* E 0:oss 0.089

0.08510 0.08610 0.08711 0.08811 0.08911

26883 84 63637 15 01260 09 39763 00 79156 23

0.08479 0.08578 0.08678 0.08777 0.08876

61745 23 89172 45 14904 84 38923 27 61208 65

0.040 0.041 0.042 0.043 0.044

0.04001 0.04101 0.04201 0.04301 0.04401

06743 54 14955 31 23578 12 32622 04 42097 16

0.03997 0.04097 0.04197 0.04297 0.04397

86871 23 70494 77 53301 05 35270 30 16382 71

0.090 0.091 0.092 0.093 0.094

0.09012 0.09112 0.09213 0.09313 0.09413

19450 15 60655 11 02781 49 45839 68 89840 07

0.08975 0.09075 0.09174 0.09273 0.09372

81741 90 00503 96 17475 79 32638 38 45972 74

0.045 0.046 0.047 0.048 0.049

0.04501 0.04601 0.04701 0.04801 0.04901

52013 56 62381 33 73210 57 84511 37 96293 83

0.04496 0.04596 0.04696 0.04796 0.04896

96618 52 75957 97 54381 30 31868 77 08400 65

0.095 0.096 0.097 0.098 0.099

'0.09514 0.09614 0.09715 0.09815 0.09916

34793 06 80709 05 27598 48 75471 75 24339 32

0.09471 0.09570 0.09669 0.09768 0.09867

57459 88 67080 87 74816 76 80648 65 84557 66

0.050

0.05002 08568 06

0.04995 83957 22

0.100

0.10016 74211 62

[ 1

0.09966 86524 91

(9’16

For use and extension of the table see Examples 21-25. For other inverse functions see 4.4 and 4.3.45. ;=1.57079 63267 95 Compilation of arcsin :c from National Bureau of Standards, Table of arcsin z. Columbia Univ. Press, New York, N.Y., 1945 (with permission).

204

ELEMENTARY

Table 4.14 X

INVERSE arcsin z

TRANSCENDENTAL

CIRCULAR

FUNCTIONS

SINES AND TANGENTS

arctan x

arcsin 2

X

arctan r

0.100 0.101 0.102 0.103 0.104

0.10016 0.10117 0.10217 0.10318 0.10418

74211 25099 77012 29961 83957

62 11 25 53 41

0.09966 0.10065 0.10164 0.10263 0.10362

86524 86531 84558 80587 74599

91 58 83 89 97

0.150 0.151 0.152 0.153 0.154

0.15056 0.15157 0.15259 0.15360 0.15461

82727 97940 14716 33066 53001

77 40 20 23 61

0.14888 0.14986 0.15084 0.15182 0.15279

99476 77989 53616 26338 96139

09 58 21 59 37

0.105 0.106 0.107 0.108 0.109

0.10519 0.10619 0.10720 0.10821 0.10921

39010 95131 52329 10617 70003

40 00 72 08 62

0.10461 0.10560 0.10659 0.10758 0.10857

66576 56498 44346 30103 13750

33 23 99 93 39

0.155 0.156 0.157 0.158 0.159

0.15562 0.15663 0.15765 0.15866 0.15967

74533 97672 22431 48819 76848

44 86 01 05 15

0.15377 0.15475 0.15572 0.15670 0.15768

63001 26906 87838 45780 00713

20 78 86 19 58

0.110 0.111 0.112 0.113 0.114

0.11022 0.11122 0.11223 0.11324 0.11424

30499 92116 54863 18752 83793

88 41 77 55 32

0.10955 0.11054 0.11153 0.11252 0.11350

95267 74637 51840 26859 99674

74 38 74 25 40

0.160 0.161 0.162 0.163 0.164

0.16069 0.16170 0.16271 0.16373 0.16474

06529 37874 70893 05599 42001

52 35 88 34 99

0.15865 0.15963 0.16060 0.16157 0.16255

52621 01487 47294 90024 29661

86 91 61 91 78

0.115 0.116 0.117 0.118 0.119

0.11525 0.11626 0.11726 0.11827 0.11928

49996 17373 85933 55688 26648

68 23 61 42 32

0.11449 0.11548 0.11647 0.11745 0.11844

70267 38620 04714 68531 30052

67 60 73 63 90

0.165 Oiibk 0.167 0.168 0.169

0.16575 Oii6677 0.16778 0.16880 0.16981

80113 19943 61505 04810 49868

10 96 87 17 19

0.16352 0.16449 0.16547 0.16644 0.16741

66188 99587 29841 56935 80850

21 25 97 49 93

0.120 0.121 0.122 0.123 0.124

0.12028 0.12129 0.12230 0.12331 0.12431

98823 72225 46865 22751 99897

95 97 07 92 22

0.11942 0.12041 0.12140 0.12238 0.12337

89260 46135 00659 52814 02582

18 12 40 72 82

0.170 0.171 0.172 0.173 0.174

0.17082 0.17184 0.17285 0.17387 0.17489

96691 45290 95678 47864 01862

29 84 23 87 19

0.16839 0.16936 0.17033 0.17130 0.17227

01571 19080 33360 44396 52169

48 34 78 07 54

0.125 oIi2k 0.127 0.128 0.129

0.12532 OIi2633 0.12734 0.12835 0.12936

78311 58006 38990 21277 04875

68 02 98 29 72

0.12435 0:125>3 0.12632 0.12730 0.12829

49945 94884 37381 77418 14977

47 45 58 71 71

i* E 0:177 0.178 0.179

0.17590 0.17692 0.17793 0.17895 0.17996

57681 15334 74832 36187 99410

64 66 75 40 13

0.17324 0.17421 0.17518 0.17615 0.17712

56664 57864 55752 50312 41528

52 43 68 74 10

0.130 0.131 0.132 0.133 0.134

0.13036 0.13137 0.13238 0.13339 0.13440

89797 76052 63651 52606 42926

03 01 45 16 95

0.12927 0.13025 0.13124 0.13222 0.13320

50040 82588 12605 40070 64968

48 96 10 89 35

0.135 0.136 0.137 0.138 0.139

0.13541 0.13642 0.13743 0.13844 Oil3945

34624 27710 22194 18087 15401

67 15 25 85 83

0.13418 0.13517 0.13615 0.13713 0.13811

87279 06986 24071 38516 50303

0.140 0.141 0.142 0.143 0.144

0.14046 0.14147 0.14248 0.14349 0.14450

14147 14334 15975 19079 23659

10 56 13 77 42

0.13909 0.14007 0.14105 0.14203 0.14301

0.145 Oil46 0.147 0.148 0.149

0.14551 Oil4652 0.14753 0.14854 0.14955

29725 37287 46358 56947 69067

04 64 19 71 22

0.14399 Oil4497 0.14595 0.14693 0.14791

0.150

0.15056 82727 77

II(792 1

0.18098 64512 47

0.17809 29382 31

0:182 ;;;; 0.183 0.184

0.18200 00402 0.18302 31505 97 20 0.18403 71212 76 0.18505 43949 25

0.17906 13858 0.18002 94941 94 59 0.18099 72613 91 0.18196 46859 59

52 49 35 25 34

0.185 0.186 0.187 0.188 0.189

0.18607 0.18708 0.18810 0.18912 0.19014

18623 95246 73830 54387 36928

31 57 71 40 36

0.18293 0.18389 0.18486 0.18583 0.18679

17662 85005 48874 09250 66119

35 94 16 85 87

59414 65832 69539 70518 68749

82 92 90 03 65

0.190 0.191 0.192 0.193 0.194

0.19116 0.19218 0.19319 0.19421 0.19523

21465 08009 96574 87169 79808

31 99 17 63 18

0.18776 0.18872 0.18969 0.19065 0.19161

19465 69270 15520 58198 97288

14 59 22 05 15

64217 56902 46789 33858 18093

09 74 00 33 19

0.195 0.196 0.197 0.198 0.199

0.19625 0.19727 0.19829 0.19931 0.20033

74501 71261 70100 71030 74061

64 85 69 03 80

0.19258 0.19354 0.19450 0.19547 0.19643

32774 64641 92873 17453 38367

60 55 18 71 38

0.14888 99476 09

0.200

0.20135 79207 90

[c-y4 1

[ C-f)3 1 ;=1.57079 63267 95

0.19739 55598 50

[c-j15 1

ELEMENTARY

TRANSCENDENTAL

205

F’UNCTIONS

INVERSE CIRCULAR SINES AND TANGENTS X

arcsin x

arctan x

X

arcsin x

Table 4.14 arctan x

0.200 0.201 0.202 0.203 0.204

0.20135 0.20237 0.20339 0.20442 0.20544

79207 90 86480 31 95890 97 07451 90 21175 10

0.19739 0.19835 0.19931 0.20027 0.20123

55598 50 69131 40 78950 44 85040 06 87384 69

0.250 0.251 0.252 0.253 0.254

0.25268 0.25371 0.25474 0.25577 0.25681

02551 42 31886 28 63988 49 98871 33 36548 08

0.24497 0.24591 0.24686 0.24780 0.24873

86631 27 96179 19 01284 51 01933 77 98113 53

0.205 0.206 0.207 0.208 0.209

0.20646 0.20748 0.20850 0.20952 0.21055

37072 61 55156 48 75438 81 97931 68 22647 22

0.20219 0.20315 0.20411 0.20507 0.20603

85968 83 80777 01 71793 81 59003 83 42391 73

0.255 0.256 0.257 0.258 0.259

0.25784 0.25888 0.25991 0.26095 0.26198

77032 07 20336 66 66475 22 15461 18 67307 97

0.24967 0.25061 0.25155 0.25249 0.25343

89810 38 77010 99 59702 05 37870 29 11502 51

0.210 0.211 0.212 0.213 0.214

0.21157 0.21259 0.21362 0.21464 0.21566

49597 58 78794 93 10251 46 43979 39 79990 96

0.20699 0.20794 0.20890 0.20986 0.21082

21942 20 97639 97 69469 83 37416 57 01465 06

0.260 0.261 0.262 0.263 0.264

0.26302 0.26405 0.26509 0.26613 0.26716

22029 08 79638 02 40148 31 03573 53 69927 28

0.25436 0.25530 0.25624 0.25717 0.25811

80585 53 45106 23 05051 53 60408 40 11163 83

0.215 0.216 0.217 0.218 0.219

0.21669 0.21771 0.21874 0.21976 0.22078

18298 42 58914 06 01850 19 47119 15 94733 28

0.21177 0.21273 0.21368 0.21464 0.21559

61600 20 17806 92 70070 19 18375 04 62706 53

0.265 0.266 0.267 0.268 0.269

0.26820 0.26924 0.27027 0.27131 0.27235

39223 20 11474 95 86696 22 64900 75 46102 31

0.25904 0.25997 0.26091 0.26184 0.26277

57304 89 98818 68 35692 33 67913 04 95468 05

0.220 0.221 0.222 0.223 0.224

0.22181 0.22283 0.22386 0.22489 0.22591

44704 97 97046 62 51770 66 08889 55 68415 75

0.21655 0.21750 0.21845 0.21941 0.22036

03049 76 39389 87 71712 05 00001 53 24243 57

0.270 0.271 0.272 0.273 0.274

0.27339 0.27443 0.27547 0.27651 0.27754

30314 67 17551 69 07827 21 01155 13 97549 38

0.26371 0.26464 0.26557 0.26650 0.26743

18344 62 36530 10 50011 84 58777 27 62813 84

0.225 0.226 0.227 0.228 0.229

0.22694 0.22796 0.22899 0.23002 0.23105

30361 79 94740 17 61563 45 30844 22 02595 07

0.22131 0.22226 0.22321 0.22416 0.22511

44423 48 60526 61 72538 37 80444 19 84229 53

0.275 0.276 0.277 0.278 0.279

0.27858 0.27962 0.28067 0.28171 0.28275

97023 92 99592 75 05269 90 14069 43 26005 45

0.26836 0.26929 0.27022 0.27115 0.27208

62109 06 56650 49 46425 71 31422 39 11628 19

0.230 0.231 0.232 0.233 0.234

0.23207 0.23310 0.23413 0.23516 0.23618

76828 63 53557 56 32794 53 14552 26 98843 48

0.22606 0.22701 0.22796 0.22891 0.22986

83879 94 79380 96 70718 22 57877 34 40844 03

0.280 0.281 0.282 0.283 0.284

0.28379 0.28483 0.28587 0.28692 0.28796

41092 08 59343 51 80773 93 05397 58 33228 75

0.27300 0.27393 0.27486 0.27578 0.27671

87030 87 57618 19 23377 99 84298 14 40366 55

0.235 0.236 0.237 0.238 0.239

0.23721 0.13824 0.23927 0.24030 0.24133

85680 94 75077 44 67045 78 61598 80 58749 37

0.23081 0.23175 0.23270 0.23365 0.23459

19604 03 94143 10 64447 07 30501 80 92293 19

0.285 0.286 0.287 0.288 0.289

0.28900 0.29004 0.29109 0.29213 0.29318

64281 74 98570 89 36110 61 76915 30 20999 43

0.27763 0.27856 0.27948 0.28041 0.28133

91571 20 37900 08 79341 26 15882 83 47512 95

0.240 0.241 0.242 0.243 0.244

0.24236 0.24339 0.24442 0.24545 0.24648

58510 39 60894 77 65915 47 73585 45 83917 73

0.23554 0.23649 0.23743 0.23837 0.23932

49807 21 03029 83 51947 10 96545 10 36809 95

0.290 0.291 0.292 0.293 0.294

0.29422 68377 49 Oi29527 19064 01 0.29631 73073 57 0.29736 30420 76 0.29840 91120 25

0.28225 0.28317 0.28410 0.28502 0.28594

74219 81 95991 65 12816 76 24683 46 31580 14

0.245 0,246 0.247 0.248 0.249

0.24751 0.24855 0.24958 0.25061 0.25164

96925 34 12621 33 31018 81 52130 88 75970 69

0.24026 0.24121 0.24215 0.24309 0.24403

72727 81 04284 90 31467 47 54261 82 72654 29

0.295 0.296 0.297 Oi298 0.299

0.29945 0.30050 0.30154 0.30259 0.30364

0.28686 0.28778 0.28870 0.28962 0.29053

33495 23 30417 18 22334 53 09235 83 91109 69

0.250

0.25268 02551 42

0.24497 86631 27

0.300

0.30469 26540 15

[y4 1

55186 70 22634 85 93479 45 67735 30 45417 24

[C-j)6 1 P2'1.57079 6326795 [F-f)4 1

0.29145 67944 78

[C-i)‘31

206

ELEMENTARY Table x

INVERSE

4.14

arcsin x

TRANSCENDENTAL

CIRCULAR

SINES

nrctan x

AND

FUNCTIONS TANGENTS

arcsin x

5

arctan x

0.300 0.301 0.302 0.303 0.304

0.30469 0.30574 0.30678 0.30783 0.30888

26540 15 11118 95 99168 60 90704 09 85740 46

0.29145 0.29237 0.29329 0.29420 0.29512

67944 78 39729 79 06453 47 68104 62 24672 09

0.350 0.351 0.352 0.353 0.354

0.35757 0.35863 0.35970 0.36077 0.36184

11036 46 88378 55 69995 85 55905 70 46125 51

0.33667 0.33756 0.33845 0.33934 0.34023

48193 a7 54100 58 54442 85 49211 al 38398 61

0.305 0.306 0.307 0.308 0.309

0.30993 0.31098 0.31203 0.31309 0.31414

a4292 78 a6376 19 92005 83 01196 91 13964 68

0.29603 0.29695 0.29786 0.29877 0.29969

76144 75 22511 55 63761 46 99883 52 30866 80

0.355 0.356 0.357 0.358 0.359

0.36291 0.36398 0.36505 0.36612 0.36719

40672 71 39564 82 42819 39 50454 05 62486 46

0.34112 0.34200 0.34289 0.34378 0.34467

21994 49 99990 70 72378 56 39149 42 00294 69

0.310 0.311 0.312 0.313 0.314

0.31519 0.31624 0.31729 0.31835 0.31940

30324 41 50291 43 73881 12 01108 aa 31990 la

0.30060 0.30151 0.30242 0.30334 0.30425

56700 42 77373 55 92875 41 03195 25 08322 38

0.360 0.361 0.362 0.363 0.364

0.36826 0.36933 0.37041 0.37148 0.37255

78934 37 99815 54 25147 84 54949 16 89237 46

0.34555 0.34644 0.34732 0.34820 0.34909

55805 a2 05674 30 49891 68 a8449 54 21339 52

0.315 0.316 0.317 0.318 0.319

0.32045 0.32151 0.32256 0.32361 0.32467

66540 50 04775 38 46710 42 92361 24 41743 51

0.30516 0.30607 0.30697 0.30788 0.30879

08246 16 02955 99 92441 31 76691 62 55696 46

0.365 0.366 0.367 0.368 0.369

0.37363 0.37470 0.37578 0.37685 0.37793

28030 75 71347 12 19204 71 71621 69 28616 34

0.34997 0.35085 0.35173 0.35261 0.35350

48553 30 70082 60 a5919 21 96054 93 00481 64

0.320 0.321 0.322 0.323 0.324

0.32572 0.32678 0.32784 0.32889 0.32995

94872 95 51765 31 12436 42 76902 11 45178 29

0.30970 0.31060 0.31151 0.31242 0.31332

29445 42 97928 14 61134 29 19053 60 71675 a4

0.370 0.371 0.372 0.373 0.374

0.37900 0.38008 0.38116 0.38224 0.38331

90206 96 56411 93 27249 69 02738 73 82897 61

0.35437 0.35525 0.35613 0.35701 0.35789

99191 23 92175 68 79426 98 60937 la 36698 38

0.325 0.326 0.327 0.328 0.329

0.33101 0.33206 0.33312 0.33418 0.33524

17280 a9 93225 91 73029 38 56707 38 44276 04

0.31423 0.31513 0.31603 0.31694 0.31784

la990 a4 60988 47 97658 63 28991 30 54976 47

0.375 0.376 0.377 0.378 0.379

0.38439 0.38547 0.38655 0.38763 0.38871

67744 96 57299 45 51579 a3 50604 92 54393 57

0.35877 0.35964 0.36052 0.36139 0.36227

06702 71 70942 35 29409 56 82096 58 28995 76

0.330 0.331 0.332 0.333 0.334

0.33630 0.33736 0.33842 0.33948 0.34054

35751 54 31150 09 30487 98 33781 50 41047 05

0.31874 0.31964 0.32055 0.32145 0.32235

75604 21 90864 60 00747 al 05244 03 04343 49

0.380 0.381 0.382 0.383 0.384

0.38979 0.39087 0.39195 0.39304 0.39412

62964 74 76337 42 94530 68 17563 64 45455 51

0.36314 0.36402 0.36489 0.36576 0.36663

70099 46 05400 09 34890 12 58562 04 76408 40

0.335 0.336 0.337 0.338 0.339

0.34160 0.34266 0.34372 0.34479 0.34585

52301 02 67559 88 86840 15 10158 39 37531 21

0.32324 0.32414 0.32504 0.32594 0.32684

98036 48 86313 34 69164 46 46580 25 la551 19

0.385 0.386 0.387 0.388 0.389

0.39520 0.39629 0.39737 0.39846 0.39954

78225 54 15893 06 58477 48 05998 24 58474 a9

0.36750 0.36837 0.36924 0.37011 0.37098

88421 al 94594 90 94920 36 a9390 92 77999 35

0.340 0.341 0.342 0.343 0.344

0.34691 0.34798 0.34904 0.35010 0.35117

68975 27 04507 29 44144 03 87902 30 35798 98

0.32773 0.32863 0.32953 0.33042 0.33131

85067 al 46120 66 01700 37 51797 60 96403 04

0.390 0.391 0.392 0.393 0.394

0.40063 0.40171 0.40280 0.40389 0.40497

15927 01 78374 28 45836 44 la333 27 95884 67

0.37185 0.37272 0.37359 0.37445 0.37532

60738 49 37601 la 08580 36 73668 96 32860 01

0.345 0.346 0.347 0.348 0.349

0.35223 0.35330 0.35437 0.35543 0.35650

87850 97 44075 25 04488 a4 69108 al 37952 29

0.33221 0.33310 0.33399 0.33489 0.33578

35507 47 69101 67 97176 49 19722 a3 36731 63

0.395 0.396 0.397 0.398 Oi399

0.40606 0.40715 0.40824 0.40933 0.41042

78510 57 66231 00 59066 02 57035 al 60160 60

0.37618 0.37705 0.37791 0.37878 0.37964

a6146 53 33521 62 74978 43 10510 12 40109 93

0.350

0.35757 11036 46

0.33667 48193 a7

0.400

0.41151 68460 67

cc-:)8)51

1C-f)6 1

[ c-t)7 1 ;=I.57079 6326795

0.38050 63771 12

[c-pa 1

ELEMENTARY INVERSE X

arcsin x

TRANSCENDENTAL

CIRCULAR

SINES

arctan x

207

FUNCTIONS

AND TANGENTS

Table

X

arcsin z

4.14

arctan z

0.400 0.401 0.402 0.403 0.404

0.41151 0.41260 0.41370 0.41479 0.41588

68460 67 81956 42 00668 29 24616 80 53822 54

0.38050 0.38136 0.38222 0.38308 0.38394

63771 12 81487 02 93250 97 99056 39 98896 72

0.450 0.451 0.452 0.453 0.454

0.46676 53390 47 0.46788 54404 09 0.46900 61761 03 0.47012 75486 20 Oi47124 95604 59

0.42285 0.42368 0.42451 0.42534 0.42617

39261 33 52156 87 58823 89 59257 92 53454 56

0.405 0.406 0.407 0.408 0.409

0.41697 0.41807 0.41916 0.42026 0.42135

88306 20 28088 50 73190 29 23632 45 79435 96

0.38480 0.38566 0.38652 0.38738 0.38824

92765 46 80656 14 62562 34 38477 69 08395 85

0.455 0.456 0.457 0.458 0.459

0.47237 0.47349 0.47461 0.47574 0.47686

22141 29 55121 50 94570 53 40513 79 92976 80

0.42700 0.42783 0.42865 0.42948 0.43031

41409 43 23118 21 98576 60 67780 36 30725 28

0.410 0.411 0.412 0.413 0.414

0.42245 0.42355 0.42464 0.42574 0.42684

40621 87 07211 31 79225 49 56685 70 39613 30

0.38909 0.38995 0.39080 0.39166 0.39251

72310 55 30215 54 82104 62 27971 64 67810 48

0.460 0.461 0.462 0.463 0.464

0.47799 0.47912 0.48024 0.48137 0.48250

51985 19 17564 68 89741 12 68540 46 53988 75

0.43113 0.43196 0.43278 0.43361 0.43443

87407 19 37821 96 81965 51 19833 80 51422 81

0.415 0.416 0.417 0.418 0.419

0.42794 0.42904 0.43014 0.43124 0.43234

28029 74 21956 53 21415 30 26427 72 37015 57

0.39337 0.39422 0.39507 0.39592 0.39677

01615 09 29379 43 51097 52 66763 44 76371 29

0.465 0.466 0.467 0.468 0.469

0.48363 0.48476 0.48589 0.48702 0.48815

46112 18 44937 02 50489 67 62796 64 81884 55

0.43525 0.43607 0.43690 0.43772 0.43854

76728 58 95747 19 08474 74 14907 40 15041 36

0.420 0.421 0.422 0.423 0.424

0.43344 0.43454 0.43565 0.43675 0.43785

53200 70 75005 03 02450 60 35559 49 74353 90

0.39762 0.39847 0.39932 0.40017 0.40102

79915 22 77389 43 68788 14 54105 66 33336 29

0.470 0.471 0.472 0.473 0.474

0.48929 0.49042 0.49155 0.49269 0.49382

07780 14 40510 26 80101 88 26582 08 79978 07

0.43936 0.44017 0.44099 0.44181 0.44263

08872 85 96398 14 77613 55 52515 43 21100 17

0.425 0.426 0.427 0.428 0.429

0.43896 0.44006 0.44117 0.44227 0.44338

18856 10 69088 44 25073 36 86833 39 54391 16

0.40187 0.40271 0.40356 0.40440 0.40525

06474 40 73514 42 34450 79 89278 00 37990 60

0.475 0.476 0.477 0.478 0.479

0.49496 0.49610 0.49723 0.49837 0.49951

40317 17 07626 82 81934 59 63268 16 51655 34

0.44344 0.44426 0.44507 0.44589 0.44670

83364 20 39303 99 88916 06 32196 95 69143 24

0.430 0.431 0.432 0.433 0.434

0.44449 0.44560 0.44670 0.44781 0.44892

27769 36 06990 78 92078 31 83054 92 79943 67

0.40609 0.40694 0.40778 0.40862 0.40946

80583 18 17050 34 47386 77 71587 18 89646 31

0.480 0.481 0.482 0.483 0.484

0.50065 0.50179 0.50293 0.50407 0.50522

47124 05 49702 34 59418 39 76300 52 00377 13

0.44751 0.44833 0.44914 0.44995 0.45076

99751 57 24018 60 41941 03 53515 61 58739 11

0.435 0.436 0.437 0.438 0.439

0.45003 0.45114 0.45226 0.45337 0.45448

82767 71 91550 28 06314 71 27084 44 53882 99

0.41031 0.41115 0.41199 0.41283 0.41366

01558 96 07319 97 06924 22 00366 64 87642 17

0.485 0.486 0.487 0.488 0.489

0.50636 0.50750 0.50865 0.50979 0.51094

31676 79 70228 19 16060 14 69201 57 29681 57

0.45157 0.45238 0.45319 0.45400 0.45480

57608 36 50120 20 36271 55 16059 33 89480 51

0.440 0.441 0.442 0.443 0.444

0.45559 0.45671 0.45782 0.45894 0.46005

86733 96 25661 07 70688 11 21838 99 79137 71

0.41450 0.41534 0.41618 0.41701 0.41785

68745 85 43672 70 12417 83 74976 36 31343 48

0.490 0.491 0.492 0.493 0.494

0.51208 0.51323 0.51438 0.51553 0.51668

97529 34 72774 22 55445 69 45573 34 43186 93

0.45561 0.45642 0.45722 0.45803 0.45883

56532 11 17211 17 71514 78 19440 06 60984 16

0.445 0.446 0.447 0.448 0.449

0.46117 0.46229 0.46340 0.46452 0.46564

42608 35 12275 10 88162 25 70294 19 58695 40

0.41868 0.41952 0.42035 0.42118 0.42202

81514 38 25484 34 63248 66 94802 67 20141 75

0.495 0.496 0.497 0.498 0.499

0.51783 0.51898 0.52013 0.52129 0.52244

48316 32 60991 55 81242 77 09100 26 44594 47

0.45963 0.46044 0.46124 0.46204 0.46284

96144 30 24917 71 47301 65 63293 45 72890 44

0.450

0.46676 53390 47

0.42285 39261 33

0.500

0.52359 87755 98

1C-J)8 1

[c-y1 ;=1.57079 6326795

[c-y1

[c-y1

0.46364 76090 01

ELEMENTARY

208 Table

4.14

INVERSE

arcsin x

X

TRANSCENDENTAL

CIRCULAR

SINES

arctan 2

AND

FUNCTIONS TANGENTS

arcsin x

arctan 2

0.550 0.551 0.552 0.553 0.554

0.58236 42378 69 0.58356 20792 89

0.50284 32109 28 0.50361 06410 37

0.58476

08688

0.50437

74226

2

76090 01 98 0.46364 51 0.46444 72889 58 91 ' 0.46524 63286 62 20 0.46604 47278 61 24863 09 54 0.46684

0.500 0.501 0.502 0.503 0.504

0.52359 0.52475 0.52590 0.52706 0.52822

87755 38615 97203 63552 37691

0.505 0.506 0.507 0.508 0.509

0.52938 0.53054 0.53170 0.53286 0.53402

19653 22 09468 69 07169 56 12787 56 26354 61

0.46763 0.46843 0.46923 0.47002 0.47082

96037 63 60799 83 19147 34 71077 82 16589 00

0.555 0.556 0.557 0.558 0.559

0.58836 0.58956 0.59076 0.59197 0.59317

29661 37 55882 10 91785 32 37411 92 92803 04

0.50667 0.50743 0.50820 0.50896 0.50972

38759 68 80629 53 16011 02 44903 52 67306 43

0.510 0.511 0.512 0.513 0.514

0.53518 0.53634 0.53751 0.53867 0.53984

47902 76 77464 20 15071 30

55678 62 88344 48 14584 38 34396 20 47777 82

0.560 0.561

14552 69

0.47161 0.47240 0.47320 0.47399 0.47478

0.59438 58000 01 0.59559 33044 41 0.59680 17978 05 0.59801 12842 95 0.59922 17681 37

0.51048 0.51124 0.51200 0.51276 0.51352

83219 17 92641 21 95572 04 92011 19 81958 22

0.515 0.516 0.517 0.518 0.519

0.54100 0.54217 0.54334 0.54451 0.54568

76492

0.47557 54727 17

0.60043

32535 81 0.60164 57448 99 0.60285 92463 89 0.60407 37623 71 0.60528 92971 89

0.51428 0.51504 0.51580 0.51655 0.51731

65412 69 42374 25 12842 52 76817 18 34297 96

0.520 0.521 0.522 0.523 0.524

0.54685 0.54802 0.54919 0.55036 0.55154

09506

0.60650 58552 13 0.60772 34408 36 0.61016 17125 74 0.61138 24076 01

0.51806 85284 57 0.51882 29776 79 0.51957 67774 41 0.52032 99277 27 0.52108 24285 22

0.525 0.526 0.527 0.528 0.529

0.577 0.578 0.579

0.61260 41480 49 69384 37 0.61505 07833 09 0.61627 56872 37 0.61750 16548 17

0.52183 0.52258 0.52333 0.52408 0.52483

42798 14 54815 96 60338 62 59366 09 51898 38 37935

60756

57

0.562

0.563 0.564

33

0.58596 06104 84 0.58716 13082 43

73

0.50514 35557 57 0.50590 90402 12

0.47636

55242

0.47715 49320 97 0.47794 36961 45 0.47873 18161 73

0.565 0.566 0.567 0.568 0.569

21007 28 40885 61 69176 11 05913 07

0.47951 0.48030 0.48109 0.48187 0.48266

92919 93 61234 17 23102 64 78523 54 27495 12

0.570 0.571 0.572 0.573 0.574

0.55271 0.55389 0.55506 0.55624 0.55742

51130 97 04864 46 67148 37 38017 69 17507 59

0.48344 0.48423 0.48501 0.48579 0.48657

70015 67

0.575

06083

0.576

0.530 0.531 0.532 0.533 0.534

0.55860 0.55978 0.56096 0.56214 0.56332

05653 43 02490 72 08055 18 22382 69 45509 33

0.48735 0.48813 0.48891 0.48969 0.49047

85795 05 89575 18 86893 19 77747 65 62137 12

0.580 0.581 0.582 0.583 0.584

0.61872 86906 72 0.61995 67994 52 0.62118 59858 34 0.62241 62545 21 0.62364 76102 44

0.52558 0.52633

0.535 0.536 0.537 0.538 0.539

0.56450 77471 34 0.56569 18305 17

0.49125 0.49203 0.49280 0.49358 0.49435

40060 25 11515 68 76502 10 35018 23 87062 83

0.585 0.586 0.587 0.588 0.589

0.62488 00577 61 0.62611 36018 60 0.62734 82473 54 0.62858

39990

0.52931 70697 19 0.53006 17765 76 0.53080 58340 23 0.53154 92420 86 0.53229 20007 93

0.540 0.541 0.542 0.543 0.544

0.57043 0.57162 0.57281 0.57400 0.57519

71094 00 56840 08 51680 58 55653 28 68796 15

0.49513 32634 68 0.49590 71732 62

0.590 0.591 0.592 0.593 0.594

0.63105 0.63229 0.63353 0.63477

88407 78 79405 66 81bb2 50 95228 17 20152 84

0.53303 0.53377 0.53451 0.53525 0.53599

41101 77 55702 73 63811 18

0.545 0.546 0.547 0.548 0.549

0.57638 0.57758 0.57877 0.57997 0.58116

91147 36 22745 29 63628 51 13835 79 73406 12

0.49899

49185 66 31328 39 06980 90 76143 74 38817 48

0.550

0.58236 42378 69

0.56687

49

46608 96 24935 25 11504 67 06350 69

68047

96

44

0.56806 26734 97 0.56924 94404 76

[IC-l)1 1

0.49668

22

50

35696 94 58854 40 75554 29

04355

48

0.49745 30502 17 0.49822 50171 59 0.49976

63362 70074

0.60894

20584

75

0.61382

87

0.62982 08619 28

0.63602

52

17477 57 0.52707 90524 63 0.52782 57076 82 0.52857 17134 28

65427

53

60552 20

71

0.595

50

0.596

0.63851

0.63726 56487 00 04281 42 0.63975 63587 17 0.64100 34455 66 0.64225 16938 57

0.53673 0.53747 0.53821 0.53894 0.53968

0.600

0.64350 11087 93

0.54041 95002 71

0.50053 70305 98 0.50130 64056 22 0.50207 51324 28

c(-y 1 f=

0.50284 32109 28

0.597 0.598 0.599

[(-;I2 1 1.570796326795

[c-y1

ELEMENTARY

INVERSE X

arcsin 5

TRANSCENDENTAL

CIRCULAR arctan x

209

FUNCTIONS

SINES

AND TANGENTS X

Table 4.14

arcsin x

arctan 22

0.600 0.601 0: 602 0.603 0.604

0.64350 0.64475 0.64600 0.64725 0.64851

11087 93 16956 07 34595 63 64059 60 05401 26

0.54041 0.54115 0.54188 0.54262 0.54335

95002 71 44700 04 87910 15 24633 69 54871 37

0.650 0.651 0.652 0.653 0.654

0.70758 0.70890 0.71021 0.71153 0.71285

44367 25 10818 82 92154 53 88447 93 99773 14

0.57637 0.57707 0.57777 0.57848 0.57918

52205 91 78870 95 99113 37 12935 07 20337 94

0.605 0.606 0.607 0.608 0.609

0.64976 0.65102 0.65228 0.65353 0.65479

58674 24 23932 51 01230 34 90622 38 92163 58

0.54408 0.54481 0.54555 0.54628 0.54701

78623 92 95892 10 06676 70 10978 51 08798 38

0.655 0.656 0.657 0.658 0.659

0.71418 0.71550 0.71683 0.71815 0.71948

26204 76 67817 97 24688 45 96892 45 84506 75

0.57988 0.58058 0.58128 0.58197 0.58267

21323 94 15895 01 04053 13 85800 31 61138 57

0.610 0.611 0.612 0.613 0.614

0.65606 0.65732 0.65858 0.65985 0.66111

05909 25 31915 05 70237 00 20931 44 84055 09

0.54774 0.54846 0.54919 0.54992 0.55065

00137 16 84995 75 63375 05 35276 01 00699 59

0.660 0.661 0.662 0.663 0.664

0.72081 0.72215 0.72348 0.72481 0.72615

87608 70 06276 21 40587 76 90622 40 56459 74

0.58337 0.58406 0.58476 0.58545 0.58615

30069 94 92596 49 48720 31 98443 49 41768 17

0.615 0.616 0.617 0.618 0.619

0.66238 0.66365 0.66492 0.66619 0.66746

59665 02 47818 67 48573 84 61988 69 88121 78

0.55137 0.55210 0.55282 0.55354 0.55427

59646 79 12118 61 58116 10 97640 33 30692 38

0.665 0.666 0.667 0.668 0.669

0.72749 0.72883 0.73017 0.73151 0.73286

38180 01 35864 02 49593 16 79449 44 25515 49

0.58684 0.58754 0.58823 0.58892 0.58961

78696 50 09230 63 33372 77 51125 11 62489 89

0.620 0.621 0.622 0.623 0.624

0.66874 0.67001 0.67129 0.67257 0.67385

27032 02 78778 71 43421 53 21020 54 11636 20

0.55499 0.55571 0.55643 0.55715 0.55787

57273 39 77384 48 91026 82 98201 62 98910 07

0.670 0.671 0.672 0.673 0.674

0.73420 0.73555 0.73690 0.73825 0.73961

87874 53 66610 44 61807 69 73551 41 01927 39

0.59030 0.59099 0.59168 0.59237 0.59306

67469 35 66065 77 58281 44 44118 66 23579 77

0.625 0.626 0.627 0.628 0.629

0.67513 0.67641 0.67769 0.67898 0.68026

15329 37 32161 29 62193 62 05488 41 62108 12

0.55859 0.55931 0.56003 0.56075 0.56147

93153 44 80932 97 62249 97 37105 74 05501 63

0.675 0.676 0.677 0.678 0.679

0.74096 0.74232 0.74367 0.74503 0.74639

47022 03 08922 43 87716 32 83492 13 96338 96

0.59374 0.59443 0.59512 0.59580 0.59649

96667 11 63383 05 23729 99 77710 32 25326 49

0.630 0.631 0.632 0.633 0.634

0.68155 0.68284 0.68413 0.68542 0.68671

32115 63 15574 24 12547 66 23100 04 47295 93

0.56218 0.56290 0.56361 0.56433 0.56504

67439 00 22919 24 71943 75 14513 97 50631 37

0.680 0.681 0.682 0.683 0.684

0.74776 0.74912 0.75049 0.75186 0.75323

26346 60 73605 52 38206 91 20242 68 19805 42

0.59717 0.597&6 0.59854 0.59922 0.59990

66580 93 01476 11 30014 52 52198 66 68031 06

0.635 0.636 0.637 0.638 0.639

0.68800 0.68930 0.69060 0.69189 0.69319

85200 35 36878 74 02396 97 81821 37 75218 73

0.56575 0.56647 0.56718 0.56789 0.56860

80297 42 03513 63 20281 53 30602 67 34478 63

0.685 0.686 0.687 0.688 0.689

0.75460 0.75597 0.75735 0.75872 0.76010

36988 49 71885 95 24592 63 95204 10 83816 68

0.60058 0.60126 0.60194 0.60262 0.60330

77514 26 80650 81 77443 31 67894 35 52006 54

0.640 0.641 0.642 0.643 0.644

0.69449 0.69580 0.69710 0.69840 0.69971

82656 27 04201 68 39923 13 89889 23 54169 09

0.56931 0.57002 0.57073 0.57143 0.57214

31911 01 22901 42 07451 52 85562 98 57237 47

0.690 0.691 0.692 01693 0.694

0.76148 0.76287 0.76425 0.76564 0.76703

90527 48 15434 36 58636 00 20231 84 00322 15

0.60398 0.60466 0.60533 0.60601 0.60668

29782 53 01224 96 66336 52 25119 88 77577 76

0.645 0.646 0.647 0.648 0.649

0.70102 0.70233 0.70364 0.70495 0.70626

32832 27 25948 84 33589 34 55824 80 92726 76

0.57285 0.57355 0.57426 0.57496 0.57567

22476 73 81282 48 33656 48 79600 51 19116 38

0.695 0.696 0.697 0.698 0.699

0.76841 0.76981 0.77120 0.77260 0.77399

99008 00 16391 29 52574 75 07661 95 81757 30

0.60736 0.60803 0.60870 0.60938 0.61005

23712 89 63528 01 97025 88 24209 28 45081 01

0.650

0.70758 44367 25 (-;I2

[

1

0.700 0.7753;-WY;66 11 (-JP [ I- ;=1.57079 6326795 [ 5 1

0.57637 52205 91

0.61072 59643 89

[c-y3 1

210

ELEMENTARY INVERSE

Table 4.14 X

arcsin x

TRANSCENDENTAL

CIRCULAR

SINES

arctan x

FUNCTIONS

AND TANGENTS arcsin z

X

arctan x

0.700 0.701 0.702 0.703 0.704

0.77539 0.77679 0.77820 0.77960 0.78101

74966 11 87394 52 19149 57 70339 20 41072 23

0.61072 0.61139 0.61206 0.61273 0.61340

59643 89 67900 75 69854 44 65507 83 54863 79

0.750 0.751 0.752 0.753 0.754

0.84806 0.84957 0.85109 0.85260 0.85413

20789 81 52355 56 10007 70 93916 63 04254 45

0.64350 0.64414 0.64477 0.64541 0.64605

11087 93 08016 53 98804 75 83456 20 61974 52

0.705 0.706 0.707 0.708 0.709

0.78242 0.78383 0.78524 0.78666 0.78807

31458 43 41608 47 71633 95 21647 44 91762 45

0.61407 37925 25 Or61474 14695 10 0.61540 85176 29 Oi61607 49371 78 0.61674 07284 52

0.755 0.756 0.757 0.758 0.759

0.85565 0.85718 0.85870 0.86024 0.86177

41195 04 04914 02 95588 84 13398 74 58524 85

0.64669 0.64733 0.64796 0.64860 0.64923

34363 37 00626 40 60767 30 14789 75 62697 45

0.710 0.711 0.712 0.713 0.714

0.78949 0.79091 0.79234 0.79376 0.79519

82093 46 92755 96 23866 39 75542 24 47901 99

0.61740 0.61807 0.61873 0.61939 0.62006

58917 52 04273 76 43356 27 76168 09 02712 26

0.760 0.761 0.762 0.763 0.764

0.86331 0.86485 0.86639 0.86794 0.86949

31150 16 31459 55 59639 86 15879 89 00370 42

0.64987 04494 12 Oi65050 40183 48 0.65113 69769 28 0.65176 93255 25 0.65240 10645 18

0.715 0.716 0.717 0.718 0.719

0.79662 41065 16 0.79805 55152 32 0.79948 90285 08 Oi80092 46586 13 0.80236 24179 26

0.62072 0.62138 0.62204 0;62270 0.62336

22991 86 37009 97 44769 70 46274 14 41526 45

0.765 0.766 0.767 0.768 0.769

0.87104 0.87259 0.87415 0.87571 0.87727

13304 26 54876 26 25283 38 24724 65 53401 29

0.65303 0.65366 0.65429 0.65492 0.65555

21942 83 27151 99 26276 46 19320 05 06286 59

0.720 0.721 0.722 0.723 0.724

0.80380 0.80524 0.80668 0.80813 0.80958

23189 33 43742 33 85965 35 49986 66 35935 64

0.62402 0.62468 0.62533 0.62599 0.62665

30529 77 13287 26 89802 10 60077 48 24116 63

0.770 0.771 0.772 0.773 0.774

0.87884 0.88040 0.88198 0.88355 0.88513

11516 69 99276 42 16888 33 64562 55 42511 51

0.65617 0.65680 0.65743 0.65805 0.65868

87179 91 62003 87 30762 31 93459 11 50098 15

0.725 0.726 0.727 0.728 0.729

0.81103 0.81248 0.81394 0.81540 0.81685

43942 88 74140 11 26660 28 01637 58 99207 37

0.62730 0.62796 0.62861 0.62927 0.62992

81922 76 33499 11 78848 95 17975 54 50882 17

0.775 0.776 0.777 0.778 0.779

0.88671 0.88829 0.88988 0.89147 0.89306

50950 00 90095 19 60166 70 61386 58 93979 43

0.65931 0.65993 0.66055 0.66118 0.66180

00683 33 45218 55 83707 72 16154 79 42563 67

0.730 or731 0.732 0.733 0.734

0.81832 0.81978 0.82125 0.82272 0.82419

19506 32 62672 31 28844 52 18163 44 30770 85

0.63057 77572 15 Oi63122 98048 79 0.63188 12315 41 0.63253 20375 38 0.63318 22232 04

0.780 0.781 0.782 0.783 0.784

0.89466 0.89626 0.89786 0.89947 0.90108

58172 34 54195 03 82279 83 42661 72 35578 41

0.66242 0.66304 0.66366 0.66428 0.66490

62938 33 77282 73 85600 83 87896 62 84174 09

0.735 0.736 0.737 0.738 0.739

0.82566 0.82714 0.82862 0.83010 0.83158

66809 86 26424 94 09761 92 16968 01 48191 83

0.63383 0.63448 0.63512 0.63577 0.63642

17888 78 07348 99 90616 06 67693 42 38584 50

0.785 0.786 0.787 0.788 0.789

0.90269 61270 38 Oi90431 19980 87 0.90593 11956 01 0.90755 37444 80 0.90917 96699 17

0.66552 74437 26 Oi66614 58690 12 0.66676 36936 71 0.66738 09181 07 0.66799 75427 24

0.740 0.741 0.742 0.743 0.744

0.83307 0.83455 0.83604 0.83754 0.83903

03583 42 83294 24 87477 24 16286 83 69878 93

0.63707 03292 76 Oi63771 61821 64 0.63836 14174 63 Oi6i900 60355 21 0.63965 00366 89

0.790 0.791 0.792 0.793 0.794

0.91080 89974 07 0.91244 17527 48 0.91407 79620 46 Oi91571 76517 23 0.91736 08485 19

0.66861 35679 28 Oi66922 89941 25 0.66984 38217 24 Oi67045 80511'32 0.67107 16827 61

0.745 0.746 0.747 0.748 0.749

0.84053 48410 98 Oi84203 52041 95 0.84353 80932 39 0.84504 35244 42 0.84655 15141 77

0.64029 34213 19 Oi64093 61897 63 0.64157 83423 76 0.64221 98795 14 0.64286 08015 33

0.795 0.796 0.797 0.798 0.799

0.91900 75795 02 Oi92065 78720 67 0.92231 17539 49 0.92396 92532 24 0.92563 03983 15

0.67168 47170 20 Oi67229 71543 22 0.67290 89950 79 0.67352 02397 05 0.67413 08886 15

0.750

0.84806 20789 81

0.64350 11087 93

0.800

0.92729 52180 02

0.67474 09422 24

;=1.57079 6326795

[ c-:15 1

[I(-f)8 1

211

ELEMENTARY TRANSCENDENTAL FUNCTIONS INVERSE X

\

arcsin x

CIRCULAR

SINES

arctan x

AND

TANGENTS

Table

arcsin x

X

4.14

arctan x

0.800 O.SOl 0.802 0.303 0.804

0.92729 0.92896 0.93063 0.93231 0.93399

52180 02 37414 22 59980 83 20178 64 18310 25

0.67474 0.67535 0.67595 0.67656 0.67717

09422 24 04009 49 92652 08 75354 19 52120 01

0.850 0.851 0.852 0.853 0.854

1.01598 1.01788 1.01979 1.02170 1.02362

52938 15 65272 25 36361 62 66824 41 57289 29

0.70449 0.70507 0.70565 0.70623 0.70681

40642 42 43293 58 40219 63 31425 16 16914 73

0.805 0.806 0.807 0.808 0.809

0.93567 0.93736 0.93905 0.94074 0.94244

54682 12 29604 66 43392 28 96363 49 88840 95

0.67778 22953 77 0.67838 87859 65 0.67899 46841 90 0.67959 99904 74 Oi68020 47052 41

0.855 0.856 0.857 0.858 0.859

1.02555 1.02748 1.02941 1.03136 1.03331

08395 76 20794 40 95147 10 32127 41 32420 77

0.70738 0.70796 0.70854 0.70912 0.70969

96692 96 70764 42 39133 73 01805 50 58784 34

0.810 0.811 0.812 0.813 0.814

0.94415 0.94585 0.94757 0.94928 0.95100

21151 54 93626 48 06601 38 60416 29 55415 87

0.68080 0.68141 0.68201 0.68261 0.68321

88289 16 23619 25 53046 96 76576 55 94212 31

0.860 0.861 0.862 0.863 0.864

1.03526 1.03723 1.03920 1.04117 1.04316

96724 81 25749 68 20218 39 80867 05 08445 30

0.71027 0.71084 0.71141 0.71199 0.71256

10074 87 55681 72 95609 52 29862 92 58446 55

0.815 0.816 0.817 0.818 0.819

0.95272 0.95445 0.95618 0.95792 0.95966

91949 40 70370 88 91039 18 54318 04 60576 23

0.68382 0.68442 0.68502 0.68562 0.68621

05958 54 11819 54 11799 62 05903 10 94134 31

0.865 0.866 0.867 0.868 0.869

1.04515 1.04714 1.04915 1.05116 1.05317

03716 61 67458 63 00463 62 03538 76 77506 61

0.71313 0.71370 0.71428 0.71485 0.71542

81365 07 98623 14 10225 41 16176 56 16481 25

0.820 0.821 0.822 0.823 0.824

0.96141 0.96316 0.96491 0.96667 0.96843

10187 64 03531 36 40991 79 22958 76 49827 60

0.68681 0.68741 0.68801 0.68860 0.68920

76497 59 52997 28 23637 73 88423 31 47358 39

0.870 0.871 0.872 0.873 0.874

1.05520 1.05723 1.05927 1.0613! 1.0633

23205 49 41489 91 33231 01 99317 03 40653 78

0.71599 0.71656 0.71712 0.71769 0.71826

11144 16 00169 99 83563 41 61329 12 33471 82

0.825 0.826 0.827 0.828 0.829

0.97020 0.97197 0.97375 0.97553 0.97731

21999 29 39880 56 03884 00 14428 17 71937 77

0.68980 0.69039 0.69098 0.69158 0.69217

00447 34 47694 55 89104 41 24681 33 54429 71

0.875 0.876 0.877 0.878 0.879

1.06543 1.06750 1.06958 1.07166 1.07376

58165 11 52793 43 25500 24 77266 67 09094 07

0.71882 0.71939 0.71996 0.72052 0.72109

99996 22 60907 02 16208 94 65906 70 10005 03

0.830 0.831 0.832 0.833 0.834

0.97910 0.98090 0.98270 0.98450 0.98631

76843 68 29583 19 30600 05 80344 64 79274 13

0.69276 0.69335 0.69395 0.69454 0.69513

78353 97 96458 54 08747 85 15226 33 15898 44

0.880 0.881 0.882 Oi883 0.884

1.07586 1.07797 1.08008 liO8221 1.08435

22004 54 17041 59 95270 75 57780 22 05681 59

0.72165 0.72221 0.72278 0.72334 0.72390

48508 65 81422 30 08750 71 30498 64 46670 83

0.835 0.836 0.837 0.838 0.839

0.98813 0.98995 0.99177 0.99360 0.99544

27852 56 26551 06 75847 95 76228 94 28187 22

0.69572 10768 63 0;69630 99841 36 0.69689 83121 11 0169748 60612 34 0.69807 32319 55

0.885 OI886 0.887 0.888 0.889

1.08649 1.08864 1.09080 1.09297 1.09515

40110 49 62227 36 73218 22 74295 43 66698 56

0.72446 0.72502 0.72558 0.72614 0.72670

57272 04 62307 01 61780 53 55697 34 44062 23

0.840 0.841 0.842 0.843 0.844

0.99728 0.99912 1.00097 1.00283 1.00469

32223 72 88847 18 98574 39 61930 35 79448 46

0.69865 0.69924 0.69983 0.70041 0.70100

98247 21 58399 85 12781 94 61398 02 04252 59

0.890 0.891 0.892 0.893 0.894

1.09734 1.09954 1.10175 1.10396 1.10619

51695 23 30581 99 04685 30 75362 43 44002 56

0.72726 0.72782 0.72837 0.72893 0.72949

26879 97 04155 34 75893 12 42098 11 02775 09

0.845 0.846 0.847 0.848 Oi849

1.00656 1.00843 1.01031 1.01220 1.01408

51670 67 79147 75 62439 41 02114 56 98751 50

0.70158 0;70216 0.70274 0.70333 0.70391

41350 19 72695 35 98292 60 18146 49 32261 58

0.895 0.896 0.897 0.898 0.899

1.10843 1.11067 1.11293 1.11520 1.11748

12027 75 80894 12 52092 94 27151 85 07636 13

0.73004 0.73060 0.73115 0.73170 0.73226

57928 87 07564 24 51686 02 90299 00 23408 01

0.850

1.01598 52938 15 (-[)7

0.70447~4'$42 42

0.900

1.11976 95149 99

[

1

[

4

1

.[

;=1.57079 6326795

I,-6)1 6

1

0.73281 51017 87

1C-l)7 1

212 Table

ELEMENTARY 4,.14

TRANSCENDENTAL

INVERSE

arcsin J

0.9”00 1.11976 95149 99

arctan .t

87 38 38 70 lb

CIRCULAR

SINES

O.&O

0.951 0.952 0.953 0.954

1.25323 1.25645 1.25970 1.26298 1.26630

AND

TANGENTS

arcsin.t

f (2)

0.901 0.902 0.903 0.904

1.12206 1.12437 1.12670 1.12903

91337 97886 lb524 49026

93 21 29 45

0.905 0.906 0.907 0.908 0.909

1.13137 1.13373 1.13610 1.13848 1.14087

97213 62953 48166 99 54823 84946 E

zl

73557 73612 73666 73721 73776

06748 01464 90715 74506 52841

0.955 0.956 0.957 0.958 0.959

1.26965 1.27304 1.27647 1.27994 1.28345

97812 97667 76222 46878 23838

42 20 92 88 00

0.76238 0.76290 0.76342 0.76395 0.76447

43244 70690 92916 09927 21729

37 08 23 81 78

1.00378 1.00370 1.00361 1.00353 1.00344

84851 34492 84523 34944 85754

78 58 57 39 69

0.910 0.911 0.912 0.913 0.914

1.14328 40618 1.14570 23976 2 1.14813 37219 91 1.15057 82610 10 1.15303 62474 12

73831 73885 73940 73995 74049

25725 93163 55160 11721 62850

0.960 0.961 0.962 0.963 0.964

1.28700 1.29059 1.29423 1.29792 1.30165

22175 57917 48124 10987 65939

87 69 14 43 20

0.76499 0.76551 0.76603 0.76655 0.76707

28327 29724 25927 16941 02769

11 78 75 02 55

1.00336 1.00327 1.00319 1.00310 1.00302

36954 88542 40518 92883 45635

10 28 88 53 89

0.915 0.916 0.917 0.918 0.919

1.15550 1.15799 1.16049 1.16300 1.16553

79206 35274 33215 75647 65266

90 19 50 25 04

0.74104 08553 83 0.74158 48835 32 0.74212 83700 10 0.74267 13153 04 0.74321 37199 05

0.965 0.966 0.967 0.968 0.969

1.30544 33771 97 1.30928 36776 35 1.31317 98896 52 1.31713 45907 19 1.32115 05615 54

0.76758 0.76810 0.76862 0.76913 0.76965

83418 58892 29196 94335 54315

33 33 53 92 49

1.00293 1.00285 1.00277 1.00268 1.00260

98775 52302 06215 60515 15201

61 33 71 39 02

0.920 0.921 0.922 0.923 0.924

1.16808 1.17063 1.17321 1.17580 1.17841

04852 97273 45487 52550 21615

14 lb 95 71 31

0.74375 0.74429 0.74483 0.74537 0.74591

99 76 25 35 97

0.970 0.971 0.972 0.973 0.974

1.32523 08092 80 1.32937 85940 93 1.33359 74601 02 1.33789 12711 79 1.34226 42528 47

0.77017 0.77068 0.77120 0.77171 0.77222

09140 58815 03345 42735 76990

20 06 05 14 34

1.00251 1.00243 1.00234 1.00226 1.00217

70272 25728 81570 37796 94406

25 74 13 07 23

0.925 0.926 0.927 0.928 0.929

1.18103 1.18367 1.18633 1.18900 1.19170

55939 58892 33953 84725 14936

97' 09 44 71 35

0.74645

68203 00

0.74699

54537

6174753 35503 92 0.74807 11107 62 0.74860 81353 36

0.975 0.976 0.977 0.978 0.979

1.34672 1.35126 1.35590 1.36064 1.36549

10414 67425 69996 80777 69629

93 45 85 70 42

0.77274 0.77325 0.77376 0.77427 0.77478

06115 30116 48996 62761 71417

63 01 45 95 51

1.00209 51400 25 1.00201 08777 78 1.00192 66538 49 1.00184 24682 01 1.00175 83208 02

0.930 0.931 0.932 0.933 0.934

1.19441 1.19714 1.19989 1.20266 1.20544

28444 29249 21492 09472 97647

77 00 75 92 69

0.74914 0.74968 0.75021 0.75075 0.75128

46246 Ob 05790 63 59991 99 08855 06 52384 76

0.980 0.981 0.982 0.983 0.984

1.37046 1.37555 1.38077 1.38614 1.39167

14844 04644 39033 32129 15119

72 29 32 70 16

0.77529 0.77580 0.77631 0.77682 0.77733

74968 73418 66774 55040 38220

12 77 45 17 91

1.00167 42116 16 1.00159 01406 08 1.00150 61077 45 1.00142 21129 93 1.00133 81563 16

0.935 0.936 0.937 0.938 0.939

1.20825 1.21108 1.21394 1.21681 1.21971

90645 93272 10524 47598 09898

07 10 70 22 74

0.75181 0.75235 0.75288 0.75341 0.75394

90586 23463 51022 73268 90205

03 79 96 49 30

0.985 0.986 0.987 0.988 0.989

1.39737 1.40326 1.40937 1.41572 1.42233

40056 84832 59766 16538 60557

99 96 46 31 98

0.77784 16321 67 0.77834 a9347 44 0.77885 57303 23 0.77936 20194 04 0.77986 78024 85

1.00125 1.00117 1.00108 1.00100 1.00091

0.940 0.941 0.942 0.943 0.944

1.22263 1.22557 1.22854 1.23153 1.23455

03055 32932 05645 27575 05382

22 59 81 05 02

0.75448 0.75501 0.75554 0.75607 0.75659

01838 08172 09212 04964 95431

34 55 86 22 57

0.990 0.991 0.992

1.42925 1.43653 1.44422 1.45240 1.46119

68534 14207 07408 56012 69689

70 77 32 67 63

0.78037 0.78087 0.78138 0.78188 0.78238

30800 78526 21207 58848 91453

67 49 32 15 98

1.00083 52139 33 1.00075 15228 31 1.00066 78695 32 1.00058 42540 02 1.00050 06762 08

0.945 0,946 0.947 0,948 0.949

1.23759 1.24066 1.24376 1.24689 1.25004

46027 56791 45292 19509 87811

74 62 24 90 06

0.75712 0.75765 0.75818 0.75871 0.75923

80619 60534 35179 04559 68681

86 05 08 90 48

0.995 0.996 0.997 0.998 0.999

1.47075 46131 83 1.48132 37665 90 1.49331 72818 71 1.50754 02279.20 1.52607 12396 26

0.78289 0.78339 0.78389 0.78439 0.78489

19029 41580 59111 71627 79133

81 64 47 31 14

1.00041 1.00033 1.00025 1.00016 1.00008

0.950

1.25323

58975 03

0.75976 27548 76

1.000

1.57079

0.78539

81633 97

c 1

55842 69089 76944 79411 76495

35

58975 03 42223 06 47250 03

arcian z

0.73281 0.73336 0.73391 0.73447 0.73502

(-i)4 For arctan ,v,.~>l seeExample

51017 73133 a9759 00900 06562

FUNCTIONS

64000 67

84259

28

0.75976 27548 76 0.76028 81166 70 0.76081 29540 28 0.76133 72674 43 0.76186 10574 14

1.00421 1.00412 1.09404 1.00395 1.00387

42513 90197 38274 86742 35601

02 55 04 15 52

63267 95

71361 36336 01689 67417 33520

$=1.570796326795

80 52 98 82 72

15 91 01 11 89

1.000000000000 (-!)5

[ 1

22.

arcsin.x=;-[2(1-:t)]tJ(x)

42376 03570 65143 27096 89428

_

ELEMENTARY

TRANSCENDENTAL

HYPERBOLIC

sinh x 0.00000 0000

'

FUNCTIONS

Table

FUNCTIONS

cash I(: 1.00000 0000

tanh x 0.00000 000

4.15

coth x 00

0.01000 0.02000 0.03000 0.04001

0167 1333 4500 0668

1.00005 1.00020 1.00045 1.00080

0000 0007 0034 0107

0.00999 0.01999 0.02999 0.03997

967 733 100 868

0.05002 0.06003 0.07005 0.08008 0.09012

0836 6006 7181 5361 1549

1.00125 1.00180 1.00245 1.00320 1.00405

0260 0540 1001 1707 2734

0.04995 0.05992 0.06988 0.07982 0.08975

838 810 589 977 779

20.01666 16.68666 14.30904 12.52665 11.14109

0.10016 0.11022 0.12028 0.13036 0.14045

6750 1968 8207 6476 7782

1.00500 1.00605 1.00720 1.00846 1.00981

4168 6103 8644 1907 6017

0.09966 0.10955 0.11942 0.12927 0.13909

800 847 730 258 245

10.03331 11 9.12754 62 8.37329 50 7.73559 23 7118946 29

0.15 0. 16 0.17 0.18 0.19

0.15056 0.16068 0.17082 0.18097 0.19114

3133 3541 0017 3576 5232

1.01127 1.01282 1.01448 1.01624 1.01810

1110 7330 4834 3787 4366

0.14888 0.15864 0.16838 0.17808 0.18774

503 850 105 087 621

6.71659 6.30324 5.93891 5.61542 5.32633

18 25 07 64 93

0.20 0.21 0.22 0.23 0.24

0.20133 0.21154 0.22177 0.23203 0.24231

6003 6907 8966 3204 0645

1.02006 1.02213 1.02429 1.02656 1.02893

6756 1153 7764 6806 8506

0.19737 0.20696 0.21651 0.22602 0.23549

532 650 806 835 575

5.06648 4.83169 4.61855 4.42422 4.24636

96 98 23 37 11

0.25 0.26 0.27 0.28 0.29

0.25261 0.26293 0.27329 0.28367 0.29408

2317 9250 2478 3035 1960

1.03141 1.03399 1.03667 1.03945 1.04234

3100 0836 1973 6777 5528

0.24491 0.25429 0.26362 0.27290 0.28213

866 553 484 508 481

4.08298 3.93243 3.79326 3.66427 3.54440

82 24 93 77 49

0.30 0.31 0.32 0. 33 0.34

0.30452 0.31498 0.32548 0.33602 0.34658

0293 9079 9364 2198 8634

1.04533 1.04843 1.05163 1.05494 1.05835

8514 6035 8401 5931 8957

0.29131 0.30043 0.30950 0.31852 0.32747

261 710 692 078 740

3.43273 3.32848 3.23094 3.13951 3.05364

84 38 55 26 59

0.35 0.36 0.37 0.38 0.39

0.35718 0.36782 0.37850 0.38921 0.39996

9729 6544 0142 1590 1960

1.06187 1.06550 1.06923 1.07307 1.07701

7819 2870 4473 2999 8834

0.33637 0.34521 0.35399 0.36270 0.37136

554 403 171 747 023

2.97286 2.89675 2.82492 2.75704 2.69280

77 36 49 28 32

0.40 0.41 0.42 0.43 0.44

0.41075 0.42158 0.43245 0.44337 0.45433

2326 3767 7368 4214 5399

1.08107 1.08523 1.08950 1.09388 1.09837

2372 4018 4188 3309 1820

0.37994 0.38847 0.39693 0.40532 0.41364

896 268 043 131 444

2.63193 2.57418 2.51933 2.46717 2.41753

24 36 32 85 52

0.45 0.46 0.47 0. 48 0.49

0.46534 0.47639 0.48749 0.49864 0.50984

2017 5170 5962 5505 4913

1.10297 1.10767 1.11249 1.11742 1.12247

0169 8815 8231 8897 1307

0.42189 0.43008 0.43819 0.44624 0.45421

901 421 932 361 643

2.37023 2.32512 2.28206 2.24092 2.20159

55 60 66 84 36

0.05 0.06 0. 07 0.08 0.09

0.52109 5305

1.12762 5965

0.46211 716

100.00333 33 50.00666 65 33.34333 27 25.01333 19 39 19 00 53 49

2.16395 34

For coth 2, x 2 .l use 4.5.67. Compilation of tanh x and coth x from National Bureau of Standards, Table of circular and hyperbolic tangents and cotangents for radian arguments, 2d printing. Columbia Univ. Press, New York, N.Y., 1947 (with permission).

214

ELEMENTARY

FUNCTIONS

HYPERBOLIC

Table 4.15 2

TRANSCENDENTAL

cash x

sinh x

FUNCTIONS tanhx

0.50 0.51 0.52 0.53 0.54

0.52109 0.53239 0.54375 0.55516 0.56662

5305 7808 3551 3669 9305

1.12762 1.13289 1.13827 1.14376 1.14937

5965 3387 4099

0.55 0.57 0. 58 0.59

0.57815 0.58973 0.60137 0.61307 0.62483

1604 1718 0806 0032 0565

1.15510 1.16094 1.16689 1.17296 1.17915

1414 0782 6245 8399 7850

0.60 0.61 0.62 0.63 0.64

0.b3665 0.64854 0.66049 0.67250 0.68459

3582 0265 1802 9389 4228

1.18546 1.19189 1.19843 1.20510 1.21188

5218 1134 6240 1190 6652

0.53704 0.54412 0.55112 0.55805

0.65 0.66 0. 67

7526 0500 4371 0370 9732

1.21879 1.22582 1.23297 1.24024 1.24764

3303 1834 2949

0.68 0.69

0.69674 0.70897 0.72126 0.73363 0.74606

0.70 0.71 0.72 0.73 0.74

0.75858 0.77117 0.78384 0.79658 0.80941

3702 3531 0477 5809 0799

0.75 0.76 0.77 0.78 0.79

8639

7557

0.46211 0.46994 0.47770 0.48538 0.49298

coth x 716 520 001 109 797

2.06023

68

2.02844 71 1.99792 1.96859 1.94039 1.91326 1.88716

13 14 39 98 42

957 710 803 222

55

955

1.86202 1.83780 1.81446 1.79194 1.77022

997 341 988 940 200

1.74926 1.72901 1.70946 1.69056 1.67229

10

5801

0.57166 0.57836 0.58497 0.59151 0.59798

1.25516 1.26281 1.27059 1.27849 1.28652

9006 7728 2733 4799 4715

0.60436 0.61067 0.61690 0.62306 0.62914

778

1.65462 1.63752 1.62098 1.60496 1.58945

lb 73

0.82231 6732 0.83530 4897 0.84837 6593 0.86153 3127 0.87477 5815

1.29468 1.30297 1.31138 1.31993 1.32862

3285 1324 9661 9138 0611

0.63514 895 0.64107 696

0.80 0.81 0.82 0.83 0.84

0.88810 0.90152 0.91503 0.92863 0.94232

5982 4960 4092 4727 8227

1.33743 1.34638 1.35546 1.36468 1.37403

4946 3026 5746 4013 8750

0.66403 677 0.66959 026 0.67506 0.68047

987

0. 85 0.86 0.87 0. 88 0.89

0.95611 0.96999 0.98397 0.99805 1.01223

5960 9306 9652 8397 6949

1.38353 1.39316 1.40293 1.41284 1.42289

0.90 0.91 0.92 0.93 0.94

1.02651 1.04089 1.05538 1.06997 1.08467

6726

0.95 0.96 0.97 0. 98 0.99

1.09948 1.11440 1.12943 I.14457 1.15982

1.00

1.17520 1194

0. 56

7362

0.50052 021 0.50797 743 0.51535 928

2.16395 34 2.12790 77 2.09336 40

0.52266

543

0.52989 561

0.56489

0.64692

683

930 535 516

945

0.65270 671 0.65840 904

59 04 70 62 67

05 16

11

38

81 83

1.57443'38 1.55987 51 1.54576 36 1.53208 17 1.51881 27

601 0.68580 906

1.50594 1.49345 1.48132 1.46955 1.45813

07 06 81 95 18

0892 1388 1201 1309 2702

0.69106 0.69625 0.70137 0.70641 0.71139

947 413 932 373

1.44703 1.43624 1.42577 1.41558 1.40569

25 99

6791

1.43308 1.44342 1.45390 1.46453 1.47530

6385 3379 4716 1444

0.71629 0.72113 0.72589 0.73059 0.73522

787 225 742 390 225

1.39606 iii8670 1.37760 1.36874 1.36013

73 82 51 95 29

4318 1794 0711 2572 8891

1.48622 1.49729 1.50851 1.51988 1.53140

5341 4680 3749

0.73978 0.74427 0.74870 0.75306 0.75736

305

1.35174 1.34358 1.33564 1.32790 1.32037

76

9155 5674 7734

[c-y1

4627

3670

5582

1.54308 0635 C-i’2

[

1

767

687

429 591 232

0.76159 416

[c-y9 1

26

98 13

60 08 50 20

1.31303 53 c-y

[

1

'

ELEMENTARY

TRANSCENDENTAL

HYPERBOLIC X

sinh x

FUNCTIONS

cash z

215

FUNCTIONS Table

tanhx

4.15

cothx

1.00 1.01 1.02 1.03 1.04

1.17520 1.19069 1.20629 1.22202 1.23788

1194 1018 9912 9437 1166

1.54308 1.55490 1.56689 1.57903 1.59133

0635 9997 4852 6398 5848

0.76159 0.76576 0.76986 0.77390 0.77788

416 202 654 834 807

1.31303 1.30588 1.29892 1.29214 1.28553

53 87 64 27 20

1. 05 1.06 1.07 1.08 1. 09

1.25385 1.26995 1.28618 1.30254 1.31902

6684 7589 5491 2013 8789

1.60379 1.61641 1.62919 1.64213 1.65524

4434 3400 4009 7538 5283

0.78180 0.78566 0.78946 0.79319 0.79687

636 386 122 910 814

1.27908 1.27280 1.26668 1.26071 1.25489

91 90 67 75 70

1.10 1.11 1.12 1.13 1.14

1.33564 1.35239 1.36928 1.38631 1.40347

7470 9717 7204 1622 4672

1.66851 1.68195 1.69556 1.70934 1.72329

8554 8678 6999 4878 3694

0.80049 0.80406 0.80756 0.81101 0.81441

902 239 892 926 409

1.24922 1.24368 1.23828 1.23301 1.22787

08 46 44 63 66

1.15 1.16 1.17 1.18 1.19

1.42077 1.43822 1.45581 1.47354 1.49142

8070 3548 2849 7732 9972

1.73741 1.75170 1.76617 1.78082 1.79565

4840 9728 9790 6471 1236

0.81775 0.82103 0.82427 0.82745 0.83057

408 988 217 161 887

1.22286 1.21796 1.21319 1.20852 1.20397

15 76 15 99 96

1.20 1.21 1.22 1.23 1.24

1.50946 1.52764 1.54597 1.56446 1.58311

1355 3687 8783 8479 4623

1.81065 1.82584 1.84120 1.85676 1.87249

5567 0966 8950 1057 8841

0.83365 0.83667 0.83965 0.84257 0.84545

461 949 418 933 560

1.19953 1.19520 1.19096 1.18683 1.18279

75 08 65 19 42

1.25 1.26 1.27 1.28 1.29

1.60191 1.62088 1.64001 1.65930 1.67875

9080 3730 0470 1213 7886

1.88842 1.90453 1.92084 1.93733 1.95402

3877 7757 2092 8513 8669

0.84828 0.85106 0.85379 0.85648 0.85912

364 411 765 492 654

1.17885 1.17499 1.17123 1.16756 1.16397

10 96 77 29 29

1.30 1.31 1.32 1.33 1.34

1.69838 1.71817 1.73814 1.75828 1.77859

2437 6828 3038 3063 8918

1.97041 1.98799 2.00527 2.02276 2.04044

4230 6884 8340 0324 4587

0.86172 0.86427 0.86678 0.86924 0.87167

316 541 393 933 225

1.16046 1.15703 1.15369 1.15041 1.14722

55 86 01 79 02

1.35 1.36 1.37 1.38 1.39

1.79909 1.81976 1.84062 1.86166 1.88288

2635 6262 1868 1537 7374

2.05833 2.07642 2.09472 2.11324 2.13196

2896 7039 8828 0090 2679

0.87405 0.87639 0.87869 0.88095 0.88317

329 307 219 127 089

1.14409 1.14104 1.13805 1.13513 1.13228

50 05 50 66 37

1.40 1.41 1.42 1.43 1.44

1.90430 1.92590 1.94770 1.96969 1.99188

1501 6060 3212 5135 4029

2.15089 2.17004 2.18941 2.20900 2.22881

8465 9344 7229 4057 1788

0.88535 0.88749 0.88959 0.89166 0.89369

165 413 892 660 773

1.12949 1.12676 1.12410 1.12149 1.11894

47 80 21 54 66

1.45 1.46 1.47 1.48 1.49

2.01427 2.03686 2.05965 2.08265 2.10586

2114 1627 4828 3996 1432

2.24884 2.26909 2.28958 2.31029 2.33123

2402 7902 0313 1685 4087

0.89569 0.89765 0.89957 0.90146 0.90332

287 260 745 799 474

1.11645 1.11401 1.11163 1.10930 1.10702

41 67 30 17 16

1.50

2.12927 9455 C-j)3

[

1

2.35240 9615 (-[)3

[

1

0.90514 825

[c-y1

1.10479 14 C-i’2

[

1

216

ELEMENTARY Table x

4.15

TRANSCENDENTAL HYPERBOLIC

sinh x

FUNCTIONS

FUNCTIONS

cash x

tanhx

coth x

1.50 1.51

1.52 1.53 1.54

2.12927 2.15291 2.17675 2.20082 2.22510

9455 0408 6654 0577 4585

2.35240 2.37382 2.39546 2.41735 2.43948

9615 0386 8541 6245 5686

0.90514 0.90693 0.90869 0.91042 0.91212

a25 905 766 459 037

1.10479 1.10260 1.10047 1.09838 1.09634

14 99 60 86 65

1.55 1.56 1.57 1.58 1.59

2.24961 2.27434 2.29930 2.32449 2.34991

1104 2587 1506 0357 1658

2.46185 2.48447 2.50734 2.53046 2.55383

9078 8658 6688 5455 7270

0.91378 0.91542 0.91702 0.91860 0.92014

549 046 576 189 933

1.09434 1.09239 1.09048 1.08861 1.08678

87 42 19 09 01

1.60 1.61 1. 62 1. 63 1.64

2.37556 2.40146 2.42759 2145397 2.48059

7953 1807 5809 2572 4735

2.57746 2.60134 2.62549 2.64990 2.67457

4471 9421 4508 2146 4777

0.92166 0.92316 0.92462 0.92606 0.92747

855 003 422 158 257

1.08498 1.08323 1.08152 1.07984 1.07819

87 58 04 18 90

1.65 1.66 1.67 1.68 1.69

2.50746 2.53458 2.56196 2.58959 2.61748

4959 5932 0366 0998 0591

2.69951 2.72472 2.75020 2.77596 2.80200

4868 4912 7431 4974 0115

0.92885 0.93021 0.93155 0.93286 0.93414

762 718 168 155 721

1.07659 1.07501 1.07347 1.07197 1.07049

13 78 77 04 51

1. 70 1. 71 1.72 1. 73 1.74

2.64563 2.67404 2.70273 2.73168 2.76091

1934 7843 1158 4749 1511

2.82831 2.85491 2.88179 2.90896 2.93643

5458 3635 7306 9159 1912

0.93540 0.93664 0.93786 0.93905 0.94022

907 754 303 593 664

1.06905 1.06763 1.06625 1.06489 1.06357

10 75 38 93 34

1.75 1.76 1.77 1.78 1.79

2.79041 2.82019 2.85026 2.88060 2.91124

4366 6265 0186 9136 6148

2.96418 2.99224 3.02059 3.04924 3.07820

8310 1129 3175 7283 6318

0.94137 0.94250 0.94360 0.94469 0.94576

554 301 942 516 057

1.06227 1.06100 1.05976 1.05854 1.05735

53 46 05 25 01

1.80 1.81 1.82 1.83 1.84

2.94217 2.97339 3.00491 3.03673 3.06886

4288 6648 6349 6545 0417

3.10747 3.13705 3.16694 3.19715 3.22767

3176 0785 2100 0113 7844

0.94680 0.94783 0.94883 0.94982 0.95079

601 la5 a42 608 514

1.05618 1.05503 1.05392 1.05282 1.05175

26 95 02 43 13

1.85 1.86 1.87 1.88 1.89

3.10129 3.13403 3.16708 3.20045 3.23414

1178 2071 6369 7378 8436

3.25852 3.28970 3.32121 3.35304 3.38522

8344 4701 0031 7484 0245

0.95174 0.95267 0.95359 0.95449 0.95537

596 884 412 211 312

1.05070 1.04967 1.04866 1.04767 1.04671

05 17 42 76 15

1.90 1.91 1. 92 1.93 1.94

3.26816 3.30250 3.33717 3.37218 3.40752

2912 4206 5754 1022 3510

3.41773 3.45058 3.48378 3.51732 3.55122

1531 4593 2716 9220 7460

0.95623 0.95708 0.95791 0.95873 0.95953

746 542 731 341 401

1.04576 1.04483 1.04393 1.04304 1.04217

53 88 14 28 25

1.95 1.96 1.97 1.98 1.99

3.44320 3.47923 3.51560 3.55233 3.58941

6754 4322 9816 6874 9168

3.58548 3.62009 3.65506 3.69040 3.72611

0826 2743 6672 6111 4594

0.96031 0.96108 0.96184 0.96258 0.96331

939 983 561 698 422

1.04132 1.04048 1.03966 1.03886 1.03808

02 55 79 72 29

2. 00

3.62686 0408 C-55)4

[

1

3.76219 5691 C-f)5

[

1

0.96402 758 (-J)4

1 1

1.03731 47 C-i’”

1 1

ELEMENTARY

TRANSCENDENTAL

HYPERBOLIC 2

sinh x

217

FUNCTIONS

FUNCTIONS

cash x

Table

4.15

coth x

tanh x

3.62686 4.02185 4.45710 4.93696 5.46622

0408 6742 5171 1806 9214

3.76219 4.14431 4.56790 5.03722 5.55694

5691 3170 8329 0649 7167

0.96402 0.97045 0.97574 0.98009 0.98367

75801 19366 31300 63963 48577

1.03731 47207 1.03044 77350 1.02485 98932 1.02030 78022 1.01659 60756

6.05020 6.69473 7.40626 8.19191 9.05956

4481 2228 3106 8354 1075

6.13228 6.76900 7.47346 8.25272 9.11458

9480 5807 8619 8417 4295

0.98661 0.98902 0.99100 0.99263 0.99396

42982 74022 74537 15202 31674

I.01356 1.01109 1.00907 1.00742 1.00607

73098 43314 41460 31773 34973

10.01787 11.07645 12.24588 13.53787 14.96536

4927 1040 3997 7877 3389

10.06766 11.12150 12.28664 13.57476 14.99873

1996 0242 6201 1044 6659

0.99505 0.99594 0.99668 0.99728 0.99777

47537 93592 23978 29601 49279

1.00496 1.00406 1.00332 1.00272 1.00223

98233 71152 86453 44423 00341

16.54262 18.28545 20.21129 22.33940 24.69110

7288 5361 0417 6861 3597

16.57282 18.31277 20.23601 22.36177 24.71134

4671 9083 3943 7633 5508

0.99817 0.99850 0.99877 0.99899 0.99918

78976 79423 82413 95978 08657

1.00182 1.00149 1.00122 1.00100 1.00081

54285 42872 32532 14040 98059

27.28991 30.16185 33.33566 36.84311 40.71929

7197 7461 7732 2570 5663

27.30823 30.17843 33.35066 36.85668 40.73157

2836 0136 3309 1129 3002

0.99932 0.99945 0.99955 0.99963 0.99969

92997 08437 03665 18562 85793

1.00067 1.00054 1.00044 1.00036 1.00030

11504 94581 98358 82794 15116

45.00301 49.73713 54.96903 60.75109 67.14116

1152 1903 8588 3886 6551

45.01412 49.74718 54.97813 60.75932 67.14861

0149 3739 3865 3633 3134

0.99975 0.99979 0.99983 0.99986 0.99988

32108 79416 45656 45517 91030

1.00024 1.00020 1.00016 1.00013 1.00011

68501 20992 54618 54666 09093

74.20321 82.00790 90.63336 100.16590 110.70094

0578 5277 2655 9190 9812

74.20994 82.01400 90.63887 100.17090 110.70546

8525 2023 9220 0784 6393

0.99990 0.99992 0.99993 0.99995 0.99995

92043 56621 91369 01692 92018

1.00009 1.00007 1.00006 1.00004 1.00004

08040 43434 08668 98333 07998

55-i 5: 8 5.9

122.34392 135.21135 149.43202 165.14826 182.51736

2746 4781 7501 6177 4210

122.34800 135.21505 149.43537 165.15129 182.52010

9518 2645 3466 3732 3655

0.99996 0.99997 0.99997 0.99998 0.99998

65972 26520 76093 16680 49910

1.00003 1.00002 1.00002 1.00001 1.00001

34040 73488 23912 83323 50092

6. 0

201.71315

7370

201.71563

6122

0.99998

77117

1.00001

22885

;*t

2:2 2.3 2.4 2.5 $2 2; 3. 0 ;=: 3:3 3. 4 E 3:7 Z t-10 4: 2 t5 4. 5 4.6 4. 7 44:: 5. 0 ;*: 5: 3 5.4 5.5

cc-y1

c 1 c-p

218 Table

ELEMENTARY

4.15

TRANSCENDENTAL

HYPERBOLIC

FUNCTIONS

FUNCTIONS

sinh x 201.71315 7370 222.92776 3607 246.37350 5831 272.28503 6911 300.92168 8157

coshx 201.71563 6122 222.93000 6475 246.37553 5262 272.28687 3215 300.92334 9715

tanhx 0.99998 77117 0.99998 99391 0.99999 17629 0.99999 32560 0.99999 44785

cothx 1.00001 22885 1.00001 00610 1.00000 82372 1.00000 67441 1.00000 55216

332.57006 367.54691 406.20229 448.92308 496.13685

4803 4437 7128 8938 3910

332.57156 367.54827 406.20352 448.92420 496.13786

8242 4805 8040 2713 1695

0.99999 0.99999 0.99999 0.99999 0.99999

54794 62988 69697 75190 79687

1.00000 1.00000 1.00000 1.00000 1.00000

45207 37012 30303 24810 20313

548.31612 605.98312 669.71500 740.14962 817.99190

3273 4694 8904 6023 9372

548.31703 605.98394 669.71575 740.15030 817.99252

5155 9799 5490 1562 0624

0.99999 0.99999 0.99999 0.99999 0.99999

83369 86384 88852 90873 92527

1.00000 1.00000 1.00000 1.00000 1.00000

16631 13616 11148 09127 07473

904.02093 999.09769 1104.17376 1220.30078 1348.64097

0686 7326 9530 3945 8762

904.02148 999.09819 1104.17422 1220.30119 1348.64134

3770 7778 2357 3680 9506

0.99999 0.99999 0.99999 0.99999 0.99999

93882 94991 95899 96642 97251

1.00000 1.00000 1.00000 1.00000 1.00000

06118 05009 04101 03358 02749

1490.47882 1647.23388 1820.47501 2011.93607 2223.53326

5790 5872 6339 2653 1416

1490.47916 1647.23418 1820.47529 2011.93632 2223.53348

1252 9411 0993 1170 6284

0.99999 0.99999 0.99999 0.99999 0.99999

97749 98157 98491 98765 98989

1.00000 1.00000 1.00000 1.00000 1.00000

02251 01843 01509 01235 01011

2457.38431 2715.82970 3001.45602 3317.12192 3665.98670

8415 3629 5338 7772 1384

2457.38452 2715.82988 3001.45619 3317.12207 3665.98683

1884 7734 1923 8505 7772

0.99999 0.99999 0.99999 0.99999 0.99999

99172 99322 99445 99546 99628

1.00000 1.00000 1.00000 1.00000 1.00000

00828 00678 00555 00454 00372

4051.54190 4477.64629 4948.56447 5469.00955 6044.19032

2083 5908 8852 8370 3746

4051.54202 4477.64640 4948.56457 5469.00964 6044.19040

5493 7574 9892 9795 6471

0.99999 0.99999 0.99999 0.99999 0.99999

99695 99751 99796 99833 99863

1.00000 1.00000 1.00000 1.00000 1.00000

00305 00249 00204 00167 00137

6679.86337 7382.39074 8158.80356 9016.87243 9965.18519

7405 8924 8366 6188 4028

6679.86345 7382.39081 8158.80362 9016.87249 9965.18524

2257 6653 9649 1640 4202

0.99999 0.99999 0.99999 0.99999 0.99999

99888 99908 99925 99939 99950

1.00000 1.00000 1.00000 1.00000 1.00000

00112 00092 00075 00061 00050

11013.23287 4703 Forx>>O,sinhx-coshx-i

11013.23292 0103 ez. Forx>lO, tanhx-l-2e-22,

0.99999 99959 * (-58)5

1.00000 00041 (-;I7

1 1 [cothx-1+2e-2zto 1 10D.

ELEMENTARY EXPONENTIAL

AND

X

erz

0. 00 0. 01

1.00000 00000 1.03191 46153 1.06484 77733

0.02 0. 03 0. 04

1.09883

19803

1.13390 07803

TRANSCENDENTAL

HYPERBOLIC e-“2

FUNCTIONS

1.00000 00000

219

FUNCTIONS

FOR

sinh zr 0.00000 00000 0.03142 10945

THE

ARGUMENT

cash ti

ti

Table

4.16

tanh ti

0.06287 0.09438 0.12599

32029 73698 47010

1.00000 1.00049 liOO197 1.00444 1.00790

59992 41813 98355 76792 32120

0.15772 0.18961 0.22168 0.25398 0.28652

63942 37699 83022 16502 56886

1.01236 1.01781 1.02427 1.03174 1.04023

23933 79512 81377 93294 89006

0.15580 0.18629 0.21643 0.24616 0.27544

03292 43856 36952 60434 21974 61929 55730 15776 92833 76928

0.96907 0.93910 0.91005 0.88191

24263 13674 72407 13783

00000 35208 45704 46105 60793

0.00000 0.06274 0.09396 0.12500

63906

00000

0.03140 55952 93000

97111

0.05 0.06 0. 07 0.08 0. 09

1.17008 87875 1.20743

17210

1.24596 64399 1.28573 09795 1.32676 45892

0.85463 0.82820 0.80258 0.77776 0.75371

0.10 0.11 0.12 0.13 0.14

1.36910 1.41280 1.45789 1.50441 1.55243

77706 23184 13610 94029 23694

0.73040 0.70781 0.68592 0.66470 0.64415

26910 31080 21659 82576 04440

0.31935 0.35249 0.38598 0.41985 0.45414

25398 46052 45975 55727 09627

1.04975 52308 1.06030 77132 1.07190 67634 2.08456

1.09829 14067

38303

0.30421 0.33244 0.36009 0.38711 0.41349

0.15 0.16 0.17 0.18 0.19

1.60197 1.65310 1.70586 1.76030 1.81648

76513 41518 23348 42750 37088

0.62422 0.60492 0.58621 0.56808 0.55051

84336 25628 37756 36059 41583

0.48887 0.52409 0.55982 0.59611 0.63298

46088 07945 42796 03346 47753

1.11310 1.12901 1.14603 1.16419 1.18349

30425 33573 80552 39405 89335

0.43919 0.46420 0.48848 0.51203 0.53484

97777 24748 66406 69673 18637

0.20 0.21 0.22 0.23 0.24

1.87445 60876 1.93427 86325 1.99601 03910

0.67048 0.70864 0.74750 0.78710 0.82747

39982 50169 54976 37973 90013

20893

22948 72203

80911 85988 93958 47001 92177

1.20397

2.05971 2.12544

0.53348 0.51698 0.50099 0.48550 0.47048

1.29796

82190

0.55689 0.57818 0.59872 0.61849 0.63751

33069 66683 05188 64181 86920

0.25 0.26 0. 27 0.28 0.29

2.19328 2.26327 2.33550 2.41004 2.48696

00507 77398 93782 62616 19609

0.45593 0.44183 0.42817 0.41492 0.40209

81278 70677 21192 97945 70227

0.86867 0.91072 0.95366 0.99755

09615 03361 86295 82336

1.32460 1.35255 1.38184 1.41248 1.44452

90893 74038 07487 80280 94918

0.65579 0.67333 0.69014 0.70624 0.72164

42026 21140 36583 19035 15276

0.30 0. 31 0. 32 0.33 0.34

2.56633 2.64823 2.73275 2.81996 2.90996

23952 59064 33366 81081 63054

0.38966 0.37760 0.36593 0.35461 0.34364

11374 98638 13069 39395 65907

1.08833 1.13531 1.18341 1.23267 1.28315

56289 30213 10148 70843 98573

1.47799 67663 1.51292 28851 1.54934 23218 1.58729

10238

0.73635 0.75041 0.76381 0.77659 0.78876

85995 03695 50706 17313 00021

0.35 0.36 0. 37 0.38 0. 39

3.00283 3.09867 3.19756 3.29961 3.40491

67606 11407 40381 30643 89460

0.33301 0.32271 0.31273 0.30306 0.29369

84355 89833 80681 58385 27474

1.33490 1.38797 1.44241 1.49827 1.55561

91626 60787 29850 36129 30993

1.66792 1.71069 1.75515 1.80133

75980 50620 10531 94514

0.80033 0.81135 0.82181 0.83175 0.84118

99933 21279 70068 52873 75743

0.40 0.41 0.42 0.43 0. 44

3.51358 3.62572 3.74143 3.86084 3.98405

56243 03579 38283 02496 74810

0.28460 0.27580 0.26727 0.25901 0.25100

95433 72607 72113 09757 03946

1.61448 1.67495 1.73707 1.80091 1.86652

80405 65486 83085 46370 85432

43239 57589 17947 19762 54241

0. 45 0.46 0. 47 0.48 0.49

4.11120 4.24241 4.37780 4.51752 4.66170

71429

0.24323 0.23571 0.22842 0.22136 0.21451

75614 48138 47266 01040 39731

2.00334 2.07469 2.14808 2.22359

07899 62194 93236 71557 61950

0.50

4.81047

47373 97717 58864 09873 73810

(-;I6 c

1

0.20787 95764 r( -p) 11 L 0 J

1.04243 24691

1.84930 58467 1.89909 75838 1.95076 38093 2.00435 2.05992 2.11752

55198 56127 89378

99617 25226 28912 35071

2.17722 2.23906 2.30311 2.36944 2.43810

23522 47756 72491 29952 74802

0.88828 0.89472 0.90081 0.90657 0.91201

89023

2.50917

84787

0.91715

[ 1 (d’3

1.62680 64481

0.85013 0.85861 0.86665 0.87426 0.88146

1.93398 47907

2.30129

li22563 36157 1.24850 48934 1.27260 84975

c-y1 [ Circular and 23357

Compiled from British Association for the Advancement of Science, Mathematical Tables, vol. 1. hyperbolic functions, exponential, sine and cosine integrals, factorial function and allied functions, Hermitian probability functions, 3d ed. Cambridge Univ. Press, Cambridge, England, 1951 (with permission). Known errors have been corrected.

220 Table

ELEMENTARY

EXPONENTIAL

4.16

AND

TRANSCENDENTAL

HYPERBOLIC

FUNCTIONS

FUNCTIONS

FOR

THE

ARGUMENT

m

0.50 0.51 0. 52 0.53 0. 54

4.81047 4.96400 5.12242 5.28590 5.45460

erz 73810 19160 61276 63869 40558

0.55 0.56 0.57 0.58 0. 59

5.62868 5.80832 5.99369 6.18497 6.38237

56460 29831 33767 97951 10460

0.17766 0.17216 0.16684 0.16168 0.15668

13694 67343 20350 20156 15832

2.72551 2.81807 2.91342 3.01164 3.11284

21383 81244 56709 88897 47314

2.90317 2.99024 3.08026 3.17333 3.26952

35077 48587 77058 09054 63146

0.93880 0.94242 0.94583 0.94904 0.95207

44259 38675 52160 97460 82009

0.60 0. 61 0.62 0. 63 0. 64

6.58606 6.79625 7.01315 7.23697 7.46794

19627 35967 34158 55091 07985

0.15183 0.14713 0.14258 0.13817 0.13390

58020 98890 92093 92710 57214

3.21711 3.32455 3.43528 3.54939 3.66701

30804 68538 21032 81191 75386

3.36894 3.47169 3.57787 3.68757 3.80092

88823 67428 13125 73901 32600

0.95493 0.95761 0.96014 0.96252 0.96477

08086 72978 69151 84417 02118

0.65 0. 66 0.67 0. 68 0.69

7.70627 7.95222 8.20601 8.46790 8.73815

72563 01304 21768 38986 37941

0.12976 0.12575 0.12186 0.11809 0.11444

43423 10461 18713 29793 06500

3.78825 3.91323 4.04207 4.17490 4.31185

64570 45422 51527 54597 65720

3.91802 4.03898 4.16393 4.29299 4.42629

07993 55883 70240 84390 72220

0.96688 0.96886 0.97073 0.97249 0.97414

01293 56859 39783 17255 52857

0.70 0.71 0.72 0.73 0. 74

9.01702 9.30480 9.60176 9.90819 10.22441

86109 36103 28381 94054 57779

0.11090 0.10747 0.10414 0.10092 0.09780

12784 13709 75422 65114 50993

4.45306 4.59866 4.74880 4.90363 5.06330

36663 61197 76480 64470 53393

4.56396 4.70613 4.85295 5.00456 5.16111

49447 74906 51901 29584 04386

0.97570 0.97716 0.97853 0.97983 0.98104

06726 35718 93563 31019 96015

0.75 0. 76 0.77 0. 78 0. 79

10.55072 10.88744 11.23491 11.59347 11.96347

40742 63743 50371 30285 42604

0.09478 0.09184 0.08900 0.08625 0.08358

02248 89025 82388 54299 77587

5.22797 5.39779 5.57295 5.75360 5.93994

19247 87359 33992 87993 32508

5.32275 5.48964 5.66196 5.83986 6.02353

21495 76384 16379 42292 10095

0.98219 0.98326 0.98427 0.98522 0.98612

33800 87071 96111 98912 31297

0.80 0. 81 0. 82 0.83 0.84

12.34528 12.73927 13.14584 13.56539 13.99832

39392 89270 81133 27988 70916

0.08100 0.07849 0.07606 0.07371 0.07143

25922 73785 96451 69955 71077

6.13214 6.33039 6.53488 6.74583 6.96344

06735 07743 92341 79017 49919

6.21314 6.40888 6.61095 6.81955 7.03488

32657 81528 88792 48972 20996

0.98696 0.98775 0.98849 Oi98919 0.98984

27033 17946 34022 b3509 53014

0.85 0.86 0.87 0. 88 0.89

14.44507 14.90608 15.38180 15.87271 16.37928

83157 74333 94795 40119 55735

0.06922 0.06708 0.06501 0.06300 0.06105

77313 66855 18571 11981 27239

7.18792 7.41950 7.65839 7.90485 8.15911

52922 03739 88112 64069 64248

7.25715 7.48658 7.72341 7.96785 8.22016

30235 70594 06683 76050 91487

0.99046 0.99103 0.99158 0.99209 0.99257

07591 90830 24938 30818 28142

0.90 0. 91 0.92 0.93 0.94

16.90202 17.44144 17.99808 18.57248 19.16521

41717 57711 28034 46925 83968

0.05916 0.05733 0.05556 0.05384 0.05217

45113 46965 14735 30919 78557

8.42142 8.69205 8.97126 9.25932 9.55652

98302 55373 06650 08003 02706

8.48059 8.74939 9.02682 9.31316 9.60869

43415 02338 21384 38922 81263

0.99302 0.99344 0.99384 0.99421 0.99456

35419 70066 48468 86036 97268

0.95 0.96 0.97 0.98 0. 99

19.77686 20.40804 21.05935 21.73145 22.42500

89693 01345 48847 60946 71560

0.05056 0.04900 0.04748 0.04601 0.04459

41212 02956 48354 62446 30738

9.86315 10.17951 10.50593 10.84271 11.19020

24240 99195 50247 99250 70411

9.91371 10.22852 10.55341 10.88873 11.23480

65453 02151 98601 61696 01149

0.99489 0.99520 0.99550 0.99577 0.99603

95797 94443 05263 39591 08084

1. 00

23.14069 26328 t-t)3

0.04321 39183 (-;I3

11.54873

93573 (-y [ 1

11.59195

X

[

1

95764 03654 99944 23136 13637

sinh 2.30129 2.38127 2.46360 2.54836 2.63563

TX 89023 57753 30666 20366 63461

cash TX 2.50917 84787 2.58272 61407 2.65882 30610 2.73754 43503 2.81896 77098

tanh 0.91715 0.92200 0.92657 0.93089 0.93496

~fl: 23357 08803 65378 34251 50714

e-rz

0.20787 0.20145 0.19521 0.18918 0.18333

1 1

32755 t-y [ 1

0.99627 20762 (-[I4

1 1

ELEMENTARY

TRANSCENDENTAL

INVERSE

221

FUNCTIONS

HYPERBOLIC

Table

FUNCTIONS

arcsinh x

4.17

arctanh z

arcsinh x 0.00000 0000

0.02 0.03 0. 04

9833 0.01999 8667 0.02999 5502 0.03998 9341

arctanh .E 0.00000 0000 0.01000 0.02000 0.03000 0.04002

0333 2667 9004 1353

o.x5o 0.51 0.52 0.53 0.54

0.48121 0.49013 0.49902 0.50788 0.51669

1825 8161 8444 2413 9824

0.54930 0.56272 0.57633 0.59014 0.60415

6144 9769 9754 5160 5603

0. 05 0. 06 0. 07 0. 08 0.09

0.04997 0.05996 0.06994 0.07991 0.08987

9190 4058 2959 4912 8941

0.05004 0.06007 0.07011 0.08017 0.09024

1729 2156 4671 1325 4188

0.55 0.56 0.57 0.58 0.59

0.52548 0.53422 0.54293 0.55159 0.56023

0448 4074 0505 9562 1077

0.61838 0.63283 0.64752 0.66246 0.67766

1313 3186 2844 2707 6068

0.10 0. 11 0. 12 0. 13 0.14

0.09983 0.10977 0.11971 0.12963 0.13954

4079 9366 3851 6590 6654

0.10033 0.11044 0.12058 0.13073 0.14092

5347 6915 1028 9850 5576

0.60 0.61 0.62 0.63 0.64

0.56882 0.57738 0.58589 0.59437 0.60282

4899 0892 8932 8911 0733

0.69314 0.70892 0.72500 0.74141 0.75817

7180 1359 5087 6144 3745

0.15 0.16 0.17 0.18 0.19

0.14944 0.15932 0.16919 0.17904 0.18887

3120 5080 1636 1904 5015

0.15114 0.16138 0.17166 0.18198 0.19233

0436 6696 6663 2689 7169

0.65 0.66 0.67 0.68 0.69

0.61122 0.61958 0.62791 0.63620 0.64445

4314 9584 6485 4970 5005

0.77529 0.79281 0.81074 0.82911 0.84795

8706 3631 3125 4038 5755

0.20 0.21 0.22 0.23 0.24

0.19869 0.20848 0.21826 0.22801 0.23775

0110 6350 2908 8972 3749

0.20273 0.21317 0.22365 0.23418 0.24477

2554 1346 6109 9466 4112

0.70 0.71 0.72 0.73 0.74

0.65266 0.66083 0.66897 0.67707 0.68512

6566 9641 4227 0332 7974

0.86730 0.88718 0.99764 0.92872 0.95047

0527 3863 4983 7364 9381

0.25 0.26 0:27 0.28 0.29

0.24746 0.25715 0.26682 0.27646 0.28608

6462 6349 2667 4691 1715

0.25541 0.26610 0.27686 0.28768 0.29856

2812 8407 3823 2072 6264

0.75 0.76 0.77 0.78 0.79

0.69314 0.70112 0.70907 0.71697 0.72484

7181 7988 0441 4594 0509

0.97295 0.99621 1.02032 1.04537 1.07143

5074 5082 7758 0548 1684

0.30 0.31 0.32 0.33 0.34

0.29567 0.30523 0.31477 0.32428 0.33376

3048 8020 5980 6295 8352

0.30951 0.32054 0.33164 0.34282 0.35409

9604 5409 7108 8254 2528

0.80 0.81 0.82 0.83 0.84

0.73266 0.74045 0.74820 0.75592 0.76359

8256 7912 9563 3300 9222

1.09861 1.12702 1.15681 1.18813 1.22117

2289 9026 7465 6404 3518

0.35 0.36 0.37 0. 38 0.39

0.34322 0.35264 0.36203 0.37140 0.38073

1555 5330 9121 2391 4624

0.36544 0.37688 0.38842 0.40005 0.41180

3754 5901 3100 9650 0034

0. 85 0.86 0.87 0.88 0.89

0.77123 0.77883 0.78640 0.79392 0.80141

7433 8046 1177 6950 5491

1.25615 1.29334 1.33307 1.37576 1.42192

2811 4672 9629 7657 5871

0. 40 0. 41 0.42 0.43 0. 44

0.39003 0.39930 0.40854 0.41774 0.42691

5320 4001 0208 3500 3454

0.42364 0.43561 0.44769 0.45989 0.47223

8930 1223 2023 6681 0804

0.90 0.91 0.92 0.93 0.94

0.80886 0.81628 0.82365 0.83100 0.83830

6936 1421 9091 0091 4575

1.47221 1.52752 1.58902 1.65839 1.73804

9490 4425 6915 0020 9345

0.45 0.46 0. 47 0. 48 0.49

0.43604 0.44515 0.45421 0.46325 0.47224

9669 1759 9359 2120 9713

0.48470 0.49731 0.51007 0.52298 0.53606

0279 1288 0337 4278 0337

0.95 0.96 0.97 0.98 0.99

0.84557 0.85280 0.86000 0.86716 0.87428

2697 4617 0498 0507 4812

1.83178 1.94591 2.09229 2.29755 2.64665

0823 0149 5720 9925 2412

0.50

0.48121 1825

0.54930 6144

1.00

0.88137 3587

co

cLxoo 0. 01 0.00999

For use of the table see Examples 26-28. Qo(x) (Legendre Function-Second

Kind)=arctanh x(/x!1) Compiled from Harvard Computation Laboratory, Tables of inverse hyperbolic functions. Harvard Univ. Press, Cambrid@, Mass., 1949 (with permission).

222

ELEMENTARY

Table 4.17

TRANSCENDENTAL

INVERSE

FUNCTIONS

HYPERBOLIC

FUNCTIONS

1.50 1.51 1.52 1.53 1.54

arcsinh x 1.19476 3217 1.20029 7449 1.20580 6263 1.21128 9840 1.21674 8362

arccosh x (S--l)+ 0.86081 788 0.85849 554 0.85618 806 0.85389 528 0.85161 706

968 798 099 841 994

1.55 1.56 1.57 1.58 1.59

1.22218 1.22759 1.23297 1.23833 1.24367

2008 0958 5390 5478 1400

0.84935 0.84710 0.84486 0.84264 0.84043

324 368 823 676 913

0.96794 0.96487 0.96182 0.95880 0.95579

529 415 625 131 904

1.60 1. 61 1.62 1.63 1.64

1.24898 1.25427 1.25953 1.26477 1.26999

3328 1436 5895 6877 4549

0.83824 0.83606 0.83389 0.83174 0.82960

520 483 788 424 376

6154 1729 5013 6208 5514

0.95281 0.94986 0.94692 0.94401 0.94111

918 146 561 139 853

1.65 1.66 1.67 1. 68 1.69

1.27518 1.28036 1.28550 1.29063 1.29573

9081 0639 9389 5495 9120

0.82747 0.82536 0.82326 0.82117 0.81909

632 179 005 097 443

1.01597 1.02235 1.02871 1.03503 1.04133

3134 9270 4123 7896 0792

0.93824 0.93539 0.93256 0.92975 0.92696

678 589 563 576 604

1.70 1.71 1. 72 1.73 1.74

1.30082 1.30587 1.31091 1.31593 1.32092

0427 9576 6727 2038 5666

0.81703 0.81497 0.81293 0.81091 0.80889

032 850 888 132 572

1.25 1.26 1.27 1.28 l-29

1.04759 1.05382 1.06002 1.06619 1.07233

3013 4760 6237 7645 9185

0.92419 0.92144 0.91871 0.91600 0.91331

624 613 550 411 175

1.75 1.76 1.77 1.78 1.79

1.32589 1.33084 1.33577 1.34068 1.34557

7767 8496 8006 6450 3978

0.80689 0.80489 0.80291 0.80095 0.79899

197 994 954 066 318

1.30 1. 31 1. 32 1.33 1. 34

1.07845 1.08453 1.09058 1.09661 1.10260

1059 3467 6610 0688 5899

0.91063 0.90798 0.90534 0.90272 0.90012

821 328 676 843 810

1.80 1.81 1.82 1.83 1.84

1.35044 1.35528 1.36011 1.36491 1.36970

0740 6886 2562 7914 3089

0.79704 0.79511 0.79318 0.79127 0.78937

701 203 816 527 328

1.35 1.36 1.37 1.38 1.39

1.10857 1.11451 1.12042 1.12630 1.13215

2442 0515 0317 2042 5887

0.89754 0.89498 0.89243 0.88990 0.88738

557 064 313 284 959

1.85 1.86 1.87 1.88 1.89

1.37446 1.37921 1.38393 1.38864 1.39333

8228 3477 8975 4863 1280

0.78748 0.78560 0.78373 0.78187 0.78002

209 160 170 231 334

1.40 1.41 1. 42 1.43 1.44

1.13798 1.14378 1.14955 1.15529 1.16101

2046 0715 2086 6351 3703

0.88489 0.88241 0.87995 0.87750 0.87507

320 348 026 336 261

1.90 1.91 1.92 1.93 1.94

1.39799 1.40264 1.40727 1.41188 1.41647

8365 6254 5083 4987 6099

0.77818 0.77635 0.77453 0.77272 0.77093

468 625 796 971 142

1. 45 1.46 1.47 1. 48 1. 49

1.16670 1.17236 1.17800 1.18361 1.18920

4331 8425 6174 7765 3384

0.87265 0.87025 0.86787 0.86550 0.86315

784 888 557 774 523

1.95 1.96 1.97 1.98 1.99

1.42104 1.42560 1.43013 1.43465 1.43915

8552 2476 8002 5259 4374

0.76914 0.76736 0.76559 0.76383 0.76208

300 437 544 612 633

1.50

1.19476 3217 t-f)4

2.00

1.44363 5475 (-i)3

0.76034

600

1. 00 1. 01 1. 02 1.03 1. 04

arcsinh x 0.88137 3587 0.88842 7007 0.89544 5249 0.90242 8496 0.90937 6928

arccosh x (227 1.00000 000 0.99667 995 0.99338 621 0.99011 848 0.98687 641

1.05 1.06 1.07 1. 08 1. 09

0.91629 0.92317 0.93001 0.93682 0.94360

0732 0094 5204 6251 3429

0.98365 0.98046 0.97730 0.97415 0.97103

1.10 1.11 1.12 1.13 1.14

0.95034 0.95705 0.96373 0.97037 0.97698

6930 6950 3684 7331 8088

1.15 1.16 1.17 1.18 1.19

0.98356 0.99011 0.99662 1.00310 1.00955

1.20 1.21 1.22 1.23 1.24

X

[ 1

0.86081 788 (-!)3 1

c 1

2

[I 1

[ 1 y2

ELEMENTARY

TRANSCENDENTAL

IUVERSE

arccosh z-ln I 0.62381 07164 0.62685 90940 0.62981 77884 0.63268 90778 0.63547 51194

t-1 0.50 0.49 0.48 0.47 0.46

arcsinh 0.75048 0.74839 0.74632 0.74428 0.74228

z-111 z 82946 16011 48341 85962 34908

0.45 0.44 0.43 0.42 0.41

0.74031 0.73836 0.73646 0.73458 0.73274

01215 90921 10057 64641 60676

0.63817 0.64079 0.64334 0.64580 0.64819

0.40 0.39 0. 38 0.37 0. 36

0.73094 0.72917 0.72743 0.72573 0.72407

04145 01001 57167 78524 70912

223

FUNCTIONS

HYPERBOLIC

Table

FUNCTIONS


2-I

arcsinh z-111 z 0.70841 81861 0.70724 57326 0.70611 72820 0.70503 32895 0.70399 41963

arccosh z-ln z 0.67714 27078 0.67842 57947 0.67965 18411 0.68082 14660 0.68193 52541

04288 23983 05002 51134 66000

0.68299 0.68399 0.68494 0.68584 0.68668

37571 74947 69555 25981 48518

z

0. 24 0.25

t 2

0.23 0.22 0.21

79566 95268 16670 61207 45429

:

0.19 0. 20

t

0.18 0.17

2

0.16

0.70300 0.70205 0.70115 0.70029 0.69948

0.65050 0.65274 0.65491 0.65701 0.65904

85051 95004 89477 81952 85249

3

0.15 3; 14 0.13 0.12 0.11

0.69872 0.69801 0.69734 0.69672 0.69615

53043 15527 56533 78946 85462

0.68747 0.68821 0.68889 0.68952 0.69010

41175 07683 51504 75836 83616

11555 72458 78974 41577 70226

3

0.10 0.09 0.08 0.07 0. 06

0.69563 0.69516 0.69474 0.69436 0.69404

78573 60572 33542 99357 59680

0.69063 0.69111 0.69154 0.69191 0.69224

77531 60018 33269 99235 59631

74382 63038 44732 27575 19258

:: 4

0.05 0.04 0.03 0.02 0. 01

0.69377 0.69354 0.69337 0.69324 0.69317

15954 69408 21047 71656 21796

0.69252 15938 0.69274 69403 0.69292 21046 0.69304 71656 0.69312 21796

4

0. 00

0.69314

71806

0.69314

0.35 0.34 0.33 0.32 0.31

0.72245 0.72086 0.71932 0.71781 0.71634

40117 91873 31846 65636 98766

0.66101 0.66290 0.66473 0.66650 0.66820

0.30 0.29 0.28 0.27 0.26

0.71492 0.71353 0.71219 0.71089 0.70963

36678 84725 48165 32154 41742

0.66984 0.67142 0.67294 0.67440 0.67580

0.25

0.70841 81861 C-56)5 [ 3

: :

3 : :

0.67714 27078 C-65) 1 [ I

0>=nearest

ROOTS

z,,, OF

integer

COL) r,, cash

:

4.73004 7.85320

46 07

i 5

10.99560 14.13716 17.27875

78 55 96

to .I’.

x,=1

For ~25, .J,,=; [2~+l]r

ROOTS

[ 1 C-56)6

a.,, OF

cos z,‘ eoah

II

%I

ii

4.69409 1.87510

41 11

3 z

7.85475 14.13716 10.99554

74 84 07

For )!>5, .z,,=: [2/t- 11~

x,,= - 1

*

4.17

71806

[ 1 t-y

Table

4.18

224

ELEMENTARY Table -A 0.00 0.05 0.10 0.15 0.20

3.14159 2.99304 2.86277 2.75032 2.65366

0.25 0.30 0.35 0.40 0.45

TRANSCENDENTAL

FUNCTIONS

ROOTS x uOF tan X )I : pt,

4.19 Xl

26

27

12.5?637 12.02503 11.70268 11.52018 11.40863

15.7%#96 15.06247 14.73347 14.56638 14.46987

18.84956 18.11361 17.79083 17.64009 17.55621

21.99115 21.17717 20.86724 20.73148 20.65782

25.;;274 24.25156 23.95737 23.83468 23.76928

28 ;;433 27:33519 27.05755 26.94607 26.88740

11.33482 11.28284 11.24440 11.21491 11.19159

14.40797 14.36517 14.33391 14.31012 14.29142

17.50343 17.46732 17.44113 17.42129 17.40574

20.61203 20.58092 20.55844 20.54146 20.52818

23.72894 23.70166 23.68201 23.66719 23.65561

26.85142 26.82716 26.80971 26.79656 26.78631

8.09616 8.07544 8.05794 8.04298 8.03004

11.17271 11.15712 11.14403 11.13289 11.12330

14.27635 14.26395 14.25357 14.24475 14.23717

17.39324 17.38298 17.37439 17.36711 17.36086

20.51752 20.50877 20.50147 20.49528 20.48996

23.64632 23.63871 23.63235 23.62697 23.62235

26.77809 26.77135 26.76572 26.76096 26.75688

4.97428 4.95930 4.94592 4.93389 4.92303

8.01875 8.00881 7.99999 7.99212 7.98505

11.11496 11.10764 11.10116 11.09538 11.09021

14.23059 14.22482 14.21971 14.21517 14.21110

17.35543 17.35068 17.34648 17.34274 17.33939

20.48535 20.48131 20.47774 20.47457 20.47172

23.61834 23.61483 23.61173 23.60897 23.60651

26.75333 26.75023 26.74749 26.74506 26.74288

4.91318

7.97867

11.08554

14.20744

17.33638

20.46917

23.60428

26.74092

4.932318 7.97867 4.90375 7.97258 4.89425 7.96648 4.88468 7.96036 4.87504 7.95422

ll'r$554 11:08110 11.07665 11.07219 11.06773

14 i:744 14:20395 14.20046 14.19697 14.19347

17.37638 17.33351 17.33064 17.32777 17.32490

20 4?917 20:46673 20.46430 20.46187 20.45943

23.6?428 23.60217 23.60006 23.59795 23.59584

26 7:092 26:73905 26.73718 26.73532 26.73345

1.93974 1.92035 1.90036 1.87976 1.85852

4.86534 4.85557 4.84573 4.83583 4.82587

7.94807 7.94189 7.93571 7.92950 7.92329

11.06326 11.05879 11.05431 11.04982 11.04533

14.18997 14.18647 14.18296 14.17946 14.17594

17.32203 17.31915 17.31628 17.31340 17.31052

20.45700 20.45456 20.45212 20.44968 20.44724

23.59372 23.59161 23.58949 23.58738 23.58526

26.73159 26.72972 26.72785 26.72598 26.72411

-0.50 -0.45 -0.40 -0.35 -0.30

1.83660 1.81396 1.79058 1.76641 1.74140

4.81584 4.80575 4.79561 4.78540 4.77513

7.91705 7.91080 7.90454 7.89827 7.89198

11.04083 11.03633 11.03182 11.02730 11.02278

14.17243 14.16892 14.16540 14.16188 14.15835

17.30764 17.30476 17.30187 17.29899 17.29610

20.44480 20.44236 20.43992 20.43748 20.43503

23.58314 23.58102 23.57891 23.57679 23.57467

26.72225 26.72038 26.71851 26.71664 26.71477

-0.25 -0.20 -0.15 -0.10 -0.05

1.71551 1.68868 1.66087 1.63199 1.60200

4.76481 4.75443 4.74400 4.73351 4.72298

7.88567 7.87936 7.87303 7.86669 7.86034

11.01826 11.01373 11.00920 11.00466 11.00012

14.15483 14.15130 14.14777 14.14424 14.14070

17.29321 17.29033 17.28744 17.28454 17.28165

20.43259 20.43014 20.42769 20.42525 20.42280

23.57255 23.57043 23.56831 23.56619 23.56407

26.71290 26.71102 26.70915 26.70728 26.70541

0.00 0.05 0.10 0.15 0.20

1.57080 1.53830 1.50442 1.46904 1.43203

4.71239 4.70176 4.69108 4.68035 4.66958

7.85398 7.84761 7.84123 7.83484 7.82844

10.99557 10.99102 10.98647 10.98192 10.97736

14.13717 14.13363 14.13009 14.12655 14.12301

17.27875 17.27586 17.27297 17.27007 17.26718

20.42035 20.41790 20.41545 20.41300 20.41055

23.56194 23.55982 23.55770 23.55558 23.55345

26.70354 26.70166 26.69979 26.69792 26.69604

0.25 0.30 0.35 0.40 0.45

1.39325 1.35252 1.30965 1.26440 1.21649

4.65878 4.64793 4.63705 4.62614 4.61519

7.82203 7.81562 7.80919 7.80276 7.79633

10.97279 10.96823 10.96366 10.95909 10.95452

14.11946 14.11592 14.11237 14.10882 14.10527

17.26428 17.26138 17.25848 17.25558 17.25268

20.40810 20.40565 20.40320 20.40075 20.39829

23.55133 23.54921 23.54708 23.54496 23.54283

26.69417 26.69230 26.69042 26.68855 26.68668

0.50 0.55 0.60 0.65 0.70

1.16556 1.11118 1.05279 0.98966 0.92079

4.60422 4.59321 4.58219 4.57114 4.56007

7.78988 7.78344 7.77698 7.77053 7.76407

10.94994 10.94537 10.94079 10.93621 10.93163

14.10172 14.09817 14.09462 14.09107 14.08752

17.24978 17.24688 17.24398 17.24108 17.23817

20.39584 20.39339 20.39094 20.38848 20.38603

23.54071 23.53858 23.53646 23.53433 23.53221

26.68480 26.68293 26.68105 26.67918 26.67730

0.75 0.80 0.85 0.90 0.95

0.84473 0.75931 0.66086 0.54228 0.38537

4.54899 4.53789 4.52678 4.51566 4.50454

7.75760 7.75114 7.74467 7.73820 7.73172

10.92704 10.92246 10.91788 10.91329 10.90871

14.08396 14.08041 14.07686 14.07330 14.06975

17.23527 17.23237 17.22946 17.22656 17.22366

20.38357 20.38112 20.37867 20.37621 20.37376

23.53008 23.52796 23.52583 23.52370 23.52158

26.67543 26.67355 26.67168 26.66980 26.66793

1.00

0.00000

4.49341

7.72525

10.90412

14.06619

17.22075

20.37130

23.51945

26.66605

6.28~19

9.4iT78

2.57043 2.49840 2.43566 2.38064 2.33208

5.99209 5.76056 5.58578 5.45435 5.35403 5.27587 5.21370 5.16331 5.12176

9.00185 8.70831 8.51805 8.39135 8.30293 8.23845 8.18965 8.15156 8.12108

0.50 0.55 0.60 0.65 0.70

2.28893 2.25037 2.21571 2.18440 2.15598

5.08698 5.05750 5.03222 5.01031 4.99116

0.75 0.80 0.85 0.90 0.95

2.13008 2.10638 2.08460 2.06453 2.04597

1.00

2.02876

x-1 -1.00 -0.95 -0.90 -0.85 -0.80

2.02876 2.01194 1.99465 1.97687 1.95857

-0.75 -0.70 -0.65 -0.60 -0.55

Xl

x3

For h-0, seejl. Sof Table 10.6.

<x>=nearest integer to X.

1: -1 1:

:

1

ELEMENTARY

TRANSCENDENTAL ROOTS

x,, OF

225

FUNCTIONS

cot

xn =Xx,,

Table

4.20

A

x1

22

53

24

0.00 0.05 0.10 0.15 0.20

1.57080 1.49613 1.42887 1.36835 1.31384

4.71239 4.49148 4.30580 4.15504 4.03357

7.85398 7.49541 7.22811 7.04126 6.90960

10.99557 10.51167 10.20026 10.01222 9.89275

14,13717 13.54198 13.21418 13.03901 12.93522

17.27876 16.58639 16.25936 16.10053 16.01066

x7 20.42035 19.64394 19.32703 19.18401 19.10552

23.5*6194 22.71311 22.41085 22.28187 22.21256

26 70354 25:79232 25.50638 25.38952 25.32765

0.25 0.30 0.35 0.40 0.45

1.26459 1.21995 1.17933 1.14223 1.10820

3.93516 3.85460 3.78784 3.73184 3.68433

6.81401 6.74233 6.68698 6.64312 6.60761

9.81188 9.75407 9.71092 9.67758 9.65109

12.86775 12.82073 12.78621 12.75985 12.73907

15.95363 15.91443 15.88591 15.86426 15.84728

19.05645 19.02302 18.99882 18.98052 18.96619

22.16965 22.14058 22.11960 22.10377 22.09140

25.28961 25.26392 25.24544 25.23150 25.22062

0.50 0.55 0.60 0.65 0.70

1.07687 1.04794 1.02111 0.99617 0.97291

3.64360 3.60834 3.57756 3.55048 3.52649

6.57833 6.55380 6.53297 6.51508 6.49954

9.62956 9.61173 9.59673 9.58394 9.57292

12.72230 12.70847 12.69689 12.68704 12.67857

15.83361 15.82237 15.81297 15.80500 15.79814

18.95468 18.94523 18.93734 18.93065 18.92490

22.08147 22.07333 22.06653 22.06077 22.05583

25.21190 25.20475 25.19878 25.19373 25.18939

0.75 0.80 0.85 0.90 0.95

0.95116 0.93076 0.91158 0.89352 0.57647

3.50509 3.48590 3.46859 3.45292 3.43865

6.48593 6.47392 6.46324 6.45368 6.44508

9.56331 9.55486 9.54738 9.54072 9.53473

12.67121 12.66475 12.65904 1?.65395 12.64939

15.79219 15.78698 15.78237 15.77827 15.77459

18.91991 18.91554 18.91168 18.90825 18.90518

22.05154 22.04778 22.04447 22.04151 22.03887

25.18563 25.18234 25.17943 25.17684 25.17453

1.00 0.86033

3.42562

6.43730

9.52933

12.64529

15.77128

18.90241

22.03650

25.17245

X5

X6

x7

25

%

29

Xl

22

53

x4

1.00 0.95 0.90 0.85 0.80

0.86033 0.84426 0.82740 0.80968 0.79103

3.42562 3.41306 3.40034 3.38744 3.37438

6.43730 6.42987 6.42241 6.41492 6.40740

9.52933 9.52419 9.51904 9.51388 9.50871

12.64529 12.64138 12.63747 12.63355 12.62963

15.77128 15.76814 15.76499 15.76184 15.75868

18.90241 18.89978 18.89715 18.89451 18.89188

22.03650 22.03424 22.03197 22.02971 22.02745

25.17245 25.17047 25.16848 25.16650 25.16452

0.75 0.70 0.65 0.60 0.55

0.77136 0.75056 0.72851 0.70507 0.68006

3.36113 3.34772 3.33413 3.32037 3.30643

6.39984 6.39226 6.38464 6.37700 6.36932

9.50353 9.49834 9.49314 9.48793 9.48271

12.62570 12.62177 12.61784 12.61390 12.60996

15.75553 15.75237 15.74921 15.74605 15.74288

18.88924 18.88660 18.88396 18.88132 18.87868

22.02519 22.02292 22.02066 22.01839 22.01612

25.16254 25.16055 25.15857 25.15659 25.15460

0.50 0.45 0.40 0.35 0.30

0.65327 0.62444 0.59324 0.55922 0.52179

3.29231 3.27802 3.26355 3.24891 3.23409

6.36162 6.35389 6.34613 6.33835 6.33054

9.47749 9.47225 9.46700 9.46175 9.45649

12.60601 12.60206 12.59811 12.59415 12.59019

15.73972 15.73655 15.73338 15.73021 15.72704

18.87604 18.87339 18.87075 18.86810 18.86546

22.01386 22.01159 22.00932 22.00705 22.00478

25.15262 25.15063 25.14864 25.14666 25.14467

2 2

0.25 0.48009 3.21910 6.32270 0.20 0.43284 3.20393 6.31485 0.15 0.37788 3.18860 6.30696 0.10 0.31105 3.17310 6.29906 0.05 0.22176 3.15743 6.29113

9.45122 9.44595 9.44067 9.43538 9.43008

12.58623 12.58226 12.57829 12.57432 12.57035

15.72386 15.72068 15.71751 15.71433 15.71114

18.86281 18.86016 18.85751 18.85486 18.85221

22.00251 22.00024 21.99797 21.99569 21.99342

25.14268 25.14070 25.13871 25.13672 25.13473

4

0.00 0.00000 3.14159 6.28319 * [c-y] ['-;'l]

9.42478 12.56637 15.70796 18.84956 21.99115 25.13274 [(-p] ['-y] ['-;"] [(-p] [y;"] ~[(-y'!

A-’

3%

59

=nearest integer to X. For h-l > .20, the maximum

error

in linear

interpolation

For A-l c .20,

I--&+&2-&3+..: *see page n.

3

is (- 4)7; five-point

interpolation

gives 5D.



: : 1 : H 2

z 3

G :oo Co

5. Exponential

Integral

WALTER

GAUTSCHI

and Related l AND WILLIAM

Functions

I?. CAHILL

2

Contents Page Mathematical Properties . . . . . . . . . . . . . . . . . . . . 5.1. Exponential Integral . . . . . . . . . . . . . . . . . .

5.2. Sine and Cosine Integrals Numerical

Methods

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . .

233 233

. . . . . . . . . . . . . . . . . . . . . . . . . .

235

5.3. Use and Extension References Table

of the Tables

5.1. Sine, Cosine and Exponential

x-‘Si(x),

x-2[Ci(x)-ln

cc-‘[Ei(x)-ln Si(x), Ci(x), Si(x), Ci(x), Table

. . . . . . . . . . . . . . . .

5.2.

Integrals

(O<sl 10) . . . .

238

x-r]

x-r],

x-‘[E,(x)+ln x+-r], x=0(.01).5, 10s 10D; Ei(x), El(x), 9D; x=.5(.01)2 10D; xP Ei(x), xez El(x), 9D; x=2(.1)10

Sine, Cosine and Exponential

Integrals

for Large Arguments

(101x1 a) . . . . . . . . . . . . . . . . . . . . . . . . . qf(x), 9D; x2g(x), 7D; ze-“Ei(x), 8D; xezEl(s), 10D f(x)=-si(x) cos x+Ci(x) sin 2, g(x)=-si(x) sin x-C;(x) co9 x x-‘=.l(-.005)0 Table

5.3.

Si(rx), Table

5.4.

228 228 231

Sine and Cosine Integrals

for Arguments

Gin(m), x=0(.1)10,

7D

Exponential

E,(x)

Integrals

243

TIZ (0 Ix 210) . . .

244

(0 5x 52) . . . . . . . . . .

245

E2(x)--2 In 2, E,(x), n=3,4, 10, 20, x=0(.01).5 E,(x), 12=2, 3, 4, 10, 20, x=.5(.01)2, 7D Table

5.5. Exponential Integrals E,(x) for Large Arg,urnents (2 Ix _< 03) a (x+n)e”E,(x), n=2, 3, 4, 10, 20, x-‘=.5(-.05).1(-.01>0, 5D

248

Table

5.6. Exponential Integral for Complex Arguments (jzj<29) zezEl(z), z=X+iy, x=-19(1)20, y=O(1)20, 6D

. .

249

(l.4 <5) .

251

Table

5.7.

Exponential

Integral

for Small Complex Arguments

ezEl(z), z=x+iy, x=-4(.5)-2, y=O(.2)1, 6D E,(z)+ln z, z=x+iy, x=-2(.5)2.5, y=O(.2)1, 6D The authors acknowledge the assistance of David S. Liepman in the preparation and checking of the tables, Robert L. Durrah for the computation of Table 5.2, and Alfred E. Beam for the computation of Table 5.6. r Guest worker, National Bureau of Standards, from the American Purdue University.) 2 National Bureau of Standards. (Presently NASA.)

University.

(Presently

227

5. Exponential

Integral Mathematical

5.1. Exponential

and Related Properties Explicit

Integral

Definitions

5.1.8

&(z)

5.1.2

5.1.3

Expressions

for

(Y,,(Z)

and

(l-l-z++ 22

a,(z)=n!z-*-le-”

/3,,(z)

. . . d-5)

5.1.9

(larg zl

f” e-’ r” edt Ei(x)=--J1 t dt=Tt 2 m

Functions

=n!2-n-1(e”

[l-z+&.

. . +(-l)“$]

(xx9 --e-z (l+~+~+

22

. . . +$I .

(x>l)

5.1.4 E,(z)=jy

e;

at

(D %(4’ s

(n=O,

1,2,

. . .; ~z>O)

‘(n=O,

1,2,

. . .; 92>0)

5.1.5

t*e-z*dt

1

5.1.6

pn(z)=J:,

t”e-“dt

(n=O, 1, 2, . . . )

In 5.1.1 it is assumed that the path of integration excludes the origin and does not cross the negative real axis. Anal& continuation of the functions in 5.1.1, 5.1.2, and 5.1.4 for n>O yields multi-valued functions with branch points at z-0 and z= a.3 They are single-valued functions in the z-plane cut along the negative real axis.* The function ii(z), the logarithmic integral, has an additional branch point at z=I. Interrelations

E1(-xfiO)=-Ei(x):)ii?r, -Ei(x)=~[~l(-x+iO)+E1(-x--iO)] [5.14], [5.16] use the entire

5.1.

y=Ei(x)

and y=&(x).

i

5.1.7

* Some authors

FIGURE

(x>O) function

: (I-e-g)dt/t 89 the basic function and denote IEm(z). We have Ein(z)=Er(z)+ln 2+7.

it by

4 Various authors define the integral z-plane cut along the by Ei(z). For 2=z>O (e.g., in [5.10], [5.25]), then used to designate Correspondingly, El(z)

_“, (et/t)& in the s positive real axis and denote it also additional notations such as %%(a$ E*(z) (in [5.2]), Ei*(z) (in [5.6]) are the principal value of the integral. is often denoted by - Ei( -z).

FIGURE

5.2.

y=&(x) n=O, 1, 2, 3, 5, 10

EXPONENTIAL

INTEGRAL

AND

RELATED

Y 4 5

229

FUNCTIONS Symmetry

n=O n=l

n.2

w3

n-4

n.5

n=6

5.1.13

am

Relation

.=Ez (2)

4 Recurrence 3

Relations

5.1.14 E.+,(z)=~[~-‘--zE.(z)]

2

5.1.15 1-s. 0

.5

I.0

I.5

2.0

2.5

FIGURE 5.3.

3.0

3.5

y=an(z)

x

(n=1,2,3,.

aY,(z)=4-z+mYn-1(2)

(n=1,2,3,.

. .)

(n=1,2,3,.

. .)

5.1.16 zSn(z)=(-l)ne’-e-“+n~,-,(z)

n=0(1)6

. .)

Inequalities

[5.8],

[5.4]

5.1.17 n+En(x)<E.+l(x)<E,(x)

(x>O;n=l,

2,3, . . .)

5.1.18 (x>O;n=1,2,3,.

J%(4<J%1wL+1(4

. .)

5.1.19 $n<ezE,(x)

I&

(x>O;n=1,2,3,.

<e%(x)
(I+:)

. .)

5.1.20

n:l \’

-151

FIGURE 5.4. Series

Ei(z)=r+lnz+FI

5.1.10

y=&,(x) n=O, 1, 2, 5, 10, 15

5.1.21 &[&]>O

(x>O;n=1,2,3,...) Continued

Expansions

a

n$

.

5.1.12

E,(z)=e-*

5.1.23

(larg 4<4

$0)=--r, W=-r+~2~ 4 56649 . . *. is Euler’s

Fraction

5.1.22

(XX)

(

--$$-n$-&.

. .)

Special

r=.57721

(x>O)

am

Values

=A

(n>l)

E,(z)=$

5.1.24

(n>l>

constant.

5.1.25

d4

=$,

@,,(,z)=~sinh z

(b-g 4-G)

230

EXPONENTIAL

INTEGRAL

AND

-dE”‘z)---E

(n==1,2,3,.

12- 1 (z)

dz

FUNCTIONS

5.1.36

Derivatives

5 . 126 .

RELATED

le --Otsin bt

. .)

t

S0

dt=arctan

k +Y&(a+ib) (a>O, b real)

5.1.27 5.1.37 ~n[e”El(z)l=&l

[e”E,(z)l

1 e”‘(l-cos

+(-W-l)!

2 3 , , ,-**

(+l 2”

Definite

and

S t bt)

Indefinite

)

5.1.38

Integrals

+WE,(-a+ib)

(a>O,b real)

bt) t +%S(a+ib) 2l-e-’ S

(a>O, b real)

1 e-“‘(l--OS

(For more [5.3],

[5.6],

involving

extensive [5.11],

tables

[5.12],

of integrals see [5.13]. For integrals

S 0

Z,(x) see [5.9].) 5.1.39

- e-“l o b+t dt=eubEl(ab)

S

5.1.28

5.1.40

5.1.29

S

op &

+Ei(a)

0

dt=$ In (1+$)-&(a)

0

- t

z e’-1 t

dt=&(z)+ln

dt=Ei

S0

(x)-In

z+y x--y

(XX)

5.1.41

dt=eciQbE,(--iab)

(a>% b>O) S

G2

dx=$

[e-“E,(-a--ix)-ee”E,(u--ix)]

5.1.30

+const.

S

om s2

eta~dt=eabEl(ub)

(a>% b>O)

5.1.42 a$ti

t+ib S

5.1.43

(a>% b>O)

m S S

e-“‘-emb’ t 0

5.1.33

[e-“I$(-a--ix)+e”E,(a--ix)]

+const.

o t2+b2 eI”ldt=e-““(-Ei(ub)+ir)

5.1.32

dx-;

S

5.1.31

dt=h

S

b a

a&

9(ef”EI(-x+ia))+const.

(a>O)

5.1.44

S

s2

mE:(t)dt=2

dx=-;

ln 2

0

Relation

5.1.34

m S

e+E,(t)dt=

dx=-L2(ef”EI(-x+ia))+const. to Incomplete

(a>O)

Gamma

Function

(see 6.5)

5.1.45

E,(z)=P-'r(l-n, 2)

5.1.46

q&)=2-*--lIyTL+1,

2)

0

q[ln

5.1.47

(1 +a) +g:

q]

(a>-1)

5.1.35

S

’ “’ SF bt dt=?r--arctan

Relation

5.1.48

i +A?Z,(-a+ib)

0

(a>% b>O)

j3,(z)=z-“-‘[r(n+l, to Spherical

d4=

-z)-Iyn+1, Bessel

Functions

z)] (see 10.2)

EXPONENTIAL Number-Theoretic

Significance

INTEGRAL

AND RELATED

li (z)

of

ao= - .57721 566 al= .999!39 193 az= - .249!Jl 055

(Assuming Riemann’s hypothesis that all nonreal zeros of t(z) have a real part of 3) 5.1.50

li (x)-7r(x)=O(@

r(x) is the number to 2.

In 2)

of primes

5.1.54

(x+=>

231

PUNG!CIONS

a3= .05519 968 a4= - .00976 004 as= .00107 857

l
less than or equal

x2+a:x+T+6 x+b

xe=J%c4=x2+b

(2)

I&> 1<5x 10-6

Y

a,=2.334733 a2= 250621 5.1.55

b1=3.330657 bz= 1.681534

lOSx<w

l&)l
I

0Y

I

400

I 600

FIGUR.E 5.5.

y=li(x)

200

Asymptotic

I 000

I 1000

lix<w

~e(x>1<2Xlo-*

L-X

al= 8.57332 87401 as= 18.05901 69730 a3= 8.63476 08925 a(= .26777 37343

and y=~(x)

Expansion

5.1.51 En(z)m e+il-n+n(n+l) 2 2 7-

b,=5.03637 b2=4.19160

n(n+l)(n+2)+ z3

bI= 9.57332 b2=25.63295 b,=2 1.09965 b4= 3.95849

23454 61486 30827 69228

5.2. Sine and Cosine Integrals

j *-*

Definitions (larg

Representation

of E,(z)

for

Large

4 <%d

5.2.1

n

5.1.52

5.2.2 ta

Ci(z)=r+ln +n(6x2-8nx+d) (x+nY

+Nn,

-.36nm4
- ’ x+n-1

Polynomial

Approximations

5.1.53

&(x)+ln

and

Rational

>

4 1

nT4 (x>O) 6

Ie(x)~<2Xlo-’ 6 The approximation 5.1.53 is from E. E. Allen, Note 169, MTAC 8, 240 (1954); approximations 5.1.54 and 5.1.56 are from C. Hastings, Jr., Approximations for digital computers, Princeton Univ. Press, Princeton, N.J., 1955; approximation 5.1.55 is from C. Hastings, Jr., Note 143, MTAC 7, 68 (1953) (with permission).

(larg 4<4

s 0 ‘Ydt

Shi(z)=

5.2.3 ‘I

‘sinh

S 0

-

t

t

dt

5.2.4 7

Chi(z)=r+ln

O<s
x=a0+aIx+azx2+a3i+a4x4+ap6+e(x)

zf

6 Some authors

z+

s 0 z coshtt-l

[5.14],

dt

(lax 4<4

[5.16] use the entire

function

*(l- cos t)dt/t as the basic function and denote it by s Coin(a). We have Gin(z)= -Ci(z)+ln 2+-r. 7 The Cinh(z) =

notations s

Sib(z)=

O’(cosh t- l)dt/t

s0

‘ sinh t dt/t,

have also been proposed (5.14.1

EXPONENTIAL

232

Auxiliary

Functions

f(z)=Ci(z)

5.2.6

g(z)=

5.2.7 and

Cosine

in

Shi(z)=~o

co9 2

cos z-si(z)

Integrals Functions

- (-l)%!~”

W) =r+ln z+zl

5.2.17

sin z--i(z)

-Ci(z)

AND RELATED FUNCTIONS 5.2.16

si(z.)=Si(z)-5

5.2.5

Sine

INTEGRAL

sin 2

Terms

5.2.8

Si(z)=i--f(z)

5.2.9

Ci(z)=f(z)

sin z-g(z)

Integral

Representations

of

co9 z-g(z)

Auxiliary

sin 2

(2n+$L+1)!

Chi(z)=r+ln

5.2.18

dgl

Symmetry

&,

Relations

Si(--z)=-Si(z),

5.2.19

2n(2n)r

Si(Z)=si(z)

5.2.20 co9 2

Ci(-z)=Ci(z)--.i7r

(O<w

z
C;(Z)=-)

I Relation

5.2.10

si(z)=-

e-2 cont cos (2 sin t)dt

Si(z)=& 5.2.11

5.2.12 5.2.13

Ci(z) +&(z)=l’

to Exponential

Integral

5.2.21 [E,(k)-EI(--iz)l+~

(kg

zl
e-’ coB’ sin (z sin t)dt

5.2.22 j(z)=S,'$$dt=l-g dt (.%'z>o) (&>O)5.2.23 g(z)= t+zt (jt= so-cos

Si(iz)=E

Ci(z)=-k

5.2.24

[Ei(z) +ITI(z)]

(x>O)

[E,(iz)+IG(--iz)] Ci(iz)=k

(larg 4-C:)

[Ei(z)--E,(x)]+i: Value

b>O)

at Infinity

lim Si (2) =i z-P-

5.2.25

Integrals

(For more extensive tables of integrals 15.31, [5.6], [5.11], [5.12], 15.131.) 5.2.26 s I

FIGURE 5.6. Series

y=Si(z)

and y=Ci(z)

5.2.15 *See page If.

&=-si

5.2.27

my s E

&=-Ci

5.2.28

a e-Wi s Cl

(t)dt=

Expansions

5.2.!29

5.2.14

OD‘!E$ *

S SOD

m e-=‘si

(2)

(2)

-&

(t)dt=--1

0

Si(z)=a

gO

JZ+t (g)

5.2.30

cos t Ci (t)dt=

0

(taris

(!urg

In (l+u*)

arctan a

a

m S

4<4

zl
(.@a>O)* (9?a>o)

sin t si (t)dl=-5

0

see

EXPONENTIAL

5.2.31

s0

5.2.32*

ODCi2 (t)dt=

INTEGRAL

AND RELATED

5.2.37

S

m si2 (t)dt=E

233

l_<s<m

0

s0

.a Ci (t) si (t)dt=

In 2

I4~>l
5.2.33

‘OS bt&=i ln(I+$) +Ci (b) s’(1-e-a’) t

0

FUNCTIONS

a,=7.547478 a~= 1.564072 5.2.38

b1=12.723684

*

bz= 15.723606

*

l
+ %?I$ (a+ ib) (a real, b>O) Asymptotic

Expansions

5.2.34 ,(z)+(l-$+$-$+.

. .)

le(z)1<5Xlo-’ 38.027264 b1= 40.021433 &-265.187033 bz=322.624911 a3=335A77320 b3=570.236280

al=

(larg zl
5.2.35

a,=

5.2.39 Rational

5.2.36

Approximations

38.102495

b,=157.105423

l<s<=J

*

152
1~(z)1<2xlo-’ a1=7.241163 b1=9.068580 &=2.463936

bz=7.157433

Numerical 5.3. Use and Extension of the Tables Example 1. Compute Ci (.25) to 5D. From Tables 5.1 and 4.2 we have

Ci (.25) - M.25) (.25)*

-Y=

_ .24g350

Ci (.25)=(.25)2(-.249350)+(-l.38629) +.577216=

I

Ei (8) =440.38. page

II.

8 From C. Hastings, Jr., Approximations computers, Princeton Univ. Press, Princeton, (with permission).

bl=

a,=302.757865

bz= 482.485984

a,=352.018498

b3= 1114.978885

a*= 21.821899

br= 449.690326

48.196927

Methods Example 3.

Compute Si (20) to 5D. Since l/20=.05 from Table 5.2 we find j(20) = .049757, g(20) = .002464. From Table 4.8, sin 20 = .912945, cos 20 = .408082. Using 5.2.8 Si(20) =;-j(20)

co9 20-g(20) sin 20 =1.570796-.022555=1.54824.

-.82466.

Example 2. Compute Ei (8) to 5s. From Table 5.1 we have ze-‘Ei (z) =1.18185 for s=8. From Table 4.4, e8=2.98096X103. Thus

*see

al = 42.242855

for digital N.J., 1955

Example 4. Compute E,(z), n=l(l)N, to 5S for z= 1.275, N= 10. If z is less than about five, the recurrence relation 5.1.14 can be used in increasing order of n without serious loss of accuracy. By quadratic interpolation in Table 5.1 we get & (1.275) = .1408099, and from Table 4.4, e-l.*” =. 2794310. The recurrence formula 5.1.14 then yields

234

EXPONENTIAL

n 1 2 3 4 5

EJ1.275)

a&(1.275)

.1408099 .0998984 .0760303 .0608307 .0504679

INTEGRAL

6 7 8 9 10

.0430168 .0374307 .0331009 .0296534 .0268469

Interpolating directly in Table 5.4 for n=lO we get E,,(1.275)= .0268470 as a check. Example 5. Compute E,(z), n= l(l)N; to 5s for 2= 10, N= 10. If, as in this example, z is appreciably larger than five and N
YfO) .41570 .38300 .355oz .33ozi .31oG .28= .27667 .25333 .25084 .22573

n 12 11 10 9 8 7 6 5 4 3 2 1

FUNCTIONS

106E,(12.3) .191038 .199213 208098 : 217793 .228406 240073 : 252951 .267234 283155 : 300998 .321117 .343953

106E,,(12.3) . 191038 . 183498 . 176516 170042 : 164015 158397 : 153144 . 148226 . 143608

n 12 13 14 15 16 17 18 19 20

From Tables 5.2 and 5.5 we find E1(12.3) = .343953 X lo-*, E,,(12.3)=.143609X lOmeas a check. Example 7. Compute (~~(2) to 6s for n= 1(1)5. The recurrence formula 5.1.15 can be used for all x rel="nofollow">O in increasing order of n without loss of From

5.1.25 we have a0(2)=f

esa

=.0676676, so we get i 1 2 3 4 5

+ .02032- .00043- .OOOOl) = 1.91038X lo-‘.

Using the recurrence relation 5.1.14, we get

RELATED

accuracy.

lw%Yo) .41570 .38302 .35488 .33041 .30898 .29005 .27325 .25822 .24472 .23253

From Table 5.2 we get zeZEl(z)= .915633 SO that E,(10)=4.15697X10-E as a check. Forward recurrence starting with E1(10)=4.1570X10-e yields the values in column (1). The underlined figures are in error. Example 6. Compute E,(z), n=l(l)N, to 5s for x=12.3, N=20. If N is appreciably larger than z, and x appreciably larger than five, then the recurrence relation 5.1.14 should be used in the backward direction to generate E,(x) for nnO, where n,=(x). From 5.1.52, with n0=12, x=12.3, we have En,(x) =e;;(l

AND

as indicated,

ffnca .0676676 .101501 .169169 .321421 .710510 1.84394

Independent calculation with 5.1.8 yields the same result for (~~(2). The functions W,(X) and q(z) can be obtained from Table 10.8 using 5.1.48, 5.1.49. Example 8. Compute p,(x), n=O(l)N to 6s for x=1, N=5. Use the recurrence relation 5.1.16 in increasing order of n if x>.368N+.184

In N+.821

and in decreasing order of n otherwise [5.5]. From 5.1.9 with n=5 we get /3,(l)= -.324297 correctly rounded to 6D. Using the recurrence formula 5.1.16 in decreasing order of n and carrying 9D we get the values in column (2). n 0 1 2 3 4 5

w 2.35040 2 -.73575 9269 .87888 3849 -.44950 9722 .55236 3499 -.32434 3774

B# 2.35040 - .73575 .87888 - .44950 .55237 -.32429

2382 888_0 4629 7385 28g 7-

Using forward recurrence instead, starting with

EXPONENTIAL

INTEGRAL

AND

p,(l)=2 sinh 1=2.350402 and again carrying 9D, we obtain column (1). The underlined figures are in error. The above shows that three significant figures are lost in forward recurrence, whereas about three significant figures are gained in backward recurrence!

RELATED

0.lw)

10

.280560 - .2oE?z .319908 -.253812 .40465

9

8 7 6

n5

- .324297 a”(l)

4 3 2 1 0

552373 - .449507 .878885 - .735759 2.350402

k

+6.8943i. From Table

9.

Compute

E,(z)

for

z=3.2578

0

1 2 3

. 059898 . 008174 -. 001859 . 000088

.095598i,

e”oE,(zo)=.059898-.107895i.

=f(zo+Az>

=f(zo>

+‘T

-. 10789% +. 012795i +. 000155i

. 059898 . 003460 -. 000094

-. 1078952’ f. 0024352’ +. 000110i

-.

-.

-.

000212i

000003

000004i

-.105354i -.022075i -.004716i

.711093 -3.784225+-12.7i .278518 .010389 -!- -1.90572+12.7i+2.0900+12.7i

- .0184106- .0736698i E,(z)z--1.87133-4.7054Oi.

From Taylor’s formula with f(z) =ezEI (z) we have

f(z)

(AZ) ff’k)(zo)/k!

Example 10. Compute E,(z) for z=-4.2 + 12.7i. Using the formula at the bottom of Table 5.6

5.6 we have for z,,=z0+iy,=3+7i

(zo) = .934958+

5.1.27

Repeating the calculation with zo=3+6i and Az=.2578+.8943,i we get the same result. An alternative ,procedure is to perform bivariate interpolation in t.he real and imaginary parts of ze“El (2).

e”E,(z) = z&~El

Thus with

f’k)(&/k!

f(z)==.063261 e-‘==.031510 El(z)= --.000332

The functions ,&(s) and &(x) can be obtained from Table 10.8 using 5.1.48, 5.1.49. Example

235

with Az=z-zo=.2578-.1057i. we get

An alternative procedure is to start with an arbitrary value for n sufficiently large (see also 15.11). To illustrate, starting with the value zero at n= 11 we get

1:

FUN(!TIONS

AZ

+jq

(A,@+.

. .

References Texts

[5.1] F. J. Corbat6, On the computation of auxiliary functions for two-center integrals by means of a high-speed computer, J. Chem. Phys. 24,452-453 (1956). [5.2] A. Erdelyi et al., Higher transcendental functions, vol. 2 (McGraw-Hill Book Co., Inc., New York, N.Y., 1953). [5.3] A. Erdelyi et al., Tables of integral transforms, ~01s. 1, 2 (McGraw-Hill Book Co., Inc., New York, N.Y., 1954).

[5.6] W.

[5.7]

[5.8]

[5.9]

[5.4] W. Gautschi, Some elementary inequalities relating to the gamma and incomplete gamma function, J. Math. Phys. 38, 77-81 (1959).

[5.10]

[5.5] W. Gautschi, Recursive computation of certain integrals, J. Assoc. Comput. Mach. 8, 21-40 (1961).

[5.11]

Grijbner and N. Hofreiter, Integraltafel (Springer-Verlag, Wien and Innsbruck, Austria, 1949-50). C. Hastings, *Jr., Approximations for digital computers (Princeton Univ. Press, Princeton, N.J., 1955). E. Hopf, Mathematical problems of radiative equilibrium, Cambridge Tracts in Mathematics and Mathematical Physics, No. 31 (Cambridge Univ. Press, Cambridge, England, 1934). V. Kourganofl’, Basic methods in transfer problems (Oxford Un:v. Press, London, England, 1952). F. Lijsch and F. Schoblik, Die Fakultiit und verwandte Funktionen (B. G. Teubner, Leipzig, Germany, 1951). N. Nielsen, Theorie des Integrallogarithmus (B. G. Teubner, Leipzig, Germany, 1906).

EXPONENTIAL

236

INTEGRAL

[5.12] F. Oberhettinger, Tabellen zur Fourier Transformation (Springer-Verlag, Berlin, Gottingen, Heidelberg, Germany, 1957). [5.13] I. M. Ryshik and I. S. Gradstein, Tables of series, products and integrals (VEB Deutscher Verlag der Wissenschaften, Berlin, Germany, 1957). [ 5.141 S. A. Schelkunoff, Proposed symbols for the modified cosine and exponential integral, Quart. Appl. Math. 2, 90 (1944). (5.151 J. Todd, Evaluation of the exponential integral for large complex arguments, J. Research NBS 52, 313-317 (1954) RP 2508. [ 5.161 F. G. Tricomi, Funzioni ipergeometriche confluenti (Edizioni Cremonese, Rome, Italy, 1954). Tables [5.17] British Association for the Advancement of Science, Mathematical Tables, vol. I. Circular and hyperbolic functions, exponential, sine and cosine integrals, etc., 3d ed. (Cambridge Univ. Press, Cambridge, England, 1951). Ei(z)-ln z, -El(z) --In z, Ci(z)-In 2, Si(z), z=O(.1)5, 11D; Ei(z), 2=5(.1)15, 10-115; &(z),z=5(.1)15, 13-14D; Si(x), Ci(z), x=5(.1)20(.2)40, IOD. [5.18] L. Fox, Tables of Weber parabolic cylinder functions and other functions for large arguments, Mathematical Tables, vol. 4, National Physical Laboratory (Her Majesty’s Stationery Office, London, England, 1960). e-ZEi(z), e=&(z), x-1=0(.001).1, 10D; f(z), Q(Z), z-‘=O(.OOl).l, 10D. [5.19] J. W. L. Glaisher, Tables of the numerical values of the sine-integral, cosine-integral and expoTrans. Roy. Sot. nential integral, Philos. London 160, 367-388 (1870). Si(z), Ci(z), Ei(z), 18D, 2=1(.1)5(1)15, 11D. -l&(x), z=O(.Ol)l, [5.20] B. S. Gourary and M. E. Lynam, Tables of the auxiliary molecular integrals A,(s) and the auxiliary functions C,,(z), The Johns Hopkins Univ. Applied Physics Laboratory, CM Report 905, Baltimore, Md. (1957). a,,(z), n!e,,(z), z= .05(.05) 15, n=O(l) 18, 9s. [5.21] F. E. Harris, Tables of the exponential integral Ei(z), Math. Tables Aids Comp. 11, 9-16 (1957). El(z), eZEl(z), Ei(z), e-*Ei(z), z=1(1)4(.4)8(1)50, 18-19s. [5.22] Harvard University, The Annals of the Computation Laboratory, ~01s. 18, 19; Tables of the generalized sine- and cosine-integral functions, parts I, II (Harvard Univ. Press, Cambridge, Mass., =1-cosu 1949). 8&z)= cZ~dz,C(o,z)= c Tax, S’ s 6D; Ss(a, z)= s= s+ sin xdx, &(a, z) = s -zsin u co8 xdx, 6D; Cs(a, 2) = c” 7 sin z&r, s-0 u s Cc(a,x)=Jyy

(l-cos

2)&r,

6D;u=&q$

0
AND RELATED

FUNCTIONS

2 l--e* u

a?, Es&, 2) = ,” F dz, Ec(a, z) = s * l-e~cosu clx, 6D; ~=&+a~, O
s- 0

s0 0 Ix< 10. 15.241 A. V. Hershey, Computing programs for the complex exponential integral, U.S. Naval Proving Ground, Dahlgren, Va., NPG Report No. 1646 (1959). -E,(-z),. z=x+iy, z=-20(1)20, y=O(1)20, 13s. t5.251E. Jahnke and F. Emde, Tables of functions, 4th ed. (Dover Publications, Inc., New York, N.Y., 1945). --El(x), Ei(s), 2=0(.01)1(.1)5(1)15, 46s; Si(s), Ci(z), 2=0(.01)1(.1)5(1)15(5)100(10) 200(100) 10s(1)7, generally 4-55; maxima and minima of Ci(z) and si(z), O<s<16, 55. [5.26] K. A. Karpov and S. N. Razumovskili, Tablitsy integral’ nogo logarifma (Izdat. Akad. Nauk SSSR., MOSCOW, U.S.S.R., 1956). ii(z), .r= O(.OOOl) 2.5 (001) 20 (.01)200 (1) 500 (1) 10000 (10) 25000, 75; Ii(z)--In /l--2], z=.95(.0001)1.05, 6D. [5.27] M. Kotani, A. Amemiya, E. Ishiguro, T. Kimura, Table of molecular integrals (Maruzen Co., Ltd., Tokyo, Japan, 1955). a(z), x=.25(.25)9(.5)19(1) 25, n=0(1)15, 11s; ,8,,(z), 2=.25(.25)8(.5)19(l) 25, n=0(1)8, 11s. [5.28] M. Mashiko, Tables of generalized exponential-, sine- and cosine-integrals, Numerical Computation Bureau Report No. 7, Tokyo, Japan (1953). E&r)+ln]z(=C&)+ln .&i&(t), z=[ee”, E=0(.05) 5, a=O”(2°)600(10)900, 6D; ze~E,(z) =A.&) exp [i&(q)],

z=-! eh, q=.O1(.01).2,

~=0”(2~)60~(1~)

900, 5-6D? [5.29] G. F. Miller, Tables of generalised exponential integrals, Mathematical Tables, vol. 3, National Physical Laboratory (Her Majesty’s Stationery Office, London, England, 1958). (z+n)e=E.(z), z=O(.Ol)l, n=1(1)8, 2=0(.1)20, n=1(1)24, z-1=0(.001).05, n=1(1)24; 8D. [5.30] J. Miller, J. M. Gerhauser, and F. A. Matsen, Quantum chemistry integrals and tables (Univ. of Texas Press, Austin, Tex., 1959). a,,(z), z=.125 (.125)25,n=0(1)16, 14S;&,(z),s=O(.125)24.875, n=0(1)16, 12-14s. [5.31] J. Miller and R. P. Hurst, Simplified calculation of the exponential integral, Math. Tables Aids Comp. 12, 187-193 (1958). e-“l%(z), Ei(z), etE,(x), El(x), z=.2(.05)5(.1)10(.2)20(.5)50(1)80; 165. [5.32] National Bureau of Standards, Tables of sine, cosine and exponential integrals, vol. I (U.S. Government Printing Office, Washington, D.C., 1940). Si(z), Ci(z), Ei(s), El(z), z=O(.OOO1)2, z=O(.l)lO; 9D. [5.33] National Bureau of Standards, Tables of sine, cosine and exponential integrals, vol. II (U.S. Government Printing Office, Washington, D.C., 1940). Si(z), a(z), Ei(z), El(z), z=O(.OO1)10, 9-10 D or S; Si(z), Ci(z), 2=10(.1)40, 1OD; Ei(z), E,(z), z= 10(.1)15, 7-11s.

EXPONENTIAL

INTEGRAL

[5.34] National Bureau of Standards, Table of sine and cosine integrals for arguments from 10 to 100, Applied Math. Series 32 (U.S. Government Printing Office, Washington, D.C., 1954). Si(r), Ci(x), z= lO(.Ol)lOO, 10D. [5.35] National Bureau of Standards, Tables of functions and of zeros of functions, Collected short tables of the Computation Laboratory, Applied Math. Series 37 (U.S. Government Printing Office, Washington, D.C., 1954). E,(z), n=0(1)20, s=O(.O1)2(.1)10, 4-9s; E&)-z In 2, x=0(.01) .5, 75; Ea(z)+~2* In 2, z=O(.Ol).l, 75. [5.36] National Bureau of Standards, Tables of the exponential integral for complex arguments, Applied Math. Series 51 (U.S. Government Printing Office, Washington, D.C., 1958). Ei(z)+ln z, 6D, 2=0(.02)1, y=O(.O2)1, 2=-l(.l)O, y= O(.l)l; E,(z), 6D, x=0(.02)4, y=O(.O2)3(.05)10, 2=0(1)20, y=O(1)20, x=-3.1(.1)0, y=O(.1)3.1, z= -4.5(.5)0, y= 0(.1)4(.5)10, z= - 10(.5) -4.5, y=O(.5)10,2= -2O(l)O, y=O(1)20; efEi(z), 6D, 2=4(.1) 10, y=O(.5) 10. [5.37] S. Oberliinder, Tabellen von Exponentialfunktlonen und-integralen zur Anwendung auf Gebieten der Thermodynamik, Halbleitertheorie und Gaskinetik (Akademie-Verlag. Berlin, Germanv.

AND

RELATED

237

FUNCTIONS

El (ff),

1-$exp($$)

T=

1000,

25(25)

exp ( --&:I,

El (g); T=

z exp (-z-l),

150(10)390,

AE=.2(.2)2, 3-4s;

z-1,

El (z-l),

e’exp (-t-l)dt, s z-1 exp (x-1) El (z-l), I.--z-l exp (z-i) El (z-1) ; z = .Ol (.OOOl) .l, 5-6s. [5.38] .V. I. Pagurova, Tables of the exponential integral E,(z)=

s

;’ e-zuu-*du.

Translated

from the Rus-

sian by D. G. Fry (Pergamon Press, New York, N.Y.; Oxfford, London, England; Paris, France, 1961). E,(z), n=0(1)20, 2=0(.01)2(.1)10, 49s; Es(z) --a In z, 2=0(.01)5, 75; E&r) +&* In z, s=O(.Ol).l, 75; eZEn(z), n=2(1)10, %=10(.1)20, 7D; e*E,(x), r=O(.l)l, z=.01(.01)7(.05)12(.1)20, 7 S or D. [5.39] Tablitsy integral’nogo sinusa i kosinusa (Izdat. Akad. Na.uk SSSR., MOSCOW, U.S.S.R., 1954). Si(z), Ci(;:), ~=0(.0001)2(.001)10(.005)100, 7D; Ci(r) --In :G,~=0(.0001).01, 7D. [5.40] Tablitsy integral’nol pokasatel’nol funktsii (Izdat. Akad. Nauk SSSR., Moscow, U.S.S.R., 1954). Ei(s), E1(~),r=0(.0001)1.3(.001)3(.0005)10(.1)15, 7D. [5.41] D. K. Trubey, A table of three exponential integrals, Oak: Ridge National Laboratory Report 2750, Oak. Ridge, Tenn. (June 1959). E,(z), R(z), E&z), ~=0(.0005).1(.001)2(.01)10(.1)20, 6s.

238

EXPONENTIAL

Table

5.1

INTEGRAL

SINE, COSINE

AND RELATED FUNCTIONS

AND EXPONENTIAL

INTEGRALS

0. 01 0.02 0.03 0.04

r-ISi 1.00000 00000 0.99999 44444 0.99997 77781 0.99995 00014 0.99991 11154

r-z[Ci(s)-ln 2--y] I -0.25000 00000 -0.24999 89583 -0.24999 58333 -0.24999 06250 -0.24998 33339

0.05 0.06 0.07 0. 08 0.09

0.99986 11215 OI99980 06216 0.99972 78178 OI99964 45127 0.99955 01094

-0.24997 -0.24996 -0.24994 -0.24993 -0.24991

39598 25030 89639 33429 56402

1.01264 1.01520 1.01777 1.02036 1.02295

0202 2272 5836 0958 7705

0.98763 0.98519 0.98276 0.98035 0.97794

75971 77714 86889 02898 25142

0.10 0.11 0.12 0.13 0.14

0.99944 0.99932 0.99920 0.99906 0.99891

46111 80218 03455 15870 17512

-0.24989 -0.24987 -0.24985 -0.24982 -0.24979

58564 39923 00480 40244 59223

1.02556 1.02818 1.03081 1.03346 1.03611

6141 6335 8352 2259 8125

0.97554 0.97315 0.97078 0.96841 0.96606

53033 85980 23399 64710 09336

0.15 0.16 0.17 0.18 0.19

0.99875 0.99857 0.99839 0.99820 0.99799

08435 88696 58357 17486 66151

-0.24976 -0.24973 -0.24969 -0.24966 -0.24962

57422 34850 91516 27429 42598

1.03878 1.04146 1.04415 1.04686 1.04957

6018 6006 8158 2544 9234

0.96371 0.96138 0.95905 0.95674 0.95443

56702 06240 57383 09569 62237

0.20 0.21 0.22 0.23 0.24

0.99778 0.99755 0.99731 0.99706 0.99680

04427 32390 50122 57709 55242

-0.24958 -0.24954 -0.24949 -0.24944 -0.24940

37035 10749 63752 96056 07674

1.05230 1.05504 1.05780 1.06057 1.06334

8298 9807 3833 0446 9719

0.95214 0.94985 0.94758 0.94531 0.94306

14833 66804 17603 66684 13506

0.25 0.26 0.27 0.28 0.29

0.99653 0.99625 0.99595 0.99565 0.99533

42813 20519 88464 46750 95489

-0.24934 -0.24929 -0.24924 -0.24918 -0.24912

98618 68902 18540 47546 55938

1.06614 1.06894 1.07176 1.07459 1.07743

1726 6539 4232 4879 8555

0.94081 0.93857 0.93635 0.93413 0.93192

57528 98221 35046 67481 94997

0.30 0.31 0.32 0.33 0.34

0.99501 0.99467 0.99432 0.99396 0.99360

34793 64779 85570 97288 00064

-0.24906 -0.24900 -0.24893 -0.24886 -0.24879

43727 10933 57573 83662 89219

1.08629 1.08316 1.08604 1.08894 1.09185

5334 5293 8507 5053 5008

0.92973 0.92754 0.92536 0.92319 0.92103

17075 33196 42845 45510 40684

0. 35 0.36 0.37 0.38 0.39

0.99321 94028 OI99282 79320 0.99242 56078 0.99201 24449 0.99158 84579

-0.24872 -0.24865 -0.24857 -0.24850 -0.24842

74263 38813 82887 06507 09693

1.09477 1.09771 1.10066 1.10363 1.10660

8451 5458 6108 0481 8656

0.91888 0.91674 0.91460 0.91248 0.91036

27858 06533 76209 36388 86582

0.40 0.41 0.42 0.43 0. 44

0.99115 0.99070 0.99025 0.98978 0.98930

36619 80728 17063 45790 67074

-0.24833 -0.24825 -0.24816 -0.24808 -0.24799

92466 54849 96860 18528 19870

1.10960 1.11260 1.11562 1.11866 1.12170

0714 6735 6800 0991 9391

0.90826 0.90616 0.90407 0.90199 0.89992

26297 55048 72350 77725 70693

0.45 0.46 0.47 0.48 0.49

0.98881 0.98831 0.98780 0.98728 0.98675

81089 88008 88010 81278 67998

-0.24790 -0.24780 -0.24771 -0.24761 -0.24751

00913 61685 02206 22500 22600

1.12477 1.12784 1.13094 1.13404 1.13716

2082 9147 0671 6738 7432

0.89786 0.89581 0.89376 0.89173 0.88970

50778 17511 70423 09048 32920

0.50

0.98621 48361

o.“oo

See Examples

[c-y1

l-2.

crl[Ei(z)-In 1.00000 1.00250 1.00502 1.00755 1.01008

-0.24741 02526

[c-y1

y=0.57721 56649

s--y1 0000 5566 2306 0283 9560

1.14030 2841 C-612 4

[

1

rl[E:l(z)+ln z+y] 1.00000 00000 0.99750 55452 0.99502 21392 0.99254 97201 0.99008 82265

0.88768 41584 c-y

[

1

EXPONENTIAL SINE,

INTEGRAL COSINE

AND

AND

RELATED

EXPONENTIAL

Si(X)

239

FUNCTIONS INTEGRALS

Ci(x)

Ei(x)

Table

5.1

El(x)

0.51 0.52 0.53 0.54

0.49310 0.50268 0.51225 0.52179 0.53132

74180 77506 15212 84228 al492

-0.17778 -0.16045 -0.14355 -0.12707 -0.11099

40788 32390 37358 07938 04567

0.45421

9905

2167 0633 0445 5931

0.55977 0.54782 0.53621 0.52495 0.51400

3595 2352 9798 1510 3886

0.55 0.56 0.57 0.58 0.59

0.54084 0.55033 0.55981 0.56926 0.57870

03951 48563 12298 92137 85069

-0.09529 -0.07998 -0.06503 -0.05044 -0.03618

95274 55129 65744 14815 95707

0.61529 0. 64667 0.67781 0.70872 0.73940

0657 74190 8642 5720 9764

0.50336 0.49301 0.48296 0.47317 0.46364

4081 9959 0034 3433 9849

0.60 0. 61 0.63 0.64

0.58812 0.59752 0.60691 0.61627 0.62561

88096 98233 12503 27944 41603

-0.02227 -0.00867 +o.O0460 0.01758 0.03026

07070 52486 59849 17424 03686

0.76988 0.80015 0.83022 0.86011 0.88983

1290 0320 6417 8716 5949

0.45437 0.44535 0.43656 0.42799 0.41965

9503 3112 1854 7338 1581

0.65 0.66 0.67 0.68 0.69

0.63493 0.64423 0.65351 0.66277 0.67200

50541 51831 42557 19817 80721

0.04264 0.05475 0.06659 0.07815 0.08946

98293 77343 13594 76659 33195

0.91938 0.94877 0.97801 1.00711 1.03607

6468 a277 9042 6121 6576

0.41151 0.40358 0.39585 0.38830 0.38095

6976 6275 2563 9243 0010

0.70 0.71 0.72 0.73 0.74

0.68122 0.69041 0.69958 0.70873 0.71785

22391 41965 36590 03430 39660

0.10051 0.11131 0.12187 0.13220 0.14229

47070 79525 a9322 32879 64404

1.06490 1.09361 1.12220 1.15068 1.17905

7195 4501 4777 40159 8208

0.37376 0.36675 0.35991 0.35323 0.34671

8843 9981 7914 7364 3279

0.75 0.76 0.77 0.78 0.79

0.72695 0.73603 0.74508 0.75411 Oi76311

42472 09067 36664 22494 63804

0.15216 0.16180 0.17123 0.18045 0.18946

36010 97827 98110 a3335 98290

1.20733 1.23551 1.26360 1.29161 1.31954

28:16 33:19 4960 2805 17!j3

0.34034 0.33411 0.32803 0.32208 0.31627

0813 5321 2346 7610 7004

0.80 0.81 0.82 0.83 0.84

0.77209 0.78105 0.78997 0.79888 0.80776

57855 01921 93293 29277 07191

0.19827 0.20688 0.21530 0.22352 0.23156

a6160 88610 45659 96752 78824

1.34739 1.37518 1.40290 1.43056 1.45816

6548 1783 1245 3978

0.31059 0.30504 0.29961 0.29429 0.28910

6579 2539 1236 9155 2918

0. a5 0.86 0.87 0.88 0.89

0.81661 0.82543 0.83423 0.84300 0.85175

24372 78170 65953 a5102 33016

0.23942 0.24709 0.25459 0.26192 Oi26907

28368 80486 69153 27264 86687

1.48571 1.51321 1.54067 1.56808 1.59546

4176 574'1 2664 8534 7036

0.28401 0.27904 0.27417 0.26941 0.26474

9269 5070 7301 3046 9496

0.90 0.91 0.92 0.93 0.94

0.86047 0.86916 0.87782 0.88645 0.89506

07107 04808 23564 60839 14112

0.27606 0.28289 0.28955 0.29606 0.30241

78305 32065 77018 41358 52458

1.62281 1.65012 1.67741 1.70467 1.73191

1714 6019 3317 6891 994.6

0.26018 0.25571 0.25133 0.24704 0.24285

3939 3758 6425 9501 0627

0.95 0.96 0. 97 0.98 0.99

0.90363 0.91218 0.92070 0.92919 0.93765

80880 58656 44970 37370 33420

0.30861 0.31466 0.32056 0.32631 0.33193

36908 20547 28495 85183 14382

1.75914 1.78635 1.81355 1.84074 1.86793

5612 6947 6941 8519 4543

0.23873 0.23470 0.23075 0.22689 0.22309

7524 7988 9890 1167 9826

1.00

0.94608 30704 c-p4

o.;o

0.62

c 1

0.337:03;229

II 6 1

0.48703 0.51953 0.55173 0.58364

19110

1.89511 7816 c-y4

[

1

0.21938 3934 C-514 5

II 1

240

EXPONENTIAL Table

5.1

SINE,

X

INTEGRAL COSINE

AND

AND

EXPONENTIAL Ci(x)

Si(X)

RELATED

FUNCTIONS INTEGRALS Ei(X)

El(x)

1.00 1.01 1.02 1.03 1.04

0.94608 Oi95448 0.96285 Oi97119 0.97949

30704 26820 19387 06039 a4431

0.33740 0.34273 0.34793 0.35300 0.35793

39229 82254 65405 10067 37091

1.89511 1.92230 1.94948 1.97667 2.00387

7816 1085 7042 8325 7525

0.21938 0.21574 0.21217 0.20867 0.20523

3934 1624 1083 0559 8352

1.05 1.06 1. 07 1.08 1.09

0.98777 0.99602 1.00423 liO1241 1.02056

52233 07135 46846 69091 71617

0.36273 0.36741 0.37196 0.37638 0.38069

66810 19060 13201 68132 02312

2.03108 2.05830 2.08554 2.11280 2.14007

7184 9800 7825 3672 9712

0.20187 0.19857 0.19533 0.19216 0.18904

2813 2347 5403 0479 6118

1.10 1.11 1.12 1.13 1.14

1.02868 1703677 1.04482 1.05284 1.06083

52187 08583 38608 40082 10845

0.38487 0.38893 0.39288 0.39671 0.40043

33774 80142 58645 86134 79090

2.16737 2:19470 2.22205 2.24943 2.27684

8280 1672 2152 1949 3260

0.18599 0.18299 0.18005 0.17716 0.17433

0905 3465 2467 6615 4651

1.15 1.16 1. 17 1.18 1.19

1.06878 1.07670 1.08459 1.09244 1.10026

48757 51696 17561 44270 29760

0.40404 0.40754 0.41093 0.41421 0.41738

53647 25593 10390 23185 78816

2.30428 2.33176 2.35928 2.38684 2.41444

8252 9062 7800 6549 7367

0.17155 0.16882 0.16615 0.16352 0.16094

5354 7535 0040 1748 1567

1.20 1.21 1.22 1.23 1.24

1.10804 1.11579 1.12351 1.13119 1.13883

71990 68937 18599 18994 68160

0.42045 0.42342 0.42629 0.42906 0.43172

91829 76482 46760 16379 98802

2.44209 2.46978 2.49752 2.52531 2.55315

2285 3315 2442 1634 2836

0.15840 0.15592 0.15347 0.15108 0.14872

8437 1324 9226 1164 6188

1.25 1.26 1.27 :'22: .

1.14644 1.15402 1.16155 1.16906 1.17652

64157 05063 88978 14023 78340

0.43430 0.43677 0.43915 0.44144 0.44363

07240 54665 53815 17205 57130

2.58104 2.60899 2.63700 2.66507 2.69320

7974 8956 7673 5997 5785

0.14641 0.14414 0.14191 0.13971 0.13756

3373 1815 0639 8989 6032

1.30 1.31 1.32 1.33 1.34

1.18395 1.19135 1.19870 1.20602 1.21331

80091 17459 88649 91886 25418

0.44573 0.44775 0.44967 0.45151 G.45326

85675 14723 55955 20863 20753

2.72139 2.74965 2.77798 2.80637 2.83484

8880 7110 2287 6214 0677

0.13545 0.13337 0.13133 0.12932 0.12735

0958 2975 1314 5224 3972

1.35 1.36 1.37 1.38 1.39

1.22055 1.22776 1.23493 1.24207 1.24916

87513 76460 90571 28180 87640

(1.45492 0.45650 0.45800 0.45941 0.46075

66752 69811 40711 90071 28349

2.86337 2.89198 2.92067 2.94943 2.97828

7453 8308 4997 9263 2844

0.12541 0.12351 0.12164 0.11980 0.11799

6844 3146 2198 3337 5919

1.40 1.41 1.42 1.43 1.44

1.25622 1.26324 1.27022 1.27717 1.28407

67328 65642 81004 11854 56658

0.46200 0.46318 0.46427 0.46529 0.46624

65851 12730 78995 74513 09014

3.00720 3.03621 3.06530 3.09448 3.12375

7464 4843 6691 4712 0601

0.11621 0.11447 0.11275 0.11106 0.10940

9313 2903 6090 8287 8923

1.45 1.46 1.47 1.48 1.49

1.29094 1.29776 1.30455 1.31130 1.31801

13902 82094 59767 45473 37788

0.46710 0.46790 0.46862 0.46927 0.46984

92094 33219 41732 26848 97667

3.15310 3.18255 3.21209 3.24172 3.27145

6049 2741 2355 6566 7042

0.10777 0.10617 0.10459 0.10304 0.10151

7440 3291 5946 4882 9593

1.50

1.32468 35312 C-j)5

[

1

0.47035 63172 c-5)2 5

ll 1

3.30128 5449 C--5)1 5

I: 1

0.10001 9582

[(-;I91

EXPONENTIAL

SINE,

COSINE

INTEGRAL

AND EXPONENTIAL

RELATED

241

FUNCTIONS

Table

INTEGRALS

Ci(x)

Si(X)

l.GO 1.32468 1.51 1.33131

AND

Ei(x’l

5.1

El(x)

35312 36664 1.33790 40489 1.34445 45453 1.35096 50245

0.47035 0.47079 0.47116 0.47146 0.47169

63172 32232 13608 15952 47815

3.30128 3.33121 3.36124 3.39137 3.42161

5449 3449 2701 4858 1576

0.10001 9582 0.09854 0.09709 0.09566 0.09426

4365 3466 6424 2786

1.55

1.56 1.57 1.58 1.59

1.35743 1.36386 1.37025 1.37660 1.38291

53577 54183 50823 42275 27345

0.47186 0.47196 0.47200 0.47197 0.47188

17642 33785 04495 37932 42164

3.45195 3.48240 3.51296 3.54363 3.57442

4503 5289 5580 7024 1266

0.09288 0.09152 0.09018 0.08887 0.08758

2108 3960 7917 3566 0504

1.60 1.61 1.62 1.63 1.64

1.38918 1.39540 1.40159 1.40773 1.41384

04859 73666 32640 80678 16698

0.47173 0.47151 0.47124 0.47091 0.47052

25169 94840 58984 25325 01507

3.60531 3.63633 3.66746 3.69871 3.73009

9949 4719 7221 9099 1999

0.08630 0.08505 0.08382 0.08261 0.08142

8334 6670 5133 3354 0970

1.65 1.66 1.67 1.68 1.69

1.41990 1.42592 1.43190 1.43784 1.44373

39644 48482 42202 19816 80361

0.47006 0.46956 0.46899 0.46837 0.46769

95096 13580 64372 54812 92169

3.76158 3.79320 3.82495 3.85682 3.88882

7:569 7456 3.310

0.08024 0.07909 0.07795 0.07683 0.07573

7627 2978 6684 8412

1.70 1.71 1.72 1.73 1.74

1.44959 1.45540 1.46117 1.46690 1.47258

22897 46507 50299 33404 94974

0.46696 0.46618 0.46534 0.46445 0.46351

83642 36359 57385 53716 32286

3.92096 3.95322 3.98562 4.01816 4.05084

3201 9462 9972 6395

0400

0.07465 0.07358 0.07253 0.07150 0.07048

4644 8518 9154 6255 9527

1.75 1.76 1.77 1.78

34189 50249 42379 09830 51872

0.46251 0.46147 0.46038 0.45924

99967 63568 29839 05471

1.79

1.47823 1.48383 1.48939 1.49491 1.50038

0.45804 97097

4.08365 4.11660 4.14970 4.18294 4.21633

3659 7647 4645 5736 2809

0.06948 0.06850 0.06753 0.06657 0.06563

8685 3447 3539 8691 8641

1.80 1.81 1.82 1.83 1.84

1.50581 1.51120 1.51655 1.52185 1.52711

67803 56942 18633 52243 57165

0.45681 0.45552 0.45419 0.45281 0.45139

11294 54585 33436 54262 23427

4.24986 4.28355 4.31738 4.35137 4.38551

7557 1681 6883 4872 7364

0.06471 0.06380 0.06290 0.06202 0.06115

3129 1903 4715 1320 1482

1.85 1.86 1.87 1.88 1.89

1.53233 32813 1.53750 78626 1.54263 94066

0.44992 0.44841 0.44685 0.44526 0.44362

47241 31966 83813 08948 13486

4.41981 4.45427 4.48888 4.52366 4.55860

6080 2746 9097 6872 7817

0.06029 0.05945 0.05862 0.05780 0.05699

4967 1545 0994 3091 7623

1.90

1.55777 53137 1.56273 42192 1.56764 98545

0.44194 0.44021 0.43845 0.43665 0.43481

03497 85005 63991 38088

4.59371 4.62898 4.66442 4.70003 4.73582

3687 62142 7249 8485 1734

0.05620 0.05542 0.05465 0.05389 0.05314

3149 3731 5927 9540

44941 72752 27288 14273 39391

4.77177 4.80791 4.84422 4.88071 4.91738

8785 1438 1501 0791

1131

0.05241 0.05169 0.05097 0.05027 0.04958

4380 0257 6988 4392 2291

4356

0.04890

1.52 1.53 1.54

1.91 1.92 1.93 1.94

1.54772

78621

1.55277 31800

1.57252 1.57735

21801

11591

1.95 1.96 1.97 1.98 1.99

1.58213 67567 1.58687 89407 1.59623 29502 1.60084 47231

0.43293 0.43101 0.42906 0.42707 0.42504

2.00

1.60541 29768

0.42298

1.59157

76810

[C-j)5 1

46388

cc-y1 08288

4.95423

6783 9528

c 1 (592

7839

4378

0511

c 1 (-!I3

242

EXPONENTIAL

Table

SINE,

5.1

INTEGRAL

AND

COSINE

AND

1.60541 1.64869 1.68762 1.72220 1.75248

Si(x) 29768 86362 48272 74818 55008

0.42298 0.40051 0.37507 0.34717 0.31729

1.77852 1.80039 1.81821 1.83209 1.84219

01734 44505 20765 65891 01946

0.28587 0.25333 0.22008 0.18648 0.15289

1.84865 1.85165 1.85140 1.84808 1:84191

25280 93077 08970 07828 39833

1.83312 1.82194 1.80862 1.79339 1.77650

FUNCTIONS

EXPONENTI.iL

INTEGRALS

se-"Ei(x) 1.34096 5420 1.37148 6802 1.39742 1992 1.41917 1534 1.43711 8315

sexI 0.72265 0.73079 0.73843 0.74562 0.75240

(x) 7234 1502 1132 2149 4829

11964 66161 48786 83896 53242

1.45162 1.46303 1.47166 1.47780 1.48174

5159 3397 2153 8187 6162

0.75881 0.76488 0.77063 0.77610 0.78130

4592 2722 6987 2123 0252

0.11962 0.08699 0.05525 +0.02467 -0.00451

97860 18312 74117 82846 80779

1.48372 1.48398 1.48274 1.48017 1.47646

9204 9691 0191 4491 8706

0.78625 0.79097 0.79548 0.79979 0.80391

1221 2900 1422 1408 6127

53987 81156 16809 03548 13604

-0.03212 -0.05797 -0.08190 -0.10377 -0.12349

85485 43519 10013 81504 93492

1.47178 1.46625 1.46003 1.45321 1.44590

2389 9659 0313 0902 5765

0.80786 0.81165 0.81529 0.81878 0.82214

7661 7037 4342 8821 8967

1.75820 1.73874 1.71836 1.69731 1.67583

31389 36265 85637 98507 39594

-0.14098 -0.15616 -0.16901 -0.17950 -0.18766

16979 53918 31568 95725 02868

1.43820 1.43020 1.42195 1.41354 1.40501

8032 0557 6813 1719 2424

0.82538 0.82849 0.83149 0.83439 0.83718

2600 6926 8602 3794 8207

1.65414 1.63246 1.61100 1.58997 1.56955

04144 03525 51718 52782 89381

-0.19349 -0.19704 -0.19839 -0.19760 -0.19477

11221 70797 12468 36133 98060

1.39641 1.38780 1.37920 1.37066 1.36219

9030 5263 9093 3313 6054

0.83988 0.84249 0.84501 0.84745 0.84982

7144 5539 7971 8721 1778

1.54993 1.53125 1.51367 1.49731 1.48230

12449 32047 09468 50636 00826

-0.19002 -0.18347 -0.17525 -0.16550 -0.15438

97497 62632 36023 59586 59262

1.35383 1.34558 1.33748 1.32953 1.32175

1278 9212 6755 7845 3788

0.85211 0.85432 0.85648 0.85856 0.86059

0880 9519 0958 8275 4348

1.46872 1.45666 1.44619 1.43735 1.43018

40727 83847 75285 91823 43341

-0.14205 -0.12867 -0.11441 -0.09944 -0.08393

29476 17494 07808 06647 26741

1.31414 1.30671 1.29947 1.29241 1.28555

3566 4107 0536 6395 3849

0.86256 0.86447 0.86633 0.86813 0.86989

1885 3436 1399 8040 5494

1.42468 1.42086 1.41870 1.41817 1.41922

75513 73734 68241 40348 29740

-0.06805 -0.05198 -0.03587 -0.01988 -0.00418

72439 25290 30193 82206 14110

1.27888 1.27240 1.26612 1.26002 1.25411

3860 6357 0373 4184 5417

0.87160 0.87327 0.87489 0.87647 0.87801

5775 0793 2347 2150 1816

1.42179 1.42581 1.43120 1.43786 1.44570

42744 61486 53853 84161 24427

+0.01110 0.02582 0.03985 0.05308 0.06539

15195 31381 54400 07167 23140

1.24839 1.24284 1.23748 1.23228 1.22726

1155 8032 2309 9952 6684

0.87951 0.88097 0.88240 0.88379 0.88515

2881 6797 4955 8662 9176

1.45459 66142 (-;)5

0.07669

52785

1.22240 8053 r( -$"l

0.88648

7675

c 1

Ci(x) 08288 19878 45990 56175 16174

RELATED

1 1 C-i’4

L

(

-I

[

C-i’6

1

EXPONENTIAL SINE,

INTEGRAL

COSINE

AND

AND

RELATED

EXPONEN’TJAL

243

FUNCTIONS

INTEGRAL!?

Table

.a-rEi(r)

:fPlr:,

3.1

1.51068 1.52331 1.53610 1.54893 1.56167

15309 37914 92381 74581 10702

Ci (.r) 0.07669 52785 0.08690 68881 0.09595 70643 0.10378 86664 0.11035 76658 0.11563 32032 0.11959 75293 0.12224 58319 0.12358 59542 0.12363 80071

1.57418 1.58636 1.59809 1.60921 1.61980

68217 66225 85106 75419 65968

0.12243 0.12001 0.11644 0.11176 0.10607

38825 66733 00055 72931 09196

1.18184 1.17849 1.17524 1.17210 1.16906

7987 2509 6676 6376 7617

0.89823 0.89927 0.90029 0.90129 0.90227

7113 7888 7306 6033 4695

1.62959 1.63856 1.64665 1.65379 1.65993

70996 96454 45309 21861 35052

1.66504 1.66908 1.67204 1.67392 1.67412 1.67446 1.67315 1.67084 1.66756 1.66338 1.65834

00758 43056 94480 95283 91725 33423 69801 45697 96169 40566 75942

0.09943 0.09193 0.08367 0.07475 0.06528 0.05534 0.04506 0.03455 0.02391 0.01325

13586 62396 93696 97196 03850 75313 93325 49134 33045 24187

1.16612 1.16327 1.16052 1.15785 1.15526 1.15275 1.15032 1.14797 1.14568 1.14347

6526 9354 2476 2390 5719 9209 9724 4251 9889 3855

0.90323 0.90417 0.90509 0.90600 0.90688 0.90775 0.90861 0.90944 0.91027 0.91107

3900 4228 6235 0459 7415 7602 1483 9530 2177 9850

+O. 00267 -0.00770 -0.01780 -0.02751

80588 70361 40977 91811

-0.03616

39563

1.14132 1.13923 1.13720 1.13524 1.13332

3476 6185 9523 1130 8746

0.91187 0.91265 0.91341 0.91416 0; 91490

2958 1897 7043 8766 7418

Si (.T) 1.45459 66142 1.46443 32441 1.47508 90554 1.48643 64451 1.49834 47533

-0.04545 64330

(2)

1.22240 1.21770 1.21316 1.20877 1.20452 1.20042 1.19645 1.19261 1.18890 1.18531

8053 9472 6264 3699 7026 1500 2401 5063 4881 7334

0.88648 0.88778 0.88905 0.89029 0.89150 0.89268 0.89384 0.89497 0.89608 0.89717

7675 5294 3119 2173 3440 7854 6312 9666 8737 4302

1.13147 0205

(-55)2

[ 1

0.91563 3339 (-46)4

[ 1 Table

SINE,

COSINE

AND

EXPONENTIAL

INTEGRALS

FOR

0351 4427 9171 1776 9405 9188 8244 3695 2682 2385

0.94885 0.95323 0.95748 0.96160 0.96557

39 18 44 17 23

.~.v*Ei (.v) 1.13147 021 1.12249 671 1.11389 377 1.10564 739 1.09773 775

0.075 0.070 0.065 0; 060 0.055

0.98191 0.98353 0.98509 0.98660 0.98803 0.98940 0.99070 0.99193 0; 99308 0.99415

0.96938 0.97302 0.97649 0.97976 0.98283

56 98 35 47 17

1.09014 087 1; 08283 054 1.137578 038 1.06896 548 1.06236 365

0.050 0.045 0.040 0.035 0.030

0.99514 0.99604 0. 99685 0.99758 0.99821

0052 3013 8722 4771 a937

0.98568 0.98830 0.99068 0.99282 0.99469

0.025 0.020 0.015 0.010 0.005

0.99875 0.99920 0.99955 0.99980 0.99995

9204 3795 1207 0239 0015

0.99629 0.99761 0.99865 0.99940 0.99985

0.000

24 52 al 12 37 57 a9 60 12 01 00

1.00000 0000 1.00000 c (-5)l 5 c (-Z)4 3 3 Si (,I,)= ; -f (.v)cos .I.-q(.r) sin .I’

1.05595 591 1.04972 640 1.04366 194 1.03775 135 I.03198 503 1.02635 451 1.02085 228 1.01547 157 1.01020 625 1.00505 077 1.00000 000 (65)5

.c-’

0.100 0.095 0.090 0.085 0.080

.r”f (I)

.J”g(.r)

; =1.57079 63268 See Exat~~ple

3.

II 1

LARGE

ARGtiMEhTS

.NEl 0. 91563 0.91925 0.92293 0.92665 0.93044 0.93427 0.93817 n. 94213 i: 94614 0.95022

(.I.) 33394 68286 15844 90998 09399 87466 424% 924% 56670 55126

0.95437 0.95858 0.96286 0.96722 0.97165 0.97616 0.98075 0.98543 0.99019 0.99504 1.00000

09099 41038 74711 35311 49596 46031 54965 08813 42287 92646 00000

c-y1 [ Ci (,c)=J (.Y)sin .~-y [I,) ~0s .L’

<* rel="nofollow">=nearest

5.2

integer to ,!,.

<s> 10 ::

:23 13 14 15 :87 :; 22; 33 40 50 lob07 200 cm

244

EXPONENTIAL Table 5.3

SINE AND COSINE

Ci

INTEGRALS

FOR ARGUMENTS

TX

Si(sx) 1.63396 48 1.63088 98 1.62211 92 1.60871 21 1.59212 99 1.57408 24 1.55635 75 1.54064 a2 1.52839 53 1.52065 96

Cin (*.r) 3.32742 23 3.36670 50 3.40335 al 3.43582 68 3.46297 82 3.48419 47 3.49941 45 3.50911 a9 3.51426 a9 3.51619 al

1.51803 39 1.52060 20 1.52794 77 1.53921 04 1.55318 17 1.56843 12 1.58344 97 1.59679 62 1.60723 30 1.61383 a5

3.51647 44 3.51674 38 3.51857 25 3.52330 06 3.53192 30 3.54500 55 3.56264 55 3.58447 72 3.60972 10 3.63727 15

1.61608 55 1.61388 08 1.60756 la 1.59785 21 1.58578 13 1.57257 88 1.55954 96 1.54794 a1 1.53885 a4 1.53309 50

3.66581 26 3.69395 05 3.72034 97 3.74385 98 3.76362 13 3.77914 01 3.79032 64 3.79749 22 3.80131 21 3.80274 91

2.80993 76 2.87498 49 2.93491 77 2.98737 63 3.03074 73 3.06427 25 3.08807 51 3.10310 38 3.11100 53 3.11393 95

1.53113 13 1.53306 26 1.53860 67 1.54713 99 1.55776 52 1.56940 54 1.58091 06 1.59117 06 1.59922 11 1.60433 29

3.80295 56 3.80315 a3 3.80453 aa 3.80812 16 3.81467 97 3.82466 68 3.83818 15 3.85496 61 3.87444 05 3.89576 52

1.49216 12 1.49599 24 1.50687 40 1.52343 40 1.54382 74 1.56593 04 1.58755 15 1.60664 04 1.62147 45 1.63080 69

3.11435 65 3.11475 a2 3.11746 60 3.12441 61 3.13699 91 3.15595 79 3.18134 a4 3.21256 74 3.24843 a5 3.28734 92

1.60607 69 1.60435 a5 1.59942 00 1.59180 91 1.58232 00 1.57191 16 1.56161 12 1.55241 46 1.54519 00 1.54059 74

3.91792 84 3.93984 77 3.96047 61 3.97890 22 3.99443 58 4.00666 94 4.01551 22 4.02119 22 4.02422 80 4.02537 29

1.63396 48 (-s3)5

3.32742 23 (-s3)6

Cin(ux) 0.00000 00 0.02457 28 0.09708 67 0.21400 75 0.36970 10 0.55679 77 0.76666 63 0.98995 93 1.21719 42 1.43932 68

1.85193 70 1.83732 28 1.79815 90 1.74191 10 1.67621 68 1.60837 27 1.54487 36 1.49103 51 1.45072 37 1.42621 05

1.64827 75 1.83737 48 2.00168 51 2.13821 22 2.24595 41 2.32581 a2 2.38040 96 2.41370 98 2.43067 75 2.43680 30

6. 0

1.41815 16 1.42569 13 1.44667 38 1.47794 03 1.51568 40 1.55583 lo 1.59441 60 1.62792 16 1.65355 62 1.66945 05

2.43765 34 2.43844 23 2.44365 73 2.45676 95 2.48004 47 2.51446 40 2.55975 53 2.61452 59 2.67647 93 2.74269 41

7. 0

1.67476 la 1.66968 11 1.65535 02 1.63369 a2 1.60721 aa 1.57870 92 1.55099 62 1.52667 49 1.50788 19 1.49612 20

a: 21” 2:: 55:: E

5: a 5. 9 z-: 6: 3 2: 6: 6 2: 6: 9 ;: 7: 3 ::5" ::7" ::9"

1.53902 91 4.02553 78 c-y (-;)7 [ I [ I Ci(m)=r+ln *+ln x-Cin(rxi) r+ln r=1.72194 55508 maximum values of si(x) if 7~>0is odd, and minimum values if n>O is even.

are

K )I NT

AND RELATED FUNCTIONS

Si(Rx) 0.00000 00 0.31244 la 0.61470 01 0.89718 92 1.15147 74 1.37076 22 1.55023 35 1.68729 94 1.78166 12 1.83523 65

[ 1

Si(nr)

INTEGRAL

2

?r ,are

odd. We have

maximum

10.0

[ 1

values

of

ci(.z)

if n>O is even, and minimum values if n>Ois

si(7z7r)mi-@$[1-n+2+&-.

. .] (n+~0)

EXPONENTIAL

INTEGRAL

EXPONENTIAL

RELATED

INTEGRALS

245

FUNCTIONS

En(z)

Table

5.4

0.01 0. 02 0.03 0. 04

&(x)-x 1.00000 0.99572 0.99134 0.98686 0.98229

0.05 0.06 0.07 0.08 0.09

0.97762 11 0.97285 08 0.96798 34 0.96301 94 0.95795 93

0.45491 0.44676 0.43883 0.43111 0.42360

88 09 27 97 96

0.30949 0.30498 0.30055 0.29620 0.29193

45 63 85 89 54

0.10503 0.10386 0.10270 0.10155 0.10042

63 24 18 44 00

0.04992 0.04940 0.04888 0.04837 0.04786

60 19 33 02 24

0. 10 0.11 0. 12 0.13 0. 14

0.95280 0.94755 0.94220 0.93676 0.93123

35 26 71 72 36

0.41629 0.40915 0.40219 0.39539 0.38876

15 57 37 77 07

0.28773 0.28360 0.27955 0.27556 0.27164

61 90 24 46 39

0.09929 0.09818 0.09709 0.09600 0. 09,493

84 96 34 95 80

0.04736 0.04686 0.04637 0.04588 0.04540

00 29 10 43 27

0.15 0. 16 0.17 0.18 0.19

0.92560 67 0.91988 70 0.91407 48 0.90817 06 0.90217 50

0.38227 0.37593 0.36974 0.36367 0.35774

61 80 08 95 91

0.26778 0.26399 0.26026 0.25660 0.25299

89 79 96 26 56

0.09387 0.09,283 0.09179 0.09077 0.08975

86 12 56 18 95

0.04492 0.04445 0.04398 0.04352 0.04306

62 47 82 66 98

0.20 0.21 0. 22 0.23 0. 24

0.89608 82 0.88991 09 0.88364 33 0.87728 60 0.87083 93

0.35194 0.34626 0.34070 0.33525 0.32991

53 38 05 18 42

0.24944 72 0.24595 63 0.24252 16 0.23914 19 0.23581 62

0.081375 0.08'776 0.08679 0. 08!582 0.08486

87 93 10 38 75

0.04261 0.04217 0.04172 0.04129 0.04085

79 07 82 03 71

0.25 0.26 0.27 0.28 0.29

0.86430 37 0.85767 97 0.85096 76 0.84416 78 0.83728 08

0.32468 0.31955 0.31453 0.30960 0.30477

41 85 43 86 87

0.23254 32 0.22932 21 0.22615 17 0.22303 11 0.21995 93

0.08392 20 0.08298 72 0.08206 30 0.08U4 92 0.08024 57

0.04042 0.04000 0.03958 0.03916 0.03875

85 43 46 93 84

0.30

0.83030 71 0.82324 69 0.81610 07 0.80886 90 0.80155 21

0.30004 0.29539 0.29083 0.28636 0.28197

18 56 74 52 65

0.21693 0.21395 0.21102 0.20814 0.20529

52 81 70 11 94

0.07935 24 0.07846 93 0.07759 60 0.07673 27 0.07587 90

0.03835 0.03794 0.03755 0.03715 0.03676

18 95 15 76 78

8% p; .

0.79415 0.78666 0.77909 0.77144 0.76370

04 44 43 07 39

0.27766 0.27344 0.26929 0.26521 0.26121

93 16 13 65 55

0.20250 0.19974 0.19703 0.19435 0.19172

13 58 22 97 76

0.07503 0.07420 0.07337 0 072!55 0:071!75

50 06 55 97 31

0.03638 0.03600 0.03562 0.03524 0.03487

22 06 31 95 98

0.40 0.41 0.42 0.43 0.44

0.75588 0.74798 0.73999 0.73193 0.72378

43 23 82 24 54

0.25728 0.25342 0.24963 0.24591 0.24225

64 76 73 41 63

0.18913 0.18658 0.18406 0.18158 0.17914

52 16 64 87 79

0.07095 0.07016 0.06938 0.06061 0. 06785

57 71 75 67 45

0.03451 40 0.03415 21 0.03379 39 0.03343 96 0.03308 89

0.45 0. 46 0.47 0.48 0. 49

0.71555 0.70724 0.69886 0.69039 0.68184

75 91 05 21 43

0.23866 0.23513 0.23166 0.22825 0.22489

25 13 12 08 90

0.17674 33 0.17437 44 0.17204 05 0.16974 10 0.16747 53

0.06710 0.06635 0.06561 0.06489 0.064117

09 58 91 07 04

0.03274 0.03239 0.03205 0.03172 0.03139

20 87 90 29 03

0.50

0.67321

0.16524

0. 06?145 83

0.03106 (-;I7

12

2 0. 00

i-33: 0: 33 0.34 0.35

&X

Examples

la x 00 22 50 87 39

AND

75 c-y 1. 1 4-6.

E3(4

E4 (x)

&o(x)

1~1oCx)

0.49027 66 0.48096 83 0.47199 77 0.46332 39

0.33333 0.32838 0.32352 0.31876 0.31408

33 24 64 19 55

0.11111 0.10986 0.10863 0.10742 0.10622

11 82 95 46 36

0.05263 0.05207 0.05153 0.05099 0.05045

16 90 21 11 58

0.50000

00

0.22160 44 C-i)5

II 1

28 (-;)I II 1

c(-92 1

c 1

246

EXPONENTIAL Table

EXPONENTIAL

5.4

Edx)

Ed4 o.“so 0.51 0.52 0. 53 0. 54 0.55 0. 56 0.51 0.58 0. 59

0.32664

39

0.31568

63

0.32110 62 0.31038 07 0.30518 62 0.30009

96

0.29511 79 0.29023 0.28545 0.28077

82 78 39

0.60 0.61 0. 62 0. 63 0. 64

0.27618

39

0.26295

35

0. 65 0.66 0. 61 0. 68 0.69

0.25455 0.25048 0.24648 0.24256 0.23872

97 44 74 67 06

0. 70

0. 71

0.23494 71 0.23124 46

0. 72 0. 73 0. 74

0.22761 0.22404 0.22054

0.75 0. 76 0. 77 0. 78 0. 79

0.21711 09 0.21373 88

0.27168 55 0.26727 61 0.25871 54

14 57 61

0.21042 0.20717 0.20398

82 77 60

0.80 0.81 0. 82 0. 83 0. 84

0.20085 0.19777

17 36

0.85 0.86 0. 87 0.88 0.89

0.18599 86 0.18318 33 0.18041 73

0.90

0.91 0.92 0.93 0.94

INTEGRAL

0.19475 04 10 0.18886 41

0.19178

0.17769 0.17502

94 87

0.17240 0.16982 0.16728

41 47 95

0.16479 77 0.16234 82

0.95 0.96 0.97 0.98 0.99

0.15757 0.15524 0.15295 0.15070

0.15994 04

1.00

0.14849

32 59 78 79

cc-y1 55

RELATED

INTEGR.4LS

44 57 18 16

0.20897

39

0.20594 0.20297 0.20004

75 15 48

0.19716 64 0.19433 53

0.16524

FUNCTIONS E,(r)

EIO(X)

E4b3

0.22160 0.21836 0.21518 0.21205

0.19155 0.18881 0.18611 0.18346 0.18085

AND

28

0.16304 30 0.16087 53 0.15873 0.15663

92 41

0.15455 0.15251

96 50

0.15050 00 0.14851 39 0.14655 65

83 42 80 96 89

0.06001 0.05935 0.05869

59 05 25

0.05804 19

0.05489 0.05428

69 89

50 12

77 33 55

0.02652 0.02624 0.02596 0.02569 0.02542

04 25 75 54 62

0.02515 0.02489 0.02463 0.02437 0.02412

98 62 53 72 19

0. 02386

92

0.16606

12

0.12678

08

0.05078

0.12513 0.12350 0.12190 0.12032

19 61 31 24

0.04966 0.04911

0.11876

38

0.15476 0.15261 0.15049 0.14840 0.14634

67 25 17 37 79

0.14432 0.14233 0.14036 0.13843 0.13653

0.11722 70

0.05192 43 0.05134 97 15 0.05021 96

0.04803 0.04750 0.04697 0.04645 0.04594

44 33 81 88 53

0.04543 0.04493 0.04443 0.04394 0.04346

15 70 33

38 07 81 55 24

0.11129 0.10985

00 67

0.10704 0.10567

93 44

0.13465 0.13281 0.13099 0.12920 0.12744

81 22 43 37

0.10431 0.10298

85 12

0.04298 0.04250 0.04203

01

0.09907

80

0.12570 0.12399 0.12230 0.12064 0.11901

30 19 63 59 02

0.04066

0.09656 0.09533 0.09411 0.09291

39 24 77 94

0.11739 0.11581 0.11424 0.11270

88 13 72 63

0.11118 80

0.09173 0.09057 0.08942 0.08828 0.08716

74 13 11 63 69

0.10969 20

0.08606

25

0.03639

[ c-y1

0.10166 22 0.10036 12 0.09781 23

40 47

0.04857 15

0.11571 0.11421 0.11274

0.10844 33

25

0.02855 01 48 18

58 94

03 60 78 49

0.02885

0.02795 0.02766 0.02737 0.02708 0.02680

0.05613 36 0.05551 18

0.17083 0.16842

0.16373 0.16143 0.15917 0.15695

70

0.02915 81

26

0.05368 0.05309 0.05250

0.17328 10

0.02946

0.05676

55 53 01 95 33

58

46 91

08

0.13538 0.13361 0.13187 0.13014 0.12845

0.17576

0.03041 34 0.03009 0.02977

0.02825

0.17829 10

0.13718 13

12 56

86

0.14462

71 53 07 28

0.03106 0.03073

0.05739

06 14 66 56 73

0.14272 0.14085 0.13900

&o(x)

0.06345 0.06275 0.06205 0.06136 0.06068

19

0.02361 91 0.02337

17

0.02288

46

76 56 91 82 28

0.02264 0.02240

49 78

29 82 89

0.04157 49

0.02148 0.02125 0.02103 0.02081

37 87 61 58

0.04111

60

0.02059

78

22

0.02038 0.02016 0.01995 0.01974

21 87 75 86

0.04021 35 0.03976

98

0.03933 11

0.02312 69

0.02217 31 0.02194 08 0.02171 11

0.03889

73

0.01954 18

0.03846

83

0.03762 0.03720 0.03679

46 98 96

0.01933 0.01913 0.01893 0.01873 0.01854

0.03804 41

c(-;I11 40

72 47 44 62 01

0.01834 60

[I(-P4 1

EXPONENTIAL

INTEGRAL

AND RELATED

1.00 1. 01 1.02 1. 03 1.04

82 (.r) 0.14849 55 0.14631 99 0.14418 04 0.14207 63 0.14000 68

EXPONENTIAL &::I(J) 0.10969 20 0.10821 79 0.10676 54 0.10533 42 0.10392 38

INTEGRALS

1.05 1. 06 1. 07 1. 08 1.09

0.13797 0.13596 0.13399 0.13206 0.13015

13 91 96 22 62

0.10253 0.10116 0.09981 0.09848 0.09717

39 43 45 42 31

0.08075 0.07974 0.07873 0.07774 0.07676

1.10 1. 11 1.12 1.13 1. 14

0.12828 0.12643 0.12462 0.12283 0.12107

11 62 10 50 75

0.09588 0.09460 0.09335 0.09211 0.09089

09 74 21 49 53

1.15 1. 16 1.17 1. 18 1.19

0.11934 0.11764 0.11597 0.11432 0.11270

81 62 14 31 08

1.20 1. 21 1.22 1.23 1.24

0.11110 0.10953 0.10798 0.10646 0.10496

1.25 1.26 1.27 1.28 1.29

Table

5.4,

0.01834 0.01815 0.01796 0.01777 0.01758

(.I.) 60 39 39 59 98

90 06 57 42 59

0.03443 0.03405 0.03367 0.03330 0.03294

28 35 85 77 10

0.01740 0.01722 0.01704 0.01686 0.01668

57 35 33 49 84

0.07580 0.07484 0.07390 0.07298 0.07206

07 83 85 12 61

0.03257 0.03221 0.03186 0.03151 0.03116

84 98 52 45 78

0.01651 0.01634 0.01616 0.01600 0.01583

37 09 99 07 33

0.08969 32 0.08850 83 0.08734 02 0.08618 88 0.08505 37

0.07116 0.07027 0.06939 0.06852 0.06766

32 22 30 53 91

0.03082 0.03048 0.03015 0.02981 0.02949

49 58 05 89 10

0.01566 0.01550 0.01534 0.01518 0.01502

76 37 14 09 21

41 25 55 27 37

0.08393 0.08283 0.08174 0.08067 0.07961

47 15 39 17 46

0.06682 0.06599 0.06516 0.06435 0.06355

42 04 75 55 40

0.02916 0.02884 0.02852 0.02821 0.02790

68 61 90 55 54

0.01486 0.01470 0.01455 0.01440 0.01425

49 94 55 32 26

0.10348 0.10203 0.10060 0.09919 0.09781

81 53 51 70 06

0.07857 0.07754 0.07653 0.07553 0.07454

23 47 16 26 76

0.06276 0.06198 0.06121 0.06045 0.05970

31 25 22 19 15

0.02759 88 0.02729 55 0.02699 57 0.02669 91 0.02640 59

0.01410 0.01395 0.01381 0.01366 0.01352

35 59 00 55 26

1.30 1. 31 1.32 1.33 1.34

0.09644 0.09510 0.09377 0.09247 0.09119

55 15 80 47 13

0.07357 0.07261 0.07167 0.07074 0.06982

63 86 42 29 46

0.05896 0.05822 0.05750 0.05679 0.05609

09 99 85 64 36

0.02611 0.02582 0.02554 0.02526 0.02498

59 91 55 51 78

0.01338 0.01324 0.01310 0.01296 0.01283

11 12 27 57 01

1.35 1. 36 1. 37 1. 38 1. 39

0.08992 0.08868 0.08745 0.08624 0.08506

75 29 71 99 10

0.06891 91 0.06802 60 0.06714 53 0.06627 68 0.06542 03

0.05539 0.05471 0.05403 0.05337 0.05271

98 51 93 22 37

0.02471 0.02444 0.02417 0.02390 0.02364

35 23 41 88 65

0.01269 0.01256 0.01243 0.01230 0.01217

59 31 17 17 31

1. 40 1.41 1.42 1.43 1.44

0.08388 0.08273 0.08160 0.08048 0.07937

99 65 04 13 89

0.06457 0.06374 0.06292 0.06211 0.06131

55 24 07 04 11

0.05206 0.05142 0.05078 0.05016 0.04954

37 22 89 37 66

0.02338 0.02313 0.02287 0.02262 0.02237

72 06 70 61 80

0.01204 0.01191 0.01179 0.01167 0.01154

58 98 52 19 99

1.45 1.46 1.47 1. 48 1.49

0.07829 0.07722 0.07616 0.07513 0.07410

30 33 94 13 85

0.06052 0.05974 0.05897 0.05822 0.05747

27 52 82 17 55

0.04893 0.04833 0.04774 0.04715 0.04657

74 61 25 65 80

0.02213 0.02189 0.02165 0.02141 0.02117

27 01 01 28 82

0.01142 0.01130 0.01119 0.01107 0.01095

91 96 14 44 86

1.50 1.51 1.52 1.53 1.54

0.07310 0.07210 0.07112 0.07016 0.06921

08 80 98 60 64

0.05673 0.05601 0.05529 0.05459 0.05389

95 35 73 08 39

0.04600 0.04544 0.04488 0.04433 0.04379

70 32 67 72 48

0.02094 0.02071 0.02048 0.02026 0.02004

61 67 97 53 33

0.01084 0;01073 0.01061 0.01050 0.01039

40 07 85 75 77

1.55 1.56 1. 57 1.58 1.59

0.06828 0.06735 0.06645 0.06555 0.06467

07 87 02 49 26

0.05320 0.05252 0.05185 0.05119 0.05054

64 83 92 92 81

0.04325 0.04273 0.04220 0.04169 0.04118

93 07 87 35 47

0.01982 0.01960 0.01939 0.01917 0.01896

38 67 21 98 98

0.01028 0.01018 0.01007 0.00996 0.00986

90 15 50 97 56

1.60

0.06380 32 C-3615

0.04990 57 C-i)3

0.04068

25

0.01876

22

0.00976 24 C-l)3

[ 1 [ 1

(A) 25 30 81 76 13

E,(z)

247

~,~(.r) 0.03639 40 0.03599 29 0.03559 63 0.03520 41 0.03481 63

.I:

0.08606 0.08497 0.08389 0.08283 0.08179

FUNCTIONS

JY4

c-y [

(-;I6 I

[

I

-f320

[ 1

248

EXPONENTIAL Table

INTEGRAL EXPONENTIAL

5.4

l.iO 1. 61

0.06380 0.06294

32 64

0.06044

97

0.06210 20 0.06126 98

1. 65 1.66 1. 61

0.05964 13

1. 70

1.71 1. 72 1.73 1.74

0.05884 0.05805 0.05728 0.05652

46 94 54 26

0.05577 0.05502 0.05429 0.05357 0.05286

06 94 88 86 86

1.75 1.76 1.77 1.78 1.79

0.05216 87 0.05147 88 86 0.05012 81

1.80 1.81 1. 82 1. 83 1. 84

0.04881 53 0.04817 27

1. 85 1. 86 1.87 1. 88

0.05079 0.04946

70

0.04753

92

0.04629

87

0.04569 0.04509 0.04450 0.04392 0.04334

15 28 24 03 63

1.90 1.91 1.92 1.93 1.94 1.95 1. 96 1.97 1.98 1. 99

0.04278 0.04222

03 22

0.04059

38

0.04006 0.03954 0.03903 0.03852 0.03802

2. 00

0.03753

1. 89

0.04691 46

57 20 67 99

0.04682 0.04622 0.04564 0.04506 0.04449

25 66 70

0.03873

64

09 84 39 72 82

0.03826 0.03779 0.03734 0.03688 0.03643

52 99 06 70 92

0.04393 0.04338

67 27

0.03599 0.03556

70 04

0.04176

45

0.03428

34

0.03386 0.03345 0.03305 0.03265 0.03225

84 86 39 44 98

0.04123

93

0.04020 0.03970 0.03920

97 51 71

0.03871 0.03823 0.03775 0.03728 0.03681

57 08 22 00 39

0.03635

40

0.03545

21

5.5

0.03329 0.03288 0.03247

86 46 59

60 55 22 59 67

0.03207

27

43

0.03013 34

(.c+3)eZE&)

1.10937

1.11329

0. 07

0.06 0.05 0.04 0. 03

0.02 0.01 0.00

1.01045

1.00861 1.00688 1.00528 1.00384 1.00258 1.00152 1.00071 1.00019

0.02823 0.02789 0.02755 0.02722 0.02690

23 30 79 70 02

0.02657 0.02625 0.02594

75 87 40

0.02563 31 0.02532 61 0.02502 28

68 38 30 43 79 37 16

1.10285 1.09185 1.08026 1.06808 1.05536 1.04222 1.02895 1.01617 1.01377 1.01147 1.00927 1.00721 1.00531 1.00361 1.00217 1.00103 1.00027

INTEGRALS

E,,(x)

(r+4)e”E,(s)

1.10937 1.10071 1.09136 1.08125 1.07031 1.05850 1.04584 1.03247 1.01889 1.01624 1.01366 1.01116 1.00878 1.00654 1.00451 1.00275 1.00133 1.00036

FOR

16 37

0.01680 0.01662 0.01644 0.01626 0.01608

79 42 24 27 50

0.01590 0.01573 0.01556 0.01539 0.01522 0.01505 0.01489 0.01473 0.01457 0.01441

92 54 34 34 53 90 45 18 10 19

0.01425 0.01409 0.01394 0.01379 0.01364

46 90 51 29 24

0.01349 0.01334 0.01320 0.01305 0.01291 0.01277 0.01263 0.01249 0.01236 0.01222 0.01209

35 63 07 67 43 34 41 64 01 54 21

LARGE

(z+lO)e=E,o(z)

1.07219 1.06926 1.06586 1.06187 1.05712 1.05138 1.04432 1.03550 1.02436 1.02182 1.01917 1.01642 :* %!8 1:00790 1.00516 1.00271 1.00081

<.r>=nearest integer to .I’.

0.00976 0.00966 0.00955 0.00945 0.00936

24 04 95 96 07

0.00926

29

0.00907 0.00897

03 56

0.00869 0.00860

72 63

0.00833 0.00825

94 22

0.00799

63

0.00916 61 0.00888 18 0.00878 90 0.00851 64 0.00842 74 0.00816 60 0.00808 07 0.00791 28 0.00783 02 0.00774 0.00766 0.00758

84 74 74

0.00750 81 97 0.00735 21

0.00742 0.00727

53

0.00712 0.00704 0.00697 0.00690 0.00683

42 98 62 33 12

0.00675 0.00668

99 93

0.00648

20

0.00719 93

[ 1

(.r+2)@&(“.)

1.04770

38

22

0.01718 0.01699

(-l)3

T-1

1.03522 1.02325 1.01240

0.02927 0.02892

0.02857 59

0.03167 46 0.03128 17 0.03089 39 0.03051 12

EXPONENTIAL

1.09750 1.08533 1.07292 1.06034

04 99

0.02999 41 0.02963 28 61

0.03371 80

0. 50 0.45

0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.09 0.08

0.03073 0.03035

0.03501 00 0.03457 37

1 1 Table

0.03187 02 0.03148 55 0.03110 56

0.03590 01

30

0.01855 0.01835 0.01815 0.01795 0.01775 0.01756 0.01737

0.03512 93 0.03470 37

0.04072 11

0.03414

0.01876

0.03921 36

0.04283 61 0.04229 67

Go (J)

J%o(.r)

0.04068 0.04018 0.03969

13

FUNCTIONS E,,(z)

E14(T)

0.04990 0.04927 0.04864 0.04802 0.04742

0.04167 18 0.04112 91

c-y

INTEGRALS

Id:< (.r)

I& (.r)

1. 62 1.63 1. 64

1.68 1. 69

AND RELATED

0.00661 95 0.00655 04 0.00641 43

[(-p1

ARGUMENTS (a+20)e~E~o(.c)

1.04270 1.04179 1.04067 t 00:;:: 1:03543 1.03249 1.02837 1.02222 1.02060 1.01883 1.01688 1.01472 1.01234 :* ~0096;; 1:00401 1.00137

St?

EXPONENTIAL

INTEGRAL

EXPONENTIAL

INTEGRAL

.f

R

-19

Y\X

9

-18

AND

RELATED

FOR COMPLEX zeZEl(2) $3 .Y-

-17

249

FUNCTIONS

Table 5.6

ARGUMENTS

4

?z -16

&

Jf

-15

1.063087 1.062827 1.062061 1.060829 1.059190

0.000001 0.004010 0.007918 0.011633 0.015079

1.067394 1.067073 1.066135 1.064636 1.062657

0.000002 0.004584 0.009032 0.013226 0.017075

1.072345 1.071942 1.070774 1.068925 1.066508

0.000006 0.005296 0.010403 0.015172 0.019486

1.078103 1.077584 1.076102 1.073783 1.070793

0.000014 0.006195 0.012118 0.017579 0.022432

1.057215 1.054981 1.052565 1.050037 1.047458

0.018202 0.020969 0.023364 0.025391 0.027066

1.060297 1.057655 1.054829 1.051905 1.048958

0.020512 0.023505 0.026044 0.028141 0.029824

1.063659 1.060510 1.057187 1.053795 1.050421

0.023272 0.026499 0.029167 0.031306 0.032960

1.067318 1.063538 1.059610 1.055664 1.051797

0.026598 0.030055 0.032823 0.034957 0.036527

1.044880 1.042345 1.039882 1.037515 1.035259

0.028412 0.029461 0.030245 0.030796 0.031148

1.046045 1.043212 1.040490 1.037901 1.035456

0.031130 0.032102 0.032781 0.033211 0.033431

1.047129 1.043967 1.040965 1.038140 1.035501

0.034183 0.035034 0.035567 0.035836 0.035888

1.048081 1.044559 1.041259 1.038192 1.035359

0.037609 0.038282 0.038616 0.038677 0.038520

0.029511 0.029445 0.029296

1.033123 1.031110 1.029222 1.027456 1.025809

0.031330 0.031368 0.031288 0.031110 0.030854

1.033162 1.031019 1.029025 1.027174 1.025459

0.033476 0.033377 0.033162 0.032855 0.032474

1.033049 1.030780 1.028685 1.026756 1.024981

0.035765 0.035502 0.035129 0.034672 0.034150

1.032754 1.030365 1.028180 1.026183 1.024360

0.038193 0.037735 0.037179 0.036552 0.035873

0.029080

1.024275

0.030534

1.023872

0.032037

1.023349

0.033582

1.022695

0.035160

1.084892 1.084200 1.082276 1.079313 1.075560

0.000037 0.007359 0.014306 0.020604 0.026075

1.093027 1.092067 1.089498 1.085635 1.080853

0.000092 0.008913 0.017161 0.024471 0.030637

1.102975 1.101566 1.098025 1.092873 1.086686

0.000232 0.011063 0.020981 0.029507 0.036422

1.115431 1.113230 1.108170 1.101137 1.093013

0.000577 0.014169 0.026241 0.036189 0.043843

1.131470 1.127796 1.120286 1.110462 1.099666

0.001426 0.018879 0.033700 0.045218 0.053451

1.071279 1.066708 1.062046 1.057448 1.053021

0.030642 0.034303 0.037117 0.039174 0.040580

1.075522 1.069960 1.064412 1.059054 1.053997

0.035599 0.039405 0.042169 0.044041 0.045176

1.079985 1.073185 1.066578 1.060352 1.054606

0.041724 0.045552 0.048115 0.049644 0.050359

1.084526 1.076197 1.068350 1.061159 1.054687

0.049336 0.052967 0.055093 0.056057 0.056158

1.088877 1.078701 1.069450 1.061235 1.054046

0.058817 0.061886 0.063225 0.063322 0.062566

1.048834 1.044928 1.041320 1.038010 1.034989

0.041444 0.041867 0.041938 0.041734 0.041321

1.049303 1.044997 1.041080 1.037537 1.034344

0.045719 0.045801 0.045531 0.044999 0.044277

1.049380 1.044674 1.040464 1.036713 1.033378

0.050452 0.050084 0.049384 0.048452 0.047365

1.048933 1.043853 1.039389 1.035473 1.032040

0.055640 0.054695 0.053465 0.052056 0.050547

1.047807 1.042417 1.037766 1.033752 1.030282

0.061249 0.059584 0.057719 0.055758 0.053773

1.032241 1.029747 1.027486 1.025437 1.023580

0.040751 0.040066 0.039301 0.038481 0.037629

1.031474 1.028895 1.026579 1.024499 1.022628

0.043422 0.042477 0.041475 0.040444 0.039401

1.030414 1.027781 1.025438 1.023352 1.021489

0.046180 0.044941 0.043679 0.042417 0.041170

1.029026 1.026377 1.024043 1.021981 1.020155

0.048991 0.047428 0.045883 0.044374 0.042912

1.027274 1.024658 1.022375 1.020375 1.018617

0.051808 0.049894 0.048049 0.046282 0.044599

1.021896

0.036759

1.020942

0.038361

1.019824

0.039950

1.018533

0.041505

1.017066

0.043001

1.152759 1.146232 1.134679 1.120694 1.106249

0.003489 0.026376 0.044579 0.057595 0.065948

1.181848 1.169677 1.151385 1.131255 1.111968

0.008431 0.038841 0.060814 0.074701 0.082156

1.222408 1.199049 1.169639 1.140733 1.115404

0.020053 0.060219 0.085335 0.098259 0.102861

1.278884 1.233798 1.186778 1.146266 1.114273

0.046723 0.097331 0.122162 0.130005 0.128440

1.353831 1.268723 1.196351 1.142853 1.105376

0.105839 0.160826 0.175646 0.170672 0.158134

1.092564 1.080246 1.069494 1.060276 1.052450

0.070592 0.072520 0.072580 0.071425 0.069523

1.094818 1.080188 1.067987 1.057920 1.049645

0.085055 0.084987 0.083120 0.080250 0.076885

1.094475 1.077672 1.064339 1.053778 1.045382

0.102411 0.099188 0.094618 0.089537 0.084405

1.089952 1.071684 1.057935 1.047493 1.039464

0.122397 0.114638 0.106568 0.098840 0.091717

1.079407 1.061236 1.048279 1.038838 1.031806

0.143879 0.130280 0.118116 0.107508 0.098337

1.045832 1.040241 1.035508 1.031490 1.028065

0.067197 0.064664 0.062063 0.059482 0.056975

1.042834 1.037210 1.032539 1.028638 1.025359

0.073340 0.069803 0.066381 0.063128 0.060070

1.038659 1.033231 1.028808 1.025171 1.022152

0.079462 0.074821 0.070524 0.066576 0.062962

1.033205 1.028260 1.024300 1.021090 1.018458

0.085271 0.079488 0.074315 0.069688 0.065542

1.026459 1.022317 1.019052 1.016439 1.014319

0.090413 0.083544 0.077561 0.072320 0.067702

:; 19

1.025132 1.022608 1.020426 1.018530 1.016874

0.054573 0.052291 0.050135 0.048106 0.046201

1.022583 1.020219 1.018192 1.016444 1.014929

0.057215 0.054559 0.052094 0.049806 0.047684

1.019626 1.017494 1.015681 1.014129 1.012790

0.059658 0.056638 0.053874 0.051341 0.049015

1.016277 1.014452 1.012912 1.011600 1.010476

0.061817 0.058460 0.055424 0.052670 0.050161

1.012577 1.011130 1.009915 1.008887 1.008009

0.063610 0.059962 0.056694 0.053752 0.051092

20

1.015422

0.044413

1.013607

0.045714

1.011629

0.046875

1.009505

0.047870

1.007254

0.048675

f

1.059090 1.059305

0.003539 0.000000

: 4

1.058456 1.057431 1.056058

0.007000 0.010310 0.013410

5 6 ;

1.054391 1.052490 1.050413 1.048217

0.016252 0.018806 0.021055 0.022996

9

1.045956

0.024637

10

1.043672

0.025993

::

1.039177 1.041402

0.027940 0.027086

:i

1.034942 1.037018

0.028581 0.029034

:z

1.032959 1.031076

0.029477 0.029326

:'8 19

1.029296 1.027620 1.026046

20

1.024570

-13

-14

Y\X

:: 14 :'b

For [21>4, linear interpolation yield about six decimals. See Example39

-10.

-10

-6

-7

-8

:i

-11

-12

-5

will yield about four decimals, eight-point

interpolation

will

250 Table

5.6 .?A

EXPONENTIAL

INTEGRAL

P

.f

9f

AND

RELATED

FOR COMPLEX .zeZEl (2) .% ./

-2

-3

FUNCTIONS

ARGUMENTS 9

92

&

-1

9

0

1.438208 1.287244 1.185758 1.123282 1.085153

0.230161 0.263705 0.247356 0.217835 0.189003

1.483729 1.251069 1.136171 1.080316 1.051401

0.469232 0.410413 0.328439 0.262814 0.215118

1.340965 1.098808 1.032990 1.013205 1.006122

0.850337 0.561916 0.388428 0.289366 0.228399

0.697175 0.813486 0.896419 0.936283 0.957446

1.155727 0.570697 0.378838 0.280906 0.222612

0.577216 0.621450 0.798042 0.875873 0.916770

0.000000 0.343378 0.289091 0.237665 0.198713

1.061263 1.045719 1.035205 1.027834 1.022501

0.164466 0.144391 0.128073 0.114732 0.103711

1.035185 1.025396 1.019109 1.014861 1.011869

0.180487 0.154746 0.135079 0.119660 0.107294

1.003172 1.001788 1.001077 1.000684 1.000454

0.187857 0.159189 0.137939 0.121599 0.108665

0.969809 0.977582 0.982756 0.986356 0.988955

0.183963 0.156511 0.136042 0.120218 0.107634

0.940714 0.955833 0.965937 0.972994 0.978103

0.169481 0.147129 0.129646 0.115678 0.104303

1.018534 1.015513 1.013163 1.011303 1.009806

0.094502 0.086718 0.080069 0.074333 0.069340

1.009688 1.008052 1.006795 1.005809 1.005022

0.097181 0.088770 0.081673 0.075609 0.070371

1.000312 1.000221 1.000161 1.000119 1.000090

0.098184 0.089525 0.082255 0.076067 0.070738

0.990887 0.992361 0.993508 0.994418 0.995151

0.097396 0.088911 0.081769 0.075676 0.070419

0.981910 0.984819 0.987088 0.988891 0.990345

0.094885 0.086975 0.080245 0.074457 0.069429

:8' 19

1.008585 1.007577 1.006735 1.006025 1.005420

0.064959 0.061086 0.057640 0.054555 0.051779

1.004384 1.003859 1.003423 1.003057 1.002747

0.065803 0.061786 0.058227 0.055052 0.052202

1.000070 1.000055 1.000043 1.000035 1.000028

0.066102 0.062032 0.058432 0.055224 0.052349

0.995751 0.996246 0.996661 0;997011 0.997309

0.065838 0.061812 0.058246 0.055066 0.052214

0.991534 0.992518 0.993342 0.994038 0.994631

0.065024 0.061135 0.057677 0.054583 0.051801

20

1.004902

0.049267

1.002481

0.049631

1.000023

0.049757

0.997565

0.049640

0.995140

0.049284

0.596347 0.777514 0.673321

0.000000 0.147864 0.1865.70

0.722657 0.747012 0.796965

0.000000 0.045686 0.078753 0.096659 0.103403

0.825383 0.831126 0.846097 0.865521 0.885308

0.000000 0.030619 0.055494 0.072180 0.081408

0.852111 0.855544 0.864880 0.877860 0.892143

0.000000 0.021985 0.040999 0.055341 0.064825

: 3 4 2 7 9" :; :: 14 :2

1

Y\X

2

4

3

0.847468 0.891460

0.165207 0.181226

0.844361 0.881036

0.131686 0.132252

0.919826 0.938827 0.952032 0.961512 0.968512

0.148271 0.132986 0.119807 0.108589 0.099045

0.907873 0.927384 0.941722 0.952435 0.960582

0.125136 0.116656 0.107990 0.099830 0.092408

0.903152 0.921.006 0.934958 0.945868 0.954457

0.103577 0.100357 0.095598 0.090303 0.084986

0.903231 0.918527 0.931209 0.941594 0.950072

0.085187 0.085460 0.083666 0.080755 0.077313

0.906058 0.918708 0.929765 0.939221 0.947219

0.070209 0.072544 0.072792 0.071700 0.069799

:: 13 14

0.973810 0.977904 0.981127 0.983706 0.985799

0.090888 0.083871 0.077790 0.072484 0.067822

0.966885 0.971842 0.975799 0.979000 0.981621

0.085758 0.079836 0.074567 0.069873 0.065679

0.961283 0.966766 0.971216 0.974865 0.977888

0.079898 0.075147 0.070769 0.066762 0.063104

0.957007 0.962708 0.967423 0.971351 0.974646

0.073688 0.070080 0.066599 0.063300 0.060206

0.953955 0.959626 0.964412 0.968464 0.971911

0.067447 0.064878 0.062242 0.059630 0.057096

15

0.987519

0.063698

:; 18 19

0.988949 0.990149 0.991167 0.992036

0.060029 0.056745 0.053792 0.051122

0.983791 0.985606 0.987138 0.988442 0.989561

0.061921 0.058539 0.055485 0.052717 0.050199

0.980414 0.982544 0.984353 0.985902 0.987237

0.059767 0.056723 0.053941 0.051394 0.049057

0.977430 0.979799 0.981827 0.983574 0.985089

0.057322 0.054644 0.052162 0.049861 0.047728

0.974858 0.977391 0.979579 0.981478 0.983135

0.054671 0.052371 0.050200 0.048160 0.046245

20

0.992784

0.048699

0.990527

0.047900

0.988395

0.046909

0.986410

0.045749

0.984587

0.044449

0.871606 0.873827 0.880023 0.889029 0.899484

0.000000 0.016570 0.031454 0.043517 0.052380

0.886488 0.888009 0.892327 0.898793 0.906591

0.000000 0.012947 0.024866 0.034995 0.042967

0.898237 0.899327 0.902453 0.907236 0.913167

0.000000 0.010401 0.020140 0.028693 0.035755

0.907758 0.908565 0.910901 0.914531 0.919127

0.000000 0.008543 0.016639 0.023921 0.030145

0.915633 0.916249 0.918040 0.920856 0.924479

0.000000 0.007143 0.013975 0.020230 0.025717

0.910242 0.920534 0.929945 0.938313 0.945629

0.058259 0.061676 0.063220 0.063425 0.062714

0.914952 0.923283 0.931193 0.938469 0.945023

0.048780 0.052667 0.054971 0.056047 0.056211

0.919729 0.926481 0.933096 0.939359 0.945154

0.041242 0.045242 0.047942 0.049570 0.050349

0.924336 0.929836 0.935365 0.940731 0.945812

0.035208 0.039123 0.041986 0.043936 0.045128

0.928664 0.933175 0.937807 0.942398 0.946833

0.030334 0.034063 0.036944 0.039060 0.040514

0.951965 0.957427 0.962128 0.966178 0.969673

0.061408 0.059735 0.057855 0.055877 0.053874

0.950850 0.955987 0.960495 0.964444 0.967903

0.055725 0.054790 0.053560 0.052146 0.050627

0.950427 0.955176 0.959421 0.963201 0.966559

0.050481 0.050135 0.049444 0.048514 0.047425

0.950535 0.954870 0.958814 0.962379 0.965591

0.045711 0.045818 0.045563 0.045038 0.044319

0.951035 0.954959 0.958586 0.961913 0.964949

0.041413 0.041861 0.041948 0.041755 0.041347

:9"

0.972699 0.975326 0.977617 0.979622 0.981384

0.051894 0.049966 0.048109 0.046332 0.044641

0.970935 0.973551 0.975940 0.978009 0.979839

0.049062 0.047489 0.045935 0.044419 0.042951

0.969539 0.972185 0.974538 0.976632 0.978500

0.046236 0.044992 0.043724 0.042456 0.041205

0.968477 0.971067 0.973393 0.975481 0.977357

0.043463 0.042516 0.041512 0.040477 0.039431

0.967710 0.970214 0.972484 0.974540 0.976402

0.040780 0.040095 0.039329 0.038508 0.037653

20

0.982938

0.043036

0.981465

0.041538

0.980169

0.039980

0.979047

0.038388

0.978090

0.036781

: :

2 i 9 10

6

Y\X 0 : 3 4 5 ; i 10 :: 13 14 15 :7"

0.000000 0.075661 0.118228

5

0.786251 0.797036 0.823055 0.853176 0.880584

0

*

INTEGRAL

-4

Y\X 0

EXPONENTIAL

7

8

10

9

If ~~10 or y>lO then (see [5.15]) eZEl(z)x 0.711093 +------+--------+e,161<3~10-6. 0.278518 0.010389 2+0.415775 z+2.29428 z+6.2900 El(iy)=-Ci(y)+i si(y) (y real) l S8e

page Ix.

EXPONENTIAL

INTEGRAL

EXPONENTIAL 9

.f

‘J

.P

11

y\x

INTEGRAL

12

AND

RELATED

251

FUNCTIONS

FOR COMPLEX zezE1 (2) 9 Y 13

Table

ARGUMENTS x

9

.A4

14

5.6 4

15

0.922260 0.922740 0.924143 0.926370 0.929270

0.000000

0.927914 0.928295 0.929416 0.931205 0.933560

0.000000

0.003972 0.007847 0.011540 0.014974

0.940804 0.941014 0.941636 0.942643 0.943994

0.000000

0.004528 0.008932 0.013098 0.016934

0.937055 0.937308 0.938055 0.939261 0.940870

0.000000

0.005212 0.010258 0.014991 0.019295

0.932796 0.933105 0.934013 0.935473 0.937408

0.000000

0.006063 0.011902 0.017321 0.022171

0.932672 0.936400 0.940297 0.944229 0.948093

0.026361 0.029857 0.032670 0.034847 0.036453

0.936356 0.939462 0.942757 0.946132 0.949500

0.023091 0.026339 0.029036 0.031205 0.032887

0.939729 0.942338 0.945140 0.948047 0.950985

0.020373 0.023378 0.025934 0.028052 0.029756

0.942816 0.945024 0.947419 0.949933 0.952502

0.018095 0.020867 0.023273 0.025315 0.027004

0.945640 0.947522 0.949582 0.951765 0.954018

0.016169 0.018725 0.020980 0.022931 0.024582

10 11 12 13 14

0.951816 0.955347 0.958659 0.961739 0.964583

0.037566 0.038261 0.038612 0.038684 0.038534

0.952792 0.955958 0.958968 0.961800 0.964447

0.034134 0.035004 0.035552 0.035833 0.035893

0.953895 0.956729 0.959454 0.962049 0.964499

0.031081 0.032068 0.032761 0.033201 0.033428

0.955075 0.957610 0.960073 0.962443 0.964702

0.028365 0.029426 0.030221 0.030781 0.031140

0.956296 0.958563 0.960787 0.962947 0.965026

0.025949 0.027052 0.027915 0.028564 0.029024

El 17 18 19

0.967199 0.969597 0.971789 0.973792 0.975621

0.038211 0.037756 0.037200 0.036572 0.035893

0.966907 0.969184 0.971285 0.973220 0.974999

0.035775 0.035515 0.035144 0.034687 0.034166

0.966799 0.968947 0.970946 0.972802 0.974521

0.033479 0.033384 0.033172 0.032865 0.032485

0.966843 0.968860 0.970752

0.031327 0.031370 0.031293

0.972521 0.974172

0.031117 0.030862

0.967011 0.968897 0.970680 0.972359 0.973936

0.029320 0.029476 0.029512 0.029448 0.029301

20

0.977290

0.035179

0.976634

0.033597

0.976112

0.032049

0.975709

0.030542

0.975414

0.029086

0.000000

0.000000

0.002290 0.004549 0.006745 0.008853

0.954371 0.954467 0.954752 0.955219 0.955856

0.000000

0.002527 0.005016 0.007430 0.009735

0.952181 0.952291 0.952619 0.953156 0.953887

0.000000

0.002804 0.005560 0.008223 0.010754

0.949769 0.949897 0.950277 0.950898 0.951741

0 : i

y\x 0

17

16

19

18

0.003512 0.006949 0.010242 0.013331

20

0.944130 0.944306 0.944829 0.945678 0.946824

0.000000 0.003128 0.006196 0.009150 0.011940

0.947100 0.947250 0.947693 0.948416 0.949395

0.948226 0.949842 0.951624 0.953527 0.955509

0.014529 0.016886 0.018994 0.020847 0.022445

0.950600 0.951995 0.953545 0.955212 0.956960

0.013121 0.015296 0.017265 0.019019 0.020555

0.952782 0.953995 0.955349 0.956815 0.958363

0.011904 0.013916 0.015753 0.017409 0.018878

0.954793 0.955853 0.957043 0.958337 0.959712

0.010847 0.012709 0.014425 0.015986 0.017387

0.956650 0.957581 0.958631 0.959779 0.961004

0.009922 0.011649 0.013253 0.014723 0.016056

0.957530 0.959559 0.961568 0.963534 0.965443

0.023797 0.024917 0.025823 0.026534 0.027070

0.958758 0.960576 0.962391 0.964181 0.965931

0.021878 0.022998 0.023927 0.024679 0.025271

0.959966 0.961598 0.963238 0.964868 0.966472

0.020163 0.021270 0.022207 0.022984 0.023616

0.961144 0.962612 0.964097 0.965582 0.967052

0.018628 0.019712 0.020645 0.021436 0.022094

0.962288 0.963611 0.964956 0.966310 0.967658

0.017250 0.018305 0.019227 0.020021 0.020694

19

0.967280 0.969038 0.970712 0.972300 0.973800

0.027453 0.027700 0.027831 0.027862 0.027809

0.967628 0.969264 0.970832 0.972328 0.973751

0.025720 0.026041 0.026249 0.526361 0.026388

0.968039 i;9695i8 0.971023 0.972430 01973775

0.024114 0.024493 0.024765 0.024943 0;0i503s

0.968496 0.969906 0.971273 0.972594 0.973863

0.022629 0.023052 0.023375 0.023607 0.023760

0.968990 0.970297 0.971571 0.972808 0.974004

0.021255 0.021712 0.022075 0.022352 0.022552

20

0.975215

0.027685

0.975099

0.026343

0.975057

0.025062

0.975079

0.023842

0.975155

: 4' 2 7

9" :1" :: 14 15 16

:i

EXPONENTIAL I

.9?

-4.0

y\r

2:

-0.359552 -0.347179 -0.333373 -0.318556 -0.303109 -0.287369

Y\"

-0.057540 -0.078283 -0.096648 -0.112633 -0.126301 -0.137768

0.636779 0.000000

2 0:8

-4.094686 -3.890531 -3.611783 -3.265262

1.260867 1.859922 2.422284 2.937296

0.2 0.4 0.6 0.8

1.0

-0.133374 -0.126168 -0.104687 -0.069328 -0.020743 +0.040177

0.5 0.000000 0.157081 0.312331 0.463961 0.610264 0.749655

-2.895820 -2.867070

COMPLEX .B?

.f

-0.094868 -0.119927

-0.141221 -0.158890 -0.173169 -0.184355

-0.494576

-0.156411

-0.462493 -0.429554 -0.396730 -0.364785 -0.334280

-0.185573 -0.208800 -0.226575 -0.239500 -0.248231

El(z)+ln -1.0

0.342700

0.679691 1.005410 1.314586 1.602372

-0.811327

0.000000

0.462804

-1.875155

0.917127 1.354712 1.767748

-2.210344

2.149077

-1.418052

0.000000 0.126210 0.251143 0.373547 0.492229 0.606074

-0.928842

0.505485 0.509410 0.521123 0.540441 0.567061 0.600568

4

-0.425168 -0.451225 -0.463193 -0.464163 -0.457088 -0.444528

0.000000 0.103432 0.205962 0.306707 0.404823 0.499516

0 0.000000 0.258840 0.513806 0.761122 0.997200

1.218731

-0.577216 -0.567232 -0.537482 -0.488555 -0.421423 -0.337404

2.0

1.5

1.0 0.219384 0.224661 0.240402 0.266336 0.302022 0.346856

-0.670483 -0.587558 -0.510543 -0.441128 -0.380013 -0.327140

-0.5 -1.147367 -1.133341 -1.091560 -1.022911

-1.895118

-2.781497 -2.641121 -2.449241

-0.257878 -0.289009 -0.310884 -0.324774 -0.332047 -0.334043

.@ -2.0

5.7

z

0.000000

-1.815717 -1.718135 -1.584591

-0.580650 -0.528987 -0.478303 -0.429978 -0.384941 -0.343719

0.022684

Table

ARGUMENTS -2.5

-1.5

-4.219228 -4.261087

Y\X 0.0

-0.420509 -0.400596 -0.379278 -0.357202 -0.334923 -0.312894

FOR SMALL

.f

.?f -2.5 -.-

-2.0

FE

1.0

INTEGRAL

.~~__

0.002085 0.004144 0.006151 0.008084

0.742048 0.745014 0.753871 0.768490 0.788664 0.814107

0.000000 0.086359 0.172075 0.256515 0.339075 0.419185

0.000000 0.199556 0.396461 0.588128 0.772095 0.946083

2.5 0.941206 0.943484 0.950289 0.961532 0.977068 0.996699

0.000000 0.073355 0.146246 0.218215 0.288822 0.357653

6. Gamma Function

and Related Functions

PHILIP

J. DAVIS 1

Contents Page Mathematical

Properties.

Gamma Function. . . . . . Beta Function . . . . . . . Psi (Digamma) Function. . . Polygamma Functions. . . . Incomplete Gamma Function. Incomplete Beta Function. .

255 258 258 260 260 263

. . . . . . . . . . . . . . . . . . . . . .

263

6.7. Use and Extension of the Tables. . . . . . . . . . . . . 6.8. Summation of Rational Series by Means of Polygamma Functions. . . . . . . . . . . . . . . . . . . . . . . . .

263

Numerical

Methods

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

264

. . . . . . . . . . . . . . . . . . . . . . . . . .

265

Gamma, Digamma and Trigamma Functions (1 Is_< 2) . .

267

References. 6.1.

. . . . . .

255

. . . . . .

6.1.

6.2. 6.3. 6.4. 6.5. 6.6.

Table

. . . . . . . . . . . . . . . . . . .

r(x), In I’(z), #(z), +‘(x)), x=1(.005)2,

10D . . .

271

6.3. Gamma and Digamma Functions for Integer and HalfInteger Values (1 In< 101) . . . . . . . . . . . . . . . . . .

272

Table

6.2. Tetragamma and Pentagamma Functions (15 ~12) $“(s),

$b3’(x), x=1(.01)2,

10D

Table

r(n),

11s

uw,

9s

r(n+3),

ddn),

1011

n!/[(27r)%P++]e?, In n+(n),

8s

8D

8D

n=l(l.>lOl Table

6.4. Logarithms of the Gamma Function (1 In < 101). . . . . . loglo r(n),

8s

log,tl r(n+i),

loglo r(n+*),

8s

klo

8s

r(n+$>,

In

r(n) -

274

8s (n- +) In nfn,

8D

n=l(l)lOl

1 National

Bureau

of Standards. 253

254

GAMMA

FUNCTION

AND

RELATED

FUNCTIONS Page

6.5. Auxiliary Functions for Gamma and Digamma Functions (66I:ZIm) . . . . . . . . . . . . . . . . . . . . . . .

Table

2!/[(2~)%P+k-~], z-l=.015 Table

(-.OOl)O,

6.6. Factorials for Large Arguments (lOO
1000) . . . . .

276

6.7. Gamma Function for Complex Arguments . . . . . . . .

277

In I’(z+iy), Table

In X+X, In z++(2)

8D

n!, ??I=100(100) 1000, Table

In r(z)-((2-i)

276

x=1(.1)2,

20s

y=O(.l)lO,

12D

6.8. Digamma Function for Complex Arguments . . . . . . . #(r+iy),

x=1(.1)2,

%‘W+iy), S?$(l+iy)-ln

y=O(.l)lO,

288

5D

10D y, y-l=.11

(-.Ol)O,

The author acknowledges the assistance the tables; and the assistance of Patricia

8D

of Mary 01-r in the preparation Farrant in checking the formulas.

and

checking

of

6. Gamma

Function

and Related

Mathematical 6.2. Gamma

6.1.1

r(z)=

s0 =k

(Factorial)

Euler’s

Integral

- t’-‘e-’

0%

s

omt’-‘em”’

Euler’s

Functions

Properties

Function

W’z>O) dt

(9?2>0,

9’k>O)

Formula

6.1.2 r(z)=lim

n!n' . . . (z+n)

n+os z&+1)

Euler’s

Infinite

(z#O,-1,-2,

. . .)

Product

-3

l+k+j!j+a+.

r= lim

m+-

=.57721

. . +;-ln

56649. . .

Y is known as Euler’s constant and is given to 25 decimal places in chapter 1. I’(Z) is single valued and analytic over the entire complex plane, save for the points z=-n(n=O, 1, 2, . . . ) where it possesses simple poles with residue (- l)“/n!. Its reciprocal i/r (2) is an entire function possessing simple zeros at the points z=-n(n=O, 1, 2, . . .). Hankel’s

6.1.4

3’ i,’ PA ,‘\ ”

n /r -5

7)X-j

&=&Jo

Contour

Integral

FIGURE 6.1. Gamma junction. -,

y=r(z),

6.1.9

r(3/2)=#=.88622

6.1.10

r(,t+)=1’5’9’13 l-(+)=3.62560

The path of integration C starts at + a~ on the real axis, circles the origin in the counterclockwise direction and returns to the starting point. Factorial

and

Integer

6.1.6

r(n+i)=i.2.3

6.1.11 r(n+#)=1a4e7’10 r(+)=2.67893

II Notations

II(Z) =2!= r(2+ 1)

6.1.5

y=l/l-(z)

69254. . . c(3)! k;l’ (4n-3) r(t)

(-t)-‘es’&

n

‘L” \-’

- - - -,

*

6.1.12

r(n+$)

6.1.13

r(n+#)=2’5’8’11

99082. , . ‘3’“’ (3n-2)

r(+)

85347. . .

=1*3*5*7 * ;; (2n-1)

r(3)

Values

. . . (7+i)n=d

‘3; (3n-1)

r(g)

6.1.7

lb ----L-E 2-ln u-4

o=

l (-n-l)!

Fractional

w =2J

0

e-,‘&=&=1.77245

r(#)=i.3541179394..

(n=O, 1, 2) . . .) Values

6.1.14

r(n+f)

=3’7’11’15

.

h;’ (4n-1)

r(j)

38509 . . . =(-$)! r&)=1.22541

-*see

page

II.

67024 . . . 255

256

GAMMA Recurrence

FUNCTION

AND

Formulas

RELATED

FUNCTIONS

i.1.29

r(i~)r(-iy)=lr(iy)[2=ysi~ny

il.30

r(d-+iy)r(t--iy)=lr(~+i21)12=~y

il.31

r(l+iy)r(l-iy)=lr(l+iy)l2=~

r(z+1)=zr(z)=z!=z(z-l)!

6.1.15 6.1.16

r(n+z)=(n-l+z)(n-2+2) =(n-l+z)!

. . . (l+z)r(l+z)

=(n-l+z)(n-2+2)

. . . (l+z)z!

Reflection

6.1.17

Formula

r(2)r(i--2)=-2r(--2)rQ)=t

csc

TZ

Power

Series

5.1.33

III r(l+2)=-h(l+z)+z(i--y) Duplication

6.1.18

r(22)=(2?r)-*22z-*

r(2) r(2++)

Triplication

c(m) is the Riemann 23).

Formula

Series

6.1.19

r(32)=(2d-138*-* Gauss’

6.1.20

Multiplication

Binomial

a

uz+l)

r(w+i)r+-w+i)

Pochhammer’s

Symbol

6.1.22 (z>c= 1,

Gamma

6.1.23 6.1.24

. . . @+?I-l)=W Function

Expansion

* for

(see chapter

1 /r(z)

6.1.34

Formula

Coefficient

’ s--z w!(z--w)! 0?V

(z).=z(z+l)(z+2)

Zeta Function

r(z)r(2++)r(2++)

r(nz)=(2rr)t"-"'~~'-l~~r(~+~)

6.1.21

ew>

+-g2 (-w-(+ll~“/~

Formula

in the

Complex

Plane

r(z)=r(2); In r(B)=ln rl.2) arg r(z+i)=arg

r(z)+arhnf

6.1.26 6.1.27

mgr b+id =8m +n$o(--&-arctan&-) (z+iy#0,-1,-2, where

~W=rWr(2)

6.1.28

r(i+iy)=iy

r(iy)

.. .)

k '1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

l.oooooc~oooo 0.57721 56649 -0.65587 80715 -0.04200 26350 0.16653 86113 -0.04219 77345 -0.00962 19715 0.00721 89432 -0.00116 51675 -0.00021 52416 0.00012 80502 -0.00002 01348 -0.00000 12504 0.00000 11330 -0.00000 02056 0.00000 00061 0.00000 00050 -0.00000 00011 0.00000 00001 0.00000 00000 -0.00000 00000 0.00000 00000 -0.00000 00000 -0.00000 00000 0.00000 00000 0.00000 00000

000000 015329 I- *-, 202538 \ I 340952' 822915'5 555443Lit 278770 ' 466630 918591 741149 823882 547807 0 934831 1 272320 2, 338417 3 160950 '-f 020075 3 812746 b 0434271 0778234 036968 A 005100 I' 000206k, 000054b 000014 i; 000001 (5

2 The coeffhxents ck are from H. T. Davis, Tables of higher mathematical functions, 2 vols., Principia Press, Bloomington, Ind., 1933, 1935 (with permission); with corrections due to H. E. Salzer.

GAMMA Polynomial

FUNCTION

AND

Approximationsa

6.1.35

FUNCTIONS Error

r(z+1)=2!=1+a,z+u~Z+u,23+a,~~+a,zbf~(~)

for

Asymptotic

Expansion

If

~e(z))115XlW~

R,(z)= 57486 46 .95123 63 aa=-. 69985 88

r (z)-(z-+)

In

In Z+z--4

al= .42455 49 a5=-. 10106 78

al=-. aa=

6.1.36

Term

257

6.1.42

Olxll

.

RELATED

-2

In (2~) 2n(iL)z

an-1

then

O<x
r(x+1)=2!=1+~,z+b,22+

. . . +b,xa+,(x)

where

(&)(~3XlO-7

bl=-.

57719 1652 bz= .98820 5891 ba=-. 89705 6937 b4= . 91820 6857 Stirling’s

bs= -. 75670 4078 b,,= .48219 9394 b,=-. 19352 7818 bs= .03586 8343

K(z) =uppe;,~oundlzz/(3+24)

1

For z real and positive, R, is less in absolute value than the first term neglected and has the same sign.

Formula

6.1.43

6.1.37

S?ln r(iy)=Sln

4

r(--iy)

ln (%I -+7y-*ln

v,

cy--++ a>

6.1.44

x1=&rxz+&p(-x+&J Asymptotic

(x>O,O
3%

r(iy)=arg =-An

r(-@)

r(iy)=-mg

q--iy)

Formulas

6.1.39 r(az+b)

$iiie-ae(uz)‘u+b-~

(lw 4<*, C-0)

6.1.40 In

(y++

6.1.45

r(2) ~(z-+) In Z--Z++ In

(274

+&

(2-m

2m(2mB>)z”-1

-1

~~21r)-“lr(z+iy)le~~1u11y11-“=1 m

in Jarg z/
For B, see chapter 23 6.1.41 In r(2) ‘v(2-*)

In z-z+&

Zt,-a r(z+a)-,+(a-b)(a+b-1) w+b)

In (2~)+&&: (z+w

in larg zl<~)

* Prom C. Hastings, Jr., Approximations computers, Princeton Univ. Press, Princeton, (with permission).

for digital N.J., 1955

+&pi”)

22

(3(a+b--1)2-a+b-l);+

...

as Z+W along any curve joining z=O and z= m, providingzf --a, --a-l, . . . ; z# -b, -b-l, . . . .

258

GAMMA Continued

FUNCTION

AND RELATED

FUNCTIONS

Fraction

6.1.48

In r(z)+z-(z-*)

In z-+ln

a0

=-

_-al z+

z+

\ i &:/’ 1

a2

---a3 z+

z+

(27r) a4

z+

a5 z+

(92

. . .

>

0)

1 53 695 1 &=---, al=-9 a2=-) a3=-, 12 30 210 371

I /

22999

10953534lOO9 48264275462

29944523

a4=ii%%’

“=19733142’ Wallis’

“=

Formula

4

6.1.49

-4 -6

FIGURE 6.2. Psi junction. y=$+)=dln

Some

Definite

Integrals

6.1.50

In r(z)=

o- (z-l) e-Qy-$J S[ =(z-&) In z-z+& In 2s

f

(g z > 0)

6.3.2

r :

nteger

Values

I

n-1 p-l

7, It(n)=--+

lw)=

x_.

rkr)/dx

. ..’

(n22>

2+ Fractional

Values

(s?~>O) 6.3.3 6.2. Beta

Function

9G) =-y-2ln

6.2.1 1

B(z,w)

=

S S

p-1 (l--t)“-’

dt=

0

Psi (Digamma)

+(z) =d[ln

6.3.4

at

#(n, +$)=--r--2ln2+2

r

(2)1/h=

’

(

l+i+...

Function r’(2)p

6Some authors write $(z) =-$ In I’(z+

Recurrence

5

functions.

Formulas

6.3.5

(2)

1) and similarly

+A) (nil)

2)

4 Some authors employ the special double factorial tion as follows: (27~) ! ! =2*4-f? . . . (2n)=2”n i (2n-1) ! ! =1*3*5. . (27&--1)=*-f 2” r(n++) the polygamma

00260 21423 . . .

(92~>0,~‘>00)

f&Y ,w)2(2)r(wLB(w r(z+w) 6.3.

6.3.1

om(l;;;z+w

4 o (sin t)2E-1 (cos t)2w-1 dt

=2

6.2.2

s

2=-1.96351

nota-

6.3.6 Hn+

for

+ &+&+w+4

GAMMA Reflection

FUNCTION

AND RELATED

FUNCTIONS

259

6.3.19

Formula

Td

6.3.7

~(1--2)=~(2)+7r

cot ?Tz

Duplication b.3.8

Formula

~@~)=W(z)++h Psi

Function

in

6.3.9

(zS9) the

bm

Complex

=ln y+&+i+l+.

2

Sin

. .

12oy4 252y6

Plane

(y+w)

=1Lo

Extremae

of r(z)

-

Zeros

of +(r)

6.3.10 -

s?~(iy)=sz~(-iy)=%?‘1L(l+iy)=5z’9(1--iy)

-

n 6.3.11

Y-$(iy)=+y-‘+;

6.3.12

Y+(i+&)

6.3.13

j$(l+iy)=-+y++

?r coth ny

>

=Yn~l(n’+Y2~ Series

_

+o.sss -3.545 +2.302 -0.888 $0.245 -XL053 +o. 009 -0.001

$1.462

=+T tanh 7~ rcoth

r (x?J

2% __

-0.504 - 1.573

my

-2.611 -3.635 -4.653 -5.667 -6.678

-1

Expansions

-

6.3.14 #(l+~)=--r+~~(-l)~{(n)~-~

-

(lzl
6.3.15 ~(l+z)=+z-‘-+

cot ?fz-(l--22)-1+1--y

-~~Ik(2n+~wn

6.3.20

xd=-n+(ln

Ma

Definite

21449

68362 gi

31944

10889

n>-‘+o[(ln Integrals

I+& \3

n>-‘1 -

6.3.21

6.3.16 @f--1,-2,-3,.

$(l+~)=-r+~$~&

. . >

+j-y[;-‘-g&q

W’z>O)

dt .

.m

1

6.3.17

=

~~(l+iy)=l--Y-&

=lnz--- S tat I[ 0

e-’

--

l222

+gl

(-l)“+‘[r(2n+l)--lly2”

n$l n-‘(n2Sy2) -l (--~
Asymptotic

$

0 (t”+z”)(P-1)

(I&E 4< ;)

(lYl<2) =-r+y2

&)z

6.3.22

Formulas

6.3.18 #(z>

--In

2 -L&-~$~

=ln

-&-&+&4-&o+.

&”

. .

(z+ m in larg zI<7r)

6 From W. Sibagaki, Theory and applications of the gamma function, Iwanami Syoten, Tokyo, Japan, 1952 (with permission).

0

260

GAMMA 6.4. Polygamma

FUNCTION

AND

Functions’

RELATED

FUNCTIONS Series

6.4.9

6.4.1

0

$‘“‘(l+z)=(-l)“+’ (n=1,2,3,

Expansions

[n![(n+i)

(n+l)! -Tl(n+2)z+@i,*{(n+3)9-... .

. . . )

1

Tifl
6.4.10 )L(“)(2),(?%=0,1, . . .), is a single valued analytic function over the cnt,irc complex plane save at, the points z=-m(m=0,1,2, . . . ) where it possesses poles of order (n + 1).

+‘“‘(z)=(-l)n+Ln!~O

(z+k)-“-1 (z#O,-1,-2,. Asymptotic

. .)

Formulas

6.4.11

0 Integer

Values

6.4.2' #‘“‘(l)=(-

l)“+‘n!{(n+

(n=l,2,3,

1)

. .)

6.4.3

(2 + 02 in 1arg 2 1
$(m)(n+l)=(-lpm! 1

i+ +2m+’

- - * +nm+l

1

. Fractional

6.4.4 ' '-' p”‘(*)

= (- l)“+‘n!(2”+‘-

#‘(n++)

=*nJ-4

Recurrence

l)l(n+

1)

(z--,-in

21-&j

I arg 2 I<*)

6.4.14

PI (2k- 1) -* (z+-

Formuh

IL(“)(2+i)=~(n)(zj+(-i)112!2--.-1

6.4.6

in 1arg

Values

(12=1,2, . .

6.4.5

(z+6.4.13

6.5.

Incomplete

Gamma

in I arg 2 I
(see also 26.4) Reflection

Formula

6.5.1

6.4.7 ~(“‘(l-z)+(-l).+lll’.‘(z)=(-l)n~~

cot *z

P(u, s) = &

s0

* e-W1 dt

6.5.2

6.4.8 *

Multiplication

P(mz)=bln

m+ &

Formula

gv)

r(a, x) =P(a,

(z+&)

n=O b=O, n>O .

s

oze-fte-l dt

6.5.3

(s&z>Ol

0

r (a, T) = r(u) +a,

b=l,

7 $’ is known as the trigamma function. $‘I, $(a), $tr) nre the t&a-, penta-, and hesagamma functions respectively. Some authors write $(z) =d[ln I’(z+ l)]/dz, and similnrly for the polygamma functions. *See page II.

2) r(u) =

T) =

S I

e-W1 dt

6.5.4 y*(u, r) =x-=qz,

r) = &$

(a, 4

y* is a single valued analytic function z possessing no finite singularities.

of a and

GAMMA

FUNCTION

AND

RELATED

261

FUNCTIONS

a t

-3

-4

FIGURE

*see page Il.

2

3

6.3. Incomplete gamma function.

y*(a += From F. G. Tricomi,

I

0

-I

-2

* e-V-W r(a) s 0

Sulla funeione gamma incompleta,

Annali

di Matematica,

IV, 33, 1950 (with permission).

4

GAMMA FUNCTION

262

AND RELATED FUNCTIONS

6.5.5 Probability

Integral

of the

6.5.16

y (3, x2)=2

6.5.17

r(+,z2)=2

+Distribution

6.5.18

6.5.6

G

S m S

oze-t2dt=J;;

erfx

e-f2dt=J;;

erfc x

2

S

x7*(+,-x2)=

‘ef2dt

0

(Pearson’s

Form

of the

Incomplete

Gamma

Function)

6.5.19

164 P> =r(pl+l) sume-v& =Np+l,

S(X,U)=~P-~

6.5.8

w]

[C(z,a)--iiS(

dp+l) Recurrence

Wu
Sz-ta-1 co5t dt

C(x,u)=

[El(x)-e-
r(a,ix)=e*rf”

6.5.20

0

6.5.7

r(--n,x)=T

P(a+l,

6.5.21

sin t dt

W’a
6.5.9

Formulas

x>=P(u,

x)-&

6.5.22

~(u+l,x)=uy(u,x)-xsae-z

6.5.23

r’(a-l,x)=xr*(u,x)+~

E, (x)=~-~-~~~-“df=x”-Lr(l--n,x) 6.5.10

Derivatives

a,(d=sm

and

Differential

Equations

6.5.24

e-=‘t”dt=x-“-lr(1+7,x) (?j$)mmo

1

ar (w) -= ax

6.5.11 6.5.25 Incomplete Gamma Hypergeometric

6.5.12

Function Function

Special

13)

ar(a2) ax

------1-=y-le-z

f$ [2-ar(a,x)]=(-l)n2-a-“r(a+n,x) (n=O, 1,2, . . .)

1+a,-x)

6.5.27

g [e%.?y* (u,x>]=e”x-r*(u-n, 2)

Values

6.5.13

P(n,X)=l-

2

6.5.26

l+a,x)

M(a,

x=-&(x)--In

as a Confluent

(see chapter

y(u,x)=u-lxae-zM(l, =u-lx=

=-lmeT-ln

(T&=0,1,2,. (

1+X+$+.

. . +&

.>

emz

6.5.28

. .>

x~+(u+l+x)~+u-y*=O

=l-e,-l(x)e-Z For relation

to the Poisson

distribution,

26.4. 6.5.14

6.5.15

Series

see

Developments

6.5.29

r*(--n, r (0, x>=

x)=x”

(2) S2me-fl-ldt=El

~*(u,z)=e+

2’ n-o r(a+n+l)

_

1 2 C-2)" r(d n-~ b+dn!

(IzI<=>

GAMMA

FUNCTION

AND

RELATED

263

FUNCTIONS

Definite

6.5.30 r(a, x+Y)-r(a, cm n=o

,e-zxa-l

m S

Integrals

6.5.36

5) (a-1)(a-2). _____

. . (4[l-e-Ye ?I

Xn

(y)J

e

0

-“‘r(b,ct)

ha

l-

y

&] (c%%+c)>O,9b>-1)

(lYl
6.5.37

trt=!%!.k!i Smp-lr(b,t)

Fraction

6.5.31

1 1-a ( &1+icp

r(u,x)=e-‘.P

a

0

1 2-u 2 1+ s+“’

(9 (a+ b) >O,

>

(x>O,bl<m> Asymptotic

6.6. Incomplete

Expansions

B,(a,b)=

6.6.1

Beta

Bu>O>

Function

S

= ta-‘(l--2)b-‘dt

0

6.5.32

r(a, Z)-p-le-r

[

,+!!$+(a-y-2)+.

6.6.2

. .]

For statistical applications, see 26.5.

(

2-m

in jarg zI
Symmetry

>

!nn(w)! and sign I?,(a,z)=sign

6.5.34

I&b)=

6.6.3

R,(u,z)=u,+,(u,z)+ . . . is t,lie remainder after R terms in this scrics. Then if a,2 arc real, WC have for n>u-2 SUppOW

6 .5 .33

I,(a,b)=B,(u,b)lB(u,b)

Relation

6.6.4

_
(a-+

in 1arg

zl<+f)

Numerical

Formulas

r,(a,b)=~:l,(a--l,b)+(l--s)r,(cl;,b-1)

6.6.6

(a+ b-m)l&,b> =u(l-x)l,(u+l,b-l)+bl,(u,b+l)

6.6.7

(a+b>Iz(a,b) =alz(a+

,

to Hypergeometric

B,(a,b)=a-‘x”F(u,l-bb;

6.6.8

*

l,b)+bIz(a,b+

1)

Function

a+l;

x)

Methods

of the Tables

Compute r(6.88) to 8s. Using the rccurrcIicc relation 6.1.16 nntl Table 6.1 WC 11nvr, r(6.R8)=[(5.38)(4.:~8)(:~.R8)(2.38)(1..78)]r(1.38) Example

see 26.1.

6.6.5

Relation

Extension

()$(1--p)‘-

a)

0 for a>1 lim !l?LkTEJz 4 for a=1 p ,1--f1 for O
Use and

Expansion

I,(a,n-a+l)=I$

Recurrence

r@ 2)--&!?2a+n __n=o (u+n)n!

6.5.35

6.7.

to Binomial

For binomial distribution,

zr,z+,(u,z).

(z-03

1--I,-,(b,a)

1.

=232.43671. Example 2. Compute ln r(56.38), using Table 6.4 and linear illtcrpolntion ill.f,. \\-c have

In I’(56.38) = (56.38---a) In (56.38) - (56.38) +J2(56.38)

The error of linear interpolation in the table of the function.fi is smaller tlian lo-’ in this region. Hence, f,(56.38)=.92041 67 and In r(56.38)= 169.85497 42. Direct interpolation in Table 6.4 of log,, I’(n) eliminates tlir necessity of employing logarithms. However, the error of linear interpolation is .002 so that log,0 r(n) is obtained wit11 a relative error of lo-“. Wee

page

11.

264

GAMMA

Example

FUNCTION

AND

3. Compute

recurrence relation

lt(6.38) to 8s. Using the 6.3.6 and Table 6.1.

$(6.38)=1+1+1+1+1+$(1.38) 5.38 4.38 3.38 =1.77275

2.38

3.38

59.

4. Compute #(56.38). Using Table 6.3 we have J/(56.38) =ln 56.38-ja(56.38). The error of linear interpolation in the table of the function f3 is smaller than 8X10-’ in this region. Hence&56.38) = .00889 53 and#(56.38) = 4.023219. Example

Example 5. Compute In I’(l-i). reflection principle 6.1.23 and In I’(l--i)=lnI’(l+i)=---.6509+.3016i.

Compute In I’(+++$. of the recurrence relation

Example

the 6.7,

Taking 6.1.15 we

r (++*i) =ln r (g+g;)-In

(4+&Q -.23419+.03467i =- (+ In *+i arctan 1) =.11239-.75073i

The logarithms from 4.1.2.

of complex

numbers

FUNCTIONS

6.8. Summation of Rational Series of Polygamma Functions

by Means

An infinite series whose general term is a rational function of the index may always be reduced to a finite series of psi and polygamma functions. The method will be illustrated by writing the explicit formula when the denominator contains a triple root. Let the general term of an infinite series have the form

2-W U”=&(n)d2(nMn) where dl(n)=(n+arl)(n+az)

6.

the logarithm have, In

From Table

RELATED

. . . (n + 4

Un)=(n+W(n+W

. . . (n+i%Y

~dn)=(n+~d3(n+rd3

. . . (n+-d3

where p(n) is a polynomial of degree m + 2r + 3s - 2 at most and where the constants ai, pi, and yc are distinct. Expand u, in partial fractions as follows are found

Example 7. Compute In I’(3+7i) using the duplication formula 6.1.18. Taking the logarithm of 6.1.18, we have

-+ In 27r= - .91894 (;+7i) In 2= 1.73287+

4.85203i

In r(++Ji)=-3.31598+ In r(Z+$i) = -2.66047+

2.32553i 293869i

In I’(3+7i)=-5.16252+10.11625i Example

8.

the asymptotic

Then,

we may express 2

u, in terms of the

75-l

constants appearing sion as follows

in this partial

fraction expan-

Compute In I’(3+7i) to 5D using formula 6.1.41. We have

ln (3+7i)=2.03022

15+1.16590

45i.

Then, (2.5$7i)

In (3+7i) =-3. 0857779+ 17.1263119i -(3+7i)=-3. ooooooo7. oooooooi + In (2~r)= . 9189385 [12(3+7i)]-‘= .0043103.0100575i -[360(3+7i)3]-1= .0000059.0000022i ---------_-__ ln I’(3+7i)=-5. 16252 +lO. 11625i

Higher order repetitions in the denominator are If the denominator contains handled similarly.

GAMMA

FUNCTION

AND RELATED

only simple or double roots, omit the corresponding lines.

265

FUNCTIONS

Therefore s=16~(1)-16~(1~)+~‘(1)+~‘(1~)=.013499.

Example 9.

Find Example 11.

1 S=% nsl (%+1)(2n+l)(4n+l)

i

we have

72 i --i Hence, al=-t a2=--, 6 6

1

1

( nii-n--2i

rel="nofollow">*

i --i as=-) ad=--’ 12 12

cY1=i, az=---i, (Y3=2i, (yq=-2i, and therefore

Example 10.

F-y

Find s= 2 ’ .)&=In2p3n+ 1)“’ Since

1 n2(8n+1)2=-x

16 16 1 +m+++;li+m

[$(l+i)-P(l-i)l+;

[#(142i)--$(l-2i)].

By 6.3.9, this reduces to

1

sf

&(l+i)-;

99(1f2i).

we have, From Table 6.8, s= .13876.

References Tables

Texts

[6.1] E. Artin, Einfiihrung in die Theorie der Gammafunktion (Leipzig, Germany, 1931). [6.2] P. E. Bohmer, Differenzengleichungen und bestimmte Integrale, chs. 3, 4, 5 (K. F. Koehler, Leipzig, Germany, 1939). [6.3] G. Doetsch, tion, vol. Switzerland,

Handbuoh der II, pp. 52-61 1955).

Laplace-Transforma(Birkhauser, Basel,

[6.4] A. Erdelyi et al., Higher transcendental functions, vol. 1, ch. 1, ch. 2, sec. 5; vol. 2, ch. 9 (McGrawHill Book Co., Inc., New York, N.Y., 1953). [6.5] C. Hastings, Jr., Approximations for digital computers (Princeton Univ. Press, Princeton, N.J., 1955). [6.61-F. L6soh and F. Schoblik, Die. Fakultiit und verwandte Funktionen (B. G. Teubner, Leipzig, Germany, 1951). [6.7] W. Sibagaki, Theory and applications of the gamma function (Iwanami Syoten, Tokyo, Japan, 1952). 16.81 E. T. Whittaker and G. N. Watson, A course of modern analysis, ch. 12, 4th ed. (Cambridge Univ. Press, Cambridge, England, 1952).

[6.9] A. Abramov, Tables of In r(z) for complex argument. Translated from the Russian by D. G. Fry (Pergamon Press, New York, N.Y., 1960). In r(z+iy), 2=0(.01)10, y=O(.O1)4, 6D. [6.10] Ballistic Research Laboratory, A table of the factorial numbers and their reciprocals from l! through lOOO! to 20 significant digits. Technical Note No. 351, Aberdeen Proving Ground, Md., 1951. [6.1 I] British Association for the Advancement of Science, Mathematical tables, vol. 1, 3d ed., pp. 40-59 (Cambridge Univ. Press, Cambridge, England, 1951). The gamma and polygamma functions. Also lf = loglo (t)!dt, z=o(.Ol)l, 10D. s0 [6.12] H. T. Davis, Tables of the higher mathematical functions, 2 ~01s. (Principia Press, Bloomington, Extensive, many place tables Ind., 1933, 1935). of the gamma and polygamma functions up to @o(x) and of their logarithms. [6.13] F. J. Duarte, Nouvelles tables de log10 n! 2133 d&imales depuis n= 1 jusqu’il n=3000 (Kundig, Geneva, Switzerland; Index Generalis, Paris, France, 1927).

266

GAMMA

FUNCTION

AND RELATED

[6.14] National Bureau of Standards, Tables of nl and P(n++) for the first thousand values of n, Applied Math. Series 16 (U.S. Government Printing Office, Washington, D.C., 1951). n!, lGS;P(n+&), 85. [6.15] National Bureau of Standards, Table of Coulomb wave functions, vol. I, pp. 114-135, Applied Math. Series 17 (U.S. Government Printing Office, Washington, D.C., 1952). &‘[F’(l+i~)/F(l

+iq],rl=O(.OO5)2

(.01)6 (.02)10(.1)

2O(.2)60(.5)110,10D;a~gF(l+i~),~=0(.01)1(.02) 3 (.05)10(.2)20(.4)30(.5)85, 8D. [6.16] National Bureau of Standards, Table of the gamma function for complex arguments, Applied Math. Series 34 (U.S. Government Printing Office, Washington, D.C., 1954). In F(z+iy),

z=9(.1)10,

y=O(.l)lO,

12D.

Contains an extensive bibliography. (6.171 National Physical Laboratory, Tables of Weber parabolic cylinder functions, pp. 226-233 (Her Majesty’s Stationery Office, London, England, 1955). RealandimaginarypartsoflnF($c+$a),k=0(1)3, a=O(.1)5(.2)20, 8D; (Ir(4+aia)/r(t+~ia)I)-“1 a=0(.02)1(.1)5(.2)20, 8D.

FUNCTIONS

(6.181 E. S. Pearson, Table of the logarithms of the complete P-function, arguments 2 to 1200, Tracts for Computers No. VIII (Cambridge Univ. Press, Cambridge, England, 1922). Loglo P(p), p=2(.1) 5(.2)70(1)1200, 10D. [6.19] J. Peters, Ten-place logarithm tables, vol. I, Appendix, pp. 58-68 (Frederick Ungar Publ. Co., New York, N.Y., 1957). nl, n=1(1)60, exact; (n!)-I, n=1(1)43, 54D; Logto( n=1(1)1200, 18D. [6.20] J. P. Stanley and M. V. Wilkes, Table of the reciprocal of the gamma function for complex argument (Univ. of Toronto Press, Toronto, Canada, 1950). x=-.5(.01).5, y=O(.Ol)l, 6D. [6.21] M. Zyczkowski, Tablice funkcyj eulera i pokrewnych (Panstwowe Wydawnictwo Naukowe, Warsaw, Poland, 1954). Extensive tables of integrals involving gamma and beta functions. For references to tabular material on the incomplete gamma and incomplete beta functions, see the references in chapter 26.

GAMMA FUNCTION GAMMA,

DIGAMMA

267

AND RELATED FUNCTIONS

AND TRIGAMMA

Table

FUNCTIONS

6.1

ti’ (x)

1.00000 0.99713 0.99432 0.99156 0.98884

00000 85354 58512 12888 42033

In r(x) 0.00000 00000 -0.00286 55666 -0.00569 03079 -0.00847 45187 -0.01121 84893

3 (.r)

1.000 1.005 1.010 1.015 1.020

-0.57721 56649 -0.56902 09113 -0.56088 54579 -0.55280 85156 -0.54478 93105

1.6449340668 1.6329941567 1.6212135283 1.6095891824 1.5981181919

0.000

1.025 1.030 1.035 1.040 1.045

0.98617 0.98354 0.98097 0.97843 0.97594

39633 99506 15606 82009 92919

-0.01392 -0.01658 -0.01921 -0.02179 -0.02434

25067 68539 18101 76511 46490

-0.53682 70828 -0.52892 10873 -0.52107 05921 -0.51327 48789 -0.50553 32428

1.5867976993 1.5756249154 1.5645971163 1.55371 16426 1.5429658968

0.025 0.030 0.035 0.040 0.045

1.050 1.055 1.060 1.065 1.070

0.97350 0.97110 0.96874 0.96642 0.96415

42656 25663 36495 69823 20425

-0.02685 -0.02932 -0.03175 -0.03414 -0.03650

30725 31868 52537 95318 62763

-0.49784 49913 -0.49020 94448 -0.48262 59358 -0.47509 38088 -0.46761 24199

1.5323573421 1.5218835001 1.51154 19500 1.5013303259 1.4912463164

0.050 0.055 0.060 0.065 0.070

1.075 1.080 1.085 1.090 1.095

0.96191 0.95972 0.95757 0.95545 0.95338

83189 53107 25273 94882 57227

-0.03882 -0.04110 -0.04335 -0.04556 -0.04773

57395 81702 38143 29148 57114

-0.46018 11367 -0.45279 93380 -0.44546 64135 -0.43818 17635 -0.43094 47988

1.4812876622 1.4714521556 1.4617376377 1.4521419988 1.4426631755

0.075 0.080 0.085 0.090 0.095

1.100 1.105 1.110 1.115 1.120

0.95135 0.94935 0.94739 0.94547 0.94359

07699 41778 55040 43149 01856

-0.04987 -0.05197 -0.05403 -0.05606 -0.05806

24413 33384 86341 85568 33325

-0.42375 49404 -0.41661 16193 -0.40951 42761 -0.40246 23611 -0.39545 53339

1.4332991508 1.4240479514 1.4149076482 1.4058763535 1.3969522213

0.100 0.105 0.110 0.115 0.120

1.125 1.130 1.135 1.140 1.145

0.94174 0.93993 0.93815 0.93641 0.93471

26997 14497 60356 60657 11562

-0.06002 -0.06194 -0.06383 -0.06569 -0.06751

31841 83322 89946 53867 77212

-0.38849 26633 -0.38157 38268 -0.37469 83110 -0.36786 56106 -0.36107 52291

1.3881334449 1.3794182573 1.3708049288 1.3622917670 1.3538771152

0.125 0.130 0.135 0.140 0.145

1.150 1.155 1.160 1.165 1.170

0.93304 0,93140 0.92980 0.92823 0.92669

09311 50217 30666 47120 96106

-0.06930 -0.07106 -0.07278 -0.07447 -0.07612

62087 10569 24716 06558 58106

-0.35432 66780 -0.34761 94768 -0.34095 31528 -0.33432 72413 -0.32774 12847

1.3455593520 1.3373368900 1.3292081752 1.3211716859 1.3132259322

0.150 0.155 0.160 0.165 0.170

1.175 1.180 1.185 1.190 1.195

0.92519 0.92372 0.92229 0.92088 0.91951

74225 78143 04591 50371 12341

-0.07774 -0.07933 -0.08089 -0.08242 -0.08391

81345 78240 50733 00745 30174

-0.32119 48332 -0.31468 74438 -0.30821 86809 -0.30178 81156 -0.29539 53259

1.3053694548 1.2976008248 1.2899186421 1.2823215358 1.2748081622

0.175 0.180 0.185 0.190 0.195

1.200 1.205 1.210 1.215 1.220

0.91816 0.91685 0.91557 0.91432 0.91310

87424 72606 64930 61500 59475

-0.08537 -0.08680 -0.08820 -0.08956 -0.09090

40900 34780 13651 79331 33619

-0.28903 98966 -0.28272 14187 -0.27643 94897 -0.27019 37135 -0.26398 37000

1.2673772054 1.2600273755 1.2527574090 1.2455660671 1.2384521360

0.200 0.205 0.210 0.215 0.220

1.225 1.230 1.235 1.240 1.245

0.91191 0.91075 0.90962 0.90852 0.90744

56071 48564 34274 10583 74922

-0.09220 -0.09348 -0.09472 -0.09593 -0.09711

78291 15108 45811 72122 95744

-0.25780 90652 -0.25166 94307 -0.24556 44243 -0.23949 36791 -0.23345 68341

1.2314144258 1.2244517702 1.2175630254 1.2107470707 1.2040028063

0.225 0.230 0.235 0.240 0.245

1.250

0.90640 24771

-0.09827 18364

-0.22745 35334

1.1973291545

0.250

f lny!

f$ In y!

Y

r(3’)

lny!

Y!

[C-56161 For a>2 see Examples

l-4.

[i-56)51

log,, e=0.43429

[(-56)71

0.005 0.010 0.015 0.020

[C-55)21

44819

Compiled from H. T. Davis, Tables of the higher mathematical functions, 2 ~01s. (Principia Press,Bloomington,Ind., 1933,1935)(with permission). Known error hasbeencorrected.

208

GAMMA

Table

6.1

$50 1.255

GAMMA,

r(x)

AND

DIGAMMA

In r(x)

RELATED

AND

FUNCTIONS

TRIGAMMA

e>

FUNCTIONS

+’(4

1.260 1.265 1.270

0.90640 0.90538 0.90439 0.90343 0.90250

24771 57663 71178 62946 30645

-0.09827 -0.09939 -0.10048 -0.10154 -0.10258

18364 41651 67254 96809 31932

-0.22745 -0.22148 -0.21554 -0.20964 -0.20376

35334 34266 61686 14193 88437

1.19732 1.19072 1.18418 1.17772 1.17131

91545 50579 94799 14030 98301

0.250 0.255 0.260 0.265 0.270

1.275 1.280 1.285 1.290 1.295

0.90159 0.90071 0.89986 0.89904 0.89824

71994 84765 66769 15863 29947

-0.10358 -0.10456 -0.10550 -0.10642 -0.10731

74224 25269 86634 59872 46519

-0.19792 -0.19211 -0.18634 -0.18059 -0.17487

81118 88983 08828 37494 71870

1.16498 1.15871 1.15250 1.14635 1.14027

37821 22990 44385 92764 59053

0.275 0.280 0.285 0.290 0.295

1.300 1.305 1.310 1.315 1.320

0.89747 0.89672 0.89600 0.89530 0.89464

06963 44895 41767 95644 04630

-0.10817 -0.10900 -0.10981 -0.11058 -0.11133

48095 66107 02045 57384 33587

-0.16919 -0.16353 -0.15790 -Oil5231 -0.14674

08889 45526 78803 05782 23568

1.13425 1.12829 1.12238 1.11654 1.11075

34350 09915 77175 27706 53246

0.300 0.305 0.310 0.315 0.320

1.325 1.330 1.335 1.340 1.345

0.89399 0.89337 0.89278 0.89221 0.89167

66866 80535 43850 55072 12485

-0.11205 -0.11274 -0.11341 -0.11404 -0.11465

32100 54356 01772 75756 77697

-0.14120 -0.13569 -0.13020 -0.12475 -0.11932

29305 20180 93416 46279 76069

1.10502 1.09934 1.09372 1.08816 1.08265

45678 97037 99497 45379 27136

0.325

:*;:: 1:360 1.365 1.370

0.89115 0.89065 0.89018 0.88973 0.88931

14420 59235 45324 71116 35074

-0.11524 -0.11579 -0.11632 -0.11682 -0.11730

08974 70951 64980 92401 54539

-0.11392 -0.10855 -0.10321 -0.09789 -0.09259

80127 55827 00582 11840 87082

1.07719 1.07178 1.06643 1.06112 1.05587

37361 68773 14226 66696 19286

0.350 0.355

1.375 1.380 1.385 1.390 1.395

0.88891 0.88853 0.88818 0.88785 0.88754

35692 71494 41041 42918 75748

-0.11775 52707 -0.11817 88209 -0.11857 62331 -0.11894 '76353 -0.11929 31538

-0.08733 -0.08209 -0.07687 -0.07168 -0.06652

23825 19619 72046 78723 37297

1.05066 1.04550 1.04040 1.03533 1.03032

65216 97829 10578 97036 50881

0.375 0.380 0.385

1.400 1.405 1.410 1.415 1.420

0.88726 0.88700 0.88676 0.88654 0.88635

38175 28884 46576 89993 57896

-0.11961 -0.11990 -0.12017 -0.12041 -0.12063

29142 70405 56559 88823 68406

-0.06138 45446 -0.05627 00879 -0.05118 01337 -0.04611 44589 -0.04107 28433

1.02535 1.02043 1.01555 1.01072 1.00593

65905 36002 55173 17518 17241

0.400 0.405 0.410 0.415 0.420

1.425 1.430 1.435 1.440 1.445

0.88618 0.88603 0.88590 0188580 0.88572

49081 62361 96587 50635 23397

-0.12082 -0.12099 -0.12114 -0.12125 -0.12135

96505 74307 02987 83713 17638

-0.03605 -0.03106 -0.02609 -0.02114 -0.01621

50697 09237 01935 26703 81479

1.00118 0.99648 0.99181 0.98719 0.98261

48640 06113 84147 77326 80318

0.425 0.430 0.435 0.440 0.445

1.450 1.455 1.460 1.465 1.470

0.88566 0.88562 0.88560 0.88560 0.88563

13803 20800 43364 80495 31217

-0.12142 -0.12146 -0.12148 -0.12148 -0.12145

05907 49657 50010 08083 24980

-0.01131 -0.00643 -0.00158 +0.00325 0.00806

64226 72934 05620 39677 64890

0.97807 0.97357 0.96911 0.96469 0.96031

87886 94874 96215 86921 62091

0.450 0.455 0.460 0.465 0.470

1.475 1.480 1.485 1.490 1.495

0.88567 0.88574 0.88583 0.88594 0.88607

94575 69646 55520 51316 56174

-0.12140 -0.12132 -0.12122 -0.12110 -0.12095

01797 39621 39528 02585 29852

0.01285 0.01762 0.02237 0.02710 0.03180

71930 62684 39013 02758 55736

0.95597 0.95166 0.94739 0.94316 0.93896

16896 46592 46509 12052 38700

0.475 0.480 0.485 0.490 0.495

1.500

0.88622 69255

-0.12078

22376

0.93480 22005

0.500

2/!

[C-56)4I *See

FUNCTION

page

II.

lng!

0.03648 99740

* dy d lny!

[c-y1

(-i)4 [ I log,, e=O.43429 44819

* $$ lny! ” (-;)9

[ 1

"0% 0:340 0.345

Ez 0:370

K%5" .

!/

GAMMA FUNCTION GAMMA, t

DIGAMMA

r(z)

AND

AND RELATED TRIGAMMA

FUNCTIONS

269

FUNCTIONS

+w

In r(r)

Table 6.1

ti’ (z)

1.500 1.505 1.510 1.515 1.520

0.88622 0.88639 0.88659 0.88680 0.88703

69255 89744 16850 49797 87833

-0.12078 -0.12058 -0.12037 -0i12Oi3 -0.11986

22376 81200 07353 01860 65735

0.03648 0.04115 0.04579 0.05041 0.05502

99740 36543 67896 95527 21146

0.93480 0.93067 0.92658 0.92252 0.91850

22005 57588 41142 68425 35265

0.500 0.505 0.510 0.515 0.520

1.525 1.530 1.535 1.540 1.545

0.88729 0.88756 0.88786 0.88817 0.88851

30231 76278 25287 76586 29527

-0.11957 -0.11927 -0.11893 -0.11858 -0.11820

99983 05601 83580 34900 60534

0.05960 0.06416 0.06871 0.07323 0.07773

46439 73074 02697 36936 77400

0.91451 0.91055 0.90663 0.90274 0.89888

37552 71245 32361 16984 21253

0.525 0.530 0.535 0.540 0.545

1.550 1.555 1.560 1.565 1.570

0.88886 0.88924 0.88963 0.89005 0.89048

83478 37830 91990 45387 97463

-0.11780 -0.11738 -0.11693 -0.11647 -0.11598

61446 38595 92928 25388 36908

0.08222 0.08668 0.09113 0.09556 0.09997

25675 83334 51925 32984 28024

0.89505 0.89125 0.88749 0.88375 0.88005

41371 73596 14249 59699 06378

0.550 0.555 0.560 0.565 0.570

1.575 1.580 1.585 1.590 1.595

0.89094 0.89141 0.89191 0.89242 0.89296

47686 95537 40515 82141 19949

-0.11547 -0.11494 -0.11438 -0.11380 -0.11321

28415 00828 55058 92009 12579

0.10436 0.10873 0.11309 0.11742 0.12174

38544 66023 11923 77690 64754

0.87637 0.87272 0.86911 0.86552 0.86196

50766 89402 18871 35815 36921

0.575 0.580 0.585 0.590 0.595

1.600 1.605 1.610 1.615 1.620

0.89351 0.89408 0.89468 0.89529 0.89592

53493 82342 06085 24327 36685

-0.11259 -0.11195 -0.11128 -0.11060 -0.10990

17657 08127 84864 48737 00610

0.12604 0.13033 0.13459 0.13884 0.14307

74528 08407 67772 53988 68404

0.85843 0.85492 0.85145 0.84800 0.84457

18931 78630 12856 18488 92455

0.600 0.605 0.610 0.615 0.620

1.625 1.630 1.635 1.640 1.645

0.89657 0.89724 0.89793 0.89864 0.89936

42800 42326 34930 20302 98138

-0.10917 -0.10842 -0.10765 -0.10687 -0.10606

41338 71769 92746 05105 09676

0.14729 0.15148 0.15566 0.15983 0.16398

12354 87158 94120 34529 09660

0.84118 0.83781 0.83446 0.83115 0.82785

31730 33330 94315 11790 82897

0.625 0.630 0.635 0.640 0.645

1.650 1.655 1.660 1.665 1.670

0.90011 0.90088 0.90166 0.90247 0.90329

68163 30104 83712 28748 64995

-0.10523 -0.10437 -0.10350 -0.10261 -0.10170

07282 98739 84860 66447 44301

0.16811 0.17222 Oil7632 0.18040 0.18447

20776 69122 55933 82427 49813

0.82459 0.82134 Oi818i2 0.81493 0.81176

04826 74802 90092 48001 45875

0.650 0.655 0.660 0.665 0.670

1.675 1.680 1.685 1.690 1.695

0.90413 0.90500 0.90588 0.90678 0.90770

92243 10302 18996 18160 07650

-0.10077 -0.09981 -0.09884 -0.09785 -0.09684

19212 91969 63351 34135 05088

0.18852 0.19256 0.19658 0.20058 0.20457

59282 12015 09180 51931 41410

0.80861 0.80549 0.80239 0.79931 0.79626

81094 51079 53282 85198 44350

0.675 0.680 0.685 0.690 0.695

1.700 1.705 1.710 1.715 1.720

0.90863 0.90959 0.91057 0.91156 0.91258

87329 57079 16796 66390 05779

-0.09580 -0.09475 -0.09368 -0iO9259 -0.09147

76974 50552 26573 05785 88929

0.20854 0.21250 0.21645 0.22037 0.22429

78749 65064 01462 89037 28871

0.79323 0.79022 0.78723 0:78427 0.78132

28302 34645 61012 05060 64486

0.700 0.705 0.710 0.715 0.720

1.725 1.730 1.735 1.740 1.745

0.91361 0.91466 0.91573 0.91682 0.91793

34904 53712 62171 60252 47950

-0.09034 -0.08919 -0.08802 -0.08683 -0.08562

76741 69951 69286 75466 89203

0.22819 0.23207 0.23594 0.23980 0.24364

22037 69593 72589 32061 49038

0.77840 0.77550 0.77262 0.76976 0.76692

37011 20396 12424 10915 13714

0.725 0.730 0.735 0.740 0.745

1.750

0.91906 25268

-0.08440

11210

0.24747 24535

0.76410 18699

0.750

Y!

[ 1 (El)3

lny!

f

[ 1 (-46)3

log,,

e=0.43429

lny!

[ 1 (-f)3

44819

d$2 lny!

[ 1 (-56)4

Y

270

GAMMA

Table

6.1

GAMMA,

2

1.750 1.755 1.760 1.765 1.770

0.91906 0.92020 0.92137 0192255 0.92376

1.775 1.780 1.785 1.790 1.795

r(x)

FUNCTION

AND

DIGAMMA

AND

RELATED

FUNCTIONS

TRIGAMMA

In r(r)

FUNCTIONS

+’(2)

d+)

25268 92224 48846 95178 31277

-0.08440 -0.08315 -0.08188 -0.08060 -0.07929

11210 42192 82847 33871 95955

0.24747 0.25128 0.25508 0.25887 0.26264

24535 59559 55103 12154 31686

0.76410 0.76130 0.75852 0.75576 0.75302

18699 23773 26870 25950 19003

0.750 0.755 0.760 0.765 0.770

0.92498 0.92622 O-92748 0:92876 0.93006

57211 73062 78926 74904 61123

-0.07797 -0.07663 -0.07527 -0.07389 -0.07249

69782 56034 55386 68509 96070

0.26640 0.27014 0.27387 0.27759 0.28129

14664 62043 74769 53776 99992

0.75030 0.74759 0.74491 0.74224 0.73960

04040 79107 42268 91617 25271

0.775 0.780 0.785 0.790 0.795

1.800 1.805 1.810 1.815 1.820

0.93138 0.93272 0.93407 0.93545 0.93684

37710 04811 62585 11198 50832

-0.07108 -0.06964 -0.06819 -0.06672 -0.06523

38729 97145 71969 63850 73431

0.28499 0.28866 0.29233 0.29598 0.29962

14333 97707 51012 75138 70966

0.73697 0.73436 0.73177 0.72919 0.72663

41375 38093 13620 66166 93972

0.800 0.805 0.810 0.815 0.820

1.825 1.830 1.835 1.840 1.845

0.93825 0.93969 0.94114 0.94261 0.94410

81682 03951 17859 23634 21519

-0.06373 -0.06220 -0.06066 -0.05910 -0.05752

01353 48248 14750 01483 09071

0.30325 0.30686 0.31046 0.31405 0.31763

39367 81205 97335 88602 55846

0.72409 0.72157 0.71907 0.71658 0.71411

95297 68426 11662 23333 01788

0.825 0.830 0.835 0.840 0.845

1.850 1.855 1.860 1.865 1.870

0.94561 0.94713 0.94868 0.95025 0.95184

11764 94637 70417 39389 01855

-0.05592 -0.05430 -0.05267 -0.05102 -0.04935

38130 89276 63117 60260 81307

0.32119 0.32475 0.32829 0.33182 0.33533

99895 21572 21691 01056 60467

0.71165 0.70921 0.70679 0.70438 0.70199

45396 52546 21650 51138 39461

0.850 0.855 0.860 0.865 0.870

1.875 1.880 1.885 1.890 1.895

0.95344 0.95507 0.95671 0.95837 0.96006

58127 08530 53398 93077 27927

-0.04767 -0.04596 -0.04424 -0.04251 -0.04075

26854 97497 93824 16423 65875

0.33884 0.34233 0.34581 0.34928 0.35273

00713 22577 26835 14255 85596

0.69961 0.69725 0.69491 0.69258 0.69027

85089 86512 42236 50790 10717

0.875 0.880 0.885 0.890 0.895

1.900 1.905 1.910 1.915 1.920

0.96176 0.96348 0.96523 0.96699 0.96877

58319 84632 07261 26608 43090

-0.03898 -0.03719 -0.03538 -0.03356 -0.03172

42759 47650 81118 43732 36054

0.35618 0.35961 0.36304 0.36645 0.36985

41612 83049 10646 25136 27244

0.68797 0.68568 0.68341 0.68116 0.67892

20582 78965 84465 35696 31293

0.900 0.905 0.910 0.915 0.920

1.925 1.930 1.935 1.940 1.945

0.97057 0.97239 0.97423 0.97609 0.97797

57134 69178 79672 89075 97861

-0.02986 -0.02799 -0.02609 -0.02419 -0.02226

58646 12062 96858 13581 62778

0.37324 0.37661 Oij7998 0.38334 0.38668

17688 97179 66424 26119 76959

0.67669 0.67448 0.67228 0.67010 0.66793

69903 50194 70846 30559 28044

0.925 0.930 0.935 0.940 0.945

1.950 1.955 1.960 1.965 1.970

0.97988 0.98180 0.98374 0.98570 0.98768

06513 15524 25404 36664 49838

-0.02032 -0.01836 -0.01639 -0.01439 -0.01239

44991 60761 10621 95106 14744

0.39002 0.39334 0.39665 0.39996 0.40325

19627 54805 83163 05371 22088

0.66577 0.66363 0.66150 0.65938 0.65728

62034 31270 34514 70538 38134

0.950 0.955 0.960 0.965 0.970

1.975 1.980 1.985 1.990 1.995

0.98968 0.99170 0.99375 0.99581 0.99789

65462 84087 06274 32598 63643

-0.01036 -0.00832 -0.00626 -0.00419 -0.00210

70060 61578 89816 55291 58516

0.40653 0.40980 0.41306 0.41631 0.41955

33970 41664 45816 47060 46030

0.65519 0.65311 0.65105 0.64900 0.64696

36104 63266 18450 00505 08286

0.975 0.980 0.985 0.990 0.995

2.000

1.00000 00000

0.64493 40668

1.000

Y!

II 3 c-y

0.00000 00000

0.42278 43351

lny!

$ In y!

[ 1 (512

log,o e=0.43429

[

c-y

44819

Y

I

GAMMA

FUNCTION

TETRAGAMMA

.,

*

#” (.I,)

AND

RELATED

AND PENTAGAMMA

+m) (.r)

.I’

271

FUNCTIONS

FUNCTIONS

Tal,le 6.2

V(T)

$(:I) (x)

1.00 1.01 1.02 1.03 1.04

-2.40411 -2.34039 -2.27905 -2.21996 -2.16303

38063 86771 42052 85963 63855

6.49393 6.25106 6.01969 5.79918 5.58891

94023 18729 49890 38573 68399

0. 00 0. 01 0.02 0.03 0.04

1.50 1.51 1.52 1.53 1.54

-0.82879 -0.81487 -0.80129 -0.78803 -0.77509

66442 76121 51399 87419 83287

1.40909 1.37489 1.34177 1.30967 1.27856

10340 70527 21104 56244 88154

0.50 0.51 0.52 0.53 0.54

1.05 1.06 1.07 1.08 1.09

-2.10815 -2.05523 -2.00419 -1.95493 -1.90737

80219 94833 19194 13213 82154

5.38832 5.19686 5.01404 4.83939 4.67247

23132 56970 67303 69702 74947

0. 05 0.06 0.07 0.08 0.09

1.55 1.56 1.57 1.58 1.59

-0.76246 -0.75012 -0.73807 -0.72630 -0.71480

41904 69793 76946 76669 85441

1.24841 1.21917 1.19082 1.16332 1.13663

46160 75841 38216 08979 77770

0.55 0.56 0.57 0.58 0.59

1.10 1.11 1.12 1.13 1.14

-1.86145 -1.81709 -1.77423 -1.73279 -1.69272

73783 75731 13035 45852 67342

4.51287 4.36020 4.21411 4.07424 3.94028

67903 88083 11755 35447 60717

0.10 0. 11 0.12 0.13 0.14

1.60 1.61 1.62 1.63 1.64

-0.70357 -0.69259 -0.68185 -0.67136 -0.66110

22779 11105 75627 44220 47316

1.11074 1.08561 1.06121 1.03752 1.01452

47490 33658 63792 76835 22608

0.60 0.61 0.62 0.63 0.64

1.15 1.16 1.17 1.18 1.19

-1.65397 -1.61647 -1.58017 -1.54503 -1.51100

01677 02206 49731 50903 36723

3.81193 3.68891 3.57095 3.45780 3.34922

80220 64540 50416 29554 38402

0.15 0. 16 0. 17 0.18 0.19

1.65 1.66 1.67 1.68 1.69

-0.65107 -0.64125 -0.63166 -0.62226 -0.61308

17793 90881 04061 96973 11332

0.99217 0.97046 0.94937 0.92886 0.90893

61290 62927 06973 81843 84502

0.65 0.66 0.67 0.68 0.69

1.20 1.21 1.22 1.23 1.24

-1.47803 -1.44608 -1.41512 -1.38510 -1.35598

61144 99765 48602 22950 56308

3.24499 3.14490 3.04875 2.95636 2.86754

48647 58422 84139 52925 95589

0.20 0. 21 0.22 0.23 0.24

1.70 1.71 1.72 1.73 1.74

-0.60408 -0.59528 -0.58667 -0.57823 -0.56997

90841 81112 29593 85490 99702

0.88956 0.87072 0.85239 0.83456 0.81722

20066 01433 48922 89940 58660

0.70 0.71 0.72 0.73 0.74

1.25 1.26 1.27 1.28 1.29

-1.32773 -1.30033 -1.27372 -1.24790 -1.22282

99375 19112 97857 32496 33691

2.78214 2.69999 2.62093 2.54484 2.47158

40092 05478 96227 97000 67746

0.25 0. 26 0.27 0.28 0.29

1.75 1.76 1.77 1.78 1.79

-0.56189 -0.55397 -0.54621 -0.53861 -0.53116

24756 14738 25238 13291 37320

0.80034 0.78392 0.76793 0.75237 0.73721

95719 47929 68005 14300 50564

0.75 0.76 0.77 0.78 0.79

1.30 1.31 1432 1.33 1.34

-1.19846 -1.17479 -1.15179 -1.12943 -1.10769

25147 42923 34794 59642 86881

2.40102 2.33304 2.26752 2.20436 2.14345

39143 08348 35032 37678 90132

0. 30 0. 31 0.32 0.33 0.34

1.80 1.81 1.82 1.83 1.84

-0.52386 -0.51671 -0.50970 -0.50283 -0.49609

57084 33630 29242 07396 32712

0.72245 0.70807 0.69407 0.68042 0.66712

45705 73565 12710 46226 61527

0.80 0.81 0.82 0.83 0.84

1.35 1.36 1.37 1.38 1.39

-1.08655 -1.06599 -1.04599 -1.02652 -1.00757

95925 75682 24073 47586 60850

2.08471 2.02802 1;97332 1.92051 1.86951

18367 97472 48830 37473 69616

0.35 E 0:38 0.39

1.85 1.86 1.87 1.88 1.89

-0.48948 -0.48300 -0.47665 -0.47042 -0.46431

70921 88813 54207 35909 03677

0.65416 0.64153 0.62921 0.61720 0.60549

50169 07680 33389 30270 04793

0.85 0.86 0.87 0.88 0.89

1.40 1.41 1.42 1.43 1.44

-0.98912 -0.97116 -0.95366 -0.93662 -0.92002

86236 53479 99322 67177 06808

1.82025 1.77266 1.72667 1.68221 1.63923

90339 81419 59295 73161 03178

0.40 0. 41 0. 42 0.43 0.44

1.90 1.91 1.92 1.93 1.94

-0.45831 -0.45242 -0.44665 -0.44098 -0.43542

28188 81007 34549 62055 37563

0.59406 0.58292 0.57205 0.56144 0.55108

66772 29238 08299 23020 95304

0.90 0.91 0.92 0.93 0.94

1.45 1.46 1.47 1.48 1.49

-0.90383 -0.88806 -0.87268 -0.85768 -0.84306

74031 30426 43070 84281 31376

1.59765 1.55743 1.51852 1.48085 1.44439

58792 77157 21649 80478 65370

0.45 0. 46 0.47 0.48 0.49

1.95 1.96 1.97 1.98 1.99

-0.42996 -0.42460 -0.41934 -0.41417 -0.40909

35876 32537 03805 26631 78630

0.54098 0.53112 0.52149 0.51208 0.50290

49774 13668 16733 91127 71324

0.95 0.96 0.97 0.98 0.99

1.50

-0.82879

66442

1.40909 10340

0.50

2.00

-0.40411

38063

0.49393 94023

1.00

Lf- In 2/! dy*

cln W

y!

Y

$ln

g!

5

In 2/!

Y

(-74)3 (-65)4 rc-4)11 L 6 J [ 1 [ 1 Compiled from H. T. Davis, Tables of the higher mathematical functions, 2 ~01s. (Principia Press, Bloomington, Ind., 1933, 1935) (with permission). *‘See page II.

*

272

GAMMA

‘l-al ,Ic 6.3

GAMMA AND DICAMMA r(u)

:54 :76 18 19 20 21 22 23 24 25 26

El 29 30

42 43

Fl’NC’lM)NS

RELATED

FOR

FUNCTIONS

lh’l’IS(;l3R

r(u+i)

l/r00

/III)

II.\I,F-I;\‘I’FXER ,I’1 (1,)

+Oc)

Vi\I,c’E:s ,1’.;01)

0)1.00000 000

-1)8.86226 93 -0.57721 56649

1.08443 755 0.57721 566

i 0i 2.00000 1.00000 00000 6.00000 ( 1)2.40000 00000

I- 1 0)11.00000 5.00000 000 1.66666 667 (- 2)4.16666 667

76684 I 0 10)1.32934 13.32335 1.16317 10 04 +0.42278 28 1.25611 43351 0.92278 ( 1)5.23427 78 1.50611 76684

1.02806 452 1.02100 1.04220 830 0.27036 712 0.17582 795 0.13017 669 285 1.01678 399 0.10332 024

(- 3)8.33333 333

( 2)2.87885 28 3)1.87125 43 4)1.40344 07

(- 6)2.75573 192

I

1.70611 76684 1.87278 43351 2.01564 14780 2.14064 14780 2.25175 25891

1.01397 285 1.01196 776 1.01046 565 1.00929 843 1.00836 536

0.08564 180 0.07312 581 0.06380 006 0.05658 310 0.05083 250

(- 7)2.75573 192

(-11)1.14707 456

( 7)1.18994 23 I 8)1.36843 37 9 1.71054 21 10 2.30923 18 11)3.34838 61

2.35175 25891 2.44266 16800 2.52599 50133 2.60291 80902 2.67434 66617

1.00760 243 1.00696 700 1.00642 958 1.00596 911 1.00557 019

0.04614 268 0.04224 497 0.03895 434 0.03613 924 0.03370 354

I -13 -14 17.64716 4.77947 733 373

85 (12)5.18999 13 8.56349 74

2.80351 2.74101 33283

539 1.00491 124 1.00522 343 0.03157 0.02970 002

-15)2.81145 725

I 15I 1.49861 21

68577 2.91789 2.86233 24133 2.97052 39922

1.00463 988 1.00439 519 0.02654 0.02803 490 657 1.00417 501 0.02520 828

(-19)4.11031 762

(19)1.10827 98

3.02052 39922 3.06814 30399 3.11359 75853 3.15707 58462 3.19874 25129

1.00397 584 1.00379 480 1.00362 953 1.00347 806 1.00333 872

0.02399 845 0.02289 941 0.02189 663 0.02097 798 0.02013 331

(-26)6.44695 029 (-27)2.47959 626 (-29 9.18368 986 i-30 3.27988 924 1-31I 1.13099 629

(25)7.87126 49 2712.08588 52 28 5.73618 43

3.23874 25129 3.27720 40513 3.31424 10884 3.34995 53741 3.38443 81327

1.00321 011 1.00309 105 1.00298 050 1.00287 758 1.00278 154

0.01935 403 0.01863 281 0.01796 342 0.01734 046 0.01675 925

(3212.65252 85981 86542 83693 76188 79904

-33)3.76998 763 -34)1.21612 504

(33)1.47092 26 (39)1.74039 42

3.41777 14660 3.45002 95305 3.48127 95305 3.51158 25608 3.54099 43255

1.00269 170 1.00260 748 1.00252 837 1.00245 392 1.00238 372

0.01621 574 0.01570 637 0.01522 803 0.01477 796 0.01435 374

47966 32679 53091 61747 82081

(-4:)9.67759 296

(-47)4.90246 976

40)6.17839 94 42)2.25511 58 43)8.45668 42 45)3.25582 34 47)1.28605 02

3.56956 57541 3.59734 35319 3.62437 05589 3.65068 63484 3.67632 73740

1.00231 744 1.00225 474 1.00219 534 1.00213 899 1.00208 546

0.01395 318 0.01357 438 0.01321 560 0.01287 530 0.01255 208

(47)8.15915 28325 i 49 52i 6.04152 51 3.34525 26613 1.40500 63063 61178

(-48)1.22561 744 I(-53)1.65521 -52 12.98931 -50 7.11740 087 083 673

48 5.20850 35 67 81 5112.16152 50 53 3.99612 90 9.18649

3.70132 73740 3.72571 29557 3.77278 3.74952 71417 76179

1.00203 455 0.01224 469 297 200 1.00193 570 1.00189 1.00198 983 606 0.01167 0.01195 668 0.01140

(54)2.65827 15748

(+5)3.76184 288

55)1.77827 64

3.79551 02284

1.00185 354 0.01115 226

I -60 -63 -58I13.86662 -57 -62 1.64397 471 1.81731 8.35965 8.05547 607 540 851 084

18 44 82 29 (63 58I 4.29046 61 60 56 1.78713 74 3.76238 8.66760 8.09115

96734 15102 15811 24506 3.83947 81768 3.81773 3.90198 3.88158 3.86074

759 803 210 321 0.01003 333 283 602 895 1.00166 1.00170 1.00181 460 1.00173 1.00177 0.01023 0.01090 879 0.01045 0.01067

(-65)3.28794 942

(65)2.16668 38

3.92198 96734

1.00163 530 0.00983 596

3.99168 00000

I1312)1.30767 2.09227 89888 43680 37057 I 14 15i 3.55687 6.40237 42810 (17)1.21645 10041 5.1OYO942172

(25)1.55112 10043 3.04888 34461

46 47 48 ;; 51

AND

( 0)1.00000 00000

11 :3

FUNCTION

(64)3.04140 93202

(11-l)! p,!= (2,)1,,4,4

l/(,1-1)!

I

I

()I -a)!

r(tr)=(2r)~,r"-~,,-)~f, (.)I)

2

Zn(tc-l)!

*

(2#=2.50662 82746 31001 +(u)=ln ~/-f:~(~~) $0)) compiled from I-I. T. Davis, Tables of the higher mathematical functions, 2 ~01s. (Principia Press, Bloomington, Ind., 1933, 1935) (with permission).

fl(II)

GAMMA FUNCTION

AND RELATED FUNCTIONS

GAMMA AND DICAMMA IWiXc'I'IOIvS FOR IKTE(;ER AND IIALF-Ih.rE(;ER VALl!& tl r(tt+3) l/rot) rot) JOI) ./I 00

73)1.26964

3.96082 3.94159 33.92198 97969 96734 75166 62103 82858

1.00160 530 100154 1.00157 1.00163 383 438 355 0.00983 0 00928 596 0.00946 0.00964 784 363 620

(1 72)4.33193 547

63

3:99821 47288

1:OOlSl 628

( 73)9.47993 5.35616 3.07979 1.80167 ( 81)1.07199

44 29 37 93 92

4.01639 4.03425 4.05179 4.06903 4.08598

65470 1.00148 36899 1.00146 75495 1.00143 89288 1.00141 80814 1.00138

919 0.00895 514 304 0.00879 758 780 0.00864 546 341 0.00849 852 984 0.00835 648

( 82)6.48559 ( 84)3.98864 2.49290 1.58299 ( 90)1.02102

51 10 06 19 98

4.10265 4.11904 4.13517 4.15105 4.16667

47481 81907 72229 02388 52388

1.00136 1.00134 1.00132 1.00130 1.00128

704 0.00821 912 498 0.00808 619 362 0.00795 750 292 0.00783 284 286 0.00771 203

( 91)6.68774 4.44735 3.00196 2.05634 1.42915

50 04 15 36 88

4.18205 4.19721 4.21213 4.22684 4.24133

98542 13693 67425 26248 53785

1.00126 1.00124 1.00122 1.00120 1.00119

341 0.00759 489 455 0.00748 125 623 0.00737 096 845 0.00726 388 118 0.00715 986

1.00755 70 4.25562 10927 7.20403 24 4.26970 55998 35 4.28359 31188 44887 3.83884 87 5.22292 4.29729 930 (108)2.85994 23 4.31080 66323

1.00117 1.00115 1.00114 1.00112 1.00111

439 0.00705 878 807 0.00696 052 675 0.00686 495 220 0.00677 197 172 0.00668 148

71 (100 1.19785 71670 (-101 8.34824 074 72 i 101I 8.50478 58857 i -102 I 1.17580 856 :i 105 4.47011 103 6.12344 54615 58377 -104 2.23707 -106 1.63306 744 868 54415

:; 78 2

40811

109)2.48091 116)8.94618

21308

(-108)3.02307

-117)1.11779

526

(110)2.15925 1.65183 12 64 4.32413 4.33729 99657 78604 1.28016 92 4.35028 48734 7.98921 28 1.00493 57 4.37576 4.36310 36140 53862 6.43131 87

81 E 84 85

126)3.31424

01346

128)2.81710

41144

0.80618 554

79 4.40060 4.41280 92931 44150 4.32425 47 5.24152 3.61075 53 4.42485 26078 127)3.05108 83 4.43675 73697

1.00100 677 1.00101 452 1.00099 255 1.00098 087

0.00603 995 0.00610 619 0.00596 419 0.00589 389

72 36 60 22 98

(157)9.33262 (II-l)

15444 (-158)1.07151 !

,I != (2T)fd’+~~-~~fl (ttj *see

p*ge II.

029

1.00108 1.00109 283 709 0.00659 0.00650 337 756 1.00106 894 0.00642 395 1.00105 220 540 0.00634 247 1.00104 0.00626 302 1.00102 933

1.56390 85 4.49424 38268

90

0:OOSll 846

4.38826 36140

'129)2.60868 05 4.44852 20756 2.25650 86 4.46014 99825 50 4.48300 4.47164 42354 78718 1.74738 38 1.97444

it

101

.f:;(l,)

I - 70)2.33924 65)3.28794 68 67 1.23979 942 6.44695 515 993 964 ( 65)2.16668 38 16 12 21 03354

75 (107)3.30788

273 'I‘;,],!,. (,.:s

(158)9.36756

l/(+1)! r (t,) = (2r) f,,J~--,>-,y, (1,)

4.50535 4.51634 4.52721 4.53796 4.54860

49379 39489 35142 62023 45002

79 4.61016 18527

(1/-i)!

*-&

Zn(tr-l)!

+()I) =ln ef:~(~l)

1.00096 946 0.00582 522 1.00095 831 0.00575 814 1.00094 676 1.00093 741 0.00562 0.00569 850 258 1.00092 635

0.00556 584

1.00091 1.00090 1.00089 1.00088 1.00087

0.00550 0.00544 0.00538 0.00532 0.00527

617 620 646 691 757

1.00082 542 [C-37)2]

(2+2.50662

457 463 598 858 239

0.00495 866 ['Ff'l]

82746 31001

274

GAMMA

LOGARITHMS

Table 6.4

log,,r (11) 0.00000000 0.00000000

0.30103 000 0.77815 125 1.38021 12

11 12 13 14 15

FUNCTION

log,, r (,l&)

AND

RELATED

OF THE

FUNCTIONS

GAMMA

log,, r ()I+;)

-0.04915 +0.07578 0.44375 0.96663 1.60345

851 023 702 576 79

-0.05245 +0.12363 0.52157 1.06564 1.71885

FUNCTION log,,

f,N

r (n-t+)

506 620 621 43 68

-0.04443 +0.17741 0.60338 1:167&S 1.83666

477 398 271 41 09

2.07918 2.85733 3.70243 4.60552 5.55976

12 25 05 05 30

2.33045 3.13208 3.99739 4.91820 5.88824

66 89 04 91 59

2.45921 3.27213 4.14719 5.07661 6.05433

95 28 41 30 66

2.58998 3.41389 4.29850 5.23635 6.22163

86 73 39 60 27

6.55976 7.60115 8.68033 9.79428 10.94040

30 57 70 03 8

6.90248 7.95684 9.04792 10.17286 11.32921

63 40 45 3 0

7.07552 8.13622 9.23313 10.36346 11.52483

59 37 38 8 6

7.24966 8.31660 9.41927 lOI 11.72126

15 83 06 3 5

'

1.00000 0.96027 0.94661 0.93972 0.93558

000 923 646 921 323

0.93281 0.93083 0.92934 0.92819 0.92726

466 524 980 400 910

0.92651 0.92588 0.92534 0.92488 Oi92444

221 137 753 990 327

12.116500 13.320620 14.551069 15.806341 17.085095

12.514847 13.727922 14.966804 16.230045 17.516352

12.715167 13.932651 15.175689 16.442861 17.732896

12.916241 14.138090 15.385245 16.656311 17.950042

0.92414 0.92383 0.92356 0.92332 0.92310

619 993 769 409 485

18.386125 19.708344 21.050767 227412494 23.792706

18.824561 20.153619 21.502573 22.870550 24.256751

19.044649 20.377088 21.729270 23.100338 24.489504

19.265313 20.601105 21.956492 23.330629 24.722740

0.92290 0.92272 0.92256 0792241 0.92227

649 615 149 055 169

25.190646 26.605619 28.036983 29.484141 30.946539

25.660444 27.080949 28.517642 29.969940 31.437301

25.896045 27.319290 28.758623 30.213468 31.683290

26.132109 27.558078 29.000035 30.457412 31.929681

0.92214 0.92202 0.92191 Oi92181 0.92171

350 481 460 198 621

32.423660 33.915022 35.420172 36.938686 38.470165

32.919221 34.415228 35.924878 37;447757 38.983473

33.167590 34.665900 36.177784 37.702829 39.240648

33.416347 34.916950 36.431055 37.958255 39.498167

0.92162 0.92154 0.92146 0.92138 0.92131

661 262 371 944 942

40.014233 41.570535 43.138737 44.718520 46.309585

40.531658 42.091963 43.664060 45.247636 46.842397

40.790876 42.353169 43.927200 45.512661 47.109258

41.050429 42.614701 44.190658 45.777995 47.376420

0.92125 0.92119 0.92113 0.92107 0.92102

329 073 146 524 182

47.911645 49.524429 51.147678 52.781147 54.424599

48.448061 50.064362 51.691044 53.327866 54.974597

48.716713 50.334761 51.963150 53.601639 55.249999

48.985659 50.605448 52.235536 53.875686 55.525670

0.92097 0.92092 0.92087 0.92083 0.92079

101 262 648 244 035

4487 49 50

56.077812 57.740570 i9:412668 61.093909 62.784105

56.631014 58.296908 59.972075 61.656322 63.349462

56.908011 58.575464 60.252157 61.937899 63.632504

57.185269 58.854276 60.532491 62.219723 631915788

0.92075 0.92071 Oi92067 0.92063 0.92060

010 156 462 919 518

51

64.483075

65.051318

65.335796

65.620510

0.92057 250

3: 33 3354

41 42 t43 45 46

log,, (+f) ! log,, (n-fj ! log,, (n-l) ! log,, (n-+) ! In r(n)=ln (n-l)!=(+) In n+l+fi(?t) In 10=2.30258 509299 log,, r(n) compiled from E. S. Pearson, Table of the logarithms of the complete r-function, arguments 2 to 1200. Tracts for Computers No. VIII (Cambridge Univ. Press, Cambridge, England, 1922) (with permission).

GAMMA

FUNCTION

LOGARITHMS

log,, r (11)

AND

RELATED

OF THE

275

FUNCTIONS

GAMMA

FUNCTION

Table 6.4

log,, r (?I++)

log,, r (n+t)

log,, r (n++)

64.483075 66.190645 67.906648 69.630924 71.363318

65.051318 66.761717 68.480496 70.207494 71.942561

65.335796 67.047603 68.767762 70.496116 72.232512

65.620510 67.333720 69.055256 70.784961 72.522683

0.92057 0.92054 0.92051 0.92048 0.92045

250 108 084 173 367

73.103681 74.851869 76.607744 78.371172 80.142024

73.685548 75.436313 77.194720 70.960637 80.733936

73.976805 75.728854 77.488522 79.255677 81.030194

74.268279 76.021606 77.782531 79.550922 81.326654

0.92042 0.92040 0.92037 0.92035 0.92032

661 051 530 095 741

81.920175 83.705505 85.497896 87.297237 89.103417

82.514493 84.302190 86.096910 87.898542 89.706978

82.811950 84.600825 86.396705 88.199479 90.009038

83.109604 84.899655 86.696691 88.500604 90.311284

0.92030 0.92028 0.92026 0.92024 0.92022

464 261 127 061 057

90.916330 92.735874 94.561949 96.394458 98.233307

91.522113 93.343845 95.172075 97.006708 98.847650

91.825280 93.648101 95.477405 97.313096 99.155080

92.128629 93.952538 95.782913 97.619659 99.462684

0.92020 0.92018 0.92016 0.92014 0.92012

115 231 401 625 900

.f&)

100.07841 101.92966 103.78700 105.65032 107.51955

100.69481 102.54810 104.40744 106.27274 108.14393

101.00327 102.85758 104.71791 106.58420 108.45636

101.31190 103.16722 \ 105.02855 106.89582 108.76895

0.92011 0.92009 0.92008 0.92006 0.92004

223 593 008 465 964

109.39461 111.27543 113.16192 115.05401 116.95164

110.02091 111.90363 113.79200 115.68594 117.58540

110.33430 112.21797 114.10727 116.00214 117.90250

110.64785 112.53246 114.42269 116.31848 118.21976

0.92003 0.92002 0.92000 0.91999 0.91998

502 078 690 338 019

118.85473 120.76321 122.67703 124.59610 126.52038

119.49029 121.40056 123.31614 125.23696 127.16296

119.80830 121.71946 123.63591 125.55760 127.48445

120.12646 122.03850 123.95583 125.87838 127.80610

0.91996 0.91995 0.91994 0.91993 0.91991

733 479 254 059 892

128.44980 130.38430 132.32382 134.26830 136.21769

129.09407 131.03025 132.97143 134.91756 136.86857

129.41642 131.35344 133.29545 135.24239 137.19421

129.73891 131.67676 133.61959 135.56735 137.51999

0.91990 0.91989 0.91988 0.91987 0.91986

752 638 550 486 446

138.17194 140.13098 142.09477 144.06325 146.03638

138.82442 140.78505 142.75041 144.72044 146.69511

139.15086 141.11228 143.07842 145.04923 147.02467

139.47743 141.43964 143.40657 145.37815 147.35435

0.91985 0.91984 0.91983 0.91982 0.91981

428 433 459 505 572

148.01410 149.99637 151.98314 153.97437 155.97000

148.67435 150.65813 152.64639 154.63909 156.63619

149.00467 150.98920 152.97820 154.97164 156.96946

149.33511 151.32039 153.31013 155.30430 157.30285

0.91980 0.91979 0.91978 0.91978 0.91977

659 764 887 028 186

158.63763

158.97163

159.30574

0.91976 361

157.97000

log,, (n-l) ! ln r(n)=ln

log,, (14) log,, (n-f) ! (n-l)!=(n-+) In n-?L+.fz(n)

!

log,, (n- 8 ! In 10=2.30258

509299

/

276 Table

GAMMA

6.5

1 o.xoi15

AUXILIARY

FUNCTION

AND

FUNCTIONS fl

FOR

RELATED

GAMMA

(x>

FUNCTIONS

AND

DIGAMMA

f2 (xl

FUNCTIONS

<X> 7”; ;;

f3 (xl

0.014 0.013 0.012 0.011

1.00125 1.00116 1.00108 1.00100 1.00091

077 735 391 050 708

0.92018 0.92010 0.92002 0.91993 0.91985

852 519 186 853 520

0.00751 0.00701 0.00651 0.00601 0.00551

875 633 408 200 008

0.010 0.009 0.008 0.007 0.006

1.00083 1.00075 1.00066 1.00058 1.00050

368 028 689 350 012

0.91977 0.91968 0.91960 0.91952 0.91943

186 853 520 187 853

0.00500 0.00450 0.00400 0.00350 0.00300

833 675 533 408 300

100 111 125 143 167

0.005 0.004 0.003 0.002 0.001

1.00041 1.00033 1.00025 1.00016 1.00008

675 339 003 668 334

0.91935 0.91927 0.91918 0.91910 0.91902

52C 187 853 520 187

0.00250 0.00200 0.00150 0.00100 0.00050

208 133 075 033 008

200 250 333 500 1000

0.000

1.00000

c(-;)l1

0.91893

853

0.00000

000

00

000

c 1 i-y

In r(x)=ln +(x)

Table n 100

6.6

=h

(x-l)!-(x-3)

In x- -z+f2(2) .

x-f3f3(x)

(2r)*=2.50662

82746 31001

<x>=nearest

integer to x.

FACTORIALS n!

91

FOR

LARGE

ARGUMENTS n!

72316 22543 07425 21544 39441 52682 6;10 1408 1.2655 200 78673 64790 50355 700 1689 2.4220 40124 75027 21799 75122 16440 63604 800 1976 7.7105 30113 35386 00414 300 400 52284 66238 95262 900 2269 6.7526 80220 96458 41584 500 36825 99111 00687 1000 i 2567 I 4.0238 72600 77093 77354 l?(n+l) Un+l> A tame of the factorial numbers and their reciprocals Compiled from Ballistic Research Laboratory, from l! to lOOO! to 20significant digits, Technical Note No. 381, Aberdeen Proving Ground, Md.(1951) (with permission). 9.3326 7.8865 3.0605 6.4034 1.2201

GAMMA GAMMA

FUNCTION

AND RELATED

FUNCTION

FOR

COMPLEX

277

FUNCTIONS Table

ARGUMENTS

6.7

z=l.O

Jln ~~~ 0:2

- 0.00819 0.00000 00000 77805 00 65 - 0.03247 62923 18

00::

- 0.12528 0.07194 93748 62509 00 21

-

0.00000 0.05732 0.11230 0.16282 0.20715

r(z)

&Tln r(z)

!I

00000 00 2940417 22226 44 0672168 58263 16

Yin r(z)

:*: 5:3 5.4

-

6.13032 6.27750 6.42487 6.57242 6.72016

41445 24635 30533 85885 21547

53 84 35 29 03

3.81589 3.97816 4.14237 4.30850 4.47650

85746 38691 74050 21885 25956

15 88 86 83 68

Pi 5:9

-

6.86806 7.01613 7.16436 7.31275 7.46128

72180 48 75979 76 7442106 12034 30 36194 29

4.64634 4.81799 4.99142 5.16659 5.34347

42978 41933 03424 19085 91013

70 05 89 37 53

-

7.60995 7.75877 7.90772 8.05680 8.20600

96929 46746 40468 35089 89631

51 55 98 04 00

5.52205 5.70228 5.88415 6.06762 6.25267

31255 61315 11702 21500 37967

15 35 39 13 05 76 12 44 02 93

5.0

o"*z

- 0.19094 0.26729 5499187 00682 14

- 0.24405 0.27274 82989 38104 91 05

0:7

- 0.35276 86908 60 - 0.44597 87835 49 - 0.54570 51286 05

- 0.29282 6351187 - 0.30422 56029 76 - 0.30707 43756 42

:*: 1:4

-

-

0.30164 0.28826 0.26733 0.23921 0.20430

1.5

- 1.23448 30515 47

::: ::g"

- 1.3593122484 1.48608 96127 65 57 - 1.61459 1.74464 53960 427617400

+

0.16293 97694 0.11546 87935 0.06219 86983 0.0034166314 0.0606128742

80 89 29 77 95

-

8.35533 65025 11 8.50478 2399125 8.65434 30931 23 8.8040151829 10 8.95379 54158 79

6.43928 6.62742 6.81707 7.00820 7.20081

16159 18579 14837 81345 01014

21"

- 2.00876 1.87607 41504 87864 31 71

2'2 214

- 2.27743 2.14258 42092 81922 96 04 - 2.41323 81411 84

0.12964 63163 0.20345 94738 0.28184 56584 0.3646140489 0.45158 81524

10 33 26 50 41

-

9.10368 06798 9.25366 79950 9.40375 45067 9.55393 74783 9.7042142849

32 15 08 21 72

7.39485 7.59032 7.78719 7.98545 8.18508

62984 36 6235184 99928 77 82004 68 20125 03

22:;

- 2.54990 2.68737 61537 68424 95 50

0.54260 44058 0.6375109190 0.73616 63516 0.83843 89130 0.94420 54730

52 46 79 96 39

- 9.85458 24074 -10.00503 94267 -10.15558 30186 -10.3062109489 -10.45692 10687

86 90 86 48 39

8.38605 8.58835 8.79196 8.99687 9.20305

30880 35709 60705 36442 97799

89 62 87 29 25

1.05335 07710 1.16576 67132 1.28135 17459 1.4000102965 1.52165 22746

69 86 32 76 73

-10.6077113103 -10.75857 96829 -10.90952 42693 -11.06054 32217 -11.21163 47589

15 95 78 92 48

9.41050 83803 9.61920 37472 9.82913 05671 10.04027 38971 10.2526191518

12 42 62 80 09

1.64619 26242 69 1.77355 09225 91 1.903651019019 2.03642 0709693 2.17179 14436 05

-11.36279 -11.51402 -11.66532 -11.81669 -11.96812

71628 87756 79970 32818 31369

04 02 81 48 01

10.46615 10.68085 10.89672 11.11373 11.33188

20903 24 88047 12 5708177 9524157 72758 53

2.30969 2.45007 2.59287 2.73802 2.88548

0.65092 31993 02 0.76078 39588 41 0.87459 04638 95 1.11186 45664 0.99177 27669 59 26

22'87 - 2.96448 2.82558 5641191 14617 89 2:9 - 3.1040154399 01 33:;

- 3.24414 3.38482 90223 42995 90 77

:*: 314

- 3.52603 3.66772 81104 43067 09 88 - 3.80988 12618 23

:::

- 3.95246 4.09546 7126189 13204 51

zl 3:9

- 4.38258 4.23884 69752 14660 71 28 - 4.52667 88647 16

21"

- 4.67109 4.81583 95934 29197 96 09

i-3' 414

- 4.96086 5.10617 37766 81606 87 63 - 5.25176 30342 30

44:; 4;i

- 5.54369 5.39760 64183 62389 84 04 - 5;69002 29483 73

i:; 5.0

03204 68 66142 39 05805 81 67844 65 0724149

80565 85299 37713 74148 56389

73 47 19 20 27

-12.1196161192 -12.27117 08338 -12.42278 59312 -12.57446 01059 -12.72619 20940

81 67 81 08 29

11.55115 11.77153 11.99300 12.21556 12.43920

62762 41183 86662 80464 06390

- 5.83657 5.98334 58655 58764 54 32

3.03519 69999 3.1871122793 3.34118 43443 3.49736 80186 3.6556199647

22 89 27 15 12

-12.87798 -13.02982 -1ji18172 -13.33367 -13.48567

06720 46547 28939 42765 77234

44 89 51 47 95

12.66389 12.88964 13.11642 13.34423 13.57307

5070128 02037 08 51346 66 91814 77 18794 55

- 6.13032 41445 53

3.81589 85746 15

-13.63773

21882 47

10.0

02 09 85 79 90

13.80291 29742 30

Linear interpolation will yield about three figures;eight-point interpolation will yield about eight figures. For z outsidethe range of the table, seeExamples S?ln r(z)=ln Ir(z) /

5-8.

.P In r (2)=arg r (2)

278

GAMMA

Table

6.7

FUNCTION

GAMMA

AND

FUNCTION

FOR

RELATED

FUNCTIONS

COMPLEX

ARGUMENTS

.r=l.l

-

gin r(z) 0.04987 24412 0.05702 02290 0.07824 35801 0.1129143470 0.16008 21257

60 38 68 17 99

-

Jln 0.00000 0.04206 0.08230 0.11905 0.15086

r(z) 00000 65443 97383 06275 79240

00 76 98 18 09

-

0.21858 0.28718 0.36464 0144978 0.54157

96764 99839 38731 83131 54093

09 43 53 87 11

-

0.1766611398 0.19566 16788 0.20740 35526 0.21167 10325 0.20843 91333

43 64 60 55 00

:G! 1:2 1.3 1.4

-

0.63908 0.74153 0.84825 0.95868 1.07235

78153 80620 85646 73364 26519

48 74 26 97 67

-

0.1978178257 0.18000 55175 0.15525 33222 0.12383 93047 0.08605 08957

67 74 12 38 00

:-z 1:7

-

1.18885 84815 1.30787 15575 1.4291103402 1.55233 58336 1.67734 40572

22 95 04 11 49

- 0.04217 34907 + 0.00751 65191 0.06275 56777 0.12329 53847 0.18890 25358

11 79 30 15 69

-

1.80395 1.93203 2.06142 2.19203 2.32375

99248 22878 99239 82866 68617

63 13 46 29 01

0.25935 0.33446 0.41402 0.49786 0758582

-

2.45649 2.59018 2.72473 2.86010 2.99622

70097 01959 65306 35591 52529

26 43 67 81 98

0.67775 04868 0.77349 56148 0.87292 80949 0.97592 26515 1.0823617859

-

3.13305 11644 3.27053 57144 3.40863 75892 3.5473192273 3.68654 63804

50 30 32 03 17

-

3.82628 77368 3.9665145962 4.10720 05882 4.24832 14278 4.38985 47017

-

4.53177 96812 4.67407 71584 4.81672 93009 4.9597195242 5.10303 23779

00~~ 0:2 0.3 0.4

::: 2:: z 214 2.5 Z 29" 3.0 ;:: ;:: 3.5 3.6 3.7 E 4.0 2: 413 4.4

44:;--t-i 4:9 5.0

514

-

Vln 5.96893 6.11415 6.25959 6.40526 6.55114

r(z) 91493 43840 93585 53566 41480

55'2 517 5.8 5.9

-

6.69722 6.84350 6.98998 7.13663 7.28347

-

7.43047 7.57764 7.72498 7.87247 8.02011

4 In r(z) 3.96198 63258 4.12446 68364 4.28888 73284 4i4552112743 4.62340 34819

60 90 80 47 04

7953189 94110 69 15495 70 77586 96 17659 19

4.79343 4.96525 5.13885 5.31419 5.49124

00232 81683 63238 39750 16322

04 67 91 77 40

76136 96383 24519 09237 01645

25 95 72 38 61

5.66997 5i85035 6.03236 6.21597 6.40116

07803 94 3832146 40835 50 56726 90 35407 92

-

8.16789 55118 88 8.31582 25159 69 8.46388 6927117 8.61208 46838 95 8.760411902172

6.58790 6.77617 6.96594 7.15720 7.34992

33956 16773 55256 27497 17993

-

8.90886 9.05744 9.20613 9.35494 9.50386

Y E :-;

6.5

93780 23 29085 79 4032150 66085 82 64745 04

52 05 61 40 20

67 32 30 24 20

48649 00129 39357 33637 51603

60 63 92 73 25

7.54408 7173966 7.93664 8i13500 8.33473

17375 09 2215113 34464 25 61862 70 17082 71

- 9.65289 63148 - 9.80203 39359 - 9.95127 52455 -10.100617572694 -10.25005 83482

29 83 81 21

8.53580 8.73819 8.94190 9.14690 9.35317

17842 86648 50606 41251 94376

76 33 84 84 01

50997 54469 70966 78390 55435

80 17 06 24 72

9.56071 9.76949 9.97950 10.19072 10.40315

49872 51583 47158 87913 28704

49 85 43 49 84

09 91 66 07 08

7.5

1.19213 1.30513 1.42127 1.54045 1.66258

51297 05 8858177 51595 43 17547 76 1463194

8.0

2:

-10.39959 -10.54922 -10.69894 -10.84875 -10.99865

25 20 64 81 40

1.78758 1.91537 2.04588 2.17904 2.31478

18092 46664 59340 52440 56943

68 26 24 32 26

8.5 8.6 8.7 8.8 8.9

-11.14863 -11.29870 -11.44885 -11.59907 -11.74937

8155138 36905 72 02353 71 59405 42 90196 53

10.61676 10.83154 11.04748 11.26457 11.48278

27802 46772 50362 06394 85664

52 22 14 86 18

84 70 83 44 21

2.45304 2.59375 2.73687 2.88232 3.03007

36058 83010 19016 91437 72080

25 13 54 48 09

E! 9:2

-11.89975 77460 43 -12.050210450183 -12.20073 5517188 -12.35133 13844 58 -12.50199 65394 43

11.70212 11.92257 12.14411 12.36673 12.59043

61836 11355 13354 49565 04241

32 62 15 33 06

5.24665 34450 28 5.3905692519 72 5.53476 7188164 5.67923 54339 89 5.82396 2896129

3.18006 3.33224 3.48657 3.64299 3.80148

55643 58288 16324 84993 37357

29 43 07 84 79

-12.65272 95175 33 -12.80352 89000 52 -12.95439 33123 60 -13.10532 14220 44 -13.25631 19372 14

12.81518 13.04099 13.26783 13.49570 13.72459

64072 18113 57709 76423 69974

43 65 12 49 44

-13.40736

13.95449 36168 27

- 5.96893 91493 52

3.96198 63258 60

E E t::

9.3 9.4 Xi Lx 9:9

10.0

36048 74

GAMMA

FUNCTION

GAMMA

FUNCTION

AND FOR

RELATED

279

FUNCTIONS

COMPLEX

ARGUMENTS

Table

6.7

1=1.2 Y In T(z)

?I

tiln r(z)

0.0

- 0.08537 40900 03

8::

- 0.09169 0.11050 75124 89067 13 86

8::

- 0.18352 0.14135 09532 07443 62 57

0.5

- 0.23614 32688 51

0":; 0.8 0.9

- 0.36884 0.29824 98509 83560 35 49 - 0.44697 73864 90 - 0.53174 22756 96

1':: :*:

- 0.71803 0.62233 46814 95313 87 44 - 0.81823 0.92237 79303 34133 78 20

1:4

- 1.0300106294

1.5 II*!

- 1.14073 5234162 - 1.25421 1.37015 22047 01536 37 39

1:8 1.9 z

55'; 5:4

-

5.8073152672 85 5.95057 66519 39 6.09410 4721191 6.23788 94064 81 6.38192 11972 10

4.10609 4.26883 4.43349 4.60005 4.76847

87 13 28 16 71

5.5 5.6 5.7 5.8 5.9

-

6.52619 11003 6.67069 06038 6.8154116425 6.96034 65682 7.10548 81209

82 24 98 97 15

4.9387143339 5.11075 23127 5.28455 29803 5.46008 61980 5.63732 28266

56 64 68 02 55

0.10119 48344 0.07868 85726 0.04983 92764 0.01483 57562 0.026111520147

90 52 14 65

E! 6:2

-

7.25082 7.39636 7.54208 7.68798 7.83406

94030 38562 52390 76072 52949

54 29 70 47 57

5.81623 5.99679 6.17897 6.36275 6.54810

46788 44733 57929 30441 14200

41 73 16 11 83

61 38 01 48 35

6.5 6.6

- 1.48829 83245 09 - 1.60844 01578 57

0.07278 23932 0.12495 51937 0.1824121090 0.24494 25273 0.31234 49712

-

7.9803128978 8.12672 52570 8.27329 74450 8.42002 47512 8.56690 26702

26 99 10 17 20

6.73499 6.92341 7.11333 7.30473 7.49759

68651 60416 62984 56416 27064

55 24 34 32 69

0.38442 0.46101 0.54192 0.62700 0.71610

80719 09100 29484 37140 23338

73 87 31 16 39

7.0

x-32 214

- 1.85397 1.73038 79144 78680 93 87 - 1.97906 72374 32 - 2.23325 2.10553 56848 013711733

-

8.71392 8.86109 9.00839 9.15583 9.30340

74 24 89 37 98

7.69188 7188759 8.08470 8728319 8.48303

67310 75313 54778 14724 69297

43 86 77 22 94

2.5 2.6

- 2.36214 55727 43 - 2.4921123232 46

El 2:9

- 2.75497 2.62307 77928 19177 95 39 - 2.88773 16568 77

0.80907 0.90579 1.00612 1.10996 1.21718

69945 69 43715 71 9056143 29987 33 49784 62

- 9.45110 - 9.59892 - 9.74686 - 9.89492 -10.04309

37743 60 47746 01 64719 23 5764138 96669 84

8.68422 8.88673 9.09055 9.29565 9.50203

37525 82 4317155 14530 96 84265 39 89238 50

Z:!

- 3.15562 3.02130 00992 57049 07 65

E 314

- 3.42636 3.29066 53170 16590 00 56 - 3.56269 77297 54

1.32769 1.44137 1.55815 1.67794 1.80064

01044 93510 91278 08829 07379

18 29 68 56 67

-10.19138 -lo:33977 -10.48828 -iO:63688 -10.78559

53082 31 9922146 08443 04 55067 01 1433166

9.70967 9.91855 10.12866 10.33998 10.55250

70361 72443 44054 37387 08134

08 36 34 77 40

3::

- 3.69962 3.83710 90860 32317 85 24

5-i 3:9

- 4.11364 3.97512 5174107 37264 61 - 4.25263 90859 57

1.92617 2.05448 2.18547 2.31908 2.45526

91533 49 0621184 33836 08 91746 67 29835 70

-10.93439 -11108329 -11.23229 -11.38138 -11.53055

62350 76070 33237 12352 92646

38 93 11 53 46

10.76620 10.98107 11.19709 11.41426 11.63257

15360 21389 91694 94790 02129

05 38 76 19 90

4.0

- 4.39208 75003 42 - 4.67225 4.53196 69393 69332 70 23 - 4.81293 84293 30 - 4.95399 36651 50

28374 96019 67976 04316 88424

55 03 01 86 26

-11.67982 54041 -11.82917 77123 -11.978614311170 -12.12813 33832 -12.27773 31694

57 44

44:: 4.3 4.4

2.59393 2.73503 2.87852 3.02434 3.17242

78 04

11.85198 12.07251 12.29413 12.51683 12.74059

88011 29482 06252 00607 97329

32 35 48 77 36

4.5 :-; 4:8 4.9

-

3.32274 3.47523 3.62985 3.78657 3.94533

25560 41545 81537 08902 04167

43 72 79 31 32

-12.42741 -12i57716 -12.72700 -12187690 -13.02688

19659 29 81225 64 0040142 61685 35 50046 68

12.96542 13.19130 13.41821 13.64615 13.87511

83615 49005 85311 86543 48849

35 92 47 64 16

5.0

- 5.80731 52672 85

-13.17693

50906 38

14.10507 70446 23

86

5.09540 60548 36 5.23716 00880 20 5.37924 5.52163 1239193 58863 97 5.66433 12381 00

-

0.00000 0.02865 0.05586 0.08025 0.10066

00000 84973 39903 91592 05658

00 21 67 09 03

5.0 5.1

-

0.11610 0.12588 0.12948 0.12663 0.11720

77219 00935 68069 80564 77278

+

9 In r(z)

x In r (2)

Y

4.10609 64053 70

6.3 6.4

fx 6:9

;::

8.8 8.9

10.0

68896 32795 78818 69016 66975

64053 70 00575 53 40204 01 23089 91 02,339 50

280

GAMMA

Table

6.7

GAMMA

FUNCTION

AND

RELATED

FUNCTION

FOR

COMPLEX

FUNCTIONS ARGUMENTS

.x=1.3 Y

:1 In r(z)

E

- 0.10817 0.11383 61080 48095 85 08

0:2

- 0.13070 20636 90

0":;

- 0.19649 0.15843 1008149 12771 78

0.5 0:8 E

- 0.24420 93680 45 - 0.30082 34434 02 - 0.36553 0.43754 39002 53407 19 27

0.9

- 0.51609 74046 40

::;

- 0.69006 0.60048 45154 62005 05 12

:*: 1:4

- 0.88259 0.78427 03001 13601 02 03 - 0.98458 61322 90

1.5 :*;

- 1.08986 76158 16 - 1.30898 1.19809 86148 54162 82 04

1:s 1.9

- 1.42227 1923714 - 1.53773 44011 63

2.0 - 1.65517 68709 10 2Il - 1.77442 7143191 s-23 - 2.01776 1.89533 34239 14331 34 28

4ln r(z)

din r(z)

Y

4 In r(z)

5.0

-

5.645414138133 5.78673 23355 5.92835 35606 6.07026 64370 6.21246 02140

37 66 51 03

4.24823 90621 27 4.41126 31957 95 4.57620 66023 67 4.74303 39118 17 4.9117110050 12

0.06126 78750 55 0.06229 79103 48 0.05805 28252 04 0.04820 73993 35 0.03257 37450 94

5.5

-

6.35492 47217 66 6ii9765 03105 97 6.64062 79133 72 617838488113 55 6.92730 48028 21

5.08220 49501 77 5.25448 39434 72 5.4285172533 50 5.60427 51684 12 5.78172 89485 09

- 0.01107 52190 48 + 010162790894 04 0.04941 70710 23 0.08822 25250 96 0.13255 01649 50

6.0 6.1

-

7.07098 80742 52 7.21489 11938 62 7.35900 70872 13 7.50332 90147 58 7.64785 0551098

5.96085 07788 45 6.1416137268 52 6.32399 1701649 6.50795 94158 99 6.69349 23498 81

0.18223 70479 17 Oi2371109920 47 0.29699 65855 44 Oi3617193463 93 0.43110 85022 51

6.5

-

7.79256 55658 27 7.93746 82058 02 8.08255 28787 24 8.2278142379 13 8.37324 7168176

6.88056 67176 38 7.06915 94350 45 7.25924 80896 76 7.45081 09123 38 7.64382 67501 64

7.0 7.1 7.2 7.3 7.4

-

8.51884 67726 68 8.66460 83606 78 8.81052 74362 48 8.95659 96875 66 9.10282 09770 73

7.83827 50411 67 8.03413 57901 50 8.23138 95458 91 8.4300173795 19 8.63000 08640 04

7.5

-

9.24918 73322 19 9.39569 49368 29 9.54234 01230 14 9.68911 93636 11 9.83602 92650 88

8.83132 20546 97 9.03396 34708 43 9.23790 80780 23 9.44313 92714 58 9.64964 08601 22

-

0.00000

00000 00

0.0167199199 34 0.03225 84033 35 0.04549 95427 81 0.05544 82296 06

55': 5:s 5.9

z*: 6:4

2:4

- 2.14159 19646 87

0.50499 87656 67 0.58323 13926 09 0.66565 47394 67 0.75212 4475930 0.84250 35670 42

2.5 2.8 2.9

-

2.2667188222 04 2.39304 70725 18 2.52049 15659 37 2.64897 56799 18 2.77843 02497 03

0.93666 21049 03 1.03447 70464 53 1.13583 18965 15 1.2406163628 56 1.34872 60013 87

3-f

- 2.90879 3.04000 60402 26554 26 06

312

- 3.1720186387 60

;::

- 3.43825 3.30478 31979 64765 94 05

1.46006 18633 96 1.57453 01525 07 1.69204 18960 57 1.81251 26335 69 1.93586 21235 97

- 9.98306 65608 89 -10.13022 8105196 -10.27751 08670 60 -10.4249119248 88 -10.57242 84612 54

9.85739 70516 25 10.06639 24378 12 10.2766119810 47 10.48804 10011 24 10.70066 51627 91

;*87

- 3.57239 88099 07 - 3.70717 37325 19 - 3.84254 3.97848 9534695 76469 59

3:9

- 4.11497 07016 98

2.0620140693 37 2.19089 58627 45 2.32243 83465 44 2.45657 55932 86 2.59324 47004 59

-10.72005 -10.86779 -11.01564 -11.16359 -11.31165

77580 15 71917 09 42292 16 64236 64 14105 63

10.91447 04638 39 11.12944 32237 30 11.34557 00727 24 11.56283 79415 00 11.78123 40512 20

2:

- 4.38944 4.25196 64012 45543 12 38

t:: 4.4

- 4.52739 4.66578 32778 37904 30 84 - 4.80459 79774 65

2.73238 56006 34 2.87394 08855 80 3.01785 56433 48 3.16407 73073 22 3.31255 55163 23

-11.45980 -11.60806 -11.75641 -11.90485 -12.05339

6904159 06939 74 06415 49 46773 52 0797849

12.00074 59040 23 12.22136 12739 31 12.44306 8198138 12.66585 49686 64 12.88971 01243 51

4.5 t::

- 4.9438171850 33 - 5.22340 5.08342 39564 19323 42 94

-12.20201 7062734 -12.35073 15923 02 -12.49953 2564949 -12.64841 82148 10 -12779738 68295 12

13.11462 2443199 13.34058 09350 03 13.56757 48342 95 13.79559 35935 62 14.02462 68767 33

-12.94643 6748034

14.25466 45529 28

t:,"

- 5.36373 5.5044110199 57615 52 31

3.46324 19848 78 3.61609 03828 59 3.77105 62237 32 3.92809 67607 19 4.08717 08902 55

5.0

- 5.64541 41381 33

4.24823 90621 27

:*; 7:8 7.9

9.4 9.5

10.0

99*! 9:8 9.9

GAMMA GAMMA

Y

din r(z)

i:: E

- 0.119612914172 0.12473 21357 76 - 0.14000 0.16515 01552 595518988

0:4

- 0.19978 93616 12

22

- 0.24337 0.29530 34438 16779 09 62

8G 0:9

- 0.35492 0.42158 4616110 20669 55 - 0.49462 85345 46

1.0 1.1 :2 1:4

-

1.5

- 1.03605 77156 27

::; 1.8 1.9

- 1.13933 1.24542 54742 63479 88 49 - 1.35407 41615 64 - 1.46505 2600714

3::

- 1.57816 1.69322 32702 14562 85 19

f-32 2:4

- 1.81008 1.92859 03838 23663 54 09 - 2.04863 37884 08

2.5 22:;

- 2.17009 23032 73 - 2.29286 2.41686 69570 69947 58 17

2.8 2.9

- 2.5420100734 84 - 2.66822 19640 86

33::

- 2.79543 2.92358 50784 79116 95 75

E 314

E 3:7

0.57345 1292103 0.65748 16506 41 0.74620 06322 98

FUNCTION

AND

RELATED

FUNCTION

FOR

COMPLEX

Pin r(z) 0.00000 0.00597 0.01097 0.01405 0.01439

00 43 66 03 49

5.0 5.1

- 0.01124 72025 - 0.0040177865 + 0.00775 78473 0.02441 65124 0.04618 11610

18 38 84 32 42

5.5 5.6

2: 5:4

:*zI 5:9

Table

ilRGUMENTS

Xln r(z)

Y

00000 40017 08056 93840 47989

-

281

FUNCTIONS

6.7

Xln r(z)

-

5.48319 5.62258 5.76231 5.90236 6.04272

80511 51037 08530 26637 85898

50 75 59 68 90

4.38842 4.55177 4.71703 4.88416 5.05313

59888 72808 54898 59286 51119

87 10 14 80 86

-

6.18339 6.32435 6.46560 6.60711 6.74889

73257 81614 09417 60288 42683

62 11 01 99 24

5.2239106968 5.3964614275 5.57075 70829 5.74676 84279 5.92446 71670

84 35 41 33 92

0.07317 0.10545 0.14300 0.18575 0.23362

82199 58409 11986 57618 80933

73 92 37 52 40

6.0

-

6.89092 7.03320 7.17572 7.31847 7.46144

69567 58135 29534 08625 23750

80 18 78 98 25

6.10382 59013 94 6.2848180874 01 6.46741 79988 09 6.65160 0690196 6.83734 19628 28

0.28650 0.34425 0.40674 0.47383 0.54537

41540 53337 45404 07041 20299

26 92 87 21 26

22 --i-87 6:9 -

7.60463 7.74802 7.89163 8.03543 8.17942

06520 91624 16647 21899 50262

25 64 23 02 34

7.0246183323 7.21340 69984 7.40368 58155 7.59543 32663 7.78862 84351

73 03 67 20 12

0.62122 0.70126 0.78534 0.87333 0.96512

82885 81 23803 49 13608 50 70735 61 6499100

7.0 7.1 7.2 7.3 7.4

-

8.32360 47045 8.46796 59849 8.61250 38438 8.7572134627 8.90209 02169

82 44 82 90 54

7.98325 8.17928 8.37669 8.57548 8.77562

09839 11291 96196 77156 71692

40 83 29 28 98

1.06059 1.15962 1.26210 li36794 1.47704

19035 08468 60952 54704 16591

92 95 18 02 47

7.5 7.6

-

9.04712 9.19232 9.33767 9.48318 9.62883

96653 75409 97419 23233 14889

17 21 53 58 78

8.97710 9.17988 9.38397 9.58934 9.79598

02057 95050 81856 97875 82575

23 80 34 68 76

19987 82510 63729 62885 16678

43 60 35 89 10

8.0

- 3.05262 3.18249 43245 29542 92 71 - 3.31314 67001 61

1.58930 1.70463 1.82296 1.94420 2.06828

- 9.77462 - 9.92055 -10.06662 -10.21282 -10.35915

35841 50889 26112 28810 27447

76 05 05 76 20

10.00387 10.21300 10.42335 10.63490 10.84764

7934191 35337 97 01372 94 31773 72 84263 58

-

2.1951197123 13 2.32465 09517 70 2.45680 90502 77 2.59153 06235 98 2.72875 5067188

-10.50560 -10.65218 -10.79888 -10.94570 -11.09264

9159110 91868 81 99915 05 88327 39 30623 27

11.06157 11.27666 11.49290 11.71027 11.92878

19846 02694 00045 82098 21916

19 74 92 57 70

-11.23969 -11.38684 -11.53411 -11.68148 -11.82895

01199 75293 28946 38972 82920

39 27 97 65 01

12.14839 95336 12.3691180877 12.59092 59658 12.81381 15312 13.03776 33912

59 89 40 39 29

39045 86282 04214 73048 73587

38 47 52 02 55

13.26277 13.48882 13.71590 13.94401 14.17313

53 45 03 46 16

87212 03

0.93588 0.83914 32199 04638 04 21

3.44454 22757 38 3.70940 25331 3.57663 98160 21 00 3.84279 64130 02 3.97678 99482 49

44'10 - 4.24646 4.11135 39012 10946 69 79

2.86842 3.01048 3.15487 3.30155 3.45047

43947 56 30870 18 7950177 79836 24 42563 18 97913 94580 98712 92973 75662

2:

3-i 7:9

2: ::i

9.0 9.1

4: 2

- 4.38208 62246 51

44::

- 4.65479 4.51820 74683 56949 47 75

22 4:7 29"

- 4.9293167880 4.79184 09340 18 70 - 5.06720 69267 30 - 5.20549 5.34416 30732 43497 23 30

3.60157 3.75482 3.91017 4.06758 4.22701

33 13 52 81 27

~~~ 9:7 Z

-11.97653 -12.12420 -12.27198 -12.41984 -12.56780

5.0

- 5.48319 8051150

4.38842 59888 87

10.0

-12.71585

;*: 9:4

03893 15982 63127 40430 45087

14.40325 7632142

GAMMA

282 Table

3 111r(z)

Y 0.0 0.1 3

0:4 0":: x*e7 0:9 1':: :*; 1:4 1.5 :*; 1:8 1.9 2.0 2.1 2.2 22:: E 217 ::t ;:: 3-c 314 3.5 ;:;

GAMMA

6.7

FUNCTION FUNCTION

AND FOR

BELATED

FUNCTIONS

COMPLEX

ARGUMENTS

.f In r(z)

9 In r(z)

9 In r(z)

-

0.12078 22376 35 0.12545 0392811 0.13938 53175 79 0.16238 37050 76 0.19412 35254 45

0.00000 00000 00 0.00378 68415 10 0.00839 39012 17 0.01460 80536 11 0.02315 34211 15

-

5.32063 00229 09 5.45809 92990 12 5.59594 21987 69 5.73414 48816 77 5.87269 42552 05

4.52667 02683 19 4.69038 4659451 4.85599 23475 89 5.02345 93914 30 5.19275 29984 42

-

0.23418 6347470 0.28208 36136 63 0.33728 34790 33 0.39923 5430120 0.46739 0870408

0.03466 89612 75 0.04969 46638 36 0.06866 64150 66 0.09191 83319 43 0.11969 06415 60

-

6.01157 79223 61 6.15078 41337 33 6.2903017435 55 6.43012 0169396 6.57022 9355139

5.36384 14702 24 5.53669 41510 65 5.71128 1379495 5.88757 44426 18 6.06554 55330 63

-

0.5412188685 47 0.6202170896 71 0.7039184698 97 0.79189 44573 28 0.88375 56946 74

0.15214 09934 52 0.18935 7309101 0.23137 07067 73 0.27816 75270 32 0.32969 99180 52

-

6.7106197369 14 6.85128 22117 36 6.99220 81085 67 7.13338 91616 09 7.27481 7485607

6.24516 77083 65 6.42641 48526 40 6.6092616403 83 6.79368 35022 65 6.97965 65928 01

-

0.97915 0939181 1.07776 48736 47 1.179315306181 1.28355 01134 19 1.39024 41643 92

0.38589 47712 67 0.44666 1020149 0.51189 54441 75 0.58148 71805 09 0.65532 1161093

-

7.41648 55529 97 7.55838 61727 29 7.7005124706 26 7.84285 7871149 7.9854160804 40

7.16715 77597 60 7.35616 45152 22 7.54665 50081 65 7.73860 79984 87 7.93200 28323 86

-

1.49919 6372585 1.61022 69592 23 1.72317 49667 28 1.83789 60327 96 1.95426 04180 71

0.73328 0681691 0.81524 92850 60 0.90111 2111692 0.99075 68430 94 1.08407 4337092

7.2 7.3 7.4

-

8.12818 10705 51 8.27114 70647 52 8.41430 85238 40 8.55766 01333 52 8.70119 67916 34

8.1268194190 02 8.32303 82082 45 8.52064 01697 48 8.71960 67728 67 8.9199199676 60

-

2.0721512706 83 2.19146 3106138 2.31210 04795 77 2.43397 68277 27 2.55701 34593 17

1.18095 90329 08 1.28130 91860 05 1.38502 69784 97 1.4920185397 98 1.60219 39035 70

7.5 7.6 7.7 7.8 7.9

-

8.84491 3598681 8.98880 5845698 9.13286 90053 22 9.27709 87224 65 9.42149 08057 13

9.12156 21668 12 9.3245162284 17 9.52876 54395 97 9.73429 35008 92 9.94108 45113 82

-

2.68113 86746 74 2.80628 69972 89 2.93239 85022 62 3.05941 82284 63 3.18729 56630 57

1.71546 69204 67 1.83175 5141118 1.95097 96800 61 2.07306 50684 28 2.19793 9101106

8.0

- 9.56604 12192 67 - 9.71074 60753 60 - 9.85560 1627136 -10.00060 42619 46 -10.14575 04950 41

10.14912 29545 01 10.35839 36845 06 10.56888 19135 53 10.78057 31993 69 10.99345 34334 60

-

3.31598 42885 64 3.44544 11840 65 3.57562 66733 10 3.70650 40135 44 3.83803 91197 27

2.32553 26824 38 2.45577 96733 92 2.588616742182 2.72398 32197 35 2.86182 0960836

8.5

-10.29103 69636 22 -10.43646 04212 40 -10.5820177325 09 -10.72770 5868109 -10.87352 19000 77

11.20750 88298 51 llii2272 59143 12 11.63909 15140 53 11.85659 27478 60 12.0752170166 56

-

3.97020 03195 93 4.10295 81356 26 4.23628 50905 75 4.37015 55336 09 4.50454 54845 89

3.00207 42115 08 3.14468 94828 47 3.2896154314 23 3.43680 2746151 3.58620 40415 07

9.0 9.1

-11.01946 -11.16552 -11.31170 -11.45800 -11.60442

29973 44 64215 28 95229 33 97367 84 45796 38

12.29495 19944 46 12.51578 56196 58 12.73770 60868 20 12.96070 18385 99 13.18476 1558147

-

4.63943 24943 00 4.77479 55187 51 4.9106148059 11 5.04687 17934 63 5i18354 90163 32

3.73777 37568 62 3.89146 80616 79 4.04724 47663 05 4.20506 32380 55 4.36488 43223 09

9.5

-11.75095 -11.89758 -12.04433 -12.19118 -12.33813

16459 94 86050 76 31977 78 32337 59 65886 95

13.40987 41617 61 13.63602 87918 31 13.86321 48100 75 14.09142 1791027 14.32063 95157 82

- 5.32063 00229 09

4.52667 02683 19

10.0

-12.48519 12016 51

14.55085 79659 84

2: 24'

2: 29"

z*: 9:4

Z E

GAMMA FUNCTION GAMMA

FUNCTION

283

AND RELATED FUNCTIONS FOR COMPLEX

ARGUMENTS

Table 6.7

.r=1.6 Y

tiln I?(z)

o":!

- 0.11687 0.11259 93076 17656 97 07

8.: 0:4

- 0.12968 0.15085 38452 70233 13 14 - 0.18012 29875 82

Go*: 0:7

- 0.21715 7659172 - 0.26155 0.31289 07142 99560 69 50

0"::

- 0.43448 0.37068 83847 55339 40 80

1'::

- 0.50382 0.57825 21960 58588 58 66

:s 1:4

- 0.65736 0.74076 95833 82809 44 61 - 0.82810 01661 20

::'b

- 0.91903 1.01326 27864 10002 52 05

119 :*i

- 1.11052 43845 66 - 1.21057 1.3131811150 08228 70 50

22':

- 1.41815 1.525315986147 60399 85

2:2

- 1.63449 89215 98

24'

- 1.74555 1.85836 85219 24696 99 22

2.5

- 1.97279 09238 15

22:;

- 2.20609 2.08873 63358 51557 24 10

f:!

- 2.32478 2.44471 44606 74052 95 94

33:;

S In r(z)

%ln r(z)

xln r(z)

0.00000 00000 0.01272 17953 0.02614 08547 0.04092 98346 0.0577147266

00 11 67 69 93

-

5.15767 38696 5.29324 00046 5.4292138858 5.56558 05247 5.70232 57347

89 70 50 67 10

4.66298 4.82709 4.99309 5.16092 5.33057

63139 40 8942123 00410 26 64732 77 61938 29

0.07705 74009 0.09944 39491 0.12527 90746 0.15488 59553 0.1885104588

90 75 90 99 87

-

5.83943 5.97689 6.11470 6.25283 6.39128

60752 88014 18170 36319 33226

49 04 24 59 66

5.50200 5.67519 5.85009 6.02670 6.20497

8200133 24850 30 99922 08 25740 71 29518 79

0.22632 0.26845 0.31495 0.36583 0.42110

8363144 42738 89 11405 00 95580 78 63293 75

-

6.53004 6.66909 6.80843 6.94806 7.08795

04959 52554 81708 02492 29088

33 28 20 33 41

6.38488 46780 6.56641 21003 6.74953 03284 6.934215201179 7.12044 32570

25

0.48071 0.54459 0.61268 0.68490 0.76114

2003131 72874 22 83586 73 11588 51 48080 60

-

7.22810 79544 7.3685175545 7.50917 42208 7.65007 07879 7.79120 03956

00 64 19 17 68

7.30819 7.49743 7.68816 7.88034 8.07395

17047 83963 18010 09808 55670

52 44 64 67 43

0.84132 0.92534 1.01310 1.10450 1.19945

42695 23984 14934 44515 56127

09 61 56 88 07

-

7.93255 8.07413 8.21592 8.35792 8.50012

64719 27171 30888 17887 32493

90 08 20 32 99

8.26898 8.46541 8.66321 8.86237 9.06288

57380 21983 61583 93155 38358

27 05 45 10 78

1.29786 1.39963 1.50467 1.61290 1.72424

13618 05453 47448 84436 91120

36 39 81 93 48

-

8.64252 21222 8.7851132665 8.92789 17384 9.07085 27813 9.21399 18168

97 62 38 87 02

9.26471 9.46784 9.67227 9.87797 10.08493

23369 78710 39098 43290 33943

30 61 48 61 44

- 2.68802 2.56582 37258 00865 46 40 - 2.81126 51983 53 - 3.06063 2.93548 64586 403316959

1.8386172327 21 1.95593 62824 65 2.07613 26817 55 2.19913 5722155 2.32487 74784 17

-

9.35730 9.50078 9.64443 9.78824 9.93220

44352 63884 35813 20648 80292

92 89 39 48 58

10.29313 10.50256 10.71321 10.92505 11.13808

57475 63937 06886 43268 33302

61 51 60 31 08

3.; 3:8 3.9

-

2.45329 27106 2.5843187608 2.71789 54457 2.85396 49506 2.9924717222

82 00 96 80 46

-10.07632 77975 -10.22059 78196 -10.3650146660 -10.50957 50232 -10.65427 56879

98 20 67 55 66

11.35228 11.56764 11.78414 12.00178 12.22054

40372 30924 74364 42963 11767

42 55 58 80 06

4.0

- 3.82845 04545 47

2:

- 3.95888 4.08994 63415 07464 67 23

t::

- 4.35379 4.22158 66759 61190 90 32

3.13336 3.27658 3.42209 3.56983 3.71976

23649 55399 18672 38320 56948

89 89 73 36 92

-10.79911 -10.94408 -11.08918 -11.23442 -11.37977

35626 56506 90522 09602 86562

11 53 76 86 21

12.44040 12.66136 12.88341 13.10652 13.33070

58504 89 63509 22 09632 56 8217040 68786 75

34062 45248 81404 48000 64388

62 92 46 81 72

9.5

-11.52525 95066 -11.67086 09597 -11.81658 05418 -11.9624158543 -12.10836 45707

64 45 21 24 60

13.55593 13.78220 14.00950 14.23781 14.46714

59442 46327 23791 88286 38298

10.0

c: 3:4 3.5

3.18665 85710 48 3.31351 44463 00 3.44115 3.56955 94046 42495 31 22 3.69866 25626 62

4.5 - 4.48654 82548 65 44'7b - 4.75358 4.6198181847 51673 33 38 4:8 4.9

- 4.88782 91705 81 - 5.02253 13317 74

3.87184 4.02602 4.18226 4.34053 4.50078

5.0

- 5.15767 38696 89

4.66298 63139 40

E E

-12.25442

44338 60

37 90 11

57 06 60 23 57

14.69746 74295 03

284 Table 6.7

GAMMA

FUNCTION

GAMMA

FUNCTION

AND

RELATED

FUNCTIONS

FOR COMPLEX

ARGUMENTS

.x=1.7 J In I'(z)

x In r(z)

Y

%ln r(z)

Y

9 In r(z)

-

4.99429 5.12797 5.26209 5.39663 5.53159

42740 31077 29486 77210 21994

24 01 79 79 12

4.79738 4.96193 5.12834 5.29658 5.46661

98064 49448 25830 04404 72692

83 06 82 85 29

-

5.66694 5.80267 5.93877 6.07522 6.21202

19505 32805 31855 93070 98903

53 14 28 61 76

5.63842 5.81196 5.98722 6.16416 6.34275

28098 55 7748103 36749 88 30480 45 91548 66

91243 32455 44205 88915 05624

57 42 39 80 82

-

6.34916 6.48662 6.62438 6.76246 6.90082

37463 02160 91385 08200 60067

25 75 04 42 27

6.52298 6.70481 6.88823 7.07320 7.25970

60784 86640 24881 38287 96365

05 24 89 20 25

0.57124 0.63832 0.70935 0.78428 0.86303

72307 60866 84280 36123 23052

84 03 02 89 04

-

7.03947 7.17840 7.31759 7.45705 7.59675

58582 98 1924147 61209 77 07120 18 82876 82

7.44772 7.63723 7.82821 8.02063 8.21449

75087 56630 29137 86480 28045

22 84 39 35 37

36116 09 46369 05 8541153 87389 10 18623 15

0.94552 1.03169 1.12144 1.21470 1.31138

91079 51 46541 37 7259194 42030 73 27144 41

-

7.7367117475 7.87690 42834 8.01732 93640 8.15798 07198 8.29885 23295

34 81 69 22 23

8.40975 8.60640 8.80443 9.00381 9.20452

58520 87697 30279 05701 37958

62 25 13 63 73

2.5 2.6 2.7

- 1.8719164452 44 - 1.98459 17246 80 - 2.09876 6557199

;:t

- 2.21434 2.33125 84448 26629 82 53

1.41140 1.51467 1.62113 1.73069 1.84327

07152 73744 35114 18813 73680

26 45 76 34 71

-

8.43993 84073 80 8.58123 33910 02 8.72273 1930122 8.86442 88760 30 9.0063192716 38

9.40655 9.60988 9.81450 10.02039 10.22753

55438 90763 80646 65738 90498

14 93 38 46 84

3.0 3.1 3.2 3.3 3.4

-

2.44940 2.56872 2.68915 2.81062 2.93309

14805 34658 28670 90603 60594

61 89 03 59 79

1.95881 2:07724 2.19847 2.32246 2.44914

7107134 0553198 95064 74 81077 41 28100 87

-

9.14839 9.29066 9.43310 9.57572 9.71851

8342151 14862 98 42680 75 24089 73 17806 54

10.43592 10.64552 10.85634 11.06835 11.28154

03060 55107 01750 01418 15743

85 28 59 23 00

3.5 3.6 3.7 3.8 3.9

-

3.05650 3.18079 3.30594 3.43189 3.55860

20770 24 9134133 27115 93 14379 84 68105 24

2.57844 2171030 2.84468 2i98150 3.12073

23336 16 76079 67 17064 22 97744 80 8955142

- 9.86146 -10.00458 -10.14786 -10.29130 -10.43489

83980 84128 81072 38884 22827

47 32 85 74 58

11.49590 09457 89 11.7114150295 52 11.92807 0889158 12.14585 58692 46 12.36475 75866 47

4.0 4.1 f-G

- 3.68605 29448 47 - 3.81419 63503 82 - 4.07245 3.94300 57284 17902 59 13

414

- 4.20250 70933 22

-10.57862 -10.72251 -10.86654 -11.01070 -11.15500

99305 35816 00900 64100 95918

96 27 14 32 83

12.58476 12.80586 13.02804 13.25129 13.47560

39218 30109 32377 32259 18323

4.5 4.6

- 4.33314 58930 01 - 4.46434 40087 52

:*s7 4:9

-11.29944 67777 -11.44401 51979 -11.5887121674 -11.73353 50824 -11.87848 14J72

28 25 47 91 43

13.70095 13.92735 14.15477 14.38320 14.61264

81399 16 14505 47 1279190 73474 23 95775 51

5.0

Oil 0.2

-

0.09580 76974 0.09977 01624 0.1116135203 0.13120 82417 0.15834 67099

07 55 43 20 43

0.00000 0.02095 0.04250 0.06524 0.08970

00000 00 5310147 9978199 48506 20 54480 34

0.5 0.6 0.7 0.8 0.9

-

0.19275 0.23410 0.28203 0.33614 0.39604

43 11 30 35 33

0.11638 0.14573 0.17810 0.21382 0.25312

82473 09476 70108 42284 66649

1.0

- 0.46133 2644119

2:

- 0.53162 0.60653 06562 43029 78 30

:::

- 0.68572 0.76884 93610 05552 37 19

0.29619 0.34317 0.39413 0.44912 0.50817

1.5 1.6 1.7 1.8 1.9

-

0.8556148134 0.94573 52538 1.03895 26210 1.13503 13039 1.23375 66975

32 42 76 83 90

2.6 2.1 2.2 2.3 2.4

-

1.3?493 1.43838 1.54394 1.65147 1.76084

0.0

44989 41754 01468 32007 36829

3.2623183125 3.40619 87555 3.55233 29614 3.70067 53013 3.8511817677

99 93 33 46 02

5.0 52 5:3 5.4

6.9 :*: 7:2 7.3

ta: 9:3 9.4 9.5

- 4.72832 4.59607 87027 85697 47 79 - 4.86107 34372 26

4.00380 99034 45 4.1585187339 90 4.31526 87017 23 4.47402 1603194 4.63474 05290 18

- 4.99429 42740 24

4.79738 98064 85

10.0

99:; E

-12.02354

j?f208 09

85 28 88 97 91

81 93 08 06 86

14.84308 80868 68

GAMMA

GAMMA

FUNCTION

AND

RELATED

FUNCTION

FOR

COMPLEX

285

FUNCTIONS

ARGUMENTS

Table

6.7

.r=1.8 .#!‘I

Y

.f In r(z)

r(z)

0"::

- 0.07108 0.07476 38729 57386 86 14

t: 0:4

- 0.10400 0.08577 55297 76857 09 32 - 0.12929 22486 30

265

- 0.20006 0.16140 82029 31015 53 52

Kl 0:9

0.00000 0.02858 0.05769 0.08782 0.11946

tiln r(z)

00000 00 6333136 29209 31 58538 91 40495 57

-

4.83045 4.96226 5.09454 5.22728 5.36046

./In r(z)

6845113 53555 54 72216 70 53433 89 35143 73

4.92989 5.09490 5.26176 5.43043 5.60088

76263 86275 50781 56009 97905

84 80 04 62 12

63619 92920 84380 06145 32737

68 13 56 63 64

5.77309 81726 5.94703 21669 6.12266 40498 6.29996 69207 6.478914668158

78 16 86 68

-

6.16796 44658 6.30383 28019 6.44003 74202 6.57656 79546 6.7134145046

02 05 92 04 23

6.65948 6.84164 7.02537 7.21065 7.39746

19384 41059 72437 80966 40550

99 65 42 53 43

- 0.24498 0.29581 08149 07721 71 51 - 0.3522150054 25

0.15304 83729 0.18897 35429 0.22758 31014 0.2691673612 0.31396 39650

82 70 17 58 50

??I --- 5.49406 5.62807 5.76248 5’78 - 5.89728 5:9 - 6.03244

::i! 1.2 1.3 lI4

-

74 52 74 43 03

0.36216 0.41389 0.46927 0.52836 0.59119

05120 86472 90315 66950 63857

09 00 88 54 23

6.0 6.1

::z :*7a

- 0.8751145440 0.78884 75850 80 57 - 0.96452 1.05684 30468 831118026

1:9

- 1.15188 37223 02

0.65777 0.72809 0.80213 0.87984 0.96117

76436 94297 42229 15616 10434

6fi 11 48 08 30

-

6.85056 76090 92 6.9880182204 65 7.12575 76814 17 7.26377 77029 87 7.40207 0344198

7.58577 31298 7.77556 39290 7.9668156346 8.15950 79813 8.35362 12360

85 39 11 46 30

f-01 2:2 ;:i

- 1.34934 1.24943 97659 28469 29 99 - 1.45143 40669 35 - 1.55556 1.661611576122 80105 11

1.04606 1.13445 1.22628 1.32148 1.41997

48267 65 96865 98 8684172 25078 65 05387 49

-

7.54062 79930 7.67944 33488 7.81850 94055 7.9578194361 8.09736 69787

63 49 06 78 03

8.54913 8.74603 8.94429 9.14390 9.34484

15 54 74 64 25

2.5 2;6 2.7

- 1.76944 28703 84 - li87895 01786 38 - 1.9900310163 61

22::

- 2.10259 2.21654 43688 12619 12 95

1.52168 li62654 1.73449 1.84544 1.95935

16884 50508 04020 85788 17594

90 69 35 28 45

-

8.23714 58220 8.37714 99935 8.51737 37469 8.6578115513 8.79845 80804

35 16 39 42 75

9.54709 9.75064 9.95547 10.16156 10.36890

7034142 48917 54 27618 74 49130 30 59844 02

3.0 Z

- 2.3318106516 27 - 2.56599 2.4483166432 45147 78 13

2.07613 36663 2.19572 97074 2.31807 70690 2.4431147704 2.57078 36890

29 49 52 17 62

8.0

-

8.93930 9.08035 9.22159 9.36303 9.50464

82029 08 69727 14 96207 08 1546181 8309120

10.57748 10.78727 10.99827 11.21046 11.42383

09733 52232 44124 45427 19293

12 56 32 62 59

2.70102 65631 2.83378 79764 2.9690143304 3.10665 38058 3.24665 63186

50 90 05 79 51

8.5

31904 52376 52667 07487 94213

38 37 34 37 31

0.41384 0.48036 0.55143 0.62673 0.70596

67690 32669 15880 30272 59713

;:'4

- 2.8046197009 2.68478 15548 41 53

;*56 3:7

- 2.92545 3.04723 78253 51190 19 42 - 3.16992 13469 31

E

- 3.29346 3.41782 06949 24159 39 89

4.0 4.1

- 3.54295 85286 89 - 3.66884 07212 13

5.; 4:4

- 3.92270 3.79543 43338 85028 26 21 - 4.05063 42744 24

4.5

- 4.17918 44552 05

2:

6.4

7.8 7.9

i:: 8.3 8.4

27” 29”

- 9.64644 56228 - 9.788419347163 - 9.93056 54816 -10.07288 01596 -10.21535 96421

63 43 06 85

11.63836 11.85404 12.07086 12.28881 12.50786

61778 40794 66893 62145 53042

3.38897 3.53355 3.68036 3.82935 3.98047

34693 93 84906 21 61916 47 29025 75 6418131

-10.35800 -10.50079 -10.64375 -10.78685 -10.93010

03128 86719 13321 50132 65382

01 24 05 67 43

12.72802 12.94927 13.17160 13.39499 13.61944

92806 69 85734 79 57894 90 9654143 91215 87

44'76 - 4.30833 4.43805 34763 72703 48 06 4:8 - 4.56833 31585 96 4.9 - 4.69913 97495 61

4.13369 4.28897 4.44626 4.60554 4.76676

59419 20315 65448 25879 44644

-11.07350 28285 -11.21704 09003 -11.3607178605 -11.50453 09034 -11.64847 73069

39 12 47 33 06

13.84494 14.07147 14.29902 14.52758 14.75715

33679 17848 39730 97368 90776

5.0

4.92989 76263 84

- 4.83045 6845113

14 17 66 92 38 10.0

-11.79255

44293 69

42 17 75 21 29

14.98772 21889 61

286 Table

GAMMA GAMMA

6.7

FUNCTION FUNCTION

AND FOR

RELATED

FUNCTIONS

COMPLEX

ARGUMENTS

.r=1.9 Y

tiln I‘(z)

::"I

- 0.03898 0.04242 42759 16648 23 18

ia; 0:4

- 0.05270 0.06974 4359613 5307116 - 0.09340 38158 25

0.5 ::"7

- 0.12349 16727 26 - 0.15978 0.20201 20244 08372 82 30

::9"

- 0.24990 0.30315 35004 95035 09 34

1.0

- 0.36147 78527 10

::: :::

- 0.42455 0.49209 86372 6462111 39 - 0.5638171504 0.63943 71834 98 20

:2

- 0.71869 0.80135 54698 82795 30 42

1:7 1.8 1.9

- 0.88717 97447 03 - 0.97595 80247 42 - 1.06749 27687 53

2.0 2.1

- 1.16160 13318 68 - 1.25811 51641 83

z-z 2: 4

- 1.45774 1.35687 89195 95259 14 72 - 1.56059 52554 63

2.5

- 1.66529 48176 11

;:;

- 1.77173 1.8798173280 64947 51 00

;::

- 2.10052 1.98944 39332 23595 80 16

3.0 3.1

- 2.21298 10520 42 - 2.32673 87919 77

;:; 3.4

- 2.55788 2.44172 77675 36468 72 15 - 2.67514 6711148

3.5 3.6 3.7 ;:9"

-

2.79346 2.91277 3.03304 3.27625 3.15421

14569 62346 29224 54337 66305

4.0 4.1 4.2

-

3.39912 3.52277 3.64718 3.77231 3.89813

4.5 416 4.7 4:8 4.9

-

4.02462 4115174 4.27948 4.40780 4.53669

5.0

- 4.66612 81728 77

X In r(z)

%ln r(z)

9 In r(z) 0.00000 0.03569 0.07184 0.10889 0.14726

00000 47077 49288 51730 87453

00 36 73 33 39

-

4.66612 81728 4.79608 44074 4.92654 53878 5.05749 30552 5.1889102823

77 24 64 47 51

5.06052 5.22603 5.39337 5.56250 5.73340

77830 70297 36626 72499 82679

38 75 27 47 93

0.18735 0.22952 0.27407 0.32128 0.37139

90383 28050 56544 97690 36389

60 02 06 64 55

-

5.32078 5.45308 5.58582 5.71896 5.85249

0812105 92008 98 07663 21 15389 41 82177 50

5.90604 6.08039 6.25643 6.43412 6.61345

80662 88340 35684 60432 07797

49 38 02 49 49

0.42457 34706 0.48097 58618 0.5407113247 0.60385 82827 0.67046 72268

81 37 70 52 81

-

5.9864181289 6.12070 91879 6.25535 98637 6.39035 91465 6.52569 65169

78 56 85 66 71

6.79438 6.97689 7.16097 7.34658 7.53371

30179 86894 43917 73625 54565

35 96 16 14 59

0.74056 0.81415 0.89123 0.97177 1.05574

4797147 76239 52 58296 55 6140147 45936 43

-

6.66136 6.79734 6.93363 7.07023 7.20711

19179 75 57285 54 87392 01 2129112 74449 04

7.72233 7:9124j 8.10397 8129695 8.49134

71224 13866 78029 64920 80626

13 57 64 80 65

1.14309 1.23379 1.32776 1.42496 1.52533

88592 01934 50714 65323 52787

34 57 39 75 28

-

7.34428 7.48173 7.61944 7.75742 7.89565

65807 17598 55170 06825 03667

56 49 18 11 87

8.68713 8.88429 9.08281 9.28267 9.48385

36229 47573 35092 23655 42409

72 07 45 74 11

1.6288105662 1.73533 09179 1.84483 46926 1.95726 05315 2.07254 77068

06 80 69 67 08

-

8.d3412 8.17284 8.31180 8.45098 8.59039

79462 70499 15468 55343 33269

62 43 79 75 14

9.68634 9.89012 10.09517 10.30148 10.50903

24629 07585 32398 43916 90590

88 45 33 76 64

2.19063 2.31146 2.43498 2.56113 2.68985

63887 78475 46022 05263 09205

13 36 00 98 60

-

8.7300194457 8.86985 86090 9.00990 57226 9.15015 58714 9.29060 43111

32 10 31 69 75

10.71782 24352 lo:92782 00504 11.1390177608 11.35140 17379 11.56495 84588

78 91 39 39 29

24 38 14 96 10

2.82109 2.95480 3.09093 3.22943 3.37025

25566 37012 41220 50808 93162

19 40 91 91 16

-

9.43124 9.57207 9.71309 9.85429 9.99566

23 85 13 97 75

11.77967 11.99553 12.21253 12.43065 12.64988

46963 75096 42358 24807 01110

13 87 42 06 27

01294 40173 27007 39057 73167

42 08 49 84 71

3.51336 3.65869 3.80622 3.95589 4.10767

10185 57993 06560 39339 52859

24 21 50 63 66

-10.137213025172 -10.27892 94790 52 -10.4208115703 58 -10.56285 5789126 -10.70505 87350 54

12.87020 13.09161 13.31410 13.53765 13.76225

52464 62520 17307 05165 16677

75 42 41 78 85

44269 84023 39577 72434 57418

53 59 56 44 38

4.26152 4.41740 4.57528 4.73511 4.89687

56312 71132 30577 79308 72979

41 72 67 60 01

-10.8474171130 -10.98992 77287 -11.13258 74849 -11.27539 33771 -11.41834 24904

13.98789 14.21456 14.44226 14.67096 14.90067

44603 83815 31243 85811 48382

16 73 75 36 65

5.06052 77830 38

-11.56143

64604 78935 43338 16464 58330

08 64 48 93 66

19955 88

15.13137 21707 60

GAMMA GAMMA

FUNCTION FUNCTION

AND FOR

RELATED

287

FUNCTIONS

COMPLEX

ARGUMENTS

Table

6.7

.,-=2.0 %ln

Y 0.0

r(z)

9 In r(z)

-

0.00000 0.00322 0.01286 0.02885 0.05107

00000 26151 59357 74027 93722

00 39 41 79 62

Ei :c!i

-

0.07937 0.11354 0.19863 0.15338

37235 77183 06626 06308

30 40 81 31

0:9

-

0.24904

17059

66

::!l

-

0.36428 0.30434

96090 77010

22 76

:*: 1:4

-

0.49700 0.42859 0.56926

21701 14442 99322

42 52 58

1.5 1.6

0.64515 0.72443 0.80688

55533 19760 50339

76 33 42

:*: 119

-

0.98053 0.89231

03476 37613

69 78

2.0

-

1.07135

98302

14

00000 57120 33372 61223 05507

00 74 06 10 97

0.21958 0.26767 0.31789 0.37051 0.42574

93100 56897 96132 53392 0726144

95 80 02 47

5.5 5.6 5.7 2::

0.48375 78429 0.5447146524 0.60872 74700 0.67588 39160 0.74624 61166

30 35 17 88 63

0.81985 0.89672 0.97687 1.06028 1.14693

67 63 07 26 53

39537 82178 35612 11909 12720

6.0 6.1

66’; 6:4 6.5 6:8 6.9

50755 88796 7022252 13522 38831

42 82

-

5.14705 5.27767 5.40874 5.54024 5.67217

-

5.80450 5.93722 6.07033 6.20381 6.33765

76 33

5.18929 5.35533 5.52318 5.69281 5.86418

93415 82031 54439 16137 81052

60 27 62 11 00

75299 60518 39987 66615 00274

57 81 03 82 24

6.03728 6.21208 6.38854 6.56665 6.74637

71248 16640 54709 30238 95048

73 30 43 56 97

07366 60439 37820 23278 05713

29 25 31 98 36

6.92770 7.11059 7.29503 7.48100 7.66847

07748 33491 43738 16040 33815

95 13 76 81 76

-

6.47183 78858 6.60636 41013 6.7412194789 6.87639 46872 7.01188 07803

22 16 19 45 50

7.85742 8.04784 8.23970 8.43299 8.62768

86143 67567 77898 22035 09788

76 00 07 86 99

8.82375 9.02119 9.21999 9.42011 9.62155

55706 78914 02960 55664 68973

27 05 14 09 45

1.23679 5034104 1.32983 65907 1.4260144920 1.52528 30352 1.62759 33595

26 94 04 36

7.0

-

7.14766 7.28375 7.42012 7.55676 7.69368

9177118 16419 02668 74543 59017

82 81 62 46

1.73289 43555 1.84113 34120 1.95225 70264 2.0662112994 2.18294 23322

35 22 63 71 91

-

7.83086 7.96830 8.10599 8.24393 8.38211

85862 87511 98924 57468 02798

69 38 36 08 83

9.82429 10.02832 10.2336151072 10.44016 10.64794

78825 25016 04128 34810

87 83 54 09 35

2.30239 2.42452 2.54926 2.67657 2.80639

65434 09185 32043 20582 71597

67 18 52 60 50

-

8.5205176753 8.65915 23247 8.79800 88177 8.93708 19330 9.07636 66296

67 82 87 47 28

10.85694 11.06716 11.27857 11.49116 11.70492

97125 48351 48933 62386 55194

60 59 86 10 45

2.93868 3.07340 3.21048 3.34989 3.49158

92920 03990 3622188 33215 50837

59 47

9.21585 9.35555 9.49544 9.63552 9.77579

80388 14566 23361 6281184 90392

55 37 92

16 57

-

11

11.9i983 12.13589 12.35308 12.57138 12.79079

96725 59137 17297 48693 33364

52 86 01 62 76

57202 32567 69172 71023 53645

41 78 45 23 81

- 9.91625 -10.05689 -10.19770 -10.33869 -10.47985

64956 46678 96994 78553 55166

49 12 20 49 49

13.01129 13.23287 13.45553 13.67924 13.90401

53818 94959 44022 90499 26078

23 63 19 21 95

-10.62117 91758 -10.76266 54322 -10.904310988175 -11: 04611 26442 -11.18806 72959

12 81 29 27

14.12981 14.35664 14.58449 14.81334 15.04319

4458193 41900 15940 66565 95540

46 42 09 92

-11.33017

27

15.27404

06485

34

-

1.2602188108 1.16463 96040 76 42 1.45772 1.35795 76568 6696157 48

2.5 2.6 c87

--

1.55940 1.66288 1.76806 1.87483

61080 49866 06566 80234

2:9

-

1.98312

8963102

2.09285

17530

93

3:2 33-i! ::'4

--

2.31629 2.20393 2.54462 2.42987

05460 48844 26813 92551

64 77 37 03

;:2 2;

-

2.66046 2.77736 3.01410 2.89525

32717 83499 62029 79709

73 84 30 78

3.9

-

3.13386

46968

42

44:!

-

3.25449 3.37595

2871145 81 29213

2:

--

3.62122 3.49820

74039 88720

59 03

4:4

-

3.74497

69383

89

3.63551 3.78164 3.92992 4.08032 4.23280

t:; 4.9

-

3.86942 77912 3.99455 19873 4.12032 31366 4.2467163216 4.37370 79930

99 65 90 20 a7

4.38732 4.54384 4.70234 4.86276 5.02509

43808 79226 08252 89562 91831

43 20 48 20 32

9:9

5.0

-

4.50127

42

5.18929

93415

60

10.0

58755

- 4.50127 - 4.62939 - 4.75805 - 4.88723 - 5.01690

i-76

2: 22::

61 52 65 17

X In r(z)

9 In r(z)

0.00000 0.04234 0.08509 0.12863 0.17335

;:‘6 x

19298

288

GAMMA

Table 6.8

FUNCTION

DIGAMMA

-A(z)

AND

FUNCTION

x-1.0

.“*(z) 0.00000

-0.57121

56649

-0.56529 -0.53073 -0.47675 -0.40786

77902 04055 48934 79442

0.16342 0.32064 0.46653 0.59770

b3572 65809

0.71269 0.81160 0.89563 0.96655 1.02628

-0.32888 -0.24419 -0.15733 -0.07088 +0.01345

61258 34022 20154

0.09465 0.17219 0.24588 0.31576 0.38196 0.44469 0.50420 0.56072 0.61448 0.66570

FOR

s ::i :-:

FUNCTIONS

COMPLEX

1.61278

1.63245 1.65175

1.79408 1.81053 1.82672 1.84265 1.85833

08018 44105 28842 45939 75219

1.48746 1.48883 1.49015 1.49143 1.49267

92858 71602 80964 87422 54582

1.49387 1.49504 1.49617 1.49727 1.49833

43346 12062 10799 45959

1.49937 1.50037 1.50135 1.50230 1.50323

2.06820

71585 35171 82397 57159 01717

1.50413 1.50501 1.50586 1.50669 1.50751

is 814

2.08074 2.09313 2.10537 2.11746 2.12941

56749 61434 53524 69410 44191

1.50830 1.50907 1.50982 1.51056 1.51127

88:;

2.14122 2.15289

11731 04718

287 8:9

2.16442

54716

2.17582 2.18710

46687

1.51197 1.51266 1.51332 1.51398 1.51462

2.19825 2.20928 2.22018 2.23097 2.24165

46616 19555 92160 90229 38740

1.51524 1.51585 1.51645 1.51703 1.51760

2.25221 2.26266 2.27301 2.28325 2.29338

61882 83093 25085 09877 58823

1.51816 1.51871 1.51925 1.51978 1.52029

2.30341

92637

1.52080

5.8 5.9

2 6:8 6.9

1.81377 1.88898 1.90396 1.91872 1.93327

15154

:*i 712 7.3 7.4

1.94761 1.96175 1.97569 1.98944 2.00300

54079 42662

0.92985 0;96803 1.00485 1.04039 1.07474

78387 70243 21252 40175 51976

1.37080 1.37849 1.38561 1.39222 1.39838

1.10798 1.1401b 1.17137 1.20164 1.23104

07107 89703 24783 84581 94107

1.40413 1.40951 1.41455 1.41928 1.42374

1.25962 1.28741 1.31446 1.34081 1.36649

36033 54995 61381 34679 26435

1.42794 1.43191 1.43566 1.43922 1.44259

1.39153 1.41597 1.43983 1.46314 1.48593

62879 41255 61892 70060 17620

1.44580 1.44885 1.45175 1.45452 1.45716

34505

1.50821 1.53001 1.55135 1.57224 1.59272

24197 88550 07370

1.45969 1.46210 1.46441 1.46663 1.46876

1.61278

48446

1.47080

36052

1.47464 1.47646 1.47820 1.47989 1.48151 1.48308 1.48459 1.48605

6.5

0.80607

1.47276

25175 90807 25988 32133

1.23772 1.25843 1.27675 1.29306 1.30766

0.84899 0.89021

1.47080

1.70751 1.72543 1.74303 1.76034 1.77735

:*z 517

79402 34618 00645 06554 39172

1.32081 1.33271 1.34353 1.35341 1.36246

20861

1.68926

28134

65515

48446 b9889


42228 67162

5:4

1.67068

b.0 6.1

20906

ARGUMENTS

.lti(z)

1.07667 1.11938 1.15580 1.187@7 1.21413

03206 05426

14328 84807

0.71459 0.76132

RELATED

t-G 6:4

::2 x 719 X:!

x 9:2 ;:i 8:: 99.87 9:9 10.0

2.01638 2.02959 2.04262 2.05549

21681

16663

92217

+(z) to 5D, computed by M. Goldstein, Los Alainos Scientific Laboratory. AUXILIARY

0.09 0.08 0.07 0.06

.54(Y) 0.00100 0.00083 0.00067 0.00053 0.00040 0.00030

FUNCTION

FOR :&(l+iy) J;(Y)

956 417 555 368 853 011

9 1110 ::

0.03 0.02 0.01

=nearest integer to y.

0.00020 0.00013 0.00007 0.00003 0.00000 0.00000

CY> 839 335 501 333 833 000

:so 53: 100

GAMMA FUNCTION DIGAMMA

FUNCTION

AND RELATED FUNCTIONS FOR C( IMPLEX

x=1.1 m(z) .fGZ) 1.61498 1.45097 1.63457 1.45332 1.65378 1.45557 ;.p; . 1.45774 1.45983

0.63764 ::2 0.73229 :*I: 0.81484 0.88630 0.94792 5:9 1.0 0.14255 1.00102 0.21327 1.04687 2: 2-7 0.28131 1.08660 6:2 1.12119 ::i F%% 1.15146 . 0.46829 1.17810 0.52507 1.20169 22 0.57930 1.22269 0.63111 1;24148 2 1:9 0.68067 1.25839 6:9 0.72813 1.27368 7.0 0.77363 1.28755 7.1 0.81730 1.30021 7.2 0.85928 1.31179 7.3 214 0.89967 1.32243 7.4

1.70933 1.46184 1.72718 ;.;z;;; * 1.46565 1.46378

1.89025 1.47857 1.87508 1.47996 1.91992 1.90519 1.48132 1.93443 1.48263 1.48391 ;-;z;;; 1.48515 1.48635 1197675 1.48752 1.99047 1.48866 2.00401 1.48977

::i

2.5 ::76

1.33224 1.34131 7.5 ::; 1.34972 1.35753 1.36482 ::: 1.37162 1.37800 i-1" 1.38398 812 1.38960 1.39489 88::

1.49085 2.01736 2.04356 1.49190 2.03054 1.49292

1.39989 1.40461 8.5 1.40907 1.41331 8:8 it; 1.41732 8.9 1.42114 1.42478 Z:! 1.42824 1.43154 z-3 1.43469 9:4 1.43771 1.44059 ;:z 1.44335 Z-I: 1.44600 1.44854 9:9

1.50024 2.14198 1.50106 ;:2 2.15363 2.17654 1.50186 2.16515 1.50265 ::; 2.18780 1.50341 3.9

::2 2

Z:! $2 ::98 3.0 ::: ;:: ::: :::, 3.9 i-t 4:2 ::: 4.5

0.93858 0.97610 1.01234 1.04736 1.08124 1.11405 1.14586 1.17671 1.20667 1.23578 1.26409 1.29164 1.31847 1.34461 1.37010 1.39496 1.41924 1.44294 1.46611 1.48876

1.51092 1.53261 t-t 1.55384 418 1.57463 4.9 1.59501 5.0

1.61498 1.45097 10.0 [c-y]

1.77893 1.46921 1.46746 1.79561 1.47090 1.81201 1.47253 1.82815 1.47411 1.84404 1.47565 1.85968 1.47713

ARGUMENTS

Table 6.8

x=1.2 %qz) .W(z) .eJ(z) !I .feC(Z) ?/ 0.0 -0.28964 o.ooodo 5.0 1.61756 1.43125 0.1 -0.28169 0.12620 5.1 1.63705 1.43396 1.65617 1.43658 0.3 -0.26014 0.2 -0.22578 0.24926 0.36640 5.2 5.3 1.67494 1.43910 0.4 -0.18064 0.47552 5.4 1.69336 1.44152

H -0 42ji5 0.00000 5.0 -0.41451 0.14258 5.1 -0.38753 0.28082 -0.34490 0.41099 :-: -0.28961 0.53042 5:s -0.22498 -0.15426 -0.08023 -0.00509 +0.06954

289

0.5 -0.12710 0.57530 -0.06753 0.66517 ;; -0.00412 0.74519 +0.06130 0.81589 0.12730 0.87806 4;.

5.5 5.6 5.7 5.8 5.9

1.71146 1.72924 1.74672 1.76390 1.78079

1.44386 1.44612 1.44829 1.45039 1.45243

0.19280 0.93260 6.0

1.79740 1.81375 1.82983 1.84567 1.86126

1.45439 1.45629 1.45813 1.45991 1.46164 1.46331 1.46493 1.46651 1.46803 1.46952

11.1" . 0.31960 0.25707 0.98046 1.02252 6.1 6.2 ::: 0.38012 1.05960 6.3 1.4 0.43846 1.09240 6.4 0.49459 0.54851 0.60028 0.64999 0.69774

1.12153 1.14752 1.17082 1.19179 1.21074

6.5 6.6 6.7 6.8 6.9

1.87661 1.89173 1.90663 1.92132 1.93579

0.74362 0.78775 212 0.83022 5:: 0.87114 0.91060

1.22794 1.24362 1.25796 1.27112 1.28323

7.0 7.1 7.2 7.3 7.4

1.95006 1.47096 1.96413 1.47236 %E 1.47372 1.47505

0.94868 0.98546 1.02103 1.05546 219 1.08881 z-i 1.12113 ::1" 1.15250 3.2 1.18295 1.21254 ::4' 1.24132

1.29442 1.30478 1.31441 1.32337 1.33173

7.5 7.b 7.7 7.8 7.9

2.01852 2.03167 2.04465 2.05746 2.07012

1.47760 1.47882 1.48001 1.48117 1.48230

1.33955 1.34688 1.35377 1.36024 1.36635

8.0 8.1 8.2 8.3 8.4

2.08262 2.09496 2.10716 2.11921 2.13111

1.48341 1.48448 1.48553 1.48656 1.48756

1.26932 1.29659 1.32315 1.34905 1.37432

1.37211 1.37756 1.38272 1.38761 1.39226

8.5 8.6 8.7 8.8 8.9

2.14288 2.15451 2.16601 2.17738 2.18862

1.48853 1.48949 ii44042 1.49133 1.49222

2.22084 1.50631 2.23161 1.50561 2'3 2.24228 1.50699 4.4

1.39898 1.42306 1.44659 1.46959 1.49209

1.39667 1.40088 1.40489 1.40871 1.41236

9.0 9.1 9.2 9.3 9.4

2.19973 2.21073 2.22160 2.23236 2.24301

1.49310 1.49395 1.49478 1.49560 1.49640

2.25283 1.50832 2.26326 1.50766 22 2.28382 yw& 2.27360 2;

1.53565 1.41586 1.51410 1.41920 9.5 9.6 1.57743 1.42240 1.55676 1.42547 9.8 9.7

2.26397 2.25354 1.49718 1.49794 2.28450 1.49943 2.27429 1.49869

2.29395 1:51021 4.9

1.59769 1.42842 9.9

2.29461 1.50015

2.05640 1.49489 2.06908 1.49392 2.08160 1.49584 2.09397 2.10619 1.49767 1.49676 2.13019 1.49855 2.11826 1.49940

zi 1:9 :-i

22

2.20995 1.50416 2.19893 1.50489 2:

2.30397 1.51082 5.O [‘-;)5]

[‘-;)I]

2:00519 1.47634

1.61756 1.43125 10.0

2.30462 1.50085

[(-p]

[‘-;)“I

[c-y]

[f-2”“]

290

GAMMA

Table 6.8

DIGAMMA

FUNCTION

AND

FUNCTION

RELATED

FUNCTIONS

FOR COMPLEX

ARGUMENTS x-1.4

x=1.3

m(z) Y 0.0 -0.16919 -0.16323 :*: -0.14567 013 -0.11748 0.4 -0.08009

A(z) 0.00000 0.11303 0.22372 0.32997 0.43011

-0.03520 +0.01541 0.07003 E 0.12718 0:9 0.18561

0.52298 0.60796 0.68491 0.75404 0.81582

0.24434 :*!? 0.30262 1:2 0.35994

6:2 g 6.3 6.4 6.5 6.6 6.7 6.8 6.9

22

Y 5.0 3 ;a'4 . 5.5 ;$ 5:8 5.9

9e4

W(Z) 1.62052 1.63990 1.65891 1.67758 1.69591

1.41163 1.41472 1.41769 1.42055 1.42331

1.71392 1.73161 174900 1'76611 1:78292 1.79947

?J 0.0

9@(z)

4e4

Y*(z) 0.00000

0.10223 0.20269 0.29974 0.39204

Y 5.0 5.1 5.2 5.3 5.4

m(z) 1.62386 1.64311 1.66200 1.68055 1.69878

1.39213 1.39559 1.39891 1:4021i 1.40519

0.1 0.2 0.3 0.4

-0.06138 -0.05646 -0.04192 -0.01844 +0.01295

1.42597 1.42853 1743101 1.43340 1.43571

0.5 0.6 0.7 0.8 0.9

0.05100 0.09436 0.14171 0.19183 0.24367

0.47862 0.55886 0.63250 0.69957 0.76033

5.5 5.6 5.7 5.8 5.9

1.71668 1.73428 1.75158 1.76860 1.78533

1.40817 1.41103 1.41380 1.41648 1.41907

1.0

0.29635

;*;;;;+ 1:84754 1.86308

1.43794 1.44011 1.44220 1.44423 1.44619

0.81517 0.86457 0.90903 0.94907 0.98517

6.0 6.1 6.2 6.3 6.4

1.80180 1.81800 1.83395 1.84966 1.86513

1.42157 1.42399 1.42634

1'::

0.34918 0.40163 0.45331 0.50395

1.87837 1.89344 1.90829 1.92293 1.93735

1.44810 1.44995 1.45174 1.45348 1.45517

1.5 1.6 1.7 1.8 1.9

0.55336 0.60144 0.64811 0.69337 0.73722

1.01778 1.04730 1.07409 1.09849 1.12075

6.5 6.6 6.7 6.8 6.9

1.88036 1.89537 1.91017 1.92475 1.93912

1.43294 1.43502 1.43702 1.43898 1.44087

7.0 7.1 7.2 7.3 7.4

1.95330 1.44271 1.96727 1.44450 1.98106 1.44625 1.99467 1.44794 2.00809 1.44959

1.3 1.4

0.41593 0.47035

0.87085 0.91983 0196341 1.00227 1.03698

1.5 1.6 1.7 :::

0.52310 0.57409 0.62333 0.67084 0.71667

1.06809 1.09605 1.12126 1.14409 1.16483

2.0 2.1 225

0.76087 0.80353 0.88447 0.84470

1.18373 7.0 1.20102 7.1 1.23148 1.21688 7.3 7.2

1.95158 1.96560 1.97944

1.45681 1.45841 ;.;fJy;;

2.0 2.1 f-3

2:4

0.92290

1.24495

7.4

2.00655 1.99309

i46294

2:4

0.77968 1.14113 0182078 1.15984 0.86058 1.17707 1.19296 0.89913 0.93647 1.20768

2.5 2.6

0.96007 0.99604

1.25743 1.26900

7.5 7.6

2.01984 2.03296

1.46438 1.46577

z-i 2:9

1.06464 1.03088 1.09739

1.28980 1.27976 1.29918

7.7 7.8 7.9

2.04591 2.07131 2.05869

1.46713 1.46845 I.46974

2.5 2.6 2.7 2.8 2.9

0.97265 1.00775 1.04179 1.07484 1.10693

1.22133 7.5 1.23402 7.6 li24585 7.7 1.25689 7.8 1.26723 7.9

3.0

1.12917 1.16004 1.19005 1.21923 1.24763

1.30797 1.31621 1.32396 1.33126 1.33814

8.0 8.1 8.2 8.3 8.4

2.08378 2.09610 2.10827 2.12029 2.13217

1.47100 1.47223 1.47342 1.47459 1.47573

3.0 3.1 3.2 3.3 3.4

1.13813 1.16846 1.19797 1.22670 1.25469

1.27693 1.28604 1.29461 1.30269 1.31032

8.0 8.1 8.2 8.3 8.4

2.08510 1.45862 2.09739 1.46000 2.10952 1.46134 2.12151 1.46266 2.13337 1.46394

1.27529 1.30223 1.32851 33'87 1.35413 319 1.37915

1.34464 1.35080 1.35663 1.36216 1.36742

8.5 8.6 8.7 8.8 8.9

2.14391 2.15552 2.16700 2.17834 2.18956

1.47685 1.47794 1.47900 1.48004 1.48106

3.5 E 3:s 3.9

1.28196 1.30855 1.33450 1.35983 1.38456

1.31753 1.32436 1.33084 1.33699 1.34283

8.5 8.6 8.7 8.8 8.9

2.14508 2.15666 2.16811 2.17943 2.19063

1.46519 1.46641 1.46760 1.46877 1.46991

4.0 4.1 4.2 4.3 4.4

1.40357 1.42744 1.45077 1.47358 1.49590

1.37242 1.37718 1.38172 1.38606 1.39020

9.0 9.1 9.2 9.3 9.4

2.20066 2.21163 2.22249 2.23323 2.24386

1.48205 1.48302 1.48397 1.48490 1.48582

4.0 4.1 4.2 4.3 4.4

1.40873 1.43235 1.45546 1.47806 1.50019

1.34840 1.35370 1.35876 1136359 1.36821

9.0 9.1 9.2 9.3 9.4

2.20170 2.21265 2.22349 2.23421 2.24481

1.47103 1.47212 1.47319 1.47423 1.47525

4.5 4.6

1.51775 1.53914 1.56010 1.58064 1.60078

1.39416 1.39795 1.40258 1.40507 1.40841

9.5 9.6 9.7 9.8 9.9

2.25437 2.26478 2.27508 2.28528 2.29537

1.48671 1.48758 1.48844 1.48927 1.49010

4.5 4.6 4.7 4.8 4.9

1.52185 1.54307 1.56387 1.58425 1.60425

1.37263 1.37686 1.38092 1.38481 1.38854

9.5 9.6 9.7 9.8 9.9

2.25531 1.47626 2.26570 1.47724 2.27598 1.47820 2.28616 1.47914 2.29623 1.48006

1.62052 [C-,,2]

1.41163 10.0

2.30537 1.49090 [‘-p”]

5.0

1.62386 [C-y]

1.39213 10.0 2.30621 1.48096 ['-;'"I ['-;'"I ['-['"I

;:: 3:: 3.5 3.6

f-i 4:9 5.0

[(-$W]

[(-y-j

::2'

2.02134 2.03442 2.04733 2.06008 2:07267

?%: .

1.45119 1.45276 1.45428 1.45576 1.45721

GAMMA

DIGAMMA

FUNCTION

AND

RELATED

FOR

COMPLEX

FUNCTION

291

FUNCTIONS

x=

x=1.5

3$(Z)

1.4 1.5 1.6 1.7

1.8 1.9

2.0 3:: 5::

92 2:7 2.8 2.9 3::

ye.4 0.00000

Y+(z)

3995 O.l-..-

0.15687 0.17976

1.35357 1.35773 1.36173 1.36558 1.36930

0.5 0.6 0.7 0.8 0.9

0.20790 0.24050 0.27674 0.31581 0.35697

0.40789 0.47942 0.54642 0.60875 0.66642

5.5 5.6 5.7 5.8 5.9

1.72313 1.74051 1.75760 1.77441 1.79095

1.37289 1.37635 1.37969 1.38293 1.38605

1.40528 1.40796 1.41055 1.41306 1.41549

1.0 1.1 :.g

0.39957 0.44305 yyw&

0.71957

6.0

o*7684o 0.81319 0.85423 0157445 0.89183

22 "6:: .

1.80724 1.82327 1.83906 1.85460 1.86992

1.38908 1.39200 1.39484 1.39759 1.40025

1.88258 1.89752 1.91225 1.92677 1.94109

1.41786 1.42015 1.42237 1.42453 1.42663

1.5 1.6 1.7 1.8 1.9

0.61757 0.66001 0.70167 0.74244 0.78228

0.92629 0.95790 0.98693 1.01363 1.03824

6.5 6.6 6.7 6.8 6.9

1.88501 1.89989 1.91455 1.92900 1.94326

1.40284 1.40534 1.40778 1.41014 1.41244

1.95521 1.96914 1.98287 1.99643 2.00981

1.42866 1.43065 ii43257 1.43445 1.43628

2.0 2:3 2.4

0.82115 0.85905 0.89597 0.93193 0.96694

1.06096 1.08197 1.10144 1.11953 1.13635

7.0 7.1 7.2 7.3 7.4

2.02301 2.03604 2.04891 2.06162 2.07417

1.43805 1.43978 1.44147 1.44312 1.44472

2.5 2.6 2.7 2.8 2.9

1.00102 1.03421 1.06653 1.09801 1.12867

1.15204 1.16668 1.18039 1.19324 1.20530

3.0 3.1 3.2

1.15856 1.18770

33::

1.21611 1.24383 1.27089

1.21664 1.22733 1.23741 1.24693 1.25594

3.5 3.6

1.29731 1.32311

1.42065 1.44373 1.4b632 1.48844

0.2 0.3 0.4

0.13189 0.16935 0.21064 0.25479 0.30091

0.44066 0.51640 0.58668 0.65144 0.71078

5.5 5.6 5.7 5.8 5.9

1.71976 1.73725 1.75445 1.77137 1.78801

1.39047 1.39364 1.39670 1.39965 1.40251

0.34824 0.39614 0.44411 0.49175 0.53878

0.76494 0.81424 0.85907 0.89980 0.93684

6.0 6.1 6.2 6.3 6.4

1.80439 1.82051 1.83638 1.85201 1.86741

0.58497 0.63018 0.67432 0.71732 0.75916

0.97054 1.00127 1.02932 1.05500 1.07855

6.5 t-7" .

1.10020 “o*ZE li12015 1.13857 0:87772

66:: 7.0 34 7:3 7.4

0.91499 0.95118

1.15563 1.17146

0.98634 1.02050 1.05370 1.08598

1.18618 1.19990 1.21271 1.22469 1.23592

;.; 7:7

1.24647 1.25639 1.26574 1.27457 1.28290

8.0 z.;

2.08657 2.09882

8:3 8.4

;-;;;;; 2:13470

1.44628 1.44781 1.44930 1.45075 1.45217

8.5 8.6 8.7 8.8 8.9

2.14638 2.15794 2.16936 2.18065 2.19182

1.45355 1.45491 1.45623 1.45753 1.45879

9.0 9.1

2.20286 2.21379

9.2 9.3 9.4

2.22460 2.23530 2.24588

1146242 1.46358 1.46471

9.5 9.6 9.7 9.8 9.9

2.25635 2.26672 2.27698 2.28714 2.29720

1.46582 1.46691 1.46798 1.46902 1.47004 1.47105 (-;I2

1.11738 1.14794 1.17769 1.20667

1.29080

1.41443 1.43779 1.46065 1.48302 1.50493

1.32464

1.52639 1.54742 1.56804 1.58826 1.60810

1.35128 1.35594 1.36041 1.36470 1.36882

5.0

ad44

1.63162 1.65057 1.66919 1.68748 1.70546

0.27432 0.35978

0.18511

1.28931 131552 1:34112 1.36612 1.39055

:*s7 4:9

Y

5.0 5.1 5.2 5.3 5.4

n i3m5 -,,5 0.1395’

;*: 3:7 3i8 3.9

4.5 4.6

-e+(z)

.%$(Z)

nn

0.09325

1.29828 1.30537 1.31212 1.31853 1.33047 1.33603 1.34134 1.34642

;a;.

6.8

1.6

0.00000 0.08566 0.17023 0.25268 0.33214

Y

1.37278 1.37658 1.38025 1.38378 1.38719

x:51 .

4.2 4.3 4.4

-@tic4

%ti(z) 1.62756 1.64667 1.66543 1.68386 1.70196

z*; 3:4

4.0 4.1

Y 5.0 5.1 5.2 5.3 5.4

0.03649 0.04062 0.05284 0.07266 0.09932

Table

ARGUMENTS

1.62756

1.37278 lo.0

2.30716

[i-y]

['-:I']

[

C-i)4

“.”

0.1

114

22.:

3.7

3.8 3.9 2::

1[

2; 414

1

.f$(1.5+iy)

I.&b”“_

1.34833 li37297 1.39707

li51012

4.5

1.53136

i-76 418 4.9

1:552i9 1.57262 1.59265 1.61232

=$ tanh KY-~- 4Y 4&l

1.95731

1.41467

1.97118

1.41684

1.98487 1.99837 2.01169

1.42101

7.5 7.6 7.7 7.8 7.9

2.02485 2.03784 2.05066 2.06332 2.07583

1.42496 1.42686 1.42871 1.43051 1.43227

8.0

2.08819 2.10040 2.11246 2.12439 2.13617

1.43398 1.43565 1.43728 1.43888 1.44043

1.26448 1.27257 1.28026 1.28757 1.29454

2.14782 2.15934 2.17073 2.18199 2.19313

1.44195 1.44344 1.44489 1.44631 1.44770

1.30117 1.30750 1.31354 1.31932 1.32485

2.20415 2.21504 2.22583 2.23650 2.24706

1.44905 1.45038 1.45168 1.45295 1.45420

1.33014 1.33522 1.34009 1.34476 1.34925

2.25751 2.26785 2.27809 2.28822 2.29826

1.45542 1.45661 1.45778 1.45892 1.46005

8.1 8.2

1.41895 1.42301

292

GAMMA

Table 6.8

DIGAMMA

FUNCTION

AND

FUNCTION

FOR

RELATED

FUNCTIONS

COMPLEX

ARGUMENTS

2=1.8

x=1.7 0'0

0 se> 20855

0"G> 00000

5:

163603 9 * (2)

133453 YG>

w

W(z)

0.0 0.1 0.2 0.3 0.4

0.28499 0.28760 0.29537 0.30809 0.32541

0.00000 0.07358 0.14644 0.21792 0.28740

5.0 5.1 5.2 5.3 5.4

1.64078 1.65939 1.67769 1.69567 1.71336

1.31566 1.32048 1.32513 1.32961 1.33393

0.5 0.6

0.34693 0.37215 0.40053 0.43155 0.46469

0.35437 0.41842 0.47928 0.53675 0.59076

5.5 5.6 5.7 5.8 5.9

1.73076 1.74787 1.76472 1.78130 1.79762

1.33810 1.34213 1.34603 1.34979 1.35344

1.81369 1.82952 1.84511 1.86047 1.87561

1.35697 1.36038 1.36369 1.36690 1.37001

1.89053 1.90525 1.91975 1.93406 1.94817

1.37303 1.37596 1.37881 1.38158 1.38426

w4

Y

a+)

‘e(z)

0:l 0.2 0.3 0.4

0:21156 0.22050 0.23511 0.25494

0:07918 0.15747 0.23407 0.30824

5:1 5.2 5.3 5.4

1:65482 1.67328 1.69142 1.70926

1:33902 1.34335 1.34752 1.35154

0.5 0.6 0.7 0.8 0.9

0.27945 0.30803 0.34001 0.37474 0.41161

0.37937 0.44701 0.51086 0.57074 0.62661

5.5 5.6 5.7 5.8 5.9

1.72680 1.74405 1.76102 1.77772 1.79416

1.35543 1.35918 1.36280 1.36630 1.36969

1.0 1.1 1.2 1.3 1.4

0.45005 0.48957 0.52973 0.57018 0.61063

0.67852 0.72661 0.77107 0.81211 0.84996

6.0 6.1 6.2 6.3 6.4

1.81034 1.82627 1.84196 1.85742 1.87266

1.37297 1.37614 1.37922 1.38220 1.38509

0.49947 0.53546 0.57226 0.60955 0.64706

0.64131

6.0

o*68847 0.73237 0.77316 0.81103

2 6:3 6.4

1.5 1.6 1.7 1.8 1.9

0.65085 0.69065 0.72990 0.76849 0.80636

0.88488 0.91710 0.94685 0.97436 0.99982

6.5 6.6 6.7 6.8 6.9

1.88767 1.90246 1.91705 1.93143 1.94561

1.38789 1.39061 1.39326 1.39582 1.39832

0.68455 0.72184 0.75879 0.79528 0.83122

0.84617 0.87877 0.90903

6.5 6.6 6.7

:*z:2 .

2:

2.0 2.1 2.2 2.3 2.4

0.84345 0.87973 0.91519 0.94981 0.98362

1.02342 1.04533 1.06570 1.08468 1.10238

7.0 7.1 7.2 7.3 7.4

1.95961 1.97342 1.98704 2.00048 2.01375

Il*to051: 1:40539 1.40762 1.40980

2.0 2.1 2.2

0.86655 0.90123 0.93523

0.98757 1.01022 1.03136

7.0 7.1 7.2

1.96210 1.97583

1.38688 1.38942 1.39189

2.3 2.4

1.00111 0.96853

1.05110 1.06957

7.4 7.3

;.lf;;;; 2:01598

;:;;:;:

2.5 2.6 2.7 2.8 2.9

1.01661 1.04879 1.08020 1.11084 1.14075

1.11893 1.13441 1.14893 1.16257 1.17539

7.5 7.6 7.7 7.8 7.9

2.02685 2.03979 2.05256 2.06518 2.07764

1.41191 1.41398 1.41599 1.41794 1.41986

2.5 2.6 2.7 2.8 2.9

1.03299 1.06416 1.09463 1.12442 1.15353

1.08687 1.10310 1.11836 1.13270 1.14622

7.5 7.6 7.7 7.8 7.9

2.02903 2.04191 2.05463 2.06719 2.07960

1.39892 1.40115 1.40332 1.40543 1.40749

3.0 3.1 3.2 3.3 3.4

1.16993 1.19842 1.22625 1.25342 1.27997

1.18747 1.19886 1.20962 1.21981 1.22945

8.0 8.1 8.2 8.3 8.4

2.08996 2.10212 2.11415 2.12603 2.13778

1.42172 1.42354 1.42531 1.42704 1.42874

3.0

1.18200 1.20982 li23703 1.26363 1.28965

1.15898 1.17103 1.18243 1.19322 1.20345

2.09187 2.10399 2.11597 2.12781 2.13952

1.40950 1.41146 1.41338 1.41525 1.41708

3.5 3.6 3.7 3.8 3.9

1.30592 1.33129 1.35610 1.38037 1.40413

1.23859 1.24727 1.25553 1.26338 1.27087

8.5 8.6 8.7 8.8 8.9

2.14939 2.16087 2.17222 2.18345 2.19456

1.43039 1.43200 1.43358 1.43513 1.43664

3.5

1.31511 1.34003 1.36441 1.38829 1.41168

1.21317 1.22241 1.23119 1.23956 1.24754

2.15109 2.16253 2.17385 2.18504 2.19611

1.41886 1.42061 1.42231 1;42398 1.42561

4.0 4.1 4.2 4.3 4.4

1.42738 1.45015 1.47246 1.49432 1.51574

1.27800 1.28481 1.29132 1.29755 1.30351

9.0 9.1 9.2 9.3 9.4

2.20555 2.21642 2.22717 2.23781 2.24834

1.43811 1.43956 1.44097 1.44235 1.44371

1.43459 1.45704 1.47904 1.50062 1.52178

1.25516 1.26243 1.26939 1.27605 1.28242

2.20707 2.21790 2.22862 2;23423 2.24974

1.42720 1.42876 1.43029 1.43178 1.43324

4.5 4.6 4.7 4.8 4.9

1.53675 1.55736 1.57758 1.59742 1.61690

1.30922 1.31470 1.31996 1.32501 1.32986

9.5 9.6 9.7 9.8 9.9

2.25877 2.26908 2.27930 2.28941 2.29942

1.44503 1.44633 1.44760 1.44885 1.45007

4.5

1.54254 1.56292 1.58291 1.60255 1.62183

1.28854 1.29440 1;30004 1.30545 1.31065

2.26013 2.27042

1.43468 1.43608 1.43745 1.43880 1.44012

5.0

1.63603 [C-i"]

1.33453 [I'-;'"]

10.0 *

2.30933 ['-y-j

1.45127 ['-['"I

5.0

1.64078 ['-;'"I

1.31566 ['-;I"]

*see page II.

8:'8 019

;:2'

33:: E 3:8 3.9

44:: 2; 414

i:; t::

2.28061

2.29069 2.30068 10.0

2.31057 [5’4]

1.44142 ['-;'"I

GAMMA

FUNCTION

DIGAMMA

AND

FUNCTION

RELATED

293

FUNCTIONS

FO IR COMPLEX

ARG IUMENTS

Table

6.8

s=1.9 :!t*(z)

0.35618 0.35847 0.36528 0.37644 0.39169 0.41071

./tic4 0.00000 0;06870 0.13681 0.20377 0.26908

z*(z) 5yo

1 tic4

0.42583 0.47874 0.52904

z.7" 5:8 5.9

1.81728 1.83300 1.84848 1.86374 1.87878

1.34107 1.34473

1.0 2:

0.57667 0.62165 0.66400 0.70380 0.74116

6.0

Z% 1:35503

0.59465 0.62468 0.65572 0.68751 0.71980

1.35826 1.36140 1.36445 1.36741 1.37029

zl 1:9

0.75239 0.78510 0.81779 0.85033 0.88262

0.77618 0.80899 0.83973 0.86853 0.89551

6.5 6.6

2: 6:9

1.89361 1.90824 1.92266 1.93688 1.95092

x 712

1.97843 1.96476 1.99192

:-::::1" 1137846

?1" 2:2

0.91459 0.94617 0.97731

2:

2.01838 2.00523

1.38355 1.38104

;::

1.03814 1.00798

0.920E: 0.94454 0.96681 0.98775 1.00743

7.5

;.;;I:;

1.38599

2.5

1.06779

::;

2:05684

1.39070 1.38838

::;

1.12548 1.09690

::;

2.08171 2.06935

1.39518 1.39297

29"

1.18102 1.15352

3.0

1.20798

::: :::

1.23442 1.26034 1.31067 1.28575

z-2 317

1.33510 1.35905 1.38254

2.14139 2.12973

1.40546 1.40350

1.32485 ::2 1.34929 1.37324 :-‘B 1.39670 3:9 1.41970

1.18823 1.19798 1.20727

2: 817 88::

2.15292 2.16432 2.17560 2.18675 2.19778

1.40738 1.40925 1.41108 1.41286 1.41461

Z:i

2.21950 2.20870

1.41800 1.41632

2'3 9:4

2.23019 2.24077 2.25124

1.41964 1.42124 1.42281

2 z-i

2.26160 2.27186 2.29207 2.28202

9:9 10.0

f-i 4:9

1.26810 1.27434 1;28033 1.28610 1.29164

5.0

1.64585

1.29698

[ 1 C-t)6

1.38522 1.38746 1.38966 1.39180 1.39389

0.51380 0.56594 0.53887

i::

1.54872 1.56885 1.58861 1.60803 1.62710

2.09613 2.10815 2.12003 2.13178 2.14339

i:; 2:

1.39734 1.39944 1.40149

t::

1.37313 1.37567 1.37815 1.38056 1.38292

1.32938 1.33730 1.33341

2.09393 2.10600 2.11793

:*:z .

2.03385 2.04661 2.05921 2.07167 2.08397

ihi 5:9

1.76868 1.78513 1.80133

21" 8;2

1.23265 1.24037 ii24775 1.25482 1.26160

1.35937 1.36227 1.36509 1.36784 1.37052

5.5

1.13119 1;14384 1.15583 1.16719 1.17798

1.44226 1.46437 1.48606

1.96761 1.98120 1.99462 2.00786 2.02094

0.31269 0.37042

1.19470 1.22184 1.24841 1.27442 1.29990

:*: 4:3 4.4

1.34358 1.34692

0.47111 0.49110

1.05588 1.07278 1.08868 1.10367 1.11782

4.0

1.89690 1.91143 1.92576 1.93990 1.95385

0.5

1.05008 1.08022 1.10975 1.13867 1.16698

:*::::i .

1.32530 1.32918 1.33295 1.33660 1.34015

1.32522 1.32092

0.98795 1.01932

::4'

1.82111 1.83671 1.85208 1.86723 1.88217

1.75197 1.73500

0.95338 0.97664 0.99840 1.01879 1.03792

3.0 3.1 3.2

1.30389 1.30846 1.31288 1.31715 1.32129

::6'

0.89031 z-10 0.92342 2:2 0.95598

z3 2:9

1.73951 1.75633 1.77290 1.78921 1.80528

1.30707 1.31185 1.31647

0.80999 0.84278 0.87335 0.90188 0.92851

2.5 2.6

1.72242

1.27849 1.28394 1.28919 1.29426 1.29916

1.68240 1.70022 1.71775

22 0.71846 0.75338 0.78814 1’2 0.82261 1:9 0.85669 ::i

::y 1.65125 1.66948 1.68742 2-G 1.70506 514

2.: 5:4

0.60749 0.65359 0.69677 0.73714 0.77483

:::

0.42480 0.43081 0.44068 0.45420

0.1

1.30212 1.29698

0.54770 0.58053 0.61431 0.64872 0.68351

1:2

Y

-@ti(z) 0.00000 0.06441 0.12833 0.19130 0.25288

1.66428 1.64585

0.43309 0.45842 0.48625 0.51614

:?i

X!b(z) 0.4i2i8

5:1

0.33229 0.39306 0.45110 0.50624 0.55838

0:7 i-2

v

010

21"

22

:::

::2

t-t. 2::

2; 6:9 7'*1" 712 :::

1.02597 1.04344 1.05992 1.07548 1.09020

7.5 7.6 7.7

1.10413 1.11733 1.12985

8.0

:%: .

:::

2: 88:: 2.15487 88’2 2.16623 817 2.17746

:*E:: 1:35639

1.39593

::i

1.42818 1.40558

1.18379 1.19310 1.20200

4.0

1.45036

1.21050

t:: 44::

1.49348 1.47212 1.53505 1.51446

1.21864 1.22643 1.24105 1.23389

2.21045 2.22121 2.23187 2.24241 2.25284

1.40548 1.40727 1.40902 1.41074 1.41241

1.42435 1.42586 1.42878 1.42733

2: 2;

1.55527 1.57514 1.59466 1.61385

2.30203

1.43020

4:9

1.63270

1.24792 1.25452 1.26086 1.26696 1.27283

2.26318 2.27340 2.28353 2.29356 2.30349

1.41406 1.41566 1.41724 1.41879 1.42030

2.31190 [ (-;)“I

1.43159 [ (-,121

5.0

1.65125 [(-951

1.27849 [ (-:)“I

2.31332 [ (-35)4]

1.42179 [ (635)3]

coth 1r,/--1+3!!? ’ Wl+!f2)

88::

2.18858 2.19957

:%z 1:40179 1.40366

7. Error

Function

and Fresnel

WALTER

GAUTSCHI

Integrals

l

Contents Mathematical Properties . . . . . . . . . . . . . . . . . . 7.1. Error Function . . . . . . . . . . . . . . . . , . . . 7.2.. Repeated Integrals of the Error Function . . . . . . . . 7.3. Fresnel Integrals . . . . . . . . . . . . . . . . . . . 7.4. Definite and Indefinite Integrals . . . . . . . . . . . .

. . . . .

. . . . .

297

297 299 300 302

. . . . . . . . . . . . . . . . . . . . . . 7.5. Use and Extension of the Tables . , . . . . . . . . . . . .

304 304

. . . . . . . . . . . . . . . . . . . . . . . . . .

308

. . . . . .

310

. . . .

312

. . . . . .

'316

Numerical

Methods

References

Table 7.1. Error

Function

and its Derivative

(2/-l;;) cz2, erf x=(2/&) Table 7.2. Derivative

1

Error Function m

2*r

(

:+I

>

in erfc

x=0(.1)5,n=1(1)6,

Table 7.5. p2

Dawson’s

. .

(2 Iz<

w).

7D

Integrals

of the Error Function

(0 Ix 15)

x=~~+T

~fl)iJ

s2

Table 7.4. Repeated

10)

e-f2dt, x-2=.25(-.005)0,

xez2erfc x= (2/J;;) xez2

n=l(l)lO,

(2 5x5

10D

ss

Table 7.3. Complementary

erfc&,

ecf2dt, 2=0(.01)2,

of the Error Function

esz2, 2=2(.01)10,

(2/J;;)

(O_
l5D

10, 11,

Integral

S

‘ef2cZt, x=0(.02) 2,

- (t-XP n.I z

317

e-12dt

6s

(O<x<

a) . . . . . . . . . . . .

319

10D

0

* 2 xeez2 ef dt, xm2=.25(-.005)0, S0 1 Guest worker, National Purdye University.)

9D

Bureau of Standards,

from The American

University.

(Presently

295

ERROR FUNCTION

296

AND FRESNEL

INTEGRALS Page

Table

(0<&2.3)

. . . . . . . . . . . . .

320

7.7.

Fresnel Integrals (O
. . . . . . . . . . . , . .

321

C(z)=l

cos (i P) dt, S(r)=lsin(g

7.6.

(3/I’(1/3))Jzemt3dt 0

z=O(.O2)1.7(.04)2.3, Table

Table

7.8.

Auxiliary

f(z) =[l-

7D

Table

7.10.

15D

T&l~,7.11.

CC&?%,

2=0(.1)3.9,y=O(.1)3, (l
S(z:)=O,n=0(1)5,

325

6D 10) . . . .

329

. . . . .

329

4D

Maxima and Minima of Fresnel Integrals (0 5n55) C(J4n+3),

.

8D

Complex Zeros of Fresnel Integrals (057~15)

z,, z:, C(z,)=O, 7.12.

z=z+iy,

Complex Zeros of the Error Function

z,,erf z,=O,n=l(l)lO,

Table

x2)

Error Function for Complex Arguments (O
ft.u(~)=e-~~ erfc (-iz),

323

sin (i x2)

cos (i r2)+[-+S(d]sin@j

2=0(.02)1,2-‘=1(-.02)0, 7.9.

7D

Functions (O<X< m) . . . . . . . . . . . . .

f?(r)] cos G x2)-[&C(x)]

g(x)=[&C(z)]

Table

t2)dt, x=0(.02)5,

S(d4n+2),

S(J4%+4),

n=O(1)5,

. . .

329

6D

The author acknowledges the assistance of Alfred E. Beam, Ruth E. Capuano, Lois K. Cherwinski, Elizabeth F. Godefroy, David S. Liepman, Mary Orr, Bertha H. Walter, and Ruth Zucker in the preparation and checking of the tables.

7. Error

Function Mathematical

7.1. Error

erf z=Z

Integrals

Properties

Function

Series

7.1.5

Definitions

7.1.1

and Fresnel

erf z=L

2

&

* eet2dt 4 lr s 0

7.1.6

a=0

7.1.3

erfc z=-

w(z) =e-“’

;

-“dt=l-erf s zme

restriction path.

arg t-+a with

(cx=:

7.1.7

is permissible

(42) is subject

lal
as t-+=

along the

w (z)=%

=Jz

ILo

For 1,-t(x),

Symmetry

S

m eet2dt s 0 Z2-t2

V~>O)

Relations

erf (--z)=-erf

7.1.10

Representation

2iz

(-1>“[12n+l/2(22)-12n+3,2(22)1

see chapter 10.

7.1.9

m emt2dt -m z--t’

22n+1

7.1.8

if L%Q2remains bounded

7.l.4 ’

n!@n+l)

to the

to the left.) Integral

792n+1

2

(l+$J’e’adt)=e-z’erfc

In 7.1.2 the path of integration

C-1)

2” e-z2 5 n=o 1.3 . . . (2n+l)

=2 &

7.1.2

Expansions

2

erf Y=erf 2

7.1.11

w(-.z)=2e-z2-w(z)

7.1.12

w(Z)=w(-2)

Y

FIQUBE

7.1.

y=e”’

P=w)6

m e-“dt.

s ..I

FIQURE

7.2.

y=e-”

z t' s 0

e dt.

p=2(1)6

297

ERROR FUNCTION

298

AND FRESNEL

INTEGRALS

7.1.15 $Jy e-'2d~-~~~~$. m z-t

Y

.. (f..

# 0)

xp) and El:“) are the zeros and weight factors of the Hermite polynomials. For numerical values see chapter 25. Value

at Infinity

7.1.16 erf z-+1 (~-+a M aximum

and

in jarg z/<:)

Inflection Integral

F(x) =e-*

Points [7.31]

for

Dawson’s

’ e”dt s0

7.1.17 Ft.92413 88730. . . )=.54104 42246. . . 7.1.18 F(1.50197 52682 . . . )=.42768

66160 . . .

Derivatives

7.1.19 dn+’ p erf 2=(-l)

*

7.1.20 u1("+2)(z)+2zw("t1) (z)+2(n+l)?P(z)=O

(n=O, 1,2, . . .) FIQURE 7.3.

Altitude

o'(z)=-2zw(z)+~

w’“‘(z)=w(z),

Chart of w(z).

(For the Hermite

polynomials

H,(z)

see chapter

22.) (7.111, [7.17]

Inequalities

Relation

to

Coniluent

7.1.21

(For other inequalities

The

see [7.2].)

Hypergeometric

Function

(see

Mean

m and

chapter 13)

Normal

Distribution Standard Deviation

Function IJ (see

With

chapter 26)

--(t-v&p

7-1-22 Continued

(D s

7.1.14 2ez2

e-z2dt-

*

----&~~,

e

%* dt=a

(l+erf

Asymptotic

Expansion

7.1.23 1

(s))

Fractions

‘I2

’

-3/2 -.2

. .

&zeJ

erfc zml+m$I

(-1)”

(27n-1)

’ *3 * (222)" *’

(z-t-,

larg zl<$)

ERROR FUNCTION

AND FRESNEL Infinite

If R,(z) is the remainder after n terms then

299

INTEGRALS Series

7.1.24

7.1.29

&(2)=(-l)”

erf (x+iy)=erf

Approximation Function

for

Complex

Error

[7.19]

2

l-3 .(2$:-l)

B=re-’

8,

(l+$)-‘+

[(l-cos

+i emz2gl ---

(larg z]
dt

x+&

For x real, R,(x) is less in absolute value than the first neglected term and of the same sign.

[fn(x,YY)+ig,(x,Y)l+~(x,y)

x cash ny cos 2xy+n sinh ny sin 2xy gn(x, y) =2x cash ny sin Bxy+n sinh ny cos 2xy le(x,y)[ -lo-l61 erf (x+;y)[

fn(x,y)=2x---2

7.2. Repeated Integrals Approximations

2 (0
(ult+azt2+a3t3) eVzz+e (x),

aI = .34802 42

p = .47047

t=1

7.2.1 1+px

i” erfc z=

m S P

Definition

in-l erfc t dt

(n=O,1,2,...)

eez2, io erfc z=erfc

a~= - .09587 98 a3= .74785 56

7.1.26 erf x=1-

of the Error Function

0~ )

7.1.25 erf x=1-

sin 2xy]

where

PI<1

Rational

2xy)+i

Differential

Equation

‘&+2z

$-2ny=O

2

(alt+azt2+a3t3+a4t4+ast5)e-.Z1+E(x), 7.2.2

t-L

1+px ]e(x))1<1.5X10-7 p= .32759 11 al= .25482 9592 a2= - .28449 6736 a,=1.42141 3741 a,=-1.45315 2027 as= 1.06140 5429

y=Ai”

erfc z+Bi”

erfc (-2)

(A and B are constants.) Expression

as a Single

Integral

7.2.3

7.1.27

Power

Series

a

7.2.4 al= .278393 a3= .000972 7.1.28

a2= .230389 u4= .078108

Recurrence

Relations

7.2.5

*

erfx=1-[l+alz+a2& ...+ag26]16+~(x)

21 ‘n-l erfc z ++n inm2erfc z

i” erfc z=-;

(n=1,2,3,.

le(z)(~3XlO-'

al= .07052 30784 a3= .00927 05272 aa= .00027 65672

a2= .04228 20123 a4= .00015 20143 aa= .00004 30638

7.2.6 2(n+l) (n+2)i”+2 erfc 2 = (2n+1+2z2)in

2 Approximations7.1.25-7.1.28 are from C. Hastings,Jr., Approximations for digital computers. Princeton Univ. Press, Princeton,

N. J., 1955 (with permission).

. .)

erfc z-f

inW2erfc 2 (n=1,2,3,

8 The terms n+4,

n+6,

. . .>

in this series corresponding to k=ni-2, are understood to be zero.

. . .

ERROR

300 Value

FUNCTION

AND

lelation

at Zero

7.2.7

INTEGRALS to

the

Confluent Hypergeometric (see chapter 13)

Function

‘.2.12

1

in erfc 0=

FRESNEL

(n=-1,0,1,2,.

. .) n erfc z=e-@

Relation

to Parabolic Cylinder chapter 19)

Asymptotic

Functions

(see

Expansion

.6

(-1)“(27+n)! n!m!(22y”

(~+a, largA
Fresnel

Integrals

Definition

FIGURE

7.4.

Repeated

y=2T(++l) n=o,

of the

Integrals

Error

7.3.1

C(z)=l

cos 6 ta) dt

7.3;2

S(z)=1

sin 6 P) dt

Function. The

inerfcz

following

functions

are also in use:

7.3.3

1, 2. 4, 8, 14, 22

Cl (x)=4;

l

zett o 4

S

cos Pdt, G(x)=-&

Derivatives

7.3.4 7.2.8

-$ in erfc z=--in-l

erfc z

(n=O,

1,2, . . .)

S,(+$~

sin Pdt, &(x)=&l

‘@$ dt

7.2.9 Auxiliary

g (era erfc

2) = (- 1) n2nn! erzin erfc 2 (n=O, Relation

7.2.11

to

Hermite

(-1)“i”erfc

Polynomials

z+i”

1,2, . . .)

.f(z)s[i-S(Z)]

s(z)+-C(Z)]

f&(dW (see

erfc (-z)=&J

COS

6

Z2)-[k-C(Z)]

sin

COS

6

Z2)+[+S(z)]sin

6

22)

6

z2)

7.3.6

(see 19.14)

in erfc z= @A-$

7.2.10 Relation

to Hh,(a)

Functions

7.3.5

chapter

*-*

Interrelations

22:

H,(iz:

7.3.7

C(x>=C1

(x&)4?2

6

x2)

ERROR

F’UNCTION

AND

FRESNEL

INTEGRALS

301

7.3.14

7.3.8

S(z)=-cos 7.3.9

c(z)=;+f(z)sin

7.3.10

(a 2)-g(z)

S(Z)=:-j(z)cos

(

522

G z2)

7.3.15

c&)=Jl,2(2)+&,2(2)+&,2(2)+

7.3~6

s2~z>=~3,2~Z~+~,,2~~~+~11,2(~~+

For Bessel functions

C(z)=$

J,+I,2(z)

~~;~;~;~~)l;

C(z) =cos (5 2) go

CPfl

l .

y& ...

+sin/.- 6 z2) SO 1 S-l”&

7.3.17

C(-2)=--c(z),

7.3.18

C(iz)=iC(z),

7.3.19

24n+l

S(-2)=--s(z) S(S)=--is(z)

Value

7.3.20

C(x)

-+-7

S(Z)=S(z) at

1

Infinity

1

S(s)+

2

'(')=go

Relations

C(Z)=C(z),

24n+3

.i

7.3.13

10.

see chapter

Expansions

7.3.12

fi

... ...

Symmetry

7.3.11

24r+3

cos G 9)

G z2)-g(z)&

Series

go 1 ynT2"+' * . . . (4n+3)

>

(-00)

2

Derivatives

(2n+1)!(4n+3) (-1,7+/2)2nf

Z4n+3

V

A lRelation

to

Error

Function

(see

7.1.1,

7.1.3)

7.3.22 C(z)+iS(z)=~

erf [$ =A${

(I--&] l-e+aW

[$

7.3.23

g(z)=$? {J&q

,l,ih]}

7.3.24

f(x)=9

(l+i)%l)

Relation

to

Confluent

{qL[f Hypergeometric chapter 13)

(l+i)z]}

Function

(see

7.3.25

.6

1.2 FIGURE

I.8 7.5. u=C(z),

Fresnel r=w

2.4

Integrals.

3.0

3.6

X

Relation

to

Spherical

7.3.26

C2(+f

Bessel

Functions

(see

chapter

10)

ERROR FUNCTION

302 Asymptotic

AND FRESNEL

7.4.2

Expansions

me-(at2+2bt+e)dt =I P e+ f:rfc& ’ 2J a S

7.3.27 nzf(+l+g

(4m-1) *3 . . * (az2)2n

(-1)“l

INTEGRALS

(9?a>O)

0

7.4.3

m S

7.3.28

e

(4m+l) az$(z)- Sri0(--lP l *3($,+i-

(ga>o,

0

mt2?le-atz& S =

LB>01

7.4.4

If Rh’) (z), RAC)(z) are the remainders after n terms in 7.3.27, 7.3.28, respectively, then 7.3.29 1 * 3 . . . (4n-1) e(,’ , (7rZ*)2n - e-rt2n-t 1 p = ,dl(Isrg rW+3) o 1+ 2 ( uz2> 7.3.30

m

7.3.31

/8”‘[<1,

S

2t ( rz2>

(S?k>O;n=O,

4<:)

7.4.6

S

me-d’ cos (2zt)dt=i

0

d-

12

%e-:

zl
(9a>O) .*

7.4.7

,dt (lwz

1,2, . . .)

L?Za>O;n=O, 1,2,. . .)

0

e-lt2n+*

o 1+

r(n+3> =2a"ff

E a

Smt*TS+le-d2&=&

f$‘(z)=(-1)“l * 3 .(;;;:+l) p, 1 e(g)=r (2n+$)

d

7.4.5

Ri”(z)=(-1)”

S

1.3...(2n-1) plan

0

S

me-d’ sin (2st)dt =i

e-I’/a

0

xl vs e%!t o

S

Wa>O)

pP’l
7.4.8

t,

For z real, R:“(z) and RLg)(x) are less in absolute value than the first neglected term and of the same sign. Rational

Approximations

’ (0
7.3.32 1+.92&x f(x~=2+l.792x+3.1042

+4x1

la(z)l52X10-3

-=L 0 &t+z)

ear erfc &

&

K@a>o, zfO,larg

A<*)

7.4.10

7.3.33 1 g(“)=2+4.142x+3.492~~+6.6702+

e(Z)

(For more accurate approximations see [i’.l].) 7.4. Definite and Indefinite

Integrals

For a more extensive list of integrals see [7.5], 1781, L7.151. .

Ioe-t$-jts S

7.4.1

S t+x -

e-D’2dtse+2

0

10,

173,

1956.1

J;; [Is

4ar,12dt

-k Ei(ax2)

0

1

*

(a>% s>O) 7.4.11

erfc 6X (a>% x>O) t2+2eaI2 S0-e-a12dLJ!. 2x

7.4.12

2

4 Approximations7.3.32, 7.3.33 are based on those given in C. Hastings, Jr., Approximations for calculating Fresnel integrals, Approximation Newsletter, April 1956, Note 10. [Seealso MTAC

mema’& S

-“t2dtu e’[l-

o1 tZS1=4 e

S

m S

(erf -\lZ)‘]

(a>01

7.4.13

yemt2dt =u 9w(x+iy) -0 (X-t)*+yZ

*See page II.

(x real, y>O)

ERROR FUNCTION

AND FRESNEL

303

INTEGRALS

7.4.24 m --(lf sin (t”> dt=; e ~ t s0 7.4.15

+; [i-s

?r * [P-(x2-y2)]e-tZdt s

(g4”+iY) y-ix

t"-z(~-y*)t*+(z*+y*)*=2

0

0

n-j

j-[&S

w(z+;Y>

7.4.17 0

e-a’ erf bt

1”’ dt=-

s 0

a

s0

{ [;-C(,/T)]coi

(ab)

(@)I

sin (ab) }

7.4.26 m ewaldt =‘$ s o &t2+b2) b

b

{[$S(dF)]cos

-[&CT(@)]

(2at)erfc bt dt=&

[l-e-‘“‘“‘*](a>O,~b>O)

sin (ab)}

7.4.27 m

S

(ab)

e-“la(t)dt=i

{[:-A

@]

-[&C(z)]

(D e -ar erfd%dt=i

S

Wb+b)>o)

0

7.4.28 m s0

7.4.20 dt=a e -2fi

(9%>0, L%?b>O)

cos (f)

0

7.4.19

Jm eeat erfc .$

(Si’a>O, Wb>O)

e4@erfc $

(la>O, lw bl
(a%>O)

y--ix

(x red y>O)

7.4.18 0

‘om$$dt=,rg

s

7.4.16

2xye-“*dt t4-2($-y2)t*+(22+y*)*=Z

(;$)1

7.4.25

(5 real, y>O)

m S

[+(;$)]

e+S(t)dt

=i { [$C’(~)] +[&

(Wa>O, ab>O)

sin (g)}

S @]

(%z>O)

cos (f) sin @}

@?a>O)

7.4.29

7.4.21 s,- e-“‘C(~~)dt=ze(~~la)r~~ (9u>O)

(Wb>o, ac>o)

7.4.30

7.4.22

m S e

--(II cos (t*)dt =&{

Jim e-“‘S(~~)dt=2~~~:a)t~~

[;-S(&/~)]cos@

0

(S?a>O) -[&C(i

#)]sing)}

(9a>O)

7.4.31

lm{

[;-C(t)]‘+[;4(t)]*}dt=;

7.4.23

m S e

7.4.32

--atsin (t*)dt= &{

[&?(q)]

cos @

0

S e

+[+B

(Ed:)]

sin (:)}

(&%>O)

erf (&z+$)+const.

(a+3

ERROR

304

FUNCTION

AND

FRESNEL

INTEGRALS

7.4.38

7.4.33

-&Lb’ e S .*dx=g

[em” erf (ox+:) S

+e- 2oberf

( ax-- i I

+const.

cos (ax2+2bx+c)dx

(a+01

=Jg

(ax+b)]

{ co9 (p)C[Jz

7.4.34

-.~~+gjx= -22 emazz2+S [w(i+iax) Se

+sin(~)S[JZ(ax+b)]}+const.

+w (-$+iax)]+const.

Serf xdx=x

7.4.35

erf x+’

(a#O) e-z2+const.

fi

7.4.39

S

sin(ax2+2bx+c)dx

7.4.36

Seaz erf

bxdx=k

[em erf bx-ee$

erf (bx-$)I +const

.

(a#O)

-sin(e)

C[g

(ax+b)]}+const.

7.4.37 Sea

erf $dx=a(e-

erf .$

+$ e+[w(&%+i

-&)+w

(-Jiiz+i

(a#@

Numerical Extension

SC(x)dx=xC(x)-isin

7.4.41

Ss(x)dz=xs(x)+~

+const.

-Jk)]}

+const.

7.5. Use and

7.4.40

Methods erf.745=.70467

of the Tables

Example 1. Compute erf .745 and e-(.‘4a)” using Taylor’s series. With the aid of Taylor’s theorem and 7.1.19 it can be shown that

58247)X

80779+(.5)(.00652

[l-(.005)(.74)+(.00000

83333)(.0952)] = .70792 8920

e--(.,w,~;

(.65258 24665) [l-.0074

+(.000025)(.0952)+(.00000

erf (x,+ph) =erf x0

+const.

co9

00833)(.74) (1.9048) 1 =.57405 7910.

+-$ e-4$

[

1--phz,+$

p2h2(2$- l)]+e

1-2phx0+p%‘(2~--1) -;p3h3q(2+3)

where (e(
1+v

Iql<3.2X10-10 if h=10m2, p=.5 and using Table 7.1

As a check the computation was repeated with xQ=.75,p=-.5. Example 2. Compute erfc x to 5s for x=4.8. We have l/22=.0434028. With Table 7.2 and linear interpolation in Table 7.3, we obtain & erfc 4.8=$8 (1.11253)(10-10)(.552669) -2=(1.1352)10-“.

ERROR

FUNCTION

AND

FRESNEL

z

Example

3.

Compute

e+’

to

e%t

INTEGRALS

305

From Table 7.1 we have $ e-(1.72)2=.058565.

5s for

s 0

Thus,

x=6.5. With

and linear

l/22=.0236686 in Table 7.5

6.6 e-WJ’ S

i erfc 1.72 = (.058565)(6.0064X

interpolation

101*)/1.0087X =3.4873X

1013 10-S

i2 erfc 1.72 = (.O58565)(1.292OX1O11)/1.OO87X1O13 =7.5013x

(.506143)/(6.5) ==.077868.

e%t=

0

10-4

i3 erfc 1.72 = (.058565) (2.6031 X 1O1o)/1.OO87X 1013

Compute i2 erfc 1.72 using recurrence relation and Table 7.1. Example 4.

=1.5114x10-4.

the Example 6.

Compute C(8.65) using Table 7.8. With x=8.65, l/x= .115607 we have from Table 7.8 by linear interpolation

By 7.2.1, using Table 7.1,

j(8.65) = .036797, g(8.65) = .000159. i-‘erfc

1.72 = .05856 50.

Using the recurrence relation i erfc 1.72= - (1.72)(.01499

From Table 4.6

7.2.5 and Table 7.1 72)f

(.5)(.05856

50)

= .0034873 i2 erfc 1.72= - (.86) (.0034873) f (.25) (.01499 72) = .0007502.

sin

.9613827

Using 7.3.9 C(8.65)=.5+(.036797)(--961382) -(.000159)(-.275218)=.46467. Example 7.

Note the loss of two significant

Compute &(l.l) to 10D. Using 7.3.8 and 7.3.10 we obtain by 6-pt interpolation in Table 7.8

digits.

Example 5. Compute i” erfc 1.72 for k=l, 2, 3 by backward recurrence. Let the sequence w~(x)(~=m, m-l, . . ., 1, 0, - 1) be generated by backward use of the recurrence relation 7.2.5 starting with u$+~.=O, w:+~= 1. Then, for any fixed k, (see [7.7]),

&(l.l)=S

(

-=w”x) w!%(x)

J;;

2

eZ2ik erfc

x

(x>O)

30169)=.31865 57172.

Example 8. Compute s2(5.24) to 6D. Enter Table 7.7 in the column headed by u.

u

.

5.20310 5.31898 6.0893801 6.43432 4.97691

With x= 1.72, m= 15 we obtain

J3

=s(.87767

Using Aitken’s lim m+w

1.1

scheme of interpolation

S:(u) 58 80

.4329l 04 .03689 .41673 97 -. 07803 .45993&3 .16061 70 .39999 44 -. 19432 11 .4699094 .2830889

42 89 99 70

.42732 691 756 674

63 63 .42718 63 60 6 52 79 9 39

.42717

71 61

.42717

67

&(5.24)=.427177 -

-

-

17 16 ::

0 i.44 4.3834

12 :i 9

13

(1) (2) 2.6399

8

(3) (4) (4) (6) (6)

-

2.1011 1.3831 9.8096 6.4143

i

L

4.1866

3

I$

::iE

(8) 8.9787 4.9570 (9) (10) 2.6031

:: -1 0

(11) 1.2920 (11) 6.0064

Example 9. Compute 5,(5.24) using Taylor’s series and Table 7.8. Using 7.3.21 we can write Taylor’s series forf, (u)

=

306

ERROR

FUNCTION

AND

f2(u)=c0++u0)+~ (u-uo)2+$(u-uo>“+*-*,

FRESNEL

By numerical

lo 1

+g (u-uo)2+$ (u-Q+

. . .],

where

clr+2= -cr+

integration,

using Table 9.1,

sYe(t)

g2(u)=-[c~+c2(u-zc,)

co=f2c7d,

INTEGRALS

$=.41826

00.

Using the fact that the remainder terms of the asymptotic expansion are less in absolute value than the first neglected terms, we can estimate

c1=--g2(u3,

1 * 3 . . . (2&l)

t--v

J%zo(2uo)k

(k=O,

1

+32 ’ 215. 52*72 5! ’ g2t-13’2 dt=7,33XlO-‘.

1, 2, . . .). Finally,

Consulting Table 7.8 we chose ~=1/.185638 =5.386819, thus having u-u,=5.24-5.386819 =--* 146819. From Table 7.8 5953819 co8 lo-sin -2688ooo m 23107 coslO+sinlO -21504oo qb

f2(u,,) = .168270, gz(u,) = .014483. Hence, applying

the series above,

f,(5.24) = .170436, g2(5.24) =.015030. Using the 4th formula

at the bottom

of Table

7.8

s2(5.24) = .5- (.170436) (.503471) -(.015030)(-.864012)=.42718.

52(2)

10.

=&/2(2)

+&2w

+&2.(2)

+e7,5,2ca

+

.

-

-

=.49129+.06852+.00297+.00006=.56284. Example

11. Compute

s

- 7Yo(t> dt by numerical 1

integration using Tables 9.1 and 7.8. [Y,,(t) is the Bessel function of thesecond kinddetkedin9.1.16.1 We decompose the integral into three parts J-

YOW ,=jy”

Yo(t> $+J,’

sYe(t) I

Compute S,(2) using 7.3.16. Generating the values of J,,++(2) as described in chapter 10 we find Example

using Tables 7.8 and 4.8.

[Yo@> -To(t)1

f

+Jy PO(t) $ where

represents the first two terms of the asymptotic expansion 9.2.2.

OD

$=.41826

10

=-.02298

78,

Hence

00- .02298 78= .39527 22.

The answer correct to 8D is .39527 290 (Table 11.2). Example 12. Compute w(.44$.67i) using bivariate linear interpolation. By linear interpolation in Table 7.9 along the x-direction at y=.6 and y=.7

w(.44+.6i)

-.6(.522246+.16788Oi)+.4(.498591 +.2026663)=.512784+.1817943

w(.44+.7i)

=.6(.487556+.147975i)+.4(.467521 +.179123i)=.479542+.160434i.

By linear x= .44

interpolation

w(.44+.67i)

along the y-direction

at

=.3(.512784+.181794i)+.7(.479542 +.160434i)= .489515+.166842i.

The correct answer is .489557+ .166889i. Example 13. Compute ~w(z) for z=.44+.61i. Bivariate linear interpolation, as described in Example 12, is most accurate if z lies near the center or along a diagonal of one of the squares of the tabular grid [7.6]. It is not as accurate for z near the midpoint of a side of a square, as in this example. However, we may introduce an auxil-

ERROR

FUNCTION

AND

FRESNEL

Example 16. Compute Using the second formula

iary square (see diagram) which contains z close to its center. Bivariate linear interpolation can then be applied within this auxiliary square. The values of w(z) needed at z=cl, and z=tz are easily approximated by the average of the four the neighboring tabular values. Furthermore parts to be used are given by Izo- &I-

w(7+2i)=(-2f7i)

( +

w(7+2i). at the end of Table

7.9

44 f;;;;y28i *

.05176536 42.27525+28i

>

=.021853+.075OlOi.

17. Compute erf (2+i). From 7.1.3, 7.1.12 we have

H=P,-P2

(Zo-tl(-Pl+Pz’

307

INTEGRALS

Example

erf z= 1 -e-Z2w(iz) = 1 -ey2-z2(cos 2xy 4 Using Tables

sin 2zy)w(y+iz)

(z=x+iy>.

7.9, 4.4, 4.6

erf (2-l-i)=1-e-3

(cos 4--i

sin 4)w(l+2i) =1.003606-.011259Oi.

where z=zO+ .l (p,+ipJ. tI=.45+.65i, S;=.45+.55i, from Table 7.9 S%‘w(~~)=$(.522246 &!w(s;) =2(.522246

Thus, with zo=.4+ .6i, p,k.4, pz=.l, we get

+.498591+.487556+.467521) = .493979

S, ((a+;) From 7.3.22, 7.3.8, 7.3.18 we have

Example

&(z)=--

Compute

18.

2’ 9

e@w [(l+i)

+.498591+.561252+.533157) = .528812

gw(z)=[l-(.4+.1)]{[1-(.4-.1)].522246 +(.4-.1).528812)+(.4+.1)X {[l-(.4-.1)] .493979+(.4-.1).498591}=.509789.

$1

-!$!

e-fz2w[(i-l)

Jz and making 7.1.12,

and Table

4).

$1.

use of 7.1.11,

7.9

The correct answer is .509756. Straightforward bivariate interpolation gives .509460. Example 14. Compute Yw(.39+.61i) using Taylor’s series. Let z=.39+.61i, zo=.4+.6i. From and using Table 7.9, we have

to 6D

----i 2

l---i e --2 cos&-isini)w(l+ii) 4 (

7.1.20, +!$ie2(

cosi+ising)w(i+$i)

w(z,)=.522246+.167880i w’(z,)=-.21634+.367383, #w”(zo)=-.215--.185i,

z-zo=(~-l+i)10-2 (z--~~)~=-2iXlO-~

~w(z)=.167880-.0021634--0036738 + .0000430= 15.

Example

From

7.1.11,

Compute

w(.4-1.3i).

7.1.12

.162086.

Example

=-.990734-.681619i. m Compute e--(1’4)L2-31cos (2t)dt s0

19.

using Table 7.9. Setting b=y+ix, 7.1.12 we find

m S

e -at-w

cos

c=O in 7.4.2 and using 7.1.3,

(2xt)(j&

2&L2w(y)

0

(a>O, 2, y real). Hence from Table Using

Tables

7.9, 4.4 and 4.6

w(.4-1.3i)=4.33342+8.042013.

7.9

m s

e-(1’4)tP-3r cos (2t)dt= 0

&$?w(2+3i)

= .231761.

ERROR

308

FUNCTION

AND FRESNEL

INTEGRALS

References Texts

[7.1] J. Boersma, Computation of Fresnel integrals, Math. Comp. 14, 380 (1960). 17.21 A. V. Boyd, Inequalities for Mills’ ratio, Rep. Statist. Appl. Res. Un. Jap. Sci. Engrs. 6, 44-46 (1959). [7.3] 0. Emersleben, Numerische Werte des FehIerintegrals fiir 6, Z. Angew. Math. Mech. 31, 393-394 (1951). [7.4] A. Erdelyi et al., Higher transcendental functions, vol. 2 (McGraw-Hill Book Co., Inc., New York, N.Y., Toronto, Canada, London, England, 1953). (7.51 A. Erdelyi et al., Tables of integral transforms, vol. 1 (McGraw-Hill Book Co., Inc., New York, N.Y., Toronto, Canada, London, England, 1954). [7.6] W. Gautschi, Note on bivariate linear interpolation for analytic functions, Math. Tables Aids Comp. 13, 91-96 (1959). [7.7] W. Gautschi, Recursive computation of the repeated integrals of t.he error function, Math. Comp. 15, 227-232 (1961). [7.8] W. Grobner and N. Hofreiter, Integraltafel (Springer-Verlag, Wien and Innsbruck, Austria, 1949-50). [7.9] D. R. Hartree, Some properties and applications of the repeated integrals of the error function, Mem. Proc. Manchester Lit. Philos. Sot. 80, 85-102 (1936). [7.10] C. Hastings, Jr., Approximations for digital computers (Princeton Univ. Press, Princeton, N.J., 1955). 17.111 Y. Komatu, Elementary inequalities for Mills’ ratio, Rep. Statist. Appl. Res. Un. Jap. Sci. Engrs. 4, 69-70 (1955-57). [7.12] E. Kreyssig, On the zeros of the Fresnel integrals, Canad. J. Math. 9, 118-131 (1957). [7.13] Th. Laible, Hohenkarte des Fehlerintegrals, Z. Angew. Math. Phys. 2, 484-486 (1951). [7.14] F. Losch and F. Schoblik, Die Fakultiit (B. G. Teubner, Leipzig, Germany, 1951). [7.15] F. Oberhettinger, Tabellen zur Fourier Transformation (Springer-Verlag, Berlin, Gottingen, Heidelberg, Germany, 1957). [7.16] J. R. Philip, The function inv erfc 8, Austral. J. Phys. 13, 13-20 (1960). [7.17] H. 0. Pollak, A remark on “Elementary inequalities for Mills’ ratio” by YQsaku Komatu, Rep. Statist. Appl. Res. Un. Jap. Sci. Engrs. 4, 110 (1955-57). [7.18] J. B. Rosser, Theory and application of *&cIz and

s,’

e-.1Jdy~’

JII

e-z’dx (Mapleton

House, Brooklyn,

N.Y., 1948). [7.19] H. E. Salzer, Formulas for calculating the error function of a complex variable, Math. Tables Aids Comp. 5, 67-70 (1951). [7.20] H. E. Salzer, Complex zeros of the error function, J. Franklin Inst. 266, 209-211 (1955). [7.21] F. G. Tricomi, Funzioni ipergeometriche confluenti (Edizioni Cremonese, Rome, Italy, 1954).

L7.221 G. N. Watson, A treatise on the theory of Bessel functions, 2d ed. (Cambridge Univ. Press, London, England, 1958). Tables

[7.23] M. Abramowitz,

Table of the integral

J

‘e-u3 du, J.

Math. Phys. 30, 162-163 (1951). :=0(.01)2.5, 8D. [7.24] P. C. Clemmow and Cara M. Munford, A table of &&r)e@ l” e-*irA2dX for complex values of p, Philos. Trans. Roy. Sot. London (A}, 245, 189211 (1952). \p\=O(.Ol).S, arg p=O”(10)450, 4D. [7.25] R. B. Dingle, Doreen Arndt and S. K. Roy, The integrals C,(X) = (P!)-~~~

ep(G+z?)-*e-de

D,(x)

eP(G+d)-*e-#de

and = (p!)-lJm

and their tabulation, Appl. Sci. Res. B 6, 155-164 (1956). C(z), S(z), z==O(1)20, 12D. [7.26] V. N. Faddeeva and N. M. Terent’ev, Tables of values of the function

u)(z) =e+

(l+zKeGdt)

for complex argument. Translated from the Russian by D. G. Fry (Pergamon Press, New York, N.Y., 1961). m(z),z=x+iy; z,y=O(.O2)3; 2=3(.1)5, y=O(.1)3; 2=0(.1)5, y=3(.1)5; 6D. [7.27] B. D. Fried and S. D. Conte, The plasma dispersion function (Academic Press, New York, N.Y. and London, England, 1961). i&(z), iJGd(z), z=x+iy; 2=0(.1)9.9, y=-9.1(.1)10; z=var. (.1)9.9, y= -lO(.l) -9.2; 6s. ’ &fx v [7.28] K. A. Karpov, Tablitsy funktsii w(z) =e+ s kompleksnoi oblasti (Izdat. Akad. NaukO SSSR., Moscow, U.S.S.R., 1954). z=x; 2=0(.001)2(.01) 10; 5D; z=pei@; e=2.5°(2.50)300(1.250)350(.6250)400; p=ps(.OOl)p; (.Ol)p;’ (.0002)5, 0 Ipe
J

v kompleksnoi oblasti (Izdat. Akad. Nauk SSSR., Moscow, U.S.S.R., 1958). z=peid; 8=45o(.3125o)48.75o(.625o)55o(1.25o)65o(2.5o)9Oo, p=pe(.OOl)pi(.Ol)p;‘, 0
y=e-23

ERROR

[7.32] W.

FUNCTION

AND

Lash Miller and A. R. Gordon, Numerical evaluation of infinite series and integrals which arise in certain problems of linear heat flow, electrochemical diffusion, etc., J. Phys. Chem. 35, 2785-2884 (1931). F(z) =e-z’

(

eL2dt; s=O(.O1)1.99,

6D;

z=2(.01)4(.05)7.5(.1)10(.2)12, 8s. [7.33] National Bureau of Standards, Tables of the error function and its derivative, Applied Math. Series 41, 2d ed. (U.S. Government Printing Office, Washington, D.C., 1954). 15D; (2/Jr)e-z*, erf 2, s=0(.0001)1(.001)5.6, (2/Jr)e-z2,

erfc 2, 2=4(.01)10,

85.

FRESNEL

309

INTEGRALS

[7.34] T. Pearcey, Table of the Fresnel integral (Cembridge Univ. Press, London, England, 1956). c(g),

S (E),

2=0(.01)50,

6-7D.

[7.35] Tablitsy integralov Frenelya (Izdat. Akad. Nauk SSSR., Moscow, U.S.S.R., 1953). C(z), S(Z), Z= 0(.001)25, 7D; S(z), r=O(.OOl) .58, 7s; C(z), 2=0(.001) .lOl, 7s. [7.36] A. van Wijngaarden and Fresnel integrals, Verh. Afd. Natuurk. Sec. I, C(z), S(Z), 2=0(.01)20, numerical values of the asymptotic expansions.)

W. L. Scheen, Table of Nederl. Akad. Wetensch., 19, No. 4, l-26 (1949). 5D. (Also contains coefficients in Taylor and

ERROR FUNCTION

310 Table

ERROR

7.1

AND FRESNEL

FUNCTION

INTEGRALS

AND ITS DERIVATIVE 2 (d-2

0. 00 0. 01 0. 02 0.03 0.04

~2 \R 1.12837 1.12826 1.12792 1.12736 1.12657

0.05 0. 06 0.07 0. 08 0. 09

1.12556 1.12432 1.12286 1.12118 1.11927

17424 43052 36333 06004 62126

0.05637 19778 0.06762 15944 0.07885 77198 0.09007 81258 0.10128 05939

0.55 0. 56 0.57 0. 58 0. 59

0.83383 0.82463 0.81536 0.80604 0.79666

66473 22395 63461 33431 75911

0.56332 0.57161 0.57981 0.58792 0.59593

33663 57638 58062 29004 64972

0.10 0. 11 0. 12 0. 13 0.14

1.11715 1.11480 1.11224 1.10946 1.10647

16068 80500 69379 97934 82654

0.11246 0.12362 0.13475 0.14586 0.15694

29160 28962 83518 71148 70331

0. 60 0. 61 0. 62 0. 63 0.64

0.78724 0.77777 0.76826 0.75872 0.74914

34317 51846 71442 35764 87161

0.60385 0.61168 0.61941 0.62704 0.63458

60908 12189 14619 64433 58291

0. 15 0.16 0.17 0.18 0.19

1.10327 1.09985 1.09623 1.09240 1.08837

41267 92726 57192 56008 11683

0.16799 59714 0.17901 18132 0.18999 24612 0.20093 58390 0.21183 98922

0. 65 0.66 0.67 0.68 0. 69

0.73954 0.72992 0.72027 0.71061 0.70095

67634 18814 81930 97784 06721

0.64202 0.64937 0.65662 0.66378 0.67084

93274 66880 77023 22027 00622

0.20 0.21 0.22 0.23 0.24

1.08413 1.07969 1.07506 1.07023 1.06522

47871 89342 61963 92672 09449

0.22270 25892 0.23352 19230 0.24429 59116 0.25502 25996 0.26570 00590

0.70 0.71 0.72 0. 73 0.74

0.69127 48604 0.68159 62792 0.67191 88112 0.66224 62838 0.65258 24665

0.67780 0.68466 0.69143 0.69810 0.70467

11938 55502 31231 39429 80779

0.25 0.26 0. 27 0.28 0.29

1.06001 1.05462 1.04904 1.04329 1.03736

41294 18194 71098 31885 33334

0.27632 63902 0.28689 97232 0.29741 82185 0.30788 00680 0.31828 34959

0.75 0.76 0.77 0.78 0. 79

0.64293 10692 0.63329 57399 0.62368 00626 0.61408 75556 0.60452 16696

0.71115 0.71753 0.72382 0.73001 0.73610

56337 67528 16140 04313 34538

0.30 0.31 0.32 0. 33 0. 34

1.03126 1.02498 1.01855 1.01195 1.00519

09096 93657 22310 31119 56887

0.32862 67595 0.33890 81503 0.34912 59948 0.35927 86550 0.36936 45293

0. 80 0. 81 0.82 0.83 0.84

0.59498 57863 0.58548 32161 0.57601 71973 0.56659 08944 3.55720 73967

0.74210 0.74800 0.75381 0.75952 0.76514

09647 32806 07509 37569 27115

0.35 0. 36 0. 37 0. 38 0. 39

0.99828 0.99122 0.98401 0.97665 0.96916

37121 10001 14337 89542 75592

0.37938 20536 0.38932 97011 0.39920 59840 0.40900 94534 0.41873 87001

0. 85 0.86 0.87 0. 88 0. 89

0.54786 97173 0.53858 07918 0.52934 34773 0.52016 05514 0.51103 47116

0.77066 0.77610 0.78143 0.78668 0.79184

80576 02683 98455 73192 32468

0.40 0.41 0.42 0. 43 0.44

0.96154 12988 0.95378 42727 0.94590 06256 0.93789 45443 0.92977 02537

0.42839 23550 0.43796 90902 0.44746 76184 0.45688 66945 0.46622 51153

0. 90 0. 91 0.92 0.93 0.94

0.50196 85742 0.49296 46742 0.48402 54639 0.47515 33132 0.46635 05090

0.79690 0.80188 0.80676 0.81156 0.81627

82124 28258 77215 35586 10190

0.45 0.46 0. 47 0.48 0.49

0.92153 20130 0.91318 41122 0.90473 08685 0.89617 66223 0.88752 57337

0.47548 17198 0.48465 53900 0.49374 50509 0.50274 96707 0.51166 82612

0.95 0.96 0.97 0. 98 0.99

0.45761 0.44896 0.44037 0.43187 0.42345

92546 16700 97913 55710 08779

0.82089 0.82542 0.82987 0.83423 0.83850

08073 36496 02930 15043 80696

0.50

0.87878 25789 C-5513

0.52049

1.00

0.41510 74974

0.84270

.I’

p-12 91671 63348 79057 40827 52040

erf 0.00000 0.01128 0.02256 0.03384 0.04511

2 00000 34156 45747 12223 11061

0.50 0. 51 0. 52 0. 53 0.54

0.87\8758 0.86995 0.86103 0.85204 0.84297

25789 15467 70343 34444 51813

erf 0.52049 0.52924 0.53789 0.54646 0.55493

.I: 98778 36198 86305 40969 92505

[ 1

98778

.I

c(-y1

See Exa~nple 1. $=

0.88622 69255

[(-y1

07929

ERROR

FUNCTION

ERROR

AND

FRESNEL

FUNCTION

G2 c--r2

AND

erf 7

311

INTEGRALS

Table

ITS DERIVATIVE

d

-2 p-2 ,G

7.1

erf .T

.l.OO 1. 01 1.02 1.03 1. 04

0.41510 0.40684 0.39867 0.39058 0.38257

74974 71315 13992 18368 98986

0.84270 0.84681 0.85083 0.85478 0.85864

07929 04962 80177 42115 99465

1.50 1.51 1.52 1.53 1. 54

0.11893 02892 0.11540 38270 0.11195 95356 0.10859 63195 0.10531 30683

0.96610 0.96727 0.96841 0196951 0.97058

1. 05 1. 06 1.07 1.08 1. 09

0.37466 0.36684 0.35911 0.35147 0.34392

69570 43034 31488 46245 97827

0.86243 61061 0.86614 35866 0.86977 32972 0.87332 61584 0.87680 31019

1.55 1. 56 1.57 1.58 1.59

0.10210 0.09898 0.09593 0.09295 0.09005

0.97162 27333 0.97262 81220 0.97360 26275 0.97454 70093 0.97546 20158

1.10 1.11 1.12 1.13 1.14

0.33647 0.32912 0.32186 0.31470 0.30764

95978 49667 67103 55742 22299

0.88020 0.88353 0.88678 0.88997 0.89308

50696 30124 78902 06704 23276

1.60 1.61 1.62 1.63 1.64

0.08722 90586 0.08447 34697 0.08178 85711 0.07917 31730 0.07662 60821

0.97634 0.97720 0.97803 0.97884 0.97962

1. 15 1.16 1.17 1. 18 1.19

0.30067 0.29381 0.28704 0.28037 0.27381

72759 12389 45748 76702 08437

0.89612 0.89909 0.90200 0.90483 0.90760

38429 62029 03990 74269 82860

1.65 1. 66 1.67 1.68 1.69

0.07414 0.07173 0.06938 0.06709 0.06487

61034 20405 26972 68781 33895

0.98037 55850 0.98110 49213 0.98181 04416 0.98249 27870 0.98315 25869

1.20 1.21 1.22 1.23 1.24

0.26734 0.26097 0.25471 0.24854 0.24248

43470 83664 30243 83805 44335

0.91031 0.91295 0.91553 0.91805 0.92050

39782 55080 38810 01041 51843

1.70 1.71 1.72 1.73 1.74

0.06271 0.06060 0.05856 0.05657 0.05464

10405 86436 50157 89788 93607

0.98379 0.98440 0.98500 0.98557 0.98613

04586 70075 28274 84998 45950

1.25 1.26 1.27 1.28 1.29

0.23652 11224 0.23065 83281 0.22489 58748 Oi21923 35317 0.21367 10145

0.92290 0.92523 0.92751 0.92973 0.93189

01283 59418 36293 41930 86327

1.75 1.76 1.77 1.78 1.79

0.05277 0.05095 0.04918 0.04747 0:04580

49959 47262 74012 18791 70274

0.98667 0.98719 0.98769 0198817 0.98864

16712 02752 09422 41959 05487

1. 30 1. 31 1.32 1. 33 1.34

0.20820 0.20284 0.19757 0.19241 0.18734

79868 40621 88048 17326 23172

0.93400 0.93606 0.93806 0.94001 0.94191

79449 31228 51551 50262 37153

1.80 1.81 1. 82 1.83 1. 84

0.04419 17233 0.04262 48543 0.04110 53185 0.03963 20255 0.03820 38966

0.98909 0.98952 0.98994 0.99034 0.99073

05016 45446 31565 68051 59476

1. 35 1.36 1.37 1.38 1. 39

0.18236 0.17749 0.17271 0.16802 0.16343

99865 41262 40811 91568 86216

0.94376 21961 0.94556 14366 0.94731 23980 0.94901 60353 0.95067 32958

1.85 1. 86 1.87 1. 88 1.89

0.03681 0.03547 0.03417 0.03292 0.03170

98653 88774 98920 18811 38307

0.99111 10301 0.99147 24883 0.99182 07476 0.99215 62228 0.99247 93184

1.40 1. 41 1. 42 1.43 1.44

0.15894 0.15453 0.15022 0.14600 0.14187

17077 76130 55027 45107 37413

0.95228 0.95385 0.95537 0.95685 0.95829

51198 24394 61786 72531 65696

1.90 1.91 1.92 1.93 1.94

0.03052 0.02938 0.02827 0.02721 0.02617

47404 36241 95101 14412 84752

0.99279 0.99308 0.99337 0.99365 0.99392

04292 99398 82251 56502 25709

1.45 1.46 1.47 1.48 1.49

0.13783 0.13387 0.13001 0.12623 Oil2254

22708 91486 33993 40239 00011

0.95969 0.96105 0.96237 Oi96365 0.96489

50256 35095 28999 40654 78648

1.95 1. 96 1.97 1.98 1.99

0.02517 96849 0.02421 41583 0.02328 09986 0.02237 93244 0.02150 82701

0.99417 0.99442 0.99466 0:99489 0.99511

93336 62755 37246 20004 14132

1.50

0.11893

[(-;)l1

0.96610

51465

2.00

0.02066

0.99532

22650

02892

[

c-y

I

86576 19506 17995 70461 65239

69854

51465 67481 34969 62041 56899

83833 68366 80884 28397 17795

c 1 C-5614

I-

s = 0.88622 69255

312

ERROR

Table

7.2

DERIVATIVE

2 F-

z

FUNCTION

OF THE

FRESNEL

ERROR

INTEGRALS

FUNCTION

0

P-J2

4

X

- 2)2.0666

r-i

X

Y=

+f

2.00 2.01 2.02 2.03 2.04

AND

985

2.50 2.51 2.52 2.53 2.54

2.05 2.06 2;07 2.08 2.09

2.55 2.56 2.57 2.58 2.59

2;13 2.14

2.60 2.61 2.62 2.63 2.64

3.00 3.01 3.02 3.03 3.04 I-

3 3 3 3I 3

1.6922 1.6079 1.5275 1.4508 1.3777

II

136 137 078 325 304

- 3 1.2416 1.1783 1.3080 455 1.0607 1.1181 075 500 090 764

I- 411.2345 698

3.50 3.51 3.52 3.53 3.54

3.05 3.06 3.07 3.08 3.09

3.55 3.56 3.57 3.58 3.59

3.10 3.11 3.12 3.13 3.14

3.60 3.61 3.62 3.63 3.64

2.15 2.16

2.65 2.66 2.67 2.68 2769

3.15 3.16 3.17 3.18 3.19

3.65 3.66 3.67 3.68 3.69

2.20 2.21 2.22 2.23 2.24

2.70 2.71 2.72 2.73 2.74

3.20 3.21 3.22 3.23 3.24

3:72 3.73 3.74

2.25 2.26 2.27 2.28 2.29

2.75 2.76 2.77 2.78 2.79

3.25 3.25 3.27 3.28 3.29

3.76 3.77 3.78 3.79

2.80 2.81 2.82 2.83 2.84

3.30 3.31 3.32 3.33 3.34

3.80 3.81 3.82 3.83 3.84

(-

3)6.5249

776

2.30 2.31 2.32 2.33 2.34 2.35 2.36 2.37 2.38 2.39 $4; 2:42 2.43 2.44

- 3)4.5088

- 3)3.7301 (- I-

3 3.5556 3.3886 3 3.2288 3 3.0760 3 I 2.9298

292

092

2.85 2.86 2.87 2.88 2.89

487 700 871 230 098

2.90 2.91 2.92 2.93 2.94

- 5)X8428 I - 5)1.7242 (- 5)1.6130

397 768 192

3.85 3.86 3.87 3.88 3.89

3.37 3.38 3.39

3.90 3.91 3.92 3;93 3.94 3.45 3.46 3.47 3.48 3.49

2.50

(-

3)2.1782

842

3.00

(-

4)1.3925

305

F~O.88622

3.50 692%

3.95 3.96 3.97 3.98 3.99 (-

6)5.3994

268

4.00

(-

6)1.1028

II

445

- 77)4.1221 3.0245 3.2689 624 3.5324 3.8162 971 796 013 867

(-

7)2.7979

(-

7 1.8896 240

I

245

- 7 I 1.6128 1.3754 1.4895 1.7459 557 458 098 135

(-

7)1.2698

235

ERROR

FUNCTION

AND

DERIVATIVE

OF THE

FRESNEL

ERROR

FUNCTION ;=2 e--Z

0

‘5 &

X

5.00 5.01 5.02 5.03 5.04

4.55 4.56 4.57 4.58 4.59

5.05 5.06 5.07 5.08 5.09

5.55 5.56 5.57 5.58 5.59

5.10 5.11 5.12 5.13 5.14

5.60 5.61 5.62 5.63 5.64

8)3.7395

5.15 5.16 5.17 5.18 5.19

5.65 5.66 5767 5.68 5.69

-10)7.2909 -10)6.6494

450 435

I

414

-11 I -11 I -11 -11 -11

4.20 (- 8 2.4632 041 4.21 4.22 4.23 I - 8 I 2.0814 1.9127 484 1.7574 2.2645 901 463 204 4724

4.70 4I71 4.72 4.73 4.74

(-lOj2.8766

694

4.25 4.26 4.27 4.28 4.29

4.75 t-lOj1.7934 4176 4.77 4.78 4.79

357

5.25 5.26 5.27 5:28 5.29

4.80 4.81 4.82 4.83 4.84

-10 -10 I -11 I -11 -11

1.1125 1.0105 9.1780 8.3337 7.5656

261 888 821 894 500

5.30 5.31 5.32 5.33 5.34

I -13 I -13 -13 -13 -13

4.35 4.36 4.37 4.38 4.39

4.85 4.86 4.87 4.88 4.89

I -11 -11 -11 -11 -11

I 6.8669 6.2315 5.6537 5.1285 4.6511

377 074 456 259 675

5.35 5.36 5.37 5.38 5.39

(-13 -13 -13 I -13 -13

4.40 4.41 4.42 4.43 4.44

4.90 4.91 4.92 4.93 4.94

5.40 5.41 5.42 5.43 5.44

4.45 4.46 4.47 4.48 4.49

4.95 4.96 4.97 4.98 4.99

5.45 5.46 5.47 5.48 5.49

4.30 4.31 4.32 4.33 4.34

4.50

I - 8I 1.6143 1.1472 994 1.2498 1.3614 1.4826 445 673 993 974 I - 98 I 9.6595 7.4517 598 8.1266 1.0528 8.8608 977 102 438 442

(- 9)1.8113

059

5.00

(-11)1.5670

866 IT-O.88622

:;670 1:4178 1.2825 1.1598 1.0487

Table 7.2 -2 e--z2 J;

x

4.50 4.51 4.52 4.53 4.54

4.60 4.61 4.62 4.63 4.64 4.15 (4.16 4.17 4.18 4.19

.t:

Q--f

313

INTEGRALS

866 169 089 820 702

5.50 5.51 5.52 5.53 5.54

I -14)8.2233 -14 I 7.3659 5.2876 6.5967 160 5.9066 480 265 906 187

I

-14 I 4.7325 4.2349 3.3891 917 3.0309 3.7888 943 422 310 585

5.70 5.71 5.72 5.73 5.74

5.50

5.75 5.76 5.77 5.78 5.79 7.1305 6.4127 5.7660 5.1835 4.6589

505 516 568 412 423

5.80 5.81 5.82 5.83 5.84

I 4.1865 3.7613 3.3786 3.0343 2.7245

979 895 913 233 096

5.85 5.86 5.87 5.88 5.89

t-1312.4458 -13 2.1952 I -13 1.9699 -13 1.7673 -13 I 1.5853

396 336 112 627 234

5.90 5.91 5.92 5.93 5.94

t-1511.5481

468

-i5 liO888 898 i -16 \ 9.6798 241

5.95 5.96 5.97 5.98 5.99 (-14)8.2233

69255

160

6.00

(-16)2.6173

012

314 Table

ERROR

FUNCTION

AND

DERIVATIVE

7.2

2 p-2

FRESNEL

OF THE

INTEGRALS

ERROR

FUNCTION

2

AL2

6.00 6.01 6.02 6.03 6.04

-16)2%73 -16 2.3211 -16 1 2.0580 -16)1.8243 -16)1.6169

012 058 187 864 533

6.50 6.51 6.52 6.53 6.54

4 e-z’ q?r -19)5.0525 -19 4.4362 -19 3.8942 -19 I 3.4178 -19)2.9990

6.05 6.06 6.07 6.08 6.09

-16)1.4328 -16 1.2693 -16 1 1.1243 -17)9.9575 -17)8.8165

188 992 934 277 340

6.55 6.56 6.57 6.58 6.59

-19)2.6310 -19 2.3078 -19 2.0238 -19 1.7744 -19 I 1.5555

6.10 6.11 6.12 6.13 6.14

-17)7.8047 -17 6.9076 -17 1 6.1124

211 453 570

6.60 6.61 6.62 6.63 6.64

7.10 7.11 7.12 7.13

7.60 7.61 7.62 7.63 7.64

6.15 6.16 6.17 6.18 6.19

6.65 6.66 6.67 6.68 6.69

7.15 7.16 7.17 7.18 7.19

7.65 7.66 7.67 7.68 7.69

6.20 6.21 6.22 6.23 6.24

6.70 6.71 6.72 6.73 6.74

7.20 7.21 7.22 7.23 7.24

7.70

6.75 6.76 6.77 6.78 6.79

7.25 7.26 7.27 7.28 7.29

7.75 7.76 7.77 7.78 7.79

6.80 6.81 6.82 6.83 6.84

7.30 7.31 7.32 7.33 7.34

6.35 6.36 6.37 6.38 6.39

6.85 6.86 6.87 6.88 6.89

7.35 7.36 7.37 7.38 7.39

7.85 7.86 7.87 7.88 7.89

6.40 6.41 6.42 6.43 6.44

6.90 6.91 6.92 6.93 6.94

t-21)2.3751

704

7.40 7.41 7.42 7.43 7.44

7.90 7.91 7.92 7.93 7.94

(-21)1.1883

540

(-22 (-22

565 967

7.45 7.46 7.47 7.48 7.49

(-22)5.9159

630

7.50

2

6.25 6.26 6.27 6.28 6.29 6.30 6.31 6.32 6.33 6.34

iI -17 -17 -18 -18 -18

1.2241 1.0801 9.5297 8.4057 7.4128

x

281 812 064 325 421

(-18)1.0982

455

6.45 6.46 6.47 6.48 6.49

(-19)9.6542

574

(-19)5.7534

461

6.95 6.96 6.97 6.98 6.99

6.50

(-19)5.0525

800

7.00

7:8244 6.8042

L

800 038 418 066 603

7.00 7.01 7.02 7103 7.04

921 100 447 651 031

7.05 7.06 7.07 7.08 7.09

T-O.88622

69255

II

t-22)2.9304 -22 1.9179 1.6645 2.2094 2.5448

l-23)3.9913

(-25)4.2013

450 736 491 450 057

893

654

7.55 7.56 7.57 7i58 7.59

t-25)1.9796 -251112573 -25)1.0803

292 541 765

t-26)2.0097

185

7.95 7.96 7.97 7.98 7.99

(-28)4.0175 -28 3.4265 -28 2.9219 -28 2.4912 I -28 I 2.1234

202 874 899 008 982

8.00

(-28)1.8097

068

ERROR

FUNCTION

DERIVATIVE

AND

OF THE

FRESNEL

315

INTEGRALS

ERROR

FUNCTION

Table

7.2

z 8.00 8.01 8.02 8.03 8.04

9.00 9.01 9.02 9.03 9.04

I -36)7.4920 -36 I 6.2572 -36 5.2249 -36 4.3620 -36 3.6409

734 800 519 651 535

9.50 9.51 9.52 9.53 9.54

-32 1.2057 541 -32 1.0155 245

9.05 9.06 9.07 9108 9.09

-36)3.0384 -36j2.5351 -36j2.1147 -36j1.7637 -36)1.4707

441 317 690 559 105

9.55 9.56 9.57 9.58 9.59

8:63 8.64

-33)510997 -33)4.2908

438 734

9.10 9.11 9.12 9.13 9.14

-36)1.2261 -36 1.0219 -37 1 8.5167 -37)7.0959 -37)5.9110

088 837 148 960 925

8.15 8.16 8.17 8.18 8.19

8.65 8.66 8.67 8.68 8.69

-33)3.6095 -33 3.0358 -33 2.5527 -33 2.1461 -33 I 1.8039

760 465 988 817 709

9.15 9.16 9.17 9.18 9.19

-37 4.9230 -37 14.0993 -37)3.4127 -37)2.8406 -37)2.3639

8.20 8.21 8.22 8.23 8.24

8.70 8.71 8.72 8.73 8.74

-33)1.5160 -3311.2737 -33'1.0700 -34 8.9869 ( -34 i 7.5464

228 818 339 668 360

9.20 9.21 9.22 9.23 9.24

-37)1.9668 -37)1.6361 -37 1.3607 -37 1.1314 -38 I 9.4066

8.25 8.26 8.27 8.28 8.29

8.75 8.76 8.77 8.78 8.79

-34)6.3355 -34j5.3178 -34j4.4627 -34)3.7444 -34)3.1411

422 836 957 525 074

9.25 9.26 9.27 9.28 9.29

8.30 8.31 8.32 8.33 8.34

8.80 8.81 8.82 8.83 8.84

8.05 8.06 8.07 8.08 8.09 8.10 8.11 8.12 8.13 8.14

8.50 8.51

t-29)8.1112

334

8.55

(-29)4.2531

077

8i58 8.59

(-29j3.6173

797

(-29j1.8891

933

(-32)2.0157

780

9.30 9.31 9.32

8.35 8.36 8.37 8.38 8.39 8.40 8.41 8.42 8.43 8.44 8.45 8.46 8.47 8.48 8.49 8.50

-31)2.5623 -31 2.1658 -31 1.8303 -31 I 1.5465 -31)1.3064

I( I

(-32)4.7280

380 657 736 399 586

139

-40)1.2918

638

9.60 9.61 9.62 9.63 9.64

-40)1.0662

907

619 592 918 437 423

9.65 9.66 9.67 9.68 9.69

(-41)4.0725

570

i-41)1.8788

710

449 251 427 847 395

9.70 9.71 9.72 9.73 9.74

-41)1.5477

017

-38)7.8186 802 -38 6.4974 888 -38 1 5.3984 710

9.75 9.76 9.77 9.78 9.79

-42)5.8524 -42 4.8150 -42 3.9608 I -42 I 3.2574 (-42)2.6784

252 968 401 873 979

(-43)8.2436

338

(-43)3.7428

271

-38j1.7684 -38)1.4672

718 880

9.80 9.81 9.82 9.83 9.84

9.35 9.36 9I37 9.38 9.39

-38 1.2171 -38 1.0094 -39 8.3703 -39 I 6.9392 -39)5.7517

545 602 932 997 311

9.85 9.86 9.87 9.88 9.89

-39 4.7664 -39 3.9491 -39 3.2713 -39 I 2.7093 -39)2.2434

456 520 439 286 186

9.90 9.91 9.92 9.93 9.94

-43)3.0708 -43 2.5189 -43 2.0658 -43 1.6939 -43 I 1.3886

096 477 489 130 628

8.90 8.91 8.92 8.93 8.94

-35)4.4873 -35 3.7552 -35 3.1420 -35 2.6283 -35 I 2.1982

418 711 030 611 476

9.40 9.41 9.42 9.43 9.44

8.95

-3511.8381

516

(-39)1.8572

574

922

-3511.0734 -36)8.9687

315 435

9.95 9.96 9.97 9.98 9.99

-43)1.1381

8;98 8.99

9.45 9.46 9.47 9:48 9.49

734

9.50

(-40)7.2007

555

10.00

(-44)4.1976

562

9.00

(-36)7.4920

$f=O.88622

69255

316

ERROR

Table 0.250 0.245 0.240 0.235 0.230

xez’ erfc x 0.51079 14 0.51.163 07 0.51247 67 0.51332 94 0.51418 90

0.225 0.220 0.215 0.210 0.205

0.51505 0.51592 0.51681 0.51769 0.51859

55 92 01 83 40

0.200 0.195 0.190

0.185 0.180

0.51949 0.52040 0.52132 0.52225 0.52318

74 85 75 45 98

0.175 0.170 0.165 0.160 0.155

0.52413 0.52508 0.52604 0.52701 0.52799

33 55 63 59 46

0.150 0.145 0.140 0.135 0.130

0.52898 0.52997 0.53098 0.53200 0.53303

25 98 67 35 02

0.125

0.53406 72 C-i)1

See Example

AND

COMPLEMENTARY

7.3

x-2

FUNCTION

2.

<x> 22

FRESNEL

INTEGRALS

ERROR

FUNCTION 2

0";25

2 :

0.100 0.095 0.090 0.085 0.080

0.53941 0.54051 0.54163 0.54276 0.54390

41 76 32 11 16

0.075 0.070 0.065 0.060 0.055

0.54505 0.54622 0.54740 0.54859 0.54980

51 19 24 69 58

3 3

0.050 0.045 0.040 0.035 0.030

0.55102 0.55226 0.55352 0.55479 0.55608

95 85 32 41 17

;

0.025 0.020

0.55738 0.55870 0.56005 0.56140 0.56278

65 90 00 99 96

22 t

2

2 :

3 3 3 3

[ 1

xez2erfc x 0.53406 72 0.53511 47 0.53617 29 0.53724 20 0.53832 23

0.015 0.010 0.005 0.000

<x> 33 33

0.56418 96 C-i)3 II 1

<x> =nearest integer to x.

n

erfc &G

: i

0.00039 88821 0.01218 27505 84803 88282 0.00000 0.00001 41444 05351 64662 02689

5

0.00000 00208 26552

n

6 87 9 10

0.00000 0.00000 0.00000 0.00000 0.00000

erfc Jn?r 00008 25422 00000 33136 00000 01343 00000 00055 00000 00002

erfc x =72 me-tzdt =1 -erf x dr Sz erfc 4% compiled from 0. Emersleben, Numerische Werte des Fehlerintegrals fiir &, Z. Angew. Math. Mech. 31,393-394, 1951 (with permissi&).

3

6 7 1: 14 m

ERROR

REPEATED

FUNCTION

AND

FRESNEL

INTEGRALS

OF THE ERROR FUNCTION

2°F l+l

(

o.“o 0.1 0.2 0. 3 0.4

317

INTEGRALS

)

i” erfc .t* 71=3

n=l

n=2

1.00000

1.00000

Table 7.4

n=4

1.00000 I - 1I 4.28565 7.62409 3.15756 5.74882

1.00000

- 1)5.56938 I - 1)4.46884 (- 1 2.29846 (- 1 1.65244 (- 2)5:68138

1. 0 :-: 1:3 1.4 1. 5

(- 3j8.02626

:*; 1: 8 1. 9

(- 3j1.26566

;:1" 9; 2: 4

(- 4j2.22250 - 6)9.52500

92 2: 7 2. 8 2.9

22 33.87 3:9

I- 9 8I 4.58945 8)5.10148 1.98190 1.04329 2.32831

4. 0 4.1

II

i-f 4: 4

-13 -14 4.02809 -12 2.35705 8.76348 1.14567 3.19826

t-6' 4: 7 4.8 4.9 5. 0

(-14)5.61169

(-13)2.62561

(-1)5.64189

See Examples 4 and 5.

58355

(-1)2.50000

(-14)1.38998

(-15)3.83592

pr (;+I)]-’ 00000

(-2)9.40315

97258

(-2)3.12500

318

ERROR

Table

7.4

FUNCTION

REPEATED

INTEGRALS

1.00000 n=5

2

0”::

AND

FRESNEL

OF THE

0:4

ERROR

FUYCTION

237 ;+I i" erfc 5 ( 1 n=6

11 =I0

1.00000

1.00000 - 1 6.28971 - 1 3.91490 - lj2:41089 1)1.46861

I I

I

i*%

INTEGRALS

I-

I- 111.65569

if: 0: 7

(-

I

i:!

2)7.95749 2)4.64127 2)2.67626 2)1.52533 3)8.59126

1. 0 1'::. 2: (- 3)1.19278

1.5

(- 4)2.89186

::;

I-

- 5j4;04407 5)2.04244

1':: 22'10 2: 2 $1'4 f:'6 ;*i 2: 9 ;:1" E 3:4 - 9)2.47236 3:: 2; 3:9

-~0)1.84200 -11)7.48503

(-11)7.30331 (-11)2.91245

4. 0 4.1 2.3' 4: 4

I

22 4:7 4. a 4. 9 5. 0

I-13)1.78294 -1416.31544 -14j2.20038

-13)3.82601 -13)1.37691 -14j4: 87328

-15 16.71719 -17 -16 2.11065 6.46126 2.10125 1.95316

(-15)1.15173

(-16)3.70336

(-18)6.51829 pr

t-3)9.40315

97258

(-3)2.60416

66667

(-18)2.61062

(;+1)]-l

(-6)8.13802

08333

(-6)

1.69609

66316

ERROR FUNCTION

AND FRESNEL

DAWSON’S 2:

0.00

0.00000

0.02 0.04 0.06 0.08

0.01999 46675 0.03995 73606 0.05985 62071 0.07965 95389

1.00 1. 02 1.04 1.06 1.08

0.10 0.12 0.14 0.16 0.18

0.09933 59924 fl.11885 46083 0.13818 49287 0.15729 70920 0.17616 19254

0.20 0.22 0.24 0.26 0.28

0.19475 0.21303 0.23099 0.24859 0.26581

319

INTEGRALS Table

7.5

xe-= s zet2c/t

<x>

INTEGRAL

e-2.2 s0=rt;lt ’ 0.53807 95069

x-2

44359 71471 50787 57454

0.250 0.245 0.240 0.235 0.230

0.6026080777 0.60046 6027 0.59819 8606 0.59588 1008 0.59351 6018

1.10 1.12 1.14 1.16 1.18

0.52620 66800 0.52291 53777 0.51935 92435 0.51555 55409 0.51152 13448

0.225 0.220 0.215 0.210 0.205

0.59110 0.58865 0.58616 0.58364 0.58109

6724 6517 9107 8516 9080

10334 68833 28865 34747 41727

1.20 1.22 1.24 1.26 1.28

0.50727 0.50282 0.49820 0.49341 0.48847

34964 85611 27897 20827 19572

0.200 0.195 0.190 0.185 0.180

0.57852 0.57593 0.57332 0.57071 0.56809

5444 2550 5618 0126 1778

0.30 0.32 0.34 0.36 0.38

0.28263 16650 0.29902 38575 0.31496 99336 0.33045 04051 0.34544 71562

1.30 1.32 1.34 1.36 1.38

0.48339 0.47820 0.47290 0.46751 0.46204

75174 34278 38898 26208 28368

0.175 0.170 0.165 0.160 0.155

0.56547 0.56287 0.56027 0.55770 0.55516

6462 0205 9114 9305 6829

0.40 0.42 0.44 0.46 0.48

0.35994 34819 Oii~739241210 0.38737 52812 0.40028 46599 0.41264 14572

1.40 1.42 1.44 1.46 1.48

0.45650 0.45091 0.44528 0.43962 0.43394

72375 79943 67410 45670 20135

0.150 0.145 0.140 0.135 0.130

0.55265 0.55018 0.54776 0.54538 0.54305

7582 7208 0994 3766 9774

0.50 0.52 0.54 0. 56 0.58

0.42443 63835 0.43566 16609 0.44631 10184 0.45637 96813 0.46586 43551

1.50 1.52 1.54 1.56 1.58

0.42824 0.42255 0.41686 0.41119 0.40555

90711 51804 92347 95842 40424

0.125 0.120 0.115 0.110 0.105

0.54079 0.53858 0.53643 0.53435 0.53233

2591 5013 8983 5529 4747

0. 60 0.62 0.64 0.66 0.68

0.47476 32037 0.48307 58219 0.49080 32040 0.49794 77064 0.50451 30066

1.60 1.62 1.64 1.66 1.68

0.39993 0.39436 0.38883 0.38335 0.37792

98943 39058 23346 09429 50103

0.100 0.095 0.090 0.085 0.080

0.53037 0.52847 0.52663 0.52484 0.52311

5810 7031 5967 9575 4393

0.70 0.72 0.74 0.76 0.78

0.51050 40576 0.51592 70382 0.52078 93010 0.52509 93152 0.52886 66089

1.70 1.72 1.74 1.76 1.78

0.37255 93490 Oi36725 83182 0.36202 58410 0.35686 54206 0.35178 01580

0.075 0.070 0.065 0.060 0.055

0.52142 0.51978 0.51817 0.51661 0.51508

6749 2972 9571 3369 1573

0.80 0.82 0.84 0.86 0.88

0.53210 0.53481 0.53702 0.53873 0.53996

17071 60684 20202 26921 19480

1.80 1.82 1.84 1.86 1.88

0.34677 0.34184 0.33700 0.33223 0.32756

27691 56029 06597 96091 38080

0.050 0.045 0.040 0.035 0.030

0.51358 0.51211 0.51067 0.50925 0.50786

1788 1971 0372 5466 5903

0.90 0.92 0.94 0.96 0.98

0.54072 0.54103 0.54090 0.54036 0.53941

43187 49328 94485 39857 50580

1.90 1.92 1.94 1.96 1.98

0.32297 43193 0.31847 19293 0.31405 71655 0.30973 03141 0.30549 14372

0.025 0.020 0.015 0.010 0.005

0.50650 0473 0.50515 8078 0.50383 7717 0.50253 8471 0.50125 9494

0.53807 95069

2.00

0.30134 03889 C-l)4

0.000

0.50000 0000

See Example

00000

3.

Compiled from J. B. Rosw,

0.53637 0.53431 0.53192 0.52921

[ 1 <x> =nearest integer

Theory and application

to 2.

of Jo” rp2dx

[c-p1

; ;

2

6 T: :: m

and so2 e-@@dy so” c~‘t/.t~.

Mapleton House, Brooklyn, N.Y., 1948; and B. Lohmander and S. Rittsten, Table of the function y=ePzz so2rt*dt,Kungl. Fysiogr. Stillsk. i Lund Forh. 28,45-52, 1958 (with permission).

320 Table

X

ERROR

FUNCTION

7.6

- 3 JZe-Qt r1 0 0 0.8000000

2

AND

FRESNEL

INTEGRALS

3 -J” e-t3at r! 0 03 3 _ J” e-Pdt r! 0 0 0.‘72276 69 0.73842 49 0.75360 34 0.76829 12 0.78247 88

54 36 70 97 49

1.50 1.52 1.54 1.56 1.58

0.99511 0.99583 0.99645 0.99699 0.99746

49 14 52 62 38

42 62 62 89 05

1.60 1.62 1.64 1.66 1.68

0.99786 0.99821 0.99850 0.99875 0.99897

63 16 65 75 03

0.90428 0.91228 0.91979 0.92683 0.93341

86 25 27 11 06

1.70

0.99914

99

1.10 1.12 1.14 1.16 1.18

0.93954 0.94525 0.95054 0.95543 0.95995

56 09 27 76 30

1.70 1.74 1.78 1.82 1.86

0.99914

99 75 05 26 14

28 66 46 80 78

1.20 1.22 1.24 1.26 1.28

0.96410 0.96791 0.97140 0.97457 0.97746

64 62 05 79 66

1.90 1.94 1.98 2.02 2.06

0.99990 0.99993 0.99996 0.99997 0.99998

01 82 24 76 69

0.63775 0.65560 0.67304 0.69006 0.70664

57 39 52 30 18

1.30 1.32 1.34 1.36 1.38

0.98008 0.98245 0.98458 0.98649 0.98820

48 07 18 52 77

2.10 2.14 2.18 2.22 2.26

0.99999 0.99999 0.99999 0.99999 0.99999

25 57 77 87 93

0.72276

69

1.40

0.98973

54

2.30

0.99999

97

69 31 72 63

0.70 0.72 0.74 0.76 0.78

0.10 0.12 0.14 0.16 0.18

0.11195 0.13432 0.15667 0.17899 0.20127

67 36 11 22 90

0.80 0.82 0.84 0.86 0. 88

0.79615 0.80932 0.82196 0.83408 0.84567

78 16 48 41 73

0.20 0. 22 0.24 0.26 0. 28

0.22352 0.24571 0.26783 0.28988 0.31184

24 24 80 71 70

0,90 0.92 0.94 0.96 0.98

0.85674 0.86728 0.87730 0.88680 0.89580

0. 0. 0. 0. 0.

30 32 34 36 38

0.33370 0.35544 0.37704 0.39850 0.41979

37 26 82 45 45

1.00 1.02 1.04 1.06 1.08

0.40 0. 42 0. 44 0. 46 0.48

0.4poso 0.46180 0.48248 0.50293 0.52312

07 52 96 51 25

0.50 0. 52 0.54 0. 56 0. 58

0.54303 0.56264 0.58194 0.60090 0.61951

0. 60 0. 62 0.64 0. 66 0. 68 0.70

00

0. 02 0. 04 0. 06 0.08

1.40 1.42 1.44 1.46 1.48

0.398973 0.99109 0.99229 0.99335 0.99429

0.02239 0.04479 0.06718 0.08957

0.

X

__3 J” eddt r! 0 0

[ 1 55)‘3

[ 1 c-:,5)7

0.99942

0.99962 0.99975 0.99984

[Ic-y1

95116 Compiled from M. Abramowitz, (with permission).

Table of the integral

Sozewu3du,J. Math.

Phys. 30,162-163,195l

ERROR

FUNCTION

AND

FRESNEL

0.00

0.00000

0.02 0.04

0.00062 0.00251

0. 06

0.00565

0.08

0.01005

00

83 33

00

0.00000

00

0.00026

80 95 76 24 38

0.00052 0.00090 0.00143 0.00214 0.00305

36 47 67 44 31

1.10 1.12 1.14 1.16 1.18

1.90066 1.97040 2.04140 2.11366 2.18717

11 29 36 70 56

0.00418 0.00557 0.00723 0.00919 0.01148

76

30 40 54 lb

1.20 1.22 1.24 1.26 1.28

2.26194 2.33797 2.41525 2.49379 2.57359

56

1.30 1.32 1.34 1.36 1.38

0.06283

19 65

0.28

0.09047 0.10618 0.12315

79 58 04

0.19992 0.21987 0.23980 0.25970 0.27957

0.30 0.32 0. 34 0.36 0. 38

0.14137 0.16084 0.18158 0.20357 0.22682

17 95 41 52 30

0.29940 0.31917 0.33888 0.35851 0.37804

10 31 06 09 96

0.01411 0.01712 0.02053 0.02435

11 68

0.02862

55

0.40 0.42 0.44 0.46 0.48

0.25132 74 0.27708 85 0.30410 62 0.33238 05 0.36191 15

0.39748 0.41678 0.43594 0.45494 0.47375

08 68 a2 40 10

0.03335 0.03858 0.04430

94 02 85

0.50 0.52 0.54

0.39269 0.42474 0.45804

91 33 42

0.56 0.58

0.49260

17

0.49234 0.51069 0.52878 0.54656 0.56401

42 69 01 30 31

0.60 0.62 0.64 0.66 0.68

0.56548 0.60381 0.64339 0.68423 0.72633

67 41 82 a9 62

0. 70 0.72 0. 74 0.76 0.78

0.76969 0.81430 0.86016 0.90729 0.95567

02 08 81 20 25

0.80 0.82 0.84 0.86 0.88

1.00530 1.05620 1.10835 1.16176 1.21642

96 35 39 10 47

0.72284 0.73312 0.74251 0.75095 0.75840

42 83 54 79 90

0.90 0.92 0.94

50 20 56 59 28

0.76482 0.77015 0.77436

30 63 72

0.98

1.27234 1.32952 1.38795 1.44764 1.50859

1.00

1.57079

63

0.77989

59

Example

8.

65 33 68 68 36 69

69 35

0.77989 0.77926 0.77735 0.77414 0.76963

34 11 01 34 03

0.43825 0.45824 0.47815 0.49788 0.51736

91 58 08 a4 86

0.76380 0.75667

67 60

0.53649

0.55517 0.57331

79 92 28

0.59079 0.60752

66 74

0.74824 0.73854 0.72759

94 68 68

0.71543 0.70211

77 76

27

0.68769 0.67223 0.65582

2.65464

58

0.63855

2.73695

2.82052 2.90534 2.99142

55

19 49 45

0.62051 0.60181 0.58259 0.56297

3.07876 3.16735 3.25720 3.3483'0 3.44067

08 37 33 95 23 17 78

68

67 33 64 62

0.62340

09

0.63831

47

78

0.65216 0.66484

34

63

0.67626

72

05

0.68633

33

11 95 73 59

0.69495 0.70205 0.70755 0.71139

62 50 67 77

0.54309 0.52310 0.50316 0.48342 0.46407

58 58 23 80 05

0.71352 0.71389 0.71248 0.70928 0.70428

51 77 78 lb 12

0.44526

12 32 99 29 96

0.69750

0.35351 0.34325 0.33481 0.32830

17 20 29 32 61 69 02 83 87 25

0.05056

42

0.05736

63

0.06473 0.08122 0.09037 0.10014

24 89 06 08 09

1.50 1.52 1.54 1.56 1.58

3.53429 3.62916 3.72530 3.82268 3.92133

60

0.11054 0.12157 0.13325 0.14557 0.15853

02 59 28 29 54

1.60 1.62 1. b4 1.66 1.68

4.02123 4.12239

86 79

0.17213 0.18636 0.20122 0.21668 0.23272

65 89 21 lb 88

1.70 1.72 1.74 1.76 1.78

4.86569

87

4.97691

11

0.32382 0.32145 0.32122 0.32318 0.32733

0.24934

14

:%i

5.08938 5.20310

01 58

0.33363 0.34203

29 39

0.26649

22

98 80 55

1:04 1.86 1.88

5.31808 5.43432 5.55182

80 70 25

0.35244 0.36476

96 35

0.37882

93

68 95

0.33977 0.35904 0.37859 0.39836 0.41827

63 93 a1 12 21

1.90 1.92 1.94 1.96 1.98

5.67057 5.79058 5.91184

0.39447 0.41148 0.42963

05 24 33

6.15814

36 91 12 99

0.44866 0.46830

69 56

34

0.43825

91

2.00

6.28318

53

0.48825

34

0.58109 54 0.59777 37 0.61400 94 0.62976 0.64499

25

0.65965 0.67370

0.68709

24 12 20

0.69977

79

0.71171

13

0.77741 0.77926

70

63

1.40 1.42 1.44 1.46 1.48

12

0.07267

0.28414 0.30227 0.32083

[ 1 [ 1 C-54)2

h

1.57079 1.63425 1.69897 1.76494 1.83217

75 39 67 41 34

0.07602

0.96

1.00 1.02 1.04 1.06 1.08

0.09999 0.11999 0.13998 0.15997 0.17995

0.20 0.22 0.24

0.52841

Table 7.7

31

0.01570 0.02261 0.03078 0.04021 0.05089

0.26

INTEGRALS

42 35 31 81

49

0.00000 0.00003 0.00011

321

INTEGRALS

0.02000 00 0.04000 00 0.05999 98 0.07999 92

0.12 0.14 0. lb 0. 18

0. 10

0.00000

FRESNEL

C-i)8

06

99

4.22481

38

4.32848 4.43341

64 56

4.53960 4.64704 4.75574

14 39 30

6.03437

47

0.42717 0.40997 0.39385 0.37895 0.36546

19 56

50

0.68898

88

0.67878 0.66697 0.65363

46

0.63888 0.62286 0.60570 0.58758

77 07 26 04

0.56867

83

0.54919 0.52934 0.50935

60 73 84

0.48946

49

0.45093 0.43280 0.41573 0.39999 0.38579

88 06 97 44 25

0.37334 0.35448 0.34838

73 37 37 30

0.34466

65

0.46990

0.36285

67

13

94

0.34341 57 C-f3

[. 1

322

ERROR

FUNCTION

Table 7.7 C(.r)=jl"

AND FRESNEL

INTEGRALS

FRESNEL INTIKRALS cos (;13) t/l

s f.1.)

C(T)

4. 04 4. 06 4.08

C(r) 0.49842 0.51821 0.53675 0.55284 0.56543

60 54 05 04 47

N (.I.) 0.42051 58 0.42301 99 0.43039 00 0.44217 81 0.45764 45

59 11 29 66 34

4.10 4.12 4.14 4.16 4.18

0.57369 56 0.57705 88 0.57527 76 0.56844 74 0.55700 75

0.47579 83 0.49545 71 0.51532 14 0.53405 87 0.55039 41

0.59334 0.58416 0.57161 0.55618 0.53849

95 97 47 06 35

4.20 4.22 4.24 4.26 4.28

0.54171 0.52362 0.50396 0.48411 0.46549

92 06 08 63 61

0.56319 89 0.57157 23 0.57491 03 0.57295 47 0.56582 05

0.51928 0.49936 0.47960 0.46084 0.44393

61 95 04 46 82

4.30 4. 32 4. 34 4.36 4. 38

0.44944 0.43712 0.42946 0.42704 0.43006

12 50 40 39 79

0.55399 0.53831 0.51990 0.50011 0.48041

59 55 77 73 08

0.43849 17 0.45514 37 0.41375 96 0.49348 70 0.51340 62

0.42964 95 0.41864 11 0.41143 69 0.40839 28 0.40967 54

4. 40 4.42 4.44 4.46 4.48

0.43833 0.45123 0.46781 0.48679 0.50671

29 59 05 41 95

0.46226 0.44707 0143599 0.42990 0.42931

80 06 33 86 16

3.50 3.52 3.54 3.56 3.58

0.53257 0.55006 0.56501 0.57668 0.58446

24 11 32 02 43

0.41524 0.42486 0.43808 0.45428 0.47265

80 72 83 17 92

4.50 4.52 4.54 4. 56 4.58

0.52602 0.54318 0.55680 0.56578 0.56936

59 11 46 27 57

0.43427 30 0.44442 34 0.45897 36 0.47676 89 0.49637 56

93 53 79 35 52

3.60 3. 62 3.64 3.66 3.68

0.58795 0.58694 0.58147 0.57178 0.55838

33 64 10 75 18

0.49230 0.51224 0.53143 0.54888 0.56366

95 12 21 15 38

4. 60 4. 62 4.64 4.66 4.68

0.56723 0.55954 0.54691 0.53039 0.51135

67 81 86 13 38

0.51619 23 0.53451 97 0.54999 67 0.56113 28 0.56702 44

0.45291 0.43518 0.42066 0.40798 0.39817

75 98 03 90 24

3.70 3.72 3.74 3. 76 3.78

0.54194 57 0.52334 49 0.50357 70 0.48371 94 0.46487 19

0.57498 0.58220 0.58492 0.58296 0.57641

04 56 61 92 91

4.70 4.72 4. 74 4.16 4.78

0.49142 0.47232 0.45572 0.44308 0.43554

65 71 30 30 28

0.56714 0.56146 0.55044 0.53504 0.51659

55 19 52 16 82

0.46749 17 0.48720 04 0.50717 21 0.52671 66 0.54538 21

0.39152 0.38828 0.38856 0.39238 0.39964

84 41 43 50 80

3.80 3.82 3.84 3.86 3.88

0.44809 0.43434 0.42443 0.41894 0.41822

49 86 43 43 16

0.56561 0.55115 0.53384 0.51466 0.49472

87 74 32 22 45

4.80 4. 82 4. 84 4.86 4.88

0.43319 0.43802 0.44786 0.46244 0.48042

66 47 69 40 90

0.49675 0.47728 0.45995 0.44637 0.43780

02 00 75 74 82

2.90 2.92 2.94 2.96 2.98

0.56237 0.57718 0.58930 0.59830 0.60384

64 78 60 19 56

0.41014 0.42353 0.43941 0.45124 0.47643

06 87 39 45 06

3.90 3.92 3.94 3. 96 3.98

0.42233 0.43105 0.44389 0.46007 0.41863

27 68 17 70 51

0.47520 0.45726 0.44198 0.43032 0.42301

24 13 92 79 17

4.90 4.92 4.94 4.96 4. 98

0.50016 0.51979 0.53747 0.55150 0.56051

10 51 34 25 94

0.43506 0.43843 0.44761 0.46175 0.47951

74 48 56 67 78

3.00

0.60572 08 [C-64)5]

0.49631 [(-y]

30

4.00

0.49842

60

0.42051

58

5.00

0.56363 [ ‘-;)‘I

12

0.49919 14 [C-$)8]

%-I

S(T)

0.34i4i 0.34467 0.34844 0.35470 0.36334

57 48 87 04 98

3.00 3.02 3.04 3.06 3.08

0.60572 0.60383 0.59823 0.58910 0.57674

08 73 78 11 01

0.49631 0.51619 0.53536 0.55311 0.56880

30 42 29 95 28

0.58156 41 0.59671 75 0.61000 60 0.62117 32 0.62999 53

0.37421

34

0.38730 0.40223 0.41880 0.43673

37 09 45 63

3.10 3.12 3; 14 3.16 3.18

0.56159 0.54421 0.52525 0.50543 0.48552

39 58 53 56 76

0.58181 0.59165 0.59791 0.60033 0.59880

2.20 2.22 2.24 2.26 2.28

0.63628 0.63990 0.64075 0.63879 0.63403

60 31 25 28 83

0.45570 0.41535 0.49532 0.51521 0.53462

46 85 41 11 03

3.20 3.22 3.24 3.26 3.28

0.46632 0.44858 0.43306 0.42040 0.41113

03 96 55 05 97

2.30 2.32 2.34 2.36 2.38

0.62656 0.61649 0.60402 0.58940 0.57293

17 45 69 65 44

0.55315 0.57041 0.58602 0.59964 0.61095

16 28 84 89 96

3.30 3.32 3. 34 3.36 3.38

0.40569 0.40431 0.40709 0.41393 0.42455

44 99 96 66 18

2.40 2.42 2.44 2.46 2.48

0.55496 0.53588 0.51612 0.49614 0.47641

14 11 29 28 35

0.61969 00 0.62562 11 0.62859 38 0.62851 43 0.62535 98

3.40 3. 42 3.44 3.46 3.48

2.50 2.52 2.54 2.56 2.58

0.45741 0.43961 0.42346 0.40939 0.39777

30 32 72 65 93.

0.61918 0.61010 0.59834 0.58415 0.56790

18 76 06 75 42

2.60 2.62 2.64 2.66 2.68

0.38893 0.38312 0.38052 0.38123 0.38525

75 73 80 50 32

0.54998 0.53087 0.51106 0.49110 0.47153

2.70 2.72 2.74 2.76 2.78

0.39249 0.40277 0.41581 0.43125 0.44865

40 39 68 85 46

2.80 2.82 2.84 2.86 2.88

2.:0 2.02 2. 04 2. 06 2.08

0.48825 0.50820 0.52782 0.54681 0.56482

2.10 2.12 2.14 2.16 2.18

For .1>5 iri

3

34 04 73 06 79

= 0.5i ( O.3183O99-T

For u>39 $#=

0.51

[ 1 [ 1 (-;I6

(-;I6

+c(2)

t@)1c3 x10-7

ERROR

FUNCTION

AND

AUXILIARY 2(sL

X

*

FRESNEL

323

INTEGRALS Table

FUNCTIONS

n

,y

7.8

f(x)= h(20

0.00000

2 00000

00000

&00062 0.00251 0.00565 0.01005

83185 32741 48667 30964

0.10 0.12 0. 14 0.16 0.18

0.01570 0.02261 0.03078 0.04021 0.05089

0.20 0.22 0.24 0.26 0.28

30718 22872 76462 91487

0.50000 0.49969 0.49880 0.49739 0.49548

00000 41196 88057 07811 44294

00000 39303 20520 66949 00553

0.50000 0.48031 0.46125 0.44281 0.42500

00000 40626 51239 99356 33536

00000 54163 79101 00196 38036

79632 94671 76080 23859 38009

67949 05847 05180 65949 88155

0.49313 0.49037 0.48724 0.48378 0.48002

18256 27777 48761 35493 21268

06624 82254 11561 31728 70713

0.40779 0.39119 0.37518 0.35976 0.34491

85545 72364 98069 55566 28197

29930 96391 99885 09573 39391

0.06283 0.07602 0.09047 0:10618 0.12315

18530 65422 78684 58316 04320

71796 16873 23386 91335 20720

0.47599 0.47172 0.46724 0.46257 0.45774

19056 22205 05176 24293 la508

49140 45221 22164 12303 40978

0.33061 0.31687 0.30365 0.29095 0.27876

91227 13200 57186 81914 42811

69034 89318 36191 92531 44593

0.30 0. 32 0.34 0. 36 0. 38

0.14137 0.16084 0.18158 0.20357 0.22682

16694 95438 40553 52039 29895

11541 63797 77490 52619 89183

0.45277 0.44768 0.44248 0.43721 0.43187

10172 05805 96860 60487 60273

56087 06203 81319 95888 53913

0.26705 0.25582 0.24505 0.23472 0.22483

92929 83796 66166 9OiO3 08578

81728 24420 57772 35799 07150

0.40 0.42 0.44 0.46 0. 48

0.25132 0.27708 0.30410 0.33238 0.36191

74122 84720 61688 05027 14736

87183 46620 67492 49800 93544

0.42648 0.42105 0.41560 0.41013 0.40466

46973 59227 24246 58491 68313

90789 36507 90070 35691 67950

0.21534 0.20626 0.19756 0.18923 0.18126

72003 34704 52322 82774 86555

95520 48744 49727 60398 47172

0.50 0.52 0.54 0.56 0.58

0.39269 0.42474 0.45804 0.49260 0.52841

90816 33267 42088 17280 58843

98724 65340 93392 82880 33803

0.39920 0.39375 0.38833 0.38294 0.37759

50585 93295 76127 71004 42617

25702 63563 15400 26771 52882

0.17364 0.16634 0.15936 0.15269 0.14631

26996 70480 86623 48414 32329

13238 39628 13733 00876 91905

0.60 0.62 0.64 0.66 0. 68

0.56548 0.60381 0.64339 0.68423 0.72633

66776 41080 81754 88799 62215

46163 19958 55190 51857 09960

0.37228 0.36702 0.36181 0.35666 0.35157

48922 41612 66571 64292 70288

35620 87842 25476 98472 80259

0.14021 0.13437 0.12880 0.12347 0.11838

la419 90361 35503 44874 13187

37684 59907 06985 03863 25611

0.70 0.72 0.74 0.76 0.78

0.76969 0.81430 0.86016 0.90729 0.95S67

02001 08158 80685 19583 24852

29499 10474 52885 56732 22015

0.34655 0.34159 0.33670 0.33188 0.32713

15463 26474 26065 33382 64271

82434 67053 33192 57734 72503

0.11351 0.10886 0.10441 0.10016 0.09611

38821 23788 73696 97688 08389

06517 79214 22082 77848 91866

0. 80 0.82 0.84 0. 86 0.88

1.00530 1.05620 1.10835 1.16176 1.21642

96491 34501 38881 09632 46754

48734 36888 86479 97506 69968

0.32246 0.31786 0.31334 0.30889 0.30452

31553 45283 12993 39917 29200

61284 60796 49704 09068 36579

0.09223 0.08852 0.08498 0.08159 0.07836

21832 57381 37656 88446 38619

05037 23702 77045 61614 62362

0. 90 0.92 0.94 0.96 0.98

1.27234 1.32952 1.38795 1.44764 1.50859

50247 20109 56343 58947 21922

03866 99200 55971 74177 53819

0.30022 0.29600 0.29186 0.28780 0.28380

82096 98149 75359 10340 98467

95385 76518 51781 91658 20271

0.07527 0.07231 0.06949 0.06679 0.06420

20035 67451 la433 13255 94813

30280 87932 26312 49021 13093

1.00

1.57079

63267 94897

0.27989

34003 76823

0.06174

08526 09645

sin uyz(u)

cos u

0. 00

0. 02 0.04 0.06 0. 08

Lm1 c-y

See Exm~ple~

6, 7, and 1 Cc,)=-+f(x) 2 1 S(X)=--fG) 2

[

c-g

I

9. sin

(

ix2

)

cos (5”)-g(x)

-8(r)

cos

t

2TTZ

sin (ixz)

1

C&(U)=!

2

s~(~)=A-~~(u) 2

+fi(u)

cos

u-g2(u)

sin

II

324

ERROR Table

7.8

x-1 1.00 0.98 0.96 0.94 0.92

0.63661 0.61140 0.58670 0.56251 0.53883

0.90 0.88 0.86 0. a4 0.82 0.80 0. 78 0.76 0.74 0.72

FUNCTION

AND FRESNEL

AUXILIARY

FUNCTIONS

INTEGRALS

g(4= k?(u)

f(1.)=h(U) 0.27989 34003 0.27597 33733 0.27197 11505 0.26788 56989 0.26371 60682

76823 36442 76851 47656 37287

0.06174 0.05933 0.05693 0.05456 0.05220

08526 31378 89827 06112 03510

09645 64174 01255 91100 52931

0.51566 20156 17741 0.49299 a3517 21455 0.47084 39836 43063 0.44919 a9113 a2565 0.42806 31349 39962

0.25946 0.25512 0.25069 0.24618 0.24157

14023 09512 40835 02994 92449

65674 a0091 25766 44393 31459

0.04986 0.04754 0.04525 0.04299 0.04075

06317 39838 30354 05078 92107

93636 94725 03048 69390 68723

0.40743 0.38731 0.36771 0.34861 0.33002

0.23689 0.23211 0.22725 0.22230 0.21726

07256 47216 14019 11393 45250

57089 24632 06110 53995 44609

0.03856 0.03640 0.03428 0.03220 0.03017

20343 19405 19524 51407 46086

27312 75704 44132 19129 88637

34743 48498 18186 74093 45674

19381 36510 13588 32978 44482

97723 96293 87822 72308 49753

66543 94695 15805 29873 36899

67581 ala25 13963 63995 31921

15252 08436 19515 48488 95354

1

2

:

;

;

I

<.r>

: 1 : 1 : :

0.70 0. 68 0. 66 0. 64 0.62

0.31194 0.29437 0.27731 0.26075 0.24471

36884 29827 15728 94587 66404

60115 42770 43318 61761 98098

0.21214 0.20693 0.20164 0.19627 0.19082

23821 57789 60404 47584 37987

60229 65521 80635 00004 55563

0.02819 0.02626 0.02439 0.02257 0.02082

0.60 0.58 0.56 0.54 0.52

0.22918 0.21415 0.19964 0.18563 0.17214

31180 88914 39606 a3256 19864

52329 24454 14474 22387 48194

0.18529 0.17969 0.17401 0.16827 0.16245

53067 17083 57076 02799 86594

79209 a6674 a9207 47273 19322

0.01913 0.01751 0.01596 0.01448 0.01307

61240 47623 29821 30628 70253

35536 30357 58470 73722 60097

0.50 0.48 0.46 0.44 0.42

0.15915 0.14667 0.13470 0.12324 0.11229

49430 71955 87438 95879 97278

91895 53491 32980 30364 45641

0.15658 0.15065 0.14466 0.13862 0.13253

43216 09597 24548 28400 62592

36302 56320 29603 34552 29647

0.01174 65939 0.01049 31590 0.00931 -_ __.__ 77420 .-~ 0.00822 09631 0.00720 30137

24659 42015 66589 . ..~ 52815 00215

0.40 0.38 0.36 0.34 0.32

0.10185 0.09192 0.08250 0.07359 0.06518

91635 78951 59224 32456 98646

78813 29879 98839 85692 90440

0.12640 0.12023 0.11403 0.10780 0.10154

69204 90456 68174 43252 55126

94864 93806 47880 41741 32988

0.00626 0.00540 0.00461 0.00390 0.00327

36346 21018 72197 73235 02912

49122 72942 27002 12822 03254

:

:i

: 3

:: 15

0.30 0.28 0.26 0.24 0.22

0.05729 0.04991 0.04303 0.03666 0.03081

57795 09901 54966 92988 23969

13082 53618 12048 a8373 a2591

0.09526 0.08896 0.08264 0.07631 0.06997

41276 36786 73969 82087 a7161

74844 39974 33180 00913 16730

0.00270 0.00220 0.00176 0.00139 0.00107

35642 41768 a7922 37442 50825

68526 84885 53708 77909 02743

i 4 4 5

0.20 0.18 0.16 0.14 0.12

0.02546 0.02062 0.01629 0.01247 0.00916

47908 64806 74661 77475 73247

94703 24710 72610 38405 22093

0.06363 0.05727 0.05091 0.04455 0.03819

11887 75644 94597 ala74 47805

04012 30652 59575 32960 44642

o.oooao a6180 a2883 0.00058 99686 10701 0.00041 45999 la234 0.00027 78633 97799 0.00017 50279 00844

0.10 0. 08 0.06 0.04 0.02

0.00636 0.00407 0.00229 o.ooioi O.QO025

61977 43665 la311 a5916 46479

23676 43153 a0523 35788 08947

0.03183 0.02546 0.01909 0.01273 0.00636

00214 44738 a5179 23855 61974

15118 95252 38105 39770 14061

o.oooio 0.00005 0.00002 0.00000 0.00000

13057 18732 la849 64845 08105

0.00

0.00000

00000 00000

0.00000

00000 00000

0.00000

00000 00000

J t-i)6 1

1

[k--1 1

C(r)=-2 +f(x) sin ;22 -g(,r) cos $9 ( > ( ) 1 s(X)=--f(x) 2

COS

5.9

(2

-g(r)

>

sin

Fz19

(2

)

94484 17470 44630 30524 69272

1

Cz(tr)=-+fz(u) sin u-gz(u) cosu 2 sz(u,=&/z(u)


co.5u-g2(u) sin u

: 2 : : : 2 z 2 I

2

:'8

76 a

2 109

10 1'; :'o m

157 245 436 982 3927 03

. ERROR

FUNCTION

ERROR

FUNCTION w(t)=e-2’

AND

FRESNEL

FOR COMPLEX erfc (-iz)

325

INTEGRALS

ARGUMENTS

Table

z=x+iy

“w(.z~z~(z)

%w(z)

.fW(Z)

.oAw(z)

x=0.3 1.000000 0.896457 0.809020 0.734599 0.670788

0.615690

0.000000 0.000000 0.000000 0.000000 0.000000

0.990050 0.888479 0.802567 0.129331

0.666463

0.112089 0.094332 0.080029 0.068410 0.058897

0.456532

0.000000 0.000000 0.000000 0.000000 0.000000

0.454731

0.030566

0.427584 0.401730 0.378537 0.351643 0.338744

0.000000 0.000000 0.000000 0.000000 0.000000

0.426044 0.400406 0.377393 0.356649 0.337876

0.021934 0.019805 0.017951

0.321585 0.305953

0.000000 0.000000 0.000000 0.000000 0.000000

0.291072

0.013648

0.278035 0.266042

0.011547

0.567805 0.525930

0.489101

0.291663 0.278560 0.266509

0.218499

0.051048 0.044524

0.523423 0.486982

0.039064 0.034465

0.320825 0.305284

0.000000 0.000000 0.000000 0.000000 0.000000

0.255396

0.245119 0.235593 0.226742

0.612109 0.564818

0.027242 0.024392

0.219753

0.913931

0.318916

0.852144

0.185252 0.157403

0.827246 0.752895

0.269600 0.229653

0.712146

0.713801 0.653680

0.134739 0.116147

0.688720 0.632996

0.197037 0.170203

0.655244 0.605295

0.584333

0.147965 0.129408 0.113821 0.100647

0.522246 0.487556

0.456579

0.089444

0.428808

0.601513

0.100782 0.087993

0.541605

Iw(z)

x-o.4

0.960789 0.864983 0.783538

0.555974

7.9

0.406153

0.777267

0.344688

0.294653 0.253613 0.219706

0.561252

0.191500 0.167880 0.147975 0.131101 0.116714

0.515991

0.077275

0.503896

0.480697 0.449383

0.068235 0.060563

0.470452 0.440655

0.421468 0.396470 0.373989

0.054014

0.413989

0.048393 0.043542

0.390028 0.368412

0.064510

0.353691

0.039336 0.035671

0.348839 0.331054

0.058329 0.052936

0.314839

0.259186

0.270346

0.053186 0.048931 0.045139

0.335294

0.079864 0.071628

0.403818

0.104380

0.381250 0.360799 0.342206 0.325248

0.093752 0.084547 0.076538 0.069538

0.016329

0.318561

0.032463

0.014905

0.303290 0.289309 0.276470 0.264648

0.029643

0.300009

0.027154

0.286406

0.024948 0.022987

0.273892 0.262350

0.048210 0.044051 0.040377 0.037118 0.034217

0.234251 0.225531 0.217404

0.021236 0.019669 0.018260 0.016991 0.015845

0.251677 0.241783 0.232592

0.031626 0.029304 0.027217

0.248844 0.239239 0.230300

0.038706 0.035968

0.216047

0.023633

0.025335

0.221963 0.214172

0.033498 0.031263

0.012536

0.010664

0.254978 0.244745 0.235256 0.226438

0.008526

0.218224

0.007949

0.009874

0.009165

0.253732 0.243628

0.224033

0.309736 0.295506

0.282417

0.063393 0.057978

0.041748

0.210557

0.007427

0.203387

0.006952

0.184602

0.000000 0.000000 0.000000 0.000000 0.000000

0.196668.0.006520 0.190360 0.006125 0.184429 0.005764

0.209813 0.202710 0.196050 0.189796 0.183912

0.014806 0.013862 0.013002 0.012216 0.011498

0.208582 0.201589 0.195028 0.188861 0.183056

0.022090 0.020687 0.019409 0.018241 0.017172

0.193613 0.187566 0.181868

0.025706 0.024168 0.022759

0.179001

0.000000

0.178842

0.005433

0.178368

0.010839

0.177581

0.016192

0.176491

0.021466

0.527292

x=0.8 0.600412

0.444858

0.210806

0.203613 0.196874 0.190549

x=0.6

x=0.5 0.778801 0.717588

0.478925 0.408474

0.691676 0.651076

0.535713

0.459665

0.612626

x=0.7 0.576042

0.663223

0.350751

0.608322 0.569238

0.396852 0.344645

0.580698 0.549739 0.520192 0.492289

0.497744 0.432442 0.377688 0.331535

0.614852 0.571717 0.533157 0.498591

0.467521 0.439512

0.414191 0.391234 0.370363

0.351335 0.333942

0.303124 0.263563

0.533581

0.230488 0.202666

0.501079 0.471453

0.264268 0.233206

0.444434

0.206787

0.419766 0.397216

0.127202 0.114460 0.103395

0.357637

0.133501

0.340241

0.120838 0.109759 0.100026

0.299804

0.091443 0.083845 0.077096 0.071081 0.065701

0.281214 0.275602 0.264718 0.254554 0.245050

0.241025

0.060876

0.232204 0.223952

0.056534

0.236152 0.227810

0.376571

0.093744

0.324229

0.085288

0.309463

0.303355 0.289866 0.277412 0.265890 0.255205

0.077851 0.071283

0.295820

0.245276

0.283192 0.271479

0.065461 0.060283

0.260598 0.250469

0.055661 0,051521 0.047804

0.184200 0.164793 0.148036

0.219347 0.211800

0.041428 0.038686

0.216219 0.208961

0.052617 0.049073 0.045859

0.204723

0.198074 0.191818 0.185924 0.180361

0.036196 0.033929 0.031859 0.029966 0.028231

0.202139 0.195717 0.189664 0.183950 0.178549

0.042936 0.040271 0.037836 0.035607 0.033561

0.175105

0.026636

0.173437

0.031680

0.227407

SeeExamples

0.466127

0.441712 0.418998

0.179123 0.159087 0.141945

0.318001

0.236031

0.300989

0.044454

0.360200 0.343375 0.327166

0.313273

Ur(-x+iy)-w(x+iy)

0.355082

0.418736 0.405763

0.429418 0.410264 0.391936 0.374518 0.358043

0.314828 0.280290 0.250532 0.224789 0.202429

0.166660

0.150681 0.136706 0.124435 0.113620

0.342511

0.182932 0.165868 0.150877 0.137661 0.125971

0.327900

0.314176 0.301294 0.289208

0.081245

0.271869 0.267228 0.251237

0.247851

0.115594 0.106355 0.098103 0.090710

0.075190

0.239027

0.084068

0.069748 0.064842

0.078085 0.072680 0.067785 0.063342

0.095563 0.088001

y

0.610142 0.536087 0.472773 0.418491 0.371813

0.392021

0.331544

0.377977 0.363957

0.296692 0.266427

0.350182

0.336799 0.323899

0.311537 0.299741 0.288519 0.277865

0.240057 0.217004 0.196783 0.178990 0.163281 0.149370 0.137012

0.249151

0.126002 0.116164 0.107348

0.240586 0.232482

0.092291

0.267166 0.258203

0.099427

0.224813

0.085845

0.217552

0.080009

0.212616

0.056391

0.230724 0.222905 0.215535 0.208581

0.205686

0.052741

0.202013

0.059298

0.210676 0.204160 0.197982

0.074712 0.069894 0.065500

0.199155 0.192992 0.187170

0.195804 0.189928 0.184362 0.179084 0.174074

0.055610 0.052238 0.04915(. 0.046315 0.043708

0.192120

0.176447

0.049417 0.046384 0.043608 0.041064 0.038728

0.181265 0.176237 0.171452

0.061486 0.057811 0.054439 0.051339 0.048485

0.171502

0.036577

0.169315

0.041306

0.166895

0.0.45851

0.219978

0.181662

w(x-iy)=2eY2-z2 erfc

0.292432

0.104054

0.469480 0.449244

0.457569 0.402194

0.060409

w(x)=e-z2+g e-22Ste@dt Jr

12-19.

w(iy)=ey2

0.397906

0.378341

0.439421 0.430271

0.489710

0.029234 0.027389

x10.9

0.522932

0.509299

0.259136 0.230646 0.206155 0.185005

0.206879 0.200039

w[(l+i)u]=e-2ia2{

(cos 2xy+i

sin Bxy)+$+j

l+(i-l)[C(&J)+iS(%)]}

0.186554

l

326

ERROR

Table

FUNCTION

ERROR

7.9 .uRw(z) .fw(z)

AND

FUNCTION

Xw(z)

FRESNEL

FOR COMPLEX

ARGUMENTS

w(z)=e- z2 erfc (-it) z= x+iy %w(z) Jw(z) %.u(z) .Iw(z) A4l(z)

x=1.1

x=1.0

INTEGRALS

x=1.3

x=1.2

0.367879 0.373170 0.373153 0.369386 0.363020

U.607158 0.538555 0.478991 0.427225 0.382166

0.298197 0.312136 0.319717 0.322586 0.321993

0.593761 0.532009 0.477439 0.429275 0.386777

0.236928 0.257374 0.270928 0.279199 0.283443

0.572397 0.518283 0.469488 0.425667 0.386412

0.184520 0.209431 0.227362 0.239793 0.247908

0.545456 0.499216 0.456555 0.417491 0.381908

0.140858 0.168407 0.189247 0.204662 0.215711

0.515113 0.476535 0.440005 0.405823 0.374110

0.354900 0.345649 0.335721 0.325446 0.315064

0.342872 0.308530 0.278445 0.252024 0.228759

0.318884 0.313978 0.307816 0.300807 0.293259

0.349266 0.316128 0.286815 0.260847 0.237800

0.284638 0.283540 0.280740 0.276693 0.271752

0.351299 0.319910 0.291851 0.266757 0.244295

0.252654 0.254784 0.254895 0.253461 0.250858

0.349611 0.320368 0.293927 0.270040 0.248462

0.223262 0.228026 0.230578 0.231385 0.230826

0.344868 0.318022 0.293453 0.271015 0.250549

0.304744 0.294606 0.284731 0.275174 0.265967

0.208219 0.190036 0.173896 0.159531 0.146712

0.285402 0.277407 0.269401 0.261476 0.253697

0.217306 0.199046 0.182742 0.168151 0.155066

0.266189 0.260213 0.253985 0.247628 0.241233

0.224168 0.206108 0.189878 0.175271 0.162100

0.247381 0.243266 0.238695 0.233813 0.228733

0.228967 0.211343 0.195398 0.180957 0.167863

0.229205 0.226767 0.223710 0.220192 0.216340

0.231897 0.214902 0.199416 0.185299 0.172423

0.257128 0.248665 0.240578 0.232861 0.225503

0.135242 0.124954 0.115702 0.107361 0.099824

0.246112 0.238752 0.231635 0.224775 0.218176

0.143305 0.132711 0.123147 0.114495 0.106650

0.234870 0.228592 0.222436 0.216428 0.210587

0.150205 0.139441 0.129684 0.120822 0.112760

0.223542 0.218309 0.213086 0.207912 0.202818

0.155975 0.145167 0.135326 0.126353 0.118158

0.212253 0.208014 0.203684 0.199315 0.194947

0.160668 0.149927 0.140103 0.131106 0.122858

0.218493 0.211816 0.205457 0.199402 0.193634

0.092998 0.086801 0.081162 0.076021 0.071324

0.211839 0.205760 0.199935 0.194356 0.189014

0.099523 0.093035 0.087116 0.081706 0.076753

0.204926 0.199452 0.194166 0.189072 0.184165

0.105411 0.098700 0.092562 0.086936 0.081773

0.197827 0.192953 0.188208 0.183599 0.179131

0.110662 0.103795 0.097495 0.091706 0.086378

0.190608 0.186324 0.182112 0.177985 0.173954

0.115286 0.108325 0.101919 0.096015 0.090567

:*; 2:9

0.188139 0.182903 0.177910 0.173147 0.168602

0.067024 0.063080 0.059456 0.056118 0.053041

0.183901 0.179008 0.174324 0.169840 0.165546

0.072208 0.068031 0.064186 0.060639 0.057363

0.179444 0.174903 0.170538 0.166342 0.162310

0.077024 0.072651 0.068617 0.064890 0.061440

0.174805 0.170623 0.166582 0.162681 0.158916

0.081467 0.076933 0.072742 0.068863 0.065266

0.170024 0.166201 0.162487 0.158883 0.155389

0.0'35532 0.080873 0.076557 0.072553 0.068834

3.0

0.164261

0.050197

0.161434

0.054331

0.158435

0.058243

0.155285

0.061926

0.152005

0.5 0.6 it: 0:9 ::: :.'3 1:4 1.5 ::; ::: ::i :*: 2:4 2.5 2.6

x-1.5

Y

x=1.6

0.065375

x-1.9

x=1.8

x=1.7

0.105399 0.134049 0.156521 0.173865 0.186984

0.483227 0.451763 0.421076 0.391665 0.363828

0.077305 0.105843 0.128895 0.147272 0.161702

0.451284 0.426168 0.400837 0.375911 0.351803

0.055576 0.083112 0.105929 0.124612 0.139717

0.420388 0.400743 0.380161 0.359313 0.338676

0.039164 0.065099 0.087090 0.105522 0.120793

0.391291 0.376214 0.359721 0.342479 0.324985

0.027052 0.051038 0.071811 0.089592 0.104641

0.364437 0.353066 0.340004 0.325873 0.311161

0.196636 0.203461 0.207990 0.210664 0.211846

0.337720 0.313397 0.290847 0.270016 0.250823

0.172820 0.181177 0.187245 0.191423 0.194049

0.328777 0.306990 0.286517 0.267378 0.249556

0.151751 0.161171 0.168379 0.173725 0.177513

0.318584 0.299261 0.280846 0.263418 0.247012

0.133288 0.143369 0.151366 0.157578 0.162268

0.307609 0.290613 0.274180 0.258431 0.243439

0.117233 0.127644 0.136134 0.142949 0.148310

0.296240 0.281392 0.266823 0.252681 0.239067

0.211837 0.210881 0.209182 0.206902 0.204177

0.233171 0.216954 0.202067 0.1'38403 0.175862

0.195407 0.195734 0.195228 0.194053 0.192347

0.233009 0.217678 0.203494 0.190384 0.178275

0.180002 0.181414 0.181938 0.181733 0.180933

0.231630 0.217253 0.203847 0.191366 0.179762

0.165667 0.167977 0.169373 0.170003 0.169997

0.229244 0.215857 0.203272 0.191471 0.180425

0.152418 0.155452 0.157569 0.158906 0.159585

0.226046 0.213656 0.201914 0.190821 0.180367

0.201115 0.197806 0.194320 0.190717 0.187043

0.164349 0.153773 0.144054 0.135113 0.126883

0.190222 0.187772 0.185073 0.182189 0.179172

0.167092 0.156765 0.147226 0.138412 0.130262

0.179651 0.177983 0.176008 0.173792 0.171390

0.168980 0.158969 0.149674 0.141045 0.133033

0.169465 0.168500 0.167183 0.165579 0.163746

0.170099 0.160457 0.151458 0.143063 0.135234

0.159709 0.159369 0.158641 0.157593 0.156282

0.170534 0.161300 0.152637 0.144516 0.136908

0.183335 0.179623 0.175930 0.172276 0.168674

0.119298 0.112302 0.105842 0.099870 0.094343

0.176064 0.172901 0.169710 0.166513 0.163330

0.122723 0.115744 0.109277 0.103280 0.097713

0.168849 0.166206 0.163493 0.160737 0.157958

0.125590 0.118674 0.112243 0.106260 0.100689

0.161733 0.159580 0.157320 0.154982 0.152591

0.127931 0.121118 0.114761 0.108827 0.103285

0.154757 0.153059 0.151224 0.149281 0.147256

0.129781 0.123108 0.116858 0.111003 0.105519

:*;: 2:9

0.165136 0.161669 0.158281 0.154975 0.151753

0.089222 0.084472 0.080061 0.075960 0.072142

0.160175 0.157060 0.153993 0.150981 0.148030

0.092541 0.087732 0.083254 0.079082 0.075191

0.155175 0.152402 0.149649 0.146927 0.144243

0.095499 0.090660 0.086143 0.081925 0.077982

0.150165 0.147722 0.145274 0.142834 0.140411

0.098107 0.093265 0.088735 0.084493 0.080519

0.145172 0.143045 0.140892 0.138725 0.136555

0.100378 0.095558 0.091037 0.086794 0.082809

3.0

0.148618

0.068585

0.145144

0.071558

0.141602-. ‘

0.074293

0.138012

0.076794

0.134391

0.079065

0.0 0.1 i-: 0:4 0.5 E 0:s 0.9 ::i :*: 1:4 1.5 :*; 1:8 1.9 2; :-: 2:4 $2

See Examples

12-19.

w(x)=&+

4 e-22 SD”,@ dt Jr

w(-x+iy)-qq

w(xpiy)=2eu2-z2

w(iy)=eY2 erfc y

w[(l+i)u]=e-2iu2{

(cos

2xy+i

sin Bxy)-w(x+iy)

l+(i-l)[C($)+iS(~$)]}

ERROR

FUNCTION

AND

FRESNEL

ERROR

FUNCTION

FOR COMPLEX

ARGUMENTS

w(t)=e-z*erfc (-iii) z=x+iy Ww(z) .Uw(z) stw(z) -@w(z) dw(z) .Yw(z) x=2.1 x=2.2 x=2.3

%w(z) Y-w(z) x-2.0

327

INTEGRALS

Table

7.9

.uRw(z) Yw(z) x=2.4

0.018316 0.040201 0.059531 0.076396 0.090944

0.340026 0.331583 0.321332 0.309831 0.297529

0.012155 0.031936 0.049726 0.065521 0.079385

0.318073 0.311886 0.303894 0.294574 0.284327

0.007907 0.025678 0.041927 0.056586 0.069655

0.298468 0.293982 0.287771 0.280232 0.271710

0.005042 0.020958 0.035728 0.049248 0.061473

0.281026 0.277795 0.272968 0.266865 0.259775

0.003151 0.017397 0.030792 0.043211 0.054585

0.265522 0.263201 0.259435 0.254478 0.248566

0.103359 0.113836 0.122574 0.129768 0.135600

0.284786 0.271881 0.259031 0.246396 0.234096

0.091422 0.101765 0.110558 0.117948 0.124081

0.273482 0.262308 0.251016 0.239772 0.228703

0.081182 0.091245 0.099943 0.107383 0.113679

0.262499 0.252844 0.242947 0.232968 0.223037

0.072408 0.082092 0.090585 0.097963 0.104309

0.251953 0.243617 0.234952 0.226111 0.217219

0.064890 0.074132 0.082345 0.089576 0.095884

0.241914 0.234714 0.227129 0.219302 0.211349

0.140240 0.143840 0.146541 0.148466 0.149725

0.222213 0.210805 0.199904 0.189529 0.179687

0.129097 0.133125 0.136286 0.138689 0.140432

0.217904 0.207442 0.197366 0.187705 0.178478

0.118941 0.123277 0.126788 0.129570 0.131709

0.213253 0.203692 0.194410 0.185446 0.176827

0.109709 0.114251 0.118019 0.121092 0.123548

0.208376 0.199660 0.191133 0.182840 0.174814

0.101336 0.105999 0.109942 0.113232 0.115935

0.203368 0.195438 0.187620 0.179965 0.172510

0.150415 0.150622 0.150418 0.149870 0.149032

0.170371 0.161572 0.153274 0.145457 0.138100

0.141604 0.142283 0.142540 0.142434 0.142021

0.169691 0.161343 0.153429 0.145938 0.138855

0.133284 0.134367 0.135021 0.135305 0.135269

0.168569 0.160680 0.153161 0.146009 0.139217

0.125454 0.126877 0.127873 0.128495 0.128792

0.167078 0.159646 0.152526 0.145721 0.139229

0.118109 0.119812 0.121096 0.122010 0.122597

0.165281 0.158299 0.151576 0.145120 0.138933

0.147953 0.146675 0.145234 0.143660 0.141982

0.131180 0.124674 0.118558 0.112810 0.107408

0.141347 0.140453 0.139375 0.138145 0.136789

0.132164 0.125849 0.119891 0.114272 0.108973

0.134959 0.134414 0.133669 0.132755 0.131699

0.132773 0.126667 0.120885 0.115413 0.110236

0.128805 0.128574 0.128130 0.127506 0.126726

0.133045 0.127161 0.121569 0.116258 0.111218

0.122897 0.122945 0.122773 0.122411 0.121884

0.133015 0.127363 0.121972 0.116834 0.111942

29"

0.140220 0.138395 0.136523 0.134619 0.132693

0.102329 0.097554 0.093062 0.088837 0.084859

0.135331 0.133791 0.132187 0.130533 0.128842

0.103977 0.099265 0.094822 0.090631 0.086677

0.130524 0.129252 0.127900 0.126483 0.125016

0.105339 0.100709 0.096330 0.092189 0.088273

0.125814 0.124792 0.123676 0.122484 0.121229

0.106436 0.101901 0.097601 0.093523 0.089658

0.121215 0.120424 0.119530 0.118548 0.117492

0.107286 0.102858 0.098648 0.094646 0.090842

3.0

0.130757

0.081113

0.127125

0.082944

0.123510

0.084568

0.119922

0.085992

0.116375

0.087227

E 0:7 0":: i-1" 1:2 ::: 1.5 1.6 :?3 1:9 ::: :*: 2:4 2.5 ::;

x-2.5

x=2.7

x-2.6

x=2.8

x=2.9

0.001930 0.014698 0.026841 0.038226 0.048773

0.251723 0.250050 0.247092 0.243042 0.238092

0.001159 I'.012635 0.023653 0.034087 0.043849

0.239403 0.238187 0.235838 0.232504 0.228337

0.000682 0.011037 0.021057 0.030626 0.039656

0.228355 0.227458 0.225569 0.222800 0.219268

0.000394 0.009778 0.018918 0.027707 0.036064

0.218399 0.217722 0.216181 0.213858 0.210843

0.000223 0.008769 0.017134 0.025225 0.032967

0.209377 0.208854 0.207577 0.205607 0.203014

0.058437 0.067205 0.075088 0.082112 0.088317

0.232420 0.226190 0.219546 0.212614 0.205504

0.052885 0.061167 0.068691 0.075467 0.081521

0.223482 0.218077 0.212247 0.206103 0.199744

0.048090 0.055890 0.063043 0.069548 0.075416

0.215093 0.210387 0.205258 0.199804 0.194111

0.043930 0.051264 0.058046 0.064266 0.069927

0.207232 0.203119 0.198594 0.193741 0.188638

0.040304 0.047194 0.053611 0.059543 0.064986

0.199873 0.196262 0.192256 0.187927 0.183344

0.093751 0.098466 0.102518 0.105960 0.108848

0.198307 0.191099 0.183943 0.176889 0.169977

0.086885 0.091598 0.095702 0.099243 0.102264

0.193255 0.186707 0.180163 0.173670 0.167270

0.080670 0.085338 0.089451 0.093044 0.096155

0.188258 0.182311 0.176328 0.170357 0.164438

0.075043 0.079632 0.083718 0.087328 0.090492

0.183354 0.177950 0.172480 0.166990 0.161519

0.069944 0.074431 0.078462 0.082059 0.085245

0.178568 0.173654 0.168651 0.163603 0.158547

0.111233 0.113165 0.114690 0.115851 0.116689

0.163237 0.156692 0.150359 0.144249 0.138368

0.104811 0.106925 0.108647 0.110016 0.111067

0.160996 0.154872 0.148918 0.143147 0.137569

0.098820 0.101076 0.102957 0.104498 0.105730

0.158604 0.152882 0.147292 0.141851 0.136571

0.093239 0.095601 0.097608 0.099288 0.100671

0.156099 0.150758 0.145518 0.140395 0.135403

0.088044 0.090482 0.092584 0.094376 0.095882

0.153515 0.148534 0.143625 0.138807 0.134094

0.117239 0.117534 0.117606 0.117481 0.117184

0.132720 0.127305 0.122121 0.117164 0.112428

0.111834 0.112347 0.112635 0.112723 0.112633

0.132191 0.127015 0.122042 0.117271 0.112699

0.106683 0.107386 0.107864 O.lOM140 0.108238

0.131459 0.126522 0.121762 0.117180 0.112775

0.101783 0.102649 0.103293 0.103737 0.104002

0.130553 0.125851 0.121303 0.116911 0.112676

0.097127 0.098133 0.098922 0.099513 0.099925

0.129498 0.125027 0.120688 0.116484 0.112419

:*: 2:9

0.116737 0.116160 0.115471 0.114685 0.113816

0.107909 0.103597 0.099487 0.095570 0.091838

0.112389 0.112008 0.111508 0.110904 0.110210

0.108322 0.104136 0.100133 0.096309 0.092657

0.108177 0.107975 0.107648 0.107213 0.106682

0.108546 0.104489 0.100601 0.096876 0.093310

0.104105 0.104066 0.103898 0.103617 0.103236

0.108597 0.104674 0.100905 0.097284 0.093810

0.100177 0.100284 0.100261 0.100122 0.099879

0.108493 0.104707 0.101058 0.097546 0.094168

3.0

0.112878

0.088283

0.109439

0.089170

0.106067 -.

0.089898

0.102767

0.090479

0.099544

0.090921

::1" :*: 1:4 22 z 1:9 ::: :: 214 ::2

see Examples

12-19.

w(x)=&+

zt e-22 Jo=et2 dt 4”

w(-x+iy)=-$i$J

w(x-iy)=2e+z2 (cos2xy+i sin Sxy)-w(xtiy)

w(iy)=e+! erfc y

w[(l+i)u]=e-zR2{

l+(i-l)[C(~$)+iS(2$]}

328

ERROR

Table 7.9

FUNCTION

ERROR

FUNCTION zu(z)=e-;2

9&u(z) Au(z)

x-3.0

au(z)

Au(z)

x -3.1

AND

FRESNEL

INTEGRALS

FOR COMPLEX erfc

z=x+iy

(-iz)

%u(z)

ARGUMENTS

&(z)

x- 3.2

&(z)

‘A4J(Z)

x-=3.3

Li%n(z) A4l(z)

x=3.4

0.000123 0.007943 0.015627 0.023095 0.030279

0.201157 0.200742 0.199669 0.197980 0.195732

0.000067 0.007254 0.014338 0.021250 0.027929

0.193630 0.193292 0.192376 0.190915 0.188951

0.000036 0.006670 0.013225 0.019639 0.025862

0.186704 0.186421 0.185630 0.184354 0.182626

0.000019 0.006167 0.012252 0.018222 0.024032

0.180302 0.180061 0.179369 0.178245 0.176715

0.000010 0.005728 0.011394 0.016966 0.022403

0.174362 0.174152 0.173542 0.172545 0.171181

0.037126 0.043598 0.049665 0.055311 0.060529

0.192984 0.189798 0.186239 0.182368 0.178243

0.034328 0.040407 0.046141 0.051509 0.056501

0.186532 0.183709 0.180534 0.177061 0.173340

0.031849 0.037565 0.042983 0.048083 0.052854

0.180484 0.177970 0.175128 0.172003 0.168637

0.029643 0.035022 0.040144 0.044989 0.049544

0.174808 0.172560 0.170006 0.167184 0.164132

0.027670 0.032738 0.037582 0.042185 0.046532

0.169475 0.167455 0.165151 0.162596 0.159821

0.065318 0.069685 0.073641 0.077202 0.080385

0.173918 0.169445 0.164866 0.160223 0.155551

0.061114 0.065350 0.069216 0.072722 0.075883

0.169418 0.165339 0.161145 0.156872 0.152553

0.057289 0.061387 0.065151 0.068589 0.071711

0.165072 0.161349 0.157502 0.153567 0.149572

0.053801 0.057757 0.061413 0.064773 0.067844

0.160886 0.157480 0.153948 0.150320 0.146623

0.050615 0.054428 0.057971 0.061246 0.064258

0.156858 0.153738 0.150490 0.147141 0.143717

0.083210 0.085697 0.087870 0.089749 0.091355

0.150880 0.146236 0.141640 0.137113 0.132667

0.078712 0.081229 0.083450 0.085394 0.087080

0.148217 0.143888 0.139588 0.135335 0.131146

0.074529 0.077055 0.079306 0.081297 0.083044

0.145545 0.141510 0.137488 0.133495 0.129548

0.070636 0.073158 0.075423 0.077445 0.079236

0.142882 0.139120 0.135357 0.131609 0.127892

0.067012 0.069518 0.071785 0.073823 0.075646

0.140239 0.136731 0.133209 0.129691 0.126192

0.092711'0.128317 0.093835 0.124071 0.094748 0.119936 0.095467 0.115919 0.096010 0.112023

0.088525 0.089749 0.090767 0.091597 0.092255

0.127031 0.123003 0.119068 0.115233 0.111503

0.084562 0.085867 0.086974 0.087900 0.088657

0.125660 0.121840 0.118099 0.114442 0.110875

0.080811 0.082182 0.083364 0.084370 0.085213

0.124219 0.120600 0.117045 0.113560 0.110153

0.077263 0.078687 0.079930 0.081004 0.081921

0.122723 0.119296 0.115919 0.112602 0.109349

::t

0.096393 0.096632 0.096739 0.096729 0.096613

0.108249 0.104600 0.101076 0.097674 0.094395

0.092754 0.093110 0.093336 0.093442 0.093442

0.107881 0;104370 0.100969 0.097680 0.094502

0.089259 0.089719 0.090050 0.090263 0.090368

0.107403 0.104027 0.100751 0.097575 0.094499

0.085905 0.086458 0.086883 0.087190 0.087391

0.106827 0.103586 0.100433 0.097369 0.094396

0.082690 0.083324 0.083832 0.084225 0.084511

0.106166 0.103057 0.100026 0.097073 0.094202

3.0

0.096402

0.091236

0.093345

0.091434

0.090375

0.091523

0.087493

0.091513

0.084700

0.091413

1.0 ::: ::: :-z 1:7 ::i 2.0 22:: ::'4 :-z 217

Y

x=3.5

x=3.6

x=3.7

x=3.8

0.000005 0.005340 0.010633 0.015846 0.020944

0.168830 0.168645 0.168102 0.167212 0.165990

0.000002 0.004995 0.009952 0.014841 0.019632

0.163662 0.163498 0.163011 0.162211 0.161111

0.000001 0.004685 0.009339 0.013935 0.018446

0.158821 0.158673 0.158235 0.157513 0.156516

0.000001 0.004406 0.008786 0.013115 0.017370

0.154273 0.154140 0.153743 0.153088 0.152183

x=:3.9 0.000000 0.149992 0.004153 0.149871 0.008282 0.149510 0.012368 0.148913 0.016389 0.148088

E

0.025897 0.030677 0.035263 0.039637 0.043785

0.164456 0.162633 0.160548 0.158227 0.155698

0.024297 0.028812 0.033158 0.037316 0.041274

0.159725 0.158075 0.156181 0.154066 0.151755

0.022847 0.027118 0.031239 0.035195 0.038974

0.155260 0.153760 0.152034 0.150102 0.147985

0.021529 0.025574 0.029486 0.033253 0.036861

0.151040 0.149672 0.148094 0.146324 0.144380

0.020326 0.024162 0.027880 0.031469 0.034916

0.147044 0.145792 0.144346 0.142721 0.140931

:*: 1:2 1.3 1.4

0.047698 0.051370 0.054798 0.057984 0.060928

0.152988 0.150124 0.147132 0.144038 0.140862

0.045023 0.048556 0.051869 0.054962 0.057835

0.149271 0.146637 0.143878 0.141014 0.138067

0.042565 0.045962 0.049161 0.052159 0.054958

0.145703 0.143277 0.140727 0.138074 0.135336

0.040301 0.043567 0.046653 0.049558 0.052279

0.142279 0.140039 0.137680 0.135218 0.132671

0.038212 0.041352 0.044328 0.047139 0.049783

0.138993 0.136922 0.134735 0.132448 0.130076

0.063637 0.066116 0.068374 0.070419 0.072260

0.137628 0.134354 0.131058 0.127755 0.124460

0.060491 0.062936 0.065176 0.067217 0.069068

0.135056 0.131999 0.128913 0.125812 0.122709

0.057557 0.059962 0.062177 0.064206 0.066058

0.132530 0.129674 0.126782 0.123869 0.120947

0.054819 0.057179 0.059362 0.061374 0.063219

0.130054 0.127384 0.124673 0.121935 0.119182

0.052260 0.054572 0.056720 0.058708 0.060540

0.127633 0.125133 0.122591 0.120016 0.117422

0.073908 0.075373 0.076666 0.077796 0.078774

0.121185 0.117940 0.114735 0.111578 0.108474

0.070736 0.072232 0.073563 0.074739 0.075770

0.119617 0.116545 0.113503 0.110500 0.107540

0.067738 0.069254 0.070615 0.071829 0.072902

0.118027 0.115120 0.112234 0.109377 0.106556

0.064903 0.066433 0.067815 0.069058 0.070166

0.116425 0.113673 0.110935 0.108218 0.105530

0.062222 0.063759 0.065156 0.066420 0.067556

0.114817 0.112212 0.109614 0.107031 0.104469

22::

0.079611 0.080316 0.080898 0.081366 0.081730

0.105431 0.102451 0.099538 0.096696 0.093927

0.076664 0.077430 0.078076 0.078612 0.079044

0.104631 0.101777 0.098981 0.096247 0.093577

0.073845 0.074663 0.075366 0.075961 0.076455

0.103777 0.101044 0.098362 0.095734 0.093162

0.071149 0.072013 0.0727'64 0.073411 0.073959

0.102875 0.100260 0.097688 0.095163 0.092688

0.068572 0.069474 0.070267 0.070959 0.071555

0.101935 0.099433 0.096968 0.094543 0.092162

3.0

0.081996

0.091230

0.079381

0.090973

0.076855

0.090649

0.074415

0.090265

0.072061

0.089826

0.0 2: 2: i-2 0:7

::2 :*i 1:9 22:: :*: 214 :s 217

0.4613135 0.09999216 0.002883894 If x>3.9 or y>3 w(z)=iz z2( 0 . 19016 35+z2-1.7844g27+z2-5.5253437 ) +r(z) +)I<2 xlo-6 If x>6ory>6 w(~)=iz(~~~.~~~~l+~~~.~~~~)f”(z) 1v(z)/
ERROR FUNCTION

329

AND FRESNEL INTEGRALS Table 7.10

COMPLEXZEROSOFTHEERRORFUNCTION

z*=xn+iyn

erf z,=O

n

xn

; :

928 2.24465 616 1.45061 2.83974 105 3.33546 074

1.88094 514 2.61657 300 3.17562 3.64617 810 438

4.15899 840 4.51631 940 5.15876 4.84797 791 031

4.43557 4.78044 144 764 804 5.10158 264 5.40333

5

3.76900 557

4.06069 723 10 5.45219 220 erf z,=erf (-z,)=erf zn=erf (--in)=0

5.68883 744

Yn

11

X7l

: 98

?/11

From H. E. Salzer, Complex zeros of the error function, J. Franklin Inst. 260,209-211, 1955 (with permission).

COMPLEX

5

OF FRESNEL

C(zn) =o

zn=.re+iyn

S(zA) =o

z;=x;+iy;I

INTEGRALS

.m 0.0000 1.7437 2.6515 3.3208 3.8759

Yn 0.0000 0.3057 0.2529 0.2239 0.2047

.r:, 0.0000 2.0093 2.8335 3.4675 4.0026

4.3611

0.1909

4.4742

MAXIMA

&= C(J4n +l) n

ZEROS

AND MINIMA

OF FRESNEL

mn=C(J411in+3)

0

Jfn 0.779893

Inn 0.321056

;

0.605721 0.640807

0.404260 0.380389

z

0.577121 0.588128

5

0.569413

Table 7.11

l Y7L 0.0000

0.2886

0.1877

INTEGRALS

M;=s(@Tz)

Table 7.12

74 = S( J4n + 4)

0.427036 0.417922

MA 0.713972 0.628940 0.600361 0.584942 0.574957

d 0.343415 0.387969 0.408301 0.420516 0.428877

0.433666

0.567822

0.435059

M !++4n+1)2-3 n- 2 ,3(4n+1)5/2

n+- 1 -- r2(4,+3)2-3

Me !+~2(4n+2)2-3 “-2 fi(4n+2)@

m;-~-16++1)2-3 2 32*3(n+1)5/2

2 *3(4n+3)5/2

(n-1

From G. N. Watson, A treatise on the theory of Besselfunctions, 2d ed. CambridgeUniv. Press,Cambridge, England, 1958(with permission).

8. Legendre IRENE

Functions

A. STEGUN 1

Contents

Page 332 332 332 333 333 333 333 334 335 335 335 335 336 337 337 337 339 339 340 342

Mathematical Properties. ................... Notation .......................... 8.1. Differential Equation .................

8.2. 8.3. 8.4. 8.5. 8.6. 8.7. 8.8. 8.9.

Relations Between Legendre Functions .......... Values on the Cut ................... Explicit Expressions .................. Recurrence Relations ................. Special Values .................... Trigonometric Expansions ............... Integral Representations ................ Summation Formulas ................. 8.10. Asymptotic Expansions ................ 8.11. Toroidal Functions .................. 8.12. Conical Functions ................... 8.13. Relation to Elliptic Integrals .............. 8.14. Integrals ....................... Numerical Methods ...................... 8.15. Use and Extension of the Tables ............ References ........................... Table 8.1. Legendre Function-First Kind P,(z) (z(1) ....... z=o(.ol)l, n=0(1)3, 9, 10, 5-8D Table 8.2. Derivative of the Legendre Function-First Kind

P;(z)

344 z=o(.ol)l, n=1(1)4, 9, lo, 5-7D 346 Table 8.3. Legendre Function-Second Kind &(s) (XI 1) ...... ~=o(.ol)l, n=0(1)3, 9, 10, 8D Table 8.4. Derivative of the Legendre Function-Second Kind &i(z) 348 (x51). ........................... 2=0(.01)1, n=0(1)3, 9, 10, 6-SD 350 Table 8.5. Legendre Function-First Kind Pn(z) (z 2 1) ....... z-1(.2)10, n=0(1)5, 9, 10, exact or 6s Ttible 8.6. Derivative of the Legendre Function-First Kind P:(z) 351 (z>l). ........................... x=1(.2)10, n=1(1)5, 9, 10, 6s 352 Table 8.7. Legendre Function-Second Kind &(z) (z> 1) ...... 2=1(.2)10, n=0(1)3, 9, 10, 6s Table 8.8. Derivative of the Legendre Function-Second Kind Q:(Z) 353 (x21). ........................... 2=1(.2)10, n=0(1)3, 9, 10, 6s The author acknowledges the assistance of Ruth E. C:apuano, Elizabeth F. Godefroy, David S. Liepman, and Bertha II. Walter in the preparation and checking of the tables and examples.

(251).

1 National

...........................

Bureau of Standards. 331

8. Legendre

Functions

Mathematical

Properties 8.1. Differential

Notation The conventions used are z=r+iy, Z, 7~ real, and in particular, z always means a real number in the interval - 1 Ix 5 + 1 with cos 0=x where B is likewise a real number; n and m are positive integers or zero; v and .u are unrestricted except where otherwise indicated. Other notations are:

d2w (l--22) G-22

ical

p:(z) for P:(z),

a(z) -Associated Harmonics)

of the

larg

l)l
for (-l)“E(z)

J

e(x)

(For P:(z), chapter 22.)

Mz>l)

8.1.2

z(n+?n)!

XX(s) for Q:(z)

Q;(z)

(zk

Legendre First and

Functions Second

larg

for e@4%)

1)”

[srF(-v,

(II-4<2) (For F(a, b; c; z) see chapter 15.)

Various other definitions of the functions occur as well as mixing of definitions. &:(a)=&*2--*-Srt

r(v+g) r(v+r+l)

see

v+1; 1-p; 2)

Q:(z)forsinsj!+ @(z)

8.1.3

*

Legendre polynomials,

p=O,

PC(z)= &

(SpherKinds

4<*

(z”- 1p= (z-lp(Z+

(2n+l)b--m)!

for (-l)R

-&21w=o

(Degree v and order cc with singularities at z=fl, 00 as ordinary branch points-p, v arbitrary complex constants.)

P&x)for(--1)“PW E(x)

dw -g+[Yb+‘) Solutions

P:(z),

c(x)

Equation

8.1.1

2 2v 2P. v+g> Pl, -+-+-7 3. 21

l+;+z

~-~-‘L-‘(~Ll)tr$3 Alternate

(I4>1)

F-s

(Additional forms may be obtained by means of geometric function, see [S.11.)

the

transformation

formulas

of

the

hyper-

(12”1<1) 8.1.4

P:(z)=%f(z2-1)-b

8.1.5

P:(z)

-

2-*-‘*-tr(-f-,)Z-v+r-l (22-l)fi’T(--v--p)

F

a+;-;,

I+;-;;

2T()+v)z”+~ +(+~)r’qy~fy-p)

8.1.6

emr@“Qf(z)=

r(i+~+~)r(-~or)(~-i)‘v(~+i)-t~ 2w+v-d +;

,+;;

,d

F -v,

r(p)(Z+l)yz-l)-~~F

z-2)

v p 1 v ----9 2 2 ----2 2

l+v;

l+p; (

-v,

l-2 2

p.l

CT-‘; 2-2 d

(1~-21<1)

>

1+v; l-/i

T)

(ll-4<2)

* The functions Y:@, ~)-~~~ R(cos 0) called surface harmonics of the first kind, tesseral for m
332

*

LEGENDRE

8.1.7

FUNCTIONS

333

e-‘rrQ:(z)=*12r(z*-1)--tp

(12*1<1)

Wronskian

(Upper and lower signs according as Yzl,O.) 8.3.2

PrI (x)=e*lifi=P:(xfiO)

*

8.3.3 =i*-‘e,-i~ff”[e-1”‘6):(s+iO)

8.1.9 8.2.

W{P,(z), Relations

Between

8.2.1

Legendre

Negative

Degree

P!y-l(z)

=Pf(z)

-e~f~rQ:(s-iO)

8.3.4

Functions

(Formulas for P:(z) and &y’(z) are obtained with the replacement of z-1 by (l--z)e*‘“, (~‘-1) by (l-~*)e*~~, z-/-l by zf 1 for z=zfiO.) 8.4.

{-7refp’cos wrP~(z) +&l’(z) Negative

sin ~~(~+dll/sin

Argument

(92

Expressions

(x=cos

SO)

P&z) = 1

8.4.1

e)

P,(x)=1

8.4.2

e-Q-sin [7r(u-tp)]Q:(z>

&o(z)=i

8.2.4

Explicit

[*b--cc)1

8.2.3 P$(--z)=eFf~‘Pf(z)-~

Q&)=f

In (s)

In (z)

Q:(-z)=-e*f~‘&:(z) Negative

] *

&:(x)=~e~f~‘[e~~fc~&:(x+iO)+e~*~”&:(x-iO)]

8.2.2 Q’-.-l(z)=

*

Q&z)} =-(2*-l)--’

=sF(+, 1; Q; cc*)

Order

8.2.5

Pip(z) =

P,(s) =2=cos 8

PI(Z) =z

8.4.3 8.4.4

Q1(i)=% In (2)-l

8.2.6 Degree

h + ) and

Order

v+ )

*

8.4.5 p&) =)(a+

Ql(x)=;

In (E)-1

P*(Z) =i(h+-

1)

az>o

1) =i(3

cos 2e+ 1)

8.4.6 Q*(z) =9*(z) 8.2.8

In (2)

-- 32 2 8.5. Recurrence 8.3.

Values

on the

(--1<2<1)

8.3.1

P~(x)=~[e~f~“P~(x+iO)+e-~f~“P~(x--iO)] *see

Q&J) =

pa23

xl.

Cut

Relations

(Both P: and &: satisfy the same recurrence relations.) Varying

Order

8.5.1

P:+‘(z) = (z’- 1)-‘{ (v-c()zPW

-(v+LoPL(4

I

334

LEGENDRE

8.5.2 (z’-l)

FUNCTIONS

8.6.9 c!!y=

(v+/.L) (v-/.tc(sl)

(z’-l)P:-‘(~)

p;J(+

2 I’* (z;-;“4 V 0?r

--Pex4 Varying

{[z+(z”-l)““]~++ -[z+(z”-l)y--v--f}

Degree

8.5.3 (Y-/.L+l)P~+I(Z)=(2~+l)@!(~)-(~+P)R!-*(~)

8.6.10 ~~(~)=~(~~)*~*(~*-1)--1~4[~+(~*-1)*~*]-~--t

8.5.4

(~“-1)

~~=,,P:(,)-(,+,)p:_I(,) 8.6.11 Varying

Order

and

Degree &,‘(4=-~(2~)~13

8.5.5

kk&1’4[2+(2-~)1/2]-V-+

*

P~+,(~)=P:-~(2)+(2~+1)(~~-1)~~~-~(~) 8.6. Special

Values

8.6.12

x=0

P!(cos 0)=(*7r)-+

8.6.1

19)-t

cos

[(v+$)e]

8.6.13

p: (0) =2%-t 8.6.2

(sin

cos [4n(v+~)]r(aY+t~+t)/r(~~-~~+l) .

(sin

8)-+

[(v+$)e]

411

8.6.14

Q:(O)= --2fi-‘7d

Qt(cos0)=-(+x)4

[3a(~+~)]r(3~+3~+3,lr(3Y-tCl+l)

sin

8.6.3

e>=(+~)-+(v+f)-l(sil~

P;+(cos

e>-* sin

[(v++>e]

e>-*

[(v++>e]

8.6.15 Q;f(cose))=(2~)

2e+%-t 8.6.4

sin

[j~(v+p)]

1(2v+l)-‘(sin

r(3Y+~~+i)/r(3Y--3E1+3)

cos

*

/A=-v 8.6.16

8.6.17

WIWx),&:(x)

lz=o= 2*rr(3v+all+l)r(3v+-2r+~) r(+fp+i)r(+-+p+$) /.L=m=l,

b=O, v=n

2,3, . . .

(Rodrigues’

Formula)

8.6.6 Pp(z)=(z*-l)fm fyx)=(-l)yl-x*)@

p,(z)-

8.6.18

dy$), dq)

8.6.7

8.6.19

Q,,(X)=:

P,(x) In E-w,+~(x)

where Q~(z)=(&l)l)”

w,

Iv,-‘(X,==~ &~(2)=(-1)yl-q~m

P!(z)=(z*-1)-“4(2a)-~~*{[Z+(~Ll)u*]v+t

+[z+(22-l)1’2]--v-~} II.

Pn--3(x)

=P&(x)+

+5(n-2)

8.6.8

page

P,-I(x)+2n-5 3(7&-l)

‘9

II-ztt

*See

1 d%*---lP 2%! dz”

W-*(z)=0

...

LEGENDRE

335

FUNCTIONS

v=o, 1 8.6.20

[IDP.!;’

e)]v;o=2

bP;‘(cos e)

8.6.21

1 1

dV

8.6.22

In (cos $0)

=-tan v=o

dP;‘(cos e> =-3 bV “3, C

46-2 cot 40 In (cos $0) tan 50 sin* +f?+sin

e In

8.7. Trigonometric

r(vfP+l) 0)p-___ r(v+g)

8.7.1

P,‘(cos 0)=r1/22r+*(sin

8.7.2

r(v+Pi-l) Q:(COSe) =7r1’*2r(siti ep ____r(v+g)

8.7.3

P,(cos O)=

8.7.4

Qn(cose)=

Expansions

(O<e
2 b+S)k(V+II+l)k k=O k! (vf$), 2 (P+3Mv+P+l)k I;=0 Wv+2h

22n+2(n!)2 nfl sin (7~+1)0+~~ a(2nSl)! 22”f’(n!)2 (2n+l)!

(cos se)

sin [(v+p+2k+1)e]

cos [(v+p+2k+W4

. sm (n+s)e+- 1’3 (n+1)(n+2) 2! (2n+3)(2n+5) 1.3 (n+l)(n+q (n+3)e+2!(2n+3)(2n+5)COS

cos (ni-l)ei-2~~3cos 8.8. Integral

S

e*‘“th2-@ r(vfp+i) 8’8’2 @(‘)= r(p+$) r(v--c(+l)

8.8.3

Q.(z;=2

1

(n+5)e+

1 1

..

..

Representations

(z not on the real axis between -1

8’8sL p~~z)=r(-v-p)r(v+~) om(z-tcosh

si,l (n+5)e+

*

and - =)

eP(--Id rel="nofollow">%>-1)

t)@-v-l (sinh t)*‘+‘dt

cash t]-‘-‘-‘(sinh

(22--l) +fl

t)*dt

(~(v&P+-l)>O)

’ -1 (z-t)-*P,(t)dt=(--l)“+‘Qn(-2)

S

(For other integral representations see [8.2].) 8.9.

Summation

Formulas

8.10.

Asymptotic

Expansions

are asymptotic expansions if z is not on the real For fixed z and v and L%I.L-+CO,8.10.1-8.10.3 axis between --03 and -1 and +a and + 1. (Upper or lower signs according as YzZO.) 8.10.1

P;(z)=

--

8.10.2

‘See

Q:(z)=fe”“”

page

I,.

e

*

LEGENDRE

336

FUNCTIONS

--

z-1 -tr F(--Y, v+1; 1+/J; ++32> ( z+1 >

1

With ccreplaced by -F, 8.1.2 is an asymptotic expansion for P;+‘(z) for fixed z and Y and 9 ~+OD if z is not on the real axis between --a~ and - 1. For fixed z and p and L%?v+w, 8.10.4 and 8.10.6 are asymptotic expansions if z is not on the real axis between - 0~ and - 1 and +a and + 1; 8.10.5 if z is not on the real axis between - 0~ and + 1. 8.10.4

r(v+P+l)

~:(2)=(27r)-~(~~--l)-~‘~

[z+ (z’-l)qy+*F(3+c(,

r(v+#)

3--/J; g+v;

+~e-r~‘[z-(22-l)~]V+~F(~+Lc,

8.10.5

&:(z)=ef~“(&r)~(z2-1)-1’4

8’10’6

&t”(‘)=

r(v+P+l) rb+g>

[?r(~-v)]

The related asymptotic 8.10.7

P:(cos 6’)=r;;;;;)

8.10.8

&$(cos e>=

3-P; 3fv;

r(v+p+l) r(v+S)

T (2

>

’

cos

>

zf(22-1) 2(z2--l)*

3+v;

4 )

-$$:‘*I}

fl-i+T]+O(v-‘)

[(v++)e+%+~l+O(v-l)

expansions, see [8.7] and [8.9]. 8.11. Toroidal

(Only special properties sections.)

Functions

are given; other

(or Ring

properties

Functions)

and representations

8.11.1

I’:-+(cosh

8.1~2

C-+(cosh ~)=r(n-m++pm~r(m+t) S

?)=[r(l-~)]-‘22~(l-ee-21)-‘e-(‘+t’lF[4-~,

8.11.3

&;-_t(cosh ~)=[r(l+v)]-1~ef~*r(~+v+~)(l-e-2~)’e-(’f~’~F(~+~,

8.11.4

@et (cash 7) =

r(n+m+t>(sinh

*See page II.

--2+(22-l)) 2(22--l)+

expansion for P!+(z) may be derived from 8.10.4 together with 8.2.1. (-&f sin Q-1 cos [(v+*)

For other asymptotic

3--c(;$+v;

~

+iefv” cos ct?r[z-(z2-1))lY-tF(3+~,

)

&P;*+v;

r(p+v)cos ~[2+(22-l)fl”-fF(3+11,3-~; M-d 1

e(~~($7r)+(22-1)-1’4 sin

[z--(ZZ-l)q”+*F(~+~,

2+(22--l)+ 2(22-l),

drn

$+v-ll;

follow

from the earlier

1-2,~; l--e-+)

* (sin CPR&J 0 (cash q+cos cpsinh v)~+~+*

cash mt dt (-i>*r(n+4> r(n-m+3> S,, (cash v+cosh t sinh

++v+r;

n)‘+

*

l-i-v; e-‘v)

*

(n>m>

LEGEBDRE

8.12. Conical

Functions 8.13.2

(Only specinl properties nre given RS other properties atid representrltions follow from earlier sections with v= -3+& (A, n real pnmmeter) and z=cos e.>

(cos 19)= 1+4*

8.13.4

s) =[+h

;I-‘K

Q&)=,/-& Q-i(cosh

(tanh

;)

K (,/&) ~)=2e-~‘2K(e-~)

P*(z) =‘, (z+ &q&T

sin2 i

+(4x2+12) (4A2+32) si,,a f+ 2242 2

...

8.13.6

(0 se<+

PI(cosh q) =i

(JlZ2)

e?12E(41 -ee2q)

8.13.7

8.12.2

P-t+rx(COS e)=P+,A(COS

8012.3 P-~+iA(cos e)=;

2

e> Qd@=~&+(&&)

cash Mdt s Q ,‘2(cos-GGT)

8.12.4 Q-~WA(COS 0) = f i sinh XT

0

(D S /--

-P(z+l)W

*

cos Atdf

0 1 ~(COSII t+c05i e)

8.13.8 8.13.9

~-t+,A(-~~~

(J&J (---l<x
8.12.5

P-,(c0s e) =i

K

e)

=c *

8.13.3

P-,(cosh

8.13.5

8.12.1 P-i+l~

337

FUNCTIONS

[Q-t+a(~~~

e>+Q+&os

8.13. Relation to Elliptic Integrals (see chapter 17) (S&>O) 8.13.1

P-,(r)=;

Jg

8.13.10

Q-,(x)=K

(@)

8.13.11

~~~~)=~[2~(J~)-K(~~)]

*

e)]

K (QEf)

8.13.12

Q,(z) =K

8.14. Integrals 8.14.1 8.14.2

J

(aP>9v>o)

,= P~(~>Q,(~)dz=[(p-v)(p+v+l)l-’ (S?(P+v)>-l,

s ,- ~~(~)Q,(~)d~=[(p--v)(p+v+l)l-'[~(p+l)-~(v+1)1

p+v+1 v, p#-1,

8.14.3 8.14.4 8.14.5 8.14.6

8.14.1

s

,- [Q4$12dz= (2v+ 1)-jv(v+

S S S

-I’ f’.W’,(x)dx=; ;l [PAz)12dz=

*v)~ #‘(v+l) r2b-t 3)

’ _,(Q”(z)l’d~=(2v+l)-‘{

-3,

. . .)

(WV>--31 {2 sinrvsinrp[#(v+l)-$(p+l)]+*sin

(rp-TV)}

+

-,’ Q~(2>Q~(~>d~=[(p-v)(p+v+l)]-‘{[~Mv+

s

-2,

1)

[(P-v)(p+v+l)]-’ r2-2(sin

#O;

I)-+(p+

I>][1 +cos pr cos ml(p+v+l#o;

Ed--c’(v+I)[I+(cosv+]}

$r sin (vr-

v, pf-1,

(vz-1.

-2,

-2,

pi)}

-3,

-3,.

*

. . .>

. .)

338

LEGENDRE

s lP,(~)Q,(z)dr=[(~--p)(p+~+l)]-l

8.14.8

FUNCTIONS

l-cm

-1

(p7r-tm)-tsin

m cos 7n+b(~+l)+(p+l)]

{ M’v>O,

8.14.9

s

(2v+l)-’

lP&)Qv(z)dz=-~

sin 2~71$‘(v+l)

(9v>O) integers)

8.14.13

8.14.10 l-(-l)‘+“(n+m)!

(z--n)(z+n+1)(?%-7n)!

s

S r

1 ~(2)Py(2)d2=0

’ [Pg(z)]Yz=(n+~)-‘(n+m)!/(n-m)!

8.14.14

Is

1 (l-~~)-‘[~(2)]%2=(n+m)!/m(n--m)!

(I #n>

-1

-1

8.14.15

1e(z)P:(z)

.

S -1

1 -‘Q(s)py(z)(&(-l)”

8.14.12

P#V>

-1

(m, n, 1 positive

8.14.11

g’p>o,

(1-22)-‘&x=0

Wm)

-1

S

**2-P-T(l+p)

l P,(x)x~dx=

0

ro+~P-3v)~(lP+3v+~)

8.14.16

S

o* (sin t)a-lP;M(cos

t)dt=

2-CI?rr(3CY+&I)r(3a-3~) rr!3~-3v)r(~~+4v+l)r(3~-3v+~)

(ab~P)>o)

8.14.17

P;qz)=(22-1)-fm

s1

I* * *

s1

zP,(z)(dz)” P

0

8.14.18

Q;“(z)=(-l)m(22-l)-tm For other integrals, 22. P”(

s/

. . . smQ.(z)C-W” s

2.0

see [8.2], [8.4] and chapter I.5

COSB) I .o

t .5

0

.2

.4

.6

.6

1.0

/ :5

-I .o

-1.5

Fmum

8.1.

P.(cob 0).

n=0(1)3.

FIQTJRE 8.2.

P,!(z).

n=1(1)3,

$51.

X

LEGENDRE P,(X)

339

FUNCTIONS

/

16'

ii/I I

2

I

I

I

I

I

I

I

I

3

4

5

6

7 PO

8

9

IO

FIQURE

8.3.

P,(z).

n=0(1)3,

z 11.

P”(X) A

- \ ‘\\,\ ‘\ ‘\\ 10-3 - ‘\ ‘Y, ‘\ ‘L, ‘\ ” \ IO-4 = ‘\ \ \\ \ \

‘.

DX

‘\

*\ . I\

1. lO-5

I

I

I

I

I

I

2

3

4

5

6

FIQURE

8.5.

Qn(z).

Numerical 8.15.

Use and

8.4.

Q,,(z).

n=0(1)3,

z
n=0(1)3,

Q2 -.

0, I‘\”

)

I

Y

8

9

IO

‘.

~>l.

Methods

Extension

Computation

FIQIJRE

I 7

. ‘.

of the Tables of P,(z)

For all values of 2 there is very little loss of significant figures (except at zeros) in using the recurrence relation 8.5.3 for increasing values of n. Example 1. Comput,eP,(x) forz=.31415 92654 and ~=2.6 for n=2(1)8.

340

LEGENDRE

rUNC’l?IONS

P,(.31415 92654) P,GW 1 0” 1 1 . 31415 92654 2. 6 2 -. 35195 59340 9. 64 3 -. 39372 32064 40.04 174.952 4 . 04750 63 122 5 . 34184 27517 786.74336 6 . 15729 86975 3604.350016 7 -. 20123 39354 16729.51005 8 -. 25617 29328 78402.55522 Computing P&) using Table 22.9 carrying ten significant figures, P,(.31415 92654)=-.256f7 2933 and P,(2.6)=78402.55526. Computation

Compute Q6(z) for x=2.6. v+2 v+1 Ten terms in the seriesfor F 292; v+$

Q.(.31415

0” 1 2 3 4 5

1 2 3 4

Computation

P&i

of P&t(z),

Q++(s)

For all values of z, P,i(x) and Q**(z) are most easily computed by means of 8.13. Example 4. Compute Q-+(z) for x=2.6. Using 8.13.3 and interpolating in Table 17.1 for K(.C$, we find

92654)

Using the results of Example 8.6.19. we find

Qm(2.6) .40546 51081 05420 928 : 00868 364 00148 95 : 00026 49 a00004 81

where Q. and Q1are obtained using 8.4.2 and 8.4.4.

.32515 34813 -. 89785 00212 -. 58567 85953 .29190 60854 .59974 26989

;4

$)

of 8.1.3 are necessary to obtain nine significant figures giving Q5(2.6) =4.8182 4468 X 10V5. Using 8.5.3 with increasing values of n carrying ten significant figures we obtain

of Q,,(z)

For xl, the recurrence relation 8.5.3 should be used only for decrea.sing values of n, after having first obtained Qn using the formulas in terms of hypergeometric functions. Example 2. Compute Qn(z) for z= .31415 92654 and n=O(l)4. With the aid of 8.4.2 and 8.4.4 we obtai7,

where W3=f

Example 3.

Q4(z) = +P4(r)ln

= (.74535 59925) (1.90424 1417) = 1.41933 7751.

1 together with (

2

>

-W,(x)

P,, giving Q4(.31415 92&X)=

On the other hand, at least nine terms in the v+2 v+1. 3. 1 expansion of F 2, 2, V-I-~, 2 of 8.1.3 are necessary to obtain comparable accuracy.

.59974 26989.

References Texts

[8.1] A. Erdelyi et al., Higher transcendental functions, vol. 1, ch. 3 (McGraw-Hill Book Co., Inc., New York, N.Y., 1953). [8.2] E. W. Hobson, The theory of spherical and ellipsoidal harmonics (Chelsea Publishing CO., New York, N.Y., 1955). [8.3] J. Lense, Kugelfunktionen (Akademische Verlagsgesellschaft, Leipzig, Germany, 1950). [8.4] T. M. MacRobert, Spherical harmonics, 2d rev. ed. (Dover Publications, Inc., New York, N.Y., 1948). [8.5] W. Magnus and F. Oberhettinger, Formulas and theorems for the special functions of mathematical physics (Chelsea Publishing Co., New York, N.Y., 1949). La.61 G. Prasad, A treatise on spherical harmonics and the functions of Bessel and Lame, Part II (Advanced) (Mahamandal Press, Benares City, India, 1932).

[8.7] L. Robin, Fonctions spheriques de Legendre e fonctions spheroidales. Tome I, II, III (GauthierVillars, Paris, France, 1957). [8.8] C. Snow, Hypergeometric and Legendre functions with applications to integral equations of potential theory, NBS Applied Math. Series 19 (U.S. Government Printing Office, Washington, D.C., 1952). [8.9] R. C. Thorne, The asymptotic expansion of Legendre functions of large degree and order, Philos. Trans. Roy. Sot. London 249, 597-620 (1957). Tables

[S.lO] H. Bateman, Some problems in potential theory, Mess. of Math., 617,52,73-75 (1922). P,(cosh u), Q,(cosh o), P:.(cosh u), Q.(cosh u); cash m=l.l, n=0(1)20, 1OD; cash u=1.2, 2, 3; n=O(l)lO, exact or 10D.

LEGENDRE

[8.11] Centre National d’fithdes des Telecommunications, Tables des fonctions de Legendre associ&es. Fonction associee de premiere espece P:(COS 0) (Editions de La Revue d’optique, Paris, France, 1952). n= -+(.l)lO, W&=0(1)5, e=o”(l”)90” (variable number of figures). [8.12] Centre National d’fitudes des .TelBcommunications, Tables numerique des fonctions associees de Legendre. Fonctions associees de premiere espece P:(cos I?) (Editions de La Revue d’Optique9 Paris, France, 1959). n= -$(.l)lO, m=0(1)2, 0=0”(1”) 180’ (variable number of figures). [8.13] G. C. Clark and S. W. Churchill, Table of Legendre polynomials P.(cos 0) for n=0(1)80 and O=O” (l”)lSO”, Engineering Research Institute Publications (Univ. of Michigan Press, Ann Arbor, Mich., 1957). [8.14] R. 0. Gumprecht and G. M. Sliepcevich, Tables of functions of the first and second partial derivatives of Legendre polynomials (Univ. of Michigan Press, Ann Arbor, Mich., 1951). Values of [XT,, -(l--22)*:]. 1W and r,lO’ for ~y=O”(100)1700 (1“)180’, n=1(1)420, 55. [8.15] M. E. Lynam, Table of Legendre functions for complex arguments TG-323, The Johns Hopkins Univ. Applied Physics Laboratory, Baltimore, Md. (1958). [8.16] National Bureau of Standards, Tables of associated Legendre functions (Columbia Univ. Press, New York,

N.Y.,

n= l(1) 10,

1945).

P:(cos

m( sn)=0(1)4,

e), $

P,D(cos .9),

~=0”(1°)900,

341

FUNCTIONS

6s;

P?(x), $ e?(x),

n=l(l)lO,

C-1) m+l%Q~(x), n=O(l)lO, 6s or exact; +P:(ti), in+2m+lQYiz), =0(1)4,

m(
i”+lm--l & Q:(k),

r=O(.l)lO,

(- l)mQ:(x),

6s;

n=o(l)lo, P,“+)(z),

x=1(.1)10, n=l(l)lO, m( In) & P,“-&(z),

(-l)mQ:-*(x), (-l)m+l 2 Qz+t, n= -1(1)4, m=0(1)4, z=l(.l)lO, 4-6s. [8.17] G. Prevost, Tables des fonctions spheriques et de leurs intAerales (Gauthier-Villars, Bordeaux and Paris, France, 1933).

P,(x),

zP,(t)dt,n=l(l)lO; .J-0 j=O(l)n, x=0(.01)1,

n=0(1)8, Pi (xl, s 0z Pi(t)& 5s. [8.18] H. Tallqvist, Sechsstellige Tafeln der 32 ersten Kugelfunktionen P,(cos e), Acta Sot. Sci. Fenn., Nova Series A, II, 11 (1938). P,(cos e), n=1(1)32, O”(10’)900; 6D. [8.19] H. Tallqvist, Acta Sot. Sci. Fenn., Nova Series A, II, 4(1937). P,(x), n=1(1)16, z=O(.OOl)l, 6D. [8.20] H. Tallqvist, Tafeln der Kugelfunktionen P,(cos 0) bis P&cos 8), Sot. Sci. Fenn. Comment. PhysMath., VI, lO(1932). P,(cos e), n=25(1)32, e=o”(lo)900,

5D.

[8.21] H. Tallqvist, Tafeln der 24 ersten Kugelfunktionen P,(cos e), Sot. Sci. Fenn. Comment. Phys-Math., P,(cos o), n=1(1)24, 0=O”(l”)90”, VI, 3(1932). 5D.

LEGENDRE

342

FUNCTIONS

LEGENDRE FUNCTION-FIRST PO(Z) = 1 PI(Z)

Table 8.1

050 0. 01 0. 02 0.03 0.04

arcco.55 90.00000 00 89.42703 26 88.85400 80 88.28086 87 87.70755 72

Pdx) -0.50000 -0.49985 -0.49940 -0.49865 -0.49760

-0.01499 -0.02998 -0.04493 -0.05984

0.05 0.06 0.07 0.08 0.09

87.13401 86.56018 85.98601 85.41143 84.83639

60 72 28 43 29

-0.49625 -0.49460 -0.49265 -0.49040 -0.48785

0.10 0.11 0.12 0.13 0.14

84.26082 83.68468 83.10789 82.53040 81.95215

95 44 74 77 37

0.15 0.16 0.17 0.18 0.19

81.37307 80.79310 80.21218 79.63024 79.04721

0.20 0.21 0.22 0. 23 0.24

Pdd 0.00000

KIND P,(x) =z PlO(4

P9(4 0.00000

000

75 00 25 00

0.02457 0.04893 0.07285 0.09614

-0.07468 -0.08946 -0.10414 -0.11872 -0.13317

75 00 25 00 75

-0.48500 -0.48185 -0.47840 -0.47465 -0.47060

-0.14750 -0.16167 -0.17568 -0.18950 -0.20314

34 38 10 02 58

-0.46625 -0.46160 -0.45665 -0.45140 -0.44585

78.46304 77.87764 77.29096 76.70292 76.11345

10 77 70 82 96

0.25 0.26 0.27 0.28 0.29

75.52248 74.92993 74.33573 73.73979 73.14204

0.30 0.31 0.32 0.33 0.34 0.35 0.36 0.37

330 045 701 188

-0.24609 -0.24474 -0.24069 -0.23400 -0.22473

37 14 84 69 64

0.11857 0.13996 0.16012 0.17885 0.19599

899 890 040 206 366

-0.21298 -0.19887 -0.18254 -0.16418 -0.14397

35 11 68 20 02

00 25 00 75 00

0.21138 0.22489 0.23637 0.24572 0.25285

764 042 363 526 070

-0.12212 -0.09887 -0.07447 -0.04918 -0.02328

50 86 93 90 12

-0.21656 -0.22976 -0.24271 -0.25542 -0.26785

25 00 75 00 25

0.25767 0.26013 0.26020 0.25785 0.25309

367 706 358 632 918

+0.00296 0.02925 0.05529 0.08080 0.10549

18 20 81 85 42

-0.44000 -0.43385 -0.42740 -0.42065 -0.41360

-0.28000 -0.29184 -0.30338 -0.31458 -0.32544

00 75 00 25 00

0.24595 0.23647 0.22472 0.21078 0.19477

712 631 407 870 914

0.12907 0.15126 0.17181 0.19047 0.20700

20 74 75 36 49

78 79 31 53 40

-0.40625 100':gg -0:38240 -0.37385

-0.33593 -0.34606 -0.35579 -0.36512 -0.37402

75 00 25 00 75

0.17682 0.15707 0.13569 0.11286 0.08879

442 305 215 642 707

0.22120 0.23287 0.24185 0.24801 0.25124

02 14 52 62 81

72.54239 71.94076 71.33707 70.73122 70.12312

69 95 51 45 59

-0.36500 -0.35585 -0.34640 -0.33665 -0.32660

-0.38250 -0.39052 -0.39808 -0.40515 -0.41174

00 25 00 75 00

0.06370 0.03780 +0.01135 -0.01539 -0.04219

038 634 691 566 085

0.25147 0.24865 0.24278 0.23389 0.22203

63 91 89 37 73

E9" .

69.51268 68.89980 68.28438 67.66631 67.04550

49 39 27 73 06

-0.31625 -0.30560 -0.29465 -0.28340 -0.27185

-0.41781 -0.42336 -0.42836 -0.43282 -0.43670

25 00 75 00 25

-0.06876 -0.09483 -0.12014 -0.14441 -0.16737

185 780 608 472 489

0.20732 0.18987 0.16988 0.14754 0.12310

00 83 48 72 73

0.40 0.41 0.42 0.43 0.44

66.42182 65.79516 65.16541 64.53243 63.89611

15 52 25 99 88

-0.26000 -0.24785 -0.23540 -0.22265 -0.20960

-0.44000 -0.44269 -0.44478 -0.44623 -0.44704

00 75 00 25 00

-0.18876 -0.20832 -0.22581 -0.24101 -0.25369

356 609 900 269 426

0.09683 0.06904 0.04006 +0.01024 -0.02002

91 71 39 69 45

0.45 0.46 0.47 0.48 0.49

63.25631 62.61289 61.96570 61.31459 60.65941

61 25 35 80 84

-0.19625 -0.18260 -0.16865 -0.15440 -0.13985

-0.44718 -0.44666 -0.44544 -0.44352 -0.44087

75 00 25 00 75

-0.26367 -0.27076 -0.27484 -0.27577 -0.27348

022 932 521 908 225

-0.05035 -0.08072 -0.10952 -0.13752 -0.16389

30 72 64 51 87

0.50

60.00000 00 (-;)5

-0.12500 (-l)4

-0.43750 00

[

1

[ 1

PZQ) =+(-1+&2)

00

[y9 1

-0.26789 856 (-i)4

[ 1

-0.18822 86 (-;)4

[

P3(z) =E (-3+5x2)

2 pg(z)=&(1260-18480$+72072&102960s6+48620z8) Plo(x)

=&4(-252+1386Ox2-12Ol2Oz*+36O36OxE-4375SOzs+l84756z1")

(n+l)P,+1(~)=(2n+l)xPn(x) For coefficientsof other polynomials,see chapter 22.

-nP,-l(X)

1

LEGENDRE

LEGENDRE FUNCTION-FIRST Po(z)=l

o.to

pzw

343

FUNCTIONS KIND P,(r)

Table 8.1

PI(X) =2

arccos z 60.00000 00 59.33617 03 58.66774 85 57.99454 51 57.31636 11

-0.12500 -0.10985 -0.09440 -0.07865 -0.06260

-0.43750 -0.43337 -0.42848 -0.42280 -0.41634

00 25 00 75 00

-0.26789 -0.25900 -0.24682 -0.23139 -0.21283

856 667 215 939 321

-0.18822 86 -0.21010 83 -0.22914 92 -0.24498 73 -0.25728 92

56.63298 55.94420 55.24977 54.54945 53.84299

70 22 42 74 18

-0.04625 -0.02960 -0.01265 +0.00460 0.02215

-0.40906 -0.40096 -0.39201 -0.38222 -0.37155

25 00 75 00 25

-0.19126 -0.16686 -0.13985 -0.11051 -0.07914

025 000 552 366 497

-0.26575 85 -0.27014 28 -0.27023 97 -0.26590 30 -0.25704 92

0.60 0.61 0.62 0.63 0.64

53.13010 52.41049 51.68386 50.94987 50.20818

24 70 55 75 05

0.04000 0.05815 0.07660 0.09535 0.11440

-0.36000 -0.34754 -0.33418 -0.31988 -0.30464

00 75 00 25 00

-0.04610 -0.01178 +0.02337 0.05890 0.09430

304 332 862 951 141

-0.24366 27 -0.22580 16 -0.20360 19 -0.17728 16 -0.14714 41

0.65 0.66 0.67 %.

49.45839 48.70012 47.93293 47.15635 46.36989

81 72 52 69 11

0.13375 0.15340 0.17335 0.19360 0.21415

-0.28843 -0.27126 -0.25309 -0.23392 -0.21372

75 00 25 00 75

0.12901 0.16248 0.19412 0.22334 0.24952

554 693 981 410 270

-0.11358 05 -0.07707 01 -0.03818 08 +0.00243 30 0.04403 37

0.70 0. 71 0.72 0.73 0. 74

45.57299 44.76508 43.94551 43.11360 42.26858

60 47 96 59 44

t ::z 0:27760 0. 29935 0.32140

-0.19250 -0.17022 -0.14688 -0.12245 -0.09694

00 25 00 75 00

0.27205 0.29036 0.30385 0.31199 0.31430

993 111 323 698 004

0.08580 58 0.12686 31 0.16625 89 0.20299 76 0.23605 08

0.75 0.76 0.77 0.78 0.79

41.40962 40.53580 39.64611 38.73942 37.81448

21 21 11 46 85

-0.07031 -0.04256 it ;8696;50 -0.01366 0:41260 +0.01638 0.43615 0.04759

25 00 75 00 75

0.31033 0.29973 0.28226 0.25777 0.22625

185 981 712 224 012

0.26437 45 0.28693 19 0..3027179 0.31078 93 0.31029 79

0.80 0. 81 0.82 0.83 0.84

36.86989 35.90406 34.91520 33.90126 32.85988

76 86 62 20 04

0.46000 0.48415 0.50860 0.53335 0.55840

0.08000 0.11360 0.14842 0.18446 0.22176

00 25 00 75 00

0.18785 0.14292 0.09201 +0.03591 -0.02431

528 678 529 226 874

0.30052 98 0.28094 87 0.25124 52 0.21139 19 0.16170 50

0. 85 0.86 0.87 0.88 0. 89

31.78833 30.68341 29.54136 28.35763 27.12675

06 71 05 66 31

0.26031 0.30014 0.34125 0.38368 0.42742

25 00 75 00 25

-0.08730 -0.15134 -0.21433 -0.27376 -0.32665

820 456 544 627 610

0.10291 23 +0.0362291 -0.03655 86 -0.11300 29 -0.18989 29

0.90 0. 91 0.92 0.93 0.94

25.84193 24.49464 23.07391 21.56518 19.94844

28 85 81 50 36

0.47250 0.51892 0.56672 0.61589 0.66646

00 75 00 25 00

-0.36951 -0.39827 -0.40826 -0.39414 -0.34981

049 146 421 060 919

-0.26314 56 -0.32768 58 -0.37731 58 -0.40457 43 -0.40058 29

18.19487 16.26020 14.06986 11.47834 8.10961

23 47 77 09 44

0.85375 : %i Of94060 0.97015

0.71843 0.77184 0.82668 0.88298 0.94074

75 00 25 00 75

-0.26842 -0.14220 +0.03750 0.28039 0.59724

182 642 397 609 553

-0.35488 03 -0.25524 34 -0.08749 40 +0.16470 81 0.52008 90

0.00000 00

1.00000 (-Z)4

1.00000 00 C-32

1.00000 000

1.00000 00 c-y

0.51 0.52 0.53 0.54

P3(4

0.34375

II 1

[

P9@)

PI,(Z)

II 1

1

Pz(z)=F(-3+5~2) 2 (1260-18480x2+7207224-10296Ch++486202*)

1'29)=4(-1+322)

Pg(x)=&

PI&Z) =~4(-252+1386~2-12O12~~+36036O.xs-43758W+l8475fW') (n+l)Pn+1@)

= (2n+l)zP&)

For coefficients of otherpolynomials, seechapter22.

-nPn-l(2)

344

LEGENDRE Table

8.2

DERIVATIVE

OF

THE

P',(x)=1

I%(x)

FUNCTIONS LEGENDRE

FUNCTION-FIRST

KIND

P:(x)

P;(x)=32

0.01 0.02 0.03 0. 04

-1.50000 -1.49925 -1.49700 -1.49325 -1.48800

-0.07498 -0.14986 -0.22452 -0.29888

25 00 75 00

Pi(x) 2.46093 2.45011 2.41773 2.36405 2.28948

Piob) 0.00000 0.27023 0.53765 0.79949 1.05299

00 41 93 17 82

0. 05 0.06 0.07 0. 08 0.09

-1.48125 -1.47300 -1.46325 -1.45200 -1.43925

-0.37281 -0.44622 -0.51899 -0.59104 -0.66224

25 00 75 00 25

2.19461 13 2.08018 11 1.94709 32 1.79639 87 1.62929 31

1.29552 1.52449 1.73750 1.93223 2.10657

05 98 05 25 29

0.10 0.11 0.12 0.13 0.14

-1.42500 1: ;g; -1:37325 -1.35300

-0.73250 -0.80170 -0.86976 -0.93655 -1.00198

00 75 00 25 00

1.44710 1.25130 1.04346 0.82528 0.59853

87 64 68 00 47

2.25858 2.38654 2.48895 2.56453 2.61230

73 80 24 90 18

0.15 0.16 0.17

-1.33125 -1.30800 -1.28325 -1.25700 -1.22925

-1.06593 -1.12832 -1.18902 -1.24794 -1.30496

75 00 25 00 75

0.36510 +0.12694 -0.11392 -0.35546 -0.59555

73 88 76 01 27

2.63150 2.62168 2.58266 2.51458 2.41783

28 25 81 04 68

-1.20000 1;. :;;g -1:10325 -1.06800

-1.36000 -1.41293 -1.46366 -1.51207 -1.55808

00 25 00 75 00

-0.83208 -1.06295 -1.28602 -1.49923 -1.70052

96 03 54 18 94

2.29315 2.14154 1.96431 1.76306 1.53966

33 35 51 37 43

-1.03125 -0.99300 -0.95325 -0.91200 -0.86925

-1.60156 -1.64242 -1.68054 -1.71584 -1.74819

25 00 75 00 25

-1.88793 -2.05954 -2.21355 -2.34823 -2.46202

72 92 15 78 63

1.29625 1.03524 0.75926 0.47115 +0.17398

99 77 26 77 30

-0.82500 -0.77925 -0.73200 -0.68325 -0.63300

-1.77750 -1.80365 -1.82656 -1.84610 -1.86218

00 75 00 25 00

-2.55347 -2.62129 -2.66437 -2.68178 ~2.67279

51 80 95 96 74

-0.12903 -0.43453 -0.73903 -1.03894 -1.33065

87 90 23 72 96

-0.58125

0:39

I:* ::;;; -0:41700 -0.35925

-1.87468 -1.88352 -1.88857 -1.88974 -1.88691

75 00 25 00 75

-2.63688 -2.57375 -2.48336 -2.36588 -2.22176

47 82 07 14 52

-1.61052 -1.87493 -2.12030 -2.34318 -2.54023

81 10 43 21 74

0.40 0.41 0.42 0.43 0.44

-0.30000 -0.23925 -0.17700 -0.11325 -0.04800

-1.88000 -1.86888 -1.85346 -1.83362 -1.80928

00 25 00 75 00

-2.05172 -1.85672 -1.63802 -1.39715 -1.13592

01 35 69 86 50

-2.70832 -2.84451 -2.94616 -3.01090 -3.03674

36 75 13 51 96

0.45 0.46 0.47 0.48 0.49

+0.01875 0.08700 0.15675 0.22800 0.30075

-1.78031 -1.74662 -1.70809 -1.66464 -1.61614

25 00 75 00 25

-0.85640 -0.56096 -0.25222 +0.06693 0.39337

91 76 53 30 29

-3.02208 -2.96573 -2.86699 -2.72566 -2.54206

63 83 80 30 98

0.50

0.37500

-1.56250

00

2

0.00

E. 0.20 E 0:23 0.24 0.25 it:; 0:28 0.29 0.30 0.33 0.34 0.35 0.36

p;.

Pi(x) 0.00000

00

[ 1

qw

59‘3

[ 1

75 64 75 34 35

0.72372 44 (-f)3

[

1

P&=+(-3+1w) PI,(z)=~2(126O-5544o22+36O36Ch?-72O'72Ox~+43758Oa+) P~o(~)=&4(277~O-48O48Ch~+216216&t?-35OO64W'+184756W) n+l [zP&) -Pn+1(41 Pi%(z)= 1-22

-2.31712 34 (-!)5

[ 1

LEGENDRE DERIVATIVE

OF

THE

LEGENDRE

345

FUNCTIONS

FUNCTION-FIRST

Pi(x)=1

KIND

P:(x)

Table

8.2

P;(x)=32

Pie(x)

x

Pi(x)

0.50 0.51 0.52 0.53 0.54

0.37500 0.45075 0.52800 0.60675 0.68700

-

1.56250 1.50360 1.43936 1.36965 1.29438

00 75 00 25 00

0.72372 1.05439 1.38160 1.70137 2.00958

44 75 24 21 86

-

0.55 0.56 0.57 0. 58 0.59

0.76875 0.85200 0.93675 1.02300 1.11075

-

1.21343 1.12672 1.03412 0.93554 0.83086

75 00 25 00 75

2.30201 2.57431 2.82213 3.04106 3.22677

29 87 05 49 77

- 0.64649 54 - 0.22698 16 + 0.21005 92 0.65868 10 1.11234 92

0.60 0. 61 0.62 0.63 0.64

1.20000 1.29075 1.38300 1.47675 1.57200

-

0.72000 0.60283 0.47926 0.34917 0.21248

00 25 00 75 00

3.37501 3.48166 3.54283 3.55487 3.51451

44 60 00 57 63

1.56397 2.00598 2.43034 2.82866 3.19230

82 31 08 68 45

0.65 0.66 0.61 0.68 0.69

1.66875 1.76700 1.86675 1.96800 2.07075

- 0.06906 25 + 0.08118 00 0.23835 25 0.40256 00 0.57390 75

3.41888 3.26561 3.05294 2.77978 2.44582

50 84 51 03 82

3.51243 3.78017 3.98677 4.12369 4.18284

07 74 13 16 84

0.70 0.71 0.72 0.73 0.14

2.17500 2.28075 2.38800 2.49675 2.60700

0.75250 0.93844 1.13184 1.33279 1.54142

00 25 00 75 00

2.05168 1.59891 1.09043 + 0.53008 - 0.07667

93 66 73 28 36

4.15678 4.03888 3.82364 3.50693 3.08626

18 45 72 03 20

0.15 0.76 0.77 0.78 0.79

2.71875 2.83200 2.94675 3.06300 3.18075

1.75781 1.98208 2.21432 2.45466 2.70318

25 00 75 00 25

-

0.72287 1.39984 2.09708 2.80201 3149987

14 93 32 52 45

2.56116 1.93351 1.20791 + 0.39215 - 0.50239

49 26 71 05 96

0.80 0.81 0.82 0.83 0.84

:* Ei 3:54300 3.66615 3.79200

2.96000 3.22521 3.49894 3.78127 4.07232

00 75 00 25 00

=. 4.17348 - 4.80308 - 5.36607 - 5.83686 - 6.18657

81 26 64 10 35

-

1.46023 2.46122 3.48002 4.48547 5.43990

77 91 97 21 91

0.85 0.86 0.81 0.88 0.89

3.91875 4.04700 4.17675 4.30800 4.44075

4.37218 4.68098 4.99880 5.32576 5.66195

75 00 25 00 75

-

68 06 97 23 54

-

6.29851 7.00851 1.50840 7.72711 7.58303

03 07 93 51 90

0.90 0.91 0.92 0.93 0.94

4.57500 4.71075 4.84800 4.98675 5.12700

6.00750 6.36249 6.72704 7.10124 7.48522

00 25 00 75 00

- 3.65335 89 - 2.02101 73 + 0.11150 20 2.81447 18 6.16433 35

+

6.98312 5.82184 3.98006 1.32394 2.29628

79 03 04 73 14

0.95 0.96 0.97

5.26875 5.41200 5.55675

1.87906

25

"0*9'9" .

2. E5" 6.00000 c-y

1.04163 13.11571 20.70612 30.04600 41.38561

58 11 01 25 43

1.00

P;(x)

8.28288 8.69677 9.12086 9.55523

P;(x)

10.24405 15.14351 20.95987 27.79800 35.77086

00 75 00 25

[I

P;(z)=;

P’lo(~)=&~

70 59 66 16 77

45.00000 00 (-y

[ 1 P;(z)=&

6.38285 6.38961 6.16612 5.66983 4.84997

(-

2.31712 2.05232 1.74978 1.41226 1.04315

55.00000 00 (-;)3

[I

60+140s2)

(1260-554&0s2+360360z4-720'720s~+437580z8) ([email protected]~+184756'h*)

P:(x)=g2 [xPn(x) -P,+,(x)]

34 40 82 67 43

346

LEGENDRE

FUNCTIONS

Table

8.3

0.00

0.00000 000 -1.00000 0.01000 0.02000 0.03000 0.04002

033 267 900 135

-0.99990 -0.99959 -0.99909 -0.99839

000 000 995 973 915

0.00000 -0.01999 -0.03998 -0.05996 -0.07991

000 867 933 399 463

0.66666 0.66626 0.66506 0.66306 0.66027

667 669 699 829 179

-0.40634 -0.40452 -0.39905 -0.38999 -0.37741

921 191 538 553 852

0.00000 -0.04056 -0.08068 -0.11993 -0.15789

000 181 584 860 513

E2 0:07 0.08 0.09

0.05004 0.06007 0.07011 0.08017 0.09024

173 216 467 133 419

-0.99749 -0.99639 -0.99509 -0.99358 -0.99187

791 567 197 629 802

-0.09983 -0.11971 -0.13954 -0.15931 -0.17902

321 169 199 602 563

0.65667 0.65229 0.64711 0.64114 0.63439

917 261 475 873 817

-0.36143 -0.34216 -0.31978 -0.29448 -0.26647

026 562 750 565 538

-0.19414 -0.22828 -0.25995 -0.28879 -0.31447

321 745 321 038 701

0.10 0.11 0.12 0.13 0.14

0.10033 0.11044 0.12058 0.13073 0.14092

535 692 103 985 558

-0.98996 -0.98785 -0.98553 -0.98300 -0.98027

647 084 028 382 042

-0.19866 -0.21821 -0.23768 -0.25705 -0.27631

264 885 596 567 958

0.62686 0.61856 0.60948 0.59964 0.58903

720 044 299 048 905

-0.23599 -0.20330 -0.16869 -0.13245 -0.09491

595 891 616 792 050

-0.33672 -0.35527 -0.36990 -0.38044 -0.38675

259 122 435 330 142

0.15 0.16 0.17 0.18 0.19

0.15114 0.16138 0.17166 0.18198 0.19233

044 670 666 269 717

-0.97732 -0.97417 -0.97081 -0.96724 -0.96345

893 813 667 312 594

-0.29546 -0.31449 -0.33339 -0.35214 -0.37075

923 610 158 699 353

0.57768 0.56558 0.55275 0.53918 0.52489

532 646 016 465 868

-0.05638 -0.01721 +0.02223 0.06161 0.10057

395 959 260 670 361

-0.38873 -0.38634 -0.37958 -0.36850 -0.35318

587 905 962 308 198

0.20 0. 21 0.22 0. 23 0.24

0.20273 0.21317 0.22365 0.23418 0.24477

255 135 611 947 411

-0.95945 -0.95523 -0.95079 -0.94613 -0.94125

349 402 566 642 421

-0.38920 -0.40748 -0.42559 -0.44351 -0.46123

232 439 062 180 857

0.50990 0.49420 0.47781 0.46074 0.44300

155 314 388 738

0.13874 0.17577 0.21130 0.24499 0.27652

395 093 336 861 557

-0.33376 -0.31043 -0.28343 -0.25302 -0.21951

565 947 378 221 969

0.25 0.26 0.27 0.28 0.29

0.25541 0.26610

281 841

0.27686

382

0.28768 0.29856

207 626

-0.93614 -0.93081 -0.92524 -0.91944 -0.91341

680 181 677 902 578

-0.47876 -0.49607 -0.51315 -0.53000 -0.54661

145 081 685 962 900

0.42461 0.40557 0.38591 0.36562 0.34474

393 719 059 819 467

0.30556 0.33182 0.35502 0.37489 0.39122

765 571 089 746 551

-0.18327 -0.14469 -0.10417 -0.06219 -0.01920

994 251 949 173 468

0.30 0.31 0.32 0.33 0.34

0.30951 0.32054 0.33164 0.34282 0.35409

960 541 711 825 253

-0.90714 -0.90063 -0.89387 -0.88686 -0.87960

412 092 293 668 854

-0.56297 -0.57906 -0.59488 -0.61041 -0.62564

466 608 256 313 662

0.32327 0.30123 0.27864 0.25551 0.23187

542 647 459 723 261

0.40380 0.41246 0.41705 0.41749 0.41371

351 080 981 822 084

+0.02428 0.06776 0.11072 0.15262 0.19295

610 975 534 723 076

0.35 0.36 0.37 i%.

0.36544 0.37688 0.38842 0.40005 0.41180

375 590 310 965 003

-0.87209 -0.86432 -0.85628 -0.84797 -0.83939

469 108 345 733 799

-0.64057 -0.65517 -0.66944 -0.68337 -0.69694

159 633 887 690 784

0.211772 0.18310 0.15802 0.13251 0.10658

970 825 883 285 256

0.40567 0.39339 0.37693 0.35638 0.33189

128 336 227 546 317

0.23117 811 0.26680 432 0.29934 337 0.32833 437 0.35334 774

0.40 0.41 0.42 0.43 0.44

0.42364 0.43561 0.44769 0.45989 0.47223

893 122 202 668 080

-0.83054 -0.82139 -0.81196 -0.80224 -0.79221

043 940 935 443 845

-0.71014 -0.72296 -0.73538 -0.74739 -0.75897

872 624 670 600 958

0.08026 0.05357 +0.02654 -0.00080 -0.02843

114 267 221 418 939

0.30363 0.27184 0.23679 0.19877 0.15815

867 811 006 461 208

0.37399 0.38991 0.40082 0.40646 0.40665

123 596 218 477 845

0.45 0.46 0.47 0.48 0.49

0.48470 0.49731 0.51007 0.52298 0.53606

028 129 034 428 034

-0.78188 -0.77123 -0.76026 -0.74896 -0.73733

487 681 694 755 044

-0.77012 -0.78080 -0.79102 -0.80074 -0.80996

243 904 336 877 804

-0.05633 -0.08446 -0.11279 -0.14128 -0.16992

524 239 034 732 027

0.11531 0.07067 +0.02471 -0.02210 -0.06923

136 773 030 100 897

0.40128 0.39028 0.37368 0.35158 0.32415

259 551 827 779 933

0.50

0.54930 [(-y3

614

-0.72534 693 [(-;)"I

-0.81866 ['-y]

327

-0.\19865 477 ['-y-j

0. 01 0.02 0. 03 0. 04

LEGENDRE

Qo(4

FUNCTION-SECOND

&I (4

Qd4

KIND

Q&3

476

t&x(%)

Qdx)

-0.11616 303 ['-;)"I

Qo(x)=t In (2) &z(x)= 3pn

Qs(~)=f(5~~-3)1n(~)-~+$

(~+1)&~+~(~)=(2~+1)xQn(x)-nQn-1(x) :arctanhx (Tab 11e 4.17) is included here for completeness. Qo(4 =

910(x)

0.29165 814 [(-;)'I

LEGENDRE

LEGENDRE

347

FUNCTIONS

FUNCTION-SECOND

Table

KIND

Q&4

8.3

QloW

0.250 0.51 0.52 0.53 0.54

0 54930 QoC4 614 0:56272 977 0.57633 975 0.59014 516 0.60415 560

-0.72534Ql(x)693 -0.71300 782 -0.70030 333 -0.68722 307 -0.67375 597

-0.81866Qdx) 327 -0.82681 587 -0.83440 647 -0.84141 492 -0.84782 014

-0.19865Q3W477 -0.22745 494 -0.25628 339 -0.28510 113 -0.31386 748

-0.11616 -0.16231 -0.20711 -0.24999 -0.29035

303 372 759 263 406

+0.29165 0.25442 0.21286 0.16748 0.11884

814 027 243 08i7 913

0.55 0.56 0.57 Oi58 0.59

0.61838 0.63283 0.64752 0.66246 0.67766

131 319 284 271 607

-0.65989 -0.64561 -0.63091 -0.61577 -0.60017

028 342 198 163 702

-0.85360 -0.85873 -0.86319 -0.86695 -0.86998

014 186 116 267 970

-0.34253 -0.37107 -0.39942 -0.42753 -0.45537

994 413 362 983 186

-0.32762 -0.36122 -0.39060 -0;41523 -0.43463

069 172 386 901 218

0.06761 +0.01449 -0.03973 -0.09422 -0.14810

47'0 441 1414 6310 594

0.60 0.61 0.62 0: 63 0. 64

0.69314 0.70892 0.72500 0.74141 0.75817

718 136 509 614 374

-0.58411 -0.56755 -0.55049 -0.53290 -0.51476

169 797 685 783 880

-0.87227 -0.87377 -0.87446 -0.87430 -0.87326

411 622 461 597 492

-0.48286 -0.50996 -0.53661 -0.56274 -0.58830

632 718 553 938 338

-0.44832 -0.45592 -0.45708 -0.45151 -0.43903

986 864 410 989 693

-0.20044 -0.25030 -0.29671 -0;33872 -0.37537

847 577 648 031 391

0.65 0.66 0.67 0.68 0.69

0.77529 871 0.79281 363 0.81074 313 0.82911 404 0.84795 576

-0.49605 -0.47674 -0.45680 -0.43620 -0.41491

584 300 211 245 053

-0.87130 -0.86838 -0.86445 -0.85948 -0.85341

380 239 768 352 027

-0.61320 -0.63739 -0.66077 -0.68327 -0.70481

855 196 634 969 480

-0.41952 -0.39296 -0.35943 -0.31915 -0.27244

271 048 834 810 363

-0.40576 -0.42904 -0.44442 -0.45121 -0.44884

815 673 606 636 377

0.70 0; 71 0. 72 0.73 0.74

0.86730 053 0.88718 386 0.90764 498 0.92872 736 0.95047 938

-0.39288 -0.37009 -0.34649 -0.32202 -0.29664

963 946 561 902 526

-0.84618 -0.83774 -0.82803 -0.81698 -0.80451

438 785 775 546 593

-0.72528 -0.74460 -0.76264 -0.77931 -0.79447

868 199 823 296 280

-0.21974 -0.16166 -0.09892 -0.03241 +0.03684

878 443 467 178 038

-0.43687 -0.41503 -0.38323 -0.34160 -0.29049

329 236 471 431 884

0.75 0.76 0.77

0.97295 0.99621 1.02032 1.04537 1.07143

507 508 776 055 168

-0.27028 -0.24287 -0.21434 -0.18461 -0.15356

369 654 763 097 897

-0.79054 -0.77498 -0.75773 -0.73868 -0.71769

669 679 539 011 507

-0.80799 -0.81973 -0.82952 -0.83721 -0.84258

424 225 866 016 586

0.10764 474 0.17866 149 0.24840 151 0.31523 275 0.37739 063

-0.23053 -0.26259 -0.08787 -0.00787 +0.07559

218 54$ 565 146 560

0.80 0.81 0.82 0.83 0.84

1.09861 229 1.12702 903 1.15681 746 1.18813 640 1.22117 352

-0.12111 -0.08710 -0.05140 -0.01384 +0.02578

017 649 968 678 575

-0.69463 -0.66934 -0.64164 -0.61130 -0.57809

835 890 264 745 671

-0.84544 -0.84555 -0.84263 -0.83641 -0.82652

435 002 849 078 589

0.43299 0.48006 0.51654 0.54037 0.54946

312 146 781 123 418

0.16037 0.24398 0.32364 0.39624 0.45844

522 96 35 17 661 913

0.54183 0.51562 0.46925 0.40147 0.31159

191 828 273 508 776

0.50669 0.53731 0.54659 0.53099 0.48727

726 190 757 253 156

E98 .

0.85 1.25615 0. 86 1.29334 0; 87 1.33307 0.88 1.37576 0.89 1.42192

281 467 963 766 587

0.06772 0.11227 0.15977 0.21067 0.26551

989 642 928 554 403

-0.54172 -0.50183 -0.45802 -0.40979 -0.35650

080 576 786 212 171

-0.81259 -0.79414 -0.77065 -0.74147 -0.70582

105 886 991 880 022

0.90 0.91 0.92 0.93 0.94

1.47221 1.52752 1.58902 1.65839 1.73804

949 443 692 002 934

0.32499 0.39004 0.46190 0.54230 0.63376

754 723 476 272 638

-0.29736 -0.23134 -Oil5708 -0.07268 +0.02458

306 775 489 272 593

-0.66270 -0.61090 -0i54880 -0.47419 -0.38399

962 890 000 336 297

0.19967 037 +0.06677 934 -0.08454 828 -0.24975 925 -0.42137 701

0.41282 0.30602 +0.16680 -0.00265 -0.19666

291 901 029 4215 273

0.95 0.96 0.97 0.98 0.99

1.83178 1.94591 2.09229 2.29755 2.64665

082 015 572 993 241

0.74019 0.86807 1.02952 1.25160 1.62018

178 374 685 873 589

0.13888 0.27707 0.45181 0.69108 1.08264

288 112 370 487 984

-0.27356 -0.13540 +0.04408 0.29436 0.70624

330 204 092 613 831

-0.58752 -0.72921 -0.81464 -0.78406 -0.48875

-0.40421 -0.60564 -0.76587 -0.81720 -0.59305

502 435 179 734 105

1.00

m

co Qo(x) =+ In rz) &2(x) = %J&!ln f:n+l)Qn+l(x)

Ql(x)=f

In ($!$-I

Q~(z)=$

(5x2-3)

= (2%+1)x&n(x)-nn&n-I(X)

In

240 201 729 452 677

348

LEGENDRE

Table

8.4

DERIVATIVE

Q;lCd

OF THE

FUNCTIONS

LEGENDRE

FUNCTION-SECOND

Q;(x)

KIND

Q:(s)

Q’,,(x)

-2.00000 -1.99959 -1.99839 -1.99639 -1.99359

000 998 968 838 487

Qj (4 0.00000 000 -0.07999 200 -0.15993 599 -0.23978 392 -0.31948 767

&k(x) 0.00000 00 0.36520 25 0.72733 83 1.08336 24 1.43027 23

-4.06349 -4.04156 -3.97600 -3.86745 -3.71697

21 71 70 44 43

0. 01 0.02 0.03 0.04

1.00000 1.00010 1.00040 1.00090 1.00160

000 001 016 081 256

Qi (x) 0.00000 000 0.02000 133 0.04001 067 0.06003 603 0.08008 546

0.05 0.06 0.07 0.08 0.09

1.00250 1.00361 1.00492 1.00644 1.00816

627 301 413 122 615

0.10016 0:12028 0.14045 0.16068 0.18097

704 894 936 662 914

-1.98998 -1.98557 -1.98035 -1.97431 -1.96746

747 401 179 766 792

-0.39899 -0.47826 -0.55725 -0.63589 -0.71414

900 951 060 347 899

1.76512 2.08508 2.38737 2.66939 2.92866

98 14 90 94 44

-3.52604 -3.29655 -3.03075 -2.73130 -2.40117

61 13 84 45 40

0.10 0.11 0.12 0.13 0.14

1.01010 1.01224 1.01461 1.01719 liO1999

101 820 039 052 184

0.20134 545 0.22179 422 0.24233 428 0.26297 462 0128372 443

-1.95979 -1.95130 -1.94198 -1.93182 -1.92081

839 431 044 094 942

-0.79196 -0.86930 -0.94609 -1.02230 -1.09787

777 001 554 373 345

3.16285 3.36984 3.54769 3.69467 3.80930

86 76 49 78 18

-2.04367 -1.66240 -1.26123 -0.84427 -0.41580

37 59 82 11 27

0.15 0.16 0.17 0.18 0.19

1.02301 1.02627 1.02976 1.03348 1.03745

790 258 007 491 202

0.30459 0.32559 0.34672 0.36800 0.38945

312 031 587 997 305

-1.90896 -1.89626 -1.88268 -1.86824 -1.85291

890 181 994 444 580

-1.17275 -1.24689 -1.32023 -1.39272 -1.46431

302 019 203 496 458

3.89031 3.93671 3.94778 3.92304 3.86234

48 92 25 76 02

+0.01970 0.45767 0.89344 1.32231 1.73958

77 92 90 56 08

0.20 0. 21 0.22 0.23 0.24

1.04166 1.04613 1.05086 1.05585 1.06112

667 453 171 471 054

0.41106 0.43285 0.45484 0.47703 0.49944

589 960 568 605 304

-1.83669 -1.81956 -1.80152 -1.78255 -1.76264

380 752 526 455 210

-1.53494 -1.60456 -1.67310 -1.74052 -1.80674

573 234 742 294 982

3.76577 3.63376 3.46700 3.26651 3.03359

54 26 84 77 33

2.14059 2.52079 2.87577 3.20128 3.49331

45 94 54 51 81

0.25 0.26 0.27 0.28 0.29

1.06666 1.07250 1.07863 1.08506 1.09182

667 107 229 944 225

0.52207 0.54495 0.56809 0.59150 0.61519

948 869 454 152 472

-1.74177 -1.71993 -1.69710 -1.67327 -1.64842

372 437 801 761 510

-1.87172 -1.93539 -1.99768 -2.05854 -2.11790

780 537 972 661 027

2.76983 2.47712 2.15764 1.81383 1.44841

31 56 35 48 22

3.74813 3.96230 4.13277 4.25684 4.33229

48 97 26 84 46

0.30 0.31 0.32 0.33 0.34

1.09890 1.10631 1.11408 1.12220 1.13071

110 707 200 851 009

0.63918 0.66350 0.68815 0.71315 0.73853

993 370 335 706 396

-1.62253 -1.59557 -1.56753 -1.53839 -1.50811

126 570 678 152 553

-2.17568 -2.23182 -2.28625 -2.33890 -2.38969

334 672 944 860 914

1.06434 0.66482 +0.25327 -0.16667 -0.59123

02 02 32 95 78

4.35733 4.33070 4.25164 4.11997 3.93608

72 22 55 79 76

FE: 0:37 0.38 0.39

1.13960 1.14889 1.15861 1.16877 1.17938

114 706 430 045 436

0.76430 0.79048 0.81711 0.84419 0.87175

415 884 039 242 994

-1.47668 -1.44406 -1.41023 -1.37516 -1.33880

292 617 606 155 960

-2.43855 -2.48539 -2.53013 -2.57269 -2.61297

378 281 394 210 926

-1.01644 -1.43822 -1.85237 -2.25465 -2.64075

63 04 43 05 25

3.70095 3.41617 3.08394 2.70708 2.28903

66 42 42 74 82

0.40 0.41 0.42 0.43 0.44

1.19047 619 1720206 756 1.21418 164 1.22684 333 1.24007 937

0.89983 0.92845 0.95764 0.98743 1.01786

941 892 831 931 572

-1.30114 -1.26213 -1.22172 -1.17988 -1.13657

509 064 641 995 597

-2.65090 -2.68637 -2.71928 -2.74954 -2.77703

420 229 520 067 216

-3.00637 -3.34725 -3.65918 -3.93806 -4.17995

81 61 35 51 45

1.83383 1.34610 0.83104 +0.29437 -0.25765

54 61 35 81 92

0.45 0.46 0.47 0.48 0.49

1.25391 1.26839 1.28353 1.29937 1.31596

1.04896 1.08077 1.11333 1.14668 1.18088

360 146 051 490 202

-1.09173 -1.04531 -0.99726 -0.94752 -0.89602

613 874 854 634 868

-2.80164 -2.82327 -2.84178 -2.85705 -2.86895

855 375 630 896 817

-4.38109 -4.53797 -4.64734 -4.70629 -4.71228

69 26 21 25 35

-0.81838 -1.38069 -1.93714 -2.48003 -3.00140

00 01 78 04 86

0.50

1.33333 333 [C-y

X 0.00

850 168 228 630 263

1.21597 281 [(-$l]

-0.84270 745 ‘-;I2

II 1

-2.87734 353 t-t14

[

1

-4.66319 54

-3.49322 79 5316

I: 1

LEGENDRE

DERIVATIVE X

Qh(x)

OF THE

LEGENDRE

Q;(x)

349

FUNCTIONS

FUNCTION--SECOND

&6(x)

KIND

Q;(x)

Qi (4

Table

Q6(4

8.4

Qio@)

0.50 0.51 0.52 0.53 0.54

1.33333 1.35153 1.37061 1.39062 1.41163

333 399 403 717 185

1.21597 1.25201 1.28905 1.32717 1.36643

281 210 905 756 680

-

0.84270 0.78748 0.73029 0.67104 0.60963

74 95 59 20 61

-

2.87734 2.88206 2.88297 2.87989 2.87266

35 72 33 70 39

-

4.66319 54 4155737 62 4.39368 94 4.17156 11 3.89102 65

-

3.493228 3.947399 4.355894 4.710854 5.004695

0.55 0.56 0.57 0.58 0.59

1.43369 1.45687 1.48126 1.50693 1.53397

176 646 204 189 760

1.40691 1.44868 1.49184 1.53648 1.58271

178 400 220 320 285

-

0.54597 0.47996 0.41147 0.34038 0.26655

91 38 39 30 35

-

2.86108 2.84497 2.82411 2.79828 2.76723

89 53 36 02 56

-

3.55277 3.15819 2.70941 2.20934 1.66171

54 61 73 79 26

-

5.230233 5.380807 5.450406 5.433812 5.326732

0.60 0. 61 0.62 0.63 0.64

1.56250 1.59261 1.62443 1.65809 1.69376

000 029 145 982 694

1.63064 1.68041 1.73215 1.78601 1.84218

718 364 259 903 458

+

0.18983 0.11006 0.02705 0.05937 0.14946

51 36 91 63 05

-

2.73072 2.68846 2.64017 2.58551 2.52414

34 75 05 08 00

- 1.07108 51 - 0.44291 60 + 0.21644 47 0.89973 10 1159875 12

-

5.125950 4.829465 4.436645 3.948368 3.367169

0.65 0.66 0.67 0.68 0.69

1.73160 1.77179 1.81455 1.86011 1.90876

173 305 271 905 121

1.90083 1.96219 2.02649 2.09399 2.16500

983 705 344 499 099

0.24343 0.34156 0.44414 0.55151 0.66402

42 40 64 17 96

-

2.45567 2.37971 2.29579 2.20342 2.10205

92 49 49 26 04

2.30438 3.00660 3.69447 4.35619 4.97914

77 55 22 14 99

+

2.697375 1.945245 1.119087 0.229371 0.711177

0.70 0.71 0.72 0.73 0. 74

1.96078 2.01653 2.07641 2.14086 2.21043

431 559 196 919 324

2.23984 2.31892 2.40266 2.49156 2.58619

955 413 159 187 998

0.78211 0.90623 1.03692 1.17478 1.32049

54 72 51 21 75

-

1.99107 1.86981 1.73752 1.59336 1.43637

23 51 72 54 96

5.54998 6.05466 6:47859 6.80675 7.02388

34 05 09 90 88

1.687501 2.682165 3.675339 4.644816 5.566082

0.75 0.76 0.77 0.78 0.79

2.28571 2.36742 2.45639 2.55362 2.66028

429 424 892 615 199

2.68724 2.79545 2.91175 3.03719 3.17305

079 751 493 894 446

1.47486 1.63879 1.81335 1.99979 2.19957

32 46 60 32 51

-

1.26549 1.07947 0.87692 0.65620 0.41542

27 65 20 16 09

7.11464 7.06387 6.85691 6.47990 5.92027

51 68 02 33 14

6.412431 7.155161 7.763836 8.206652 8.450921

0. 80 0. 81 0.82 0.83 0. 84

2.77777 2.90782 3.05250 3.21440 3.39673

778 204 305 051 913

3.32083 3.48236 3.65986 3.85608 4.07443

451 488 997 883 439

2.41444 2.64650 2.89827 3.17286 3.47409

73 26 40 02 64

- 0.15235 72 + 0.13562 04 0.45165 68 0.79955 16 1.18395 08

5.16720 18 4121227 67 3.05023 28 1.67989 36 + 0.10532 57

8.463693 8.212559 7.666669 6.798024 5.583115

0.85 0.86 0.87 0.88 0.89

3.60360 3.84024 4.11353 4.43262 4.81000

360 578 352 411 481

4.31921 4.59595 4.91185 5.27647 5.70283

588 604 380 688 015

3.80679 4.17707 4.59287 5.06465 5.60654

33 50 14 07 69

1.61061 2.08677 2.62171 3.22751 3.92032

19 72 45 63 16

- 1.66270 - 3.60489 - 5.69098 - 7.87652 -10.09858

0.90 0. 91 0.92 0.93 0.94

5.26315 5.81733 6.51041 7.40192 8.59106

789 566 667 450 529

6.20906 6.82129 7.57861 8.54217 9.81365

159 988 025 980 072

6.23815 6.98747 7.89613 9.02883 10.49236

05 73 09 27 44

4.72224 5.66456 6.79318 8.17876 9.93658

63 11 58 62 04

-12.26944 98 -14.26758 89 -15.92348 54 -16.99643 22 -17113329 84

- 9.045801 -12.315713 -15.495090 -18.304274 -20.319071

0.95 0.96 0.97 0.98 0.99

10.25641 12.75510 16.92047 25.25252 50.25125

026 204 377 525 628

11.57537 14.19080 18.50515 27.04503 52.39539 m

057 811 528 467 613

12.47698 56 15.35932 33 20.00905 43 29.00735 14 55.11181 39 co

12.26978 15.57616 20.76422 30.50045 57.80864 00

50 37 38 90 53

-15.78782 -12.04072 - 4.11777 +12.32933 54.86521

-20.873659 -18.851215 -12.140718 + 4.242107 49.428990 m

00

85 91 02 81 18

62 38 87 89 05

+ -

4.005017 2.056070 0.258625 2.916594 5.871760

350

LEGENDRE Table

LEGENDRE

8.5

FUNCTION-FIRST

PO(X) = 1

pm

22:20 %*Z 2:a 3.0 3-g 3:6 3. a ::2" z! 4: a 5. 0 55:: 2:: 6. 0 2:: 2:: ;:2" 764 7: a a. 0 a. 2 i-2 a: a 9.0 z:: 9':;

FUNCTIONS KIND

P1(x)=x

P4W

p3w

P,(x)

p5w

1.00 1.66 2.44 3.34 4.36

1.00 2.52 4.76 7. a4 il.88

1.00000 4.04700 9.83200

1.00000 6.72552

5.50 % 9:64 11.26

17.00 23.32 30.96 40.04 50.68

13.00 14.86 16.84 la.94 21.16

63.00 77.12 93.16 111.24 131.48

23.50 25.96 28.54 31.24 34.06

154.00 178.92 206.36 236.44 269.28

3 3 4 4 II 4

7.51150 9.65154 1.22500 1.53765 1.91071

37.00 40.06 43.24 46.54 49.96

305.00 343.72 385.56 430.64 479.08

4 4 4 4 II 4

2.35250 2.87205 3.47916 4.18440 4.99917

53.50 57.16 60.94 64.84 68.86

531.00 586.52 645.76 708.84 775.88

73.00 77.26 81.64 86.14 90.76

847.00 922.32 1001.96 1086.04 1174.68

5 5 5 5 II 5

1.29367 1.49122 1.71215 1.95846 2.23227

95.50 100.36 105.34 110.44 115.66

1268.00 1366.12 1469.16 1577.24 1690.48

5 5 5 5 II5

2.53583 2.87149 3.24171 3.64912 4.09643

121.00 126.46 132.04 137.74 143.56

lao9.00 1932.92 2062.36 2197.44 2338.28

5 5 5 5 II5

4.58649 5.12230 5.70699 6.34383 7.03621

p9w

1.00000

PlO(X) 1.00000

I 234 I 1.06544 1.13789 2.alllo 6.65436 1 1 2 2 II 2

5.53750 8.47120 1.23927 1.74952 2.39887

II 67 1.59814 3.01437 5.46578 9.57313 8.09745

II II II t/ 78 3.41632 7.90944 5.25060 2.17406 1.16994

8a9 4.33189 1.05524 2.68690 6.82993 1.62597

9 2.38657 1.60047 3.50362 7.23884 5.06985

a9 9.01781 2.81890 2.14858 1.62372 1.21596

II II 10 1.23283 2.92387 1.91848 2.37430 1.54212

11 1.16898 12 6.10897 7.62030 1.43817 9.45994

10. 0 149.50 2485.00 (5)7.78769 (10)9.29640 (4)4.33754 (12)1.76laa From National Bureau of Standards, Tables of associated Legendre functions. Columbia Univ. Press, New York, N.Y., 1945 (with permission).

LEGENDRE DERIVATIVE

OF

THE

LEGENDRE

FUNCTIONS

FUNCTION-FIRST

Pi(x)

KIND

P.i (xl

Pi (2)

1 1 2 2 II2

Pk(z)

Table

8.6

P&)=32

P;(z)=1 6.000 9.300

351

Pie(x)

P6 (2)

1.50000 4.57230 1.01688 1.92723 3.30168

4.50000 t ;70::; 1174282 5.33445

1 5.50000 3 1.53586 4 I 1.13477 4 5.24824 5)1.85808

I II I II I II II I II 3 3 3 3 3

2.95500 3.86184 4.96025 6.27516 7.83305

5 5 5 6 6

1.39531 3.25362 6.94480 1.38132 2.59296

6 6 7 7 7

4.63721 7.95819 1.31805 2.11632 3.30652

78 7.52431 1.10110 5.04229 2.23988 1.58313

4 4 4 4 4

2.39550 2.80816 3.27172 3.79020 4.36775

4 4 4 4 4

5.00869 5.71746 6.49870 7.35714 8.29772

5 5 5 5 5

1.59602 1.76260 1.94187 2.13445 2.34099

5 5 5 5 5

2.56215 2.79860 3.05102 3.32013 3.60663

tl II II 4 1.35580 1.63974 1.54109 1.44647 1.26900

6 5.50068 7 5 7.29317 1.42939 1.48267 3.36028

8 8 9 9 9

4.19097 6.57653 1.00955 1.51918 2.24508

10 10 10 10 10

1.72421 2.32397 3.10217 4.10354 5.38214

(2)7.485 (4)1.74250 (5)3.91127 (10)8.40642 (12)1.77028 From National Bureauof Standards,Tablesof associated Legendrefunctions. ColumbiaUniv. Press, New York,N.Y., 1945(with permission).

352

LEGENDRE FUNCTIONS Table

8.7

LEGENDRE

FUNCTION-SECOND

KIND

Q&Z)

33'20 3:4 it:",

II

5.0

(-1 2.02733

II -5 9.56532

z-24 5: 6 5.8

-1 1.94732 1.87347 -1 1.80507 -1 1.74153

-5 8.14823 6.98500 -5 6.02278 -5 5.22117

7. 0 II -1 -1 -1 ;:fI (-1 -1 8.0

i:: i:: 9.0 E 9:6 9. 8

1.43841 1.39792 1.35967 11:28915 32346

II II -3 -3 -3 -3

6.88725 6.50550 6.15475 5.83171 5.53353

Ii -5 -5 -5 -5

2.43500 2.17277 1.94497 1.57242 1.74631

-3 5.25771

II -5 1.41968

-3 4.76469 5.00208

-5 1.28507 1.16606

-3 4.54386 4.33807

-6 1.06054 -5 9.66707

II I

-13 5.37876

-13 4.19350 3.28941 -13 2.59524 2.05891 -13 I 1.64205 -13 1.06011 1.31620

-14 8.57794 -14 6.97159

10.0 (-1)1.00335 (-3)3.35348 (-4)1.34486 (-6)5.77839 (-14)5.69010 (-15)2.71639 From National Bureau of Standards, Tables of associatedLegendre functions. Columbia Univ. Press, New York,N.Y., 1945 (with permission).

LEGENDRE

DERIVATIVE X

-

OF THE

QhCd

LEGENDRE

FUNCTIONS

FUNCTION-SECOND

-Q;(x)

-&i(x)

KIND

- Qi (x)

Q;(z)

Table

8.8

-Qhdx)

-&h(x)

1.0 1.2 ?Z 1: 8

I

- 8 4.51200 - 8 1 2.11821

220 4: 4 4.6

II

-5 7.82792 5.27543 3.66172 4.38019 6.40058

I -10 -11 I 1.28985 5.43056 2.43819 3.61188 8.29696

t-20 6: 4 7: Pi! 8

8:4

-12 1.36497 -11 3.40566 5.31340 2.21848 8.43598

-13 3.19817 -12 4.59703 9.83782 1.46703 6.68395

::t

Ei

II II I

II -3 -3 -3 -4

1.32691 1.23104 1.14421 1.06538 9.93646

II II -4 -4 -4 -4

1.71573 1.53040 1.36949 1.10651 1.22923

-5 -5 -5 -5

9.98765 9.03846 8.19960 7.45601 6.79498

10.0 (-2)1.01010 (-4)6.74753 (-5)4.05782 (-6)2.32430 (-14)5.72014 From National Bureau of Standards, Tables of associated Jxgendre functions. Columbia York, N.Y., 1945 (with permission).

-13 -13 -13 -14

2.24909 1.59779 1.14602 I 6.05494 8.29452

(-15)3.00374 Univ. Press, New

9. Bessel Functions F.

w.

of Integer J.

OLVER

Order

1

Contents Page

Mathematical Notation. Bessel

....................

Properties

358

..........................

358

Functions J and Y. .................. 9.1. Definitions and Elementary Properties ......... 9.2. AsymptoCc Expansions for Large Arguments ...... 9.3. Asymptotic Expansions for Large Orders ........ 9.4. Polynomial Approximations. ............. 9.5. Zeros. .......................

Modified 9.6. 9.7. 9.8. Kelvin

Numerical

358 358 364 365 369 370

Bessel Functions I and K. .............. Definitions and Properties .............. Asymptotic Expansions. ............... Polynomial Approximations. .............

374 374 377 378

Functions. ...................... 9.9. Definitions and Properties .............. 9.10. Asymptotic Expansions ............... 9.11. Polynomial Approximations .............

379 379 381 384

Methods ...................... 9.12. Use and Extension of the Tables.

References. Table 9.1.

Table 9.2.

1 National

..........

385 385 388

.......................... Bessel Functions-Orders 0, 1, and 2 (0 5x5 17.5) .... Jo@), 15D, JIW, JzP), Y&3, YIW, 1011 Y&J>, 8D x=0(.1)17.5 Bessel Functions-Modulus and Phase of Orders 0, 1, 2 (lO
Jo(z) In z,

x=0(.1)2,

8D

2[Yl(z)--f

JI(z)

on leave from the National

396

397

In 21

Bessel Functions-Orders 3-9 (0 52_<20) ........ n=3(1)9 J&t ynw, 5D or 5s x=0(.2)20,

Bureau of Standards,

390

398

Physical Laboratory. 355

BESSEL

356 Table 9.3.

FUNCTIONS

OF INTEGER

ORDER

Bessel Functions-Orders 10, 11, 20, and 21 (0 1x_<20) . . x-‘“J~o(x), x-llJIT,,(z), Z’“Y~O(Z> x=0(.1)10, 8s or 9s JlOb), Jll@), YlO(X> x= 10(.1)20, 8D x-‘“J*&), lc-21J21(.x)) 2mY20(4 x=0(.1)20, 6s or 7s Bessel Functions-Modulus and Phase of Orders 10,11,20, and21 (2O<x
Table

9.4.

Bessel Functions-Various Orders (0
Table

9.5.

Zeros and Associated Values of Bessel Functions and Their Derivatives (OIn18, 1 <s<20) . . . . . . . . . . . 5D (10D for n=O) j?w J?x.L.A ; L, J&b,,), 5D (8D for n=O) YY,,, YXYw); Yyb,7 Y,(yk,,), s=1(1)20, n=0(1)8

406

407

409

Table

9.6.

Bessel Functions Jo(&x), x=0(.02)1, 5D

. . . . . . . . . . .

413

Table

9.7.

Bessel Functions-Miscellaneous Zeros (s= 1(1)5) . . . . . sth zero of 5 J1 (2) - xJo(x) x, x-‘=0(.02) .l, .2(.2)1, 4D sth zero of &(x) - We(x) x=.5(.1)1, X+=1(-.2).2, .l(-.02)0, 4D sth zero of Jo(x) Yo(hx) - Yo(x)Jo(Ax) X-‘=.8(-.2) .2, .l(-.02)0, 5D (8D for s=l) sth zero of J,(x) Yl (AZ)- Yl (x)J1(Xx) X-l=.8(-.2) .2, .l(-.02)0, 5D (8D for s=l) sth zero of J1(z) Yo(Xx)- Yl(x) Jo(~) X-l=.8(-.2) .2, .l(-.02)0, 5D (8D for s=l)

414

Table

9.8.

Modified Bessel Functions of Orders 0, 1, and 2 (0 Ix 120) e-zIo(x), e”Ko(x), e-“II(x), e%(x) x=0(.1)10 (.2)20, 10D or 10s

.

416

Table for Large . . . . . . . .

422

Table for Small . . . . . . . .

422

x-212

(4,

39K2

s= l(l)5

, . . . .

Page 402

(4

x=0(.1)5, lOD, 9D e-z12(x), e”K2 (2) x=5(.1)10 (.2)20, 9D, 8D Modified Bessel Functions-Auxiliary Arguments (202x< a) . . . . . . . x+emzl,(x), 7r-‘x*e”K,(x), n= 0, 1, 2 x-l= .05(- .002)0, 8-9D Modified Bessel Functions-Auxiliary Arguments (Oix12). . . . . . . . K,(x) + lo(x) Inx, x{ K (5) -II(x) lnx} x=0(.1)2, 8D

BESSEL

FUNCTIONS

OF INTEGER

ORDER

357 Page

Table 9.9.

Modified Bessel Functions-Orders emZIn(x), eZKn(x), n=3(1)9 x=0(.2)10(.5)20, 5s

3-9 (0 <x120)

. . .

Table 9.10. Modified Bessel Functions-Orders 10, 11, 20 and 21 (O, em2111 (21, e”Kl0(x) lOD, 10D, 7D 2=10(.2)20, 3J-2°120(~>,

cr2112,

(2))

423

425

z20K20(z)

2=0(.2)20, 5s to 7s Modified Bessel Functions-Auxiliary Tabie for Large Arguments (205x5 ~0). . . . . . . . . . . . . . . In { x+e-~IIo(x) } , In { x+e-‘II1 (x) } , In {a-‘x~e”KIo(x) } ln{xie-“Izo(x)}, In{ x~e-z121(x)}, In{7r-1xfe”K20(x)} s-‘=.05(-.001)0, 8D, 6D

427

Table 9.11. Modified Bessel Functions-Various Orders (0 In 5 100) . In(x), K,(x), n=0(1)20(10)50, 100 z=l, 2, 5, 10, 50, 100, 9s or 10s

428

Table 9.12. Kelvin Functions-Orders 0 and 1 (0 1215) . . . . . . ber x, bei x, beq x, beil x ker x, kei x, ker, x, kei, x x=0(.1)5, iOD, 9D Kelvin Functions-Auxiliary Table for Small Arguments (O<x
430

Mob),

cob-3,

No(4,

40(x>,

M(x), N(x),

430

432

4(x> h(x)

z=O(.2)7, 6D Kelvin Functions-Modulus and Phase for Large Arguments (6.6 5x5 a). . . . . . . . . . . . . . . . . x+e-“‘~Zi140(x),O,(x) - (x/G), x*e-z’dM~ (51, 4 (2) - (x/@> xWJzNO(x), 40(x> + (x/-\/z>, xte”‘dZN~(x), 91(x>+ (x/43 x-‘=.15(-.Ol)O, 5D

432

The author acknowledges the assistance of Alfred E. Beam, Ruth E. Capuano, Lois K. Cherwinski, Elizabeth F. Godefroy, David S. Liepman, Mary Orr, Bertha H. Walter, and Ruth Zucker of the National Bureau of Standards, and N. F. Bird, C. W. Clenshaw, and Joan M. Felton of the National Physical Laboratory in the preparation and checking of the tables and graphs.

9. Bessel Functions Mathematical

of Integer

Order

Properties Bessel

Notation

The tables in this chapter are for Bessel functions of integer order ; the text treats general orders. The conventions used are: z=z+iy; 5, y real. n is a positive integer or zero. Y, p are unrestricted except where otherwise indicated; Yis supposed real in the sections devoted to Kelvin functions 9.9, 9.10, and 9.11. The notation used for the Bessel functions is that of Watson [9.15] and the British Association and Royal Society Mathematical Tables. The function Y”(z) is often denoted NV(z) by physicists and European workers. Other notations are those of: Aldis, Airey: Gn(z) for -+rYn(z),K,(z)

for (-)nK,(z).

Clifford: C,(x) for z+Jn(2&). Gray, Mathews Y&9

and MacRobert

for 37rY&)+

[9.9]:

(ln ~--TV&),

F”(z) for ?revri sec(v?r)Y,(z), G,(z) for +tiH!l)

9.1. Definitions

and Elementary Differential

9.1.1

J and Y

Functions

CPW 22=+2

Properties

Equation

clw ~+(22-v2)w=o

Solutions are the Bessel functions of the first kind J&z), of the second kind Y”(z) (also called Weber’sfunction) and of the third kindH$,“(z), Hb2)(z) (also called the Hankel functions). Each is a regular (holomorphic) function of z throughout the z-plane cut along the negative real axis, and for fixed z( #O) each is an entire (integral) function of v. When v= &n, J”(z) has no branch point and is an entire (integral) function of z. Important features of the various solutions are as follows: J”(z) (g v 2 0) is bounded as z-0 in any bounded range of arg z. J”(z) and J-,(z) are linearly independent except when v is an integer. J”(z) and Y”(z) are linearly independent for all values of v. W!l)(z) tends to zero as Jz/-+co in the sector O<arg z
(2). Relations

Jahnke, Emde and Losch [9.32]:

Between

Solutions

A,(z) for l?(~+l)(~z)-vJV(z). Jeffreys: Hsy(2) for HP(z),

Hi”(z) for Hi2)(z), Kh,(z)

for (2/a)&(z).

Heine: K,(z) for--&Y,(z). Neumann: Y"(z) for +sY,(z)+ Whittaker

(ln 2--y)Jn(z).

and Watson 19.181: EG(z) for cos(vr)&(z).

358

The right of this equation is replaced limiting value if v is an integer or zero.

by its

9.1.3

H:“(z)=Jv(z)+iY,(z) =i csc(m) {e-Y”tJv(z) - J--Y(z)] 9.1.4

HS2’(z)=J&)-iYv(z) =i CSC(VT){J--Y(~)-eYTtJp(z)} 9.1.5

J-,(z)= (-)nJn(z)

9.1.6

iY$~(z)=ev”*H~“(z)

y-m= (-->“Yn (4 ~?,!(z)=e-v”f~i2’(z)

BESSEL

FUNCTIONS

OF

INTEGER

ORDER

359

t

,I’ ,‘l,,lXl I

I’

: /

FIGURE 9.2.

Jlo(z),

M1cl(x)=JJ?&+ FIGURE 9.1.

Jo(x),

Y&i), J,(x),

Ydz),

and

Edx).

Yl(z).

t

”

FIGURE 9.3.

Jv(lO)

and Y,(lO).

-DX

Contour lines of the modulus and phase of the Hankel Function HA” (x+iy)=M&Q. From E. Jahnke, F. Emde, and F. L&h, Tables of higher functions, McGraw-Hill Book Co., Inc., New York, N.Y., 1960 (with permission).

FIGURE 9.4.

360

BESSEL Limiting

Forms

for

Small

FUNCTIONS

OF

Arguments

INTEGER

ORDER, Integral

9.1.18

When v is fixed and z-0 9.1.7 Jr(z) - &>“Ir(v+ 9.1.8

1)

cos (2 sin s)de=i -2,

(vz-1,

Ye(z) --iHAl)

Representations

-3,

-~iH$~‘(z) -(2/r)

. . .)

T cos (2 co9 0)&J

9.1.19 Y&> =f

In z

s0

J’

cos (2 cos 0) {r+ln

(22 sin2 0) } a%

0

9.1.20

9.1.9 Y&) --iH:l)(z)

-iH:2’(z)

--

(l/~)r(v)(~z)Wv>O)

Ascending

9.1.10

JY(z)=(w”

=**;y&Jy

Series

2

(1-t2F

cos (zt)dt

@?v>--4)

9.1.21

&$;;1)

9.1.11 *-?a T =- a dr ~080 cos r s0

Y,(&+ ,(3W n-1(n-k--l)! (fz”>” -a k3 k! 9.1.22

2 +- a In (3z)Jn (2)

J.,z,=~J

COS(Z

Y,(z) =; J

J,,(z) = 1

-~~m(e~‘+e-.’

9.1.13 2 Yo(z>=- ?r Iln (3z>+rVo(z)+41-t-4)

2 $2” I-(1!)2

e-ve)de

F+(l+i+t)

S~II(Z

sin

e-zslDh

r-vcdt (la-g .zl<&r)

e--ve)de

cos (v7r)}e-*s**tdt

(jarg2/<$7r)

9.1.23

*

fg-

J,(I)=:lrnsin(z

* * *I

9.1.14 J,(W,(4

sin

-- sin(m) ?r s 0

where #(n) is given by 6.3.2. 9.1.12

(ne)de

cash t)dt (s>O)

cos(s cash t)dt (s>O) 9.1.24

= (-)“uv+P+2k+l)

(iz”)”

k=Or(V+k+l)r(C(+k+l)r(V+CC+k+l)

k!

9.1.15 9.1.25 W(J”(Z),

9.1.16

mu&

J-,(z)

1 =J”+l(z)J-,(z)+J”(~)J-~,+l,(~)

= -2 sin (wr)/(?rz)

Y”c4 1=J”+lM y&9 -J&9 Yv+1(d =2/(m)

HI”(z)=$ H:)(z)=-:

-+*i ezslnht--r~dt (jargzl<$r) _ s (D m-r: _ ezsl*+-vtdt (largzl<+r) s OD

9.1.26

9.1.17 W{Hj”(z),

H~2’(z)}=H~~~(z)H~2~(~)-H~1~(z)H~,2!~(~) = -4i/(rz)

In the last integral the path of integration must lie to the left of the points t=O, 1, 2, . . . .

BESSEL

FUNCTIONS

OF INTEGER

ORDER

361

and 9.1.27

4 pySY-pJY=- u2ab

9.1.34

Analytic

Continuation

In 9.1.35 to 9.1.38, m is an integer.

u:(z,=-Y”+&,+~

W”(4

9.1.35

$T denotes J, Y, WI’, ZF2’ or any linear combination of these functions, the coefficients in which are independent of z and Y.

9.1.36

9.1.28

9.1.37

J;(z)=-Jl(z)

Y;(z)=-Yy,(z)

Jv(ze”f)=em”r*

Yv(zem*f)=e-m’“* Y,(z)+2i

J”(z)

sin(mva) cot(v?r) J,(z)

If fi(z) = zP%~(W) where p, p, X are independent of Y, then

sin(~?r)H!~)(ze”“~) = -sin { (m- l)~?r}H!~)(z)

9.1.29 fY-l(Z) +fY+1@) = WWYf,(~)

9.1.38 sin(vr)HP)(ze”*f)=sin{

(p+q)fv-l(z) + (P--?2)fv+&) Z,fl(~> =Mz*fv-l(z)

+ev** sin(mv7r)H”)(z) Y 9.1.39 H~‘)(ze~f)=-e-~~fH$)(z)

(P+vdfY(Z)

Formulas

for

(m+1)v~}H~2)(z)

= cw~)cfl(~)

+ (P- df4z>

zf:(z>=-xaz”fY+l(z>+

--e-v”* sin(mv7r)HP)(z)

H!2)(ze-“f)=-e’“LH~1’(z)

Derivatives

9.1.30 9.1.40

(

f$ k{z-‘$f”(z)}=(-)kz-‘-k%v+k(~> > (k=O, 1,2,

J&9 =m H!“(Z)=H:‘(z)

(2)

Generating =;

{ sf”-k(Z)

-(f)

. . . +(-)k~v+r(~)

(k=O, 1,2, . . .) Relations

(Y real)

Function

and

Associated

Series

for

eW-l/t)=

5

tk Jk(z)

@*O)

kx-.m

1

+G>

Recurrence

H:Q(Z) =H!“(z)

UV-k,2(4

9.1.41 q”-k+,(z)-

= Yl(z)

. . .)

9.1.31 g:“)

Y,(B)

9.1.42

cos (z sin O)=J,,(z)+2

9.1.43

sin (z sin 0)=2 ‘& Jsn+I(z) sin { (2k+l)B}

g

J&Z)

cos (2M)

Cross-Products

If 9.1.32

p,= J&)Yv(b) - Jv(b)Yh) qv=J,(@Y:(b) - J:@)Y&) rv= J:(Wv(b) - Jv(bP’:W ev= J:(a)Y:(b)

9.1.44 cos (z cos e)=J&)+2

- J:(b)Yl(a)

k$ (-)kJ&)

cos (2ke)

9.1.45

then 9.1.33

sin (z cos 0)=2 g 9.1.46

(-)‘J:Jnn+I(z) cos { (2k+l)O}

1=Jo(z)+2J&)+2J&)+2J&)+

. . .

9.1.47 cos z=J,,(z)-2J&)+2J&)-2J&)+

1 a,=- 2 p.+,+;

p.-1-;

P,

9.1.48

sin z=2J,(z)-2J&)+2J&)-

. . . . . .

362

BESSEL

Other

Differential

FUNCTIONS

OF

Equations

INTEGER

ORDER

9.1.63

9.1.49 tuff+(XZ-y)W=o,

w=z~w"(xz) Derivatives

With

Respect

to Order

9.1.64 9.1.50 w~~+(~-!$)w=o,

w=z%T"(x2*) $ J,(z)=J,

9.X.51

(2) In ($2)

w=dWR,,,(2XdP/p)

w”+A22v-2w=o,

OD

-(32)”

9.1.52

gl

A! +b+k+l)

caz”)”

(---I r(v+k+l)

k!

9.1.65

WI'-- 2v-1 w~+x2w=o, 2

w=z"w~(xz) $ y. (2) =cot

9.1.53 ~~w”+(1-2p)zW’+(x’q%2~+p’-v2q2)20=0,

(ml {;

-csc

J, (z) -‘IFY, (z) 1

(VT) $ J-v(z)---rJ,

w= zpw~(xz~)

9.1.54

(z)

(VZO, 51, 62, . . . >

9.1.66 w”+(X2ezr-v’)w=O,

w= %Yv(Ae*>

9.1.55 9.1.67

zyz2-v2)w"+z(z2-3v2)w' + { (2-vy(22+v2)}w=o, 9.1.56 w(23=

(-)fppW,

w=Vl(z) 9.1.68

w=z%f,(2xad)

where (Y is any of the 2n roots of unity. Differential

Equations

for

Products

In the following 8= z& and V,(z), g,(z) cylinder

functions

are any

Expressions

of Hypergeometric

= JP(z)=r(v+l)

{ 64--2(va+$)82+

w=~v(z)~p(z)

Aqv++,

(k4”

~(t?2-4vz)w+4zz(~+l)w=o,

w=&?“(2)~“(2) (v20),

2* o<J,(v)<3*Iy+)vt

9.1.61

IJ,(z)l
9.1.62

- r (v+l) II.

--limF

X,p;

v+l;

-&)

(

15.)

Bounds

IJY(~)111/@

2v+l,2iz)

as X, P+=J through real or complex values; z, v being fixed. (,F1 is the generalized hypergeometric function. For M(a, b, z) and F(a, b; c; z) see chapters 13 and

23W”‘+2(4Z2+1-44y2)W’+(4+-1)W=0,

9.1.60 jJ”(2)jll

J’(Z)-r(v+l)

w=W,(z>L9v(~>

9.1.59

Upper

-&!‘)

9.1.70

9.1.58

page

,F,(v+l;

=&Ye-“* w+u

(v2-/2)2}w

+422(6+1)(9+2)w=o,

Functions

9.1.69

of orders v, P respectively.

9.1.57

*see

in Terms

Connection

(v11)

(v>O)

With

Legendre

Functions

If cc and x are fixed and Y+Q) through positive values

9.1.71 (v2--3)

*

(x>O)

real

BESSEL

FUNC!l’IONS

OF

For P;’

(cos f)} =-$rY,(r)

(2>0)

Continued

J&4

Fractions

1 2(v+2)z-‘-

1 2(V+l)z-‘-

-=A

X,

v sincr=w

sin x

the branches being chosen so that W-W and x+0 as z-0. 0;’ (cos CX)is Gegenbauer’s polynomial (see chapter 22).

and Q;‘, see chapter 8.

9.1.73

363

ORDER

In 9.1.79 and 9.1.80, w=~(?.4~+&--2uv cos CY), u-v cos a=w cos

9.1.72 lim (#Q;”

INTEGER

x

*’ ’ ”A

Multiplication

LdYY

Theorem

9.1.74 ~v(AZ)=Afv 2 (v(A2-1)k(w

Gegenbaue?‘s addition

$f?“*,n(z)

k!

ka0

(IA”-ll If %‘= J and the upper signs are taken, the restriction on X is unnecessary. This theorem will furnish expansions of %?,(rete) in terms of 5ZVflll(r). Addition

Theorems

If u, v are real and positive and 0 +Y 5 r, then w, x are real and non-negative, and the geometrical relationship of the variables is shown in the diagram. The restrictions Ive*‘al< 1~1 are unnecessary in 9.1.79 when %= J and v is an integer or zero, and in 9.1.80 when %Z= J. Degenerate

Neumann’s

theorem

Form

(u=

m):

9.1.81 eir “““~=I’(~)($v)-~

‘& (u+k)inJ,+r(v)C~“(c~s (YZO,

The restriction Ivj
9.1.76

Neumann’s

Expansion Series

of an of Bessel

9.1.82 f(~)=~~~(2)+2

1= JiC4+2k$ Jib)

Arbitrary Functions

a) -1,

Function

. . .) in

(IK4

& U&(Z)

where c is the distance of the nearest singularity off(z) from z=O,

9.1.77

l

o=E= (-YJd4 Jad4

+2 2 J&> J2n+d4 b2 1) 9.1.83 ak=L2az

9.1.78

J,(24=$o J&> Jn-n(z)+2 $ (--YJd4 Jn+nk)

@
fwkwt s Irl=e’

and On(t) is Neumann’s polynomial. is defined by the generating function

The latter

9.1.84

&=JoW&)+~ O,(t) is a polynomial

of degree n+ 1 in l/t; 00(t) * l/t,

9.1.85

Gegenbauer’s

o&)=; g

9.1.80

GC?” (4 -&(y+k) %$dv -=2q7(4 WY

w4tl>

kg J&)0&)

a

(Y#O,-1,

C’;’

(cos

. . ., lveif~l

a)

+--k-l)!

kf

2 n-2k+1

(T)

(n=1,2,.

The more general form of expansion

9.1.86

f(z>=hJ.(z>+2

g1 %Jv+&)

. .)

a

BESSEL

364

FUNCTIONS

OF

INTEGER

9.2.6

also called a Neumann expansion, is investigated in [9.7] and [9.15] together with further generalizaExamples of Neumann expansions are tions. Other 9.1.41 to 9.1.48 and the Addition Theorems. examples are

9.2.7

9.1.87

~~lyz)=~~j{~(v,

&)“=g

(v+2Qr(v+k) k!

k=O

ORDER

{P(v, 2) sin X+ Q(v, 2) cos X}

Y,(z) =42/(7rz)

Wg zl
z)+~&(v,

(- r<arg

J,+2n(z) (VfO, -1,-q.

. .>

9.2.8

H;yz)=Jqz){P(v,

9.1.88

+z{ln(32)~$(n+l> lJ&> -3 g

(-I”

(n+2k)J,+&) k(n+k)

z<2?r)

z)-iQ(v, z)}e-*x (--2*<arg z
where X=Z-(+v+$)uand, 9.2.9

(II-1)(P-99) Pb, ego(-1”~g$i=l2!(&)2

+G-l)(p--9)(~--25)(p-49)_. 4! (82)4

where G(n) is given by 6.3.2.

..

9.2.10

2k+ 1) Q(v, 4-got-1”(v, -r--l (P-l)(P-9h-25)+

9.1.89

(2z)2x+1

9.2. Asymptotic

Principal

Expansions Arguments Asymptotic

for

Forms

9.2.1 {cos (z-)~-~~)+e’~“O(lzl-‘)}

9.2.2

YY(z)=Jm{sin

Asymptotic

(larg zl
. (z-)v,-t~)+e’~“O(IZI-‘)}

(--*
2<27r)

9.2.4

With the conditions ceding subsection

(-2n<arg

-R(v,

J:(z)=J2/(*2){

of Derivatives

and notation

of the pre-

2) sinx--S(v,

2) co9 x.}

(kg 4
9.2.12

Y:(z) =JG)

H~z)(Z),J~)e-“‘-1’“-1”’

Expansions

9.2.11

(Ias 4<4

9.2.3 H~l)(z)‘VJ~e”‘-t~~-t~)

’ ’ ’

If v is real and non-negative and z is positive, the remainder after k terms in the expansion of P(v, z) does not exceed the (k+l)th term in absolute value and is of the same sign, provided that k rel="nofollow">tv-a. The same is true of Q(v,z) provided that k>!p---f.

When v is fixed and [z]+o,

Jy(z)=~q(Fg

3!(8~)~

82

Large

{ R(v,

2) cos x-

S(V,

2) sin x}

z<7r)

(be 4-C~) 9.2.13

Hankel’s

Asymptotic

Expansions

H!‘)‘(z)=J2/(?rz){iR(v,

z)-S(V,

When v is fixed and Iz]-+ao

(--?r<arg

z<27r)

9.2.14

9.2.5 JY(z)=~2/(m){P(v,

z)}efx

z) COSX-Q(v, z) sinx}

(larg Kr>

H$2)‘(z)=J2/(az){

--iR(v,

z)--S(v,

z)}ebfX

(--2?r<arg

.~

BESSEL

FUNCTIONS

OF

INTEGER

ORDER

365

9.2.29

9.2.15

(fv+$)~+~ P-1

b--2=I-(P-1)(P+15)+ 2!(82)2

+i’l-~~~~)~25)+(ii--l)(r2-114p+1073)

**-

5(4x)”

9.2.16

s(v,z,-go(-)”4~~+4(2k+l)~-l dv2-

-PLf3 82

b-1)

G-9) (rfW+ 3!(82)3

Modulus

+ (p-1) (5cr3- 1535pz+54703p-375733) + . .. 14(4x)’

(v, 2kfl) (22yk+’

(4,7+1)2

9.2.30

. . .

2 Nf”z{1-2

and

1 p-3 1.1 (p-l)(p-45) ~-(zx)4 (2x)2 2.4

-

*’ .1

Phase

The general term in the last expansion is given by

For real v and positive x

- 1 . 1 .3 . . . (2k-3) 2.4 - 6 . . . (2k)

9.2.1’7 MY=IH~l)(x))=~{J~(x)+Y~(x)}

&=arg Hi’)(z) =arctan

{ YV(z)/Jy(z)}

x,(P-1)b9).

. . w-3;~l

b-@k+1)@k--1)21

9.2.18 N”=py(x)I=~{J:2(x)+y:2(x)}

Ip,=arg Hz’)‘(z)=arctan

9.2.31

{ Yi(z)/Ji(z)}

9.2.19

Jy(z)=My

cos &,

Yp(x)=A4v

sin B,,

9.2.20

J:(x)=Nv

cos (p,,

Y:(x) = NV sin cpV.

p+3 “+2(4x)+

4Px-(+t)

p2+46p--63 6(4~)~

p3+185p2-2053p+1899 +

In the following relations, primes denote differentiations with respect to x. 9.2.21

M39:=2/(?rx)

A$p~=2(~-v”)/(7rx3)

9.2.22

Nf=M;2+A4;fI:2=M:2+4/(~xM,)2

9.2.23

($-v2)M~M:+x2N,N;+xN;=o

9.3.

Asymptotic Principal

9.2.25

ZM;‘+xM:+

Expansions

for

Asymptotic

Large

Orders

Forms

In the following equations it is supposed that through real positive values, the other variables being fixed.

9.2.26 (4v2- l)w=O,

. . .

V-+m

(x2-S)M,--4/(fM;)=O

i7?w”‘+x(45c2+ 1-422)w’+

+

If v 2.0, the remainder after k terms in 9.2.28 does not exceed the (k+ 1) th term in absolute value and is of the same sign, provided that k>v-3.

9.2.24 tan ((py-ey)=M,BI/M:=2/(*xM,M:) M,N,sin ((py-ev)=2/(1rx)

5(4x)5

w=xw

9.3.1

9.2.27 Asymptotic

When

v

Expansions

of Modulus

and

Phase

is fixed, x is large and positive, and

p=4v2

9.3.2

9.2.28 Jy(v sech ff) md2~~ 1.3.5 +2.4.6

(/.l-1)(/L-9)(p-25)+ (2x)6

tanh (Y

Yv (v se& 4 - -J3?Tv tanh a!

* * *I *see

page

II.

b>O)

(a>@

*

366

BESSEL

FUNCTIONS

OF

INTEGER

ORDER

9.3.3

9.3.9

J, (v set 13)=

q(t)=1 v,(t)= (3t-5ta;/24 uz(t)=(81t2-462t4+385te)/1152 u3(t)= (30375t3-3 69603ts+7

J2/(7rv tan p) {cos (vtanp-v/3-&r)+O

(v-l)} KKP<3*)

Yy(v set @)= 1/2/ (7rv t,an p) {sin (v tan /3-v/3--in)

+O(v-‘)

1

65765t’ -4 25425t9)/4 14720 u4(t)=(44 65125t4-941 21676te+3499 22430P -4461 85740t1°+1859 10725t12)/398 13120

(O For I

9.3.4 J,(v+zvH)=2%-H Y”(v+zvfs~=

Ai(--2%)+O(v-l) --2%-B

Bi( -2%)

and u,(t) see [9.4] or [9.21].

9.3.10 +

O(V-‘)

~~+~(t)=:t”(1-t’)zl;(t)+~~~

(1--5P)uJt)dt 0

(k=O,

9.3.5

1, . . .)

Also 9.3.11 JL (v sech a) -

9.3.6

9.3.12 YL(v sech CX)

where 9.3.13

vo(t)= 1 In the last two equations 9.3.39 below.

Debye’s

{ is given by 9.3.38 and

Asymptotic

(i) If LY:is fixed and positive positive 9.3.7

Expansions

and

v

is large and

q(t‘l=(-9t+7tS)/24 v,(t)=(-l35t2+594t4--455t~)/1152 v3(t) = (-42525t3+4 51737P--8 835752’ $4 75475t9)/4 14720 9.3.14 vh(t)=u,(t)+t(t2-1){ ~uM(t)+tu;-,(t)} (k=l, 2, . . .) (ii) If /3 is fixed, O
42/(7rv tan p){L(v,

P) cos \k +Mb, P) sin *I

p)=J2/(7rv tan /3){L(v, 8) sin + --M(v,

9.3.8

where

Y,(v sech CY)-

9.3.17

\k=v(t#an

a> cos *}

p-/3)--&r

L(v, p> ‘u 2 u=yt

0)

k=O

=l-81 where

cot2 p+462

cot4 fit385 1152~~.

cot’ S+

.. .

BESSEL

FUNCTIONS

9.3.18

OF INTEGER

ORDER

367

9.3.26

=3 cot 8-l-5 cot3 /3 -... 24v

17 1 z3+70 70

g1(2)=--

Also

9.3.19 set fl)=J(sin

&(v

2/3)/(m){ -N(v,

P) sin \k -O(v, a> cos \E}

9.3.20

where

The corresponding expansions for HJ’) (V + ZV) and IP(v+zv~‘~) are obtained by use of 9.1.3 and 9.1.4; they are valid for -+3n<arg v<#?T and -#?r<arg v<&r, respectively.

9.3.21

9.3.27

Y:(v set P)=J(sin

2@)/(m){N(v,

N(v, /I) 'v 2 v2ky? =1+

/3) cos * -O(V, p) sin \E}

l-9

Ji(v+d3)

Ai’ (-21132) {1+x - h&+ k=l vZki3

--$

135 cot2 /3+594 cot4 b-j-455 cot6 /3 -... 11529

+g

Ai (-21/32)

9.3.22 O(,,, ,,j)+

vzk+d.h;t

8)=g cot b-t-1 Cot3 8-.

..

Expansions

in

the

Transition

When z is fixed, Iv/ is large and jarg

- ‘$ Bi’ (-21j32) {1+x - h&)) -

kc1 vZkf3

Regions

-$

VI<+

9.3.23

Bi (-21132) g0 $$

where

Jv(v+d~3)-~

Ai (-21132) {l+eja} k-1 +f

Ai’ (-2%)

9.3.29

Vs’3

g

$$

h,(z)=--;

z

h&)=-&o

2+;

h,(z)=%

A2g

h,(z)=&

SO-+0

where 9.3.30

9.3.25 x++

f3(.2)

In(z) vZkJ3

9.3.28 Y~(v+zv”~)

Asymptotic

5 kc0

lo(z) =-

=-

957 7000

28

.x$3--

3150

1

225

3 1 z3-5 5

2 z3+go z7+z

z4--

1159 115500

z

BESSEL

368

FUNCTIONS

OF INTEGER

ORDER

9.3.37 Ai

(e2rt/3y2/3~) v1/3

e2*1/3& +

r (e2~1/3y2/3t) v5/3

When v++ m, these expansions hold uniformly with respect to z in the sector larg zls?r--~, where e is an arbitrary positive number. The corresponding expansion for HZ2) (vz) is obtained by changing the sign of i in 9.3.37. Here

where p/3

a=----=.44730 3"3r(g) 22/3

b= ----=.41085 3'W(Q) cxo=l, a2=.00069

73184,

3ia=.77475

01939,

$b= .71161 34101

1 a~=--=-.004, 225 3735 . . .,

j30=j$=.01428

90021

’

equivalently,

ff,=--00035

38 . * *

I213 -.00118 48596..., 10 23750= f13=-.00038 . . . /92=.00043 78 . . ., Yo"1,

9.3.39 5

57143 . . .,

p,=-

r,=~o=.00730

9.3.38

(-a3/2=l*F

&=~-arccos

($)

the branches being chosen so that { is real when z is positive. The coefficients are given by 9.3.40

15873 . . ., ak(l)=g

y3=.00044 40 . . . 7300 . . .) 60+ (&=--.-- 947 =- .00273 30447 . . ., 3 46500 &=.OOO6O 47 . . ., 63=-.00038 . . .

C(8f-3a’2U2k-8{

(1-z2)-tj

y2=-.00093

Uniform

Asymptotic

Expansions

These are more powerful than the previous expansions of this section, save for 9.3.31 and 9.3.32, They but their coefficients are more complicated. reduce to 9.3.31 and 9.3.32 when the argument equals the order. 9.3.35

2k+l r-‘Z

b(c)=-

XJ1-38’2U2k-1+*I(1-22)-tj

where uk is given by 9.3.9 and 9.3.19, A,,=&=1 and 9.3.41

x =(2~+1)(2~+3)...(6s-1) 8 s!(144)"

' I&=--gq

6sfl

x

I

Thus a&) = 1, 9.3.42

+*i’(v”“s) g+ a,(s)) v5/3

k=O

v

9.3.36

goty Y&z)ti- ( E2 >1’4{Bi$y3r) +Bi’(v2/31) 2 a,(r)) v5/3

k=O

3k

b,(c)= -~+~{24(15~2)3,2-s(1181)lj =--

5

4852

+’

5 (-~)~~24(za-l)312+8(~2-l)~

Tables of the early coefficients are given below. For more extensive tables of the coefficients and for bounds on the remainder terms in 9.3.35 and 9.3.36 see [9.38].

BESSEL

Uniform

Expansions

With the conditions

of the

FUNCTIONS

OF

INTEGER

ORDER

For f>lO

Derivatives

369

use

of the preceding subsection a,(+;

9.3.43

p-.104p-2, p+.146{-

+Ai’

(3’“~) 5 $13

k=O

dx(p)) VXk

For {<-lo

(v213[) .& a&&), g/3

k=O

co(r)--$

Hp’(Vz)--

Ai

(e2*U3&3{)

{

z

a1(~)=.000,

l&31=.0008, (r
9.4.1

9.3.46

ld,Wl=.~ol

cl(t):)-.0035-i

9.4. Polynomial

where

as f-++m.

Approximations

2

-35x13

&(x)=1-2.24999

2k+l 2

d,(f)=.OOO.

l-l-1.33(-[)-5’2,

Mr)l=.003,

#I3

h(~)l=.OO8

c&-)=--p

d,(l)=.OOs.

values of higher coefficients:

Maximum t

,

3’”

9.3.4s &pi/3

1

use

bo(S‘)-~r2, +Bi’

a,(3-)=.003,

97(x/3)‘+1.26562

08(~/3)~

~,~-3s’2uZk-~+~~(1---z)-*~

-.31638

66(~/3)~+.04444 79(x/3)”

-.00394 44(x/3)‘0f.00021

OO(x/3)‘2fE

lt]<5X10-8

and & is given by 9.3.13 and 9.3.14. For bounds on the remainder terms in 9.3.43 and 9.3.44 see [9.38].

r

=

=

--

--

boW

-~

0

0.0180

B

: 0278 0351

: :

: 0366 0352 : 0311 0331

ii

: 0278 0294

1:

-. 004 -. 001 +. 002 .003 . 004 . 004 . 004 . 004 .004 . 004

: 0265 0253

=

-I

ho(r)

--l--0 1 E

0.0180 .0109 : 0044 0067

4

.0031

6” 7 s8

: 0022 0017 .0013 : 0009 0011

10

--

.0007

I

= al(r)

-

-0. -. -. -.

004 003 002 001

-. -. -. -. -. -. -.

001 000 000 000 000 000 000

--

. 005 . 004 . 003 . 003 . 003 . 003 . 003 .003

=

cow

d,(r)

_---

691 384(x/3)4

+.25300 117(x/3)‘-.04261

214(x/3)’

+.00427 916(x/3)1o-.OOO24 846(x/3)12+e lel<1.4X10-B 9.4.3

3<x
f,=.79788

cos e,

Yo(x)=x-*f,

456-.OOOOO 077(3/x)-.00552

sin 0, 740(3/x)”

-.00009 512(3/~)~+.00137 237(3/x):)’ -.00072 805(3/~)~+.00014 476(3/x)6+e

0. 007 . 004 . 002 . 001 . . . . . . .

-

h(jx)Jo(x)+.36746

+.60559 366(~/3)~-.74350

0.007 . 009 .007

0. 1587 . 1323 . 1087 .0903 . 0764 . 0658 . 0576 .0511 . 0459 . 0415 .0379

-

di W

.- ---

0. 1587 . 1785 . 1862 . 1927 . 2031 . 2155 . 2284 . 2413 . 2539 . 2662 . 2781

-0.004

o<x53

Yo(x)=(2/r)

=

COG-1

9.4.2

001 000 000 000 000 000 000

(r(<1.6XlO-a 2 Equations 9.4.1 to 9.4.6 and 9.8.1 to 9.8.8 are taken from E. E. Allen, Analytical approximations, Math. Tables Aids Comp. 8, 240-241 (1954), and Polynomial approximations to some modified Bessel functions, Math. Tables Aids Comp. 10, 162-164 (1956)(with permission). They were checked at the National Physical Laboratory by systematic tabulation; new bounds for the errors, C, given here were obtained as a result.

BESSEL

370

.

8,=x-.78539

816-.04166

FUNCTIONS

lej<1.3X

lel
lo-’

Yl(x) =x-*jl

sin e1

10-B

449+.12499

* * '

2
* - *

vij~,~
612(3/x)

+ .00005 650(3/~)~- .00637 879(3/x)3 +.00074 348(3/x)*+.00079 824(3/x)5 -.00029 166(3/~)~+e 10-B

For expansions of Jo(x), Ye(x), Jl(x), and Y1(x) in series of Chebyshev polynomials for the ranges O<x<8 and O<S/x
%‘v(z)= J”(z) cos(d)+ Y”(z) sin(d)

9.5.3 t

is a parameter, %K(P. rel="nofollow">

=

then

VP-1

(P,)

=

-

v"+l(P">

If uVis a zero of %‘i (z) then 9.5.5

U,(u,,=~

%c,(u.)=~

Vv+l(G)

The parameter t may be regarded as a continuous variable and pr, u, as functions dt), u,(t) of t. If these functions are fixed by 9.5.6

P"(O)

40)

=o,

=jL,

1

then 9.5.7

Yv,1=PAS-%

jv,8=ds),

(s=l,2,

. . .)

(s=l,

2, * . .)

9.5.8

ji,s=uv(s-l>, 9.5.9

y:,s=QY(s-~)

u:(,J=($ $)-+, w7,,=($$ $)-*

Zeros

When v is real, the functions JP(z), J:(z), Y,(z) and Y;(z) each have an infinite number of real zeros, all of which are simple with the possible exception of z=O. For non-negative v the sth positive zeros of these functions are denoted by

. . .

The positive zeros of any two real distinct cylinder functions of the same order are interlaced, as are the positive zeros of any real cylinder function Q?‘“(z), defined as in 9.1.27, and the contiguous function %?V+l(~). If pu is a zero of the cylinder function

9.5.4

f,= .79788 456+ .OOOOO156(3/x)+ .01659 667(3/x)’ + .00017 105(3/x)3- .00249 511(3/x)* +.00113 653(3/x):)“-.00020 033(3/~)~+a

Real

. .)

according to the inequalities

Yv,1
where

35x<=

lel<9X

.


xY~(x)=(2/s)xIn($x)J1(x)-.63661 98 +.22120 91(x/3)2+2.16827 09(x/3)* -1.31648 27(x/3)6+.31239 51(x/3)* - .04009 76(~/3)‘~+.00278 73(x/3)12+e

8,=x-2.35619

(s=2, 3,

j:,,=j1,+1

10-S

o<x13

Itl<4X

j&,=0,

j.,*<jY+l.l<jY,2<jr+1.2<j".3<

x-l J,(x)=+.56249 985(x/3)2+.21O93 573(x/3)’ -. 03954 289(2/3)~f.00443 319(x/3)8 -. 00031 761(x/3)1o+.OOOO1 109(x/3)12+e

J~(x)=x-+$ cos 01,

9.5.1

9.5.2

-35x53

9.4.6

except that z=O YY:, J respectively, as the first zero of J;(z). Since J;(z)=-Jl(z), it follows that s and

is counted

The zeros interlace

le1<7X 10-s

9.4.5

ORDER

yy,s,3, s, Y,

397(3/x)

-. 00003 954(3/x)2+.OO262 573(3/x)3 - .00054 125(3/x)*- .00029 333(3/x)5 + .00013 558(3/~$~+ e

9.4.4

OF INTEGER

Infinite

9.5.10 9.5.11

Products

z2 ,!I ( l-x >

&I” J’(“)‘r(vfl)

J;(z)=%

ii (1-g)

r-1

(v>O)

BESSEL

FUNCTIONS

McMahon’s

When Y is fixed, s>>v

OF

Expansions

INTEGER for

ORDER

Large

371

Zeros

and p=4v2

9.5.12 p-1 3”. s, Y”. 8-fl --_ &3

4(P-l)(7w”-31) 3(8N3

-- 32(/J-l)

(83p2-9fi2p+377g) 15(8/3)’ -64(cr-

1) (6949p3- 1 53855p2+15

85743p--62

77237) -.

105(fq)’

where P=(s++$)~ forj,,,, /3=(s+&-$)a asymptotic expansion of p"(t) for large t.

for Y”,~. With

. .

the right of 9.5.12 is the

P=(t++v-i)?r,

9.5.13 ‘, s, yy, I ~-~‘-~8~,3-4(7~2+82a--9)_32(83r3+2075~2-3o39~+353~~ J,,,

15(8~?)~

3 (8P’)3

64(6949r4+2 96492p3-12 48002p2f74 1438Op-58 53627) -. 105(8@‘)’

where P’=(s+%v--f)s forjl,,, P’=(s++-2)~ 9.5.12 and 9.5.13 see [9.4] or [9.40]. and

Asymptotic Associated

Expansions Values for

for Y:,~, P’=(t+$v+t)s

of Zeros Large Orders

Uniform

9.5.14 jv,l~v+1.85575 71~“~+1.03315 Ov-“3 - .00397v-‘- .0908v-5/3+ .043v-7’3f

9.5.15 yp,l-v+

. . .

9.5.22

for

For higher terms in

u,,(t).

Asymptotic Expansions Associated Values for Large

j.,S-vz([)+z

of Zeros Orders

and

m fkW ,zrc-l with {=v-2i3a8

9.5.23 .93157 68vlt3+ .26035 l~-“~ +.01198v-‘-.0060v-5’3-.001v-“3+

. . .

Jxj”.J---$~

{1+2

k=l

9.5.16 j:, l -v+

..

.80861 65~“~$ .07249 OV-"~ - .05097v-‘+.oo94v-5’s+

with . . .

9.5.24

9.5.17 y:,1~+1.82109

80~“~+.94000 7~-“~ -.05808v-‘-.0540v-5’3+

. . .

J:(j,,)--1.11310 28~-~‘~/(1+1.48460 6vT2j3 + .43294v-4f3- .1943v-2+ .019v-s’s+ . . . )

9.5.19 y:(y,, I>w.95554 86~-~‘~/(1+ .74526 l~-~‘~ .003v-8’3+

. . .)

c=vq2f3aa

- ,zr-l gk(i-) with r=v+J3a:

jl,,-vz(l)+zI

9.5.25 J Y(j’ “, J)-&

9.5.18

+.10910v-4’3-.0185v-2-

Y}

(a’) ,

ho

+3

Gk(r) T}

{ l+e

with l=vS2J3a:

k=l

where a,, a.: are the sth negative zeros of Ai( Ai’ (see 10.4), z=z({) is the inverse function defined implicitly by 9.3.39, and 9.5.26 Mf)=14u(1--2)I~

9.5.20 Jy(j:, I) m-67488 51v-1’3(1-.16172 3~-~‘~ + .02918v-4’3-.0068v-2+

fi(r)=32(r>Ih(~)j2b,(r) . . .)

9.5.21 Y,(y:, 1)w-.57319 40~-“~(1- .36422 OV-~‘~ + .09077v-4’s+ .0237v-2-c . . . )

Corresponding expansions for s=2, 3 are given in [9.40]. These expansions become progressively weaker as s increases; those which follow do not suffer from this defect.

mw =%--‘4l){W)

12COW

where b,(l), co([) appear in 9.3.42 and 9.3.46. Tables of the leading coefficients follow. More extensive tables are given in [9.40]. The expansions of yy, S, YV(yy,J, y:, Sand Y,(y:, 3 corresponding to 9.5.22 to 9.5.25 are obtained by changing the symbols j, J, Ai, Ai’, a, and a: to y, Y, -Bi, -Bi’, 6, and b: respectively.

372

BESSEL

FUNCTIONS

OF

INTEGER

ORDER

= -1.000000 1. 166284 1.347557

FI(I)

0. 0143 .0142 .0139 .0135

1. 25992 1.22070 1. 18337 1. 14780 1. 11409 1.08220

:. E% 1: 978963

f,W

0: E

(-ShllW

-0.007 -. 005 2-E: -. 003 -0.002

-0. -. -. -. -. -0.

1260 1335 1399 1453 1498 1533

(-SMS) -0.010 -. 010 -. 009 -. 009 -. 008 -0.008

“:88: .004 .005

0: 8:x

=

z(S)

h(S)

1. 978963 2.217607 2. 469770 2.735103 3. 013256

: EG

--

PI W

flW 0.0126

:E

1: 02367 0.99687 .97159

. 0110 .0105

5: 661780 6.041525 6.431269

5. 8

iti 6: 8

0619 0573

-. -. -.

;g; 0464 0436 0410

0.0062

8.968548 9. 422900 9.885820 10.357162 10. 836791

0.70836

11.324575 11. 820388 12.324111 12. 835627 13.354826

0. 65901 . 65024

: FE% .0065

. 67758 .66811

:ii$

001

.002

-0.001

0.001 . 001 . 001 . 001 0. 001

r: -.

;;;g 0311

-I-“: -. -. -.

;;g 0270 0258 0246

Complex

fl(S)

Sl(S) I-

1.528915 1. 541532 1.551741 k . KfEr:

1. 62026 1.65351 1: ; y3; 71607

0.0040 ..0029 : y; 0006

-2

0. 15

1.568285

1.72523

0.0003

-0.0014

.E . 00

1. 570703 1.570048 1.570796

1. 73002 :. . %ii

Values

-.

-

1

I

Maximum

0.006

-l-o:8;: -:E

0.61821

(--r)+W

-I

-I

G,(I)

- 0.0386 -. 0365

.2E

-

z(r)-8(-r):

O. 40 :% 3

-. -.

. Ei: .73115 .71951

-

c-r)-+

-0.0807

: E1

0. 76939

13.881601

7. 0

1533 1301 1130 0998 0893

. 90397

0.0078 .0075

4.4 4. 6 4.8

001 001 001

-0. -. -. -. -.

I: Kg

0. 84681 .82972 . 81348 . 79806 .78338

2X

-. -. -. -0.001 -0.001

0.94775 92524

BXi%!?

-.. g22

Q-J w

s1w

of Higher

. 0001 0000 . 0000

w;

-. 0033 1: ym;

I: -*

Coefficients

lf*(!3I=.OOl, I~2(!31=.0~4 (Oh-<4 lga(r)l=.ool, IG2(s)I=.ooo7 Cl<---r
Zeros

of J,(s)

When u rel="nofollow"> - 1 the zeros of J”(z) Y< - 1 and Y is not an integer the plex zeros of J”(z) is twice the t-v) ; if the integer part of (-v) these zeros lie on the imaginary If ~20, all zeros of J:(z) are

are all real. If number of cominteger part of is odd two of axis. real.

g;‘: 0000 Complex

Zeros

of Y”(a)

When Yis real the pattern of the complex zeros of Y”(z) and Y:(z) depends on the non-integer part of Y. Attention is confined here to the case v=n, a positive integer or zero.

BESSEL

FUNCTIONS

1

i(na+b)

'\

'. -__

/'.

_*- .' -i(no+b)

t FICWRE 9.5.

Zeros of Y,(z) and Yh(z) . . . 1arg 215x.

Figure 9.5 shows the approximate distribution of the complex zeros of Y,(z) in the region larg zj<x. The figure is symmetrical about the real axis. The two curves on the left extend to infinity, having the asymptotes Az=f$ln3=&.54931

. . .

There are an infinite number of zeros near each of these curves. The two curves extending from z=--12 to z=n and bounding an eye-shaped domain intersect the imaginary axis at the points fi(na+b), where a;=-=.66274

b=$,/m

T..--m-----UYI’J!iti&kc unlJr;n

C” ,I

are given by the right of 9.5.22 with v=n and {=n-2/3& or n-2i3&, where 8,, pS are the complex zeros of Bi(z) (see 10.4). Figure 9.5 is also applicable to the zeros of Y;(z). There are again an infinite number near the infinite curves, and n near each of the finite curves. Asymptotic expansions of the latter for large n are given by the right of 9.5.24 with v=n and {=n+l”PL or r~-~‘~&; where @j and &! are the complex zeros of Bi’(z). Numerical values of the three smallest complex zeros of Y,(z), Yllz) and Y;(z) in the region 0< arg Z
CUT

-e--q.

and 4=1.19968 . . . is the positive root of coth t =t. There are n zeros near-each of these curves. Asymptotic expansions of these zeros for large n

Hankel

-n

Functions

n

,..a -+ilnz

\ \

FIGURE . . .

of the

The approximate distribution of the zeros of H:)(z) and its derivative in the region larg zll?r is indicated in a similar manner on Figure 9.6.

. . .

In 2=.19146

Zeros

r/n

7 1’

.

rino

Zeros of HA’)(z) and IQ)‘(z)

9.6.

(arg 21Ix. The asymptote of the solitary given by ys=--)ln2=-.34657

infinite

. . . curve is

. . .

Zeros of Ye(z) and Valufs of YI (2) at the Zeros 3 Zero

Real -2.40301 -5.51987 -8.65367

6632 6702 2403

Yl

Real Imag. +. 53988 2313 +. 10074 7689 +. 54718 0011 -. 02924 6418 +. 54841 2067 ’ +. 01490 8063

Imag.

-. 88196 7710 f. -.

58716 46945

9503 8752

Zeros of Yl(z) um! Values of YO(z) at the Zeros Zero yo Real -0.50274 3273 -3.83353 5193 -7.01590 3683

Imag. +. 78624 3714 +. 56235 6538 +. 55339 3046

-. f. -.

Real 45952 7684 04830 1909 02012 6949

Imag. +l. 31710 1937 -0.69251 2884 +O. 51864 2833

Zeros of Y:(z) und Vuhes of Yl (2) at the Zeros ZWO Yl Real +O. 57678 5129 -1.94047 7342 -5.33347 8617 * From Columbia

Imag. +. 90398 4792 +. 72118 5919 +. 56721 9637

Real -. 76349 7088 +. 16206 4006 -. 03179 4008

National Bureau of Standards, Tables of the Bessel functions Univ. Press, New York, N.Y., 1950 (with permission).

Zmag. f .58924 4865 -. 95202 7886 +. 59685 3673

Ye(a) and

Y1(z) for complex

arguments,

BESSEL

374

FITNCTIONS

There are n zeros of each function near the finite curve extending from z=-n to z=n; the asymptotic expansions of these zeros for large n are given by the right side of 9.522 or 9.5.24 with p=n and f=e-2rg/k-2/aa8 or pe-2+*&-2&:. Zeros

the zeros of the

J”(Z) Y”(XZ)---J”(XZ) Y”(Z)

If X>l, are real and simple. expansion of the sth zero is P fl+s+

9.5.28

n-PyQPd-2P3 /33

06

the asymptotic

+*-*

where with 4v2 denoted by cc, 9.5.29

Modified

--?L-1 8X

Differential

9.6.1

6(4X)3(X-1) 211

The asymptotic expansion of the large positive zeros (not necessarily the sth) of the function J:(z) Yp(Xz) --J:(xz) Y;(z)

(A>l)

LO

is given by 9.5.28 with the same value of & but instead of 9.5.29 we have 16

9.5.31

P=x’

k4+3

g&2+46~--63)(~3-1) 6(4X)3(X-l)

r=(p3+185~2-2053p+1899)(X”-l) 5(4X)6(X- 1)

1.2

The asymptotic expansion of the large positive zeros of the function 9.5.32

dzw 22p+Z

I and K

and Properties Equation

dw --(z2+v2)w=o d2

Solutions are I&z) and K(z). Each is a regular function of z throughout the z-plane cut along the negative real axis, and for fixed z( #O) each is an entire function of v. When v= f n, I,(z) is an entire function of 2. Iv(z) ($3’~ 2 0) is bounded as 2+0 in any bounded range of arg 2. Iv(z) and I-42) are linearly independent except when v is an integer. K(z) tends to zero as jzj-+ao in the sector jarg 21<337, and for all values of v, I"(2) and KY(z) arelinearly independent. I"(z), K(2) are real and positive when Y>-1 and z>O.

,=(lr-l)(~--25~(x3-l)

T,(p-1)(p2-l14/l+1073)(x6-l) 5(4X)yh- 1)

9.5.30

Bessel Functions

9.6. Definitions

jT3=sr/(X- 1) ‘-

ORDER

of Cross-Products

If Y is real and X is positive, function 9.5.27

OF INTEGER

A

Jl(z)Y”(xz)-Y:(z)J,(xz)

is given by 9.5.28 with 9.5.33

B= b--%)7+--l)

.4

,&+3)X-w) 8X(X-- 1) ,=(~~+46~-63)~~-(p-1)(~-25) 6(4X)3(X-l) 5(4X)s(X-l)r=(p3+185~2-2053p+1899)X6 -(/b-l)

(/.&-114c(+1073)

(

BESSEL

FUNCTIONS

OF INTEGER

ORDER

375

9.6.5 Yv(zet*f)=e*(Y+l)rfl,(z)-

(2/7r)e-fv”tK,(z) (--a<arg

9.6.6

I-n(z>=In(z>,

zIh>

K-,(z)=K,(z)

Most of the properties of modified Bessel functions can be deduced immediately from those of ordinary Bessel functions by application of these relations.

Limiting

Forms

for

Small

Arguments

When v is fixed and z+O FIGURE 9.8.

e-Zlb(2),e-ZI~(2),e"Ko(~)

and e"&(x).

9.6.7 (v# -1,

~“(~)~(3~)“/~(v+l) 9.6.8 9.6.9

Ko(z)m-In K(z)-J~r(v)(42)-”

odd="

9.6.11 K,(z)+($z)-"

FIGURE

9.9.

1,(5)

Wv>O)

9.6.2

K”(z)=3*

Between

Series

go s k(r;;;;l) .

ns (n-;yl)! (-iz">" k-0 * + (-->"+I ln WI&)

where #(n) is given by 6.3.2.

and KJ5).

9.6.12 Relations

. . .)

2

Ascending

9.6.10

-2,

Ldz)=1+;$+ ---

(;z2)2

(;z2)3

c2!J2+ c3!12+. . .

9.6.13

Solutions

I4(4--lr(Z) sin (~)

Kohl=-

The right of this equation is replaced limiting value if v is an integer or zero.

by its

Ih (3z)+r)Io(z>+-

42" (1!)2

+(l+i)~+(l+a+3)$$+...

9.6.3 I,(~)=e-fvr~J,(zet*~) ~,r(~)=e~~*~~~J,(~-~~*‘~)

(-r<arg

253~)

&<mz

z
K,(~)=3?rieC”~H~‘(ief”%) (---
zi3d 25~)

9.6.4

Wronskians

9.6.14 W{ I"(Z), I+(z)) =I"(z)I~(,+l)(z)--I"+l(z)l-,(z) =-2 sin (vT)/(~z) 9.6.15 W{K"(Z),

I"(Z)}=I"(Z)K"+I(Z)+l"+~(Z)K"(Z)=l/Z

BESSEL

376 Integral

FUNCTIONS

OF

INTEGER

3, denotes Iv, e”**K, or any linear combination of these functions, the coefficients in which are independent of z and v.

Representations

9.6.16

9.6.27 9.6.17

Koiz)=-~sor

9.6.18

e*2cone {r+ln

Set2

(32)” l’(Z)=?rv(Y+f)

= 0

ca2)”

=Ayv+g

s

COI

l (1-P) -1

I.(z)ll

9.6.19 9.6.a

0

ORDER

(2zsid

I;(z)=l~(z),

K;(z)=-K,(z)

e)ja% Formulas

sin2V

0

&

for

Derivatives

9.6.28

v-fe*rr&

-+

(> %“!E’ (2) =fi~,-kc4 +(;)z”-k+2cz, 9--v-,+4(4 +**’+s”+kk) 1 5 -g *{ 2-“~(2)}

e’ Eoaecos (n@7!8

=z-“-k%o”+k(z)

(k=0,1,2,.

* .)

(k=O,1,2,.

. .)

9.6.29

I”(z)=: s,

e’ cone cos (vO)d49

-- sin (VT) ODe-2 oontll-v’& ‘1F 0

S

(b-g 4<+4

cos (x sinh t)dt=- cos* Ko(x)=s0 s 0 JP+1

&

9.6.21

Analytic

(x>O)

9.6.22

9.6.30

Iv(zemwf)

Continuation

(m an integer)

==emvrfIv(z)

9.6.31 Om) s,- cos (x sinh t) cash (A)&

Kh)=sec

=csc (&J?f)

S

Q sin (x sinh

0

t) (vt)dt

Ky(ze”LrO=e-mylfKI(z)--?ri sin (WI) csc (v?~)I,.(z) (m an integer)

sinh

(I~‘yI
XX)

9.6.32

I,(;)

K,(H) =Kx

=Ir(Z),

(Y real)

9.6.23

?d(~z)” o=e-rco*r sinh2? dt K&)=r(v+t) s

d(&2)” ‘r
-

S

Generating

dt

emrr(t2-1)‘+

WV>--4,

law 4<44

Function

and

Series

9.6.33

e~‘(‘+“‘)=

9.6.34

e’ cOse=Io(~) +2 2 In(z) cos(ke)

5 tkIk(z) kas-oa

O#O)

k=l

Kv(,z)=~me-‘coa’ cash (vt)dt ((arg 21<$T) 9.6.35

9.6.25 4X"

s*

Recurrence

x>O, kg

Relations

9.6.26 ~“~lo-~“+l~z)=~~“(Z) \ a,&-;

g

(-)k12k+l(z)

sin{ (2k+i)e}

(t2+z2)"+'

(SV>--3,

s@;(z)=

e2a1ne=IO(z)+2

- cos txtjdt

K”(xz)=r(V+~)(22)”

2i?@“(Z)

~“--1(2)+~“+1(z)=2~:(2)

+2

4<&)”

& I

(-)%(2)

9.6.36

l=I,(z)

9.6.37

e2=Io(z)+211(z)+212(z)+212(2)+

9.6.38

e-2=lo(z)-211(z)+212(z)-21,(z)+

-212(2) +214(z) -2&(2)

c0sWe)

+ . . . ... .. .

9.6.39 cash 9.6.40

*See page 11

Associated

2=lo(2)

+212(2) +21,(2) +21,(z) + . . .

sinh 2=211(2)+21,(2)

+21,(2)+

.. .

BESSEL Other

Differential

FUNCTIONS

Equations

OF

INTEGER

9.6.50

The quantity X2 in equations 9.1.49 to 9.1.54 and 9.1.56 can be replaced by --X2 if at the same time the symbol ‘% in the given solutions is replaced by Iz”.

ORDER

377

lim { v-rem@&: (cash f)} =K,,(z)

For the definition

of P;’

and Q:, see chapter 8.

Multiplication

Theorems

9.6.51

9.6.41

zW’+2(1f2z)w’+(fz--~)w=o,

w=eT2%ry(z)

Differential equations for products may be obtained from 9.1.57 to 9.1.59 by replacing z by iz. Derivatives

With

Respect

If %“=I and the upper signs are taken, the restriction on X is unnecessary. 9.6.52

to Order

9.6.42 Neumann

Series

for

K.(s)

9.6.53

9.6.43

K,(z)=(--)a-l{ln $ K,(z)=3

u csc(vu) {$ r-.(z)-; -u

cot(vu)K”(z~

($2)~$(n+l)]I,(z)

I”(Z)} (v#O,fl,f2,

* * .>

9.6.44

+(-)”

5

(7JSWIn*2r(z)

k-l

C--P

[g/w]

p-1

=

9.6.54

Ko(z)=-

kb+k)

(In (~z)+~)Io(2)+2

8 ‘q m

Zeros

9.6.45

9.6.46

Expressions

in Terms

of Hypergeometric

Functions

9.6.47 -.i!@- 1) OF, (V+l; lv(z)=r(v+

9.6.48

$2”)

Properties of the zeros of II(z) and K,(z) may be deduced from those of J”(z) and W)(z) respectively, by application of the transformations 9.6.3 and 9.6.4. For example, if v is real the zeros of IV(z) are all complex unless -2k
K~(z)=($vo,.(22) 9.7.

(oFI is the generalized hypergeometric function. For Ma, b, z), MO,.(z) and Wo.y(z) see chapter 13.) Connection

With

Legendre

If /1 and z are fixed, &‘z>O, real positive values 9.6.49

Functions

and

v--m

Asymptotic

When

v

Asymptotic

Expansions

Expansions

for

Large

Arguments

is fixed, (21is large and I.LCC=~V~

9.7.1 through

cc-1 x+

(w-l>G---9) 2f(&,)2

~(rc--l)wocP--25)+ 3!(82)3

* *

.I

(lawl<W

378

BESSEL

FUNCTIONS

OF

INTEGER

ORDER

9.7.10

J-c(vz)--J 72; (1+22)"4 2 9.7.3

When v++ 03, these expansions hold uniformly with respect to z in the sector (arg 21 <&r-e, where e is an arbitrary positive number. Here

rf3 GL- 1) 01+15) 82 + 2! (82) ’

m)“&11-

JP-l)oc--9)Gc+w 3!(8~)~

e--(l+$+)ky}

+ ..4

Wg4
9.7.11

t=l/&p,

~=~+ln

K:(z) --

&e-y

J

1+x+cc+3

01-l) (rfl5) 2! (82) 2

+(p-1)Gc-g)b+35)+ 3!(8~)~

* - .)

and z&), vk(t) are given by 9.3.9, 9.3.10, 9.3.13 and 9.3.14. See [9.38] for tables of II, uk(t), vk(t), and also for bounds on the remainder terms in 9.7.7 to 9.7.10.

(larg zl<#~)

The general terms in the last two expansions can be written down by inspection of 9.2.15 and 9.2.16.

9.8. Polynomial

In equations

If Y is real and non-negative and z is positive the remainder after k terms in the expansion 9.7.2 does not exceed the (k+ 1)th term in absolute value and is of the same sign, provided that k_>v-3. 9.7.5

b-w-9)~ &I4

lt\<1.6XlO-’

. . *)

9.7.6

3.75 5x<= .39894 228 + .01328 592 t-l +.00225 315t-2-.00157 565t-a t .00916 281t-4-.02057 706t-s +.02635 537t-6-.01647 633t-1 +.00392 377t-8fc \e~
9.8.3 (r-45)

+

(22)4

. .

) *

The general terms can be written inspection of 9.2.28 and 9.2.30. Uniform

9.7.7 9.7.8

Asymptotic

I.(vzg-

j!G

Expansions

ey’

(1+22)1’4

for

jl+gl

Large

Y}

-3.75 sx 53.75 =$+ .87890 594t2+.51498 869t4 + .15084 934te + .02658 733t8 +.00301 532Pf.00032 411t’*+a (e)<8XlO-’

x-‘I,(x)

down

by 9.8.4 xk=I,(x)

Orders

5xs3.75

29t2+3.0899424t4+1.20674 92te + .26597 32t8+ SO360768t’O+ .00458 13P2+ t

(la%?4<+7d

-- 1.- 1 b-1) 2.4

*

&,(z)=1+3.51562

xhPIo(x) 2.4

Approximations

9.8.1 to 9.8.4, t=x/3.75. -3.75

9.8.1

9.8.2 ; l-3

L-

1+4+9

9.7.4

3.75 <x
4 See footnote

2, section 9.4.

BESSEL

9.8.5 Ir,(x)=-In

FUNCTIONS

o<x<2

INTEGER

Differential

]cl
w=ber,

bei, x,

her-, x+i bei-, x, kei, x, ker-, x+i kei-, x

3.9.4

]t]<1.9X

10-7

(1+2v2) (22w”-xw’) +(v4-4v2+x4)w=o,

w=ber*, Relations

x, bei+, x, ker,, x, ke&

Between

ber-V x=cos(m)

In (x/2)1,(x)+1 +.15443 144(~/2)~ -.67278 579(x/2)4-.18156 897(x/2)0 -.01919 4O2(x/2)8-.OO11O 404(x/2)‘O -.00004 686(~/2)‘~+s ]e]<8XlO+

Solutions

berY x+sin(va)

bei+ x=-sin(m)

bei, x + (2/7r) sin(vr) kerY x

berY X+COS(V?T) bei, x + (2/7r) sin(m)

kei, x

9.9.6 ker-V

2<x<m

x*e%,(x)=1.25331 414+.23498 619(2/x) -.03655 620(2/~)~+.01504 268(2/x):)” - .00780 353 (2/~)~+ .00325 614(2/x)6 - .00068 245(2/@+ e

ker,

x=cos(v?r)

kei-, x=sin(m)

kei, r

x-sin(v?r)

ker, z+cos(va)

kei, x

9.9.7

her-, z= (-)”

ber, 2, bei-, r= (-)”

bei, x

9.9.8

ker-, a= (-)”

ker, 2, kei-, x= (-)”

kei, x

]a]<2.2x10-7 For expansions of 1o(x), Ko(x), II(x), and K,(x) in series of Chebyshev polynomials for the ranges Osx18 and OSSjxSl, see [9.37]. Kelvin

Ascending

9.9.9

and

Series

{(sv+3bl (tx2”>” l-m x=(W~~-E 0x3 krr(v+k+l) . - sin{ (+++k)r} b& ~=(tx)“~~ k,r(v+k+l) (ix2Y

Functions

9.9. Definitions

Properties 9.9.10

In this and the following section v is real, x is real and non-negative, and n is again a positive integer or zero.

(ix”)” --* herx=1 (tx”)” (2!)2 +m-(+xy (+xy”- * * * bei x=ax* -- (3!)2 +m

Definitions

9.9.1

berY xfi

9.9.11 n-1

bei, x=Jy(xe3*f’4) =ey**JV(xe-*f’4) =etv”i~v(xe”‘“)

ker, x=$($x)-” x(7L-k-1)! k!

kei, x=e+nfKy(xeri’4) =$.&$;I)

(xe3ri/4)

When v=O, su&es

2

cos { (~wl-$k)~j

,e3v*i/21v(xe-3W4)

9.9.2 ker, x+i

x

9.9.5

o<x52

9.8.8

xfi ker, x+i

x*eZKo(x)=1.25331 414-.07832 358(2/x) +.02189 568(2/x)*-.01062 446(2/~)~ +.00587 872(2/~)~--00251 540(2/~)~ +.00053 208(2/x)e+e

x&(x)=x

Equations

E2W’~-+2W’-+x2+v2)w=0,

&+~$253w’!-

25x
9.8.7

379

ORDER

9.9.3

(x/2)1&)-.57721 566 +.42278 420(x/2)2f.23069 756(~/2)~ +.03488 59O(x/2)6+ .00262 698(x/2)* +.OOOlO. 75O(x/2)1o+.OOOOO 740(x/2)12+e

9.8.6

OF

=

(tx2)k-ln

(ix) ber, x++n

bei, x

+3(3x>” F. ~0s I (9n+#>*l -$~,-v*iH;2,

(xe-*i/4)

are usually suppressed.

x Mk+;,;“:“k,‘“+” .n

!

1 +z)” 4

BESSEL FUNCTIONS

380 kei, x=-$(3x)-”

($8)k-ln

k! +MY

ORDER

9.9.16

ngia sin { ($n+t&} B

x(n-k-l)!

SF INTEGER

er’ x=ber,

ab

(3x) bei, x-5

her, x

x+bei,

112 bei’ x=-berl

x+be& x

9.9.17 l/z ker’ x=ker,

go sin { (Sn+34*1 x I+(k+lk)r;~ktk+l)

x

x+keil

@ kei’ x=-kerl

1 oti>” !

Recurrence

x

x+kei,

Relations

for

x

Cross-Producta

If where #(n> is given by 6.3.2. 9.9.18 9.9.12 ker x=-ln

(3x) ber x+$t

+go kei x=--In

p,=bee q,=ber,

bei x t-1” :rk;j2

(3x) bei x+r ber x .a) +g l-1” {$y-$

x+beif x x bei: x-her:

rV=berr x her: x+bei, .s,=be$ x+beiia x

(t’)”

of Negative

x bei: x

then w)“+’

9.9.19 P.+l=P”-1-T

Functions

x bei, x

rr

Argument

qv+1= -;

In general Kelvin functions have a branch point at x=0 and individual functions with arguments xe*‘: are complex. The branch point is absent however in the case of berY and bei, when Y is an integer, and

P”+r,=--q,4+2r,

(v+l) ----z&I-

T”+l=

sv=;

p.+,+;

P”+l+qv a.&$

p,

and

9.9.13 ber,(-x)

= (-)” her, x, Recurrence

be&,(-x) = (-) * bei, x

9.9.20

pd.= 19i- d

The same relations hold with ber, bei replaced throughout by ker, kei, respectively.

Relations

9.9.14

Indefinite

j”+l+j”-l=-@

x

fi=&

(.frgv~ cf”+1+g”+1T~“-1-!7J”-1)

Integrals

In the following jy, gV are any one of the pairs given by equations 9.9.15 and jf, g: are either the same pair or any other pair. 9.9.21

j+=+ jI+;f”

U”+l+g”+1)

S

xl+“j~~=2c”

=-’ Jz (f”-l+g”-l)

“+l--g”+J=--~ Jz

I+”

(5

S.-d)

(j

9.9.22

(;9.+g:> Sx*-"@x,x~ @(j"-l-g"-l)=xl-'

where 9.9.15 f,=ber,

x

j,=bei,

x

g,=bei,

x1

g.= -berV x 1

9.9.23

S

x(j”g:-g”fl)dx=~

-s:(j”+l-

2Jgq vxf”+l+s”+l)

g”+1)-j”(~+l+gF+1)+g”(j~+~-g~+l)

=; x(flft-j”~‘+g:gf-g”s:‘)

1

BESSEL

FUNC’I’IONS

OF

INTEGER

Zeros

9.9.24 s

ORDER

z(j”g:+gvjz)dz=;

381

of Functions

ber x

~‘(2j”s~-j47~+1

of Order

=

=

bei x

1st 2nd 3rd 4th 5th

-j”+lg2-1+2g”fr-g~-lff+l-g,+l~-l~

zero zero zero zero zero

2.84892 7. 23883 11. 67396 16. 11356 20. 55463

6

=

ker x

--

9.9.25

Zero

kei x

.5. 02622 9.45541 13. 89349 18.33398 22. 77544

_1. 71854 6. 12728 10.56294 15. 00269 19.44381

3, 8. 12, 17. 21.

91467 34422 78256 22314 66464

Sx(f".+gay)dx=x(j"g:-f:gl) =-(x/:/1I2)(frf~+l+g"g"+l--f,g~+l+f,+lg~) ~2(2j~g"-j~-lg~+,-j"+lg,_l) Sxj"gdx=; =

=

ber’ x

bei’ z

ker’ x

--

9.9.26

1st zero 2nd zero 3rd zero sl::r:

9.9.27

6.03871 10. 51364 14.96844 19.41758 23. 86430

3. 77320 8.28099 12. 74215 17. 19343 21. 64114

9.10. Asymptotic

for

Asymptotic

Cross-Producta

--

f 11: 16. 20.

4.93181 9.40405 13.85827 18. 30717 22. 75379

%i 63218 08312 53068

-

~(~-j"-lj~+l-g3+g"-lg"+l) Sx(-E-g:)dx=; Series

kei’ x

.-

-

Ascending

=

Expansions

-

Expansions for

Large

Arguments

When v is fixed and x is large

9.9.28

berf, x+beit

x=

9.10.1 0

(ix)2’ 3

r (v+k+l)

1 r (v+2k+

WS>“” 1) k!

ber, x=zx{

j,(x) cos a+gv(x) sin a}

-k {sin(2~74 km.

9.9.29

her, x bei: x-b& =(*x)2*+1

x bei, x 2

(24

kei, x)

(24

kei, x)

9.10.2

r (v+k+l)

1 r (v+2k+2)

WYk k!

bei, x==~e/d (j.(x) sin cr-g”(x)

9.9.30

+;

her, x her: x+bei,

x+cos

x bei: x

1

{ cos (24

co8 CX}

ker, x-sin

9.10.3

ker, x=dme-*/d2{j,(-x)

cos S-gl(-x)

sin PI

9.10.4

her? x+bei?

kei, x=~~e-z~~3a(-j,(-x)

x OD

=(4xP-2 Expansions

3

(2k2+2vk+fv2) r(v+k+l)r(v+2k+l)

in Series

of Bessel

W”>‘” k!

where 9.10.5 ~=bMa++-~>~,

0

w

cos S}

Fuxwtions

9.9.32

her, x+i bei, x=E

sin 8-g.(-x)

e(8r+kw4~Jv+k(x)

2*k k!

a=(x/m+(3v+H~=~+tn

and, with 49 denoted by p, 9.10.6 jv(*

4

-,+&+-l%--9) k-l

* *.b4k-v~cos k! (8x)”

h

04

6 From British Association for the Advancement of Science, Annual Report (J. R. Airey), 254 (1927) with permission. This reference ah30 gives 5-decimal values Of the next five zeros of each function.

BESSEL FUNCTIONS

382

OF INTEGER ORDER

9.10.16

-$, (Wn (cc--l)h-9)

. . .{P--(2k-l)“] k! (8x)n

sin kr 0 T-

The terms” in ker. x and kei, x in equations 9.10.1 and 9.10.2 are asymptotically negligible compared with the other terms, but their inclusion in numerical calculations yields improved accuracy. The corresponding series for her: x, b ei: x, ker: x and kei: x can be derived from 9.2.11 and 9.2.13 with z=xe3ri14; the extra terms in the expansions of her: x and bei: x are respectively -(I/W)

{sin(%vr)ker:

x+cos@v?r)kei: x-sin(2vr)kei:

9.10.17 x~M~‘+~M:--~M,=~~~~?

&(rM2le:)/&=dC,

9.10.18 N,=d(ke83.+kei?x),

+.= arctan (kei, x/ker, x)

9.10.19 ker,x=N,

cos &,

kei, x=N,

x}.

6 4

Modulus

and

I

Phase

x+beil, s),

9.10.9 ber, x=M, 9.10.10

M-,=

1.04 - .02

9.10.8 MV=d(bee

sin 4”

x}

and (l/r){cos(2u?r)ker:

M;=M, cos te,-e,-+) e;=(M,/M,) sin (&-&,-$w)

ey= arctan (bei, x/her, r)

cos 8,,

bei, x=Mv

sin 0,

e-,=e,--nq

M,,,

9.10.11 her: jc= 3 MP+* cos (o,+1-&)-4 MP-1 cos (e,-,-+) = (Y/X)M, co9 e,+ M,+l cos (e,+l- 47r) = - (v/x)M, cos e,-M,-, cos (e,-l-$d

-6-10 L

\

FIGURE 9.10.

ber x, bei x, ker x and kei x.

9.10.12 bei: x= $M,+, sin (8,+1-+n)

- $M,-, sin (&-I - $r)

= (V/S)M,sin e,+ M,+, sin (&+,-$a) = - (u/x) M, sin &-- Mvml sin (e,-,- ia)

9.10.13 ber’ x=M,

cos (o~--~T),

bei’ 2 = Ml sin (e,- tr)

9.10.14 2M:=(v/x)M,+M,+~cos (e,,-e,-$T) = - (v/~)M,- M,-, cos (e,-,--e,-tT)

9.10.15 e;= (M,+,IM,)

sin (fI,+I-&-$?r)

= - (MJM,)

sin (e,-,-e,-)lr)

6 The coefficients of these terms given in [9.17] are incorrect. The present results are due to Mr. G. F. Miller.

FIGURE 9.11.

In MO(x), b(x), In NO(~) and 40(x).

9.10.11 to 9.10.17 hold with the symbols b, M, e replaced throughout by k, N, 4, respectively. In place of 9.10.10

Equations

9.10.20

N-p= Ny,

4-Y=&+ wr

BESSEL FUNCTIONS Asymptotic

Expansions

of Modulus

and

OF INTEGER

ORDER

383

9.10.29

Phase

When Y is fixed, x is large and r=49

ber x ber’ x+bei

9.10.21

x bei’ x-e

L-3

212

!.

42

8x

15 1 45 1 315 -- 6442 x2 --- 512 ?+819242

1 p+

’ - *

1 2475 1 --+. x4 i?+S192

. .

_ G- 1)w+ 14/J-399) 614442

9.10.22 In My=?-+

In (27rx)-- r-l

42

--1

842 5

(p-1)(/?-25) 38442

1 ?i?

JrW-13)

75 +25642

9.10.31

k&x+kei2 x-&e--r42

128

l--!-!.

9.10.23

?f+jj-$-;+ p-l

1 --p-l 1 16 Z

-G-w--25) 38442

’ ’

442 x+64 2

+ 25642 - 33 2--1 1 >+o

8192 x4 1797 -+ 1

9.10.32 ($5)

ker x kei’ x-ker’

x kei x--g

e-zd2 L-!. 42

9.10.24

N,=

. . .>

+-

,-WT{I+% ;+&I&? 2

9 --1 39 s+slaaJa 1 --L+. 75 6442 x2 512

i

8x

. .

x4

9.10.33

+(P-w2+14P-399) 614442

$+ik

ker x ker’ x+ kei x kei’ x m -Fx e -zd2

(

9.10.25 1 (cc-1)(/e-25) h N,=-g+f ln 0& +-cc-1 ;+ 384,i2

1 i?

9.10.34

JP-l)(P--13) 128

ker’2 x+kei’2

x-g

em242 1 +&t

+& f

9.10.26

+k-l)kW 38442 Asymptotic

Expansions

Asymptotic

f@)-‘-1

If& -m

9.10.28 x bei 2-c

166

I cc-1 ; b--l)(5P+lg) 3262 153663

where ~=43. 33

ber x bei’ x-ber’

Zeros

9.10.35

9.10.27 xm2g

of Large

Let

of Cross-Products

If 5 is large

ber2 x+bei2

Expansions

1 1797 1 s-8192 p+

* * *>

I 3(rU2 51264

; .. .

Then if .s is a large positive integer

9.10.36 Zeros of her, z*&{G-f(8)},

6= (s-*Y-~),

Zeros of bei, xw &{ S-~(S) },

s=(s--)Y+$)*

Zeros of ker, x-&{~+f(-s)},

s=(S-+--Q)7r

Zeros of kei, x-@{~+f(-Qj,

s=(s-&-*)*

384

BESSEL

FUNCTIONS

OF

For v=O these expressions give the 6th zero of each function; for other values of v the zeros represented may not be the sth. Uniform

Asymptotic

Expansions

for

Large

INTEGER

ORDER

9.11.3

O<x58

ker x=-In

(h) ber x-&r

-59.05819 -60.60977

Orders

When v is large and positive

bei x-.57721

744(x/8)4+171.36272 451(x/8)12+5.65539

133(x/8)8 121(x/8)”

- .19636 347 (x/S)‘O+ .00309 699 [x/8)24

9.10.37

-.00002

ber,(vx) +i bei,

566

458(x/8)2*+a

161<1 x 10-n

-

9.11.4 9.10.38 ker, (~x)++i kei, (vx)

0<218

kei x---ln($x)bei -142.91827

x-&r ber s-j-6.76454 936(x/8)2 687(x;/8)‘+124.23569 650(x/8)l”

-21.30060 -.02695

904(x/8)“+1.17509 875(x/8)22+.OOO29

9.10.39 her: (vx)+si bei: (vx)

064(r/8)‘8 532(~/8)‘~+c

(tj<3x10-9

9.11.5 9.10.40 ker: (vx)+i

-8<x<8 aher’ ~=~[-4(x/8)~+14.22222 kt:iI (vx)

-6.06814 -.02609

222(x/8)’

810(~/8)‘~+.66047

849(x/8)”

253(~/8)‘~+.00045

957(x/8)22

-.OOOOO 394(x/8)20]+c where

~e~<2.1x10-*

9.10.41

[=&FT? and u,(t), c*(l) are given by 9.3.9 and 9.3.13. fractional powers take their principal values. 9.11. Polynomial

All

9.11.6

-812_<8

bei’ z=z[$-

Approximations

10.66666 SS~(X/S)~

+11.37777 9.11.1

-85x18 ber x=1-64(2/8)‘+113.77777 -32.36345 -.08349

+.14677

772(~/8)~-2.31167 204(x/8)“--00379

774(x/8)*

652(x/8)12+2.64191 609(x/8)“+.00122

514(x/8)12 386(x/8)”

+.00004 397(x/8)“’

609(x/8)24]+c

IcI<7xlO-*

552(x/8)“’

- .OOOOO 901 (x/S)“+t (cl
9.11.7 ker’ x=--In

-8Sx_<8 113.77777 774(x/8)e

+72.81777 +.52185

742(s/8)*O-10.56765 615(x/8)‘*.-.01103 +.OOOll ~c~<SX~O-~

O<x<8 (4%) ber’ z--2+

ber s+t~

bei’ x

779(x/8)”

+x[-3.69113 734(~/8)~.+21.42034 017(x/8)’ -11.36433 272(~/8)‘~+1.41384 780(x/8)‘”

667(~/8)~~

-.06136

358(~/8)~~+.00116 -.OOOOl

346(x/8)2e+c Icl<SXlO-*

137(~/8)~’ 075(x/S)““]+b

BESSEL FUNCTIONS

kei’ x=--In

(ix) bei’ x-x-l

+x[.21139

385

where

O<x<8

9.11.8

OF INTEGER ORDER

bei x-tr

217-13.39858

9.11.11

ber’ x

846(a/8)4

+19.41182 758(x/8)‘-4.65950 823(x/8)12 +.33049 424(x/8)"--00926 707(~/8)'~ +.00011 997(z/8)*4]+e

19(x)=(.00000 00-.39269 91;) +(.01104 86-.01104 85$(8/x) +(.OOOOO 00-.00097 6%)(8/~)~ +(-.00009 +(-.00002 +(-.ooooo

06-.00009 52+.00000 34+.00000

Oli)(8/~)~ OOi)(8/x)' 51i)(8/x)'

+(.OOOOO OS+.00000 9.11.12

ker’ x+i

8<x<=

9.11.9

ker x+i kei x=f(x)

85x<m kei’ x=-f&)$(-x)

19i)(8/x)'

(1 +ta)

l~al<2XlO-’

(1 +eJ

81x<m ber’ x+i bei’ x-i ’ (ker’x+ikei’r)=g(+$(x)(l+ti)

9.11.13

j(x)=Gx

exp [-$

x+0(-x)]

(e4~<3x10-' where 9.11.14

9.11.10

=

81x<

her x+i bei x-z

(ker xfi

9(x>=kx

t#~(x)=(.70710 68+.70710 68;) +(-.06250 Ol-.OOOOO Oli)(8/x) +(-.00138

kei x)=g(x)(l+cJ

exp 19 * x+e(x)

1 Numerical

Methods n

of the Tables

Trial valuea

9

Example 1. To evaluate . ., each to 5 decimals. The recurrence relation

Jn-l(4

+Jn+1(4

J&.55),

n=O,

1li)(8/x)2

+(.OOOOO 05+.00024 52i)(8/~)~ +(.00003 46+.00003 38i)(8/~)~ +(.OOOOl 17-.OOOOO 24i)(8/x)" +(.OOOOO 16-.OOOOO 32i)(8/~)~

Icl<3XlO--7

9.12. Use and Extension

13+.00138

1, 2,

= (W4J,(4

can be used to compute Jo(x), 51(z):), J&c), . . ., successively provided that n<x, otherwise severe accumulation of rounding errors will occur. Since, however, J,,(x) is a decreasing function of n when n>x, recurrence can always be carried out in the direction of decreasing n. Inspection of Table 9.2 shows that J,,(l.55) vanishes to 5 decimals when n>7. Taking arbitrary values zero for Jo and unity for Ja, we compute by recurrence the entries in the second column of the following table, rounding off to the nearest integer at each step.

8 7 6 6 4 3 2 1 0

0

1 10 89 679 4292 21473 78829 181957 166954

541.66) .ooooo

.oooOO .00003 .00028 .00211 .01331 .06661 .24453 .56442 .48376

We normalize the results by use of the equation 9.1.46, namely JO(X)+~J~(X)+~J~(X)+

This yields the normalization l/322376=.00000

. . . =I

factor 31019 7

386

BESSEL

FUNCTIONS

and multiplying the trial values by this factor we obtain the required results, given in the third As a check we may verify the value of column. J,(1.55) by interpolation in Table 9.1. (i) In this example it was possible Remarks. to estimate immediately the value of n=N, say, at which to begin the recurrence. This may not always be the case and an arbitrary value of Nmay have to be taken. The number of correct significant figures in the final values is the same as the number of digits in the respective trial values. If the chosen N is too small the trial values will have too few digits and insufficient accuracy is obtained in the results. The calculation must then be repeated taking a higher value. On the other hand if N were too large unnecessary effort would be expended. This could be offset to some extent by discarding significant figures in the trial values which are in excess of the number of decimals required in J,,. (ii) If we had required, say, Jo(1.55), J1(1.55), each to 5 significant figures, we . . ., Jlo(l.55), would have found the values of J,,(l.55) and J11(1.55) to 5 significant figures by interpolation in Table 9.3 and t,hen computed by recurrence being required. Jet Je . . ., Jo, no normalization Alternatively, we could begin the recurrence at a higher value of N and retain only 5 significant figures in the trial values for nx, because in the latter event Y,,(z) is a numerically increasing function of n. We therefore compute Y,(1.55) and Y1(1.55) by interpolation in Table 9.1, generate YZ(l .55), Ya(1.55), . . .) Y,,(1.55) by recurrence and check YlO(l .55) by interpolation in Table 9.3. n Y,(f 56) n Y,(l.M) 0

+O. 40225

1 2 3 4 5

-0. 37970 -0.89218 - 1.9227 -6. 5505 - 31.886

6 7 8 9 10

-

1.9917x loa 1. 5100x 103 1. 3440 x 10’ 1.3722X lob 1.5801 x 10’

OF

INTEGER

ORDER

Remarks. (i) An alternative way of computing YO(x), should J,,(x), Jz(r), J&c), . . ., be availtble (see Example l), is to use formula 9.1.89. The other starting value for the recurrence, Y1(z), can then be found from the Wronskian :elation Jl(z) Y,,(x) - J,,(x) Y1(x) =2/(7rx). This is a :onvenient procedure for use with an automatic :omputer. (ii) Similar methods can be used to compute the modified Bessel function K,(x) by means of the recurrence relation 9.6.26 and the relation 9.6.54, except that if z is large severe cancellation will occur in the use of 9.6.54 and other methods for evaluating K,,(Z) may be preferable, for example, use of the asymptotic expansion 9.7.2 or the polynomial approximation 9.8.6. Example 3. To evaluate J,(.36) and Y,(.36) each to 5 decimals, using the multiplication theorem. From 9.1.74 we have

go (X z) =x

m

ak%Yk(z) , where aR = WW~l)‘(W.

k-0

We take z= .4. Then X= .9, (X2- 1) (32) = -.038, and extracting the necessary values of Jk(.4) and Yn(.4) from Tables 9.1 and 9.2, we compute the required results as follows: k 0 1 2 3 4 5

ak

$1.0 +0.038 +0.7220X +0.914x +0.87X +0.7x10-0

akJ

k(d)

akyk(..b)

-

+ .96040 + .00745 + .00001

lo-” 10-S lo-’

J,(.36)

= + .96786

Y&36)

Remark. This procedure

polating

is equivalent by means of the Taylor series

.60602 .06767 .00599 .00074 .OOOll .00002

= -.68055

to inter-

Gfo(z+h) =Foa ; . go’*(z) at z=.4, and expressing the derivatives %?e’(z) in terms of qk(z) by means of the recurrence relations and differential equation for the Bessel functions. Example 4. To evaluate J”(x), J:(z), Y,(z) and Y:(x) for v=50, x=75, each to 6 decimals. We use the asymptotic expansions 9.3.35, 9.3.36, 9.3.43, and 9.3.44. Here z=x/v=3/2. From 9.3.39 we find arccos i= + .2769653.

BESSEL

FUNCTIONS

387

ORDER

we find

Hence {=-.5567724

4{ li4 =+1.155332. and -l-22 ( >

as= -7.944134,

Ai’(

+ .947336.

Hence

Next,

~~‘~[=-7.556562.

~“~=3.684031, Interpolating Ai

OF INTEGER

in Table 10.11, we find that

= + .299953,

Bi(v2/3{)= -.160565,

Ai’(v213{) = + .451441, Bi’(v2/3[)=

+.819542.

As a check on the interpolation, we may verify that Ai Bi’-Ai’Bi=l/?r. Interpolating in the table following 9.3.46 we obtain b,(l) = + .0136,

c&)=+.1442.

Interpolating obtain

in the table following 9.5.26 we

z(c)= +2.888631, .fiw=+.0107,

h(l) = + .98259, F,(l)=-.OOl.

The bounds given at the foot of the table show that the contributions of higher terms to the asymptotic series are negligible. Hence jlo,s=28.88631+.00107+

. . . =28.88738,

The contributions of the terms involving a,({) and d,(r) are negligible, and substituting in the asymptotic expansions we find that x(1-.00001+ r&(75) = + 1.155332(5o-‘fix

+50-6”X.451441X.0136)=+.094077, &(75j = - (4/3)(1.155332)-1(5O-4/3X .299953 X.1442+5O-2/3X.451441)=-.O38658,

As a check we may verify that JY’-

. . .)=-.14381.

.299953 Example 6. To evaluate the first root of Jo(x)Y&x)-Yo(x)Jo(Ax)=O for X=Q to 4 significant figures. Let CX~’denote the root. Direct interpolation in Table 9.7 is impracticable owing to the divergence of the differences. Inspection of 9.5.28 suggests that a smoother function is (X-l)@. Using Table 9.7 we compute the following values l/X 0. 4

(A- l)cQ(1) 3.110

0. 6

3.131

J’Y=2/(75s).

Remarks. This example may also be computed using the Debye expansions 9.3.15, 9.3.16, 9.3.19, and 9.3.20. Four terms of each of these series are required, compared with two in the computations above. The closer the argument-order ratio is to unity, the less effective the Debye expansions become. In the neighborhood of unity the expansions 9.3.23, 9.3.24, 9.3.27, and 9.3.28 will furnish results of moderate accuracy; for high-accuracy work the uniform expansions should again be used. Example 5. To evaluate the 5th positive zero of Jlo(x) and the corresponding value of Jio(x), each to 5 decimals. We use the asymptotic expansions 9.5.22 and 9.5.23 setting v=lO, s=5. From Table 10.11

0. 8 1. 0

3.140 3.142(x)

6

+21 +9

62

-12 -7

+2

Interpolating for l/X=.667, we obtain (x-l)a:“=3.134 and thence the required root @b=6.268. Example 7. To evaluate ber, 1.55, bei, 1.55, n=o, 1, 2, . * ., each to 5 decimals. We use the recurrence relation

taking arbitrary values zero for Jg(xe3*t/4) and l+Oi for J8(xe3ri/4) (see Example 1).

.

BESSEL

388

FUNCTIONS

OF INTEGER

= n

Real ial valuer

t-y -7

C

ber,,z

Imag. is1 valuer

$50: - 4447 + 14989 +11172 - 197012 +2s1539

- 475 - 203 + 17446 - 88578

$106734

+ 207449

1/(294989-22011i)=(.337119+.025155i)x10-6, be&,x

obtained

--

x +8i

-: -. +. -. +.

. 00000 :?A!; 00003 00181 01494 04614

$_: 8;;;‘: +. 91004

ORDER

--

. xX138: -. 00003 2: gyg -. 00180 +. 06258 -. 29580 +. 36781 +. 59461 +. 72619

f. 30763

-

The values of ber,,x and bei,,x are computed by multiplication of the trial values by the normalieing factor

from the relation

jo(marf/4) +2Ja(dy

+2J4(~3rf’4) + . . . = 1.

Adequate checks are furnished by interpolating in Table 9.12 for ber 1.55 and bei 1.55, and the use of a simple sum check on the normalization. Should ker’s and kei,x be required they can be computed by forward recurrence using formulas 9.9.14, taking the required starting values for n=O and 1 from Table 9.12 (see Example 2). If an independent check on the recurrence is required the asymptotic expansion 9.10.38 can be used.

References Texts [9.1] E. E. Allen, Analytical approximations, Math. Tables Aids Comp. 8, 246-241 (1954). [9.2] E. E. Allen, Polynomial approximations to some modified Bessel functions, Math. Tables Aids Comp. 10, 162-164 (1956). [9.3] H. Bateman and R. C. Archibald, A guide to tables of Bessel functions, Math. Tables Aids Comp. 1, 205-308 (1944). [9.4] W. G. Bickley, Bessel functions and formulae (Cambridge Univ. Press, Cambridge, England, 1953). This is a straight reprint of part of the preliminaries to [9.21]. [9.5] H. S. Carslaw and J. C. Jaeger, Conduction of heah in solids (Oxford Univ. Press, London, England, 1947). [9.6] E. T. Copson, An introduction to the theory of functions of a complex variable (Oxford Univ. Press, London, England, 1935). [9.7] A. Erdelyi et al., Higher transcendental functions, ~012, ch. 7 (McGraw-Hill Book Co., Inc., New York, N.Y., 1953). [9.8] E. T. Goodwin, Recurrence relations for crossproducts of Bessel functions, Quart. J. Mech. Appl. Math. 2, 72-74 (1949). [9.9] A. Gray, G. B. Mathews and T. M. MacRobert, A treatise on the theory of Bessel functions, 2d ed. (Macmillan and Co., Ltd., London, England; 1931). [9.10] W. Magnus and F. Oberhettinger, Formeln und S&e fiir die speziellen Funktionen der mathematischen Physik, 2d ed. (Springer-Verlag; Berlin, Germany, 1948). [9.11] N. W. McLachlan, Bessel functions for engineers, 2d ed. (Clarendon Press, Oxford, England, 1955). [9.12] F. W. J. Olver, Some new asymptotic expansions for Bessel functions of large orders. Proc. Cambridge Philos. Sot. 48, 414-427 (1952).

[9.13] F. W. J. Olver, The asymptotic expansion of Bessel functions of large order. Philos. Trans. Roy. Sot. London A247, 328-368 (1954). [9.14] G. Petiau, La theorie des fonctions de Bessel (Centre National de la Recherche Scientifique, Paris, France, 1955). [9.15] G. N. Watson, A treatise on the theory of Bessel functions, 2d ed. (Cambridge Univ. Press, Cambridge, England, 1958). [9.16] R. Weyrich, Die Zylinderfunktionen und ihre Anwendungen (B. G. Teubner, Leipzig, Germany, 1937). [9.17] C. S. Whitehead, On a generalisation of the functions ber x, bei z, ker x, kei x. Quart. J. Pure Appl. Math. 42, 316-342 (1911). [9.18] E. T. Whittaker and G. N. Watson, A course of modern analysis, 4th ed. (Cambridge Univ. Press, Cambridge, England, 1952). Tables [9.19] J. F. Bridge and S. W. Angrist, An extended table of roots of 5;(z) Yi(&r) -J:(&r) Y;(z) =O, Math. Comp. 16, 198-204 (1962). [9.20] British Association for the Advancement of Science, Bessel functions, Part I. Functions of orders zero and unity, Mathematical Tables, vol. VI (Cambridge Univ. Press, Cambridge, England, 1950). [9.21] British Association for the Advancement of Science, Bessel functions, Part II. Functions of positive integer order, Mathematical Tables, vol. X (Cambridge Univ. Press, Cambridge, England, 1952). [9.22] British Association for the Advancement of Science, Annual Report (J. R. Airey), 254 (1927). [9.23] E. Cambi, Eleven- and fifteen-place tables of Bessel functions of the first kind, to all significant orders (Dover Publications, Inc., New York, N.Y., 1948).

BESSEL

FUNCTIONS

[9.24] E. A. Chistova, Tablitsy funktsii Besselya ot deistvitel’nogo argumenta i integralov ot nikh (Izdat. Akad. Nauk SSSR, Moscow, U.S.S.R., 1958). (Table of Bessel functions with real argument and their integrals). [9.25] H. B. Dwight, Tables of integrals and other mathematical data (The Macmillan Co., New York, N.Y., 1957). This includes formulas for, and tables of Kelvin functions. [9.26] H. B. Dwight, Table of roots for natural frequencies in coaxial type cavities, J. Math. Phys. 27, 8449 (1948). This gives zeros of the functions 9.5.27 and 96.39 for n=0,1,2,3. [9.27] V. N. Faddeeva and M. K. Gavurin, Tablitsy funktsii Besselia J,(z) tselykh nomerov ot 0 do 120 (Izdat. Akad. Nauk SSSR, Moscow, U.S.S.R., 1950). (Table of J.(z) for orders 0 to 120). [9.28] L. Fox, A short table for Bessel functions of integer orders and large arguments. Royal Society Shorter Mathematical Tables No. 3 (Cambridge Univ. Press, Cambridge, England, 1954). [9.29] E. T. Goodwin and J. Staton, Table of J&o,J), Quart. J. Mech. Appl. Math. 1, 220-224 (1948). [9.30] Harvard Computation Laboratory, Tables of the Bessel functions of the first kind of orders 0 through 135, ~01s. 3-14 (Harvard Univ. Press, Cambridge, Mass., 1947-1951). [9.31] K. Hayashi, Tafeln der Besselschen, Theta, Kugelund anderer Funktionen (Springer, Berlin, Germa.ny, 1930). [9.32] E. Jahnke, F. Emde, and F. Loach, Tables of higher functions, ch. IX, 6th ed. (McGraw-Hill Book Co., Inc., New York, N.Y., 1960). [9.33] L. N. Karmazina and E. A. Chistova, Tablitsy funktsii Besselya ot mnimogo argumenta i integralov ot nikh (Izdat. Akad. Nauk SSSR, Moscow, U.S.S.R., 1958). (Tables of Bessel

OF INTEGER

[9.34] [9.35]

[9.36] [9.37]

[9.38]

[9.39]

[9.40]

[9.41]

19.42)

ORDER

389

functions with imaginary argument and their integrals). Mathematical Tables Project, Table of f.(z)=nl(%z)-nJ.(z). J. Math. Phys. 23, 45-60 (1944). National Bureau of Standards, Table of the Bessel functions Jo(z) and J1(z) for complex arguments, 2d ed. (Columbia Univ. Press, New York, N.Y., 1947). National Bureau of Standards, Tables of the Bessel functions YO(z) and Yi(z) for complex arguments (Columbia Univ. Press, New York, N.Y., 1950). National Physical Laboratory Mathematical Tables, vol. 5, Chebyshev series for mathematical functions, by C. W. Clenshaw (Her Majesty’s Stationery Office, London, England, 1962). National Physical Laboratory Mathematical Tables, vol. 6, Tables for Bessel functions of moderate or large orders, by F. W. J. Olver (Her Majesty’s Stationery Office, London, England, 1962). L. N. Nosova, Tables of Thomson (Kelvin) functions and their first derivatives, translated from the Russian by P. Basu (Pergamon Press, New York, N.Y., 1961). Royal Society Mathematical Tables, vol. 7, Bessel functions, Part III. Zeros and associated values, edited by F. W. J. Olver (Cambridge Univ. Press, Cambridge, England, 1960). The introduction includes many formulas connected with zeros. Royal Society Mathematical Tables, vol. 10, Bessel functions, Part IV. Kelvin functions, by A. Young and A. Kirk (Cambridge Univ. Press, Cambridge, England, 1963). The introduction includes many formulas for Kelvin functions. W. Sibagaki, 0.01 % tables of modified Bessel functions, with the account of the methods used in the calculation (Baifukan, Tokyo, Japan, 1955).

390

BESSEL Table

9.1

FUNCTIONS

BESSEL

OF INTEGER

FUNCTIONS-ORDERS

ORDER 0,

1 4ND

2

66040 39576 38296 59563

+I, (I) 0.00000 00000 0.04993 75260 0.09950 08326 0.14831 88163 0.19602 65780

tJ&) 0.00000 00000 0.00124 89587 0.00498 33542 0.01116 58619 0.01973 46631

98072 48634 08886 73527 37981

40813 97211 07405 50480 22545

0.24226 0.28670 0.32899 0.36884 0.40594

84571 09881 57415 20461 95461

0.03060 0.04366 0.05878 0.07581 0.09458

40235 50967 69444 77625 63043

:*'3 1:4

0.76519 0.71962 0.67113 0.62008 0.56685

76865 20185 27442 59895 51203

57967 27511 64363 61509 74289

0.44005 0.47090 0.49828 0.52202 0.54194

05857 23949 90576 32474 77139

0.11490 0.13656 0.15934 0.18302 0.20735

34849 41540 90183 66988 58995

E 1:7 1.8 1. 9

0.51182 0.45540 0.39798 0.33998 0.28181

76717 21676 48594 64110 85593

35918 39381 46109 42558 74385

0.55793 0.56989 0.57776 0.58151 0.58115

65079 59353 52315 69517 70727

0.23208 0.25696 0.28173 0.30614 0.32992

76721 77514 89424 35353 57277

0.22389 0.16660 0.11036 0.05553 +0.00250

07791 69803 22669 97844 76832

41236 31990 22174 45602 97244

0.57672 0.56829 0.55596 0.53987 0.52018

48078 21358 30498 25326 52682

0.35283 0.37462 0.39505 0.41391 0.43098

40286 36252 86875 45917 00402

-0.04838 -0.09680 -0.14244 -0.18503 -0.22431

37764 49543 93700 60333 15457

68198 97038 46012 64387 91968

0.49709 0.47081 0.44160 0.40970 0.37542

41025 82665 13791 92469 74818

0.44605 0.45897 0.46956 0.47768 0.48322

90584 28517 15027 54954 70505

-0.26005 -0.29206 -0.32018 -0.34429 -0.36429

19549 43476 81696 62603 55967

01933 50698 57123 98885 62000

0.33905 0.30092 0.26134 0.22066 0.17922

89585 11331 32488 34530 58517

0.48609 0.48620 0.48352 0.47803 0.46972

12606 70142 77001 16865 25683

-0.38012 -0.39176 -0.39923 -0.40255 -0.40182

77399 89837 02033 64101 60148

87263 00798 71191 78564 87640

0.13737 0.09546 0.05383 +0.01282 -0.02724

75274 55472 39877 10029 40396

0.45862 0.44480 0142832 0.40930 0.38785

91842 53988 96562 43065 47125

-0.39714 -0.38866 -0.37655 -0.36101 -0.34225

98098 96798 70543 11172 67900

63847 35854 67568 36535 03886

-0.06604 -0.10327 -0.13864 -0.17189 -0.20277

33280 32577 69421 65602 55219

0.36412 0.33829 0.31053 0.28105 0.25008

81459 24809 47010 92288 60982

i-7" 4: 8 4. 9

-0.32054 -0.29613 -0.26933 -0.24042 -0.20973

25089 78165 07894 53272 83275

85121 74141 19753 91183 85326

-0.23106 -0.25655 -0.27908 -0.29849 -0.31469

04319 28361 07358 98581 46710

0.21784 0.18459 0.15057 0.11605 0.08129

89837 31052 30295 03864 15231

5. 0

-0.17759

-0.32757 91376 i-t)5

0.04656

0.0 i:: 00143 E is’8 0: 9 ::1"

2.0 ::: $2 ::z 1'78 2:9 3.0 3.1 ?-'3 3: 4 3.5 :*7" 3: 8 3.9 ::1" 2; 4:4 4. 5

1.00000

Jo (.c) 00000 00000

0.99750 0.99002 0.97762 0.96039

15620 49722 62465 82266

0.93846 0.91200 0.88120 0.84628 0.80752

67713 14338 rc-4x1

L‘ li

_I

c 1

51163 (-;I3 [ 1

Compiled from British Association for the Advancement of Science,Besselfunctions, Part II. Functions of positive integer order, Mathematical Tables, vol. X (CambridgeUniv. Press,Cambridge,England, 1952) and Harvard Computation Laboratory, Tables of the Besselfunctions of the first kind of orders 0 through 135, ~01s.3-14 (Harvard Univ. Press,Cambridge, Mass., 1947-1951)(with permission).

BESSEL FUNCTIONS BESSEL

x ::1” is: 0: 4 8:: 00-i 0:9 1. 0

OF INTEGER

FUNCTIONS-ORDERS

Ye(x)

ORDER

0, 1 AND

391

2

Table

9.1

Yz(x)

Yl(X)

00

-1.5342; -1.08110 -0.80727 -0.60602

86514 53224 35778 45684

-6.458; -3.32382 -2.29310 -1.78087

10947 49881 51384 20443

-127.64478 - 32.15714 - 14.48009 - 8.29833

324 456 401 565

-0.44451 87335 -0.30850 98701 -0.19066 49293 -0.08680 22797 +0.00562 83066

-1.47147 -1.26039 -1.10324 -0.97814 -0.87312

23927 13472 98719 41767 65825

-

5.44137 3.89279 2.96147 2.35855 1.94590

084 462 756 816 960

0.08825 0.16216 0.22808 0.28653 0.33789

69642 32029 35032 53572 51297

-0.78121 -0.69811 -0.62113 -0.54851 -0.47914

28213 95601 63797 97300 69742

-

1.65068 1.43147 1.26331 1.13041 1.02239

261 149 080 186 081

0.38244 0.42042 0.45202 0.47743 0.49681

89238 68964 70002 17149 99713

-0.41230 -0.34757 -0.28472 -0.22366 -0.16440

86270 80083 62451 48682 57723

-

0.93219 0.85489 0.78699 0.72594 0.66987

376 941 905 824 868

0.51037 0.51829 0.52078 0.51807 0.51041

56726 37375 42854 53962 47487

-0.10703 -0.05167 +0.00148 0.05227 0.10048

24315 86121 77893 73158 89383

-

0.61740 0.56751 0.51943 0.47261 0.42667

810 146 175 686 397

0.49807 0.48133 0.46050 0.43591 0.40791

03596 05906 35491 59856 17692

0.14591 0.18836 0.22763 0.26354 0.29594

81380 35444 24459 53936 00546

-

0.38133 0.33643 0.29188 0.24766 0.20381

585 556 692 928 518

0.37685 0.34310 0.30705 0.26909 0.22961

00100 28894 32501 19951 53372

0.32467 0.34962 0.37071 0.38785 0.40101

44248 94823 13384 29310 52921

+

0.16040 0.11753 0.07535 0.03402 0.00627

039 548 866 961 601

0.18902 0.14771 0.10607 0.06450 +0.02337

19439 00126 43153 32467 59082

0.41018 0.41539 0.41667 0.41411 0.40782

84179 17621 43727 46893 00193

0.04537 Or08306 0.11915 Oil5345 0.18576

144 319 508 185 256

-0.01694 -0.05609 -0.09375 -0.12959 -0.16333

07393 46266 12013 59029 64628

0.39792 0.38459 0.36801 0.34839 0.32597

57106 40348 28079 37583 06708

0.21590 0.24370 0.26899 0.29163 0.31150

359 147 540 951 495

t*i 4:9

-0.19470 -0.22345 -0.24938 -0.27230 -0.29205

50086 99526 76472 37945 45942

0.30099 0.27374 0.24450 0.21356 0.18124

73231 52415 12968 51673 66920

0.32848 0.34247 0.35343 0.36128 0.36603

160 962 075 928 284

5. 0

-0.30851

76252

0.14786 31434 2n

:*: 1:s 1.4 1.5 :*; 1:8 1.9 21" z 214 2. 5 z 2:8 2.9 3. 0 33': 3: 3 3.4 ;:2 z.78 3:9 4. 0 2:: 2:: t:2

Yn+l(x)=~Yn(x)-Y,-l(X)

0.36766 288

392

BESSEL Table

FUNCTIONS

BESSEL

9.1

OF INTEGER

ORDER 0, 1 AND

FUNCTIONS-ORDERS

Jo(x)

2

JlC4

52(z)

-0.17759 -0.14433 -0.11029 -0.07580 -0:04121

67713 14338 47470 60501 04397 90987 31115 85584 01012 44991

-0.32757 -0.33709 -0.34322 -0.34596 -0.34534

91376 72020 30059 08338 47908

0.04656 +0.01213 -0.02171 -0.05474 -0.08669

51163 97659 84086 81465 53768

-0.00684 +0.02697 0.05992 0.09170 0.12203

38694 17819 08846 85114 00097 24037 25675 74816 33545 92823

-0.34143 -0.33433 -0.32414 -0.31102 -0.29514

a2154 28363 76802 77443 24447

-0.11731 -0.14637 -0.17365 -0.19895 -0.22208

54816 54691 60379 35139 16409

0.15064 0.17729 0.20174 0.22381 0.24331

52572 50997 14222 42744 72229 48904 20061 32191 06048 23407

-0.27668 -0.25586 -0.23291 -0.20808 -0.18163

38581 47726 65671 69402 75090

-0.24287 -0.26118 -0.27688 -0.28987 -0.30007

32100 15116 15994 13522 23264

0.26009 0.27404 0.28506 0.29309 0.29810

46055 81606 33606 24146 47377 10576 56031 04273 20354 04820

-0.15384 -0.12498 -0.09534 -0.06521 -0.03490

13014 01652 21180 86634 20961

-0.30743 -0.31191 -0.31352 -0.31227 -0.30821

03906 61379 50715 75629 85850

0.30007 0.29905 0.29507 0.28821 0.27859

92705 19556 13805 01550 06914 00958 69476 35014 62326 57478

-0.00468 +0.02515 0.05432 0.08257 0.10962

28235 32743 74202 04305 50949

-0.30141 -0.29196 -0.27997 -0.26559 -0.24896

72201 59511 97413 49119 78286

0.26633 0.25160 0.23455 0.21540 0.19436

96578 80378 18338 49976 91395 86464 78077 46263 18448 41278

0.13524 0.15921 0.18131 0.20135 0.21917

84276 37684 27153 68728 93999

-0.23027 -0.20970 -0.18746 -0.16377 -0.13887

34105 34737 49278 78404 33892

0.17165 08071 37554 0.14751 74540 44378 0.12221 53017 84138 0.09600 61008 95010 0.06915 72616 56985

0.23463 0.24760 0.25799 0.26573 0.27078

63469 77670 85976 93020 62683

-0.11299 -0.08637 -0.05928 -0.03197 -0.00468

17204 97338 88146 25341 43406

0.04193 +0.01462 -0.01252 -0.03923 -0.06525

92518 42935 29912 78741 27324 49665 38031 76542 32468 51244

0.27312 0.27275 0.26971 0.26407 0.25590

19637 48445 90241 37032 23714

+0.02232 oio4aao 0.07452 0.09925 0.12275

47396 83679 71058 05539 93977

-0.09033 -0.11423 -0.13674 -0.15765 -0.17677

36111 82876 92326 83199 83707 64864 51899 43403 15727 51508

0.24531 0.23243 0.21740 0.20041 0.18163

17866 07450 86550 39278 22040

0.14484 0.16532 0.18401 0.20075 0.21541

73415 2912y 11218 49594 67225

-0.19392 -0.20897 -0.22179 -0.23227 -0.24034

87476 87422 87183 68872 54820 31723 60275 79367 11055 34760

0.16126 0.13952 0.11663 0.09284 0.06836

44308 48117 86479 00911 98323

0.22787 0.23804 0.24584 0.25122 0.25415

91542 63875 46878 29849 31929

-0.24593 57644 51348 ';14'4

0.04347 27462

ll 1

[

(6’4

5,+1(z)

=c

1

J,,(z) - Jn-I(Z)

0.25463 03137

[

( y4

1

BESSEL FUNCTIONS BESSEL FUNCTIONS-ORDERS

520 55:: Z:i ::2 :2 5: 9 2:: 2: 6: 4 i-2 6: 7 29" 7. 0 ::: 77:: 7.5 3 5 8.0 i-21 8:3 8. 4 256 Ei 8: 9 ;*10 9:2 E x-2 9:7 82 10. 0

393

OF INTEGER ORDER 0,lAND

2

Table 9.1

-0.30851 -0.32160 -0.33125 -0.33743 -0.34016

76252 24491 09348 73011 78783

0.14786 0.11373 0.07919 0.04454 +0:01012

31434 64420 03430 76191 72667

Ydx) 0.36766 288 0.36620 498 0.36170 876 0.35424 772 0.34391 872

-0.33948 -0.33544 -0.32815 -0.31774 -0.30436

05929 41812 71408 64300 59300

-0.02375 -0.05680 -0.08872 -0.11923 -0.14807

82390 56144 33405 41135 71525

0.33084 0.31515 0.29702 0.27663 0.25417

123 646 614 122 029

-0.28819 -0.26943 -0.24830 -0.22506 -0.19994

46840 49304 99505 17496 85953

-0.17501 -0.19981 -0.22228 -0.24224 -0.25955

03443 22045 36406 95005 98934

0.22985 0.20392 0.17660 0.14815 0.11883

790 273 555 715 613

-0.17324 -0.14522 -0.11619 -0.08643 -0.05625

24349 62172 11427 38683 36922

-0.27409 -0.28574 -0.29445 -0.30018 -0.30291

12740 72791 93130 68758 76343

0.08890 0.05863 to.02829 -0.00185 -0.03154

666 613 284 639 852

-0.02594 +0.00418 0.03385 0.06277 0.09068

97440 17932 04048 38864 08802

-0.30266 -0.29947 -0.29342 -0.28459 -0.27311

72370 88746 25939 43719 49598

-0.06052 -0.08854 -0.11535 -0.14074 -0.16449

661 204 668 495 573

0.11731 0.14242 0.16580 0.18722 0.20652

32861 85247 16324 71733 09481

-0.25912 -0.24280 -0.22431 -0.20388 -0.18172

85105 10021 84743 50954 10773

-0.18641 -0.20632 -0.22406 -0.23950 -0.25252

422 353 617 540 628

0.22352 0.23809 0.25011 0.25951 0.26622

14894 13287 80276 49638 18674

-0.15806 -0.13314 -0.10724 -0.08059 -0.05348

04617 87960 07223 75035 45084

-0.26303 -0.27096 -0.27627 -0.27893 -0.27895

660 757 430 605 627

0.27020 0.27145 0.26999 0.26587 0.25915

51054 77123 91703 49418 57617

-0.02616 +0.00108 0.02801 0.05435 0.07986

86794 39918 09592 55633 93974

-0.27636 -0.27120 -0.26355 -0.25352 -0.24120

244 562 987 140 758

0.24"3 0.23833 0.22449 0.20857 0.19074

66983 59921 36870 00676 39189

0.10431 0.12746 0.14911 0.16906 0.18713

45752 58820 27879 13071 56847

-0.22675 -0.21032 -0.19207 -0.17221 -0.15092

568 151 786 280 782

0.17121 0.15018 0.12787 0.10452 0.08037

06262 01353 47920 70840 73052

0.20317 0.21705 0.22866 0.23789 0.24469

98994 89660 00298 32421 24113

-0.12843 -0.10495 -0.08072 -0.05597 -0.03094

591 952 839 744 449

yocd

Ylk)

0.05567 11673 C-i)4

[

1

0.24901 54242 (-;I4

[ 2n 1

Y,+1(2)=zY,(z)-Y,-1(2)

-0.00586 808 c-t)4

[ 1

394

BESSEL Table

9.1

FUNCTIONS BESSEL

Job)

OF

INTEGER

ORDER

FUNCTIONS-ORDERS

0, 1 AND

2

J&4

Jl(4

lo” 0 10:1 10.2 10.3 10.4

-0.24593 -0.24902 -0.24961 -0.24771 -ii24337

57644 96505 70698 68134 17507

51348 80910 54127 82244 14207

0.04347 +0.01839 -0.00661 -0.03131 -0.05547

27462 55155 57433 78295 27618

0.25463 0.25267 0.24831 0.24163 0.23270

03137 23269 98653 56815 39119

10.5 10. 6 10.7 10. 8 10.9

-0.23664 -0.22763 -0.21644 -0.20320 -Oil8806

81944 50476 27399 19671 22459

62347 20693 23818 12039 63342

-0.07885 -0.10122 -0.12239 -0.14216 -0.16034

00142 86626 94239 65683 96867

0.22162 0.20853 0.19356 0.17687 0.15864

91441 53000 43429 48248 02851

11. 0 11.1 11.2 11. 3 11.4

-0.17119 -0.15276 -0.13299 -0.11206 -0.09021

03004 82954 19368 84561 45002

07196 35677 59575 09807 47520

-0.17678 -0.19132 -0.20385 -0.21425 -0.22245

52990 82878 31459 50262 05864

0.13904 0.11829 0.09658 0.07414 0.05118

75188 47301 95894 72125 80816

11. 5 11.6 11.7 11.8 11.9

-0.06765 -0.04461 -0.02133 +0.00196 0.02504

39481 56740 12813 71733 94416

11665 94438 88500 06740 99590

-0.22837 -0.23200 -0.23330 -0.23228 -0i22898

86207 04746 02408 47343 32497

0.02793 +0.00461 -0.01854 -0.04133 -0.06353

59271 55923 91017 74673 40215

12.0 12.1 12.2 12.3 12.4

0.04768 0.06966 0.09077 0.11079 0.12956

93107 67736 01231 79503 10265

96834 06807 70505 07585 17502

-0.22344 -0.21574 -0.20598 -0.19425 -0.18071

71045 89734 20217 88480 02469

-0.08493 -0.10532 -0.12453 -0.14238 -0.15870

04949 77609 76677 47549 78405

12.5 12.6 12.7 12.8 12.9

0.14688 0.16260 ii17658 0.18870 0.19884

40547 72717 78885 13547 24371

00421 45511 61499 80683 36331

-0.16548 -0.14874 -0.13066 -0.11143 -0.09124

38046 23434 22290 15593 82522

-0.17336 -0.18621 -0.19716 -0.20611 -0.21298

14634 71675 46175 25359 94530

13.0 13.1 13.2 13.3 13.4

0.20692 0.21288 0.21668 0.21829 0.21772

61023 81975 59222 80903 51787

77068 22060 58564 19277 31184

-0.07031 -0.04885 -0.02706 -0.00517 +0.01659

80521 24733 67028 74806 90199

-0.21774 -0.22034 -0.22078 -0.21907 -0i21524

42642 65904 69378 66588 77131

13.5 13.6 13.7 13.8 13.9

0.21498 0.21013 0.20322 0.19433 0.18357

91658 31613 08326 56352 98554

80401 69248 33007 15629 57870

0.03804 0.05896 0.07914 0.09839 0.11652

92921 45572 27651 05167 48904

-0.20935 -0.20146 -0.19166 -0.18007 -0.16681

22337 19030 71443 61400 36842

14. 0 14.1 14.2 14.3 14.4

0.17107 0.15695 0.14136 0.12448 0.10648

34761 28770 93846 76852 41184

10459 32601 57129 83919 90342

0.13337 0.14878 0.16261 0.17472 0.18503

51547 43513 07342 90520 16616

-0.15201 98826 -0.13584 87137 -0.11846 64643 -0.10005 00556 -0iO8078 52766

14.5 14.6 14.7 14.8 14.9

0.08754 0.06786 0.04764 0.02708 +0.00639

48680 40683 18459 23145 15448

10376 23379 01522 85872 90853

0.19342 0.19985 Or20425 0.20659 0.20687

94636 26514 12683 55672 61718

-0.06086 -0.04048 -0.01985 +0.00083 0.02137

15.0

-0.01422

44728 26781 '$13

[ 1

0.20510 40386 C-f)3

0.04157 16780 c-:)3

[ 1

[ 1

J n-l.1(x) =p Jn(x) -Jn

49420 69928 25577 60053 70688

l(X)

BESSEL FUNCTIONS OF INTEGER ORDER BESSEL x

FUNCTIONS-ORDERS

I’o(x)

0, 1 AND

2

Yl(X)

395 Table

9.1

Y2b)

10. 0 10.1 10.2 10. 3 10.4

0.05567 0.03065 +0.00558 -0.01929 -0.04374

11673 73806 52273 78497 86190

0.24901 0.25084 0.25018 0.24706 0.24155

54242 44363 58292 99395 05610

-0.00586 +0.01901 0.04347 0.06727 0.09020

808 478 082 260 065

10. 5 10. 6 10.7 10.8 10. 9

-0.06753 -0.09041 -0.11218 -0.13263 -Oil5158

03725 51548 58897 83844 31932

0.23370 0.22362 0.21144 0.19728 0.18131

42284 92892 47763 90905 85097

0.11204 0.13260 0.15170 0.16917 0.18485

546 936 828 340 264

11. 0 11.1 11.2 11.3 11. 4

-0.16884 -0.18427 -Oil9773 -0.20910 -0.21829

73239 57716 28675 34295 37073

0.16370 55374 0.14463 71102 0.12431 26795 0.10294 21889 OiO8074 39654

0.19861 0.21033 0.21993 0.22732 0.23245

197 651 156 329 932

11.5 11. 6 11.7 11.8 11.9

-0.22523 -0.22986 -0i23218 -0.23216 -0.22983

21117 97260 05930 17790 32139

0.05794 0.03476 +0.01144 -0.01178 -0.03471

25471 64663 60113 90120 14983

0.23530 0.23586 0.23413 0.23016 0.22399

908 394 718 364 935

12.0 12.1 12.2 12. 3 12.4

-0.22523 -0.21843 -0.20952 -0.19859 -0.18577

73126 83806 18128 30946 66153

-0.05709 -0.07873 -0.09941 -0.11894 -0.13714

92183 69315 84171 84033 43766

0.21572 0.20542 0.19322 0.17925 0.16365

078 401 371 189 655

12.5 12. 6 12. 7 12.8 12.9

-0.17121 -0.15506 -0.13749 -0.11870 -0.09887

43068 41238 83780 19463 03702

-0.15383 -0.16887 -0.18212 -0.19347 -0.20281

82565 79186 85528 38454 69743

0.14660 0.12825 0.10881 0.08847 0.06742

019 810 672 166 588

13.0 13.1 13.2 13.3 13.4

-0.07820 -0.05692 -0.03523 -0.01336 +0.00848

78645 52'68 78771 34191 02072

-0.21008 -0.21521 -0.21817 -0.21895 -0.21755

14084 15060 29066 27145 94728

0.04588 0.02406 +0.00218 -0.01956 -0.04095

765 854 138 180 177

13.5 13.6 13.7 13. 8 13.9

0.03007 0.05121 0.07168 0.09129 0.10985

70090 50115 83040 90143 91895

-0.21402 -0.20839 -0.20074 -0.19115 -0.17975

29303 36044 21453 85095 09511

-0.06178 -0.08186 -0.10099 -0.11900 -0.13572

411 113 373 315 264

14.0 14.1 14.2 14.3 14.4

0.12719 0.14313 0.15754 0.17027 0.18123

25686 62286 20895 82640 02411

-0.16664 -0.15198 -0.13591 -0.11861 -0.10026

48419 13335 58742 65967 25924

-0.15099 -Oil6469 -0.17668 -0.18686 -0.19515

897 386 517 800 560

14.5 14.6 14.7 14.8 14. 9

0.19030 0.19741 0.20251 0.20556 0.20654

18912 62858 63238 51604 64347

-0.08104 -0.06115 -0.04078 -0.02016 +0.00052

20909 05609 87536 07059 82751

-0.20148 -0.20579 -0.20806 -0.20828 -0.20647

011 307 581 958 553

15.0

0.20546 42960 C-i!3

[ 1

0.02107 36280 c-:13

[ 1 2n

Y~+l(x)=~Y,(x)-Y,~l(x)

-0.20265 448 c-;13

[ 1

396

BESSEL FUNCTIONS Table

9.1

BESSEL

X

Job4

OF INTEGER ORDER

FUNCTIONS-ORDERS

0,

1 AND

Jl(4

2

J2(4

15. 0 15.1 15.2 15.3 15.4

-0.01422 -0.03456 -0.05442 -0.07360 -0.09193

44728 18514 07968 75449 62278

26781 55565 44039 51123 62321

0.20510 0.20131 0.19554 0.18787 0.17840

40386 02204 54359 94498 02717

0.04157 0.06122 0.08015 0.09816 0.11510

16780 54568 04595 69502 50943

15.5 15.6 15.7 15.8 15.9

-0.10923 -0.12532 -0.14007 -0.15332 -0.16497

06509 59640 02118 57477 04994

00050 22481 29049 60686 85673

0.16721 0.15443 0.14021 0.12469 0.10802

31804 95871 57469 13334 78901

0.13080 0.14512 0.15793 0.16910 0.17855

65451 59111 20904 94608 89133

16. 0 16.1 16.2 16. 3 16.4

-0.17489 -0.18302 -0.18927 -0.19360 -0.19597

90739 36924 49469 23723 48287

83629 65310 77945 28377 91007

0.09039 0.07197 0.05296 0.03353 +0.01389

71757 94186 14991 50765 46807

0.18619 0.19196 0.19581 0.19771 0.19766

87209 52352 34037 71056 93020

16. 5 16.6 16.7 16.8 16.9

-0.19638 -0.19482 -0.19134 -0.18597 -0.17878

06929 78558 35295 38653 33878

36861 05566 25189 47601 91219

-0.00576 -0.02524 -0.04436 -0.06292 -0.08074

42137 71116 24008 32177 92543

0.19568 0.19178 0.18603 0.17848 0.16922

20004 60351 06671 30061 72631

17. 0 17.1 17.2 17.3 17. 4

-0.16985 -0.15928 -0.14719 -0.13370 -0.11895

42521 53315 11467 06470 58563

51184 32265 66030 75764 36348

-0.09766 -Oil1351 -0.12814 -Oil4142 -0.15321

84928 88483 97057 33355 61760

0.15836 0.14600 0.13229 0.11735 0.10134

38412 82733 00182 11285 48016

17.5

-0.10311 03982 28686

-0.16341 99694

0.08443 38303

[ 1

C-92 2n J,+.~(z)=~ Jn(z)-Jn-I(Z)

Table

9.1

BESSEL

FUNCTIONS-MODULUS

J&)=&(z)

AND

PHASE

OF

ORDERS

0,l

Y,(x)=&(x)

cos h(z)

AND

2

sin e,(x)

x-1 0.10 0. 09 0. 08 0.07 0.06

x”Mo(x) 0.79739 375 0.79748 584 0.79756 868 0.79764 214 0.79770 609

eo(4 --z -0.79783 499 -0.79660 186 -0.79536 548 -0.79412 617 -0.79288 426

zfJf1 (x) 0.79936 575 0.79908 654 0.79883 586 0.79861 398 0.79842 116

-2.31885 -2.32256 -2.32627 -2.33000 -2.33372

508 201 732 016 965

0.80542 0.80398 0.80269 0.80156 0.80058

555 367 711 472 549

-3.73985 -3.75850 -3.77717 -3.79586 -3.81456

605 527 539 377 786

0.05 0. 04 0.03 0.02 0.01

0.79776 0.79780 0.79783 0.79786 0.79787

-0.79164 -0.79039 -0.78914 -0.78789 -0.78664

0.79825 0.79812 0.79801 0.79794 0.79789

-2.33746 -2.34120 -2.34494 -2.34869 -2.35244

488 495 891 580 465

0.79975 0.79908 0.79855 0.79818 0.79795

851 299 829 387 937

-3.83328 -3.85201 -3.87075 -3.88949 -3.90824

521 346 034 363 117

0.00

0.79788 456 [‘-y]

040 498 975 463 957

009 402 641 764 810

-0.78539 816 [(-I)43

761 353 908 438 952

0.79788 456 [(-I)“] =nearest

01(x)--2

x+x2

-2.35619 449 [q”] integer to 2.

(4

0.79788 456 [‘I)“]

e2(4--x

-3.92699 082 [(-i)3]

BESSEL FUNCTIONS OF INTEGER ORDER BESSEL

FUNCTIONS-ORDERS

0,l

Ye(r)

.x

397

AND 2

Table

9.1

15.0 15.1 15.2 15.3 15.4

0.20546 0.20234 0.19722 0.19018 Oil8128

42960 32292 76821 15001 71741

0.02107 0.04127 0.06093 0.07985 0.09786

36280 35340 08736 51269 41973

Y2(:r) -0.20265 448 -0.19687 654 -0.18921 046 -0.17974 292 -0.16857 754

15.5 15.6 15.7 15.8 15.9

0.17064 0.15837 0.14459 0.12947 0.11315

49112 15368 92412 41833 49657

0.11478 0.13046 0.14474 0.15749 0.16860

61425 07959 12638 52835 64314

-0.15583 -0.14164 -0.12616 -0.10953 -0.09194

380 579 086 807 661

16.0 16.1 16.2 16. 3 16. 4

0.09581 0.07762 0.05876 0.03944 0.01985

09971 07587 99918 98249 48596

0.17797 0.18551 0.19117 0.19490 0.19667

51689 97173 67538 19240 01648

-0.07356 -0.05457 -0.03516 -0.01553 +0.00412

410 483 792 548 931

16.5 16. 6 16.7 16.8 16.9

+0.00018 -0.01937 -0.03862 -0.05736 -0.07543

12325 53254 14147 78596 15476

0.19647 0.19433 0.19027 0.18434 0.17663

58378 26715 35142 99015 14431

0.02363 0.04278 0.06140 0.07931 0.09633

402 890 866 428 468

17.0 17.1 17.2 17.3 17.4

-0.09263 -0.10881 -0.12382 -0.13750 -0.14973

71984 90473 24237 52134 91883

0.16720 0.15617 0.14365 0.12978 0.11470

50361 39131 65362 53467 53859

0.11230 0.12708 0.14052 0.15250 0.16292

838 500 667 930 372

17.5

-0.16041 11925

Yl

(x>

0.09857 27987

%+1(x)=$

0.17167 666

Yn(2)-FL-1(~)

Table BESSEL

FUNCTIONS-AUXILIARY

fl@> 070 -0.07380 430 -0.07202 984 0”:: -0.06672 574 -0.05794 956 ::34 -0.04579 663 0.5 0":; !I:: 1.0

-0.03039 -0.01192 +0.00942 0.03341 0.05979

904 435 612 927 263

0.08825 696 [(-p]

TABLE

f2 64

FOR SMALL X

-0.63661 -0.63857 -0.64437 -0.65382 -0.66660

977 491 529 684 964

1. 0 1'::

-0.68228 -0.70029 -0.71998 -0.74059 -0.76130

315 342 221 789 792

1'::

:::

11-i 1: 9

9.1

ARGUMENTS

fd4

flk)

0.08825 0.11849 0.15018 0.18296 0.21647

696 917 546 470 200

-0.78121 -0.79936 -0.81476 -0.82642 -0.83332

282 142 705 473 875

0.25033 0.28416 0.31758 0.35020 0.38166

233 437 436 995 415

-0.83449 -0.82895 -0.81583 -0.79427 -0.76356

074 780 036 978 508

-0.78121 282 [C-74)5]

Ye(x)=f 1(z)+z Jo(x) In

:c

Yl(x)=~f2(x)+~J1(x)

In z

398

BESSEL

FUNCTIONS

OF

INTEGER

BESSEL FUNCTIONS-ORDERS

Table 9.2

Jz(:I$ 0.0000

J4 (.1.) 0.0000

(1;

Js (1.) 0.0000

J6 (29 0.0000

Et 2:8

3-9 Ji

(2)

0.0000

0.0000

0.0000

g.;;;;

-3 9:0629 -2 1.4995 II-2 23197 ;:2"

ORDER

- 5)8.5712

0.12894 0.16233 0.19811 0.23529 0.27270

- 6)2.4923

0.30906 0.34307 0.37339 0.39876 0.41803

0.13203 0.15972 0.18920 0.21980 0.25074

0.43017 0.43439 0.43013 0.41707 0.39521

0.28113 0.31003 0.33645 0.35941 0.37796

0.13209 0.15614 0.18160 0.20799 0.23473

0.36483 0.32652 0.28113 0.22978 0.17382

0.39123 0.39847 0.39906 0.39257 0.37877

0.26114 0.28651 0.31007 0.33103 0.34862

0.13105 0.15252 0.17515 0.19856 0.22230

0.11477 +0.05428 -0.00591 -0.06406 -0.11847

0.35764 0.32941 0.29453 0.25368 0.20774

0.36209 0.37077 0.37408 0.37155 0.36288

0.24584 0.26860 0.28996 0.30928 0.32590

0.12959 0.14910 0.16960 0.19077 0.21224

-0.16756 -0.20987 -0.24420 -0.26958 -0.28535

0.15780 0.10509 +0.05097 -0.00313 -0.05572

0.34790 0.32663 0.29930 0.26629 0.22820

0.33920 0.34857 0.35349 0.35351 0.34828

0.23358 0.25432 0.27393 0.29188 0.30762

0.12797 0.14594 0.16476 0.18417 0.20385

-0.29113 -0.28692 -0.27302 -0.25005 -0.21896

-0.10536 -0.15065 -0.19033 -0.22326 -0.24854

0.18577 0.13994 0.09175 +0.04237 -0.00699

0.33758 0.32131 0.29956 0.27253 0.24060

0.32059 0.33027 0.33619 0.33790 0.33508

0.22345 0.24257 0.26075 0.27755 0.29248

0.12632 0.14303 0.16049 0.17847 0.19670

-0.26547 -0.27362 -0.27284 -0.26326 -0.24528

-0.05504 -0.10053 -0.14224 -0.17904 -0.20993

0.20432 0.16435 0.12152 0.07676 +0.03107

0.32746 0.31490 0.29737 0.27499 0.24797

0.30507 0.31484

E! 9:8

-0.18094 -0.13740 -0.08997 -0.04034 +0.00970

8'ZY 0:32318

0.21488 0.23266 0.24965 0.26546 0.27967

10.0

0.05838

-0.21960

-0.23406

-0.01446

0.21671

0.31785

0.29186

3.0 3.2 xi 3:8 t:2" t-46 418 E! 514 55:: 6.0 6.2 246 6:8 7.0 7.2 2 718

9.0 9.2

I - 43 I 1.3952 9.3860 2.0275 4.0270 2.8852

Compiled from British Association for the Advancement of Science, Bessel functions, Part II. Functions of positive integer order, Mathematical Tables, vol. X (Cambridge Univ. Press, Cambridge, England, 1952) and Mathematical Tables Project, Table of .f,(x) = n!(G)-nJ,(:x). J. Math. Phys. 23, 45-60 (1944) (with permission).

‘399

BESSEL FUNCTIONS OF INTEGER ORDER BESSEL X

Y3h)

Y4(5)

4 3 2 1

-5.8215 -3.5899 -2.4420 -1.7897 -1.3896

FUNCTIONS-ORDERS Ys(x)

Y5b)

3-9 Y7(2)

Table Ys(x)

9.2

Y!)(Z)

:1;162 -1.2097 -2.4302 -7.8751

II

-9.4432 -5.8564 -3.9059

II

-1.1278 -0.94591 -0.81161 -0.70596 -0.61736

-2.7659 -2.0603 -1.6024 -1.2927 -1.0752

-9.9360 -6.5462 -4.5296 -372716 -2.4548

-0.53854 -0.46491 -0.39363 -0.32310 -0.25259

-0.91668 -0.79635 -0.70092 -0.62156 -0.55227

-1.9059 -1.5260 -1.2556 -1.0581 -0.91009

-5.4365 -3.9723 -2.9920 -2.3177 -1.8427

-9.3044 -6.6677 -4.9090

-0.18202 -0.11183 -0.04278 +0.02406 0.08751

-0.48894 -0.42875 -0.36985 -0.31109 -0.25190

-0.79585 -0.70484 -0.62967 -0.56509 -0.50735

-1.5007 -1.2494 -1.0612 -0.91737 -0.80507

-3.7062 -2.8650 -2.2645 -1.8281 -1.5053

:5 516 5.8

0.14627 0.19905 0.24463 0.28192 0.31001

-0.19214 -0.13204 -0.07211 -0.01310 +0.04407

-0.45369 -0.40218 -0.35146 -0.30063 -0.24922

-0.71525 -0.64139 -0.57874 -0.52375 -0.47377

-1.2629 -1.0780 -0.93462 -0.82168 -0.73099

-2.8209 -2.2608 -1.8444 -1.5304 -1.2907

-7.7639 -5.8783 -4.5302 -3.5510 -2.829l5

2; 6:4 6.6 6.8

oo%E 0:333s3 0.32128 0.29909

0.09839 0.14877 0.19413 0.23344 0.26576

-0.19706 -0.14426 -0.09117 -0.03833 +0.01357

-0.42683 -0138145 -0.33658 -0.29151 -0.24581

-0.65659 -0.59403 -0.53992 -0.49169 -0.44735

-1.1052 -0.95990 -0.84450 -0.75147 -0.67521

y;;

;-2o 714

0.26808 0.22934 0.18420 0.13421 0.08106

0.29031 0.30647 00%2B~ 0:30186

0.06370 0.11119 0.15509 0.19450 0.22854

-0.19931 -0.15204 -0.10426 -0.05635 -0.00886

-0.40537 -0.36459 -0.32416 -0.28348 -0.24217

-0.61144 -0.55689 -0.50902 -0.46585 -0.42581

-0.992:20 -0.87293 -0.77643 -0.69726 -0.63128

+0.02654 -0.02753 -0.07935 -0.12723 -0.16959

0.28294 0.25613 0.22228 0.18244 0.13789

0.25640 0.27741 0.29104 0.29694 0.29495

+0.03756 0.08218 0.12420 0.16284 0.19728

-0.20006 -0.15716 -0.11361 -0.06973 -0.02593

-0.38767 -0.35049 -0.31355 -0.27635 -0.23853

-0.57528 -0.52673 -0.48363 -0.444140 -0.40777

;:i

-0.20509 -0.23262 -0.25136 -0.26079 -0.26074

0.09003 +0.04037 -0.00951 -0.05804 -0.10366

0.28512 0.26773 0.24326 0.21243 0.17612

0.22677 0.25064 0.26830 0.27932 0.28338

+0.01724 0.05920 0.09925 0.13672 0.17087

-0.19995 -0.16056 -0.12048 -0.07994 -0.03928

-0.37271 -0.33843 -0.30433 -0.26995 -0.23499

10.0

-0.25136

-0.14495

0.13540

0.28035

0.20102

+0.00108

-0.19930

2.0

2: 2:6 2.8 3.0 3.2 z-46 3:8 4.0 4.2 2: 4:8 5.0

;:i 8.0 E 8:6 8.8 %*20 9:4

1 1 1 1

-4.6914 -2.7695 -1.7271 -1.1290 -7.6918

( l)-1.1471 -8.3005 -6.1442 -4.6463 -3.5855

-1:5713 -1.33011 -1.1414

400

BESSEL

Table

FUNCTIONS

BESSEL

9.2

OF

INTEGER

FUNCTIONS-ORDERS

ORDER

3-9

JT(z)

Js (x)

X

J3 (xl

10.0 10.2 10.4 10.6 10.8

0.05838 0.10400 0.14497 0.17992 0.20768

-0.21960 -0.18715 -0.14906 -0.10669 -0.06150

-0.23406 -0.25078 -0.25964 -0.26044 -0.25323

-0.01446 -0.05871 -0.10059 -0.13901 -0.17297

0.21671 0.18170 0.14358 0.10308 0.06104

0.31785 0.30811 0.29386 0.27515 0.25210

0.29186 0.30161 0.30852 0.31224 0.31244

11.0 11.2 11. 4 11. 6 11.8

0.22735 i* x: 0:23359 0.21827

-0.01504 to.03110 0.07534 0.11621 0.15232

-0.23829 -0.21614 -0.18754 -0.15345 -0.11500

-0.20158 -0.22408 -0.23985 -0.24849 -0.24978

to.01838 -0.02395 -0.06494 -0.10361 -0.13901

0.22497 0.19414 0.16010 0.12344 0.08485

0.30886 0.30130 0.28964 0.27388 0.25407

12.0 12.2 12.4 12.6 12.8

0.19514 0.16515 0.12951 0.08963 0.04702

0.18250 0.20576 0.22138 0.22890 0.22815

-0.07347 -0.03023 +0.01331 0.05571 0.09557

-0.24372 -0.23053 -0.21064 -0.18469 -0.15349

-0.17025 -0.19653 -0.21716 -0.23160 -0.23947

0.04510 +0.00501 -0.03453 -0.07264 -0.10843

0.23038 0.20310 0.17260 0.13935 0.10393

13.0 13.2 13.4 13.6 13.8

+0.00332 -0.03984 -0.08085 -0.11822 -0.15059

0.21928 0.20268 0.17905 0.14931 0.11460

0.13162 0.16267 0.18774 0.20605 0.21702

-0.11803 -0.07944 -0.03894 +0.00220 0.04266

-0.24057 -0.23489 -0.22261 -0.20411 -0.17993

-0.14105 -0.16969 -0.19364 -0.21231 -0.22520

0.06698 to.02921 -0.00860 -0.04567 -0.08117

14. 0 14.2 14.4 14.6 14.8

-0.17681 -0.19598 -0.20747 -0.21094 -0.20637

0.07624 +0.03566 -0.00566 -0.04620 -0.08450

0.22038 0.21607 0.20433 0.18563 0.16069

0.08117 0.11650 0.14756 0.17335 0.19308

-0.15080 -0.11762 -0.08136 -0.04315 -0.00415

-0.23197 -0.23246 -0.22666 -0.21472 -0.19700

-0.11431 -0.14432 -0.17048 -0.19216 -0.20883

15. 0 15.2 15.4 15.6 15.8

-0.19402 -0.17445 -0.14850 -0.11723 -0.08188

-0.11918 -0.14901 -0.17296 -0.19021 -0.20020

0.13046 0.09603 0.05865 +0.01968 -0.01949

0.20615 0.21219 0.21105 0.20283 0.18787

+0.03446 0.07149 0.10580 0.13634 0.16217

-0.17398 -0.14634 -0.11487 -0.08047 -0.04417

-0.22005 -0.22553 -0.22514 -0.21888 -0.20690

16.0 16.2 16.4 16.6 16.8

-0.04385 -0.00461 +0.03432 0.07146 0.10542

-0.20264 -0.19752 -0.18511 -0.16596 -0.14083

-0.05747 -0.09293 -0.12462 -0.15144 -0.17248

0.16672 0.14016 0.10913 0.07473 0.03817

0.18251 0.19675 0.20447 0.20546 0.19974

-0.00702 +0.02987 0.06542 0.09855 0.12829

-0.18953 -0.16725 -0.14065 -0.11047 -0.07756

17.0 17.2 17.4 17.6 17.8

0.13493 0.15891 0.17651 0.18712 0.19041

-0.11074 -0.07685 -0.34048 -0.00300 +0.03417

-0.18704 -0.19466 -0.19512 -0.18848 -0.17505

+0.00072 -0.03632 -0.07166 -0.10410 -0.13251

0.18755 0.16932 0.14570 0.11751 0.08571

0.15374 0.17414 0.18889 0.19757 0.19993

-0.04286 -0.00733 +0.02799 0.06210 0.09400

18.0 la.2 la.4 la.6 18.8

0.18632 0.17510 0.15724 0.13351 0.10487

0.06964 0.10209 0.13033 0.15334 0.17031

-0.15537 -0.13022 -0.10058 -0.06756 -0.03240

-0.15596 -0.17364 -0.18499 -0.18966 -0.18755

0.05140 +0.01573 -0.02007

0.19593 0.18574 0.16972 0.14841 0.12253

0.12276 0.14756 0.16766

19.0 19.2 19.4 19.6 19.8

0.07249 0.03764 +0.00170 -0.03395 -0.06791

0.18065 0.18403 0.18039 0.16994 0.15313

+0.00357 0.03904 0.07269 0.10331 0.12978

-0.17877 -0.16370 -0.14292 -0.11723 -0.08759

-0.11648 -0.14135 -0.16110 -0.17508 -0.18287

0.09294 0.06063 +0.02667 -0.00783 -0.04171

0.19474

20.0

-0.09890

J4 (x)

J5 (d

Js(x9

1: . g:;:

Js (z)

it. :9812;97

: :E; 0:16869 0.14916

0.13067 0.15117 -0.05509 -0.18422 -0.07387 0.12513 c-p (-$11 6;I1 C-,4)8 (-y-J C-,4)8 [ 1 [ 1 c 1 II 1 [ 1 c 1 cc-y1

BESSEL

FUNC!!CIONS

BESSEL

OF

INTEGER

401

ORDER

FUNCTIONS-ORDERS

3-9

Table

9.2

lo”0 10:2

y4cd

y5w

yS(x)

y7cd

10.4 10.6 10.8

Ydx) -0.25136 -0.23314 -0.20686 -0.17359 -0.13463

-0.14495 -0.18061 -0.20954 -0.23087 -0.24397

0.13540 0.09148 +0.04567 -0.00065 -0.04609

0.28035 0.27030 0.25346 0.23025 0.20130

0.20102 0.22652 0.24678 0.26131 0.26975

0.00108 0.04061 0.07874 0.11488 0.14838

-0.19930 -0.16282 -0.12563 -0.08791 -0.04793

11.0 11.2 11.4 11.6 11.8

-0.09148 -0.04577 +0.00082 0.04657 0.08981

-0.24851 -0.24445 -0.23203 -0.21178 -0.18450

-0.08925 -0.12884 -0.16365 -0.19262 -0.21489

0.16737 0.12941 0.08848 0.04573 +0.00238

0.27184 0.26750 0.25678 od . zsx

0.17861 0.20496 0.22687 0.24384 0.25545

-(LO1205 +0.02530 0.06163 0.09640 0.12906

12.0 12.2 12.4 12.6 12.8

0.12901

-0.15122 -0.11317 -0.07175 -0.02845 +0.01518

-0.22982 -0.23698 -0.23623 -0.22766 -0.21163

-0.04030 -0.08107 -0.11875 -0.15223 -0.18052

0.18952 0.15724 0.12130 0.08268 0.04240

0.26140 0.26151 0.25571 0.24409 0.22689

0.15902 0.18573 0.20865 0.22728 0.24122

13.0 13.2 13.4 13.6 13.8

0.22420 0.21883 0.20534 0.18432 0.15666

0.05759 0.09729 0.13289 0.16318 0.18712

-0.1887d -0.15987 -0.12600 100' . 00:;;;

-0.20279 -0.21840 -0.22692 -0.22813 -0.22204

+0.00157 -0.03868 -0.07722 -0.11296 -0.14489

0.20448 0.17738 0.14625 0.11185 0.07505

0.25010 0.25369 0.25184 0.24454 0.23190

14.0 14.2 14.4 14.6 14.8

0.12350 0.08615 0.04605 +0.00477 -0.03613

0.20393 0.21308 0.21434 0.20775 0.19364

-0.00697 +0.03390 0.07303 q.10907 0.14080

-0.20891 -0.18921 -0.16363 -0.13305 -0.09850

-0.17209 -0.19380 -0.20939 -0.21842 -0.22067

+0.03682 -0.00186 -0.03994 -0.07640 -0.11024

0.21417 0.19170 0.16501 0.13470 0.10149

15. 0 15.2 15.4 15.6 15.8

-0.07511 -0.11072 -0.14165 -0.16678 -0.18523

0.17261 0.14550 0.11339 0.07750 +0.03920

k 11:::70 0120055 0.20652 0.20507

-0.06116 -0.02228 +0.01684 0.05489 0.09059

-0.21610 -0.20489 -0.18743 -0.16430 -0.13627

-0.14053 -0.16644 -0.18723 -0.20234 -0.21134

0.06620 +0.02969 -0.00710 -0.04322 -0.07775

16.0 16.2 16.4 16.6 16.8

-0.19637 -0.19986 -0.19566 -0.18402 -0.16547

-0.00007 -0.03885 -0.07571 -0.10930 -0.13841

0.19633 0.18067 0.15873 0.13135 0.09956

0.12278 0.15038 0.17250 0.18843 0.19767

-0.10425 -0.06928 -0.03251 +0.00487 0.04164

-0.21399 -0.21025 -0.20025 -0.18432 -0.16297

-0.10975 -0.13838 -0.16286 -0.18253 -0.19685

17. 0 17.2 17.4 17.6 17.8

-0.14078 -0.11098 -0.07725 -0.04094 -0.00347

-0.16200 -0.17924 -0.18956 -0.19265 -0.18846

0.06455 +0.02761 -0.00990 -0.04663 -0.08123

0.19996 0.19529 0.18387 0.16616 0.14282

0.07660 0.10864 0.13671 0.15991 0.17752

-0.13688 -0.10686 -0.07387 -0.03895 -0.00320

-0.20543 -0.20805 -0.20464 -0.19533 -0.18039

18.0 18.2 18.4 18.6 18.8

+0.03372 0.06920 0.10163 0.12977 0.15261

-0.17722 -0.15942 -0.13580 -0.10731 -0.07506

-0.11249 -0.13928 -0.16067 -0.17593 -0.18455

0.11472 0.08289 0.04848 +0.01272 -0.02310

0.18897 0.19393 0.19229 0.18414 0.16980

+0.03225 0.06629 0.09782 0.12587 0.14955

-0.16030 -0.13566 -0.10722 -0.07586 -0.04252

19.0 19.2 19.4 19.6 19.8

0.16930 0.17927 0.18221 0.17805 0.16705

-0.04031 -0.00440 +0.03131 0.06546 0.09678

-0.18628 -0.18111 -0.16930 -0.15134 -0.12794

-0.05773 -0.08993 -0.11857 -0.14267 -0.16139

0.14982 0.12490 0.09595 0.06399 +0.03013

0.16812 0.18100 0.18782 0.18838 0.18270

-0.00824 +0.02593 0.05895 0.08979 0.11750

20.0

0.14967

EKi92971 0:20959 0.22112

Ys(4

Y9@)

0.14124 0.12409 -0.10004 -0.17411 -0.00443 0.17101 c-p (-;I1 C-$)1 (-;)9 (-;I8 (-j+)l 1 [ 1 [ 1 [ 1 [ 1 [ 1 [ 1 II‘-$18

BESSEL Table

BESSEL

9.3

10'ki-'oJlo(cz) 2.69114 446 2.69053 290 0.1 2.68869 898 2.68564 500 Fl*: 0: 4 2.68137 477 z 0. 0

0. 5

2.67589 2.66920 2.66132 2.65226 2.64201

E ::9” 2.63061 2.61806 ::1” 2.60437 :-: 2.58958 1:4 2.51368

FUNCTIONS

OF INTEGER

FUNCTIONS-ORDERS

10,

ORDER 11,20

AND

21

1.22324 748 1.22299 266 1.22222 850 1.22095 588 1.21917 626

-0.11831 -0.11841 -0.11857 -0.11880

049 335 200 661 750

3.91990 3.91944 3.91804 3.91571 3.91244

9.33311 9.33205 9.32886 99. . '3::::

-0.406017 -0.406071 -0.406231 -0.406499 -0.406873

362 838 738 043 878

1.21689 1.21410 1.21081 1.20703 1.20276

169 481 883 750 518

-0.11910 -0.11946 -0.11990 -0.12040 -0.12097

510 998 282 444 581

3.90825 3.90314 3.89710 3.89015 3.88228

9.30663 9.29500 9.28128 9.26546 9.24758

-0.407355 -0.407945 -0.408644 -0.409452 -0.410369

512 358 963 012

1.19800 1.19276 1.18705 1.18087 1.17422

675 764 385 185 867

-0.12161 -0.12233 -0.12312 -0.12398 -0.12492

801 229 002 273 212

3.87350 3.86383 3.85325 3.84179 3.82945

9.22162 9.20562 9.18157 9.15550 9.12743

-0.411397 -0.412536 -0.413788 co.415153 -0.416632

323

-0.11828

2.55670 2.53867 2.51960

842 639

:-ii 1:9

2.49952 2.47846

907 955 207

1.16713 1.15958 1.15160 1.14320 1.13437

182 931 961 168 488

-0.12594 -0.12703 -0.12821 -0.12948 -0.13084

004 852 977 616 030

3.81624 3.80216 3.78723 3.77146 3.75485

9.09737 9.06534 9.03137 8.99546 8.95766

-0.418228 -0.419940 -0.421771 -0.423122 -0.425795

2: $3 2:4

2.45643 2.43346 2.40959 2.38483 2.35922

192 545 000 384 612

1.12513 1.11550 1.10548 1.09508 1.08431

904 438 152 144 551

-0.13228 -0.13382 -0.13545 -0.13719 -0.13903

497 319 821 351 284

3.73742 3.71918 3.70015 3.68032 3.65973

8.91797 8.87643 8.83306 8.78790 8.74096

-0.427992 -0.430315 -0.432764 -0.435344 -0.438056

2.5

2.33279 2.30557

682 074 976

1.07319 1.06173 1.04994 1.03783 1.02541

540 312 098 155 767

-0.14098 -0.14303 -0.14521 -0.14751 -0.14994

022 997 672 543 141

3.63831 3.61627 3.59344 3.56989 3.54564

8.69228 8.64189 8.58981 8.53609 8.48076

-0.440902 -0.443885 -0.447007 -0.450272 -0.453682

770 836 598 517 085

1.01271

242

0.99972 0.98648 0.97298 0.95924

906

:-: 3: 4

2.15873 2.12745 2.09561 2.06325

-0.15250 -0.15519 -0.15804 -0.16103 -0.16419

037 840 206 836 482

3.52071 3.49510 3.46885 3.44195 3.41444

8.42385 8.36539 8.30542 8.24397 8.18110

-0.457241 -0.460951 -0.464816 -0.468840 -0.473027

::2 ;:I.

2.03039 1.99709 1.96336 1.92926 1.89481

820 260 956 467 352

-0.16751 -0.17102 -0.17470 -0.17859 -0.18268

951 110 889 286 376

3.38633 '3*3':;;: 3129855 3.26821

8.11682 8.05119 7.98424 7.91600 7.84653

-0.477379 -0.481902 -0.486600 -0.491476 -0.496537

1.86005 1.82501 1.78973 1.75425 1.71860

168 462 765 588 416

-0.18699 -0.19153 -0.19631 -0.20136 -0.20667

314 346 812 159 950

3.23736 3.20601 3.17419 3.14192 3.10921

1.77586 7.70403 7.63108 7.55707 1.48202

-0.501786 -0.507229 -0.512872 -0.518719 -0.524777

1.68281 1.64692 1.61097 1.57498 1.53899

701 860 267 249 084

0.79652 0.78103 0.76550 0.74992

852

873 757 582 498 840

3.07608 3.04256 3.00866

0.73431

-0.21228 -0.21820 -0.22445 -0.23105 -0.23802

7.40598 7.32900 7.25112 7.17238 7.09282

-0.531051 -0.537549 -0.544276 -0.551240 -0.558448

1.50302

991

0.71869

942

-0.24540 147 (-55)5

2.90490

7.01250 (93

-0.565907 (6;)3

1.5 1.6

Z ::t 3. 0 3.1

2.21759 2.24889 2.21948 2.18942

613 732

108 213 599

0.94528

659

0.93111

794 415

0.91675 0.90220 0.88749

939

785

0.87263

375

0.85763 0.84250 0.82726 0.81193

469 806

130 548 093

829 130

351

[ 1 C-55)6

[ 1

5.. Eltl"

Y,+1(@

[ 1 c 1 E

Yn(z)-Yn-l(X)

Compiledfrom British Associationfor the Advancement of Science,Besselfunctions, Part II. Functions of positive integer order, Mathematical Tables, vol. X (CambridgeUniv. Press,Cambridge, England, 1952),L. Fox, A short table for Besselfunctions of integer ordersand large arguments. Royal Society Shorter Mathematical Tables No. 3 (CambridgeUniv. Press,Cambridge,England, 1954), and Mathematical Tables Project, Table of fm(z)=n!(%-c-V,(a). J. Math. Phys. 23, 45-60 (1944) (with permission).

BESSEL FUNCTIONS BESSEL

OF INTEGER

FUNCTIONS-ORDERS

403

ORDER

10,

11, 20 AND

21

Table

9.3

lo'%-'oJ~o(z) 1 50302 991 1:46713 132 1.43132 603 1.39564 431 1.36011 571

lolL-"Jll(s) 0.71869 0.70307 0.68747 0.67188 0.65634

942 931 104 722 019

1o-%'oYlo(z) -0.24540 147 -0.25320 186 -0.26145 975 -0.27020 813 -0.27948 304

2.90490 2.86969 2.83421 2.79846 2.76248

7.01250 6.93145 6.84971 6.76734 6.68437

-0.565907 -0.573626 -0.581612 -0.589875 -0.598423

1 32476 1'28963 1'25473 1'22009 1:18574

904 229 264 642 907

0.64084 0.62540 0.61003 0.59475 0.57957

205 463 945 774 041

-0.28932 -0.29977 -0.31088 -0.32269 -0.33528

400 431 154 795 105

2.72628 2.68988 2.65330 2.61656 2.57967

6.60085 6.51682 6.43233 6.34742 6.26213

-0.607266 -0.616414 -0.625876 -0.635663 -0.645788

i::

1.15171 1 11801 1'08468 1'05172 1:01917

513 822 098 510 129

0.56448 0.54952 0.53467 0.51997 0.50540

805 091 890 158 814

-0.34869 -0.36300 -0.37829 -0.39464 -0.41215

413 693 631 698 232

2.54267 2.50556 2.46837 2.43111 2.39381

6.17651 6.09059 6.00443 5.91806 5.83152

-0.656261 -0.667094 -0.678301 -0.689895 -0.701890

E-2 6:7 6.8 6.9

0 98703 0'95534 0:92411 0.89335 0.86308

926 769 427 563 740

0.49099 0.47674 0.46266 0.44876 0.43504

740 781 745 400 477

-0.43091 -0.45104 -0.47267 -0.49594 -0.52098

524 907 855 084 648

2.35647 2.31913 2.28179 2.24448 2.20721

5.74485 5.65810 5.57131 5.48451 5.39775

-0.714300 -0.727140 -0.740427 -0.754178 -0.768410

77': 7:3 7.4

0.83332 0 80407 0'77536 0:74719 0.71957

414 941 570 450 626

0.42151 0.40818 0.39505 0.38214 0.36943

665 616 943 216 970

-0.54798 -0.57710 -0.60855 -0.64254 -0.67930

051 346 234 159 390

2.17000 2.13286 2.09582 2.05888 2.02206

5.31106 5.22448 5.13805 5.05181 4.96579

-0.783140 -0.798389 -0.814177 -0.830524 -0.847452

7.5 7.6 7. 7 7.8 7.9

0.69252 0.66603 0.64012 0.61480 0.59007

040 536 854 640 439

0.35695 0.34469 0.33266 0.32087 0.30930

696 850 845 058 826

-0.71909 -0.76217 -0.80884 -0.85940 -0.91419

088 356 258 807 914

1.98539 1.94887 1.91252 1.87635 1.84038

4.88002 4.79455 4.70940 4.62461 4.54021

-0.864985 -0.883147 -0.901963 -0.921460 -0.941665

8.0 8.1 8.2 8.3 8.4

0.56593 0.54239 0.51945 0.49712 0.47539

704 791 967 408 201

0.29798 0.28690 0.27606 0.26546 0.25512

448 187 265 873 162

-0.97356 -1.03786 -1.10747 -1.18278 -1.26419

279 231 485 826 685

1.80462 1.76908 1.73378 1.69874 1.66395

4.45624 4.37272 4.28968 4.20716 4.12518

-0.962608 -0.984319 -1.006831 -1.030178 -1.054394

8.5 8.6 8.7

0.45426 0.43373 0.41381 0.39448 0.37575

352 779 323 748 740

0.24502 0.23517 0.22557 0.21621 0.20711

250 220 121 969 750

-1.35209 -1.44687 -1.54891 -1.65856 -1.77613

608 598 312 097 854

1.62944 1.59521 1.56128 1.52765 1.49434

4.04377 3.96296 3.88277 3.80323 3.72436

-1.079518 -1.105589 -1.132647 -1.160736 -1.189902

;': 9:3 9.4

0.35761 0 34006 0'32309 0:30670 0.29088

917 823 939 683 411

0.19826 0.18965 0.18130 0.17318 0.16532

418 897 082 839 010

-1.90191 -2.03610 -2.17882 -2.33011 -2.48986

706 452 801 366 396

1.46136 1.42872 1.39641 1.36447 1.33288

3.64619 3.56873 3.49201 3.41606 3.34088

-1.220192 -1.251657 -1.284351 -1.318328 -1.353647

i:;

0.24676 422 0.26091 0.27562 963 227

0.15030 409 0.15769 0.14316 825 025

-2.83359 -3.01652 -2.65783

602 353 251

9:8 9.9

0.23314 362 0.22005 470

0.13624 751 0.12956 726

-3.20574 -3.40010

283 421

1.30166 1.27082 1.24036 1.21029 1.18061

3.26651 3.19294 3.12022 3.04834 2.97733

-1.390372 -1.428567 -1.468301 -1.509646 -1.552680

10.0

0.20748 611 ['-y]

0.12311 653 [(-;)"I

-3.59814 ['-y]

152

5'0 511 :3 514 ;:; :'i 5: 9 6.0 66::

7.0

88:: 9.0

-1.597484 c-y II 1 c 1 [ 1

1.15134 (-l)5

2.90720

c--4)2 4

404

BESSEL

Table

BESSEL

9.3

FUNCTIONS

OF

FUNCTIONS+RDERS

10:1 10. 2 10. 3 10. 4

0.20748 0.21587 0.22413 0.23223 0.24011

611 417 707 256 699

0.12311 0.13041 0.13787 0.14549 0.15324

653 285 866 509 123

YlO(X) -0.35981 415 -0.34383 078 -0.32793 809 -0.31207 433 -0.29618 615

10. 5 10.6 10.7 10.8 10. 9

0.24774 0.25507 0.26205 0.26863 0.27477

554 240 109 466 603

0.16109 0.16902 0.17701 0.18503 0.19304

407 861 780 266 230

-0.28022 -0.26416 -0.24795 -0.23159 -0.21505

11. 0 11.1 11. 2 11.3 11.4

0.28042 0.28554 0.29007 0.29398 0.29722

823 479 999 925 944

0.20101 0.20891 0.21670 0.22434 0.23181

401 340 446 974 048

11.5 11.6 11. 7 11.8 11.9

0.29975 0.30153 0.30253 0.30270 0.30203

923 946 345 737 061

0.23904 0.24601 0.25268 0.25899 0.26492

12.0 12.1 12.2 12.3 12.4

0.30047 0.29802 0.29464 0.29033 0.28507

604 036 445 357 771

12.5 12. 6 12. 7 12. 8 12.9

0.27887 0.27171 0.26361 0.25458 0.24462

13. 0 13.1 13.2 13. 3 13. 4 13. 5 13. 6 13. 7 13.8 13. 9

JlO(Z)

Jll(4

INTEGER

ORDER

10, 11, 20 AND 21 10~5x-2tJ2o(x)

1027x-2'52l(X)

10-23~2oY~~(~)

1.151337 1.122469 1.094012 1.065970 1.038347

2.907199 2.837961 2.769629 2.702215 2.635729

-

1.59748 1.64414 1.69275 1.74339 1.79618

819 276 949 513 324

1.011148 0.984374 0.958030 0.932118 0.906639

2.570182 2.505582 2.441939 2.379259 2.317550

-

1.85121 1.90861 1.96848 2.03097 2.09619

-0.19832 -0.18140 -0.16429 -0.14700 -0.12955

403 409 620 917 753

0.881596 0.856989 0.832821 0.809092 0.785801

2.256817 2.197065 2.138299 2.080523 2.023738

-

2.16430 2.23544 2.30977 2.38746 2.46870

680 789 218 761 183

-0.11196 -0.09424 -0.07644 -0.05858 -0.04071

142 628 263 580 566

0.762950 0.740539 0.718565 0.697029 0.675930

1.967947 1.913152 1.859352 1.806548 1.754740

-

2.55367 2.64257 2.73563 2.83307 2.93513

0.27041 0.27542 0.27992 0.28386 0.28720

248 744 508 459 623

-0.02287 -0.00511 to.01251 0.02995 0.04716

631 577 441 946 182

0.655266 0.635035 0.615236 0.595866 0.576923

1.703925 1.654102 1.605267 1.557418 1.510551

-

3.04208 3.15419 3.27175 3.39509 3.52453

175 575 509 064 889

0.28991 0.29194 0.29326 0.29385 0.29366

166 422 923 431 968

0.06406 0.08059 0.09670 0.11231 0.12737

154 668 381 845 554

00.KE: 0:522625 0.505359 0.488504

1.464660 1.419743 1.375791 1.332800 1.290762

-

3.66044 3.80321 3.95323 4.11095 4.27684

0.23378 0.22206 0.20952 0.19617 0.18208

201 793 032 859 776

0.29268 0;29088 0.28824 0.28474 0.28037

843 684 464 526 612

0.14180 0.15555 0.16855 0.18073 0.19204

995 698 286 529 392

0.472056 0.456011 0.440365 0.425114 0.410252

1.249671 1.209520 1.170299 1.132001 1.094617

-

4.45140 4.63518 4.82874 5.03272 5.24778

0.16729 0.15186 0.13585 0.11932 0.10235

840 646 302 411 036

0.27512 0.26899 0.26198 0.25410 0.24534

884 942 851 149 866

0.20242 0.21181 0.22016 0.22743 0.23357

090 137 393 118 014

0.395776 0.381681 0.367961 0.354612 0.341628

1.058137 1.022552 0.987853 0.954028 0.921067

-

5.47464 5.71407 5.96691 6.23405 6.51646

14. 0 0.08500 14.1 0.06737 14.2 0.04952 14. 3 0.03156 14.4 +0.01356

671 200 862 199 013

0.23574 0.22531 0.21407 0.20206 0.18931

535 197 407 238 275

0.23854 0.24231 0.24486 0.24616 0.24620

273 614 329 313 100

0.329005 0.316736 0.304816 0.293240 0.282001

0.888960 0.857694 0.827260 0.797644 0.768835

-

6.81520 7.13138 7.46624 7.82110 8.19739

14.5 14. 6 14.7 14. 8 14.9

-0.00438 -0.02218 -0.03974 -0.05697 -0.07378

689 745 898 854 344

0.17586 0.16176 0.14707 0.13182 0.11609

611 836 028 729 931

0.24496 0.24246 0.23869 0.23367 0.22742

888 568 741 730 597

0.271095 0.260516 0.250257 0.240312 0.230676

0.740821 0.713590 0.687129 0.661426 0.636467

- 8.59667 - 9.02062 - 9.47109 - 9.95006 -10.45971

15.0

-0.09007 181 ['-y]

0.21997 141 72

0.221343 ( -;I6

0.612240

-11.00239

lo”0

0.09995 048 ['-y-j

[( 1

[I 1 [(-;I11

[ 1 c-j,4

BESSEL FUNCTIONS BESSEL

FUNCTIONS-ORDERS

JIOG)

ls”0 -0.09007 181 15:1 -0.10575 330

OF INTEGER 10,

11,

20 AND

21

405 Table

1027~-215~~(~)

wh-“oJzo(~)

YlO(X)

Jll(Z)

c

ORDER

9.3

lo-?iGoY~~(z)

0.09995 0.08344 0.06666 0.04967 0.03256

048 886 618 738 035

0.21997 0.21134 0.20160 0.19077 0.17893

141 904 159 902 834

0.22134 0.21230 0.20356 0.19510 0.18691

33 71 16 08 87

0.61224 0.58873 0.56593 0.54382 0.52239

04 25 06 12 14

-

11.0024 11.51807 12.1974 12.8555 13.5585

15.2 15.3 15.4

-0.12073 964 -0.13494 535 -0.14828 828

15.5 15.6 15.7 15.8 15.9

-0.16069 -0.17207 -0.18238 -0.19154 -0.19949

032 791 269 204 958

+0.01539 -0.00173 -0.01874 -0.03555 -0.05207

539 513 731 621 632

0.16614 0.15246 0.13797 0.12276 0.10691

338 453 838 733 918

0.17900 0.17136 0.16398 0.15685 0.14997

91 62 38 60 67

0.50162 0.48151 0.46204 0.44319 0.42496

76 66 52 99 74

-

14.3098 15.1136 15.9742 16.8962 17.8849

16.0 16.1 16.2 16.3 16.4

-0.20620 -0.21161 -0.21570 -0.21842 -0.21978

569 797 160 977 394

-0.06822 -0.08390 -0.09905 -0.11357 -0.12738

215 874 224 046 344

0.09052 0.07368 0.05650 0.03907 0.02150

660 666 016 110 600

0.14334 0.13694 0.13077 0.12482 0.11910

00 00 08 65 14

0.40733 0.39028 0.37381 0.35789 0.34253

43 75 35 93 16

-

18.9460 20.0855 21.3104 22.6279 24.0462

16.5 16. 6 16.7 16.8 16.9

-0.21975 -0.21833 -0.21554 -0.21139 -0.20590

411 905 637 267 350

-0.14041 -0.15258 -0.16383 -0.17409 -0.18329

403 841 668 338 797

+0.00391 -0.01359 -0.03091 -0.04793 -0.06454

319 786 729 557 431

0.11358 0.10828 0.10318 0.09827 0.09356

96 55 34 77 30

0.32769 0.31338 0.29957 0.28626 0.27343

75 39 78 66 76

-

25.5740 27.2209 28.9975 30.9150 32.9859

17.0 17.1 17.2 17.3 17.4

-0.19911 -0.19106 -0.18181 -0.17141 -0.15993

332 538 155 203 505

-0.19139 -0.19833 -0.20407 -0.20858 -0.21182

539 646 831 485 701

-0.08063 -0.09610 -0.11086 -0.12479 -0.13782

696 960 170 683 343

0.08903 0.08468 0.08051 0.07650 0.07266

37 45 02 53 49

0.26107 0.24917 0.23771 0.22669 0.21608

81 57 82 32 89

-

35.2237 37.6429 40.25'94 43.0904 46.1543

17.5 17.6 17.7 17.8 17.9

-0.14745 -0.13405 -0.11983 -0.10487 -0.08928

649 943 363 499 492

-0.21378 -0.21443 -0.21378 -0.21183 -0.20858

318 935 944 538 727

-0.14985 -0.16081 -0.17062 -0.17922 -0.18654

544 304 321 038 691

0.06898 0.06545 0.06207 0.05884 0.05575

37 69 96 68 39

0.20589 0.19609 0.18668 0.17764 0.16896

33 48 17 27 66

-

49.4711 53.06822 56.9506 61.1611 65.7197

18.0 18.1 18.2 18.3 18.4

-0.07316 -0.05663 -0.03980 -0.02279 -0.00571

966 961 852 278 052

-0.20406 -0.19829 -0.19130 -0.18314 -0.17386

341 032 265 307 213

-0.19255 -0.19720 -0.20045 -0.20229 -0.20271

365 030 582 875 742

0.05279 0.04996 0.04726 0.04468 0.04222

63 93 85 96 83

0.16064 0.15265 0.14500 0.13767 0.13064

24 91 62 32 97

-

70.6543 75.9946 81.7717 88.0182 94.7683

18.5 18.6 18.7 18. 8 18.9

+0.01131 0.02817 0.04474 0.06090 0.07654

917 711 490 579 556

-0.16351 -Oil5217 -0.13990 -0.12679 -0.11291

793 591 845 446 893

-0.20171 -0.19928 -0.19546 -0.19026 -0.18373

011 520 113 637 930

0.03988 0.03764 0.03550 0.03347 0.03154

04 17 84 64 21

0.12392 0.11749 0.11133 0.10545 0.09982

57 14 69 28 98

-102.0574 -109.9219 -118.3992 -127.5270 -137.3432

19.0 19.1 19.2 19.3 19.4

0.09155 0.10582 0.11925 0.13174 0.14321

333 247 134 416 168

-0.09837 -0.08325 -0.06765 -0.05168 -0.03544

240 039 283 334 863

-0.17592 -0.16688 -0.15669 -0.14540 -0.13312

797 985 143 785 231

0.02970 0.02795 0.02628 0.02470 0.02320

16 15 80 79 78

0.09445 0.08933 0.08443 0.07977 0.07532

89 lo 76 01 03

-147.8850 -159.1885 -171.2882 -184.2155 -197.9980

19.5 19.6 19.7 19.8 19.9

0.15357 0.16275 0.17068 0.17731 0.18259

193 089 305 198 079

-0.01905 -0.00262 +0.01374 0.02994 0.04584

771 120 948 285 818

-0.11992 -0.10591 -0.09119 -0.07587 -0.06006

560 538 555 548 922

0.02178 0.02043 0.01915 0.01794 0.01679

44 46 54 37 67

0.07108 0.06704 0.06319 0.05953 0.05606

01 16 71 92 06

-212.6582 -228.2122 -244.6678 -262.0226 -280.2622

20.0

0.18648 256 542

[

1

0.06135 630

[

C-212

1

-0.04389 465 (542

[ 1

0.01571 16 C-j)4

[

1

-299.3574 (--j’9 [ 1 [c-p1

0.05275 42

406

Y

BESSEL Table 9.3 BESSEL

FUNCTIONS

FUNCTIONS-MODULUS

OF

AND

PHASE

INTEGER

OF

ORDER

ORDERS

10,

11,

20

AND

21

Y&j =Nn(z) sin &(.r)

.r*Mlo(z)

0.048 0.046 0.044 0.042

0.85676 0.85136 0.84633 0.84164 0.83727

701 682 336 245 251

-13.94798 -14.05389 -14.15926 -14.26413 -14.36853

864 581 984 968 333

0.87222 0.86513 0.85857 0.85250 0.84689

790 271 314 587 281

811 (z)-.c -14.96758 -15.09771 -15.22701 -15.35552 -15.48330

686 672 466 901 635

<s> 20 21 22 23 24

0.040 0.038 0.036 0.034 0.032

0.83320 0.82942 0.82590 0.82264 0.81962

419 012 472 403 546

-14.47247 -14.57600 -14.67912 -14.78187 -14.88428

807 035 589 967 611

0.84170 0.83689 0.83246 0.82836 0.82459

044 917 283 826 496

-15.61039 -15.73682 -15.86265 -15.98791 -16.11265

144 771 679 896 291

25 26 28 29 31

0.030 0.028 0.026 0.024 0.022

0.81683 0.81427 0.81191 0.80976 0.80780

775 076 546 370 825

-14.98636 -15.08815 -15.18965 -15.29090 -15.39191

880 085 477 253 569

0.82112 0.81794 0.81503 0.81237 0.80997

469 133 056 970 751

-16.23689 -16.36068 -16.48405 -16.60703 -16.72967

620 504 469 912 149

33 36 38 42 45

0.020 0.018 0.016 0.014 0.012

0.80604 0.80446 0.80305 0.80183 0.80077

267 127 902 156 512

-15.49271 -15.59332 -15.69375 -15.79403 -15.89418

527 192 598 741 589

0.80781 0.80588 0.80416 0.80267 0.80139

410 079 997 505 036

-16.85198 -16.97400 -17.09577 -17.21731 -17.33865

406 835 505 438 590

50 56 63 71 83

0.010 0.008 0.006 0.004 0.002

0.79988 0.79916 0.79860 0.79820 0.79796

647 297 244 323 417

-15.99422 -16.09416 -16.19402 -16.29383 -16.39360

093 168 726 652 832

0.80031 0.79943 0.79875 0.79827 0.79798

114 341 398 039 093

-17.45982 -17.58086 -17.70178 -17.82262 -17.94340

880 166 301 084 316

100 125 167 250 500

0.000

0.79788

456

-16.49336

143

0.79788

456

-18.06415

776

00

.r- 1 0.050 0.048 0.046 0.044 0.042

.2X20(~) 1.474083 1.320938 1.211667 1.131459 1.070845

02”(Z) -.l’ -21.047407 -21.606130 -22.149524 -22.676802 -23.188535

1.791133 1.525581 1.347435 1.224460 1.136653

-21.290925 -21.927545 -22.550082 -23.154248 -23.738936

0.040 0.038 0.036 0.034 0.032

1.023762 t 9985%; 0:930635 0.909513

-23.685951 -24.170500 -24.643620 -25.106640 -25.560748

1.071741 1.022171 0.983229 0.951902 0.926211

-24.304948 -24.853951 -25.387848 -25.908478 -26.417500

0.030 0.028 0.026 0.024 0.022

0.891605 0.876293 0.863121 0.851743 0.841895

-26.006988 -26.446280 -26.879433 -27.307159 -27.7300+8

0.904821 0.886799 0.871483 0.858385 0.847145

-26.916369 -27.406346 -27.088527 -28.363869 -28.833211

0.020 0.018 0.016 0.014 0.012

0.833375 0.826019 0.819702 0.814321 0.809796

-28.148822 -28.563847 -28.975650 -29.384666 -29.791303

0.837487 0.829198 0.822114 0.816105 0.811069

-29.297299 -29.756800 -30.212318 -30.664405 -31.113569

5" 51,

0.010 0.008 0.006 0.004 0.002

0.806062 0.803071 0.800781 0.799165 0.798204

-30.195941 -30.598942 -31.000652 -31.401404 -31.801522

0.806925 0.803612 0.801081 0.799297 0.798237

-31.560285 -32.005000 -32.448139 -32.890109 -33.331307

1"" iis 167 250 500

0.000

0.797885 C-73)5

-32.201325

0.797885

-33.772121

0~

!C-l 0.050

[ 1

ho@+-r

GM,

[<2>:nearest 1 C-f)2

1(z)

:,? iI (.I.)

[(-P1

.921(x)--R.



20 21 22: 24 ;56 2 31 33 5~

2 42 45

76: a3

Integer to 5. Compiled from L. Fox, A short table for Bessel functions of integer orders and large arguments. Royal Society Shorter Mathematical Tables No. 3(Cambridge Univ. Press, Cambridge, England, 1954) (with permission).

BESSEL FUNCTIONS BESSEL

OF INTEGER ORDER

FUNCTIONS-VARIOUS

Jr4) (-

-

(-

ORDERS

J&V

1)2.23890

Table

9.4

J,(5)

7791

3)2.47663 8964

9)5.24925 0180

407

I-

3)7.03962 9756

-

6)2.49234 3435

I II

1-

(-

l)-1.77596 2 1 -3.27579 +4.65651 3.64831 7713 2306 1628 1376 1 3.91232 3605

(-

1) 2.61140 5461

-

3139 7318 1665 3 5.52028 2 1 5.33764 1016 1.31048 1.84052

1.19800 6746 16)6.88540 8200

II

- 25 60 2.90600 80 42 3.48286 4948 1.10791 3.87350 5851 9794 3009

(-189)8.43182 8790

I - 10 11 7)2.51538 9 8 I 2.30428 1.07294 5149 1.49494 1.93269 6448 4758 6283 2010

II -

19 3.91897 33 3.65025 48 1.19607 65 3.22409

2805 6266 7458 5839

(-158)1.06095 3112

Jn(I0)

II

- 45 33 2.77033 21 11 2.67117 0052 2.29424 8.70224 7616 1617 7278

(-119) 6.26778 9396

Jn(50)

J,UOO)

+5.58123 2767 -9.75118 2813 - 2)+7.08409 7728

I- 1 2I +2.16710 -2.34061 -1.44588 3.17854 5282 1268 9177 4208

II

(- 1) 2.91855 6853

(- 2)-2.71924 6104

(- 1) 2.07486 1066

(- l)-1.13847

I -7 1 2I 2.89720 6.33702 5497 1.23116 1.19571 6324 8393 5280

- 1 2 +1.04058 +6.04912 -8.71210 4770 -8.14002 0126 5632 2682

-2 -7.41957 -2 -3.35253 -2 +7.01726 -2 +4.33495 -2)-6.32367

jl

3696 8314 9099 5988 6141

8491

- 2)-6.98335 2016

I - 43 5i 1.52442 5.05646 4853 4.50797 4.31462 1.56675 7752 3144 6697 6192

I -30 -215I 1.78451 --12 6.03089 6078 1.15133 1.55109 3608 5312 6925

I

- 2 1I +4.84342 +1.21409 -1.16704 2812 -1.38176 5725 3528 0219

II

-2 +8.14601 -3.86983 5850 +7.27017 +6.22174 3973 5482 2958

(-89) 6.59731 6064

(-21)+1.11592 7368

( -2) +9.63666

7330

408

BESSEL

Table

9.4

FUNCTIONS

BESSEL

Y7s)

OF

INTEGER

ORDER

FUNCTIONS-VARIOUS

ORDERS

Y,,(2) I - 10/ +5.10375 -1.07032 4315 -1.12778 -6.17408 6726 3777 1042

Yn(5) -3.08517 6252

-1.65068 2607

i10 12 13 14

(

l)-3.32784

2303

I

2 -2.60405 3 -2.57078 4 -3.05889 5 -4.25674 6I -6.78020

8666 0243 5705 6185 4939

(

0 -2.76594 3226

-7.76388 3188

-2.42558 0081

-9.27525 -3.82982 -1.74556 -8.69393

-1.27536 1870 13)-3.31061 6748 t lOI-2.98102 3646

15 16

(

5719 1416 1722 8814

41-4.69404 9564

:87 19 20 200 50 100

n

62)-1.97615 0576 (185)-3.77528 7810

(155)-3.00082 6049

I'n(50) -9.80649 9547 -5.67956 6856

Y,(lO)

-2.51362 6572

+6.44591 2206 - 2)-8.80580 7408 (- 2)-7.85484 1391 - 2 +7.23483 9130 - 2 ~9.59120 2782 2 -4.54930 2351 I-- 1I -1.10469 7953

10 11 12 13 14

II 8 18 29 42

-5.93396 -4.02856 -9.21681 -2.78883

5297 8418 6571 7017

(115)-5.08486 3915

Y,(lOO)

-2)-7.72443

1337

(-2)-7.54301

1992

-2)-4.89199 9608 -2)+5.83315 7424 -4.50025 8583

- 2)-9.15700 8429

:56 17 18 19

I-

2 +4.04128 0205 1 +1.15817.7655 2 +3.37105 6788 2 -9.28945 7936 1I -1.00594 6650 2 +1.64426 1 -1.16457 2 -4.53080 1I -2.10316

20 30 40 50

( 3)-1.59748 3848

I-

3395 2349 1120 5546

100

(85)-4.84914 8271

(+18)-3.29380 0188

-2)+7.09440 9807 -2)+5.12479 7308

(-l)-1.66921

4114

BESSEL

ZEROS

AND

FUNCTIONS

ASSOCIATED

VALUES

8. b5372

11.79153 14.93091 18.07106 21.21163 24.35247 27.49347 30.63460

:76 :ff 20

48065

77086

-0.20654

39679 bb299

17132 83717

49.48260 52.62405 55.76551

98974

18411 07550 39261 91902

58.90698 b2.04846

FUNCTIONS

8

-0.40276

64331

7.01559 10.17347 13.32369 16.47063

+0.30012 -0.24970 +0.21836 -0.19647

+0.18772 -0.17326 +0.16170 -0.15218 +0.14416

88030 58942 15507 12138 59777

19. b1586 22.76008 25.90367 29.04683 32.18968

+0.18006 -0.16718

-0.13729 +O. 13132 -0.12606 +0.12139 -0.11721

69434 46267

11989

35.33231 38.47477 41.61709 44.75932 47.90146

-0.13421

94971

+0.11342 -0.10999 +O.lOb84 -0.10395 +0.10129

91926 11430 78883 95729 34989

-0.27145 +0.23245

57646

46.34118

j,

74973

+O. 34026

0213b 83537

36.91709

:: 15

-0.51914

b4684

40.05842 43.19979

BESSEL

22999 98314

86248

': ::;:: +0:11925 -0.11527 '2 :fg; +0:10537 -0.10260 +0.10004

j4, 8

J303,8)

'-; :;g; +0:14061

J14G4,

9.76102 13.01520 lb.22347 19.40942

-0.29827 +0.24942 -0.21828 +0.19644 -0.18005

7.58834 11.06471 14.37254 17.61597 20.82693

22.58273 25.74817 28.90835 32.06485 35.21867

+0.16718 -0.15672 +0.14801 -0.14060 +0.13421

24.01902 27.19909 30.37101 33.53714 36.69900

38.37047 41.52072 44.66974

-0.12862 +0.12367 -0.11925

39.85763

::

-0.12607 +0.12140 -0.11721

2

4570. . wi

To": :::;:

:;

54.11162 57.25765 60.40322

+0.10839 -0.10537

6.38016

4 a 1: 11

:t 20

';

63.54840 bb. b9324

B’ 190

13

1':

::g:

68.21417

:; . :g; +0.10685 -0.10396 :; g's;; +0:09652

3: t2323:24 39:60324

': :;;:: +0:12606 -0.12139 +0.11721 -0.11343 +0.10999 -0.10685

44.21541 47.39417 50.56818 53.73833 56.90525

-0.11924 +0.11526 -0.11166

'-; :00:%8 +0:09882

668' K 66: 39141 69.54971 72.70655

lb

58.59961 61.75682

:i 19 20

64.91251

68.06689 71.22013

-0.09652

+0.09438

14.79595

+O. 20654

21.11700 24.27011 27.42057 30.56920 33.71652

+0.17326

17.95982

-0.33967

-0.18773

-0.lb170 +0.15218 -0.14417 +0.13730

36.86286

-0.13132

46.29800

'-: ::g +0:11721 -0.11343

40.00845 43.15345 49.44216 52.58602 55.72963 58.87302 62.01622 65.15927

k

8

+0.10999 -0.10685 +0.10396 -0.10129 +0.09882

J’dk, d

+0.15669 -0.14799 +0.14059 -0.13420 +0.12861

30.03372

4; YE: 52127945 55.44059

+0.27138 -0.23244

:t t238:: 31:81172 34.98878 38.15987

+0.14792 -0.14055 +0.13418 -0.12859 +0.12365

1

J’z(.iz, 8)

5.13562 8.41724 11.61984

-0.17323

28.19119 31.42279 34.63709 37.83872 41.03077

n,s

ha s

Table DERIVATIVES

+0.16168 -0.15217 +0.14416 -0.13729 +0.13132

+0.23188 -0.20636 +O. 18766

+0.15212 -0.14413

26.82015

d

THEIR

-0.24543 +0.21743 -0.19615 +0.17993 -0.lb712

J17 (j7, d -0.21209 +0.19479 -0.17942 +0.16688 -0.15657

Y6(j,

AND

8.77148 12.33860 15.70017 18.98013 22.21780

11.08637 14.82127 18.28758 21.64154 24.93493

42.77848 ::

55.62165 58.77084 61.91925 b5.06700

d

-0.26536

-0.22713 +0.20525 -0.18726 +0.17305 -0.16159

h, n 9.93611 13.58929 17.00382 20.32079 23.58608 b

12 10::87; 49:32036 52.47155

+O: 09765

409

ORDER

J’1 (A, R)

55577 81103 79129 44391

15308 91320

33.77582

::

OF

INTEGER

J’o(jo, 8)

jo, s 2.40482 5.52007

OF

:: . 110058;; +0.10260 -0.10003 +0.09765 -0.09543 +0.09336

1k z;i 47: 64940 50.80717 53.96303 57.11730 60.27025 63.42205 66.57289

69.72289

j, s 12.22509 16.03777 19.55454 22.94517 26.26681

-0.12366 +0.11925 -0.11527 '; . :'og; +0.10537 -0.10260 +0.10003 -0.09765 +0.09543

J’s(j8,d

-0.19944 +0.18569 -0.17244 +0.16130 -0.15196 +0.14404 -0.13722 +0.13127 -0.12603 +0.12137

tc t23z3 52:00769 55.18473 58.35789 61.52774 64.69478

67.85943 :: . EY

-0.11719 +0.11342 -0.10998 +0.10684 -0.10395 '-if g;; +0:09652 -0.09438 +0.09237

9.5

410

BESSEL Table 9.5 ZEROS AND s : i

5 76 : 10 :: :43 15 :; :t 20

; : 4 5 ; a 1; :: :: 15 :i :t 20

s 1

ASSOCIATED YO. s

O.iSi57 3.95767 7.08605 10.22234 13.36109

697

lb. 50092

244 970

19.64130 22.78202 25.92295 29.06403

842 lob 504 747

805 765

025

32.20520 35.34645 38.48775 4l.b2910 44.77048

412 231

41.91189 51.05332 54.19477 57.33624 60.47772

633

bb5

447 661

855 936 570 516

FUNCTIONS

VALUES

OF

OF

Y’o(?/o,J +0.87942 080 -0.40254 267 +0.30009 761 -0.24970 124 +0.21835 830

BESSEL Yl, d

THEIR 1/z, s

Y’Z(Y2, s)

+0.20655

lb.37897

-0.16718 +0.15672 -0.14801

18.04340 21.18807 24.33194 27.47529 30.61829

-0.18773 to.17327 -0.16170 +0.15218 -0.14417

19.53904 22.69396 25.84561 28.99508 32.14300

-0.18006 +0.16718 -0.15672 +0.14801 -0.14061

+0.14060 -0.13421 +0.12861 -0.12366 +0.11924

578 123 6bl 795 981

33.76102

+0.13730 -0.13132

+0.13421 -0.12862

46.33040

-0.12140 +0.11721

35.28979 38.43573 41.58101 44.72578 47.87012

-0.11527 +0.11167 -0.10838 +0.10537 -0.10260

369

49.47251 52.61455 55.75654 58.89850 62.04041

-0.11343 +0.10999 -0.10685 +0.10396 -0.10129

51.01413 54.15785 .57.30135 60.44464 63.58777

-0.11167

+O. 18006

049 535 405 057

Y’3(?/8,s) ‘-; ::;g

36.90356

40.04594 43.18822

-0.23246

+O. 12607

3.38424

DERIVATIVES

494 318 450 493 108

-0.19646

6.79381

10.02348 13.20999

'-; ::;;: +0:11527

tfl- :g:; +0:10260 -0.10004

+0:23232 -0.20650 +0.18771

Y4, s 5.64515 9.36162 12.73014 15.99963 19.22443

+0.28909 -0.24848 +0.21805 -0.19635 +0.18001

10.59718 14.03380 17.34709 20.60290

+0.25795 -0.23062 +0.20602 -0.18753 +0.17317

20.99728 24.16624 27.32880 30.48699 33. b4205

-0.17326 +0.16170 -0.15218 +0.14416 -0.13730

Z2.42481 t5.61027 ta.70509 31.95469 35.11853

-0.16716 +0.15671 -0.14800 +0.14060 -0.13421

23.82654 27.03013 30.22034 33.40111 36.57497

-0.16165 +0.15215 -0.14415 +0.13729 -0.13132

36.79479

+0.13132 -0.12607 +0.12140 -0.11721 +0.11343

38.27867

+0.12861 -0.12367 +0.11925 -0.11527 +0.11167

39.74363 42.90825 46.06968

49.22854 52.38531

+0.12606 -0.12140 +0.11721 -0.11343 +0.10999

57.19635 io.34513 i3.49320

i4.04673

-0.10838 +0.10537 -0.10260 +0.10003

55.54035 58.69393 61.84628 64.99759

-0.10685 +0.10396 -0.10129 +0.09882

ib. b4065

-0.09765

68.14799

-0.09652

4.52702 8.09755 11.39647 14.62308 17.81846

39.94577 43.09537 46.24387

49.39150 52.53840 55.68470 58.83049 61.97586 65.12086

?/Ii, s

-0.10999 +0.10685 -0.10396 +0.10129 -0.09882

I”e(wl, d)

Il. 43596

14.59102 17.74429 SO.89611

Y7, d

: 4 5 ; a 9 10

25.20621 28.42904 31.63488 34.82864 38.01347

-0.15664 +0.14796 -0.14058 +0.13419 -0.12860

it6.56676 ;!9.80953

41.19152 44.36427 47.53282 50.69796 53.86031

+O. 12366

:t 20

AND

Y’l (VI, ,“)

:: ;;;;: +0:24967 -0.21835 +0.19646

8.91961 L3.00771 .6.57392 L9.97434 t3.29397

:76

FUNCTIONS

+0.52079 -0.34032 +0.27146

+0.23429 -0.21591 +0.19571 -0.17975 +0.16703

:: 14 15

ORDER

2.19714 5.42968 8.59601 11.74915 14.89744

7.83774 11.81104 15.31362 18.67070 21.95829

11

INTEGER

57.02034 60.17842

63.33485 bb. 48986 69.64364

-0.11924 +0.11527 -0.11167 +0.10838 -0.10537 +0.10260 -0.10003 +0.09765 -0.09543

Y’4(2/4,8)

Y'i (T/7, s)

+0.21556 -0.20352 +0.18672 -0.17283 +0.16148

12.62391 15.80544

i2.15369 i5.32215 is.48767

LA. 65071 ti4.81164

17.97075 71.12830

$3, d

9.99463 14.19036 17.81789 21.26093 24.61258 27.91052 31.17370 34.41286 37.63465 40.84342

$3.03177 S6.23927 19.43579

118.98171

!J5, Y

6.74718

to.12138 -0.11720 +0.11342 -0.10999 tO.10684

44.04215 47.23298 50.41746 53.59675 56.77177

-0.10396 +0.10129 -0.09882 +0.09652 -0.09438

59.94319 63.11158 66.27738 69.44095 72.60259

Y’5(!/5,

Y'S(Y/s,S) 'Fj- gg;

+0:17aao -0.16662

+0.15643 -0.14785 +0.14051 -0.13415 +0.12857 -0.12364 +0.11923 -0.11526 +O. 11166

-0.10838 +0.10537 -0.10260 '-; g;: +0:09543 -0.09336

s)

BESSEL

FUNCTIONS

OF

INTEGER

411

ORDER Table

ZEROS

6 7 t 10 :: 13 :z :76 18 :'o

AND

ASSOCIATED

VALUES

OF

BESSEL

.Jo(j’o, s)

FUNCTIONS

AND

I’z, *

.JI(I”I, v)

j’n, 0.00000 3.83170 7.01558 10.17346 13.32369

x 00000 59702 66698 81351 19363

+1.00000 -0.40275 +0.30011 -0.24970 +0.21835

00000 93957 57525 48771 94072

1.84118 5.33144 8.53632 11.70600 14.86359

+0.58187 -0.34613

16.47063 19.61585 22.76008 25.90367 29.04682

00509 85105 43806 20876 85349

-0.19646 +0.18006 -0.16718 +0.15672 -0.14801

53715 33753 46005 49863 11100

32.18967 35.33230 38.47476 41.61709 44.75931

99110 75501 62348 42128 a9977

+0.14060 -0.13421 +0.12861 -0.12366 +0.11924

47.90146 51.04353 54.18555 57.32752 60.46945

08872 51836 36411 54379 78453

-0.11527 +0.11167 -0.10838 +0.10537 -0.10260

.JI( j’:;, ,)

THEIR

.i’l, 3

.I:( j’z, .v)

TJ-g::;; +0:20701

3.05424 6.70613 9.96947 13.17037 16.34752

+0.48650 -0.31353 +0.25474 -0.22088 +0.19794

18.01553 21.16437 24.31133 27.45705 30.60192

-0.18802 +0.17346 -0.16184 +0.15228 -0.14424

19.51291 22.67158 25.82604 28.97767 32.12733

-0.18101 +0.16784 -0.15720 +0.14836 -0.14088

57982 12403 66221 79608 98120

33.74618 36.88999 40.03344 43.17663 46.31960

+0.13736 -0.13137 +0.12611 -0.12143 +0.11724

35.27554 38.42265 41.56893 44.71455 47.85964

+0.13443 -0.12879 +0.12381 -0.11937 +0.11537

36941 04969 53489 40554 05671

49.46239 52.60504 55.74757 58.89000 62.03235

-0.11345 +0.11001 -0.10687 +0.10397 -0.10131

51.00430 54.14860 57.29260 60.43635 63.57989

-0.11176 +0.10846 -0.10544 +0.10266 -0.10008

.J,(j’,, <)

., J i, \ 6.41562 10.51986 13.98719 17.31284 20.57551

s

f:,, s

1 2 3 4 5

4.20119 8.01524 11.34592 14.58585 17.78875

to.43439 -0.29116 +0.24074 -0.21097 co.19042

5.31755 9.28240 12.68191 15.96411 19.19603

+0.39965 -0.27438 +0.22959 -0.20276 to.18403

20.97248 24.14490 27.31006 30.47027 33.62695

-0.17505 +0.16295 -0.15310 +0.14487 -0.13784

22.40103 25.58976 28.76784 31.93854 35.10392

-0.16988 +0.15866 -0.14945 +0.14171 -0.13509

23.80358 27.01031 30.20285 Z%l

-0.16533 +0.15482 -0.14616 +0.13885 -0.13256

36.78102 39.93311 43.08365 46.23297 49.38130

+0.13176 -0.12643 +0.12169 -0.11746 +0.11364

38.26532 41.42367 44.57962 47.73367 50.88616

+0.12932 -0.12425 +0.11973 -0.11568 +0.11202

39.73064 42.89627 46.05857 49.21817 52.37559

+_;-:::20: +0:11790 -0.11402 +0.11049

52.52882 55.67567 58.82195 61.96775 65.11315

-0.11017 +0.10700 -0.10409 +0.10141 -0.09893

54.03737 57.18752 60.33677

-0.10868 +0.10563 -0.10283 +0.10023 -0.09783

55.53120 58.68528 61.83809 64.98980 68.14057

-0.10728 +0.10434 -0.10163 +0.09912 -0.09678

7.50127 11.73494 15.26818 18.63744 21.93172

Jdi'B. 8) +0.35414 -0.25017 +0.21261 -0.18978 +0.17363

.1 i, * 8.57784 12.93239 16.52937 19.94185 23.26505

I;*:;;;;

+0:20588 -0.18449 +0.16929

.I s. * 9.64742 14.11552 17.77401 21.22906 24.58720

to.32438 -0.23303 +0.19998 -0.17979 +0.16539

25.18393 28.40978 31.61788 34.81339 37.99964

-0.16127 +0.15137 -0.14317 +0.13623 -0.13024

26.54503 29.79075 33.01518 36.22438 39.42227

-0.15762 +0.14823 -0.14044 +0.13381 -0.12808

27.88927 31.15533 34.39663 37.62008 40.83018

-0.15431 to.14537 -0.13792 +0.13158 -0.12608

41.17885 44.35258 47.52196 50.68782 53.85079

+0.12499 -0.12035 +0..11620 -0J1246 +0.10906

42.61152 45.79400 48.97107 52.14375 55.31282

+0.12305 -0.11859 '_"o-::g; +0:10771

44.03001 47.22176 50.40702 53.58700 56.76260

+0.12124 -0.11695 +0.11309 -0.10960 +0.10643

57.01138 60.16995 63.32681 66.48221 69.63635

-0.10596 +0..10311 -0.10049 +0.09805 -0-09579

58.47887 61.64239 64.80374 67.96324 71.12113

-0.10471 co.10195 -0.09940 +0.09704 -0.09484

59.93454 63.10340 66.26961 69.43356 72.59554

-0.10352 +0.10084 -0.09837 +0.09607 -0.09393

11 :'3 14 15 :; 18 :;:

s

6 7 i 10 11 :: 14 15 :t :t 20

.I J

6,

N

j'4,

\

6636'% .

,Ji(j’7.

4

9.5

DERIVATIVES

.fr,(j',, x) +0.37409 -0.26109 +0.22039 -0.19580 +0.17849

.JY(~‘K,s)

412

BESSEL Table ZEROS

FUNCTIONS

OF

INTEGER

ORDER

FUNCTIONS

AND

9.5 AND

ASSOCIATED

VALUES

OF

?/‘I, *

YI(?/‘I,8)

THEIR 9’2, 8

DERIVATIVES

Y2(?/'2,n)

711

3.68302 6.94150 10.12340 13.28576 16.44006

+0.41673 -0.30317 +0.25091 -0.21897 +0.19683

5.00258 8.35072 11.57420 14.76091 17.93129

';$;;g +0:23594 -0.20845 +0.18890

-0.18772 +0.17326 -0.16170 +0.15218 -0.14416

909 604 163 126 600

19.59024 22.73803 25.88431 29.02958 32.17412

-0.18030 +0.16735 -0.15684 +0.14810 -0.14067

21.09289 24.24923 27.40215 30.55271 33.70159

-0.17405 +0.16225 -0.15259 +0.14448 -0.13754

780 532 464 810 925

+0.13729 -0.13132 +0.12606 -0.12139 +0.11721

696 464 951 863 120

35.31813 38.46175 41.60507 44.74814 47.89101

+0.13427 -0.12866 +0.12370 -0.11928 +0.11530

36.84921 39.99589 43.14182 46.28716 49.43202

+0.13152 -0.12623 +0.12153 -0.11732 +0.11352

568 077 488 617 115

-0.11342 +0.10999 -0.10684 +0.10395 -0.10129

920 115 789 957 350

51.03373 54.17632 57.31880 60.46118 63.60349

-0.11169 +0.10840 -0.10539 +0.10261 -0.10005

52.57649 55.72063 58.86450 62.00814 65.15159

-0.11007 +0.10692 -0.10402 +0,10135 -0.09887

2.19714 5.42968 8.59600 11.74915 14.89744

133 104 587 483 213

+0.52078 -0.34031 +0.27145 -0.23246 +0.20654

641 805 988

18.04340 21.18806 24.33194 _..~ 21.41529 30.61828

228 893 257 498 649

33.76101 36.90355 40.04594 43.18821 46.33039 49.47250 52.61455 55.75654 58.89849 62.04041

Y’3,*

BESSEL

YO(Y’O,.J

Y’O, 8

8)

171

s

~44(!i4,8)

6.25363 9.69879 12.97241 16.19045 19.38239

+0.33660 -0.26195 +0.22428 -0.19987 +0.18223

7.46492 11.00517 14.33172 11.58444 20.80106

to.31432 -0.24851 to.21481 -0.19267 to.17651

?I $8 8.64956 12.28087 15.66080 18.94974 22.19284

+0.29718 -0.23763 +0.20687 -0.18650 to.17151

22.55979 25.72821 28.89068 32.04898 35.20427

-0.16867 +0.15779 -0.14881 +0.14122 -0.13470

23.99700 27.17989 30.35396 33.52180 36.68505

-0.16397 to.15384 -0.14543 ~0.13828 -0.13211

25.40907 28.60804 31.79520 34.97389 38.14631

-0.15980 +0.15030 -0.14236 +0.13559 -0.12973

38.35728 41.50855 44.65845 47.80725 50.95515

+0.12901 -0.12399 +0.11952 -0.11550 +0.11186

39.84483 43.00191 46.15686 49.31009 52.46191

to.12671 -0.12193 :"o*:::;; +0:11031

41.31392 44.47779 47.63867 50.79713 53.95360

+0.12458 -0.12001 +0.11591 -0.11221 +0,10885

54.10232 57.24887 60.39491 63.54050 66.68571

-0.10855 +0.10552 -0.10273 +0.10015 -0.09775

55.61257 58.76225 61.91110 65.05925 68.20679

-0.10712 to.10420 -0.10151 +0.09901 -0.09669

57.10841 60.26183 63.41407 66.56530 69.71565

-0.10578 +0.10295 -0.10035 +0.09793 -0.09568

Y:&‘a,

lF(!/‘O. J

!/‘4,

Y:,(w’s,

8)

9.81480 13.53281 16.96553 20.29129 23.56186

+0.28339 -0.22854 +0.20007 -0.18111 +0.16708

10.96515 14.76569 18.25012 21.61275 24.91131

I'i(?/'i, v) +0.27194 -0.22077 +0.19414 -0.17634 +0.16311

!I s.s 12.10364 15.98284 19.51773 22.91696 26.24370

YR(i8, R) +0.26220 -0.21402 +0.18891 -0.17207 +0.15953

26.79950 30.01567 33.21697 36.40752 39.59002

-0.15607 +0.14709 -0.13957 +0.13313 -0.12753

28.i7105 31.40518 34.62140 37.82455 41.01785

-0.15269 +0.14417 -0.13700 +0.13085 -0.12549

29.52596 32.77857 36.01026 39.22658 42.43122

-0.14962 +0.14149 -0.13463 +0.12874 -0.12359

42.16632

45.93775 49.10528 52.26963 55.43136

+0.12260 -0.11822 +0.11428 -0.11072 +0.10748

44.20351 47.38314 50.55791 56.89619

'-;*::;;; +0:11275 -0.10931 +0.10618

45.62678 48.81512 51.99761 55.17529 58.34899

+0.11904 -0.11497 +0.11131 -0.10798 +0.10494

58.59089 61.74857 64.90468 68.05943 71.21301

-0.10451 +0.10177 -0.09925 +0;09690 -0.09471

60.06092 63.22331 66.38370 69.54237 72.69955

-0.10330 +0.10065 -0.09820 +0.09592 -0.09379

61.51933 64.68681 67.85185 71.01478 74.17587

-0.10216 +0.09958 -0.09720

Y’O, *

?/‘7.

1

53.72810

+;.;;g .

BESSEL

FUNCTIONS

BESSEL

INTEGER

ORDER

Table

FIJSCTIONS--.luO’U,,x)

9.6

Jd &?I

JoG,.:~~

Jo(jo,J)

Jo(jo,52)

0.99942 0.99769 0.99480 0.99077

1.00000 0.99696 0.98785 0.97276 0.95184

1.00000 0.99253 0.97027 0.93373 0.88372

1.00000 0.98614 0.94515 0.87872 0.78961

1.00000 0.97783 0.91280 0.80920 0.67388

0.10 0.12 0.14 0.16 0.18

0.98559 0.97929 0.97186 0.96333 0.95370

0.92526 0.89328 0.85617 0.81429 0.76800

0.82136 0.74804 0.66537 0.57518 0.47943

0.68146 0.55871 0.42632 0.28958 0.15386

0.51568 0.34481 0.17211 .tO.O0827 -0.13693

0.20 0.22 0.24 0.26 0.28

0.94300 0.93124 0.91844 0.90463 0.88982

0.71773 0.66392 0.60706 0.54766 0.48623

0.38020 0.27960 0.17976 +0.08277 -0.00942

+0.02438 -0.09404 -0.19716 -0.28155 -0.34466

-0.25533 -0.34090 -0.39013 -0.40225 -0.37917

0. 30 0.32 0. 34 0.36 0.38

0.87405 0.85734 0.83972 0.82122 0.80187

0.42333 0.35950 0.29529 0.23126 0.16795

-0.09498 -0.17226 -0.23986 -0.29664 -0.34171

-0.38498 -0.40207 -0.39653 -0.36998 -0.32493

-0.32527 -0.24698 -0.15223 -0.04980 +0.05137

0.40 0.42 0.44 0.46 0.48

0.78171 0.76077 0.73908 0.71669 0.69362

0.10590 +0.04562 -0.01240 -0.06769 -0.11983

-0.37453 -0.39482 -0.40264 -0.39835 -0.38259

-0.26467 -0.19304 -0.11431 -0.03289 +0.04684

0.14293 0.21767 0.27011 0.29684 0.29671

0.50 0.52 0. 54 0.56 0.58

0.66993 0.64565 0.62081 0.59547 0.56967

-0.16840 -0.21306 -0.25349 -0.28941 -0.32062

-0.35628 -0.32056 -0.27678 -0.22648 -0.17130

0.12078 0.18527 0.23725 0.27445 0.29541

0.27086 0.22252 0.15667 +0.07960 -0.00168

0.60 0. 62 0.64 0.66 0.68

0.54345 0.51685 0.48992 0.46270 0.43524

-0.34692 -0.36821 -0.38441 -0.39551 -0.40152

-0.11295 -0.05320 +0.00622 0.06363 0.11745

0.29959 0.28731 0.25977 0.21892 0.16735

-0.08007 -0.14891 -0.20259 -0.23697 -0.24965

0. 70 0.72 0. 74 0. 76 0. 78

0.40758 0.37977 0.35186 0.32389 0.29591

-0.40255 -0.39871 -0.39019 -0.37721 -0.36003

0.16625 0.20878 0.24399 0.27107 0.28945

0.10814 +0.04470 -0.01945 -0.08082 -0.13618

-0.24019 -0.21003 -0.16237 -0.10179 -0.03389

0.80 0. 82 0.84 0.86 0.88

0.26796 0.24009 0.21234 0.18476 0.15739

-0.33896 -0.31433 -0.28652 -0.25591 -0.22293

0.29882 0.29915 0.29063 0.27374 0.24914

-0.18270 -0.21808 -0.24067 -0.24957 -0.24461

+0.03525 0.09960 0.15369 0.19306 0.21464

0.90 0. 92 0.94 0.96 0.98

0.13027 0.10346 0.07698 0.05089 0.02521

-0.18800 -0.15157 -0.11411 -0.07605 -0.03787

0.21774 0.18059 0.13891 0.09399 0.04722

-0.22637 -0.1961: -0.15580 -0.10779 -0.05486

0.21694 0.20021 0.16630 0.11854 0.06138

1.00

0.00000

X 0. 00

0.02 0.04 0.06 0.08

From Math.

Jo(jo,,x) 1.00000

OF

0.00000 C-i)1 [(-fjP 1 [ 1 E. T. Goodwin and J. Staton,

0.00000

.

[C-y]

Table of .Io(,&,r), 1, 220-224 (1948) (with pernksion).

0.00000 (-;I5 _ _ Quart. J. Mech. Appl.

0.00000

['$'"I

[

1

414

BESSEL FUNCTIONS OF INTEGER ORDER Table

9.7

BESSEL

FUSCTIONS-MISCELLANEOUS

ZEROS

$11Zero ofdl (x)-XJo(5) X\S 0. 00 0. 02 0.04 0. 06 0. 08 0.10

1 0.0000 0.1995 0.2814 0.3438 0.3960 0.4417

3.8;17 3.8369 3.8421 3.8473 3.8525 3.8577

7.03156 7.0184 7.0213 7.0241 7.0270 7.0298

10.:735 10.1754 10.1774 10.1794 10.1813 10.1833

13.;237 13.3252 13.3267 13.3282 13.3297 13.3312

0.20 0.40 0. 60 0.80 1.00

0.6170 0.8516 1.0184 1.1490 1.2558

3.8835 3.9344 3.9841 4.0325 4.0795

7.0440 7.0723 7.1004 7.1282 7.1558

10.1931 10.2127 10.2322 10.2516 10.2710

13.3387 13.3537 13.3686 13.3835 13.3984

A- ‘\s

1. 00 0. 80 0. 60 0. 40 0.20

1 1.2558 1.3659 1.5095 1.7060 1.9898

2 4.0795 4.1361 4.2249 4.3818 4.7131

7.15358 7.1898 7.2453

lo.??710 10.2950 10.3346

13.33984 13.4169 13.4476

1

7.3508 7.6177

10.4118 10.6223

13.6786 13.5079

;

0.10 0.08 0.06 0. 04 0. 02 0. 00

2.1795 2.2218 2.2656 2.3108 2.3572 2.4048

5.0332 5.1172 5.2085 5.3068 5.4112 5.5201

7.9569 8.0624 8.1852 8.3262 8.4840 8.6537

10.9363 11.0477 11.1864 11.3575 11.5621 11.7915

13.9580 14.0666 14.2100 14.3996 14.6433 14.9309

21

:; :; 50 01

stll Zero of J~(~)--krJo&) 1 0.0000 1.1231 1.4417 1.6275 1.7517 1.8412

2 5.1356 5.2008 5.2476 5.2826 5.3098 5.3314

3 8.4172 8.4569 8.4853 8.5066 8.5231 8.5363

4 11.6198 11.6486 11.6691 11.6845 11.6964 11.7060

1. 00 0. 80 0.60 0.40 0.20

1 1.8412 1.9844 2.1092 2.2192 2.3171

2 5.3314 5.3702 5.4085 5.4463 5.4835

0.10 0.08 0. 06 0. 04 0.02 0. 00

2.3621 2.3709 2.3795 2.3880 2.3965 2.4048

5.5019 8.6421 11.7830 5.5055 8.6445 11.7847 5.5092 8.6468 11.7864 5.5128 8.6491 11.7881 5.5165 8.6514 11.7898 11.7915 5.5201 8.6537 =nearestintegerto A.

Xk’\S

3 8.5363 8.5600 8.5836 8.6072 8.6305

4 11.7060 11.7232 11.7404 11.7575 11.7745

5 14.7960 14.8185 14.8346 14.8467 14.8561 14.8636

14.8:36 14.8771 14.8906 14I9041 14.9175 14.9242 14.9256 14.9269 14.9282 14.9296 14.9309

1 21 I 5 10 13 17 25 50 co

Compiled from H. S. Carslaw and J. C. Jaeger, Conduction of heat in solids (Oxford Univ. Press, London, England, 1947) and British Association for the Advancement of Science, Bessel functions, Part I. Functions of orders zero and unity, Mathematical Tables, vol.VI (Cambridge Univ. Press, Cambridge, England, 1950)(with permission).

BESSEL BESSEL

0.80 0.60 0.40 0.20

1 12.55847 4.69706 2.07322 0.76319

031 410 886 127

0.10 0.08 0.06 0.04 0.02 0. 00

0.33139 0.25732 0.18699 0.12038 0.05768 0.00000

387 649 458 637 450 000

A- ‘\S *

FUNCTIONS.

OF

INTEGER

415

ORDER

FUNCTIONS-MISCELLANEOUS

ZEROS

stll Zero of Jo(z)Y&z)Y&)Jo(xZ) 2 3 4 25.12877 37.69646 50.26349 9.41690 14.13189 18.84558 4.17730 6.27537 8.37167 1.55710 2.34641 3.13403 0.68576 0.53485 0.39079 0.25340 0.12272 0.00000

1.03774 0.81055 0.59334 8*EZ 0: 00000

1.38864 '0: pa; 0:51759 0.25214 0.00000

Table

9.7

5 62.83026 23.55876 10.46723 3.92084



1.73896 1.35969 0.99673 0.64923 0.31666 0.00000

10

:

3 5

*

:: 25 50 00

Stll Zero of JI (2) Yl (kr) - Yl(4 Jl (u) *

A-‘\S 0. 80 0.60 0.40 0.20

1 12.59004 4.75805 2.15647 0.84714

151 426 249 961

2 25.14465 9.44837 4.22309 1.61108

37.7:706 14.15300 6.30658 2.38532

4 50.27145 18.86146 8.39528 3.16421

5 62.83662 23.57148 10.48619 3.94541

0.10 0. 08 0. 06 0.04 0.02 0. 00

0.39409 0.31223 0.23235 0.15400 0.07672 0.00000

416 576 256 729 788 000

0.73306 0.57816 0.42843 0.28296 0.14062 0.00000

1.07483 0.84552 0.62483 0.41157 0.20409 0.00000

1.41886 1.11441 0.82207 0.54044 0.26752 0.00000

1.76433 1.38440 1.02001 0.66961 0.33097 0.00000

4 44.02544 16.53413 7.36856 2.78326

5 56.58224 21.23751 9.45462 3.56157

* 00

8th Zero ofJ1(z)Yo(Ar)-Yl(z)Jo(b) * *

A-‘\s 0. 80 0. 60 0.40 0.20

1 6.56973 2.60328 li24266 0.51472

310 1% 626 663

2 18.94971 7.16213 3.22655 1.24657

3 31.47626 11.83783 5.28885 2.00959

0.10 0.08 0. 06 0.04 0.02 0. 00

0.24481 0.19461 0.14523 0.09647 0.04813 0.00000

004 772 798 602 209 000

0.57258 0.45251 0.33597 0.22226 0.11059 0.00000

0.90956 0.71635 0.53005 0.34957 0.17353 0.00000

1.25099 0.98327 0.72594 0.47768 t* 203088:

1.59489 1125203 0.92301 0.60634 0.29991 0.00000



1 3 5 :30 :sT 50 00

<X> =nearest integer to A. ’ Compiled from British Association for the Advancement of Science, Bessel functions, Part I. Functions of orders zero and unity, Mathematical Tables, vol.VI (Cambridge Univ. Press, Cambridge, England, 1950) (with permission). *see page II.

*

416

BESSEL Table

9.8

FUNCTIONS

MODIFIED

BESSEL

OF

INTEGER

FUNCTIONS-ORDERS

ORDER 0, 1 AND

2

s-do(x) 1.00000 00000 0.90710 09258 0.82693 85516 0.75758 06252 0.69740 21705

e-d1 0.00000 0.04529 0.08228 0.11237 0.13676

(cc) 00000 84468 31235 75606 32243

2-212 0.12500 0.12510 0.12541 0.12594 0.12667

(z) 00000 41992 71878 01407 50222

0.64503 0.59932 0.55930 0.52414 0.49316

52706 72031 55265 89420 29662

0.15642 0.17216 0.18466 0.19449 0.20211

08032 44195 99828 86933 65309

0.12762 0.12879 0.13018 0.13180 0.13365

45967 24416 29658 14318 39819

0.46575 0.44144 0.41978 0.40042 0.38306

96077 03776 20789 49127 25154

0.20791 0.21220 0.21525 0.21729 0.21850

04154 16132 68594 75878 75923

0.13574 0.13809 0.14069 0.14356 0.14670

76698 04952 14455 05405 88837

0.36743 0.35331 0.34051 0.32887 0.31824

36090 49978 56880 19497 31629

0.21903 0.21901 0.21855 0.21772 0.21661

93874 94899 28066 62788 19112

0.15014 0.15389 0.15795 0.16235 0.16711

87192 34944 79288 80900 14772

0.30850 0.29956 0.29131 0.28369 0;27662

83225 30945 73331 29857 23231

0.21526 0.21374 0.21208 0.21032 0.20848

92892 76721 77328 30051 10887

0.17223 0.17775 0.18368 0.19006 0.19690

71119 56370 94251 26964 16460

0.27004 0.26391 0.25818 0.25280 0.24775

64416 39957 01238 55337 57304

0.20658 0.20465 0.20269 0.20073 0.19877

46495 22544 90640 74113 72816

0.20423 0;2120< 0.22050 0.22951 0.23915

45837 20841 71509 53938 52213

0.24300 0.23851 0.23426 0.23024 0.22643

03542 26187 88316 79845 14011

0.19682 0.19489 0.19297 0.19109 0.18922

67133 21309 86229 01727 98511

0.24946 0.26049 0.27229 0.28490 0.29839

80490 85252 47757 86686 61010

0.22280 0.21934 0.21604 0.21290 0,20988

24380 62245 94417 01308 75279

0.18739 0.18560 0.18383 0.18210 0.18041

99766 22484 78580 75810 18543

0.31281 0.32823 0.34472 0.36235 0.38121

73100 72078 57467 83128 61528

0.20700 0.20423 0.20157 0.19902 0.19656

19211 45274 73840 32571 55589

0.17875 0.17712 0.17553 0.17397 0.17245

08394 44763 25260 46091 02337

0.40138 0.42296 0.44605 0.47075 0.49719

68359 47539 16629 72701 98689

0.19419 0.19191 0.18971 0.18758 0.18552

82777 59151 34330 62042 99721

0.17095 0.16949 0.16807 0.16667 0.16530

88223 97311 22681 57058 92936

0.52550 0.55581 0.58827 0.62304 0.66030

70272 63319 61978 67409 07270

0.18354

08126

0.16397 22669 c-y [. I

0.70022

45988

[ 1 (6$3

In+1 (x) = - -” In (z) +I+ 1(cc) Compiled from British Association for the Advancement of Science, Bessel functions, Part I.Functions of orders zero and unity, Mathematical Tables, vol. VI , Part II. Functions of positive integer order, Mathematical Tables, vol. X(Cambridge Univ. Press, Cambridge, England, 1950,1952)and L. Fox, A short table for Bessel functions of integer orders and large arguments. Royal Society Shorter Mathematical Tables No. 3 (Cambridge Univ. Press, Cambridge, England, 1954) (with permission).

BESSEL

MODIFIED

070 i:: 0”:; 8-Z 0: 7 0":: 1.0 11.: 1:3 1. 4 E 11*: 1: 9 f:! z 214

BESSEL

FUNCTIONS

OF

INTEGER

FUNCTIONS-ORDERS

@Ko(x> 2.682m3261023 2.14075 73233 1.85262 73007 1.66268 20891

417

ORDER

0, 1 AND

exK1(z)

2

Table

9.8

x2 K2 (x)

10.89%8 5.83338 4.12515 3.25867

2683 6037 7762 3880

2.00000 1.99503 1.98049 1.95711 1.92580

0000 9646 7172 6625 8202

1.52410 1.41673 1.33012 1.25820 1.19716

93857 76214 36562 31216 33803

2.73100 2.37392 2.11501 1.91793 1.76238

97082 00376 13128 02990 82197

1.88754 1.84330 1.79405 1.74067 1.68401

5888 9881 1681 2762 1992

1.14446 1.09833 1.05748 1.02097 0.98806

30797 02828 45322 31613 99961

1.63615 1.53140 1.44289 1.36698 1.30105

34863 37541 75522 72841 37400

1.62483 1.56385 1.50167 1.43886 1.37590

8899 0953 3576 2011 4446

0.95821 0.93094 0.90591 0.88283 0.86145

00533 59808 81386 35270 06168

1.24316 1.19186 1.14603 li10480 1.06747

58736 75654 92462 53726 09298

1.31322 1.25119 1.19011 1.13026 1.07184

5917 2681 6819 0897 2567

0.84156 82151 Or82301 71525 0.80565 39812 0.78935 61312 0.77401 81407

1.03347 1.00236 0.97377 0.94737 0.92291

68471 80527 01679 22250 36650

1.01503 0.95999 0.90680 0.85556 0.80633

9018 1226 7952 9487 1113

0.75954 0.74586 0.73290 0.72060 0.70890

86903 82430 71515 41251 49774

0.90017 0.87896 0.85913 0.84053 0.82304

44239 72806 18867 00604 20403

0.75912 0.71396 0.67085 0.62977 0.59070

6289 9565 9227 9698 3688

0.69776 15980 0: 68713 11010 0.67697 51139 0.66725 91831 0.65795 22725

0.80656 0.79100 0.77628 0.76232 0.74907

34800 30157 02824 42864 20613

0.55359 0.51840 0.48508 0.45358 0.42382

4126 5885 7306 1550 7789

0.64902 0.64045 0.63221 0.62429 0.61665

63377 59647 80591 15812 73147

0.73646 0.72446 0.71300 0.70206 0.69159

75480 06608 65010 46931 88206

0.39576 0.36931 0.34443 0.32103 0.29905

2241 9074 1194 0914 0529

0.60929 0.60219 0.59533 0.58871 0.58230

76693 65064 89889 14486 12704

0.68157 0.67196 0.66274 0.65387 0.64535

59452 61952 24110 98395 58689

0.27842 0.25908 0.24096 0.22399 0.20813

2808 1398 1165 8474 1411

0.57609 67897 Oi57008 72022 0.56426 24840 0.55861 33194 0.55313 10397

0.63714 0.62924 0.62161 0.61425 0.60714

97988 26383 69312 66003 68131

0.19329 0.17944 0.16651 0.15445 0.14320

9963 6150 4127 0249 3117

0.54780 75643

0.60027 38587

0.13272 3593

418

BESSEL

Table 9.8

MODIFIED

FUNCTIONS

BESSEL

OF

INTEGER

ORDER

FUNCTIONS-ORDERS

0, 1 AND

2

5. 0 5.1 5.2 5.3 5. 4

0.18354 0.18l.61 0.17974 0.17794 0.17618

08126 51021 94883 08646 63475

0.16397 0.16266 0.16138 0.16012 0.15890

22669 38546 32850 97913 26150

e-zZz(x) 0.11795 1906 0.11782 5355 0.11767 8994 0.11751 4528 0.11733 3527

z-2 5: 7 5.8 5. 9

0.17448 0.17282 0.17122 0.16965 0.16813

32564 90951 15362 84061 76726

0.15770 0.15652 0.15537 0.15424 0.15313

10090 42405 15922 23641 58742

0.11713 0.11692 0.11670 0.11647 0.11622

7435 7581 5188 1384 7207

0.16665 0.16521 0.16381 0.16244 0.16110

74327 59021 14064 23718 73175

0.15205 0.15098 0.14994 0.14892 0.14792

14593 84754 62978 43212 19595

0.11597 0.11571 0.11544 0.11516 0.11488

3613 1484 1633 4809 1705

0.15980 0.15853 0.15729 0.15608 0.15489

48490 36513 24831 01720 56090

0.14693 0.14597 0.14502 0.14409 0.14318

86457 38314 69866 75991 51745

0.11459 0.11429 0.11400 0.11369 0.11339

2958 9157 0845 8525 2660

0.15373 0.15260 0.15149 0.15041 0.14935

77447 55844 81855 46530 41371

0.14228 0.14140 0.14054 0.13969 0.13886

92347 93186 49809 57915 13353

0.11308 0.11277 0.11245 0.11214 0.11182

3678 1974 7913 1833 4046

0.14831 0.14729 0.14630 0.14532 0.14436

58301 89636 28062 66611 98642

0.13804 0.13723 0.13644 0.13566 0.13489

12115 50333 24270 30318 64995

0.11150 0.11118 0.11086 0.11054 0.11021

4840 4481 3215 1268 8852

0.14343 0.14251 0.14160 0.14072 0.13985

17818 18095 93695 39098 49027

0.13414 0.13340 0.13267 0.13195 0.13124

24933 06883 07705 24362 53923

0.10989 0.10957 0.10925 0.10892 0.10860

6158 3368 0645 8142 6000

0.13900 0.13816 0.13734 0.13653 0.13573

18430 42474 16526 36147 97082

0.13054 0.12986 0.12918 0.12852 0.12786

93551 40505 92134 45873 99242

0.10828 0.10796 0.10764 0.10732 0.10700

4348 3305 2983 3481 4894

0.13495 0.13419 0.13343 0.13269 0.13196

95247 26720 87740 74691 84094

0.12722 0.12658 0.12596 0.12534 0.12473

49839 95342 33501 62139 79145

0.10668 0.10637 0.10605 0.10574 0.10542

7306 0796 5437 1294 8428

0.13125 0.13054 0.12985 0.12916 0.12849

12609 57016 14223 81248 55220

0.12413 0.12354 0.12296 0.12238 0.12182

82477 70154 40258 90929 20364

0.10511 0.10480 0.10449 oIio419 0.10388

6893 6740 8015 0759 5010

33371 ( -p c 1

0.12126

26814

0.10358

0801

2

6. 0 2: 2: 6. 5 2:: 66:; 7.0 7.1 7. 2 ;:i

7.5 ;*7" 7:8 7. 9 88:: E 8: 4

e-xZo(x)

0.12783

e-xZl(x)

[ 1 C--W

5

[ 1 92

BESSEL

MODIFIED

BESSEL

FTJNCTIONS

OF

INTEGER

FUNCTIONS-ORDERS

419

ORDER

0, 1 AND

2

Table 9.8

0.54780 0.54263 0.53760 0.53271 0.52795

75643 53519 73540 69744 80329

0.60027 0.59362 0.58718 0.58095 0.57490

38587 50463 86062 36085 98871

e=Kz (z) 0.78791 711 0.77542 949 0.76344 913 0.75194 475 0.74088 762

0.52332 0.51881 0.51441 0.51012 0.50594

47316 16252 35938 58183 37583

0.56904 0.56335 0.55783 0.55246 0.54725

79741 90393 48348 76495 02639

0.73025 0.72001 0.71014 0.70063 0.69145

127 128 511 190 232

0.50186 0.49787 0.49399 0.49019 0.48647

31309 98929 02237 05093 73291

0.54217 0.53723 0.53243 0.52774 0.52318

59104 82386 12833 94344 74101

0.68258 0.67402 0.66574 0.65773 0.64997

843 358 225 001 339

0.48284 0.47929 0.47582 0.47242 0.46910

74413 77729 54066 75723 16370

0.51874 0.51440 0.51017 0.50604 0.50200

02336 32108 19097 21421 99471

0.64245 0.63517 0.62811 0.62126 0.61461

982 753 553 350 177

0.46584 0.46265 0.45953 0.45646 0.45346

50959 55657 07756 85618 68594

0.49807 0.49422 0.49046 0.48678 0.48318

15749 34737 22755 47842 79648

0.60815 0.60187 0.59577 0.58983 0.58405

126 345 030 426 820

0.45052 0.44763 0.44480 0.44202 0.43930

36991 71996 55636 70724 00819

0.47966 0.47622 0.47285 0.46955 0.46631

89336 49486 33995 18010 77847

0.57843 0.57295 0.56762 0.56242 0.55735

541 955 463 497 522

0.43662 0.43399 0.43141 0.42887 0.42638

30185 43754 27084 66329 48214

0.46314 0.46004 0.45699 0.45401 0.45108

90928 35709 91615 39001 59089

0.55241 0.54758 0.54287 0.53827 0.53378

029 538 592 757 623

0.42393 0.42152 0.41916 0.41683 0.41454

59993 89433 24781 54743 68462

0.44821 0.44539 0.44262 0.43991 0.43724

33915 46295 79775 18594 47648

0.52939 0.52510 0.52091 0.51681 0.51280

797 909 604 544 410

0.41229 0.41008 0.40790 0.40575 0.40364

55493 05783 09662 57809 41245

0.43462 0.43205 0.42952 0.42703 0.42459

52454 19116 34301 85204 59520

0.50887 0.50503 0.50127 0.49759 0.49398

894 704 562 202 369

xi 9: 9

0.40156 0.39951 0.39750 0.39551 0.39355

51322 79693 18313 59416 95506

0.42219 0.41983 0.41751 0.41522 0.41297

45430 31565 06989 61179 84003

0.49044 0.48698 0.48358 0.48025 0.47698

819 321 651 597 953

10. 0

0.39163

19344

0.41076 65704 c-t)3

0.47378

525

5”o 55:: 25 5.5 E 5: 8 5. 9 6. 0 2: 6: 3 6. 4 2: 2; 6: 9 ;-1" 7: 2 77:: ::2 x3 7:9 t:: Ez 8: 4 E EG 8: 9 9. 0 8:: '9:: E

eZKo(z)

[I 1 c-y

eZKl(2)

[

1

[

C-t)6

1

420

BESSEL

Table 9.8

MODIFIED

FUNCTIONS

BESSEL

OF

INTEGER

ORDER

FUNCTIONS-ORDERS ecxZ1(x)

e-“Zo(x)

0, 1 AND

2

e-zZ2(X)

lo” 0 10:2 10. 4 10.6 10.8

0.12783 0.12653 0.12528 0.12406 0.12288

33371 91639 35822 47082 07840

0.12126 0.12016 0.11909 0.11805 0.11704

26814 64024 89584 91273 57564

0.10358 0.10297 0.10237 0.10178 0.10120

0801 7124 9936 9401 5644

11. 0 11. 2 11.4 11.6 11. 8

0.12173 0.12061 0.11952 0.11846 0.11743

01682 13250 28165 32942 14923

0.11605 0.11509 0.11415 0.11323 0.11233

77582 41055 38276 60059 97710

0.10062 0.10005 0.09949 0.09893 0.09839

8758 8806 5829 9845 0853

12.0 12.2 12. 4 12.6 12.8

0.11642 0.11544 0.11449 0.11355 0.11264

62212 63616 08594 87206 90074

0.11146 0.11060 0.10977 0.10895 0.10815

42993 88096 25611 48501 50080

0.09784 0.09731 0.09678 0.09626 0.09574

8838 3770 5608 4300 9787

13. 0 13.2 13.4 13. 6 13.8

0.11176 0.11089 0.11004 0.10921 0.10840

08338 33621 57995 73954 74378

0.10737 0.10660 0.10585 0.10512 0.10440

23993 64190 64916 20685 26267

0.09524 0.09474 0.09424 0.09375 0.09327

2003 0874 6323 8268 6622

14. 0 14.2 14. 4 14.6 14.8

0.10761 0.10684 0.10608 0.10533 0.10461

52517 01959 16613 90688 18671

0.10369 0.10300 0.10232 0.10166 0.10101

76675 67148 93142 50311 34506

0.09280 0.09233 0.09186 0.09141 0.09096

1299 2208 9257 2352 1401

15.0 15.2 15.4 15.6 15.8

0.10389 0.10320 0.10251 0.10184 0.10118

95314 15618 74813 68351 91887

0.10037 0.09974 0.09913 0.09852 0.09793

41751 68245 10348 64572 27574

0.09051 0.09007 0.08964 0.08921 0.08879

6308 6980 3321 5238 2637

16. 0 16. 2 16.4 16. 6 16.8

0.10054 0.09991 0.09929 0.09868 0.09808

41273 12544 01906 05729 20539

0.09734 0.09677 0.09621 0.09566 0.09511

96147 67216 37828 05145 66444

0.08837 0.08796 0.08755 0.08715 0.08675

5426 3511 6802 5210 8644

17. 0 17.2 17: 4 17. 6 17. 8

0.09749 0.09691 0.09634 0.09579 0.09524

43005 69938 98277 25085 47546

0.09458 0.09405 0.09353 0.09303 0.09252

19107 60614 88542 00560 94423

0.08636 0.08598 0.08559 0.08522 0.08484

7017 0242 8235 0911 8188

18. 0 18.2 18. 4 18. 6 18.8

0.09470 0.09417 0.09365 0.09314 0.09264

62952 68703 62299 41336 03503

0.09203 0.09155 0.09107 0.09060 0.09014

67968 19113 45848 46237 18411

0.08447 0.08411 0.08375 0.08340 0.08305

9984 6221 6819 1701 0793

19. 0 19.2 19.4 19. 6 19. 8

0.09214 0.09165 0.09117 0.09070 0.09023

46572 68400 66923 40151 86167

0.08968 0.08923 0.08879 0.08835 0.08792

60569 70968 47929 89829 95099

0.08270 0.08236 0.08202 0.08168 0.08135

4020 1309 2590 7792 6848

20. 0

0.08978 03119 c-p5

0.08750 62222 C-l'4 [ 1

0.08102

[ 1

9690 t-y [ 1

BESSEL

MODIFIED

BESSEL

X

FUNCTIONS

OF

INTEGER

FUNCTIONS-ORDERS

0, 1 AND e=Ki(x)

eZKo(x)

421

ORDER

2

Table 9.8 eZKz(x)

10.0 10.2 10.4 10.6 10. 8

0.39163 0.38786 0.38419 0.38063 0.37716

19344 02539 55846 29549 77125

0.41076 0.40644 0.40225 0.39819 0.39425

65704 68479 98277 88825 78391

0.47378 0.46755 0.46155 0.45576 0.45017

525 571 324 482 842

11.0 11. 2 11.4 11. 6 11. 8

0.37379 0.37051 0.36731 0.36419 0.36115

54971 22156 40243 73076 86616

0.39043 0.38671 0.38309 0.37958 0.37616

09362 27920 83725 29618 21391

0.44478 0.43956 0.43452 0.42964 0.42491

294 807 427 265 496

12. 0 12.2 12.4 12.6 12.8

0.35819 0.35530 0.35247 0.34972 0.34703

48784 29318 99643 32746 03081

0.37283 0.36958 0.36642 0.36334 0.36033

17534 79032 69191 53438 99192

0.42033 0.41589 0.41158 0.40739 0.40333

350 111 108 714 342

13.0 13.2 13.4 13.6 13.8

0.34439 0.34182 0.33931 0.33684 0.33444

86455 59943 01806 91405 09142

0.35740 0.35454 0.35175 0.34902 0.34635

75702 53922 06397 07143 31558

0.39938 0.39554 0.39181 0.38817 0.38463

443 499 028 572 702

14. 0 14.2 14.4 14. 6 14. 8

0.33208 0.32977 0.32751 0.32530 0.32312

36383 55402 49332 02091 98364

0.34374 0.34119 0.33870 0.33626 0.33387

56322 59314 19539 17039 32858

0.38119 0.37783 0.37455 0.37136 0.36824

016 131 687 346 785

15.0 15.2 15.4 15.6 15.8

0.32100 0.31891 0.31687 0.31486 0.31289

23534 63655 05405 36051 43424

0.33153 0.32924 0.32700 0.32480 0.32264

48949 48132 14043 31080 84361

0.36520 0.36223 0.35933 0.35650 0.35373

701 805 826 503 592

16. 0 16.2 16. 4 16. 6 16.8

0.31096 0.30906 0.30720 0.30537 0.30357

15880 42269 11919 14592 40487

0.32053 0.31846 0.31643 0.31443 0.31248

59682 43471 22766 85164 18807

0.35102 0.34838 0.34579 0.34325 0.34077

858 081 049 562 427

17.0 17.2 17.4 17. 6 17. 8

0.30180 0.30007 0.29836 0.29668 0.29504

80193 24678 65276 93657 01817

0.31056 0.30867 0.30682 0.30500 0.30321

12340 54888 36027 45765 74518

0.33834 0.33596 0.33363 0.33134 0.32910

464 497 361 898 956

18. 0 18.2 18.4 18. 6 18.8

0.29341 0.29182 0.29025 0.28870 0.28718

82062 26987 29472 82654 79933

0.30146 0.29973 0.29803 0.29637 0.29472

13089 52642 84697 01096 94003

0.32691 0.32476 0.32264 0.32057 0.31854

391 064

843

19. 0 19. 2 19.4 19.6 19.8

0.28569 0.28421 0.28276 0.28133 0.27993

14944 81554 73848 86117 12862

0.29311 0.29152 0.28996 0.28842 0.28691

55877 79458 57766 84068 51886

0.31654 0.31458 0.31266 0.31077 0.30891

577 565 076 008 262

20.0

0.27854

48766

0.28542

I:(-y 1

[(-p1 54970

602 218

0.30708 743 C-$)3

c 1

422

BESSEL

Table

MODIFIED

9.8

BESSEL

FUNCTIONS

OF

INTEGER

FUNCTIONS-AUXILIARY

ORDER

TABLE

FOR

LARGE

T--lde~Ko(x)

ARGUMENTS

a-lx*e~Kl

(x)

<x> :1” ::

R- ~xbKiz(x)

0.050 0.048 0.046 0.044 0.042

xfe-zZo(x) 0.40150 0.40140 0.40129 0.40119 0.40108

9761 4058 8619 3443 8526

xf PZ1 0.39133 0.39164 0.39195 0.39226 0.39257

(2) 9722 8743 7336 5502 3245

xfd2(x) 0.36237 0.36380 0.36523 0.36667 0.36811

579 578 854 408 237

0.39651 0.39661 0.39670 0.39680 0.39689

5620 0241 5057 0069 5278

0.40631 0.40601 0.40572 0.40543 0.40514

0355 9771 8854 7604 6017

0.43714 0.43558 0.43403 0.43247 0.43092

666 814 211 858 754

0.040 0.038 0.036 0.034 0.032

0.40098 0.40087 0.40077 0.40067 0.40056

3868 9466 5319 1424 7781

0.39288 0.39318 0.39349 0.39380 0.39410

0567 7470 3958 0032 5695

0.36955 0.37099 0.37244 0.37389 0.37534

342 722 375 302 502

0.39699 0.39708 0.39718 0.39727 0.39737

0686 6293 2101 8110 4322

0.40485 0.40456 0.40426 0.40397 0.40368

4094 1832 9230 6286 2998

0.42937 0.42783 0.42628 0.42474 0.42321

901 299 949 850 003

25

0.030 0.028 0.026 0.024 0.022

0.40046 0.40036 0.40025 0.40015 0.40005

4387 1241 8340 5684 3270

0.39441 0.39471 0.39502 0.39532 0.39562

0950 5798 0243 4286 7929

0.37679 0.37825 0.37971 0.38118 0.38264

973 716 729 012 564

0.39747 0.39756 0.39766 0.39776 0.39785

0738 7359 4186 1221 8465

0.40338 0.40309 0.40280 0.40250 0.40221

9365 5386 1058 6380 1349

0.42167 0.42014 0.41860 0.41708 0.41555

410 070 984 153 576

33

0.020 0.018 0.016 0.014 0.012

0.39995 0.39984 0.39974 0.39964 0.39954

1098 9164 7469 6009 4785

0.39593 0.39623 0.39653 0.39683 0.39714

1176 4028 6487 8556 0236

0.38411 0.38558 0.38705 0.38853 0.39001

385 474 830 453 342

0.39795 0.39805 0.39815 0.39824 0.39834

5918 3583 1460 9551 7857

0.40191 0.40162 0.40132 0.40102 0.40073

5965 0226 4130 7674 0858

0.41403 0.41251 0.41099 0.40947 0.40796

256 191 383 833 540

0.010 0.008 0.006 0.004 0.002

0.39944 0.39934 0.39924 0.39914 0.39904

3793 3033 2503 2202 2128

0.39744 0.39774 0.39804 0.39834 0.39864

1530 2440 2968 3116 2886

0.39149 0.39297 0.39446 0.39595 0.39744

496 91.5 599 546 756

0.39844 0.39854 0.39864 0.39874 0.39884

6379 5119 4077 3256 2657

0.40043 0.40013 0.39983 0.39953 0.39924

3679 6136 8226 9949 1300

0.40645 0.40494 0.40344 0.40193 0.40043

505 730 214 958 962

0.000

0.39894

2280

0.39894

2280

0.39894

228

0.39894

2280

0.39894

2280

0.39894

228

2-l

[ 1 q3)3

For

interpolating

['-;I51 near

[c-;'"]

x--1 =0 note

that

[q"]

if f&-l) <x>

=zk~Z,(z) =nearest

integer

['-;'"I thenf,(

-2-l)

24

g 31

a; 45

1’00 125 167

~~00 00

['-;I"]

=r-de~K~(z),

to x.

Compiled from L. Fox, A short table for Bessel functions of integer orders and large arguments. Royal Society Shorter Mathematical Tables No. 3 (Cambridge Univ. F’ress, Cambridge, England, 1954) (with permission). MODIFIED

2

Ko(x)+Zo(z)

BESSEL

ln x

FUNCTIONS-AUXILIARY

s[Kl(x)

-II(X)

TABLE

In 4

FOR

SMALL

+ZO(X)

Ko(x)

ARGUMENTS

In x

x[ZG (x) -II

(2) In xl

::1”

0.11872 0.11593

152 387

0.99691 1.00000

180 000

1:1 lx0

0.49199 0.42102

896 444

0.49390 0.60190

723 093

z-3’ 0: 4

0.12713 0.14124 0.16121

128 511 862

0.98754 0.97158 0.94852

819 448 090

:3

0.66373 0.57261 0.76632

444 364 938

0.21236 0.36514 +0.03176

944 381 677

0. 5 E

0.18726 0.21967 0.25879

857 734 579

0.91759 0.87784 0.82804

992 659 980

1.5

;:i

0.88149 1.15456 1.01045

436 879 200

-0.18096 -0.43076 -0.72326

553 976 964

0:8

0. 9

0.30504 0.35892

682 957

0.76669 0.69201

810 997

1: 9

1.31536 1.49454

786 429

-1.06486 -1.46281

242 214

1.0

0.42102

444

0.60190

723

2.0

1.69398

200

-1.92535

914

[(-;)l1

c 1 (-;)2

1: 4

[ 1 c-p3

ii598 1

BESSEL

FUNCTIOKS

OF

INTEGER

423

ORDER

‘I‘nlh s ;; 0:s

r-qx) 0.0000 (-4)1.3680 (-4)9.0273 (--3)X5257 (-3)4.9877

c - ,’I, (.I.) 0.0000

(-6)3.4182 (-5)4.5047 4)1.8858 4)4.9483

r-~qs) 0.0000 (-8)6.8341 (-6)1.7995 (-5)1.1281 (-5)3.9377

(-3)1.0069

r.rf,;(.v) 0.0000 * (-9)1.1388 (-8)5.9925 (-735.6286 (-6)2.6152 (-6)8.2731 (-5)2.0544

r-JfJ(.V) 0.0000 -11)1.6265 - 9)1.7109

i - 8)2.4084 (-

7)1.4902

(((((-

7 5.8832 6I 1.7497 6 4.2831 6)9.0974 5)1.7349

;;

.

(-2)5.6454 (-2)5.8893

(-3)7.2431 (-3)8.2288

4.0 4.2 (-2)3.1221 (-2)3.2854

-3)9.2443 -2 1 1.0283 -2 1.1337 -2 1 1.2402 -2 1.3471

(-3)2.4106

(-

4)6.1640

(-3)2.8291 (-3)3.2785

c -.rL,(x) 0.0000

c-.rl,(.v) 0.0000 (-13)2.0328 (-11)4.2750 (-10)9.0201 (- 9)7.4343

(-1532.2585 C-13)9.4957 (-11)3.0037 (-10)3.2983

((-

(((((-

9)2.0301 9)8.6707 8)2.8797 8)7.9596 7)1.9131

(-

7)4.1199

8)3.6643 ;pm;

I- 7 9:0178 (- 6 1 1.9302 I - 6)6.7325 6)3.7487 5)1.1339 5)1.8099 [I 5)2.7609

(-2)1.4234

‘J.9

(((((-

5)4.0512 5)5.7482 5)7.9208 4 1.0638 41 1.3965

(((-

5)1.4507 5)2.0556 5)2.8380

(((((-

4)1.7968 4)2.2703 4)2.8224 4)3.4578 4)4.1806

((-

5)8.3667 4)1.0508

(-

331.0866

(((-

3)1.9399 3)2.0808 3)2.2260

-2'1.4540 -2 i 1.5605 -2)1.6662 -2)1.7707 -2)1.8738 -2)1.9752 -2 1 2.0747 -2 2.1723 (-2)4.5567

(-

(-2)1.0849

22” 8.4

(-2 I 1.5854 (-2 1.6453 (-2 1.7045

2;

10.0

(-2)7.9830

(-2)5.5683

10.5 11.0 11.5 ;;.; .

-2)7.9687 -2)7.9465 I -2)7.9182 (-2)7.8848 (-2)7.8474

(-2)5.6549

3)3.1156

(-2 3.2269 (-2 1 3.2915

(-2)1.7627 (-2)1.8201

(-2)3.52'34

(-2)2.0398

-

3)6.7449 3 7.1440 3 I 7.5464 3 7.9513 3)8.3582

II ((-

- 3 1.0484 1.3351 1.1870 3)1.4924 3)1.6587

(-

3)2.8292

- 3)9.5839 - 3)9.9924 (- 2)1.0400 (- 2)1.0806

- 2)1.3775 - 2)1.4722 - 2)1.5642

(-

3)5.2694

(-

3)2.3753

(((-

3)5.9380 3)6.6192 3)7.3082

(((((-

3)2.7653 3)3.1769 3)3.6073 3)4.0537 3)4.5134

(((-

3)4.9837 3)5.4622 3)5.9469

(((-

3)7.4171 3)7.9071 3)8.3947

(-2)2.9779 (-2)3.0538 (-2)4.4726

(-2)6.0119

I -2 I 3.1251 3.2543 3.1918 -2)3.3128 -2)3.3675

((-

2)1.4543 2)1.5125

Compiled from British Association for the Advancement of Science, Besselfunctions, Part II. Functions of positive integer order, Mathematical Tables, vol. X (Cambridge Univ. Press, Cambridge, England, 1952) (with permission).

424

BESSEL

Table 9.9

FUNCTIONS

MODIFIED

e"K3(x)

BESSEL

OF

INTEGER

FUNCTIONS-ORDERS

e%(x)

@K4(x)

@Kc(X)

4 3.6;20 3 2.7602 2 I 6.5506 2 2.4743 1 1 0 0 II 0

1.9303 1.2984 9.4345 7.2438 5.7946

ORDER

3-9

e"K7(n>

@Kg(x)

e"Ks(x)

9 4.7897 7 4.1679 6 I 2.9548 5 4.7618

2)1.2024 1 6.8382 1 I 4.3280 1 2.9585 1 2.1426

Ii 45 1.3160 1.2610 2.0785 3.6055 6.6436

2: 1-z 4:8

0 0 0 0 II 0

1.6317 1.5303 1.4414 1.3629 1.2931 0)2.2646

::2" :*: 5:8

1)1.1973

II 0 2.1186 1.8746 1.9895 1.7720

II 01 1.0645 7.7717 9.5285 8.5813

6.0 2:

II 0 1.3902 1.4528 1.5213 1.5967 1.6798

1.1467 1.3978 1.2630 1.7387 II 1i 1.5547

28"

-1 -1 -1 -1 -1

;:2" 99-z 9:8 10.0

(-1)6.0028

9.4941 9.2354 8.9918 8.7620 i a.5449

( -1)8.3395

0 1.5047 0 1.4505 0 I 1.4001 0 1.3529 0)1.3088

(

0)1.2674

1 2.3486 1 I 2.1529 1 1.9794 1 1.8251 1)1.6873

(

0)2.1014

10.5 11.0 11.5 12.0 12.5

( 0)3.7891

(

0)7.4062

oj1.9059

13.0 13.5 14.0 14.5 15.0 15.5 16.0 16.5 17.0 17.5

-1)3.8191

-1)5.1982 -1 5.0414 -1 4.8959 -1 4.7605 -1 I 4.6343

18.0 18.5 19.0 19.5 20.0

-1)3.7411 -1 i 3.6674 -1 3.5976 -1 3.5313 -1 3.4684

-1 I 4.5162 -1 4.4055 -1 4.3015 -1 4.2037 -1)4.1114

0 1.4319 0 1.3480 0 1.2729 0 I 1.2053 0)1.1442

-1 I 6.8656 -1 6.6036 -1 6.3633

-1 6.1420 -1)5.9376

-1)7.7097 -1 7.4176 -1 I 7.1482 -1 -1

6.8990 6.6679

(-q9.1137

0 2.2561

0 0 0I 0

2.0964 1.9552 1.8299 1.7181

(

1)1.5639

BESSEL MODIFIED

BESSEL

1osx-‘ozlo(~)

FUNCTIONS

OF INTEGER

FUNCTIONS-ORDERS

10,

425

ORDER 11,

20

AND

21

Table

9.10

0.26911 0.26935 0.27009 0.27132 0.27305

445 920 468 457 504

lO".r"Zll(X) 1.22324 1.22426 1.22733 1.23245 1.23965

748 724 125 366 820

1.85794 1.85588 1.84970 1.83947 1.82524

idd 251 867 021 326

0.391990 0.392177 0.392738 0.393674 0.394988

0.933311 0.933736 0.935008 0.937136 0.940123

6.37771 6.37435 6.36429 6.34757 6.32424

0.27529 0.27805 0.28135 0.28519 0.28961

480 517 012 648 396

1.24897 1.26045 1.27414 1.29011 1.30843

831 740 918 798 932

1.80713 1.78527 1.75981 1.73095 1.69887

290 169 781 297 992

0.396684 0.398766 0.401239 0.404112 0.407392

0.943974 0.948703 0.954321 0.960843 0.968285

6.29437 6.25807 6.21545 6.16665 6.11184

0.29462 0.30025 0.30653 0.31350 0.32118

538 682 784 170 565

1.32920 1.35250 1.37845 1.40718 1.43883

036 061 262 285 260

1.66381 1.62600 1.58569 1.54314 1.49861

982 944 822 529 645

0.411087 0.415209 0.419768 0.424778 0.430253

0.976669 0.986016 0.996351 1.007703 1.020101

6.05118 5.98488 5.91314 5.83620 5.75428

0.32963 0.33888 0.34899 0.36002 0.37203

121 455 681 459 039

1.47355 1.51153 1.55295 1.59803 1.64700

907 657 782 551 388

1.45238 1.40471 1.35587 1.30613 1.25574

126 020 192 075 432

0.436209 0.442662 0;449632 0.457139 0.465205

1.033581 1.048178 1.063935 1.080893 1.099102

5.66764 5.57655 5.48128 5.38210 5.27932

0.38508 0.39925 0.41464 0.43132 0.44940

316 889 125 237 362

1.70012 1.75766 1.81995 1.88733 1.96016

064 896 978 435 700

1.20496 1.15402 1.10314 1.05255 1.00243

150 052 736 442 944

0.473853 0.483111 0.493006 0.503569 0.514832

1.118613 1.139481 1.161768 1.185538 1.210861

5.17321 5.06408 4.95224 4.83197 4.72159

0.46899 0.49022 0.51322 0.53813 0.56513

655 387 061 536 169

2.03886 2.12388 2.21572 2.31490 2.42204

82 83 08 71 09

0.95298 0.90435 0.85670 0.81016 0.76484

465 626 405 129 483

0.526830 0.539601 0.553186 0.567630 0.582979

1.237813 1.266475 1.296933 1.329281 1.363622

4.60339 4.48367 4.36272 4.24084 4.11830

0.59438 0.62610 0.66050 0.69781 0.73832

965 759 400 972 033

2.53777 2.66282 2.79796 2.94406 3.10208

36 00 48 93 00

0.72085 0.61827 0.63718 0.59762 0.55964

532 767 161 235 137

0.599284 0.616599 0.634984 0.654501 0.675219

1.400061 1.438715 1.479709 1.523176 1.569259

3.99537 3.87234 3.74945 3.62695 3.50507

0.78229 0.83007 0.88201 0.93850 0.99998

881 854 663 764 773

3.27303 3.45808 3.65847 3.87560 4.11098

69 34 74 29 38

0.52326 0.48851 0.45539 0.42389 0.39401

729 672 529 854 295

0.697210 0.720554 0.745333 0.771639 0.799570

1.618113 1.669904 1.724808 1.783016 1.844734

3.38405 3.26411 3.14543 3.02821 2.91264

1.06693 1.13989 1.21945 1.30625 1.40103

936 641 007 534 829

4.36629 4.64339 4.94432 5.27132 5.62688

90 88 35 42 64

0.36571 0.33898 0.31377 0.29004 0.26776

690 159 202 783 418

0.829231 0.860735 0.894204 0.929769 0.967571

1.910180 1.979593 2.053225 2.131351 2.214264

2.79887 2.68705 2.57733 2.46983 2.36466

1.50460 1.61784 1.74175 1.87744 2.02612

429 713 933 369 620

6.01375 6.43496 6.89386 7.39417 7.93999

48 31 57 36 51

0.24687 0.22732 0.20905 0.19202 0.17616

251 134 690 382 568

1.007764 1.050510 1.095988 1.144389 1.195919

2.302281 2.395741 2.495011 2.600488 2.712593

2.26193 2.16172 2.06411 1.96916 1.87692

0.16142 553

1.250800

2.831786

[c-y1

2.18917 062

[ 1 (-;)2

8.53588 02

[ 1 C-l)6

[c-p1

[I 1 (-;)4

1.78744

2n

Kn+1(2)=~Kn.(z)+Kn-l(2)

Compiled from British Associationfor the Advancement of Science,Besselfunctions, Part II. Functions of positive integer order, Mathematical Tables, vol. X (CambridgeUniv. Press,Cambridge,England, 1952)and L. Fox, A short table for Besselfunctions of integer ordersand largearguments. Royal Society Shorter Mathematical Tables No. 3 (CambridgeUniv. Press,Cambridge,England, 1954) (with permission).

426 Table

BESSEL FUNCTIONS MODIFIED

9.10

lo”0 10:2

BESSEL

ORDER 10, 11, 20 AND

FUNCTIONS-ORDERS

21

1024r20120(2)

eZKlo(z)

e-21l1(2)

e-zllo(z)

OF INTEGER

10.4 10.6 10.8

0.00099 0.00107 0.00115 0.00124 0.00132

38819 29935 52835 06973 91744

0.00038 0.00042 0.00046 0.00050 0.00054

75284 45861 37417 50080 83934

35.55633 32.60759 29.98423 27.64297 25.54714

91 68 91 29 23

1.25080 1.30927 1.37160 1.43806 1.50895

2.83179 2.95856 3.09345 3.23703 3.38992

1.787443 1.700753 1.616873 1.535814 1.457578

11.0 11.2 11.4 11.6 11.8

0.00142 0.00151 0.00161 0.00171 0.00181

06490 50508 23051 23339 50559

0.00059 0.00064 0.00069 0.00074 0.00079

39013 15309 12768 31298 70766

23.66558 21.97172 20.44277 19.05917 17.80405

79 20 46 72 56

I.58462 1.66540 1.75169 1.84390 1.94249

3.55278 3.72634 3.91139 4.10876 4.31937

1.382160 1.309546 1.239714 1.172637 1.108279

12.0 12.2 12.4 12.6 12.8

0.00192 0.00202 0.00213 0.00225 0.00236

03870 82412 85303 11650 60548

0.00085 0.00091 0.00097 0.00103 0.00109

31003 11805 12937 34132 75097

16.66281 15.62277 14.67293 13.80364 13.00649

24 97 16 34 01

2.04795 2.16080 2.28162 2.41105 2.54975

4.54421 4.78434 5.04093 5.31521 5.60856

1.046601 0.987556 0.931095 0.877164 0.825703

13.0 13.2 13.4 13.6 13.8

0.00248 0.00260 0.00272 0.00284 0.00297

31086 22347 33415 63375 11314

0.00116 0.00123 0.00130 0.00137 0.00144

35512 15035 13301 29926 64509

12.27407 11.59989 10.97821 10.40394 9.87258

71 74 07 07 79

2.69846 2.85799 3.02921 3.21306 3.41058

5.92244 6.25845 6.61832 7.00393 7.41731

0.776652 0.729947 0.685520 0.643305 0.603230

14.0 14.2 14.4 14.6 14.8

0.00309 0.00322 0.00335 0.00348 0.00361

76327 57518 53999 64894 89341

0.00152 16634 0.00159 85870 0.00167 71776 0.00175 73898 0.00183 91776

9.38015 8.92308 8.49819 8.10265 7.73392

52 36 79 95 53

3.62289 3.85121 4.09686 4.36131 4.64613

7.86068 8.33644 8.84722 9.39585 9.98543

0.565225 0.529218 0.495137 0.462910 0.432464

15.0 15.2 15.4 15.6 15.8

0.00375 0.00388 0.00402 0.00416 0.00429

26491 75510 35583 05908 85705

0.00192 0.00200 0.00209 0.00218 0.00227

24942 72921 35235 11403 00942

7.38971 31 7.06797 04 6.76684 87 6.48467 94 6.21995 46

4.95305 5.28394 5.64087 6.02608 6.44202

10.61932 11.30119 12.03503 12.82520 13.67643

0.403728 0.376630 0.351101 0.327070 0.304470

16.0 16.2 16.4 16.6 16.8

0.00443 0.00457 0.00471 0.00485 0.00500

74209 70675 74378 84612 00690

0.00236 0.00245 0.00254 0.00263 0.00273

03366 18192 44936 83118 32259

5.97130 5.73750 5.51741 5.31001 5.11438

87 35 43 78 19

6.89137 7.37705 7.90228 8.47055 9.08571

14.59389 15.58322 16.65059 17.80271 19.04691

0.283235 0.263299 0.244598 0.227071 0.210658

17.0 17.2 17.4 17.6 17.8

0.00514 0.00528 0.00542 0.00557 0.00571

21947 47735 77427 10418 46119

0.00282 0.00292 0.00302 0.00312 0.00322

91884 61523 40709 28982 25887

4.92965 4.75506 4.58989 4.43349 4.28527

63 40 42 60 20

9.75197 10.47392 11.25663 12.10562 13.02697

20.39124 21.84444 23.41611 25.11674 26.95781

0.195301 0.180944 0.167532 0.155012 0.143336

18.0 18.2 18.4 18.6 18.8

0.00585 0.00600 0.00614 0.00629 0.00643

83964 23403 63909 04971 46098

0.00332 0.00342 0.00352 0.00362 0.00373

30977 43808 63948 90969 24450

4.14467 4.01119 3.88437 3.76378 3.64903

40 75 85 89 41

14.02734 15.11406 16.29515 17.57946 18.97668

28.95188 31.11272 33.45541 35.99648 38.75407

0.132454 0.122321 0.112891 0.104124 0.095978

19.0 19.2 19.4 19.6 19.8

0.00657 0.00672 0.00686 0.00701 0.00715

86817 26672 65226 02059 36768

0.00383 0.00394 0.00404 0.00415 0.00425

63982 09161 59590 14885 74667

3.53974 3.43559 3.33626 3.24146 3.15093

93 74 62 65 00

20.49749 22.15363 23.95803 25.92489 28.06989

41.74804 45.00024 48.53460 52.37745 56.55768

0.088414 0.081397 0.074892 0.068865 0.063285

20.0

0.00729 68965 C-l)4

3.06440 75 (-;I4

30.41029

61.10706 c-2)5 5

0.058124 c-:)4

II 1

0.00436 38567 C-l)3

[

1

II 1 [ 1 [ c-y

1 [I 1

BESSEL FUNCTIONS MODIFIED

BESSEL

FUNCTIONS-AUXILIARY

TABLE

FOB

LARGE

ARGUMENTS

2-l In [A~Z~o(z)l In rGk-zlll(~)l In [T- lzLKl~(~)] In [&-~Z2o(z)] 0.050 -3.42244 062 13.93653 ‘263 1.47299 048 -10.434749 0.049 -3.37318 689 -3.87762 888 1.42771 939 -10.263511 0.048 -3.32386 306 -3.81861 524 1.38232 785 -10.091302 0.047 -3.27447 055 -3.75949 454 1.33681 644 - 9.918126 0.046 -3.22501 139 -3.70026 938 1.29118 575 - 9.743983

In [$k-~Z~l(~)] -11.346341 -11.160467 -10.973471 -10.785351 -10.596108

0.045 0.044 0.043 "0-K .

-3.17548 -3.12590 -3.07625 -2.97678 -3.02655

0.040 %Z .

-2.92697 559 -2.82719 -2.87711 539 002

0.037 0.036

766 147 496 033 979

427

OF INTEGER ORDER Table

In [r-lrtezZ&(z)] 8.250182 8.088946 7.926737 7.763551 7.599386

9.10



:: 2’:

-3.64094 -3.58151 -3.52199 -3.46237 -3.40267

242 639 408 835 211

1.24543 1.19956 1.15358 1.10748 1.06126

642 910 449 332 635

-

9.568876 9.392809 9.215785 9.037810 8.858889

-10.405744 -10.214259 -10.021658 - 9.827944 - 9.633121

7.434240 7.268110 7.100994 6.932893 6.763806

-2.77723 405 -2.72722 837

-3.34287 -3.28300 -3.22304 -3.16300 -3.10288

833 006 039 246 949

1.01493 0.96848 0.92192 0.87525 0.82847

437 822 874 686 349

-

8.679029 8.498236 8.316519 8.133888 7.950352

-

9.437195 9.240173 9.042063 8.842873 8.642612

6.593733 6.422673 6.250630 6.077603 5.903597

0.035 0.034 0.033 0.032 0.031

-2.67718 -2.62709 -__ -. -2.57696 -2.52681 -2.47661

076 365 _-948 074 992

-3.04270 -2.98245 -2.92213 -2.86175 -2.80131

472 146 308 298 461

0.78157 0.73457 0.68746 0.64024 0.59291

961 624 441 520 975

-

7.765923 7.580613 7.394434 7.207403 7.019533

-

8.441293 8.238927 8.035529 7.831113 7.625695

5.728614 5.552659 5.375732 5.197843 5.018998

0.030 0.029 0.028 0.027 0.026

-2.42639 -2.37615 -2.32588 -2.27558 -2.22527

955 216 032 659 356

-2.74082 -2.68027 -2.61968 -2.55904 -2.49837

147 709 504 894 243

0.54548 0.49795 0.45031 0.40257 0.35474

920 475 764 915 059

-

6.830842 6.641348 6.451070 6.260027 6.068243

-

7.419294 7.211929 7.003620 6.794389 6.584261

4.839203 4.658466 4.476796 4.294202 4.110696

0.025 0.024 0.023 0.022 0.021

-2.17494 -2.12460 -2.07424 -2.02388 -1.97351

384 002 475 063 031

-2.43765 -2.37691 -2.31613 -2.25533 -2.19451

918 291 733 620 329

0.30680 0.25876 0.21063 0.16241 0.11409

331 871 822 332 551

-

5.875738 5.682539 5.488669 5.294155 5.099025

-

6.373261 6.161416 5.948754 5.735305 5.521102

3.926290 3.740995 3.554826 3.367799 3.179929

0.020 0.019 0.018 0.017 0.016

-1.92313 -1.87276 -1.82238 -1.77201 -1.72165

643 162 853 979 806

-2.13367 -2.07281 -2.01195 -1.95107 -1.89020

239 731 186 986 514

0.06568 +0.01718 -0.03139 -0.08007 -0.12883

636 745 959 306 128

-

4.903309 4.707035 4.510235 4.312943 4.115190

-

5.306177 5.090565 4.874302 4.657427 4.439978

:*z::; 2:611440 2.420383 2.228582

0.015 0.014 0.013 0.012 0.011

-1.67130 -1.62096 -1.57064 -1.52033 -1.47004

595 610 113 365 626

-1.82933 -1.76846 -1.70760 -1.64675 -1.58592

153 286 295 564 472

-0.17767 -0.22659 -0.27559 -0.32467 -0.37383

247 485 659 581 061

-

3.917011 3.718443 3.519520 3.320281 3.120763

-

4.221995 4.003521 3.784599 3.565272 3.345586

2.036059 1.842840 1.648949 1.454415 1.259264

0.010 0.009 0.008 0.007 0.006

-1.41978 -1.36954 -1.31933 -1.26914 -1.21900

154 207 040 908 063

-1.52511 -1.46432 -1.40356 -1.34284 -1.28214

400 725 824 072 841

-0.42305 -0.47235 -0.52172 -0.57116 -0.62066

904 911 881 608 881

-

2.921004 2.721043 2.520921 2.320676 2.120350

-

3.125587 2.905322 2.684833 2.464184 2.243408

1.063526 0.867231 0.670412 0.473099 0.275328

100 111 125 143 167

0.005 EE 0:002 0.001

-1.16888 -1.11881 -1.06877 -1.01878 -0.96883

754 229 735 514 808

-1.22149 -1.16088 -1.10031 -1.03980 -0.97934

499 414 949 463 314

-0.67023 -0.71986 -0;76954 -0.81929 -0.86908

489 215 834 138 886

-

1.919982 1.719613 1.519284 1.319036 1.118907

-

2.022558 1.801685 1.580838 1.360065 1.139416

+0.077133 -0.121451 -0.320388 -0.519640 -0.719170

0.000

-0.91893 853

- 0.918939 C-44) 1 [ 1 integer[ to :c.1 +>=nearest

-0.918939

200 250 333 500 1000 m

-0.91893 853

-0.91893 853 c-y

- 0.918939

24

:z :; 28 29 :i 31 32

50 5563 59 63

Royal Compiled from L. Fox, A short table for Bessel functions of integer orders and large arguments. Society Shorter Mathematical Tables No. 3 (Cambridge Univ. Press, Cambridge, England, 1954) (with permission).

428

BESSEL

Table 9.11

FUNCTIONS

MODIFIED I

-

Z?%(l) 011.26606 5878 1I Sib5159 1040 1 1.35747 6698 2 2.21684 2492 3 2.73712 0221

OF

BESSEL

II

ORDER

FUNCTIONS-VARIOUS

ORDERS Zn(5) 112.72398 7182

II II

I

10 1.75056 5.10823 4214 1.03311 2.43356 4764 1497 5017

II -

f- lOj2.75294

INTEGER

f

5346 2.76993 1.60017 9323 3.04418 2.24639 9.82567 1420 3364 5903 6951

oj2.15797

-

4547

12 2.56488 7.41166 6690 7192285 1.93157 3216 1882 9417

8040

- 11 14 1.24897 13 16 1.99563 7.11879 8308 5.19576 1678 1153 0054

20 ioo 50 100

-

25 3.96683 42 3.53950 60 1.12150 80i 2.93463

(-189)8.47367

5986 0588 9741 5309 4008

II

3 2.67098 1.22649 6628 2.28151 2.81571 1.75838 0538 8968 0717 8304

- .9 4.31056 0576 - 33 3.89351 9664 - 48 1.25586 9192 - 65 3.35304 2830

II

(-158)1.08217

( 2012.49509

1475

(-119)7.09355

f4116.49897

II

::

1489

552

41 4.01657 4.59832 209 ii84924 5.21214 700 794 227

:34 3.07376 I 191912.25869

11: 17 18 19

100

- 11 21 5.02423 45 32 2.93146 9358 1.18042 3.99784 6980 9647 4971

894

10

20 30 40 50

II

4 1.25079 112 7.78756 -20 2.04212 -30 I 4.75689 (-88)1.08234

9736 9783 3274 4561 4202

18 16 13 +lO I

5.44206 4.27499 6.00717 1.76508

(-16)2.72788

455 581

840 365 897 024 795

41 40 38 36

1.44834 1.20615 3.84170 4.82195

II

(21)4.64153

613 487 550 809 494

BESSEL

MODIFIED

FUNCTIONS

BESSEL

KnU)

OF

INTEGER

FUNCTIONS-VARIOUS

II

429

ORDER

ORDERS

Table

9.11

K,(5)

G(2)

-10 6.47385 2.19591 8728 2.53759 1.39865 1.13893 3909 8818 5927 7546

I

2033 4099 5896 1230 52347I 6.22552 1.00050 8312 4.42070 3.65383 3.60960

II

2130 5.21498 9.75856 2921 9.65850 1.92563 4.15465 3277 2913 2829 4995

20 Ip 100

II 22 39 58 77

6.29436 4.70614 1.11422 3.40689

9360 5527 0651 6854

(185)5.90033

3184

II

6853 9886 1740 45 30 16 62 4.27112 2.97998 5755 5.77085 9.94083

(155)4.61941

Kn(lO)

5978

(115)7.03986

0193

Kn(100)

Kn(50)

2 87 9 10 :: 13 14 15 1176 ii

-1 -1 0 1 II 1

2.65656 8.81629 3.08686 1.13769 4.40440

3849 2510 9988 8721 2395

20

(85)4.59667

4084

-44 7.65542 -45 1.23283 963 1.07829 9.52475 8.49696 148 044 206 797

-21 4.21679 -22 1.16980 8.17096 5.81495 3.11621 235 398 523 828 117

-21 -19 -16 -13

El 50 100

II

1.70614 2.00581 1.29986 4.00601

II

(+13)1.63940

I II

838 681 971 347 352

645 763 531 371 I -44 1.42348 2.79144 325 2.32445 1.95464 1.65987

-44 -43 -41 -40

3.38520 3.97060 1.20842 9.27452

II

(-25)7.61712

541 205 080 265 963

430

BESSEL Table

:I 0.0 2: fJ:fi 0.5 it76 0”::

9.12

FUNCTIONS KELVIN

ber P 1.00000 00000

OF

INTEGER

ORDER

FUNCTIONS-ORDERS

bei z 0.00000 00000

0 AND

berl .X 0.00000 00000

0.99999 0.99997 0.99987 0.99960

84375 50000 34319 00044

0.00249 0.00999 0.02249 0.03999

99996 99722 96836 82222

-0.03539 -0.07106 -0.10725 -0.14423

95148 36418

0.99902 0.99797 0.99624 0.99360 0.98975

34640 51139 88284 11377 13567

0.06249 0.08997 0.12244 0.15988 0.20226

32184 97504 89390 62295 93635

0.98438 ii97713 0.96762 0.95542 0.94007

17812 79732 91558 87468 50567

0.24956 0.30173 0.35870 0.42040 0.48673

0.92107 0.89789 0.86997 0.83672 0.79752

21835 11386 12370 17942 41670

0.75173 0.69868 0.63769 0.56804 0.48904

1

bei, z 0.00000 00000

08645

0.03531 0.07035 0.10486 0.13857

11265 65360 83082 41359

-0.18224 -0.22153 -0.26233 -0.30485 -0.34931

31238 37177 33470 87511 01000

0.17119 0.20244 0.23202 0.25962 0.28491

51797 39824 24623 00070 16898

60400 12692 44199 59656 39336

-0.39586 -0.44469 -0.49591 -0.54964 -0;60594

82610 19268 45913 13636 56099

0.30755 0.32719 0.34345 0.35593 0.36421

66314 65305 43903 34649 64560

0.55756 0.63272 0.71203 0.79526 0.88212

00623 56770 72924 19548 23406

-0.66486 -0;72639 -0.79050 -0.85709 -0.92601

54180 98786 51846 05470 39357

0.36786 0.36641 0.35939 0.34630 0.32660

49890 93986 88584 18876 72722

41827 50014 04571 89261 77721

0.97229 1.06538 1.16096 1.25852 1.35148

16273 81608 99438 897iii 54165

-0.99707 76519 -1 07002 37462 -1.14452 92997 -1.22020 15903 -1.29657 31717

0.29977 0.26525 0.22246 0.17082 0.10976

54370 03092 17120 a3322 13027

0.39996 0.30009 0.18870 +0.06511 -0.07136

84171 20903 63040 21084 78258

1.45718 1.55687 1.65514 1.75285 1.84717

20442 77737 24073 05638 61157

-1.37309 -1.44914 -1.52398 -1.59680 -1.66670

68976 09315 37854 94413 26139

+0.03866 -0.04304 -0.13594 -0.24062 -0.35762

84440 07916 96285 74875 26713

-0.22138 -0.38553 -0.56437 -0.75840 -0.96803

02496 14550 64305 70121 a9953

1.93758 2.02283 2.10157 2.17231 2.23344

67853 90420 33881 01315 57503

-1.73264 -1.79350 -1.84805 -1.89492 -1.93265

42211 71373 23125 53482 36306

-0.48745 -0.63060 -0.78750 -0.95851 -1.14396

41770 25952 00586 92089 11510

-1.19359 -ii43530 -1.69325 -1.96742 -2.25759

81796 51217 99843 32727 94661

2.28324 2.31986 2.34129 2.34543 2.33002

99669 36548 77145 30614 18823

-1.95964 -1.97418 -1.97443 -1.95842 -1.92410

41313 19924 00262 92665 07174

-1.34404 -1.55888 -1.78847 -2.03271 -2.29130

23731 06139 96677 31257 70630

-2.56341 -2.88430 -3.21947 -3.56791 -3.92830

65573 57320 98323 08628 66215

2.29269 2.23094 2.14216 2.02364 1.87256

03227 27803 79867 70694 37958

-1.86924 -1.79156 -1.68863 -1.55794 -1.39689

84590 42730 39648 55649 95997

-2.56382 -2.84963 -3.14790 -3.45759 -3.77737

16886 19932 74393 07560 59182

:*i 4:9

-4.29908 -4.67835 -5.06388 -5.45307 -5.84294

65516 69372 55867 61749 24419

1.68601 1.46103 1.19460 0.88365 0.52514

72036 68359 07968 68537 68109

-1.20282 -0.91297 -0.70458 -0.39486 -0.04099

16315 72697 98649 10961 46681

-4.10568 -4.44064 -4.78006 -5.12141 -5.46179

54084 68813 93721 92170 58790

5. 0

-6.23008

24787

0.11603

43816

+0.35977

66668

-5.79790

79018

:*10

1:2 ::: ::2 :-; 1:9

::1" 2: 2:4 2.5 ::; 2: 21" z-23 3:4 ::2 :'78 3:9 44:10 44.: 4: 4 42

KELVIN

2: 0. 0 0.1

[(-;I6 1

FUNCTIONS-AUXILIARY

ker x+ber .z’In .g

t: 0:4

0.11593 0.11789 0.12374 0.13339 0.14669

0. 5

0.16343

1516 2485 5076 8210 9682

kei .I,+bei .I’In .I’ -0.78539 -0.78260 -0.77421 -0.76019 -0.74045

8163 7108 9267 0919 0212

TABLE

41168

[(-;I6 1 FOR

SMALL

[(92 1

ARGUMENTS

,r(kerl .c+berl .CIn .r) .@eil .r+beil .XIn .G) -0.70710 -0.70651 -0.70486 -0.70248 -0.69994

6781 7131 2164 3157 6658

-0.70710 -0.70215 -0.68733 -0.66272 -0.62851

6781 4903 0339 8003 1738

C-$)1 C-f)1 [ 1 I [ 1 I [ I Bureau of Standards, Tables of the Bessel functions JO(Z) and J,(z) for complex 5574

-0.71489

8693

-0.69804

1049

-0.58492

2770

(78

Compiled from National arguments, 2d ed. (Columbia Univ. Press, New York, N.Y., 1947) and National Bureau of Standards, Tables of the Bessel functions I’s(z) and YI(z) for complex arguments (Columbia Univ. Press, New York, N.Y., 1950) (with permission).

BESSEL KELVIN kern:

FUNCTIONS

OF INTEGER

FUNCTIONS-ORDERS kei z

431

ORDER

0 AND 1 ker, 2

Table 9.12 kei, .X

2.4200047 1.73314 1.333721 1.06262

3980 2752 8637 3902

-0.78539 -0.77685 -0.75812 -0.73310 -0.70380

8163 0646 4933 1912 0212

-7.14628 -3.63868 -2.47074 -1.88202

1711 3342 2357 4050

16.94074 -3.32341 -2.08283 -1.44430

2153 7218 4751 5150

0.85590 0.69312 0.56137 0.45288 0.36251

5872 0695 8274 2093 4812

-0.67158 -0.63744 -0.60217 -0.56636 -0.53051

1695 9494 5451 7650 1122

-1.52240 -1.27611 -1.09407 -0.95203 -0.83672

3406 7712 2943 2751 7829

-1.05118 -0.78373 -0.59017 -0.44426 -0.33122

2085 8860 5251 9985 6820

0.28670 0.22284 0.16894 0.12345 0.08512

6208 4513 5592 5395 6048

-0.49499 -0.46012 -0.42616 -0.39329 -0.36166

4636 9528 3604 1826 4781

-0.74032 -0.65791 -0.58627 -0.52321 -0.46718

2276 0729 4386 5989 3076

-0.24199 -0.17068 -0.11325 -0.06683 -0.02928

5966 4462 6800 2622 3749

0.05293 0.02602 +0.00369 -0.01469 -0.02966

4915 9861 1104 6087 1407

-0.33139 -0.30256 -0.27522 -0.24941 -0.22514

5562 5474 8834 7069 2235

-0.41704 -0.37195 -0.33125 -0.29442 -0.26105

4285 1238 0485 5803 9495

+0.00100 0.02530 0.04461 0.05974 0.07137

8681 6776 5190 7779 3592

-0.04166 -0.05110 -0.05833 -0.06367 -0.06737

4514 6500 8834 0454 3493

-0.20240 -0.18117 -0.16143 -0.14313 -0.12624

0068 2644 0701 5677 1488

-0.23080 -0.20337 -0.17850 -0.15599 -0.13563

5929 3135 9812 6054 6638

0.08004 0.08624 0.09035 0.09271 0.09361

9398 3202 1619 2940 7161

-0.06968 -0.07082 -0.07097 -0.07029 -0.06893

7972 5700 3560 6321 9052

-0.11069 -0.09644 -0.08342 -0.07157 -0.06082

6099 2891 1858 0648 5473

-0.11725 -0.10069 -0.08580 -0.07246 -0.06052

6136 5314 8451 1339 9755

0.09331 0.09201 0.08991 0.08716 0.08391

3788 8037 5810 7762 2666

-0.06702 -0.06467 -0.06198 -0.05903 -0.05589

9233 8610 4833 2916 6550

-0.05112 -0.04239 -0.03458 -0.02761 -0.02144

1884 5446 2313 9697 6287

-0.04989 -0.04045 -0.03211 -0.02476 -0.01832

8308 9533 3183 5662 9556

0.08027 0.07634 0.07222 0.06797 0.06367

0223 3451 0724 7529 7999

-0.05263 -0.04931 -0.04597 -0.04264 -0.03937

9277 5556 1723 6864 3608

-0.01600 -0.01123 -0.00707 -0.00348 -0.00041

2568 1096 6704 6665 0809

-0.01272 -0.00787 -0.00370 -0.00014 +0.00285

3249 0585 0576 7138 1155

0.05937 0.05511 0.05093 0.04687 0.04294

6256 7592 9514 2681 1728

-0.03617 -0.03308 -0.03010 -0.02726 -0.02455

8848 4395 7574 1764 6892

+0.00219 0.00438 0.00619 0.00766 0.00882

8399 5818 3613 1269 5624

0.00535 0.00740 0.00906 0.01037 0.01136

1296 6063 4226 0752 6998

0.03916 0.03556 0.03213 0.02889 0.02585

6011 0272 5235 8142 3229

-0.02199 -0.01959 -0.01734 -0.01524 -0.01330

9875 5024 4409 8188 4899

0.00972 0.01037 0.01082 0.01109 0.01120

0918 8865 8725 7399 9526

0.01209 0.01257 0.01285 0.01296 0.01291

0904 7182 7498 0651 2753

0.02300 0.02034 0.01787 0.01559 0.01349

2160 4409 7607 7847 9960

-0.01151

1727

0.01118

7587

0.01273

7390

0.01157

7754

KELVIN

FUNCTIONS-AUXILIARY

ker z+ber z In J

TABLE FOR SMALL ARGUMENTS

kei .c+bei L In .c z(kerl z+berl z In .v)r(keil z+beil r In .J)

0.16343 0.18332 0.20604 0.23116 0.25823

5574 9435 1279 6407 4099

-0.71489 -0.68341 -0.64584 -0.60204 -0.55182

8693 3456 9920 5231 2327

-0.69804 -0.69777 -0.70035 -0.70720 -0.71993

1049 1567 3648 4389 1903

-0.58492 -0.53229 -0.47105 -0.40176 -0.32512

2770 1460 2294 2012 0736

0.28670

6208

-0.49499

4636

-0.74032

2276

-0.24199

5966

[(-;)I1

432

BESSEL Table

9.12

her z=&h~

FUNCTIONS

KELVIN cos 80 h) ’ ’ bei r‘=:llo(x) Mn(x) 1.000000 1.000025 :* t%2 1: 006383

berl

0.10 0. 09 0.08 0. 07 0. 06

PHASE

cos h(.r) beil .X =YI

(x) sin 81 (2:)

Xl(T) 0.000000 0.100000 0.200013 0.300101 0.400427

81(.r) 2.356194

2.481086 2.535872 2.600386 2.674406 2.757605

1.015525 1.031976 1.058608 1.098431 1.154359

0.248294 0.354999 0.471755

0.501301

0.613860

0.813585 0.924407

1.229006

0.912639

s- xt 2:401189 2.436166

0.603235 0.706982

0.759999

1.301211 1.449780 1.614838

2.849536 2.949617 3.oEJ7139 3.171285 3.291160

1.833156 1.979784 2.124854 2.268771

1.799908 2.008844 2.245840 2.515453 2.822653

3.415839 3.544415 3.676044 3.809981 3.945601

3.439118 3.867032 4.351791 4.901189 5.524209

2.411887 2.554483 2.696771 2.838893 2.980942

3.172896 3.572227 4.027393 4.545990 5.136619

4.082407 4.220023 4.358179 4.496691 4.635441

b. 231163

3.122970 3.265002 3.407044 3.549094 3.691142

5.809060

4.774362 4.913417 5.052589 5.191872 5.331267

lb.736836 18.986208

3.833179 3.975197 4.117190 4.259152 4.401083

10.850182 12.312791 13.978402 15.875614 18.037122

6.029884

21.547863

4.542982

20.500302

b.lb9913

1.442891 1.585536 1.754059

:* m 11380379

1.950193 2.176036 2.434210 2.727979 3.061341

1.684559

FUNCTIONS-MODULUS z~e-=~J~,Mo(z) 0.40418

0.40383 0.40349 0.40315 0.40281

AND Be(z)-(z/&j)

-0.40758 -0.40644

-0.40534 -0.40427 -0.40323

0.40246

0.40211

-0.40221 -0.40119

0.40176

-0.40019

0.40141 0.40106

6.574474 7.445618

8.437083 9.565568

PHASE

FOR

.xt,-@&(T) 0.38359 0.38457 : ;8856:; 0:38755 0.38957 0.39060

0.39162

-0.39728

0.39369

0.40071 0.40035 0.40000 0.39930

-0.39541 -0.39449 -0.39359

0. 00

0.39894 ['-y]

-0.39270 ['-j"]

-0.39634

a>=nearest

5.470172 5.610390 5.750117 5.889950

LARGE

ARGUMENTS

@lk) -(x/&2) 1.22254 1.21922 1.21598

<.I’> 5 8

1.20968 1.21280

98

0.38856

-0.39921 -0.39824

0.05 0. 04 0.03 0. 02 0.01

0.39965

1.041167 1.165949

1.533667

11.500794 13.025757 14.761257

0.14 0.13 0.12 0.11

ORDER AND

.c =Af~(.r)

sin @O(Z) h&e) 0.000000 0.010000 0.039993 0.089919 0.159548

7.033841 7.945700 8.982083 10.160473

i-1 0.15

INTEGER

FUNCTIONS-MODULUS

1.324576

KELVIN

OF

0.39266

0.39474

0.39894 ['-;I33 integer

to z.

1.20660 :- :kE 1:19762 1.19471

:10

1.18901 1.19184

:50

1.18622 1.18348 1.18077

:; 100

1.17810

m

:i 17

BESSEL FUNCTIONS KELVIN

OF INTEGER

FUNCTIONS-MODULUS

433

ORDER Table 9.12

AND PHASE

kerl :c =NI (.c)cos 41(z) ker z=No (2) cos do (z) kei ~=No(x) sin do(x) kei 1 z =NI (cc)sin $1(x) x 0.0

No(4

@l(X)

Wl (x)

$0(4 0.000000 -0.412350 -0.584989 -0.743582 -0.896284

4.927993 2.372347 1.497572 1.050591

-2.356194 -2.401447 -2.487035 -2.590827 -2.704976

0.778870 0.597114 0.468100 0.372811 0.300427

-2.825662 -2.950763 -3.078993 -3.209526 -3.341804

0”:;

1.8G702 1.274560 0.941678 0.725172

:$I

0.572032 0.458430

:::

is ;E!i 0:249850

-1.045803 -1.193368 -1.339631 -1.484977 -1.629650

0.206644 0.171649 0.143095 0.119656 0.100319

-1.773813 -1.917579 -2.061029 -2.204225 -2.347212

0.244293 0.200073 0.164807 0.136407 0.113353

-3.475437 -3.610143 -3.745715 -3.881994 -4.018860

0.084299 0.070979 0.059870 0.050578 0.042789

-2.490025 -2.632692 -2.775236 -2.917672 -3.060017

0.094515 0.079039 0.066264 0.055677 0.046873

-4.156217 -4.293990 -4.432118 -4.570551 -4.709250

8. %!i 0:026095 0.022174 0.018859

-3.202283 -3.344478 -3.486612 -3.628692 -3.770724

0.039530 ii*E:; 0:023918 0.020280

-4.848179 -4.987312 -5.126623 -5.266093 -5.405705

0.016052 0.013674 0.011656 0.009942 0.008485

-3.912712 -4.054662 -4.196576 -4.338460 -4.480314

0.017213 0.014624 0.012435 0.010583 0.009013

-5.545443 -5.685295 -5.825250 -5.965298 -6.105430

286

0.007246 0.006191 0.005292 0.004526 0.003872

-4.622142 -4.763947 -4.905730 -5.047493 -5.189238

:* EE: 0:005590 0.004773 0.004077

-6.245638 -6.385917 -6.526260 -6.666662 -6.807119

7.0

0.003315

-5.330966

0.003485

-6.947625

0”::

1:4 2. 0 2.2 :-z 2: a

::2” ::“6

3. 8 2: :"b 4: 8 5.0 ::: ::i i-20 6: 4

KELVIN 2-I

0.15 0.14 0.13 0.12 0.11

FUNCTIONS-MODULUS xbZI~No(x)

AND PHASE

do(x)+(@2)

FOR LARGE

x+e?/@N, (x)

1.23695 1.23802 1.23909

1;. ::1"1;

1.30377

-0:38217

ARGUMENTS

01 (x)+(x/a)

<x>

is :E

-1.99943 -1.99725 -1.99505

7 ii

1.24017 1.24125

-0.38291 -0.38367

1129363 1.29024

-1.99281 -1.99055

i:

0.10 0. 09 0.08 0.07 0.06

1.24342 1.24233 1.24451

-t&38522 -0.38444 100':tg

1.28349 1.28687 1.28012

-1.98825 -1.98592 -1.98357

:1" 13

1.24560 1.24670

-0:38761

1.27675 1.27339

-1.98118 -1.97876

:7"

0.05 0. 04 0.03 0.02 0.01

1.24779 1.24889 1.25000

-0.38926 -0.38843 -0.39010

1.26667 1.27002 1.26332

-1.97630 -1.97381 -1.97128

:5" 33

1.25110 1.25221

-0.39096 -0.39182

1.25998 1.25664

-1.96872 -1.96613

1.25331

-0.39270

1.25331

-1.96350

0. 00

[(-y]

[ (-$“I

[q”]

=nearest integer to 2.

[ (-$“I

1z 00

10.

Bessel Functions

of Fractional

Order

H. A. ANTOSIEWICZ~ Contents Page Mathematical Properties . . . . . . . . . . . . . . . . . . . . 10.1. Spherical Bessel Functions . . . . . . . . . . . . . . . 10.2. Modified Spherical Bessel Functions . . . . . . . . . . .

10.3. Riccati-Bessel Functions . . . . . . . . . . . . . . . . 10.4. Airy Functions . . . . . . . . . . . . . . . . . . . . Numerical 10.5.

Methods

. . . . . . . . . . . . . . . . . . . . . .

Use and Extension

References

of the Tables

. . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . .

437 437 443 445 446 452 452 455

10.1. Spherical Bessel Functions-Orders sd4, Y&d n=O, 1, 2; %=0(.1)5, 6--S, x=5(.1)10,

0, 1, and 2 (0 _<x110)

.

457

Table

Spherical Bessel Functions--Orders j&), YnO) n=3(1)8; 2=0(.1)10, 5s 2-njn(z), z”+‘Y”(4 n=9, 10; x=0(.1)10, 7-8s

3-10 (0 Iz 5 10) . . .

459

Table

10.3. Spherical Bessel Functions--Orders 2-B exp (2/(4n+2)lj,(~)

20 and 21 (0 Iz 225)

463

Table

10.2.

z”+l exp (--2/(4+2))y,(z) n=20, 21; x=0(.5)25,

5s

6-8s

10.4. Spherical Bessel Functions-Modulus and Phase--Orders 9,10,20 and 21 . . . . . . . . . . . . . . . . . . . . . . . ai%,++ 09, en++(4 -z where Ad4 = am&+&> CO8k+t (4 ~44 =-\13+%++ (4 sin en++(4 n=9, 10; z-‘=.l(-.005)0, 8D n=20, 21; z-*=.04(-.002)0, 8D

Table

464

Table

Spherical Bessel Functions-Various Ah), Y&d n=0(1)20, 30, 40, 50, 100 2=1, 2, 5, 10, 50, 100, 10s

Orders (0 In 5 100) .

465

Table

10.6. Zeros of Bessel Functions of Half-Integer Order (0 -, Y&z> and Values of JX.& & YL(y,.,)

467

10.5.

v=n+$,

n=0(1)19,

6-7D

10.7. Zeros of the Derivative of Bessel Functions of Half-Integer Order (OIn519). . . . . . . . . . . . . . . . . . . . . . . Zeros jL, ,, yi. t of J:(z), Y:(z) and Values of J&j:, d), Y,(yl, ,)

Table

v=n+$, 1 National 637472

n=0(1)19,

6D

Bureau of Standards. (P resently, University O-64-20

468

of Southern

California.) 435

436

BESSEL

FUNCTIONS

OF

FRACTIONAL

ORDER

T&Ie 10.8. ModiGed Spherical Bessel Functions-Orders (0<2<5)........................... Awn+* (4 , lmK+t 0) n=o, I, 2;2=0(.1)5, 4-9D Table 10.9. Modified Spherical Bessel Fuuctions-Crders (O
4%

0, 1 and 2

Page

469

9 and

10 470

exp [2-n(n+l>l(2z)lK,+t(2)

n=9,

10;z-1=.1(-.005)0, 7-S Table 10.10. Modified Spherical Bessel Functions-Various Orders (O~n~lOO) . . . . . . . . . . . . . . . . . . . . . . . . . l&d&+1 cd, ma+* (4 n=0(1)20, 30, 40, 50, 100 z=l, 2, 5, 10, 50, 100, 10s Table 10.11. Airy Functions (OlzS w) . . . . . . . . . . . . . . Ai( Ai’( Bi(z), Bi’(z)

z=o(.ol)l,

473,

475

sD

Ai(--2), Ai’( Bi(-z), Bi’(-z) ~=O(.Ol)l(.l)lO, 8D Auxiliary Functions for Large Positive Arguments Ai (2) = jz-1/4e-y( - i) ; Bi (2) =z-1/4ey([) Ai’(z&~“~e-~g(-[); Bi’(z)=z1/4etg(t) f(h[), g(kl); i=$i’2, [-‘=1.5(-.1).5(-.05)0, 6D Auxiliary Functions for Large Negative Arguments Ai(--s)=z-1/41f,(t) cos t+fi(t) sin ,$I Bi(--z)=z-1’41fi([) cos c-fi([) sin .$I Ai’(-z)=s”4[g1([) sin E-g&$) cos [] Bi’(-z>=z1/41g2(t) sin t+gl(t) cos [I flc9,AW,

m(E>,

t-'=.05(-.Ol)O, Table 10.12. Integrals s0

9269;

t=iw2

6-7D of Airy Functions

’ Ai(t)dt,

x=0(.1)7.5;

’ Bi(t)&,

2=0(.1)2;

s0

’ Ai(--t)dt,

(OIz
. . . . . . . . 7D

’ Bi(--t)dt, z=O(.l)lO, 7D s Table 10.13. Zeros and Assoiiated Values of Airy Functions and Their Derivatives (11~510) . . . . . . . . . . . . . . . . . . . . Zeros a,, ai, b,, b: of Ai( Ai’( Bi(z), Bi’(sc) and values of Ai’( AWL Bi’&), Bi(b:) s=l(l)lo, 8D Complex Zeros and Associated Values af Bi(z) and Bi’(z) (lG55) Modulus and Phaseof e-rf’3p6, e-r(la@:, Bi’(@J, Bi(&) .3=1(1)5, 3D s0

478

The author acknowledges the assistance of Bertha H. ,Walter and Ruth Preparation and checking of the tables and graphs.

478

Zucker in the

10. Bessel Functions

of Fractional

Mathematical 10.1. Spherical

Order

Properties WronddanI3

Bessel Functions

Definitions DiBerential

10.1.6

Equation

Wjn(z),

10.1.7

10.1.1

W{h:‘(q,

z*W”+2zW’+[~-n(n+l)]w=o (n=O,fl,

f2,

Particular solutions are the Spheric& junctions of the$rst kind

. . .)

Representations

Bessel

by Elementary

10.1.8 j&)=2-'[P(n+$,

2) sin (z-+&f)

10.1.9

of th-e second kind

y,(z)=(-l)“+‘z-~[P(~+~,z)

Yd4=dmYn+t(& Bessel junctions

CO8(z++n7r)

-Q(n+3, 4 sin b+h)l

of the th&d

p cn+3, 4=1-2!

hp’(z)=j,(z)+iy,(z)=~~~~:‘:t(z),

(n+2)! I$+

h~~‘(z)=j,(z)--iy,(z)=~~~~,(z).

series

(see 9.1.2,

= !!I! ,” (-l)“(n+$,

9.1.10)

Q(n++,

10.1.2 -=1.3.5

.2”. . (2n+l)

(24‘2

b+4)! (n-3)

+4!r

The pairs i&9, ~44 and hf’(z), hP’(z) are linearly independent solutions for every n. For general properties see the remarks after 9.1.1. Aecendlng

F’unctions

+Q(n+3, 4 CO8(z-k/L1F)I

Bessel junctiom

and the Spherical kind

(n=O, 1, 2, . . .)

h:‘(z)}=-2irs

ia(4=dmJ~+to, the Spherical

I =2-a

Y&)

z) =#$

2k)(2z)-4X

(22) -l-,gF;)

.

(22) -s

(n+5)1 +51r (n-4)

1 l-l!(z3) +2!(2n+3)

(2n+5) - ’ ’ ’

= It7

...

(&a)-‘-

(-l)“(n+),

..

(22)-L.

2k+1)(2z)+-’

10.1.3 y,(+-1*3s5

*;&

(2n-1)

lDl, 1

.

(n=O, 1,2, . . .)

(lf"a2n) -

(n+k):)l (n+3, k)‘k!r(n-k+l)

w)* +2!(1-2n)(3-2n)-

’ *’

(n=O, 1,2, . . .) Limiting

10.1.4

Pj,(z)-*

Valuer

n

k

1 1.3.5 . . . Pn+l)

2

3

4

5

-:,

26

34

Ei 30

6

. (2n-1)

1

as z-0

10.1.5 2”+$/&)+-1.3*5..

\

Ei 180 420

120 3::

1680 15120

30240

(n=O, 1, 2, . . .)

437

438

BESSEL

FUNCTIONS

OF

FRACTIONAL

ORDER

10.1.10 j,(z)=fn(z)

sin z+(-l)n+lf-n-~(z)

cos 2

f~(z)=z-2 f&9 =2-l, fn-,(~)+fn+l(~)=~2~+w-‘fn(4 (n=O, The

10.1.11

Functionsj,(z),

y,(z)

for

fl,

n=O,

f2,

. . .)

1, 2

j,(z)==s’nz

j,(Zl=~

sin 2

-7

cos 2

j2(z)=($-i)

sin 27$

*

cos 2

10.1.12 y&)=-j-&)=-~ yl(z)+-2(+-y-~

FIGURE

Y2(Z)=-j-~(z)=(-,~+~)cos

2-I

sin

2

10.2.

y,,(z).

n=0(1j3.

*

J,(x) t

* * ‘\ \ Y, lx) \ \ \ \

FIGURE

* Pokon’s

10.1.13

Integral

10.3.

j,,(z),

and

y.(z).

Gegenbauer’s

jn(z)=&Jr

z=lO.

Generalization

cos (2 cos0) sin*“+49 d13 ’ ’

(See 9.1.20.) 10.1.14 =f

-.+ FIQURE

*see page II.

10.1.

j,,(z).

n=0(1)3.

(-i)%‘e

u coBeP,,(cos 13)sin 0 de (n=O, 1, 2, . . .)

BESSEL Spherical

Remel

Functions

of

the

Second

F-UNCTIONS and

OF

FRACTIONAL

ORDER

Third

439

Modulus

and

Phase

Rind jr&>

10.1.15 yn(z)=(--l)“+‘j-.-l(z)

(n=O, fl,

*t,

=4&M&+*(4

YnW=&ZK+d4

. . .)

cos

ht+,c4,

sin

&+t(z>

(See 9.2.17.)

10.1.16 hl”(z)=i-f-‘z-‘e’~~

(n+&

10.1.27

k)(-2iz)-”

10.1.17 hyz)=i”+‘z-‘e-‘$

(n+t,

k) (2iz)-k

*

(See 9.2.28.) 10.1.28

10.1.18 h~~_,(z)=i(-l)“h~)(z)

10.1.29

h~~-,(z)=--i(-l)nh~)(z) Elementary Recurrence f.(z)

:j,c7!,

Y&f>,

h?(z),

10.1.30

Properties Relations

(%r/z)M~,2(2)=j:(z)+y:(z)=2-2+32-'+92-~

h?(z)

cross Products

10.1.19

L(z)

10.1.20

nfn-l(z)-((7l+l)fn+l(Z)=(2n+l)

f2,

. . .)

+fn+*(~)=(2~+l)~-1f,(~)

n+

fn(z> +g

10.1.31

j,(z>y,~~(z)-j,-~~z)y,(z)=2--2

10.1.32 2

fn(4

L+d~)Yn-l(+jn-l(~)Y,+l(~)=(2~+l)z-a 10.1.33

fn(4=fn-1W

jo(4jn(z> +Y&)Yd4

(See 10.1.23.) 10.1.22

(3?r/z)~~,,(z)=j:(z)+y~(z)=z-~+z-'

(n=O, 1, 2, . . .)

(n=O, fl,

10.1.21

(~/z)M:,,(z)=j;(z)+y;(z)=z-2

;r. (4 -g

=2 -2 q

f" (4 =fn+1(4

(-1)82”-ti

(k+Q-,,

(g

(See 10.1.24.)

(n=O, 1, 2, . . .) Differentiation

f&9

:ide,

Y&>,

W(d,

Analytic

Formulas

l.0.1.34

PC4

(n=O, fl, 10.1.23 10.1.24

( (

52, . . .)

5 -g m[zR+l~~(z)]=z.-m+lfn-,(z) > ;g

>

m[z-“fn(z)l=(-l)“z-“-nf.+,(z) (m=l,

Rayleigh’e

2, 3, . . .)

Continuation

j,(zemrf) =emnrfjn(z)

10.1.35

yn(ze”rf)=(-l)“e”nrfy,(z)

10.1.36

h~'(zP(h+l)*f)=(-l)fh~)(g)

10.1.37

h:‘(ze (~+y=(-l)tlgyz)

10.1.38

hi8’(ze”“)=hil)(z) (Z=l,

Formulae

10.1.25

Generating

10.1.39 i sin&&ZZ=~ 10.1.26

Y&)=-z”

z2r-n

-

2; m, n=O, 1,2, , . .)

Functions

(-t)"

7

y,-l(z)

@lKl4

0

--&1 d ncos 7 2 ( >

(n=O, 1, 2, . . .)

10.1.40

i cos JiiGZt=$$

jn-l(z)

440

BESSEL Derivatives

With

Respect

FUNCTIONS

OF

FRACTIONAL

ORDER

to Order

Fresnel

Integrals

10.1.41

,-0= (3+J IWW

sin x-Si(2x)

co8 2)

10.1.42

=Jzbs

(&f/x) [$ j.bl],w-l=

{ Ci(2x) co8 x+Si(2x)

(-1W2*+4(34

sin x}

+sin 33 I$ (-1W2n+3d341

10.1.43 [

3s 2g

10.1.54

$1/v(x) ,_o~Od4 I

{Ci(W

co8 x+[Si(Bx)--?r]

sin 2)

10.1.44 =JZb (+7r/x) {Ci(2x) sin x-[Si(22)-7r] Addition

Theorems

r, p, 8, x arbitrary

and

Degenerate

33 T

co8 2)

--cos b I$ (--lPJ*n+3,*(341.

Forms

complex; R=d(r2+p2-2rp

cos 0)

(See also 11.1.1, 11.1.2.) Zeros and Their

10.1.45

f$$=g

0

*10.1.46-c~=~

(2n+l)j&) (2n+l)

0

j,(Xp)P,(cos

efzcons=$

j,(Xr)y,(Xp)P,(cos

(2n+l)e~n"fj,(z)P,(cos

13)

e)

10.1.43 Jo(z sin e)=$

e2

Duplication

10.1.49 j&z)

(4n+l)

Asymptotic

Expansions

0)

Ire*‘el
10.1.47

(- l>"Jzn+t(34

j2,(z)P2,(cose)

The zeros of j,(x) and y,,(x) are the same as the zeros of J,++(x) and Y,+t(x) and the formulas for J?.$ and yp.# given in 9.5 are applicable with v=n+#. There are, however, no simple relations connecting the zeros of the derivatives. Accordingly, we now give formulas for a:,#, bh,,, the s-th positive zero of jk( z), y:(z), respectively; z=O is counted as the first zero of j;(z). (Tables of a;,,, L, [ 10.311.)

j,(&J,

y,(L)

are given in

Formula Elementary

=

fn(z)=j,(z)

co8 m?+y,(z)

Relations

sin art

(t a real parameter, If T. is a zero of fk (z) then

* Some

10.1.50

Ix&kite

Serier,

$I @n+l)

Involving

*see page n.

19.1.55 fn(7d=[rn/(n+ (See 10.1.21.)

l>lfn-1(7n)

jfi(z>=l

10.1.51 10.1.52

j:(z)

OD Si(22) T jX(+ -- 2z

10.1.56 (See 10.1.22.)

= bn/n>fn+l (7,)

10.1.57

=

5 [3,-n(n+l)] 1

S}-’

0 1111)

BESSEL McMahon’s

Expansions

for

FUNCl’IONS

n Fixed

and

OF

8 Large

10.1.58

FRACTIONAL

ORDER,

Uniform

Asymptotic Associated

441 Expansions of Zeros Values for it Large

and

10.1.63

4 .I, Gl **Iv 8- (P+7) (8/3)-l a:.* -(n+3) -;

(7$+

{z[(n+3)-2’3u:l

154pL+95) (s/3)--”

+gl hnt(n+))-“‘“a:l(n+3>-~~ -$

(85/.?+3535pz+3561p+6133)(8/3)-s

-g5

(6949p4+474908p3+330638j.i2

10.1.64 m

+904678Op-5075147)(8/V)-‘-

/3=7r(s++3)

...

for a:,,, fl=r(s+#n)

10.1.65

for bi,,; p=(2n+ly

Asymptotic

Expansions for

of Zeros n Large

and

Associated

Values

j, (d. .,I -&Wd> (n+3)+ h[(n+#-%:] (z[(n+~)-%:])-“2 OD

I1 +c H,[(n+))-2’3a:l(n+3)-2k }

10.1.59

k=l

a;.1-(n+~)+.8086165(n+#>Y3-.236680(n+~)-1’3 -.20736(n+~)-1+.0233(n+~)-5’3+

10.1.66 ...

10.1.60

Y,@L, .> - -&WU h[(n+$)-VQ


a:,,-(n+~)+1.8210980(n+~p3 {I+&

+.802728(n++)-‘la-.1174O(n+$)-’ +.0249(n++)-5’3+

...

10.1.61 j,(u:J

-.8458430(n+#-5’E

+.38081(n+~)-4’a-.2203(n+~)-2+

. . .}

10.1.62 y,(b:,,)-.7183921(n+~)-“‘6{

1-l.274769(n++)-2’3

+1.23038(n+))-4’3-1.0070(n+~)-2+

See [lo.311 6=2, 3.

for corresponding

. . .)

expansions

for

~k[(~+3)-2’3b:l(,+3$-2kj

A,(,$), z(t) are defined as in 9.5.26, 9.3.38, 9.3.39. a:, b: s-th (negative) real zero of Ai’( Bi’(z) (see 10.4.95, 10.4.99.) Complex

{ 1-.566032(~1++)-~‘~

I

Zeros

of h:‘(Z),

hY(Z)

f@ (2) and h$ (ze”“**), m any integer, have the same zeros. /&I) (z) has n zeros, symmetrically distributed with respect to the imaginary axis and lying approximately on the finite arc joining z=-n and z=n shown in Figure 9.6. If n is odd, one zero lies on the imaginary axis. hc)‘( z) has n+ 1 zeros lying approximately on the If n is even, one zero lies on the same curve. imaginary axis.

442

BESSEL FUNCTIONS

--I

-b

(-w-z(r)

(-i-h(l)

OF FRACTIONAL

ORDER

(--b)ha(r)

X:i E 0: 8

-.4409724 -.4572444 -.4702250 -.4802184 -.4875705

-.122500 --.114201 -.107243 -.101318 -.096159

-.06806 -.05986 -.05279 -.04674 -.04160

. 000000 : EE

. 00000

.065677 .078255

: z: .01592 .01983

:Ei :.0075 8%

1. 0

-.4926355

-.091561

-.03725

.087587

.02290

.0085

h(r)

HI(~)

ha(t)

Hz(b)

-.4926355 -.4131280 -.3551700 -.3108548 -.2757704

-.09156 -.05056 -.03043 -.01950 -.01310

-.2472521 -.2235898 -.2036314 -.1865701 -.1718217

--.00914 -.00658 -.00485 --.00366 -.00280

-.1589519 -.1476304 -.1376005 -.1286601 -.1206469

--.00219 -.00173 -.00138 -.00112 --.OOOQl

-.1134296 -.1069004 -.1009699 -.0955634 -.0906180

-.00075 -.00062 -.00052 -.00044 -.00037

.005753 .005111

-.0860804 -.0819049 -.0780523 -.0744888 -.0711850

-.00032 -.00027 -.00023 -.00020 -.00018

.003313 .002998 .002722

-.0681152 -.0652570 -.0625905 -.0600985 -.0577653

-.00015 -.00013 -.00012 -.00010 -. 00009

.002070

-.0555773

-.00008

.001375

t-u-+ 0. 40 2: .3: .20

-.037 -.014 -.006 -.003 -.OOl

.0229 .0121 : :xi; .0027 : 88:; .0008 .0006 .0004 .011217 .009701

: 88:;

:.006505 E%Y

: 8Ez .OOOl

: 8X:X:8 .003672

: ::322" : Et941 : xX:1::

hl(I) -.0645731 -.0487592 -. 0352949 -.0242415 --.0155683

:i :OS .04

-.0091416 -.0047276 -.0020068 -.0005965 -.0000747

. 00

-. 0000000

I

Mf) -.00013 -.00005 -.00002 -.00001

HIW

BESSEL

Modified

10.2.

Spherical

Bessel

FUNCTIONS

Functions

OF

FRACTIONAL

4-43

ORDER Wronskians

10.2.7

Definitions Differential

~{ma,+&),

Equation

10.2.1

&q&&z)}

=(--l)n+F

10.2.8

zw’+2zw’-[22+n(n+1)]w=o (n=O, fl,

f2,

Particular solutions are the Modijied Bessel junctions of the jirst kind,

. . .)

Spherical

by

Elementary

Functions

10.2.9 J~In+~(z)=(22)-1[R(n+~,--z)ez

10.2.2

~In+t(z)=e-n”“2j,(ze”f’2)

(--?r<arg

=e3n*t/2jn(,e-3rU2)

of

Representations

2<+1r>

(h<arg

2 54

-(-l)“R(n+$,a)e-“1 10.2.10 ~3nIJ_,-~(z)=(22)-'[R(n+~,-z)eZ

the second kind,

+(--lPR(~+3,a>e-‘l 10.2.11

10.2.3 ~I-n-f(Z)=e3(n+1)rr'2yn(ze'f'2

R(n+$f, (--?r<arg

=e

z)=l+#

(22)-l

2 244

(n+2)! +2!p(n-1)

-(n+l)*f/2yn(Ze-3rf/2) (h<arg

z<4

(22)-S+

...

n

of the third kind,

=C

10.2.4

0

(n+3, k)@4-n (n=O, 1,2, . . .)

(See 10.1.9.) 10.2.12

The pairs

&&n++(z)=g,,(z>

sinh z+g-,-l(z)

cash z

and vmL+d4 2lm&+,(4 are linearly independent solutions for every n. Most properties of the Modified Spherical Bessel functions can be derived from those of the Spherical Bessel functions by use of the above relations. Ascending

g n _ 1 (z) - g A ~~~;~;;~~;;~-=g;~)-2 1

The

Functions

J&&I*

Series

&i7iIl,2(z)

3.5 . 2” . . &+I) (32”)

2

* * *)

10.2.6

n=O,

1, 2

=+

sinh z

&&,&)=-~+y

1+,!(:+3)+2!(27&+3)(2n+5)+

(n+*) (2))

&2, . . .)

10.2.13

12

{

n

(n=O, fl,

10.2.5

dmn+t(4=l.

1

~I~,2(z)

=(s+k)

cash z

sinh s-3

cash z

10.2.14 J3?rlzI...l,2(z> -559

12

&qa-3,2(z)

=q-=g

(322>2

‘+l!(l?Zn)+2!(1-2n)(3-2n)+

’’’ (n=O, 1,2, . . ,)

&&I-~,~(z)=-$ *See

page

11.

sinh z+(-$+i)

cash s

444

BESSEL

Maed

Spherical Bessel Functions

FUNCTIONS

of $he

Third

OF

FRACTIONAL

ORDER

Rind

10.2.15 &$~K,+t(z)

=3aie(~+1)r~‘2h~1)(zel”) (- 7r<arg 2 I+r) =- 4 Irie-'"+l'"'/2h~2'(Ze-fri) 25 4

(h<arg

n = (&r/z)e-z c (n+ 3, k) (2.4

--L

0

10.2.16 (n=O, 1,2, . . .)

Kn+t (2) =L&) The

10.2.17

Functions

d$&K,+&),n=O,

&J&I,2(z)= &&X2,2(2>

1, 2

($r/z)e-’ = (3dz)e-z(1 +z-‘1

~~~12(~)=(3?r/z)e~‘(l+32~‘+32~2)

Elementary Recurrence J&(4:

d&L+&>,

Properties Relations

FIGURE 10.4. &L++(x),

&

IL+&).

n=0(1)3.

(--l)n+l&&&+tM

(n=O, fl,

10.2.18

fn-l(z)-fn+,(z)=(2n+l)z-‘f,(z)

10.2.19

nfn-l(z)+(n+l)f.+l(z)=(2nfl)$f.(z)

zk2,. . .)

10.2.20

G fdz> +-& fn(4 =fn-I(4 (See 10.2.22.)

10.2.21

32-

-~~~cz,+$f.(z)=fn+l(z)

(See 10.2.23.) DilTerentiation

Formulae IO

fn(4:

vmL+&>,

(-1)“+‘4mIL+,(4

s-

(n=O, fl,

10.2.22

(

I4-

*2, . . . )

z-

; -$ ~[zl+tfn(z)l=z”-“+%-01(2) > I

10.2.23

2

l[z-Iyn(z)l=z-.-mf.+m(z) (m=l,

2,3,.

..)

FIQURE 10.5.

3

4

5

6

7

f K,,+&).

x=10.

BESSEL Formulas

of Rayleigh’s

10.2.24 &&I,+&)=z~(,

FUNCTIONS

OF

FRACTIONAL

Type

$

Addition

Theorems

r, p, 8, h arbitrary

+

445

ORDER and

Degenerate

complex;

Forms

R=&*$p*-2rp

cos 8

10.2.35 10.2.25

J~Ln-*(z)=2” (; g >*C=p (n=O, Formulas

for

IX++(Z)

-IT,-+

1,2,

. . .)

hmm”+t(~P)l~n(eos

0)

10.2.36 eEcase=$2 @n+l) hmL+t~4lcdcos

(2)

10.2.26

0)

10.2.37

2$ -$ (-l)k+’ (2n-4) ! (2n--2k)! k! [(n-k)

(n=O,

10.2.27 10.2.28

(22)a-2n

e--y$

(-1)“(2n+l)[~~~+t(z)]P,(cos

e>

!]’

1,2,

. . .)

&r/2)[1:*(2)-12-~,*(2)]=-2-*

Duplication

Formula

10.2.38 &+tcw= nf,-+2n++

(&t-/2)[1f,2(2)-12_a,2(2)]=2-2-2-4

2

(-1)‘(2n-2k+1) k!(2n--k+l)!

0

10.2.29

K:

r++(2) -

10.3. Riccati-Bessel Functions

(3?r/2)[I~,,(2)--II,,,(2)]=-2-*+32-4-92-~ Generating

Differential

Equation

10.3.1

Functions

2w’+[z2-n(?&+

16.2.30

(n=O,

- (-it)" 7

k sinhdm=C

0

Pairs of linearly

[,4&I-,,++(2)]

cw
10.2.31 - n! (it)” [,&51,&z)]

5 cosh,/mt=T With

Respect

to Order

10.2.32

f2,

. . .)

independent &M,

zy&)

2l@(2),

zhf'(2)

solutions

are

Functions

zj,(z),

q/.(z),

n=O,

1, 2

10.3.2 c&(2) =sin

[g IJx)]~~~=-&~

fl,

All properties of these functions follow directly from those of the Spherical Bessel functions. The

Derivatives

l)]w=O

2,

2j,(2)=2-'

sin z-co9

2j2(2)=(3X-2-1)

[Ei@x)e-“-I$(--2x)4

2

sin 2-32-l

COS 2

*

CO9 2

*

10.3.3 10.2.33

2&l(2) = -CO8

2,

q/~(2)=-sin

Zy&)=

[

&I”(X)

1

v=-,=&

10.2.34 [$K.(x)]vmht=

[Ei(2x)e-“+E,(-2x)eZ] T-@%Ei(-2x)ez

-32-l

*see page n.

Sin

2-(32-2-1)

co9 2

wronskians

10.3.4

W{ zj,(z),

zy,(z)}

10.3.5 W(zh~~'(z), zhjf’(2)) For E,(x) and Ei(x), see 5.1.1, 5.1.2.

2-2-l

=l

=-2i

(n=O,

1, 2, . . .)

446

BESSEL

10.4. Definitions

Airy

and

FUNCTIONS

Functions

Elementary

Differential

Properties

OF

FRACTIONAL

ORDER

10.4.12

W’{Ai (z), Ai (ze-2*1’3)} =&rm1eTi16

10.4.13

W{Ai

(ze 2rf/3),

Ai

(ze-2*1/3)}

=.$in-l

Equation

w”-zw=o

10.4.1

Pairs of linearly independent solutions Ai (z), Bi (z), Ai (z), Ai (ze22*1’3), Ai (z), Ai (ze-2ff’3). Ascending

are

Series

10.4.2

Ai (z)=cJ(z)-c2g(z)

10.4.3

Bi (z) = & [cJ(z) +c2g(z)] 1 1.4 1.4.7 f(z)=l$g 23+w zs+ 9! z9+ * * * =q

3 (gk

.a -

&

g(z)=z+;,l+y. z’+$ ,10+. . . =$ 3”(gk&$ 1 (a+- 30>=l

-.6

-

-3

-

-1.0

I-

”

FIGURE 10.6. Ai (h),

Ai’ (3s~).

1 3” a+- 3 x=(3a+l)(3~+4) . . . (3a+3k-2) (CXarbitrary; k=l, 2, 3, . . .)

( >

(See 6.1.22.)

2.2 c

10.4.4

2.0 -

cI=Ai

(0) =Bi

(O)/&=3-2/3/I’

(2/3) =.35502

10.4.5 c2=-Ai’

(@)=Bi’

10.4.6

Bi (z) =e*“’

80538 87817

I.6 -

(0)/q?j=3-1/3/~(1/3) =.25881

Relations

. LB -

Between

94037 92807

Solutions

Ai (ze*~*~) +e+”

Ai (~e-~**/~)

10.4.7 Ai (z)+,+tn

Ai (~2~ffl)+e-2~“~

Ai (~-2~*/3)=()

Bi (~2~*fl)+e-2*f”

Bi (~--2*~f/3)=0

10.4.8 Bi (z)+e2rifi

10.4.9

Ai (ze*2rg/3)=+e*r~13[Ai

(z) ri

Bi (z)]

Wroxlskians 10.4.10 10.4.11

W(Ai

(z), Bi (z)} =?r+

W{ Ai (z), Ai (ze2*1’3) 1 =+r-le-+l/*

-.6

-

-.s

-

-1.0 -

FIGURE 10.7. Bi (kts), Bi’ (~3).

BESSEL Rep~~ntations

in Terms

FUNCTIONS

Beed Functions

of

+323/3

OF

FRACTIONAL

447

ORDER

10.4.30 I~2/3(f)=(fl/2z)[fJ3

Ai’(z)+Bi’(z)]

10.4.31

Ai' (4

= -ddW)

K+tdO

10.4.14

Integral

Ai

10.4.15

(3~)~I% Ai [f (3a)-%]= Ai (--2)=~~~[Jl,a(r)+5-1,3(r)l

+e-*lW~S(f)l

=j~z/3[e**‘“Hlfb(O

Bi [ f (3~)~*%] =

* -Ai’(z)=~z[I-3,3(T>-~3,3(~)]=*-‘(Y~)~3,3(~)

10.4.17 Ai' (-2) =

s The

-~zLJ--3,3(r~

10.4.34

-l-11,&)]

s0

10.4.35

’

Ai ( f t)dt,

10.4.20 Bi’ (2) = (z/@)[I-2&)

10.4.36

s0

’ Bi ( f t)&

+Jdt)l

in

Terms

of Airy

‘Bi (t)&=$

Jr

s0

S

‘Bi (--t)a!t=$

Jr

+~,/3(G3d~

LJ--1,3(~)-J1,a(Ol~~

for

So’Ai(il)at,So’Bi(ft)dl

* Ai (t)dt=c&)-c&(z)

S

‘Ai (--t)dt=-c$(--z)+c&(-z)

0

(-2)--i

10.4.24 H3,3({) = eirf”m[Ai

r [I-1,3(0

S 10.4.2.)'

10.4.39

Ai (--z)TBi

= e=W’&i&ii

v-l,a(~)+Jl,a(~)l~

0

Series

10.4.38 (See

-lua(Qld

0

Ascending

; 1 2J3 >

10.4.22 JA1t3(t)=3J3Tz[fi

v-l,,(t)

0

0

of Bessel Functions Functions

(

s0

10.4.37

=3~(2/~)[e-~‘mH~~b(~)-e*f’eH~:~(~)]

2=

Jr

s ’ Ai (--t)&=a

-I-b(C)l

10.4.21 Bi' C-4 = (d&LJ--~dt)

’ Ai (t)d!=~

0

Bi <-z>=~[J-1,3(f)-J1/3(r)l =3;~[e*“‘H~Jb(~)-e-~*“H~~~(f)l

(-z)] Bi (-z)]

(- z) +i Bi (- z)]

10.4.25 1A1,3({)==3@[F 4 Ai (z)+Bi

(z)]

10.4.40

’ Bi (t)dt=fi[cJ’i’(z)

s0 (See 10.4.3.) 10.4.41

S

+sG(z>l

‘Bi (--t)dt=-&$Q’(-z)+c&(-z)l

0

10.4.26

K*, /3(f) =4G Ai (4 10.4.27 J~2,3({)=(~/2z)[~~Ai' (-z)+Bi’(-z)] 10.4.28 =er*l”(fi/z)[Ai’

10.4.29 Hi;&) =ez*fflH!!&,(~) =e -““‘(fl/z)[Ai

(-2)-i

Bi’ (-z)]

G(4=s =$

(-2) +i Bi’ (-z)]

1

w=z+a

Hi;&({) =e-“f’3H!‘:,3(f)

*seepagen.

s0

+3z3f3

10.4.19

H&(f)

Integrals

(ut3fd)]&

+e~lW%(f)l

Bi (2) =&%1-1,3(s)

Representations

om [exp (--at3fzt)+sin

-Jwa(f,l

=)(z/&)[e+WifW

10.4.18

- co9 (ut3fzd)dt

s0

10.4.33 (3a)++r

10.4.16

30.4.23

Representations

10.4.32

(z)=~~[I-1,3(Z)-l~,3(~)1=~-‘~~,,3(r)

1

z*+y

z’+$j

2

3k (:)k

1.4

z’+

1.4.7 zlO+ . . . 101

2.5p,+2.5.8 -yp--2. z6+ 81

1, +...

&

The constants cl, c2 are given in 10.4.4, 10.4.5.

BESSEL

448 Tbe Functions

FUNCTIONS

OF

FRACTIONAL

ORDER

Differential

Gi(z), Hi(z)

10.4.42

Equations

10.4.55

Gi (z), Hi (z)

for

-pw=--a-’

W”

Gi (z)=s-l~-sin(~P+zt)dt w(0) =i =gBi

(z)+s’[Ai

(z) Bi (t)-Ai

w’(O)=aBi’

10.4.43 (z)+S’[Ai’(z)

Bi (t)-Ai

(0)=-l

55424 78

10.4.44

Ai’ (0)=.14942

8

-zwq(-l

W”

w(O)=: Bi (O)=-$ Ai (0)=.40995

Hi(z)=7rS11

expi-i

Pfzl)

(z)+J’[Ai

9452449

w(z)=Gi(z)

(t) Bi’(z)]dt

10.4.56

=iBi

Ai (0)=.20497

(t) Bi (z)]dt

0

Gi’ (z)=iBi’

Bi (O)=$

10849 56

dt

(t) Bi (z)-Ai

Bi(t)]dt

w’(O) =f Bi’ (0) =--$

Ai’ (0) =.29885 89048 98

0

w(z) =Hi

10.4.45 Differential

Hi’(z)=iBi’

(z)+S’[Ai

(t) Bi’(z)-Ai’

Bi (t)]dt

0

10.4.46

Gi (z)+Hi

Representations

10.4.47 s0

of

(z)=Bi

s by Gi (*i),

‘

(z)

Ai( f t)dt,

s0

‘ Bi( f t)dt

10.4.57

*Ai (t)&=&r[Ai’

of

Airy

Functions

are

Ai

W “‘--4~w~-~w=o

Wronskian

(z)Gi(z)-Ai

(z)Gi’(z)]

for

Asymptotic cO=l,

(2) Hi (z)-Ai

ck=

(z) Hi’ (z)]

solutions

Producta

of Airy

(z),

Functions

-Ai

(-2)

Gi’ (-z)]

10.4.50 (-2)

-Ai

(--z)Hi’(-z)]

ck

(k=l,

10.4.53 (-2) Gi (-2) -Bi

(-2)

Gi’ (-z)]

(-2) Hi (-2) -Bi (-2) Hi’ (-z)]

r-1'2z-1'4f?-r $ (-l)kckl-k

10.4.60 Ai (-z) ,.,~-Wg-1/4

’ Bi (t)dt=?r[Bi’ (z) Gi (z)-Bi (z) Gi’ (z)] s0 10.4.52 =-?r[[Bi’ (z) Hi (z)-Bi (z) Hi’(z)]

=r[Bi’

[a I Large

2,3, . . .)

10.4.59 kii (+a

Hi (-2)

10.4.51

’ Bi (-t)dt=-r[Bi’

for

Gi (-2)

‘Ai (--t)& =-$r[Ai’(-z)

=i+r[Ai’

Expansions

r(3k+3) -(2k+1)(2k2+&..(6k-J), 54%!r(k+3)-

&=l,dk=-6g

10.4.49

10.4.54

Products

10.4.58 W{AP (z), Ai (z) Bi (z), Bi2 (2)) =2ca

=-$r[Ai’

s0

for

Linearly independent Ai (z) Bi (z), Bi2 (2).

Hi (fz)

10.4.4%

s0

Equation

(z)

10.4.61 Ai'(-+T-

((mg zl
BESSEL

FUNCTIONS

OF

FRACTIONAL

449

ORDER

10.4.70

10.4.62 Ai’ (-Z)N-T

-+

z+ [cos

(5+;)

I$

(-l)%,r-“:

Ai’ (-z)=N(x)

+sin(C+$ I$ (--Ukd2k+l Fml]

cos 4(z), Bi’ (-z)=N(z)

N(z)=J[Ai’*

(--2)+Bi’2 $+)=arctan

Differential

10.4.63

Priiea

Bi (z)*n-

+,z-f& 2

ckJ-k

(Ia% 4<3 4

0

Equations

+sin (C tr)

$

1

(--1)kc2k+~F-’

(larg 4<3

for

Modulus

and

N2=M~2+jjpe’2=Mf2

10.4.73 10.4.74

NN’=

tan ($4)

4

10.4.65

10.4.75

Bi (zefrtja)

10.4.76

Phaw

--r-lx + ,rr-2M-2

*

-xMM’

= Me’/M’= - (rMM’)-l, MN sin (+-e)=fl

M" +xM-,+&f-a=O (M2)“!+4z(M’)‘--UP=0

10.4.77

e'2+3(e"'/e'>-g(e"/e')2=2

Asymptotic

Expansions

of Modulus Large 2

and

(-1)” 10.4.78. M2(z) -t z-1/2 T- m

Phase

2” (;)

10.4.66 *

Bi’ (z)*n%fef

$

d,t-“: (jarg

(-x)1

with respect to x

&pet= - *- 1, p#=

10.4.72

10.4.64

(-s)], [Bi’ (--z)/Ai’

denote differentiation

10.4.71

sin 6(z)

for

(2x)-“”

3k

zj<)r)

10.4.67 Bi’

$’

(--z)vr-f2f

-COS

(

,+a

>

$

(-l)kd,r{-a

*2?g

(-11)u,+~~-‘-‘]

(22)-o+'28,2,03;e525(22)-"-

. . .]

10.4.80

(larg A<3 4 10.4.68 10.4.81

Bi’ (zefrila)

;c3’2[ 1+;

4(x) +-; +

hl 2 $ (-l)“d,k+,r-“-l-j

4+

(2x)-3-

+22)-'

(2s)-9-2065,~~(22)-'p+

. ..I

>

(lw 4<3 4 Modulus

and

Phase

10.4.82

10.4.69 Ai (-2) =M(s) M(z)=J[Ai2

*see

page II.

cos e(s), Bi (-5) =M(z)

Asymptotic

sin e(z)

(--s)+BP (-z)], B(z)=arctan [Bi (--s)/Ai

(-s)]

10.4.83

Forms

s0

of

s

‘Ai 0

'Ai (t)&,.,i-i

( f 1) dt,

s

“Bi ( f f) dt tot

Large

0

~-U2~-3/4 exp (-Ei/l)

3:

450

BESSEL

S= S' Bi

10.4.84

Bi

FUNCTIONS

exp

(t)dt-s-1/2z--3/4

0

10.4.85

(- t)&-~-1&--3/4

2

(

3 x312 >

sin (i x3/,,:)

0

AsymptoticFormsofGi(rtz),Gi’(*z),Hi(fz),Hi’(f~) for Large

10.4.86

OF

FRACTIONAL

ORDER

10.4.104

10.4.105 Bi (&)=(-l)a-l&e*~‘Egl

(21 sufficiently

Gi (2) &x-*x-’

10.4.87 Gi (-x) ~~~~~~~~~~~ cos (; .3,2+g

f(z)-22’3

1+$

z-2-4

10.4.89 Gi’ (-x) ~1r-1f2x1/4 sin (; x3,2+;) 10.4.90

Hi (x) ~~~~~~~~~~~ exp (zx”‘“)

10.4.91

Hi (-z) WFW~

fI(Z) wr-“%z1’6 Their

Asymptotic

10.4.94

a*=--j[3*(4s-1)/8]

10.4.95

a:= -g[3*(4s-3)/8]

b:= -g[3a(4s-

10.4.102 /3a=e*f'3f $ (&-l)+$ln [

‘See

page

II.

23 97875 z-e-ee. 6 63552 7 -2 1673 m4 * 96' +6144’ 843 94709 -26542080 '--'+ ' - *

-1/22-1/6

g&IN*

I-(

Formal ential Points

and Asymptotic Equations of

Solutions Second

of Ordinary Order With

DifferTurning

X)W'+b(z, X)W=O

1)/8]

10.4.101 Bi (b:) = (- l)8gI[3r(4s-

10.4.103 /3i=e1i13g

5 2-2-46o8 1525 2 -4

in which X is a real or complex parameter and, for fixed X, a(~, X) is analytic in z and b(z, X) is continuous in z in some region of the z-plane, may be reduced by the transformation

10.4.107 W(z)=w(z)

Bi’ (bJ= (- l)+‘f[3~(4s-3)/8]

10.4.100

. . .)

+

10.4.106 W'+a(z,

1)/8]

10.4.97 Ai (a:)= (- 1)8-1g1[37r(4s-3)/8]

10.4.99

- ..* >

An equation

10.4.96 Ai’ (a,)= (- l)*-‘f[3~(4s--

b,=-j[3*(4s-3)/8]

1 +s

Expansions

Ai (z), Ai’ (z) have zeros on the negative real axis only. Bi (z), Bi’ (z) have zeros on the negative real axis and in the sector $r<] arg z]<+x. a,, a:; &, b: s-th (real) negative zero of Ai (z), Ai’ (2); Bi (z), Bi’ (z), respectively. & ~3:; s,, 5: s-th complex zero of Bi (z), Bi’ (z) in the sectors &r<arg z<+r, -$r<arg z<-f?r, respectively.

10.4.98

3344 30208

9 11458 84361 z-*0+ 1911 02976

Hi' (-5) A, -5 ~-1x-2 and

z-6

186 83371 --8 + 12 44160 ’

Hi’ (x) ~~~~~~~~~~ exp ($x3/2)

Zeros

z-4+%

181223 --B z-2+g8 c4-- 207360 2

g(z)-22’3 (1-i

10.4.93

large

16 23755 96875 z-Io

+

10.4.92

ln 2

1080 56875 z-8 69 67296

7 Gi’ (z) mss x-1x12

10.4.88

1

$ (JS-~)+:

2

exp (-aJ’u(t,

h)dt)

to the equation

1)/8]

1

2

10.4.108 w"+Q(z, X)w=O

cp(z,A)=&

A,-; u2(2,A)-f $ u(z, A).

BESSEL

If &,

X) can be written

10.4.109

FUNCTIONS

A>

where a(z, X) is bounded in a region R of the zplane, then the zeros of p(z) in R are said to be turning points of the equation 10.4.108. The

Special

Case

w”+[X*z+p(z,

X)Jw=O

Let X=IXle*” vary over a sectorial domain S: lXll&,(>O), wl<w_<wz, and suppose that n(z, X) is continuous in z for Izl
0

qn(z)Xmn as X+w in S. Formal

Series

FRACTIONAL

451

ORDER

10.4.114

in the form

&, v=~2PM+!&

OF

y&)=Ai

(-X2~x)[1+O(~-1)J

yl(x)=Bi

(-X2%)[l+O(X-1)]

For further representations and details, WQ refer to [10.4]. When z is complex (bounded or unbounded), conditions under which the formal series 10.4.110 yields a uniform asymptotic expansion of a solution are given in [lo.121 if q(z, X) is independent of X and /Xl+= with fixed o, and in 110.14] if X lies in any region of the complex plane. Further references are [10.2; 10.9; lO.lO].

Solution

10.4.110 w(z)=u(z) q $9n((Z)h-"+X-1u'(z) q Sn(z)X-" u"+x2zu=o co, cl constants

(N-j-=>

The

General

Case

w”+[Vp(z)+p(z,

h)]w=O

Let X=IXle’” where IXj>b(>O) and -r_
10.4.11s

t (g2=P(z)

t

yields the special case (n-0, Uniform

Asymptotic

Expansions

1, 2, . . .)

Y”+[A2x+q(x,

*

of Solutions

a, x)=(~)-2*k A)-($)-* $ [(gy

For z real, i.e. for the equation

10.4.111

10.4.116

Exumple:

UlY=O

where zcvaries in a bounded interval a Iz 5 b that includes the origin and where, for each fixed X in S, a(s, X) is continuous in x for a <x
10.4.112

Consider the equation

10.4.117

y"+[x2-(x2-~)x-2]y=o

for which the points x=0, = are singular points and x=1 is a turning point. It has the functions x~J&x), x+Y&x) as particular solutions (see

9.1.49). The equation

y,,(x)=Ai yl(x)=Bi and, uniformly

10.4.115 becomes

(-X2/“z)[l+O(X-1)] (-X2/3x)[1+O(X-1)] in x on 0 lxlb

whence 8 (-,$)“I”=

10.4.113 y,,(x)=Ai

(-X2Dz>[1+O(X--l)]+Bi

(-X2%)O(X-1),

yl(x)=Bi

(-X2/“;E>[1+O(X-‘)]+Ai

(-X2flx)O(X-1) 0-J)

(ii) If $%‘X20, JX 20, there are solutions yo(x>, yl(x) such that, uniformly in x on alx_
-m+ln

~~‘(1 +Ji-F”) @<xS (15x<

Thus

10.4.118

v&) =(gy4

y(x)

a>.

1)

BESSEL

452

FUNCTIONS

OF

FRACTIONAL

ORDER

satisfies the equation

It follows from 10.4.120 that uniformly o<x< c0

10*4.119

10.4.125

which is of the form 10.4.111 with x replaced by 5 and n(E, X) independent of X. Suppose 3”x>O, 3%#0. By the first equation of 10.4.114 there is a solution v,,(t) of 10.4.119, i.e., a solution y,,(x) of 10.4.117 for which the representation

Jx (xx)

10.4.120 %B> =

Numerical

c

1’4y,,(z)=Ai(-~2~3[)[I+O(~-1)]

Spherical

holds uniformly in x on O<x
2

Hence, using

4.6 and 4.8, $Ol.+W=-

exp (; xw/‘)

-,-112~-1/3(-5)11*

Functions

From 10.1.11, jI(x)=y-‘T. Tables

( rel="nofollow"> x2-1

Bessel

of the Tables

To computej,(x), y,(x), n=O, 1, 2, for values of x outside the range of Table 10.1, use formulas 10.1.11, 10.1.12 and obtain values for the circular functions from Tables 4.6-4.8. Example 1. Compute j,(x) for x= 11.425.

YCM =x

Methods

10.5. Use and Extension ( X”E >

in x on

.90920 500 .41634 873 c11.425j2 11.425 = -.00696

54535-

.03644 1902

= - .04340 7356. 1 C-r 2

-1,2x-l/6

(k$)-“’

exp

(;

x(+)3,2)

Let now X be fixed and x+0 in 10.4.121. results 10.4.122

yO(x) -3 ,-1’2A-1“‘x”2

There

(+ x)” eh.

On the other hand, y,,(x) is a solution of 10.4.117 and therefore it can be written in the form 10.4.123

ye(x) =~“~[c,Jx(Xx)

+sYx(Xx)]

where, from 9.1.7 for X fixed and x+0

w> x cot Xa- w>+ Y,(Xx) -r(x+l) r(l--x)

csc ha .

Thus, letting x+0 in 10.4.123 and comparing the resulting relation with 10.4.122 one finds that c2=0 and 10.4.124

y,(x)=

$7r- 1’2h-X-1’6eXr (A+ 1)~“~ J,(xx).

To compute j,(x), 11 In 120, for a value of x within the range of Table 10.3, obtain from Table 10.3, directly or possibly by linear interpolation, j,,(x), j,,,(x) and use these as starting values in the recurrence relation 10.1.19 for decreasing n. An alternative procedure which often yields better accuracy and which also applies to computations of j,(x) when both n and x are outside the range of Table 10.1 is the following device essentially due to J. C. P. Miller [9.20]. At some value N larger than the desired value n, assume tentatively FN+I=O, F,=l and use recurrence relation 10.1.19 for decreasing N to obtain the sequence FN--l, . . . , Fo. If N was chosen large enough, each term of this sequence up to F, is proportional, to a certain number of significant figures, to the corresponding term in the sequence j,-,(z), . . ., j,(z) of true values. The factor of proportionality, p, may be obtained by comparing, say, F. with the true value j,(x) computed separately. The terms in the sequence pF, are then accurate to the number PFO, . . . of significant figures present in the tentative values. If the accuracy obtained is not sufficient, the process may be repeated by starting from a larger value N.

BESSEL

OF

for x=24.6. Interpolation in Table 10.3 yields for x=24.6 ~-~~ez)‘*~j~~(~)=(-28)3.934616 ~-“~e=“*~j,,(x)= (-27)9.48683 whence . Example

Compute

FUNCTIONS

2.

recurrence

-

.00890 .02484 .04628 .04099 .00871

relation

10.1.19

[.00890 [ - .02485 [ - .04628 [ - .04099 [-.00871

67660 93173 17554 87086 65122

there 701 901 161 881 671

For comparison, the correct values are shown in brackets. To compute j,,(z) for x=24.6 by Miller’s device, take, for example, N=39 and assume Fa=O, Fs8=1. Using 10.1.19 with decreasing N, i.e., F,-,=[(2N+l)/x]F,-FAT+,, N-39, 38, . . ., 1, 0, generate the sequence Fag, F3,, . . . ,’ F1: Fo, compute from Table 4.6, j,(24.6)= (sin 24.6)/24.6 = -.02064 620296, and obtain the factor of proportionality p=jo(24.6)/Fo=

.OOOOO 03839 17642.

The value pF,b equals j,,(24.6) to 8 decimals. The final part of the computations is shown in the following table, in which the correct values are given for comparison. N

FN

-22704.71107 + 78178.88236 13 f114866.80811 +47894 44353 :: -66193. 59317 10 - 109782. 76234 -27523.39903 : + 88524.85252 7 + 88699.11017 -34440.02929 t - 106899. 12565 - 13360.39272 3” f 102011. 17704 2 + 42387.96341 -93395.73728 A - 53777.68747

:t

j, (24.6)

PFN

I

-. +. +. +. -. -. -. +. +. -. -. -. +. +. -. -.

00871 03001 04409 01838 02541 04214 01056 03398 03405 01322 04104 00512 03916 01627 03585 02064

67391 42522 93941 75218 28882 75392 67185 62526 31532 21348 04602 92905 38905 34870 62712 62030

-. f. f t. -. -. -. + +. -. -. -. +. +. -. -.

00871 03001 .04409 01838 02541 04214 01056 .03398 03405 01322 04104 00512 03916 01627 03585 02064

674 425 939 752 289 754 672 625 315 213 046 929 389 349 627 620

It may be observed that the normalization of the sequence FN, FN--I, . . ., F. can also be obtained from formula 10.1.50 by computing the sum c=e (2k+l)fl and finding p=l/& This 0

yields, in the case of the example, .ooooo 03839 177.

ORDER ModiCed

Jo

j,,(24.6) = .05604 29, j,,(24.6) = .03896 98. From the results j,,(24.6) = j,,(24.6) = j,,(24.6) = j1,(24.6) = j,6(24.6) =

FRACTIONAL

To

compute

Spherical

453 Bessel

&d~~~+t(xl,

Functions &@L+t(x),

p=

0,

1,2,

. . . for values of x outside the range of Table 10.8, use,formulas 10.2.13, 10.2.14 together with 10.2.4 and obtain values for the hyperbolic and exponential functions from Tables 4.4’ and 4.15. In those cases when &~/XI,,+(X) and &T/XI-~-~(X) are nearly equal, i.e., when x is sufficiently large, compute &JZ&+,(x) from formula 10.2.15, for which the coefficients (n + 4, %) are given in 10.1.9. Example 3. Compute &/x1&x), &~/xX&,c) for x= 16.2. From 10.2.13, &$JI~,~(x) = (3+1) sinh al/SC”,3 cash X/S?; from Table 4.4, cash 16.2=(6)5.4267 59950 and this equals the value of sinh 16.2 to the same number of significant figures. Hence &?r/16.2&,,(16.2)=(.06243 402371 -.01143 118427)[(6)5.4267 =338814.4594-62034.29298 =276780.1664. To compute obtain

&a/16.2&~(16.2)

599501

use 10.2.17

=(-7)2.8945

38069[.036932

=(-8)1.0690

28283.

and

604001

To compute &$~I,,+i(x), 3In18, far a value of x within the range of Table 10.9, obtaiin from Table 10.9, ~+r/~l~~,~(x), &/xI~~,~(x) for the desired value of x and use these as starting values in the recurrence relation 10.2.181 for decreasing n. To compute &&K,+*(x) for some integer n outside the range of Table 10.9, obtain from 10.2.15 or from Table 10.8, &&d&(z), &$cK~,~(x) for the d esired value of x and use these as starting values in the recurrence relation 10.2.18 for increasing n. If x lies within the range of Table 10.9 and n>lO, the recurrence may be started with &7r/~K,~~~(2), &&K~&c) obtained from Table 10.9. Example 4. Compute &$JK,,,(x) for x*3.6. Obtain from Table 10.8 for x=3.6 &&KID(x)

= .01192 222

&&K&x)

= .01523 3952

p=l/&=

454

BESSEL

The recurrence relation

yields successively

10.2.18

-&?r/3.6&2(3.6)=-.Oll92

222 -&

=-.02461 J&J575K,,2(3.6)

FUNCTIONS

=.01523

(.01523 3952)

OF

FRACTIONAL

ORDER

y’(x), respectively, the following formulas may be used, in which d, d’ denote approximations to c, c’ and u=y(d)/y’(d), v=y’(d’)/d”y(d’). c=d--u--ad

$2

$24d2

5;

718 +88d $-

3952 +;6

(88+72Od3) $

+5856d2 ;-(16640d+40320d4);+

(.02461 718)

...

= .04942,4480 -Jj$i3Kg,2(3.6)=-.~246i

718 -A =-.12072

&/3.6K&3.6)

-(105+76d’3+24d’8);

(.04942 4480)

-(945+756d’“+272d$-.

034

= .04942 4480

+;6

y’(c)=y’(d)

As a check, the recurrence can be carried out until n=9 and the value of &/3.6K,,,,(3.6) so obtained can be compared with the corresponding value from Table 10.9. To compute &&In++(x) when both n and x are outside the range of Table 10.9, use the device described in [9.20].

Hence, from Table thus Ai (4.5)

1029, t-l=.15713 10.11, f(-[)=I&348

.

.

;

y(c’) =y(d’)

{ l-df2

;-dr3

$-

24 and

=a(4.5)-V4(.55848 24) exp (-6.36396 1029) =$(.68658 905) (.55848 24) (.00172 25302) = .00033 02503. TO compute the zeros c, c’ of a solution y(x) of the equation y”-xy=O and of its derivative

Example 6. Compute -Bi(x) near d= - .4. From Table 10.11,

c=-.4+.03395

. . .}

the zero of y(x)=Ai(x)

y( - .4) = .02420 467, y’(--.4)=-.71276 whenceu=y(--.4)/y’(-.4)=-.03395 the above formulas

. . .}

(3d’3+3d’e)$

-(105d’s+101d’“+45d’4)~-

48403.

$

-(1432d+1575d4)$+

Functions

To compute Ai( Bi(x) for values of x beyond 1, use auxiliary functions from Table 10.11. Example 5. Compute Ai for x=4.5. First, for x=4.5, t=$x3/2=6.36396

$+14d

$+$-3d2

- (14+45d$+471da

= .35122 533.

Airy

l-d

(.12072 034)

. .}

627 8776. From

8776-.OOOOO 5221 +.ooooo 0111+.00000 0001 = - .36604 6333. y’(c)=(-.71276 627) { 1+.00023 0640 - .OOOOO6527- .OOOOO0027f .OOOOO0002} =(--.71276 627)(1.00022 4088) = -.71292 599.

BESSEL

FUNCTIONS

OF

FRACTIONAL

455

ORDER

References Texts

[lO.l] [10.2] (10.31 [10.4] [IO.51 [10.6] 110.71 [10.8]

[10.9]

[lO.lO]

[lO.ll]

[10.12]

[10.13] [10.14]

[10.15]

[10.16]

H. Bateman and R. C. Archibald, A guide to tables of Bessel functions, Math. Tables Aids Comp. 1, 205-308 (1944), in particular, pp. 229-240. T. M. Cherry, Uniform asymptotic formulae for functions with transition points, Trans. Amer. Math. Sot. 68, 224257 (1950). A. Erdelyi et al., Higher transcendental functions, vol. 1, 2 (McGraw-Hill Book Co., Inc., New York, N.Y., 1953). A. Erdelyi, Asymptotic expansions, Caliiornia Institute of Technology, Dept. of Math., Technical Report No. 3, Pasadena, Calif. (1955). A. Erdelyi, Asymptotic solutions of differential equations with transition points or singularities, J. Mathematical Physics 1, 16-26 (1960). H. Jeffreys, On certain approximate solutions of linear differential equations of the second order, Proc. London Math. SOC. 23, 428-436 (1925). H. Jeffreys, The effect on Love waves of heterogeneity in the lower layer, Monthly Nat. Roy. A&r. Sot., Geophys. Suppl. 2, 101-111 (1928). H. Jeffreys, On the use of asymptotic approximations of Green’s type when the coefficient has zeros, Proc. Cambridge Philos. Sot. 52, 61-66 (1956). R. E. Langer, On the asymptotic solutions of differential equations with an application to the Bessel functions of large complex order, Trans. Amer. Math. Sot. 34, 447-480 (1932). R. E. Langer, The asymptotic solutions of ordinary linear differential equations of the second order, with special reference to a turning point, Trans. Amer. Math. Sot. 67, 461-490 (1949). W. Magnus and F. Oberhettinger, Formeln und Siitze fiir die speziellen Funktionen der mathematischen Physik, 2d ed. (Springer-Verlag, Berlin, Germany, 1948). F. W. J. Olver, The asymptotic solution of linear differential equations of the second order for large values of a parameter, Philos. Trans. Roy. Sot. London [A] 247, 307-327 (1954-55). F. W. J. Olver, The asymptotic expansion of Bessel functions of large order, Philos. Trans. Roy. SOC. London [A] 247, 328-368 (1954). F. W. J. Olver, Uniform asymptotic expansions of solutions of linear second-order differential equations for large values of a parameter, Philos. Trans. Roy. Sot. London [A] 250,479-517 (1958). W. R. Wasow, Turning point problems for systems of linear differential equations. Part I: The formal theory; Part II: The analytic theory. Comm. Pure Appl. Math. 14, 657-673 (1961) ; 15, 173-187 (1962). G. N. Watson, A treatise on the theory of Bessel functions, 2d ed. (Cambridge Univ. Press, Cambridge, England, 1958).

Tables

[lo.171 H. K. Crowder and G. C. Francis, Tables of spherical Bessel functions and ordinary Bessel functions of order half and odd integer of the first and second kind, Ballistic Research Laboratory Memorandum Report No. 1027, Aberdeen Proving Ground, Md. (1956). [lO.lS] A. T. Doodson, Bessel functions of half int,egral order [Riccati-Bessel functions], British Assoc. Adv. Sci. Report, 87-102 (1914). [10.19] A. T. Doodson, Riccati-Bessel functions, British Assoc. Adv. Sci. Report, 97-107 (1916). [19.20] A. T. Doodson, Riccati-Bessel functions, British Assoc. Adv. Sci. Report, 263-270 (1922). [10.21] Harvard University, Tables of the modified Hankel functions of order one-third and of their derivatives (Harvard Univ. Press, Cambridge, Mass., 1945). [10.22] E. Jahnke and F. Emde, Tables of functions, 4th ed. (Dover Publications, Inc., New York, N.Y., 1945). [IO.231 C. W. Jones, A short table for the Bessel functions In+i(z), (2/a)&++(z) (Cambridge Univ. Press, Cambridge, England, 1952). [10.24] J. C. P. Miller, The Airy integral, British Assoc. Adv. Sci. Mathematical Tables, Part-vol. B (Cambridge Univ. Press, Cambridge, England, 1946). [10.25] National Bureau of Standards, Tables of spherical Bessel functions, ~01s. I, II (Columbia Univ. Press, New York, N.Y., 1947). [10.26] National Bureau of Standards, Tables of Bessel functions of fractional order, ~01s. I, II (Columbia Univ. Press, New York, N.Y., 1948-49). [10.27] National Bureau of Standards, Integrals of Airy functions, Applied Math. Series 52 (U.S. Government Printing Office, Washington, D.C., 1958). [10.28] J. Proudman, A. T. Doodson and G. Kennedy, Numerical results of the theory of the diffraction of a plane electromagnetic wave by a conducting sphere, Philos. Trans. Roy. Sot. London [A] 217, 279-314 (1916-18), in particular pp. 284288. [10.29] M. Rothman, The problem of an infinite plate under an inclined loading, with tables of the integrals of Ai (+z), Bi (&.z), Quart. J. Mech. Appl. Math. 7, l-7 (1954). [10.30] M. Rothman, Tables of the integrals and differential coefficients of Gi (+z), Hi (--z), Quart. J. Mech. Appl. Math. 7, 379-384 (1954). [10.31] Royal Society Mathematical Tables, vol. 7, Bessel functions, Part III. Zeros and associated values (Cambridge Univ. Press, Cambridge, England, 1960).

BESSEL

456 [10.32] R. S. Scorer, Numerical the form

and the tabulation Gi (z) = (l/r)h-

evaluation

FUNCTIONS of integrals

OF FRACTIONAL of

of the function sin (uz+ 1/3ua)du,

Quart. J. Mech. Appl. Math.

3, 107-112

(1950).

ORDER

[10.33] A. D. Smirnov, Tables of Aiiy functions (and special confluent hypergeometric functions). Translated from the Russian by D. G. Fry (Pergamon Press, New York, N.Y., 1960). [10.34] I. M. Vinogradov and N. G. Cetaev, Tables of Bessel functions of imaginary argument (Izdat. Akad. Nauk SSSR., Moscow, U.S.S.R., 1950). [10.35] P. M. Woodward, A. M. Woodward, R. Hensman, H. H. Davies and N. Gamble, Four-figure tables of the Airy functions in the complex plane, Phil. Mag. (7) 37, 236-261 (1946).

BESSEL FTJNCTIONS OF FRACTIONAL SPHERICAL x

&(4

i:! EE

0:4

1.0 ::: ::.i 1.5 i-7" 1:8 1.9 2.0 2: ::: 2.5 ::; 2: z-1" 3:2 33:: ::6' :-; 3:9 4.0 4.1 :-: 4:4 4.5 t:; 29" 5.0

j,(x) 0.00000 0000

BESSEL

ORDER

FUNCTIONS-ORDERS

& (4

Y,(X)

457

0.1 AND 2

Table 10.1

?/I (4

Y,(X)

1.00000 0.99833 0.99334 0.98506 0.97354

000 417 665 736 586

0.033300012 0.066400381 0.099102888 0.13121215

0.00000 619061 -9.95;04 17 0.00066000000

0.95885 0.94107 0.92031 0.89669 0.87036

108 079 098 511 323

0.16253703 0.19289196 0.22209828 0.24998551 0.27639252

0.016371107 0.023388995 0.031538780 0.040750531 0.050945155

-1.75516 51 -1.37555 94 -1.09263 17 -0.87088 339 -0.69067 774

-4.46918 13 -3.23366 97 -2.48121 34 -1.98529 93 -1.63778 29

-25.059923 -14.792789 -9.54114 00 -6.57398 92 -4.76859 87

0.84147 0.81018 0.77669 0.74119 0.70389

098 851 924 860 266

0.30116868 0.32417490 0.34528457 0.36438444 0.38137537

0.062035052 0.073924849 0.086512186 0.099688571 0.11334028

-0.54030 231 -0.41236 011 -0.30196 480 -0.20576 833 -0.12140 510

-1.38177 33 -1.18506 13 -1.02833 66 -0.89948 193 -0.79061 059

-3.60501 76 -2.81962 54 -2.26887 66 -1.86995 92 -1.57276 05

0.66499 0.62473 0.58333 0.54102 0.49805

666 350 224 646 268

0.39617297 0.40870814 0.41892749 0.42679364 0.43228539

0.12734928 0.14159426 0.15595157 0.17029628 0.18450320

-0.04715 8134 +0.018249701 0.075790879 0.12622339 0.17015240

-0.69643 541 -0.61332 744 -0.53874 937 -0.47090 236 -0.40849 878

-1.34571 27 -1.16823 87 -1.02652 51 -0.91106 065 -0.81515 048

0.45464 0.41105 0.36749 0.32421 0.28144

871 208 837 966 299

0.43539778 0.43614199 0.43454522 0.43065030 0.42451529

0.19844795 0.21200791 0.22506330 0.23749812 0.24920113

0.20807342 0.24040291 0.26750051 0.28968523 0.30724738

-0.35061 200 -0.29657 450 -0.24590 723 -0.19826 956 -0.15342 325

-0.73399 142 -0.66408 077 -0.60282 854 -0.54829 769 -0.49902 644

0.23938 0.19826 0.15828 0.11963 0.08249

886 976 884 863 9769

0.41621299 0.40583020 0.39346703 0.37923606 0.36326136

0.26006673 0.26999585 0.27889675 0.28668572 0.29328784

0.32045745 0.32957260 0.33484153 0.33650798 0.33481316

-0.11120 588 -0.07151 1067 -0.03427 3462 +0.00054 2796 0.032953045

-0.45390 450 -0.41208 537 -0.37292 316 -0.33592 641 -0.30072 380

0.04704 +0.01341 -0.01824 -0.04780 -0.07515

0003 3117 1920 1726 9148

0.34567750 0.32662847 0.30626652 0.28475092 0.26224678

0.29863750 0.30267895 0.30536678 0.30666620 0.30655336

0.32999750 0.32230166 0.31196712 0.29923629 0.28435241

0.062959164 0.090555161 0.11573164 0.13847939 0.15879221

-0.26703 834 -0.23466 763 -0.20346 870 -0.17334 594 -0.14424 164

-0.10022 -0.12292 -0.14319 -0.16101 -0.17635

378 235 896 523 030

0.23892369 0.21495446 0.19051380 0.16577697 0.14091846

0.30501551 0.30205107 0.29766961 0.29189179 0.28474912

0.26755905 0.24909956 0.22921622 0.20814940 0.18613649

0.17666922 0.19211667 0.20514929 0.21579139 0.22407760

-0.11612 829 -0.08900 2337 -0.06287 8964 -0.03778 7773 -0.01376 91'l2

-0.18920 -0.19957 -0.20751 -0.21306 -0.21627

062 978 804 185 320

0.11611075 0.091522967 0.067319710 0.043659843 +0.020695380

0.27628369 0.26654781 0.25560355 0.24352220 0.23038368

0.16341091 0.14020096 0.11672877 0.093209110 0.069848380

0.23005335 0.23377514 0.23531060 0.23473838 0.23214783

+0.009129107 0.030854018 0.051350236 0.070561855 0.088434232

-0.21722 -0.21601 -0.21274 -0.20753 -0.20050

892 978 963 429 053

-0.00142 95812 -0.02257 9838 -0.04262 9993 -0.06146 5266 -0.07898 2225

0.21627586 0.20129380 0.18553900 0.16911850 0.15214407

0.046843511 0.024380984 +0.002635886 -0.01822 8955 -0.03806 3749

0.22763858 0.22132000 0.21331046 0.20373659 0.19273242

0.10491554 0.11995814 0.13351972 0.14556433 0.15606319

-0.09508 9408 0.13473121 (-$)3

-0.05673 2437

0.18043837

0.16499546

-0.19178 485 (-$)4

c 1

[ /-- 1

0.0026590561 -4.90033 29 0.0059615249 -3.18445 50 0.010545302 -2.30265 25

-,0,,",875 - 30;5:0125 -25.495011 -377.52483 -11.599917 -112.81472 --6.73017 71 -48.173676

Compiledfrom National Bureau of Standards,Tablesof sphericalBesselfunctions, ~01s.I, II. Univ. Press,New York, N.Y., 1947 (with permission).

Columbia

BESSEL FUNCTIONS Table

SPHERICAL

10.1

BESSEL

OF FRACTIONAL

ORDER

FUNCTIONS-ORDERS

-1 -1 -1 -1 -1

0,l

AND

2

-1.2885 -1.3849 -1.4644 -1.5268 -1.5720

II

-2 -2 -2 -2 -1

-1 -2 -2 -2 -2

II

-2.9542 -1.6303 -3.2520 +9.4953 2.1829

-2 -1.1105 -1 -a.5177 -9.4473 -1.0313 -7.5334

-2 -2 -2 -2 Ii-1

-1 r: -1 -1

-2 -2 -2 -2 -2

-2 6.0883 -2 4.7295 -2 3.3674 -2 2.0132 -3 +6.7812

(-2)

-1 -1 -1 -1 -1

-1.3123 -1.3171 -1.3092 -1.2891 -1.2571

1.8188 3.0067 4.1360 5.1973 6.1820

a.1877 a.8851 9.4810 9.9723 1.0357

1.0632 :* oOts0: ioa13 1.0663

1; +F -2 -1.0349 211498 3.1395 yfl

-1 1.0124 -1 1.0415 -1 1.0596 -1 1.0669 -1 1.0635 -1 -1 -2 -2 -2

(-2)-5.4402

II

II II II II II II II II II II II II

-2 -2 -3 -3 Ii-2

II-2 -3 +2.6357 1.3382 2.4227 3.5066 4.5791

-1.0770 -9.6415 -a.4493 -7.2065 -5.9263

-5.6210 -6.9018 -8.0795 -9.1466 -1.0097

7.8467 I-

(-2)

7.7942

-3.4542 -2.3621 -1.2710 -1.9101 +8.6782

(-2)

6.2793

-6.2736 -1.8929 -3.1089 -4.2662 -5.3561

-2 -2 -2 -2 -2

-6.3711 -7.3040 -8.1487 -8.8997 -9.5527

-1 -1 -1 -1 -1

-1.0104 -1.0551 -1.0892 -1.1126 -1.1253

1.0497 1.0257 9.9213 9.4941 8.9817

(-2) 8.3907 Y&)=

-2 -2 -2 -3 -3

-3 -2 -2 -2 -2

d

+/xyn+*

I--

(x)=(-‘)“+l-\l~“ix~-~~+*~(x)

(-2)-6.5069

BESSEL

FUNCTIONS

SPHERICAL

BESSEL

OF

FRACTIONAL

459

ORDER

FUNCTIONS-ORDERS

Table

3-10

j7 (4 0.0000

10.2

i.0000 2.9012 7.4212 1.8995 1.8938

1.52734 1.52698 1.52589 1.52407 1.52154

93 7.27309-i9 56 7.27151 10 53 7.26677 00 96 7.25887 47 09 7.24783 46

1.1261 4.8282 1.6515 4.7873 1.2228

1.51828 1.51430 1.50962 1.50423 1.49815

26 7.23366 88 7.21637 48 7.19599 66 7.17254 12 7.14605

29 65 61 61 44

2.8265 6.0254 1.2013 2.2640 4.0669

1.49137 1.48392 1.47579 1.46700 1.45757

65 7.11655 11 7.08407 48 7.04866 80 7.01035 18 6.96919

26 57 21 39 61

61 6.6832 6 9.7670 5 1.4009 5)1.9754 5)2.7420

1.38804 1.37444 1.36030 1.34565 1.33050

63 35 78 67 81

6.66533 6.60575 6.54379 6.47951 6.41301

28 19 07 98 19

3.7516 5.0647 6.7532 8.9013 1.1607

1.31488 1.29879 1.28226 1.26531 1.24796

05 28 44 50 48

6.34434 22 6.27358 74 6.20082 63 6.12613 95 6.04960 91

;::

1.4983 1.9160 2.4283 3.8056 3.0520

1.21214 41 1.23023 38 5.97131 5.89135 85 26 1.19371 48 5.80979 75 1.15592 1.17496 82 54 5.72674 5.64226 82 00

3.5 ::7"

4.7098 5.7875 7.0639

1.13660 79 1.09723 52 1.11703 73

5.55647 05 5.46943 47 5.38125 61

'3.: .

1.0325 8.5665

1.05702 31 1.07722 33

5.20181 62 5.29201 05

4.0 4.1

1.2372 1.4743

1.03665 63 1.01614 44

5.11072 78 5.01885 80

:s 414

2.0603 1.7473 2.4174

0.99550 0.97477 88 06 0.95395 10

4.83311 07 4.92629 51 4.73942 00

::2

3.2814 2.8229

0.91215 06 0.93307 01 4.55082 4.64529 34 25

i-i 419

3.7976 4.3763 5.0226

0.89120 0.87026 94 97 4.45609 4.36119 35 18 0.84934 88 4.26620 13

-14)4.9319 -12)6.3072

1.0

II-3 9.0066 -2 1.1847 -2 1.5183

::i

-2 1.9033 2.3411

II- 54 1.4786 3.3461 2.2643 9.2561 4.7963

I

-2 I 1.4079 -2 1.6788 -2 1.9817 -2 2.3176 -2 2.6872

-

2'8 2:9

‘3.: 3:2

5.0

(-1)2.2982

(-1)1.8702

(- 1)1.0681

(- 2)4.7967

&Cd =&IxJ~+~

(- 2)1.7903

(4

(- 3)5.7414

0.82846 70

[ (-yJ]

Compiledfrom National Bureau of Standards,Tablesof sphericalBesselfunctions, ~01s.I, II. Univ. Press,New York, N.Y., 1947 (with permission).

4.17120 50

[‘-y] Columbia

460

BESSEL Table

FUNCTIONS

SPHERICAL

10.2

OF

BESSEL

FRACTIONAL

ORDER

FUNCTIONS-ORDERS

3-10

Y&4

Y,W

15 :2.;277 12 -3.9643 11 I -1.0329 9)-7.7739 6 5I 5 4 4

-1.3458 -3.7747 -1.2907 -5.1035 -2.2552

4I 3 3 3 3

-1.0881 -5.6378 -3.0988 -1.7901 -1.0790

1.5

/I I 543 -1.4045 -1.7686 -9.8790 -3.3227 -6.6058

i-76 1:s 1.9 ::1" ;*: 2:4 2.5 2.6 z-i 2:9

(-2)-1.5443

(-l)-1.8662

(-l)-3.2047

90 14 86 18 23

-0.34713 -0.34826 -0.34960 -0.35115 -0.35291

86 48 12 04 56

-0.65905 -0.66096 -0.66323 -0.66586 -0.66885

23 47 28 06 29

6 5 5 5 5

-2.0959 -8.9515 -4.1224 -2.0227 -1.0477

-0.35490 -0.35710 -0.35954 -0.36221 -0.36512

04 89 56 57 46

-0.67221 -0.67595 -0.68007 -0.68458 -0.68949

50 30 37 47 42

4 4 4 4 3I

-5.6859 -3.2143 -1.8835 -1.1395 -7.0931

-0.36827 -0.37168 -0.37534 -0.37928 -0.38348

87 46 96 17 96

-0.69481 -0.70054 -0.70670 -0.71331 -0.72036

14 60 90 20 75

-9.7792 -7.0870 -5.2238 -3.9108 -2.9702

3 -4.5301 3 -2.9613 3 1-1.9771 3 -1.3458 2 1-9.3247

-0.38798 -0.39277 -0.39786 -0.40327 -0.40901

26 08 50 71 97

-0.72788 -0.73589 -0.74439 -0.75340 -0.76294

93 19 11 38 81

1 1 1 1 0

-2.2859 -1.7812 -1.4041 -1.1189 -9.0069

2 -6.5676 g I 1:;;;; 2 -2:5025 2 1 -1.8615

-0.41510 -0.42155 -0.42837 -0.43558 -0.44320

62 14 10 18 20

-0.77304 -0.78371 -0.79497 -0.80685 -0.81937

34 06 18 08 31

-0.45125 -0.45975 -0.46872 -0.47818 -0.48818

11 01 14 95 03

-0.83256 -0.84645 -0.86108

59 82 11

-0.87646

78

-0.89265

39

-0.49872

20

-0.90967

72

-0.50984

49

-0.92757

84

I I

-2.9528 -2.5201 -2.1660 -1.8743 -1.6325

(e.I)-5.1841 F

lo-%llY,o(z) -0.65472 -0.65490 -0.65541 -0.65628 -0.65749

1 1 1I 1 1

0 0 0 0 0

5.0

10-8~10YJ~) -0.34459 42 -0.34469 56 -0.34499 99 -0.34550 77 -0.34622 02

II I 0 -9.8471 -4.7139 -5.6086 -6.7182 -8.1040

( 0)-1.0274

1 1 1 1 1

-3.9249 -3.1246 -2.5070 -2.0265 -1.6498

( 0)-2.5638

-0.52158

17

-0.53396

75

05

-0.94640 -0.96619 -0.98699

-0.64247

43

-1.13650

10

-0.75092

23

-1.30156

80

-0.54704

[c-p]

[C-$)2]

10 15 97

BESSEL FUNCTIONS SPHERICAL

OF FRACTIONAL

Table

10.2

FUNCTIONS-ORDERS

3-10

j,(x)

j,(x) 5.7414 6.5379 7.4172 8.3843 9.4443

logz-gjg(x) 0.82846 70 0.80764 29 0.78689 50 0.76624 10 0.74569 86

1.0602 1.1862 1.3229 1.4707 1.6299

0.72528 0.70501 0.68490 0.66497 0.64523

47 58 78 60 54

3.69892 3.60555 3.51270 3.42045 3.32886

98 18 30 23 66

21” 2: 614

1.8010 1.9842 2.1797 2.3877 2.6084

0.62570 0.60638 0.58729 0.56845 0.54987

01 37 93 94 57

3.23801 3.14794 3.05873 2.97043 2.88309

06 66 50 34 73

6:1

2.8417 3.0876 3.3461 3.6168 3.8996

0.53155 0.51352 0.49577 0.47831 0.46116

94 10 04 68 89

2.79677 2.71153 2.62739 2.54443 2.46266

98 12 98 09 76

4.1940 4.4994 4.8154 5.1412 5.4759

0.44433 0.42782 0.41163 0.39578 0.38026

45 11 52 30 97

2.38215 2.30291 2.22500 2.14844 2.07326

03 70 27 05 03

-2 I 5.8188 -2 6.1686 -2 6.5244 -2 6.8849 -2 7.2486

0.36510 0.35027 0.33580 0.32169 0.30793

02 86 85 28 39

1.99948 1.92715 1.85627 1.78687 1.71897

99 45 66 63 14

-2 7.6143 -2 7.9804 -2 8.3451 -2 8.7069 -2)9.0640

0.29453 0.28149 0.26881 0.25649 0.24453

36 30 29 33 39

1.65257 1.58770 1.52436 1.46257 1.40232

72 64 97 53 92

j,(x)

j, (4

&44 -2 4.7967 -2 -2 5.2015 -2 -2 5.6221 -2 -2 6.0573 -2 -2 I 6.5057 II-2

5.2 5.3 5.4

:2 517

BESSEL

461

ORDER

-1 -1 -1 -1 II-1

1.7903 -3 1.9908 -3 2.2061 -3 2.4365 -3 2.6821 II-3

2.0078 2.0150 2.0147 2.0069 1.9913

22 -1 I 1.5077 -1 1.5217 -1 1.5312 -1 1.5360 -1)1.5361

9':: z-: 9:4 ;:z

II-2 -2.2842 -1.3689

Ki 9:9

-2 -3.1654 -2 -4.8048 -4.0072

10.0 (-2)-3.9496

(-l)-1.0559

(-2)-5.5535

-2 -2 -2 '-2 -2

I

I II I II II 8.5853 7.8016 6.9921 6.1608 5.3120

(-2)4.4501 j,(x) =

10llx-lojlo(x) 4.17120 50 4.07628 42 3.98151 88 3.88698 72 3.79276 59

-1 1.3795 1.3746

-1 1.1000 1.1270 0.18025 0.17076 78 84 1.07343 1.02401 42 72

-1 1.3655 1.3520 -1 1.3341

-1 1.1747 1.1520 0.16161 0.15280 93 62 0.92973 0.97612 24 83 -1 1.1949 0.14432 46 0.88485 16

-1 -1 -1 -1 -1

-1 -1 -1 -1 -1

1.3117 1.2849 1.2536 1.2180 1.1780

(-1)1.1339

1.2126 1.2275 1.2394 1.2482 1.2537

(-1)1.2558

0.13616 0.12833 0.12081 0.11360 0.10670

93 53 68 83 35

0.84144 0.79950 0.75902 0.71996 0.68231

75 99 10 20 26

0.10009 64 0.64605 15 [(-:)5] [(-;)“I

462 Table

BESSEL 10.2

FUNCTIONS

SPHERICAL

BESSEL

OF

FRACTIONAL

FUNCTIONS-ORDERS

ORDER 3-10

-0.89751 -0.93284 -0.97058 -1.01093 -1.05412

90 85 31 09 18

-1.51723 -1.56801 -1.62178 -1.67873 -1.73913

25 75 08 97 16

-7.9262 -7.2128 -6.5889 -6.0416 -5.5598

-1.10040 -1.15007 -1.20342 -1.26079 -1.32256

93 32 16 38 26

-1.80321 -1.87128 -1.94363 -2.02062 -2.10262

67 02 49 45 69

-3.6163 -3.3996 -3.2032 -3.0246 -2.8613

-1.81128 -1.91762 -2.03285 -2.15774 -2.29311

11 85 95 75 31

-2.72637 -2.85777 -2.99896 -3.15082 -3.31433

44 73 17 08 45

-2.43982

13

-3.68071 -3.49057 -3.88603

53 56 37

-4.73689 -5.06881 -5.42169 -5.79550 -6.18991

09 69 35 68 88

-6.26721 -6.68628 -7.13987 -7.63051 -8.16074

41 70 95 13 96

-6.60420 -7.03717 -7.48710 -7.95166 - 8.Q2777

33 -8.73317 50 -9.35034 95 -10.01475 19 -10.72873 38 -11.49443

65 96 2 2 4

56 63 58 3 4

-12.31371 -13.18805 -14.11841 -15.10518 -16.14793

5 0 9 2 9

9

-17.24536

7

5.5 5.6 55'87 5:9

I

-1.6331 -1.5471 -1.4627 -1.3795 -1.2973 9.0

-2 -2.8097 -2 -3 7880 9:s ii -2 -515782 ,9-i -4'7130 9.4

II-1

1.1748 1.1976 1.1900 1.1899

Ii -2

8.1384 7.5427 5.4540 6.8948 6.1976

-2 -6.3774

-8.91157 -9.39828 -9.88210 -10.35610 -10.81210

9.5 ij -2 -7.1053 9.6 -2 -7.7572 9.7 -2 -8.3288 9.8 -2 -8.8169 9.9 (-2 -9.2189 10.0 (-2)-9.5327

(-3)-1.6599

(-2)

9.3834

y,(+

(-1)

1.0488

(-2)

4.2506

(-2) -4.1117 -11.24057

-cn+&)

[ 1 (73

BESSEL FUNCTIONS SPHERICAL X

BESSEL

lo26.f20(4

OF FRACTIONAL

FUNCTIONS-ORDERS

1027.f21 (4

463

ORDER 20 AND

21

Table

10.3

lo-25921(4

lo-24920(x~

7.62597 90 -7.62705 91 7.63028 29 7.63560 15 7.64293 25

1.77348 35 1.77371 23 1.77439 56 1.77552 32 1.77707 85

-0.31983 -0.31988 -0.32003 -0.32028 -0.32065

10 11 25 86 49

-1.31130 -1.31149 -1.31205 -1.31300 -1.31436

70 33 61 70 61

7.65215 99 7.66313 22 7.67566 19 7.68952 28 7.70444 90

1.77903 78 1.78137 03 1.78403 80 1.78699 49 1.79018 73

-0.32113 -0.32175 -0.32250 -0.32342 -0.32450

96 30 82 08 98

-1.31616 -1.31842 -1.32121 -1.32457 -1.32856

11 87 43 29 95

7.72013 23 7.73621 95 7.75231 00 7.76795 28 7.78264 38

1.79355 29 1.79702 05 1.80050 95 1.80392 94 1.80717 91

-0.32579 -0.32730 -0.32907 -0.33112 -0.33350

69 79 24 44 34

-1.33328 -1.33879 -1.34521 -1.35264 -1.36123

02 33 03 77 89

7.79582 23 7.80686 80 7.81509 84 7.81976 53 7.82005 32

1.81014 64 1.81270 77 1.81472 70 1.81605 56 1.81653 14

-0.33625 -0.33943 -0.34309 -0.34731 -0.35216

47 07 23 02 70

-1.37113 -1.38251 -1.39557 -1.41055 -1.42771

69 67 96 73 82

10.0 10.5 11.0 11.5 12.0

7.815076 7.803876 7.785428 7.758627 7.722309

1.815979 1.814208 1.811016 1.806185 1.799482

-0.35776 -0.36420 -0.37164 -0.38023 -0.39019

04 59 20 59 23

-1.447374 -1.469891 -1.495697 -1.525305 -1.559325

12.5 13.0 13.5 14.0 14.5

7.675238 7.616116 7.543601 7.456316 7.352841

1.790664 1.779472 1.765639 1.748885 1.728929

-0.40176 -0.41527 -0.43113 -0.44987 -0.47223

53 46 22 76 40

-1.5b8497 -1.643728 -1.696143 -1.757166 -1.828625

15.0 15.5 16.0 16.5 17.0

7.231764 7.091689 6.931265 6.749220 6.544411

1.705481 1.678251 1.646956 1.611324 1.571096

-0.49918 -0.53209 -0.57279 -0.62378 -0.68821

70 15 98 79 72

-1.912922 -2.013273 -2.134049 -2.281228 -2.462936

17.5 18.0 18.5 19.0 19.5

6.315851 6.062784 5.784739 5.481584 5.153621

1.526041 1.475960 1.420698 1.360155 1.294299

-0.76981 49 -0.87240 01 -0.99883 14 -1.149171 -1.317987

-2.689957 -2.975953 -3.336925 -3.789188 -4.344958

20.0 20.5 21.0 21.5 22.0

4.801647 4.427041 4.031843 3.618830 3.191590

1.223178 1.146936 1.065826 0.98022 63 0.89065 46

-1.490982 -1.641599 -1.728777 -1.697442 -1.483467

-5.004711 -5.745922 -6.508927 -7.182333 -7.592679

22.5 23.0 23.5 24.0 24.5

2.754567 2.313103 1.873442 1.442686 1.028721

0.79777 92 0.70243 25 0.60561 45 0.50849 80 0.41242 27

-1.024223 -0.274630 co.773430 2.072631 3.508629

-7.504782 -6.640003 -4.717888 -1.52185 +3.01816

25.0

0.640055

0.31888 30

4.901591

8:: :-i

2:o 9.:

3:5 44::

::; 66’!

7:o

c&,(d = 1f,x” exp ( - S/4n+2) [ 1 yn(x)=gnx-(n-+1) (-l)3

+8.74251

(-$7

exp (~2/4n+2) Compiled from National Bureau of Standards, Tables of spherical Bessel functions, ~01s. I, II.ColumbiaUniv. Press, New York, N.Y., 1947(with permission).

464

BESSEL FUNCTIONS Table 10.4 SPHERICAL

I *700

BESSEL

OF FRACTIONAL ORDER

FUNCTIONS-MODULUS

AND

9, 10, 20 AND 21

PHASE-ORDERS

W(x) --z

eygx) -x

<x>

01095 0.090 0.085 0.080

&GM1 3’2(x) 1.50513 630 1.41043 073 1.33509 121 1.27462 197 1.22560 809

0.075 0.070 0.065 0.060 0.055

1.18548 1.15231 1.12467 1.10147 1.08190

011 423 134 221 340

0.38142 +0.12729 -0.12255 -0.36849 -0.61090

613 416 277 087 826

1.25559 223 1.20514 049 1.16476 186 1.13202 416 1.10519 883

-0.35524 -0.67664 -0.99107 -1.29911 -1.60143

574 889 278 571 947

0.050 0.045 0.040 0.035 0.030

1.06534 781 1.05133 389 1.03949 892 1.02956 235 1.02130 658

-0.85018 -1.08669 -1.32077 -1.55274 -1.78293

673 229 114 891 175

1.08304 588 1.06466 562 1.04939 746 1.03675 104 1.02635 931

-1.89870 -2.19155 -2.48055 -2.76627 -3.04920

678 009 907 814 936

20 22 25 29 33

0.025 0.020 0.015 0.010 0.005

1.01456 1.00920 1.00512 1.00226 1.00056

304 210 574 240 327

-2.01160 -2.23905 -2.46552 -2.69127 -2.91655

832 224 469 701 326

1.01794 1.01130 1.00628 1.00276 1.00068

637 529 277 864 866

-3.32981 -3.60853 -3.88577 -4.16191 -4.43732

737 532 070 106 935

40 27" 100 200

0.000

1.00000

000

-3.14159

265

1.00000

000

-4.71238 '#9 c

898

co

1 1 (5312

1.72311 1.44562 1.17232 0.90378 0.64017

121 029 718 457 615

1.84157 1.65174 1.50947 1.40190 1.31955

[

6$6

799 534 539 550 792

1

II 1 (-;I6

1.35401 461 1.00196 372 0.65310 249 +0.30984 705 -0.02643 915

by/,(x) -x

1.31126 1.25741 1.21433 1.17917 1.15001

605 042 612 949 033

1.12909 207 0.61321 135 +0.11048 098 -0.38066 745 -0.86163 915

1.37979 868 1.30763 025 1.25205 767 1.20806 627 1.17245 178

+0.54348 -0.04056 -0.60729 -1.15885 -1.69717

547 472 830 172 688

0.030 0.028 0.026 0.024 0.022

1.12549 1.10467 1.08687 1.07157 1.05838

256 736 488 283 371

-1.33366 -1.79783 -2.25507 -2.70621 -3.15199

819 172 118 373 149

1.14310 1.11857 1.09787 1.08027 1.06523

153 851 629 122 083

-2.22398 -2.74075 -3.24876 -3.74910 -4.24275

514 480 024 503 239

0.020 0.018 0.016 0.014 0.012

1.04700 1.03721 1.02883 1.02170 1.01571

987 972 137 104 485

-3.59305 -4.03000 -4.46335 -4.89362 -5.32124

805 220 928 072 187

1.05235 1.04134 1.03195 1.02400 1.01735

561 092 154 423 560

-4.73055 -5.21325 -5.69154 -6.16604 -6.63731

105 651 843 479 350

0.010 0.008 0.006 9.004 0.002

1.01078 282 1.00683 452 1.00381 592 1.00168 705 1.00042 044

-5.74664 -6.17024 -6.59240 -7.01351 -7.43392

872 356 995 707 365

1.01189 1.00753 1.00420 1.00185 1.00046

351 093 153 654 253

-7.10588 -7.57224 -8.03687 -8.50021 -8.96270

196 522 285 498 770

0.000

1.00000

-7.85398

164

1.00000

000

-9.42477

796

(2)

000 (-p [ 1

841,/,(x)

d3?rzM4%

--2

(2)

[<x> = nearest 1 integer[ to 2. 1 (-92

‘32

11 1':

I

1 0.?40 0.038 0.036 0.034 0.032

df?rsM4$/,

ii!

<x> 25 22: 5; 33 ;$ 45

100 125 167 250 500 m

[ 1 53P

Compiled from National Bureau of Standards, Tables of spherical Bessel functions, ~01s. I, II. Columbia Univ. Press,NewYork,N.Y.,1947 (with permission).

465

BESSEL FUNCTIONS OF FRACTIONAL ORDER SPHERICAL

BESSEL

x=1 I-

123I 9.00658 1.01101 6.20350 5808 3.01168 8.41470 5201 1117 6789 9848

FUNCTIONS-VARIOUS

ORDERS

Table

II f-

x=2 114.54648 7134 2 4.35397 1 6.07220 9276 1.40793 1.98447 9766 9491 7750 2.29820 6182 1.87017 6553

II

f- 816.82530 0865 - 10 12 11 9 4.81014 3.58145 8901 2.48104 5.97687 1612 9119 1402

ii! :: 14 I - 22 21I,.1.55670 24 19 18 3.08874 2364 1.20385 4.45117 5.13268 5742 7504 8271 6115

1.08428 0182 1.67993 9976

II 100

(-190)7.44472

- l)-1.05589

7742

- 20 66 7.63264 49 34 4.01157 1101 1.66097 5.83661 5290 8779 7888

I - 12 33I 2.85747 46 22 1.21034 7583 4.28273 5.42772 9350 0217 6761

(-160)9.36783 25?1

(-120) 5.53565 0303

x=50 (- 3)-5.24749 7074

x=100 -3)-5.06365 6411

2851 f- 21-2.00483 0056

2 7 98 10

I - 23I 7.46558 1.72159 4477 2.94107 8342 9974

(I-

2 -1.50392 3 +9.90845 2 +1.95971 4 -1.09899 2I -1.96564

2146 4236 1041 0300 5589

3.23884 7439 8.89662 7269

I-

2 -1.12908 2 +1.26561 2 +1.96438 3 +1.09459 2I -1.88338

4539 3175 9234 2888 9360

I- 21 6.46051 5449 (- 2 3155744 1489

11: ii :z 1187

19

-3)-9.29014 -3 +3.15754 -3 +9.70062 -3 i -1.70245 (-3)-9.99004

I

8935 5454 9844 0977 6510

2 -1.57850 2990 - 3 -1.49467 3454 - 2 -2.60633 6952 I-- 2I +1.88291 0737

20 30 40 50 100

10.5

(-90) 5.83204 0182

(-22)+1.01901 2263

(-2)+1.08804 7701

466

BESSEL Table

10 11 12

10.5

FUNCTIONS

SPHERICAL

ll)-3.23191

1198 $i 50 100

BESSEL

3629

1143 :65 17

OF

I

FRACTIONAL

ORDER

FUNCTIONS-VARIOUS

5 -3.55414 6 -3.69396 7 -4.21251 8 -5.22870 9I -7.01663

7201 5631 9003 9098 2092

15 -6.29800 17 -1.95020 18 -6.42938 20 -2.24833 I 21 I -8.31241

7233 7734 7516 5423 1677

11 -1.01218 I 12 -1.56186 13 -2.56695 14 -4.47655 I 15I -8.25596

2944 6932 8608 8894 4368

23 -3.23959 40 -2.94642 58 -8.02845 I 78 I ~2.73919

2219 8547 0851 2285

63)-1.23502

1944

(186)-6.68307

9463

(156)-2.65595

5830

ORDERS

( -l)-3.20465 -1 I -5.18407 0 -1.02739 0 -2.56377 0 -7.68944

0467 5714 4639 6345 4934

(116)-1.79971

3983

-3 +3.72067 8486 -3 +9.49950 2019 -3 -2.48574 3224 -3 I -9.87236 3502 -4)+8.07441 4285

2 7 98 :10 12 13 14 112 :; 19

II

20

-2 -4.19000 -5 -2.24122 Cl.37595 t4.97879

zoo 50 100

(85)-8.57322

6309

(+18)-1.12569

7221 6812 0150 3130

2891

-3 t6.25864 -3 1-6.78002 -3 -8.49604 -3 t3.80640 -3 1c9.90441

1510 3635 9309 6377 9669

-5 t5.63172 -3 -5.41292 -4 -7.04842 -2 I tl.07478

9379 9349 0407 2297

(-2)-2.29838

5049

BESSEL ZEROS

OF

FUNCTIONS BESSEL

OF FRACTIONAL

FUNCTIONS

OF

J&,)=0 Y

s

l/2

1 32 4 5 6 7 8

3/2

512

1 :

9/2

11/2

13/2

Jib, s)

?h,, (-l)“+lY~~w,.,~) y

3.141593

-0.45015

82

1.570796

-0.63661

98

9.424778 6.283185 12.566370 15.707963 18.849556 21.991149

+0.31830 -0.25989 +0.22507 -0.20131 +0.18377 -0.17014

89 99 91 68 63 38

4.712389 7.853982 10.995574 14.137167 17.278760 20.420352 23.561945

+0.36755 -0.28470 +0.24061 -0.21220 +0.19194 -0.17656 +0.16437

26 50 97 66 81 66 45

-0.36741 +0 28469 -0'24061

35 69 20

2.798386 6.121250 9.317866

+0.44914 +0.25989 -0.31827

84 33 37

4 14:066194 5 17.220755 6 20.371303 7 23.519452

+0:21220 -0.19194 +0.17656 -0.16437

57 77 64 44

12.486454 15.644128 18.796404 21.945613

-0.22507 +0.20131 -0.18377 +0.17014

76 63 61 37

;

-0.31710 +0.25973 -0.22503 +0.20130 -0.18376 +0.17014

58 30 59 14 96 05

3.959528 7.451610 10.715647 13.921686 17.103359 20.272369 23.433926

-0.36184 +0.28430 -0.24053 +0.21218 -0.19193 +0.17656 -0.16437

68 75 93 15 81 19 21

3 4 5 6 7

7/2

J., 8

Y”(V”,

1 2 3 4 5 6 7

1 2 3 i 5 6

1 2 3 4 5 6

1 2 3 4 5

4.493409 10'904122 7 725252

5.763459 9.095011 12.322941 15.514603 18.689036 21.853874

6.987932 Xi417119 13.698023 16.923621 20.121806 23.304247

8.182561 11.704907 15.039665 18.301256 21.525418 24.727566

9.355812 12.966530 16.354710 19.653152 22.904551

10.512835 14.207392 17.647975 20.983463 24.262768

-0.28223 +0.24019 -0.21208 +0.19189 -0.17654 +0.16436

-0.25620 +0.22432 -0.20107 +0.18367 -0.17009 +0.15912

-0.23580 +0.21109 -0.19155 +0.17639 -0.16428

-0.21926 +0.19983 -0.18321 +0.16988 -0.15902

71 23 02 90 40 28

49 53 12 44 46 86

60 29 58 49 83

48 04 82 82 21

5.088498 8.733710 12.067544 15.315390 18.525210 21.714547 24.891503

6.197831 9.982466 13.385287 16.676625 19.916796 23.128642

7.293692 11.206497 14.676387 18.011609 21.283249 24.518929

8.379626 12.411301 15.945983 19.324820 22.628417

+0.30882 -0.25896 +0.22485 -0.20124 to.18374 -0.17012 +0.15914

-0.27236 +0.23908 -0.21179 +i;19179 -0.17649 +0.16433

+0.24538 -0.22293 +0.20067 -0.18352 +0.17002 -0.15909

-0.22441 +0.20946 -0.19106 +0.17619 -0.16419

36 77 68 01 36 77 62

25 76 27 35 69 89

14 49 86 21 38 15

70 65 59 60 26

467

ORDER

HALF-INTEGER

ORDER

Table

10.6

,I=0

.JX.i, ,)

y, ,~ (- l)“+‘T(!/“, s)

s

h,,

15/2

1 2 3 4 5

11.657032 15.431289 18.922999 22.295348

-0.20550 +0.19008 -0.17582 +0.16402

46 87 99 38

9.457882 13.600629 17.197777 20.619612 23.955267

+0.20754 -0.19801 +0.18264 -0.16964 +0.15890

83 01 01 44 14

17/2

1 2 3 4

12.790782 16.641003 20.182471 23.591275

-0.19382 +0.18155 -0.16922 +0.15870

82 15 10 04

10.529989 14.777175 18.434529 21.898570

-0.19361 +0.18810 -0.17517 +0.16373

38 92 27 75

19/2

1 2 3 4

13.915823 17.838643 21.428487 24.873214

-0.18376 +0.17398 -0.16326 +0.15383

12 80 17 84

11.597038 15.942945 19.658369 23.163734

+0.18186 -0.17944 +0.16849 -0.15837

42 10 33 45

21/2

1 2 3 4

15.033469 19.025854 22.662721

-0.17496 +0.16722 -0.15785

82 59 09

12.659840 17.099480 20.870973 24.416749

-0.17179 +0.17176 -0.16247 +0.15347

22 97 13 56

23/2

1 2 3

16.144743 20.203943 23.886531

-0.16720 +0.16113 -0.15290

39 25 87

13.719013 18.247994 22.073692

+0.16304 -0.16491 +0.15700

06 86 50

25/2

1 2 3

17.250455 21.373972

-0.16028 +0.15560

44 47

14.775045 19.389462 23.267630

-0.15534 +0.15875 -0.15201

97 20 34

27/2

1 2 3

18.351261 22.536817

-0.15406 +0.15056

88 00

15.828325 20.524680 24.453705

+0.14852 -0.15316 +0.14743

56 36 15

29/2

1 2

19.447703 23.693208

-0.14844 +0.14593

69 21

16.879170 21.654309

-0.14242 to.14806

04 91

31/2

1 2

20.540230 24.843763

-0.14333 +0.14166

12 70

17.927842 22.778902

+0.13691 -0.14340

88 05

33/2

1 2

21.629221

-0.13865

11

18.974562 23.898931

-0.13192 +0.13910

99 20

35/2

1

22.715002

-0.13434

93

20.019515

+0.12738

05

37/2

1

23.797849

-0.13037

81

21.062860

-0.12321

13

39/2

1

24.878005

-0.12669

81

22.104735

+0.11937

34

Values to greater accuracy and over a wider range are given in [10.31]. From National Bureau of Standards, Tables of spherical Bessel functions, ~01s. I, II. Columbia Univ. Press, New York, N.Y., 1947 (with permission).

468

BESSEL Table

10.7

FUNCTIONS

ZEROS

OF

OF FRACTIONAL

THE DERIVATIVE OF HALF-INTEGER

OF

J:G:,,)=o Y

5

l/2

1 2 3 4

3/2

5/2

., 3”. 8 1.165561 4.604217 7.789884 10.949944 14.101725 17.249782 20.395842 23.540708

1 :

2.460536 9.261402 6.029292

z

15.611585 12.445260

76

18.769469 21.922619

1 2 : 2

3.632797 7.367009 10.663561 13.883370 20.246945 17.072849

7

23.412100

J” K, J +0.679192 -0.369672 +0.285287 -0.240870 +0.212340 -0.192029 +0.176620 -0.164412

+0.525338 -0.328062 +0.263295 -0.226711 +0.202245

!/I,, 2.975086 6.202750 9.371475 12.526476 15.676078 18.822999 21.968393

w;,

(-l)“+ly”(v;,s) -0.456186 +0.319331 -0.260267 +0.225258 -0.201419 +0.183841 -0.170188

-0.184363 +0.170542

4.354435 7.655545 10.856531 14.029845 17.191285 20.346496 23.498023

+0.388891 -0.290138 +0.242910 -0.213417 +0.192678 -0.177046 +0.164709

+0.457398 -0.301449 +0.247304 -0.215670 +0.194015 -0.177917 +0.165314

5.634297 9.030902 12.278861 15.480655 18.661309 21.830390 24.992411

-0.350669 +0.270006 -0.229783 +0.203956 -0.185432 +0.171262 -0.159953

9/2

11/2

13/2

1 ;

4.762196 12.018262 8.653134

4 z

15.279081 18.496200 21.690284

7

24.870602

1 2 3 4 2

5.868420 9.904306 13.337928 16.641787 19.888934 23.105297

+0.415533 -0.282237 +0.234875 -0.206685 +0.187103 -0.172377 +0.160741

+0.386006 -0.267385 +0.224788 -0.199151 +0.181169 -0.167534

1 32

6.959746 14'630406 11 129856

:

21'256291 17'977886

+0.363557 -0 255385 +0'216349 +0'175987 -0'192692

6

241496327

-0:163244

1 2 3

8.040535 12.335631 15.901023

+0.345649 -0.245384 +0.209127

54

22:602185 19 291967

;:::%;:

6.863232 10.356373 13.656304 16.891400 20.095393 23.281796

8.060030 11.646354 14.999624 18.270330 21.500029 24.705942

9.234274 12.909478 16.315912 19.623229 22.879980

10.391621 14.151399 17.610124 20.954335 24.238863

+0.324651 -0.254849 +0.219318 -0.196124 +0.179270 -0.166245

-0.305246 +0.242810 -0.210673 +0.189472 -0.173929 +0.161826

to.289946 -0.232895 +0.203344 -Oil83714 +0.169229

-0.277420 +0.224513 -0.197009 +0.178651 -0.165043

BESSEL ORDER

FUNCTIONS

,)==O

”

d,, (-1)“+‘y”(Y:,3

s

15’23: 4 5

9.113402 13.525575 17.153587 20.587450 23.929631

+0.330874 -0.236854 to.202841 -0.182077 +0.167294

11.535731 15.376058 18.885886 22.266861

+0.266883 -0.217283 to.191447 -0.174147

17'2

: 3 4

10.180054 14.702493 18.390930 21.866965

+0.318378 -0.229449 +0.197291 -0.177623

12.669130 16.586323 20.145940 23.563314

-0.257833 to.210950 -0.186505 to.170098

19/2

1

11.241675 15.868463 19.615227 23.132584

to.307606 -0.222927 +0.192335 -0.173605

13.793646 17.784362 21.392422 24.845689

+0.249935 -0.205332 to.182067 -0.166427

12.299124 17.025072 20.828186 24.385974

to.298179 -0.217118 +0.187870 -0.169950

14.910648 18.971857 22.627032

-0.242951 +0.200296 -0.178048

13.353045 18.173567 22.031181

+0.289825 -0.211893 +0.183813

16.021196 20.150142 23.851147

to.236710 -0.195742 to.174383

:

14.403937 19.314945 23.225333

to.282348 -0.207156 to.180103

17.126125 21.320300

-0.231081 +0.191594

27'2

: 3

15.452196 20.450018 24.411571

+0.275596 -0.202830 to.176690

18.226109 22.483219

to.225965 -0.187792

29/2

:

16.498138 21.579459

to.269455 -0.198856

19.321702 23.639641

-0.221286 +0.184287

31/2

1 2

17.542024 22.703832

+0.263833 -0.195187

20.413362 24.790191

+0.216981 -0.181040

33/2

1 2

18.584071 23.823614

+0.258658 -0.191783

21.501477

-0.213000

3512

1 2

19.624460 24.939214

to.253871 -0.188612

22.586374

to.209303

37/2

1

20.663347

to.249423

23.668335

-0.205855

39/2

1

21.700865

to.245275

24.747606

to.202629

: 4

21/2

1 : 4

23/2

1 :

7/2

ORDER

25/2

1

-Values to greater accuracy and over a wider range are given in [10.31]. From National Bureau of Standards, Tables of spherical Bessel functiomqvols. I, II. Press, New York, N.Y., 1947 (with permission).

Columbia Univ.

BESSEL MODIFIED

x

SPHERICAL

BESSEL

00.:

1.01506 1.00668 764 001

0:4

1.02688 081

0.5 0.6

1.04219 061 1.06108 930

oo*; 1.11013 1.08369 100 248 0:9 1.14057 414

FRACTIONAL

469

ORDER

FUNCTIONS--ORDERS

0, 1 AND

2

kl(4

Table

10.8

0.03336 668 0.06693 370 0.10090 290 0.13547 889 0.17087 071 0.20729 319 0.24496 858 0.28412 808 0.32501 361

0.01696 6360 0.02462 3348 0.03382 5678 0.04465 2156 0.05719 5452

1.90547 226 1.43678 550 1.11433 482 0.88225 536 0.70959 792

5.71641 679 3.83142 801 2.70624 170 1.98507 456 1.49804 005

36.203973 20.593926 12.712514 8.32628 49 5.70306 48

0.36787 944 0.41299 416 0.46064 259 0.51112 785 0.56477 365

0.07156 2871 0.08787 7251 0.10627 7995 0.12692 2227 0.14998 6112

0.57786 367 0.47533 880 0.39426 230 0.32930 149 0.27668 115

1.15572 735 0.90746 4974 0.72281 4219 0.58261 0332 0.47431 0537

4.04504 57 2.95024 33 2.20129 78 1.67378 69 1.29306 09

0.62192 665 0.68295 906 0.74827 140 0.81829 550 0.89349 778

0.17566 6332 0.20418 1728 0.23577 5138 0.27071 5433 0.30929 9770

0.23366 136 0.19821 144 0.16879 918 0.14425 049 0.12365 360

0.38943 5596 0.32209 3595 0.26809 2818 0.22438 9655 0.18873 4440

1.01253 25 0.80213 693 0.64190 415 0.51823 325 0.42165 535

0.97438 274 1.06149 681 1.15543 247 1.25683 283 1.36639 653

0.35185 6089 0.39874 5868 0.45036 7165 0.50715 7959 0.56959 9849

0.10629 208 0.09159 719 0.07911 327 0.06847 227 0.05937 476

0.15943 8124 0.13521 4906 0.11507 3847 0.09824 2824 0.08411 4246

0.34544 927 0.28476 135 0.23603 215 0.19661 508 0.16451 757

0.63822 2102 0.71360 6125 0.79639 0365 0.88727 5704 0.98703 1387

0.05157 553 0.04487 256 0.03909 858 0.03411 437 0.02980 354

0.07220 5736 0.06213 1241 0.05357 9539 0.04629 8067 0.04008 0625

0.13822 241 0.11656 246 0.09863 140 0.08371 944 0.07126 626

il(X)

1.00166 1.00000 750 000

OF

iz(x) ko(4 0.00000 0000 0.00066 7143 14.2&5 0.00267 4294 6.43029 0.00603 8668 3.87891 0.01078 9114 2.63234

io (2)

0"::

FUNCTIONS

0.00000

000

293 156.30044682 630 38.58177 78 513 16.80863 22 067 9.21319 233

kz(4 4704.:536 585.15696 171.96524 71.731283

::1"

119 1.17520 497 1.21422

:*; 1:4

1.25788 446 1.30644 803 1.36021 536

::6'

1.41951 997 1.48472 964

i-s7 1:9

1.63454 1.55625 408 127 1.72008 574

2.0

1.81343 020

;::

690 2.02595 988 1.91516

f:i

551 2.27759 513 2.14650

2.5

2.42008 179

;:;

2.74306 701 2:57489 041

;:;

3.12398 513 2.92568 658

1.48488 308 1161311877 1.75200 304 1.90251 546 2.06572 335

3.0

3.33929 164 3.57304 872 3.82683 875 4.10238 723 4.40157 747

2.24279 012 2.43498 437 2.64368 983 2.87041 631 3.11681 153

1.09650 152 1.21661 224 1.34837 954 1.49291 787 1.65144 965

0.02606 845 0.02282 681 0.02000 910 0.01755 635 0.01541 841

0.03475 7931 0.03019 0302 0.02626 1944 0.02287 6452 0.01995 3243

0.06082 638 0.05204 323 0.04462 967 0.03835 312 0.03302 422

3.5 316

4.72646 494 5;0792'iJ316 5.46251 092

3:9 z-i

6.33105 5.87879 128 220

3.38467 421 3.67596 831 3.99283 865 4.33762 799 4.71289 572

1.82531 562 2.01598 623 2.22507 418 2i45434 813 2.70574 780

0.01355 255 0.01192 222 0.01049 611 0.00924 735 0.00815 280

0.01742 4712 0.01523 3952 0.01333 2903 0.01168 0862 0.01024 3262

0.02848 802 0.02461 718 0.02130 658 0.01846 908 0.01603 223

4.0 iI21

6.82247 930 060 7.93706 374 7.35655

Ei

8.56816 9.25438 571 538

5.12143 838 5.56631 208 6.05085 704 6.57872 451 7.15390 628

2.98140 051 3.28363 932 3.61502 300 3.97835 791 4.37672 200

0.00719 253 0.00634 934 0.00560 833 0.00495 661 0.00438 300

0.00899 0668 0.00789 7961 0.00694 3650 0.00610 9316 0.00537 9136

0.01393 554 0.01212 834 0.01056 808 0.00921 893 0.00805 059

10.00066 914 7.78076 689 10.81241 998 8.46407 908 '11.69554 012 9.20906 250 12.65647 789 10.02142 620 13.70227 889 10.90741 515

4.81349 122 5.29236 840 5.81741 513 6.39308 652 7.02426 961

0.00387 777 0.00343 248 0.00303 975 0.00269 318 0.00238 716

0.00473 9498 0.00417 8666 0.00368 6506 0.00325 4257 0.00287 4331

0.00703 744 OiOO615769 0.00539 284 0.00472 709 0.00414 695

5.0 14.84064 212 11.87386 128 ['-;"I

7.71632 535 [‘-;‘“I

0.00211 679

0.00254 0146

0.00364 088

33:: 2:

4.5 4.6 4.7 4.8 4.9

c(-$11

,-

I_

in(x)= d f

,/XI

*+p

kn(x)

= j/1

r/xK,,+,(x)

470

BESSEL

Table X

10.9

MODIFIED

1o%-%g (z)

FUNCTIONS

OF

SPHERICAL

FRACTIONAL

BESSEL

lolOz-‘Oi,o(x)

ORDER

FUNCTIONS-ORDERS

9 AND

10-‘x’Okg(z)

10-9x”~,&)

38 21 99 70 92

1.02844 1.02817 1.02736 1.02601 1.02412

10

1.52734 1.52771 1.52880 1.53062 1.53317

93 30 46 54 79

0.72730 0.72746 0.72794 0.72873 0.72984

92 73 19 35 30

5.41287 5.41128 5.40650 5.39856 5.38746

1.53646 1.54049 1.54526 1.55078 1.55706

54 23 36 57 60

0.73127 18 0.73302 17 0.73509 47 0.73749 33 0.74022 04

5.37323 5.35590 5.33549 5.31206 5.28564

85 33 79 23 31

1.02170 47 1.01875 42 1.01527 95 1.01128 67 1.00678 27

1.56411 1.57193 1.58054 1.58994 1.60016

27 49 32 87 42

0.74327 0.74667 0.75040 0.75448 0.75891

93 38 79 62 37

5.25629 5.22406 5.18902 5.15123 5.11078

13 45 48 93 01

1.00177 0.99627 0.99028 0.98382 0.97689

53 31 56 30 61

1.61120 1.62308 1.63581 1.64941 1.66390

30 02 13 38 60

0.76369 0.76883 0.77434 0.78023 0.78649

58 83 76 05 43

5.06772 5.02215 4.97414 4.92379 4.87119

38 07 57 68 57

0.96951 0.96169 0.95345 0.94478 0.93572

68 72 03 97 94

;-: 2:4

1.67930 1.69563 1.71292 1.73118 1.75044

73 90 33 39 59

0.79314 68 0.80019 63 0.80765 17 0.81552 21 0.82381 79

4.81643 4.75961 4.70083 4.64019 4.57780

66 72 65 67 09

0.92628 0.91646 0.90629 0.89579 0.88495

41 88 89 04 95

z-2 2:7 2.8 2.9

1.77073 1.79208 1.81451 1.83806 1.86277

63 32 64 76 03

0.83254 0.84172 0.85136 0.86147 0.87206

94 78 49 30 54

4.51375 4.44816 4.38113 4.31277 4.24318

41 23 22 10 63

0.87382 0.86239 0.85069 0.83874 0.82655

25 63 78 39 20

$4 .

1.88865 1.91577 1.94414 1.97382 2.00485

96 24 79 74 39

0.88315 0.89475 0.90688 0.91956 0.93279

57 86 95 42 97

4.17248 4.10077 4.02816 3.95475 3.88064

53 50 19 12 76

0.81413 0.80152 0.78872 0.77574 0.76262

92 28 01 83 45

2.03727 2.07113 2.10648 2.14337 2.18187

33 33 43 94 40

0.94661 0.96102 0.97605 0.99171 1.00804

40 55 38 97 44

3.80595 3.73076 3.65519 3.57933 3.50326

33 99 70 16 88

0.74936 0.73598 0.72250 0.70894 0.69531

56 84 95 53 19

2.22202 68 2.26389 90 2.30755 54 2.35306 35 2.40049 43

1.02505 1.04276 1.06120 1.08040 1.10038

08 26 45 28 47

3.42710 3.35091 3.27481 3.19885 3.12314

13 95 07 96 76

0.68162 0.66790 0.65415 0.64039 0.62664

50 02 25 66 70

44'76 4: 8 4.9

2.44992 2.50142 2.55508 2.61099 2.66924

1.12117 1.14281 1.16532 1.18874 1.21310

91 58 63 39 29

3.04775 2.97275 2.89821 2.82421 2.75081

39 34 88 90 98

0.61291 75 0.59922 16 0.58557 24 0.57198 25 0.55846 39

5. 0

2.72991 40

2.67808 38

0.54502 82

::1” E

0:4 i:: i-l 0: 9 ::1" :*: 1:4 ::i :*i 1:9 2. 0 2.1

::: 3.5 :*; 3: 8 3.9 4. 0 4.1 i.23 4:4 4. 5

27 71 99 74 03

1.23843 97

(-;)3 [

c 1 (-;)4

I

in(x)= d ; ,/XIn+*(x)

60 54 41 35 59

[(-y 1

Compiled from C. W. Jones, A short table for the Bessel functions Zn++(z), (~/T)K~+P), Cambridge Univ. Press, Cambridge, England, 1952 (with permission).

BESSEL

MODIFIED

SPHERICAL

FUNCTIONS

BESSEL

OF

FRACTIONAL

FUNCTIONS-ORDERS

ORDER

471 9 AND

10

Table

10.9

II(-4 -4 1.56545 1.43285 1.30831 1.19157 1.08240

II(-4 -4 2.35684 1.85569 1.70632 2.18075 2.01376

(-3)1.55045

(-4)6.25963

(1)1.48772

(1)3.50537

472

BESSEL

Table

FUNCTIONS

OF

FRACTIONAL

ORDER

10.9 MODIFIED

1

SPHERICAL fQ@)

BESSEL

FUNCTIONS-ORDERS

flO(@

gQ @>

9 AND

glow

0.095 0.090 0.085 0.080

1.10630 1.08238 1.06167 1.04394 1.02899

573 951 683 741 406

1.21411 1.17260 1.13650 1.10534 1.07872

149 877 462 464 041

0.65502 0.68557 0.71563 0.74502 0.77352

364 030 676 124 114

0.56777 0.60351 0.63926 0.67473 0.70961

303 931 956 612 813

0.075 0.070 0.065 0.060 0.055

1.01661 1.00662 0.99883 0.99304 0.98907

895 998 728 985 251

1.05626 1.03762 1.02248 1.01055 1.00151

085 412 982 159 009

0.80093 0.82707 0.85175 0.87480 0.89608

667 483 354 587 425

0.74360 0.77639 0.80768 0.83717 0.86461

745 538 018 510 675

0.050 0.045 0.040 0.035 0.030

0.98670 0.98573 0.98593 0.98707 0.98892

320 080 357 842 100

0.99506 0.99091 0.98874 0.98822 0.98900

643 634 519 421 824

0.91546 0.93284 0.94817 0.96140 0.97253

455 978 344 216 769

0.88977 0.91245 0.93251 0.94985 0.96444

340 301 041 358 830

0.025 0.020 0.015 0.010 0.005

0.99120 0.99367 0.99605 0.99807 0.99947

680 323 259 595 760

0.99073 0.99302 0.99549 0.99774 0.99937

519 746 538 259 316

0.98161 0.98871 0.99394 0.99744 0.99939

804 764 654 863 894

0.97632 0.98556 0.99231 0.99679 0.99925

121 077 623 434 415

0.000

1.00000 000 (-;I4

o.;io

[

1.00000 000

1 &%119(x)

1.00000 000 C-$3 [ I =fQ(x)ez-4k-1

-5 @z121(x)

=flo(x)e+55~-1 Y

@&&9(x)

= gQ(x)e-z+452-1 -5

<x> = nearest integer to 5.

10

1.00000 000 'y3

c 1

13

s”o” 1El 200 Co

BESSEL

MODIFIED

FUNCTIONS

SPHERICAL

OF

BESSEL

I-

tI

FRACTIONAL

473

ORDER

Table 10.10 ORDERS

FUNCTIONS-VARIOUS

43 7I 3.58484 6 5 8.59805 8301 7.09794 8.24936 5.40595 3854 9394 4523 2086

II

- 10 12 11 9 8.12182 8 2.82275 9483 5.57826 4.11114 7.01394 9636 2138 8275 3211

I-

23I 2.56465 1 5.33186 8690 7.83315 2.14704 7.45140 3294 9422 4364 1251

II (-

3j1.20941 3702

-

46 5 2.52325 8.74937 7454 1.46862 4.87152 7470 7330 8858

- 18 24 1.60182 21 22 19 3.16500 2725 4.57312 5.29060 1.23512 3796 2995 7153 0086

I-I

- 26 61 7.71514 81 43 3.65054 7565 1.55685 5.65589 5122 5412 8686

(-190)7.48149 1755

II

- 20 66 8.37672 34 49 6.21921 8478 1.74298 4.17042 9214 6176 4440

(-160)9.55425 1030

II (

x=10 3)1.10132 3287 2 9.91190 3.91520 9985 5.89207 8.03965 9633 4237 9640

1I

- 12 33 9.70826 22 46 1.63577 9664 3.64245 6.36889 1994 3001 6441

(-120)6.26113 6933

x=50

I 19I 4.23682 5.18470 4844 5.08101 4.59302 4.87984 1073 6934 1418 5529

II

5

7

40 1.15601 41 8.55360 0470 9.36222 1.01451 1.08840 6574 3425 8456 0111

:10 12 13 14

II II

- 2 3 2.23450 4 4.23421 1.66914 1.02488 9437 6.26543 6979 3574 7720 8379

20 30

5400 100

-125 2.37154 -31 --21 2.81471 5.88991 3577 1.22928 6154 5830 4325

(-90)9.54463 8661

II

18 3.40719 1.17158 1.70426 2.43274 1747 4.68149 8938 7856 6870 3423

II

(40)1.64074 7551

+17 12 15 9 2.00489 7.90430 5.67659 4104 7.34905 8633 3929 8082

(-17)2.34189 3740

(20) 3.73598 8741

474

BESSEL

Table

FUNCTIONS

OF

FRACTIONAL

10.10 MODIFIED

SPHERICAL

II

x=1

’

BESSEL

I-

FUNCTIONS-VARIOUS

3I 1.40478 4 1.56063 5 2.04287 6 3.07991 7 5.25629

:i

II

Ii - 3i 2.11678 2.54014 6.18102 a479 3.64087 6184 2359 6175

6594 6427 5221 9195 1384

6814 7227 0417 9783 3701

20 30 40 50

23)4.95991 7633

100

( a7)1.04451 3645

x=10 -6 7.13140 4291 -6 7.84454 4720 -6 9.48476 7707 -5 1.25869 2857 I-5 I 1.82956 1771

:: 12 13 14 15 16 :; 19

x=5

(- 211.22943 0749

14 13 11 10 9 1.00177 1.21727 4.86068 2.10898 3.29151 4384 9443 1836 5282 5179 15I 9.55756 17 2.96613 la 9.79781 20 3.43219 22 1.27089

ORDERS

x=2 111.06292 0829

- 1 i02 5.77863 4.04504 6749 1.53711 2.13809 1.15572 7350 5724 7375 5597

::: 12

ORDER

9.90762 2.50109 6.74327 1.93592 5.90133

2914 2290 4558 7868 2701

1.90497 6.49556 2.33403 a.81868 3.49631

9270 9007 5699 1848 5854

fi

II I

9 4.79605 a 7 6 5 4.52287 9.50401 1652 6.99881 5.56068 2999 9354 0749 7078

12 16I 2.15637 14 13 11 6.27234 9105 1.16395 3.57187 1.38508 6139 7368 6330 0704

( 17)2.27598 6819

(156)4.08894 4237

I

1 1.11621 1 4.96235 2 2.39430 3 1.24677 3I 6.97201 4 4.16844 5 2.65415 6 1.79342 7 1.28194 7I 9.66570

I

7817 0604 3059 5036 5499 6493 6981 a072 1220 7838

8)7.66744 6235 18)7.97979 3303

(116)2.49323 8041

x=50

I -24 I 6.05934 6.18053 6.82355 7.38547 6353 6.43017 1115 5506 8350 3280

3.63628 4300

-46 -46 -46 -46 -46

6.78387 7.20097 7.71999 a.35897 9.14102

0523 0973 6750 0485 1732

-45 -45 -45 -45 -45

1.00957 1.12611 1.26858 1.44325 1.65826

6461 3230 2504 8856 2396

II II

t-22)3.67748 3017

(-45)4.68935 4218

(+12)5.97531 1344

(-25)1.48279 6529

it 100

(a5)8.14750 7624

BESSEL

FUNCTIONS

OF

AIRY x

Ai

Bi(x)

Ai’

FRACTIONAL

475

ORDER

FUNCTIONS

Bi’(x)

IX

Table

Ai

Ai’

10.11

Bi’ (x)

Bi(x)

0.00 0.01 0.02 0.03 0.04

0.35502 0.35243 0.34985 0.34726 0.34467

805 992 214 505 901

-0.25881 -0.25880 -0.25874 -0.25866 -0.25854

940 174 909 197 090

0.61492 0.61940 0.62389 0.62837 0.63286

663 962 322 808 482

0.44828 0.44831 0.44841 0.44856 0.44878

836 926 254 911 987

0.50 0.51 0.52 0.53 0.54

0.23169 0.22945 0.22721 0.22499 0.22279

361 031 872 894 109

-0.22491 -0.22374 -0.22257 -0.22138 -0.22018

053 617 027 322 541

0.85427 0.85974 0.86525 0.87081 0.87641

704 431 543 154 381

0.54457 0.54890 0.55334 0.55789 0.56257

256 049 239 959 345

0.05 0.06 0.07 0.08 0.09

0.34209 0.33951 0.33693 0.33435 0.33177

435 139 047 191 603

-0.25838 -0.25819 -0.25797 -0.25772 -0.25744

640 898 916 745 437

0.63735 0.64184 0.64634 0.65084 0.65534

409 655 286 370 975

0.44907 0.44942 0.44984 0.45033 0.45088

570 752 622 270 787

0.55 0.56 0.57 0.58 0.59

0.22059 0.21841 0.21624 0.21408 0.21193

527 158 012 099 427

-0.21897 -0.21775 -0.21653 -0.21529 -0.21404

720 898 112 397 790

0.88206 0.88776 0.89350 0.89930 0.90515

341 152 934 810 902

0.56736 0.57227 0.57730 0.58246 0.58774

532 662 873 311 120

0.10 0.11 0.12 0.13 0.14

0.32920 0.32663 0.32406 0.32150 0.31894

313 352 751 538 743

-0.25713 -0.25678 -0.25641 -0.25600 -0.25557

042 613 200 854 625

0.65986 0.66438 0.66890 0.67343 0.67798

169 023 609 997 260

0.45151 0.45220 0.45297 0.45381 0.45472

263 789 457 357 582

0.60 0.61 0.62 0.63 0.64

0.20980 0.20767 0.20556 0.20347 0.20138

006 844 948 327 987

-0.21279 -0.21153 -0.21025 -0.20898 -0.20769

326 041 970 146 605

0.91106 0.91702 0.92303 0.92910 0.93524

334 233 726 941 011

0.59314 0.59867 0.60433 0.61012 0.61603

448 447 267 064 997

0.15 0.16 0.17 0.18 0.19

0.31639 0.31384 0.31130 0.30876 0.30623

395 521 150 307 020

-0.25511 -0.25462 -0.25411 -0.25356 -0.25300

565 724 151 898 013

0.68253 0.68709 0.69167 0.69625 0.70085

473 709 046 558 323

0.45571 0.45677 0.45791 0.45912 0.46041

223 373 125 572 808

0.65 0.66 0.67 0.68 0.69

0.19931 0.19726 0.19521 0.19318 0.19116

937 182 729 584 752

-0.20640 -0.20510 -0.20380 -0.20248 -0.20117

378 500 004 920 281

0.94143 0.94768 0.95399 0.96037 0.96681

066 241 670 491 843

0.62209 0.62827 0.63460 0.64106 0.64766

226 912 222 324 389

0.20 0.21 0.22 0.23 0.24

0.30370 0.30118 0.29866 0.29615 0.29365

315 218 753 945 818

-0.25240 -0.25178 -0.25114 -0.25047 -0.24977

547 548 067 151 850

0.70546 0.71008 0.71472 0.71938 0.72405

420 928 927 499 726

0.46178 0.46324 0.46477 0.46638 0.46808

928 026 197 539 147

0.70 0.71 0.72 0.73 0.74

0.18916 0.18717 0.18519 0.18322 0.18127

240 052 192 666 478

-0.19985 -0.19852 -0.19719 -0.19585 -0.19451

119 464 347 798 846

0.97332 0.97990 0.98655 0.99327 1.00006

866 703 496 394 542

0.65440 0.66129 0.66832 0.67549 0.68282

592 109 121 810 363

0.25 0.26 0.27 3.28 0.29

0.29116 0.28867 0.28619 0.28372 0.28126

395 701 757 586 209

-0.24906 -0.24832 -0.24756 -0.24677 -0.24597

211 284 115 753 244

0.72874 0.73345 0.73818 0.74292 0.74769

690 477 170 857 624

0.46986 0.47172 0.47367 0.47571 0.47783

119 554 549 205 623

0.75 0.76 0.77 0.78 0.79

0.17933 0.17741 0.17549 0.17360 0.17171

631 128 975 172 724

-0.19317 -0.19182 -0.19047 -0.18912 -0.18777

521 851 865 591 055

1.00693 1.01387 1.02088 1.02798 1.03516

091 192 999 667 353

0.69029 0.69792 0.70571 0.71365 0.72174

970 824 121 062 849

0.30 0.31 0.32 0.33 0.34

0.27880 0.27635 0.27392 0.27149 0.26906

648 923 055 064 968

-0.24514 -0.24429 -0.24343 -0.24254 -0.24164

636 976 309 682 140

0.75248 0.75729 0.76213 0.76699 0.77187

559 752 292 272 782

0.48004 0.48235 0.48474 0.48722 0.48980

903 148 462 948 713

0.80 0.81 0.82 0.83 0.84

-0.16984 0.16798 0.16614 0.16431 0.16249

632 899 526 516 870

-0.18641 -0.18505 -0.18369 -0.18232 -0.18096

286 310 153 840 398

1.04242 1.04976 1.05719 1.06470 1.07230

217 421 128 504 717

0.73000 0.73842 0.74701 0.75576 0.76468

690 795 380 663 865

0.35 0.36 0.37 0.38 0.39

0.26665 0.26425 0.26186 0.25947 0.25710

787 540 243 916 574

-0.24071 -0.23977 -0.23881 -0.23783 -0.23684

730 495 481 731 291

0.77678 0.78172 0.78669 0.79169 0.79671

917 770 439 018 605

0.49247 0.49524 0.49810 0.50106 0.50412

861 501 741 692 463

0.85 0.86 0.87 0.88 0.89

0.16069 0.15890 0.15713 0.15536 0.15362

588 673 124 942 128

-0.17959 -0.17823 -0.17686 -0.17549 -0.17413

851 223 539 823 097

1.07999 1.08778 1.09566 1.10363 1.11170

939 340 096 385 386

0.77378 0.78304 0.79249 0.80211 0.81191

215 942 282 473 759

0.40 0.41 0.42 0.43 0.44

0.25474 0.25238 0.25004 0.24771 0.24539

235 916 630 395 226

-0.23583 -0.23480 -0.23376 -0.23270 -0.23163

203 512 259 487 239

0.80177 0.80686 0.81198 0.81714 0.82233

300 202 412 033 167

0.50728 0.51053 0.51389 0.51736 0.52092

168 920 833 025 614

0.90 0.91 0.92 0.93 0.94

0.15188 0.15016 0.14845 0.14676 0.14508

680 600 886 538 555

-0.17276 -0.17139 -0.17003 -0.16866 -0.16730

384 708 090 551 113

1.11987 1.12814 1.13651 1.14499 1.15357

281 255 496 193 539

0.82190 0.83207 0.84243 0.85298 0.86373

389 615 695 891 470

0.45 0.46 0.47 0.48 0.49

0.24308 0.24078 0.23849 0.23621 0.23394

135 139 250 482 848

-0.23054 -0.22944 -0.22833 -0.22720 -0.22606

556 479 050 310 297

0.82755 0.83282 0.83812 0.84346 0.84885

920 397 705 952 248

0.52459 0.52837 0.53225 0.53625 0.54035

717 457 956 338 729

0.95 0.96 0.97 0.98 0.99

0.14341 0.14176 0.14012 0.13850 0.13689

935 678 782 245 066

-0.16593 -0.16457 -0.16321 -0.16185 -0.16050

797 623 611 781 153

1.16226 1.17106 1.17998 1.18901 1.19815

728 959 433 352 925

0.87467 0.88581 0.89716 0.90871 0.92046

704 871 253 137 818

0.50

0.23169

361

-0.22491

053

0.85427

704

0.54457

256

1.00

0.13529

242

-0.15914

744

1.20742

359

0.93243

593

[‘-y]

AIRY

[‘-f)4]

R-r)

[‘y72]

[c-y]

FUNCTIONS-AUXILIARY

x

r-1

[‘-;‘“I

FUNCTIONS

f(r)

d-0

g(r)

I-1

[‘-pl]

FOR LARGE x

[‘-y-j

POSITIVE

[‘a’“]

ARGUMENTS

j--r)

f(r)

d-4

gw

:*: 1:3 :::

1.047069 1.000000 1.100099 1.160397 1.229700

0.527027 0.528783 0.530601 0.534448 0.532488

0.620335 0.619912 0.620327 0.619799 0.618649

0.619954 0.617156 0.614275 9.608239 0.611305

0.478728 0.479925 0.481658 ;.;;;1";; .

0.45 0.50 0.40 0.35 0.30

2.231443 2.080084 2.413723 2.638450 2.924018

0.549584 0.548230 0.550980 0.552421 0.553912

0.589451 0.593015 0.585855 0.578985 0.582330

0.585235 0.587245 0.583174 0.581056 0.578878

0.530678 0.526011 0.535345 0.544235 0.539902

1.0 0.9

1.310371 1.405721 1.520550

0.536489 0.538618 0.540844

0.616764 0.614022 0.610309

0.605068 0.601782 0.598372

;.;;;qO;: 0:501859

1.662119 1.842016

0.545636 0.543180

0.599723 0.605543

0.594823 0.591120

;.:1";;;; .

0.25 0.20 0.15 0.10 0.05

3.301927 3.831547 4.641589 6.082202 9.654894

0.555456 0.557058 0.558724 0.560462 0.562280

0.575908 0.573135 0.570636 0.568343 0.566204

0.576635 0.574320 0.571927 0.569448 0.566873

0.548255 0.551930 0.555296 0.558428 0.561382

0:6 E

Om5[(-i)7] 2.080084 Ai(

[(-45)2] 0.548230 [(-911 0.593015 f x -i

e-rf(

-0

[(-45)2] 0.587245 F;‘j .

Bi(x)=x-’

erf(r)

0*0° Ai’(.-ix’

e-fg(-r)

Bi’(.z)=x’erg(r)

From J. C. P. Miller, The Airy integral, British Assoc. Adv. Sci. Mathematical Tables, Part-vol. B. Cambridge Univ. Press, Cambridge, England, 1946 (with permission).

”

476 Table x

BESSEL

10.11

Ai( - x) 0.355‘02 0.35761 0.36020 0;36279 0.36537

iO5 b19 397

0.05

0.36796 0.37054 0;37312

149 416 460

0.37570 0.37827

243 725

0.10 0.11 Oil2 0.13 0.14

0.38084 0.38341 0;38597 0.38853 0.39109

867

843 213

0.15 O.lb 0.17 0.18 0.19

0.39364 0.39618 0.39871 0.40124 0.40376

037 269 868 789 987

0.20

0.40628

0;07 0.08 0.09

102 b99

628 961

Ai ‘c-x) -0.258il

771 731 986

-0.25836

484

-0.25816 -0.25731

173 001 918 872

0.59249 0.58800 0.58351 0.57901 0.57450

841

-0.25695 -0.25655 -0;25611 -0.25563

811 685 443 033

-0.25510

406

0.56999 0.56548 0.56096 0.55643 0.55189

904 397 268 466 940

-0.25453 -0;25392

511 297

-0.25326

716

642 523 536 b34

771

0.45470 0.45554 0.45643 0.45737 0.45835

903

0.45938 0.46045

-0.25880 -0.25874 -0.25865 -0.25852

-0.25792 -0.25763

0.41872 0.42118 0.42363

461 319 082 701 126

-0.24531 -0.24419 -0.24303 -0.24181

310 200

-0.24054 -0.23922

0.35 0.36 0.37 0;38

0.39 0.40 0.41

0.42 0.43 0.44

0.44856 0.44877

963

0.44903 0.44936 0.44974

0.52450

-0.24741

0.32 0.33 0.34

524 863

267

-0.24838

0.43330

0.60147 0.59698

-0.25103 -0.25019 -0.24931

653

0.43090

0.44831 0.44841

419 038 798 557

0.30

0.60596

364 005

250

0.41377

0.31

0.44828

-0.25182

-0.25256

0.41625

0.42849

bb3

0.54735 0.54280 0.53824 0.53367 0.52909

0.23 0.24

0.42606

Bi’(-x)

0.61492 0.61044

0.40879 0.41128

0.27 0.28 0.29

Bi( -x)

940 157

0.21 0.22

0.25 0.26

OF

AIRY

0.00 0.01 0.02 0;03 0.04

0.06

FUNCTIONS

-0.24638

716

720 559 737 206

0.51990 0.51529 0.51067 0.50604

767 218 261

986 977 835 518

0.45017 0.45066 0.45121 0.45180

0.51 0.52 0.53 0.54

0;48174 0.48369

138 089 487

0.48562 0.48752 0.48939

274 389 774

976

0.55 0.56 0.57 0.58 0.59

336

0.60

015 104 074 833 293 364

955

0.45390

546 355 047 530

713 503 806

0.50

0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68

0.50333 0.50493 0.50650 0.50803

007 395

511 295

0.51649

336

0.51777 0.51901

258 296

0.72

279 740

0.73 0.74

0.71

919 828 888 053

0.50139

987

0.46515

148

0.75

0.49674

203

0.49207 0.48738

408 423

0.76 0.77

0.52021

390

276

0.48268

127 722 953

0.46642 0.46773 0.46908 0.47046

095 327

0.78 0.79

0.52137 0.52249

479 501

513 719 851

0.47797 0.47325 0.46851

784 181 112

0.47188 0.47333 0.47481

022 081 405

0.80 0.81 0.82

0.52357 0.52461

395 101

0.38035

266

0.50593

371

-0.20408

lb7

[ C-j)7

[t-i’“]

-0.14464 -0.14103 -0.13736 -0.13363 -0.12984

521

0.53525

282

0.53721

0.29153 0.28612

084 932

0.53917

768

0.28070

835

0.54306

618 575 714

129

0.27526

801

0.54499

366

0.26980

0.26432

0.54692 0.54883

912

479

840 964

185 516

0.55260

464

0.25883

322

0.25331

-0.12599 -0.12207 -0.11810 -0.11406 -0.10996

157 538 815

-0.10580 -0.10159 -0.09731

999 101 134

0.54112

0.55072

048 000 642 852

055

0.24777

973

0.55447

665

0.24222 0.23665

0.55632

0.55815

0.2310b 0.22545

571 329 265

398

0.56176

506 480 647 884 063

0.21982 0.21418 0.20852

751 345 204

745 994 679 671 845 071

0.55996

059

906

0.53195 0.53254 0.53308

995 502 163

-0.06091 -0.05609

100

0.16263

895

0.57974

926

407

-0.05121

879

420 518

0.53356

920

-0.04628

549

0.53400

715

-0.04129

452

0.15683 0.15101 0.14518 0.13933

0.58119 0.5R7hn 0.58397 0.58530

484 171 309 317

0.58659

217 879 174 973 145 563

0.95 0.96

809

046 733 964

0.30227 0.29691

0.17996 0.17420 0.16842

0.41531

0.47572

218 007 701 290

854

0.90 0.91 0.92 0.93 0.94

0.50

-0.16176 -0.15846 -0.15509 -0.15167 -0.14818

852

486 652 062 611 193

-0.20643

022

0.53329

925

0.48773 0.48946 0.49122

965 548

0.53132

0.31294

0.30761

-0.07036

094 495 153

0.47367

438 780

0.31824

-0.06566

0.43002 0.42513 0.42023

702

341 392 399 345

676

141 723 826

031 070 713 850

115

700

307 670

0.49844 0.50029 0.50215 0.50403

0.52540 0.52737 0.52934

0.53064

0.48434 0.48602

0.49660

927

184 796 435 101 795

0.53132

500

156 973

457 934 570 348 251

0.52342

0.32879 0.32352

424 991

0.87

0.48268

0.43488

0.40541 0.40043 0.39544 0;39043 0.38540

260

0.51949 0.52145

0;88 0.89

667

llb

894 721 893 379 147

-0.17736 -0.17436 -0.17130 -0.16818 -0.16500

lb6

0.57200 0.57362

-0.23023

-0.21525 -0.21313 -0.21095 -0;20872

339

630 813

-0.22682

864 066

853

0.51753

0.19143 0.18570

937 631

0.46523 0.46738

0.51557

033 520 043 599

979 904

0;44968 0.45196

0.45 0.46 0.47 0;48 0.49

589

0.34965 0.34446 0.33926 0.33403

-0.08410 -0.07958

970

354

0.49479

0.35481

824 678

0.47944

0.48105

154

050 999 016 078

0.52832

784

534

0;49299

132 OiO

0.52914 0.52991

0.45419

0.44939

047

123

0.51169 Oi51363

0.85

368

773

0.41037

0.50976

853 193

0.86

-0.23344

412

579

0.36508 0.35996

354

-0.23186

447

0.37019

449 192

820

817 477 535

-0;21933

846

0.20284

0.44275 0.44508 0.44739

-0.21732

3il 166

0.19714

450

918

0.50784

U3

895

0.47787

233 567

409

0.505i3

055

0.47632

443

-0.22503 -0.22318 -0.22128

-0.19146 -0.18875 -0.18600 -0.18318 -0.18030

Bi’(-x) 2kb 379

-0.08857

543

0.45898

0.45422 0.45646 0.45868 i46089 0.46307

-0.20167 -0.19920 -Oil9668 -0.19410

Bit-xl 0.38035 0.37528

479

0.46375

0.44457 0.43974

lb7

0.52746

865

893 687

Ai’( -x) -0.20408

0.56353 0.56527 0.56699 0.56869 0.57036

718

[t-j”]

0.50170

040 882 087 591

-0.23785 -0.23643

[ C-f’31

953 821 b59 408

0.50953

-0.23496

119

0.49484 0.49660 0.49833 0.50003

0.51100 0.51242 0.51382 0.51517

747 900

0.46950 0;47159

369 115

0.69

607

561

0.49306

0.70

529 578 860

0.43568 0.43805

675

0.49124

809 692

685 620

0.44041

-0;22855

Ai( -x) 0.47572 0.47775 0.47976

945 712

0.46156 0.46272 0.46391

ORDER

FUNCTIONS

x 836 896

0.45245

0.45315

FRACTIONAL

0.83

0.84

0.52560 0.52655

557 703

982

-0.09297

-0.07500

399 419

226

585

720 lb5 777

0.53439

490

-0.03624

628

0.97 i;ii 0.99

0.53473 0.53501 0.53525 0.53543

189 754 129 259

-0.03114 -0.02597 -0.01548

116 957 197 880

0.13347 0.12760 0.12171 0.11582 0.10991

587

0.58783 0.5R904 0.59019 0.59131

1.00

0.53556

088

-0.01016

057

0.10399

739

0.59237

[C-:)7]

-0.02076

[‘-j’“]

[C-t’“]

634 415 971

0.57520 0.57675 0.57826

346

[C-j+]

BESSEL

FUNCTIONS

OF

AIRY x

Ai

Ai’( -x)

Bi( -x)

FRACTIONAL

477

ORDER

FUNCTIONS

Bi’(-x)

x

Table

Ai( -x)

Ai’( -n)

Bi( -x)

10.11

Bi’( -x)

1.0 1.1 1.2 1.3 1.4

0.53556 0.53381 0.52619 0.51227 0.49170

088 051 437 201 018

-0.01016 +0.04602 0.10703 0.17199 0.23981

057 915 157 181 912

+0.10399 +0.04432 -0.01582 -0.07576 -0.13472

739 659 137 964 406

0.59237 0.60011 0.60171 0.59592 0.58165

563 970 016 975 624

5.5 5.6 5.7 5.8 5.9

+0.01778 -0.06833 -0.15062 -0.22435 -0.28512

154 070 016 192 278

0.86419 0.85003 0.78781 0.67943 0.52962

722 256 722 152 857

-0.36781 -0.36017 -0.33245 -0.28589 -0.22282

345 223 825 021 969

+0.02511 -0.17783 -0.37440 -0.55300 -0.70247

158 760 903 203 952

1.5 1.6 1;7 1.8 1.9

0.46425 0.42986 0;38860 0.34076 0.28680

658 298 704 156 006

0.30918 0.37854 0.44612 0.50999 0.56809

697 219 455 763 172

-0.19178 -0.24596 -0.29620 -0.34140 -0.38046

486 320 266 583 588

0.55790 0.52389 0.47906 0.42315 0.35624

810 354 134 137 251

6.0 6.1 6.2 6.3 6.4

-0.32914 -0.35351 -0.35642 -0.33734 -0.29713

517 168 107 765 762

0.34593 +0.13836 -0.08106 -0.29899 -0.5014:

549 394 856 161 985

-0.14669 -0.06182 +0.02679 0.11373 0.19354

838 255 081 701 136

-0.81289 -0.87622 -0.88697 -0.84276 -0.74461

879 530 896 110 387

2.0 2.1 2.2 2.3 2.4

0.22740 0.16348 0.09614 +0.02670 -0.04333

743 451 538 633 414

0.61825 0.65834 0.68624 0.70003 0.69801

902 069 482 366 760

-0.41230 -0.43590 -0.45036 -0.45492 -0.44905

259 235 098 823 228

0.27879 0.19168 +0.09622 -0.00581 -0.11223

517 563 919 106 237

6.5 6.6 6.7 6.8 6.9

-0.23802 -0.16352 -0.07831 +0.01210 0.10168

030 646 247. 452 800

-0.67495 -0.80711 -0.88790 -0.91030 -0.87103

249 925 797 401 106

0.26101 0.31159 0.34172 0.34908 0.33283

266 995 774 418 784

-0.59717 -0.40856 -0.19009 +0.04437 0.27926

067 734 878 678 391

2.5 2.6 2.7 2.8 2.9

-0.11232 -0.17850 -0.24003 -0.29509 -0.34190

507 243 811 759 510

0.67885 0.64163 0.58600 0.51221 0.42118

273 799 720 098 281

-0.43242 -0.40500 -0.36709 -0.31929 -0.26258

247 828 211 389 500

-0.22042 -0.32739 -0.42989 -0.52445 -0.60751

015 717 534 040 829

7.0 7.1 7.2 7.3 7.4

0.18428 0.25403 0.30585 0.33577 0.34132

084 633 152 037 375

-0.77100 -0.61552 -0.41412 -0.18009 +0.07027

817 879 428 580 632

0.29376 0.23425 0.15821 +0.07087 -0.02159

207 088 739 411 652

0.49824 0.68542 0.82650 0.90998 0.92812

459 058 634 427 809

3.0 3.1 3.2 3.3 3.4

-0.37881 -0.40438 -0.41744 -0;41718 -0.40319

429 222 342 094 048

0.31458 0.19482 +0.06503 -0.07096 -0.20874

377 045 115 362 905

-0.19828 -0.12807 -0.05390 +0.02196 0.09710

963 165 576 800 619

-0.67561 -0.72544 -0.75412 -0.75926 -0.73920

122 957 455 518 163

7.5 7.6 7.7 7.8 7.9

0.32177 0.27825 0.21372 0.13285 +0.04170

572 023 037 154 188

0.31880 0.54671 0.73605 0.87115 0.94004

951 882 242 540 300

-0.11246 -0.19493 -0.26267 -0.31030 -0.33387

349 376 007 057 856

0.87780 0.76095 0.58474 0.36122 +0.10670

228 509 045 930 215

3.5 3.6 3.7 3.8 3.9

-0.37553 -0.33477 -0.28201 -0.21885 -0.14741

382 748 306 598 991

-0.34344 -0.46986 -0.58272 -0.67688 -0.74755

343 397 780 257 809

0.16893 0.23486 0.29235 0.33904 0.37289

984 631 261 647 058

-0.69311 -0.62117 -0.52461 -0.40581 -0.26829

628 283 361 592 836

8.0 8.1 8.2 8.3 8.4

-0.05270 -0.14290 -0.22159 -0.28223 -0.31959

505 815 945 176 219

0.93556 0.85621 0.70659 0.49727 +0.24422

094 859 870 679 089

-0.33125 -0.30230 -0.24904 -0.17550 -0.08751

158 331 019 556 798

-0.15945 -0.41615 -0.64232 -0.81860 -0.92910

050 664 293 044 958

4.0 4.1 4.2 4.3 4.4

-0.07026 +0.00967 0.08921 0.16499 0.23370

553 698 076 781 326

-0.79062 -0.80287 -0.78221 -0.72794 -0.64085

858 254 561 081 018

0.39223 0.39593 0.38346 0.35494 0.31122

471 974 736 906 860

-0.11667 +0.04347 0.20575 0.36320 0.50858

057 872 691 468 932

8.5 8.6 8.7 8.8 8.9

-0.33029 -0.31311 -0.26920 -0.20205 -0.11726

024 245 454 445 631

-0.03231 -0.30933 -0.56297 -0.77061 -0.91289

335 027 685 301 276

+0.00775 0.10235 0.18820 0.25778 0.30483

444 647 363 240 241

-0.96296 -0.91547 -0.78882 -0.59221 -0.34136

917 918 623 371 475

4.5 4.6 4.7 4.8 4.9

0.29215 0.33749 0.36736 0.38003 0.37453

278 I 598 748 668 635

-0.52336 -0.37953 -0.21499 -0.03676 +0.14695

253 391 018 510 743

0.25387 0.18514 0.10794 +0.02570 -0.05774

266 576 695 779 655

0.63474 0.73494 0.80328 0.83508 0.82721

477 444 926 976 903

9.0 9.1 9.2 9.3 9.4

-0.02213 +0.07495 0.16526 0.24047 0.29347

372 989 800 380 756

-0.97566 -0.95149 -0.84067 -0.65149 -0.39986

398 682 107 241 237

0.32494 0.31603 0.27858 0.21570 0.13293

732 471 425 835 876

-0.05740 +0.23484 0.50894 0.73928 0.90348

051 379 402 028 537

5.0 5.1 5.2 5.3 5.4

0.35076 0.30952 0.25258 0.18256 0.10293

101 600 034 793 460

0.32719 0.49458 0.63990 0.75457 0.83122

282 600 517 542 307

-0.13836 -0.21208 -0.27502 -0.32371 -0.35531

913 913 704 608 708

0.77841 0.68948 0.56345 0.40555 0.22307

177 513 898 694 496

9.5 9.6 9.7 9.8 9.9

0.31910 0.31465 0.28023 0.21886 0.13623

325 158 750 743 503

-0.10809 +0.19695 0.48628 0.73154 0.90781

532 044 629 486 333

+0.03778 -0.06091 -0.15379 -0.23186 -0.28738

543 293 421 331 356

0.98471 0.97349 0.86898 0.67936 0.42147

407 918 388 774 209

5.5

0.01778

154

0.86419

722

-0.36781

345

0.02511

158

10.0

0.04024

124

0.99626

504

-0.31467

983

0.11941

411

[q’“]

[q’“] AIRY

[‘-a”]

FUNCTIONS-WXILIARY

9.654894 11.203512 13.572088 17.784467 28.231081

Ai(-x)=x-’ A;Y-x)-x’

0.39752 0.39781 0.39809 0.39838 0.39866

[(-93)4]

FUNCTIONS ARGUMENTS f,
.f#>

X

f-l 0.05 0.04 0.03 0.02 0.01

[y]

21 14 83 24 38

[fi(r) cos ,+f,(r) sin r]

0.40028 0.40002 0.39975 0.39949 0.39921

87 58 97 03 79

[qy]

FOR

LARGE

L&> 0.40092 0.40052 0.40012 0.39972 0.39933

31 06 11 48 19



0.39704 0.39741 0.39779 0.39817 0.39855

87 99 49 37 62

cos r-fl(r)

:05 53; 100

sin r]

[g,(r) cos r+g,(r) sin r] to t.

[(-;Lo)1]

NEGATIVE &?(d

Bi(-x)=x-a[f,(r)

Bi’(--x)=x’ [gl(r) sin r-g,(r) cos I] 3 ,=yxz =nearest integer

[(-;)“I

478

BESSEL Table

10.12

FUNCTIONS INTEGRALS

zs

dt Ji Ai(-t)dtJi 0.00000 00 0.00000 00

5 Ai (t)

06 09

-0.03679 -0.07615 -0.11802 -0.16229

54 70 51 44

0.16801 0.14595

79 33

-0.25736 -0.20880

07 95

;*i 0:9

0.18795 0.20589 0.22196

45 52 97

-0.30768 -0.35944 -0.41225

05 15 56

1.0

0.23631

73

-0.46567

40

:-: 1:3 1.4

0.24907 0.26037 0.27034 0.27910

33 12 09 66

-0.51918 -0.57224 -0.62421 -0.67447

94 05 79 31

0.28678 0.29349

67 24

-0.72232 -0.76709

88 26

:*i 1:9

0.29932 0.30438 0.30876

75 82 29

-0.80807 -0.84459 -0.87602

24 41 06

2.0

2:

0.31253 0.31577 0.31854 0.32091 0.32292

28 11 43 19 74

-0.90177 -0.92135 -0.93435 -0.94050 -0.93967

28 09 56 97 67

2.5

0.32463

80

-0.93187

78

::;

0.32730 0.32608

57 74

-0.89633 -0.91730

20 54

::;

0.32919 0.32833

55 83

-0.83758 -0.86951

3.0 3.1

0.32992 0.33052 0.33102 0.33144 0.33178

04 31 49 15 65

-0.80146 -0.76220 -0.72100 -0.67915 -0.63802

3.5

0.33207

15

:*! 3:s 3.9

0.33230 0.33249 0.33265 0.33278

63 93 76 70

4.0 4.1 4.2

0.33289 0.33297 0.33304

i:: 4.5 4.6

OF

FRACTIONAL AIRY

s

i Bi(-t)dl

0.00000 00

85 50 36 82 64

-0.25006 -0.28553 -0.31575 -0.34052 -0.35966

28 62 56 58 27

0.87276 0.99838 1.13466 1.28318 1.44579

91 41 38 00 42

-0.37300 -0.38042 -0.38185 -0.37726 -0.36673

50 77 43 99 34

1.62470 1.82252 2.04231 2.28772 2.56304

81 33 52 12 90

-0.35038 -0.32847 -0.30132 -0.26939 -0.23325

81 24 67 97 04

2.87340

83

-0.19354 -0.15106 -0.10667 -0.06132 -0.01603

74 46 18 23 45

7.0

77 37

+0.02812 0.07009 0.10878 0.14317 0.17234

94 01 06 88 20

29 32 37 91 56

0.19544 0.21180 0.22092 0.22252 0.21655

25 21 49 61 57

-0.59897

71

-0.56335 -0.53242 -0.50730 -0.48892

61 25 05 77

0.20321 0.18296 0.15652 0.12485 0.08914

27 86 84

-0.47800 -0.47496 -0.47992

75 79 95

0.33315 0.33310

07 50

-0.51269 -0.49268

28 51

0.33318 0.33321

76 73

-0.53908 -0.57068

35 59

2; 4:9

0.33324 0.33326 0.33327

11 02 54

-0.60606 -0.64358 -0.68146

63 51 70

5.0

cLL~3zz;~

15 01

2:

0.09497 0.12164

E

a.

67 45 68 70

0.36533 0.45356 0.54773 0.64845 0.75649

0.33331 0.33332 0.33332 0.33332 0.33332

97 27 50 69 83

-0.82151 -0.81897 -0.80797 -0.78914 -0.76354

82 90 96 06 19

-0.01617 +0.02038 0.05518 0.08625 0.11181

86 99 54 18 25

0,33332 0.33333 0.33333 0.33333 0.33333

95 03 10 16 20

-0.73267 -0.69836 -0.66268 -0.62781 -0.59592

53 93 96 93 62

0.13038 0.14086 0.14262 0.13555 0.12011

11 00 05 73 15

0.33333 0.33333 0.33333 0.93333 0.33333

23 25 27 29 30

-0.56902 -0.54883 -0.53667 -0.53334 -0.53906

35 59 65 74 98

0.09726 0.06847 0.03562 +0.00088 -0.03340

08 29 42 80 40

:::

0.33333 0.33333 0.33333 0.33333 0.33333

31 31 32 32 33

-0.55345 -0.57549 -0.60365 -0.63593 -0.66999

17 72 96 60 96

-0.06491 -0.09147 -0.11121 -0.12273 -0.12521

67 36 47 90 80

7.5

0.33333

33

::9"

-0.70336 -0.73355 -0.75830 -0.77575 -0.78453

19 34 99 13 65

-0.11847 -0.10300 -0.07997 -0.05114 -0.01872

31 57 85 35 22

if 8:2 8.3 8.4

-0.78398 -0.77413 -0.75578 -0.73041 -0.70011

26 57 55 93 70

+0.01475 0.04664 0.07440 0.09577 0.10902

64 84 43 a7 22

50 47 33 43 28

-0.66739 -0.63499 -0.60566 -0.58192 -0.56584

21 08 32 70 22

0.11303 0.10749 0.09285 0.07039 0.04205

86 35 98 64 63

+0.05076 +0.01121 -0.02788 -0.06494 -0.09837

01 78 79 00 02

-0.55881 -0.56148 -0.57358 -0.59403 -0.62093

97 12 51 00 76

+0.01033 -0.02196 -0.05192 -0.07682 -0.09439

04 26 24 93 a7

-0.12673 -0.14876 -0.16347 -0.17018 -0.16857

04 50 66 59 74

-0.65181 -0.68375 -0.71373 -0.73889 -0.75680

01 25 85 84 07

-0.10300 -0.10183 -0.09101 -0.07157 -0.04539

27 70 44 33 57

21" t: 6:4

:::

::;

4..~;2j

10.13

3

C.06373 0.13199 0.20487 0.28256

0.00000 00 87 10 57 89

0.03421 0.06585

ORDER

FUNCTIONS

-0.05924 -0.11398 -0.16411 -0.20952

i.1” 0:2

2:

Table

Bi(t)dt

OF

ZEROS &‘(a,)

AND

ASSOCIATED VALUES OF AND THEIR DERIVATIVES Ai(a’,) a’, h

AIRY

FUNCTIONS

b’,

Bi’(b,)

Bi

(b’,)

1 2

- 2.33810 - 4.08794

741 944

+0.70121 -0.80311

082 137

- 1.01879 - 3.24819

297 758

+0.53565 -0.41901

666 548

- 1.17371 - 3.27109

322 330

+0.60195 -0.76031

789 014

- 2.29443 - 4.07315

968 509

-0.45494 +0.39652

438 284

i 5

:6'78670 5 52055 - 7:94413

983 809 359

+0.86520 -0.91085 +0.94733

403 074 571

- 4.82009 6.16330 - 7.37217

736 921 726

-0.35790 +0.38040 +0.34230

647 794 124

- 4.83073 6.16985 - 7.37676

784 213 208

+0.83699 -0.88947 +0.92998

990 101 364

- 5.51239 6.78129 - 7.94017

573 445 869

+0.34949 -0.36796 -0.33602

916 912 624

6 7 8 9 10

- 9.02265 -10.04017 -11.00852 -11.93601 -12.82877

085 434 430 556 675

-0.97792 +1.00437 -1.02773 +1.04872 -1.06779

281 012 869 065 386

- 8.48848 - 9.53544 -10.52766 -11.47505 -12.38478

673 905 040 663 837

-0.33047 +0.32102 -0.31318 +0.30651 -0.30073

623 229 539 729 083

- 8.49194 - 9.53819 -10.52991 -11.47695 -12.38641

885 438 351 355 714

-0.96323 +0.99158 -1.01638 +I.03849 -1.05847

443 637 966 429 184

- 9.01958 -10.03769 -11.00646 -11.93426 -12.82725

336 633 267 165 831

+0.32550 -0.31693 +0.30972 -0.30352 +0.29810

974 465 594 766 491

AUXILIARY

TABLE-COMPLEX Bi (z) e-438,

1”

M~d$ue

p,e

ZEROS AND

Bi’(B.) M$i&ts I’tai;

AND Bi’(z)

ASSOCIATED

VALUES

OF

Mc+l&us

Bi(o’.) zv6;e

e-A/38’,

M+tte

pO*

:

5.524 4:093

0.027 0:042

1.224 1:136

-0:513 +2.625

4.824 31257

0.033 0:059

0.538 0:592

-21632 +0.515

4 5

6.789 7.946

0.020 0.015

1.288 1.340

-0.519 +2.622

6.166 7.374

0.023 0.017

0.506 0.484

-2.624 to.519

From J. C. P. Miller, The Airy integral, British Assoc. Adv. Sci. Mathematical Tables Part-vol. B. Cambridge Univ. Press, Cambridge, England, 1946 and F. W. J. Olver. The asymptotic expansion of Bessel functions of large order. Philoa. Trans. Roy. Sot. London [A] 247, 32&368,1954(with permission).

11. Integrals

of Bessel Functions YTJDELL L. LUKE~

Contents Page

Mathematical Properties . . . . . . . . . . . . . . . . 11.1. Simple Integrals of Bessel Functions . . . . . . . 11.2. Repeated Integrals of Jn(z) and K,(z) . . . . . . 11.3. Reduction Formulas for Indefinite Integrals . . .

. . . .

. . . .

480 480 482 483 485

Numerical Methods . . . . . . . . . . . . . . . . . . . . . . 11.5. Use and Extension of the Tables . . . . . . . . . . . .

488 488

References

490

11.4. Definite

Table

Integrals

. . . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

11.1 Integrals

l

of Bessel Functions JoW,~

Y,(t)&

e-z~&(t)&, Table 11.2 Integrals

7D

of Bessel Functions

e --z = [r,(t)-lldt:,J, t s0

1

z=“(*l)lo

. . . . . . . . . . . . .

m Y,wt, s

t

,&

494

8D

t

;

492

1oD

e’j3(~)dt,

= [l-W)ldl, s 0

. . . . . . . . . . . . .

- K&h? t sz

6-,-,

2=0(.1)5

The author acknowledges the assistance of Geraldine Coombs, Betty Kahn, Marilyn Kemp, Betty Ruhlman, and Anna Lee Samuels for checking formulas and developing numerical examples, only a portion of which could be accommodated here.

1 Midwest of Standards.)

Research

Institute.

(Prepared

under

contract

with

the National

Bureau 479

11. Integrals Mathema 11.1. Simple

Integrals

of Bessel

of Bessel Functions ! ical Properties

Functiops

11.1.8

S

z.zo(t)dt=xzo(x) ++rxr -Lo(z)Z,(~)+L,(~)Zo(~~

S

atkL(t)dt

I

0

0

ZIP(x) =AI,(z)

+Be*YrKr(z),,v=O,l

A and B are constants.

J&(Z) and chapter 12).

L”(z)

are Struve

functions

be

11.1.9

,

S 0

w4

+‘k&

2k

(k!)2(2k+1)2

(92(P+~+1)>0) 11.1.2

S

&)&=a

0

11.1.3

jyJ2.(t)dkSo'J,(t)dt-2

11.1.4

s0

y (Euler’s constant)=.57721

go J”+2k+,(z>(9’v>-l) .-

Recurrence

In this and all other integrals of 11.1, x is real and positive although all the results remain valid for extended portions of the complex plane unless stated to the contrary.

~J,,,,(z)

g~~,+l(t)dt=1-Jo(z)-2

g J&) B

11.1.10

-t-z J’Yo@)ch SzJo(t)dt+i; S K,(t)dt=;

Relations

0

0

11.1.5

Asymptotic

S

aJ,+l(t)dt=

0

s

ogJ.4(t)dt-2J.(z)

S

kl(t)dt=l--Jo(z)

11.1.6

56649 . . .

b>O>

0

Expansions

11.1.11

Jm [J,(t)+iY,(l)]dt-(~)~ z

0

x[go 11.1.12

e’(‘-*‘*)

1

(--)‘a2k+lx -ak-i+i 5kIO (-)ka2kx-ak

,,-r(k+f)

5 us+*) r-0 2d r (3)

r(i)

11.1.7

Soz~o(t)dt=x~o(x)+~~x{Ho(~)~(~)--~~~)~(~)~ 11.1.13

~(x>=AJ"(x)+BY"(x),),V=0,1 A and B are constants. 480

2(k+l)ak+l=3

(k+!j)

(k+i)

ak

INTEGRALS

11.1.14 de-"

s0

Xlo(t)dt-(2~)-~

go

OF BESSEL

481

FUNCTIONS

11.1.18

75x<=

a,x-"

de”

where the ak are defined as in 11.1.12.

S

OD K~(t)dt=$~ (-)“ek(x/7)-“+c(x) I 2 Ic(x)l<2xlO-7

11.1.15 de" (DK,(t)&sz

0

f ' PO (-)bkxwk -

k

0

where the ak are defined as in 11.1.12. Polynomial

Approximations

11.1.16 s

1.2533: 0.11190 . 02576 .00933 . 00417 .00163 .00033

1 2 3 4 5

*

81X<=

414 289 646 994 454 271 934

im [Jo(t)+iYo(t)Mt =xTje*(z-*l*) go (-)kak(x/8)-w-1 7

11.1.19 1

s0 [e(x)1 12x10-9

=22-l

ak 00233 47304

i :

.: 00404 00100 03539 89872 : 00053 00039 92826 06169

.79788

-al)1

J2k+2(~)

=1-2x-V,(x)

bk

0

PO I (2;+3)w+2)

+2x-’

45600

: 00178 01256 42405 70944 : 00067 00041 00676 40148

go

(2k+5)[~(k+3)-~(1)--1lJ2k+l(~)

For #(z), see 6.3.

11.1.20 sz 11.1.17

Slxl=

21emzl

IO(t)&=&

d,(x/8)-“+.2(x)

[e(x)1<2x10-6 k 0

di . 39894 23

1

. 03117 34

2

. 00591 91

3

. 00569 56

4

-. 01148 58

5

* 01774 40

6

-. 00739 95

2 Approximation 11.1.16 ls from A. J. M. Hitchcock. Polynomial approximations to Bessel functions of order zero and one and to related functions, Math. Tables Aids Comp. 11, 86-88 (1957) (with permission).

{ $(k+l)

11.1.23 --ia

K,(t)& i* S S --iz t=z

ODJ,(t)& z T----Z

?r

*&-ln

- Yo(t)dt z t

S

i}

482

INTEGRALS

Asymptotic

OF

BESSEL

JTUNCTIONS

11.1.30

Expansions

45xs

ODK,(t)& 2 Qez - t sz

=&

k 0

1 2 3 4 5 6

S

11.1.25

go(x)=2$ 2

11.1.26 aY2e2 "~O(W~ t z

S

where

11.1.27

0

; 4&)kckx-k

11.1.31

k

[I,(t);lldt-(~,&f$o

ckz-k

0

cnis defined as in 11.1.27. Polynomial

11.1.29

00

0

S

where

dk

52x2

0 1 2 3 4 5 6 7 8 Q 10

Approximations

55x1 aJ

S2- ~~=2gl(~)~Pox))_go(x>~Px) t X2

2

(;)-kt4xl

1.25331 41 0.50913 39 .32191 84 .26214 46 .20601 26 11103 96 : 02724 00

S

~(~+l~2+~~c~-,+,c~-~

11.1.28 33/ze-z +

(-lkd,

x*e-z =~~ow-ll~t =8x t

co=l,cl=~

2(k+l)ck+l=

co

($)-I +4x)

fk

0.39893 13320 -: 04938 1.47800 -8.65560 28.12214 -48.05241 40.39473 -11.90943 -3.51950 2.19454

14 55 43 44 13 78 15 40 95 09 64

11.2. Repeated Integrals ofJ,(z) andKo(z)

where

Repeated gO(x)=$o

(-)kak(x/5)-2k+e(x),

Integrals

of J,,(Z)

Let

11.2.1

.

I

k

ak

b

11.2.2

-

k:5999 .10161 .13081 .20740 : 28330 27902 .17891 .06622 .01070

2815 9385 1585 4022

ki i1998 .30485 52324 1: 03702

5629 8155 6341 0112

0508 9488 5710 8328 2234

1.95320 1.69980 1.43132 0. 59605 .10702

6413 3050 5684 4956 2336

Then

11.2.3

(9-?Kp>O) S.&)=+) s,(z-t)‘-‘Jn(t)dt

INTEGRALS Recurrence

OF

BESSEL

11.3.2

Relations

11.2.5 r(r--l)j”,l,

t&4=2@--1M.

483

FUNCTIONS

J”(Z), I”(Z) K(z)

nw

-[(1-+--n2+~21f,-l,

Y”(Z),

Z”W H?(z),

n(z)

11.2.6

1 1

1

-1

11.3.3 pg,, &z)=

-e-pzz~Zv(z) +(P+~h-l.

j,+l. n+l (4 =fi+,. n-1(4 -%, Repeated

Integrals

of

v(~>-W,,v+1(~)

11.3.4

nc4

pgr, v+l(z>=-ee-PE~~Zv+l(~)

I&(5)

Let

+(~-v-l)g,-,,v+l(z)+bg,,v(z)

11.2.8

11.3.5

K&(z) =&(a),

(p2+ab)g,,v(z)=ae-P”zrZv+l(z) +(p-v-l)ePzz~-lZ”(z)-pe-pzz~Z”(z)

S

K&(z)=

mK,(t)&,

I

S

m Ki,-,(t)dt z

. . ., K&(z)=

Ki-, (z)= (-)’

11.2.9

g7 K,,(z)

+P(2r--l)g,-l,“(~)+[~2-~(cc--1)219r-2.”~~>

11.3.6

Then

a(v-~)gp,“+l(z)=-2ve-pzz~Z”(z)-2vpg~,”(z)

11.2.10

+WCc+vh.v--1(z)

=

- e-’ ooabf &a

S 0

cash’

Case

t

(Wz>O,

wr>o,

szz>o,

r=O)

p.2.11

Ki(4

=&

11.2.12

Jm (t--z)‘-‘K,(t)& z Iii,,(O)

= rr(Fii$)

(WzlO,

ar>o>

W’r>O)

11.2.13

p2+ab=0,

1:

11.3.7

g”, .(z,=eG

11.3.8

g-,.&9=-

v= f

{

e-pzz-“+l 2v-l

1

z”+&)}

z”(z)+;

Z”&)}

0

’ e%f”J”(t)dt

=gg

[J”(z)-iJy+l(z)]

11.3.10

11.2.14

S

z e”t-“J”(t)(jt=-e~ V

0

Formulas Integrals

for Indefinite

S

” e-%!@Z”(t)dt

where Z,(z) represents any of the Bessel functions of the first three kinds or the modified Bessel The parameters a and b appearing in functions. the reduction formulae are associated with the particular type of Bessel function as delineated in. the following table.

[J”(z)+iJ”--I(2)1 (v+3)

+2”-Y2A)r(v) 11.3.11 E

Let 6&)=

Z”(4 -;

(2’y>-%

Ki27+l(0) =

11.3. Reduction

(r-l)

11.3.9 S

11.3.1

b

1 -1

Tj,+l.O(Z)=Zf,O(Z)-(~--l)j~-l.O(~)+zf~-2,o(~)

Ki&)

a

T&M +(2r--3)zf,-a,.(z)--zzf,-,

11.2.7

W’(z)

S

ef~t”Y”(t)dt-2v+l-eirZY+1 [Y”(Z) -iYy+l(z)]

0

- i2y+‘r (Yfl) 42v+lJ

(WV>-3)

11.3.12

S

’ e*“t”I”(t)dt=

gg

V”(Z) ‘FIv+1(41

0

@v>-t)

‘.

484

INTEGRALS

OF

11.3.13

BESSEL

FUNCTIONS

11.3.25

St~I,-,(t)dt=z%(z) St-~I,+,(t)dt=z-~‘I,(z)z

Wv>O)

0

s0

’ e-lr,(t>dt=ze-‘[r,(z)+Il(z)l 11.3.26

+nIe-.lo(z)-11+2e-‘~~

g

l 2’Iyvfl)

0

(n--kV~(z) 11.3.27

11.3.14

Szt”K”-l(t)&= S t-YK”+l(t)dt=z-YK”(z) --2+Ky(2)

s0

’ e*lt-‘l,(t)dt=-e~vz~~+l

[l,(z) 1 F24(2V-l)r(v)

b#3>

11.3.28

==g

’ e*‘t’K,(t)dt

@v>O)

Integrals

of

Products

of

Beeeel

Functiona

Let %$(z) and .Qy(z) denote any two cylinder functions of orders cc and v respectively.

[K”(Z) fKv+1(z)l

11.3.29

F2”r(v+l) 2v+1

King’s integral

+)

E

Indefinite

11.3.15

s0

+2v

0

F~F-I(~>I

vi?v>--3) s{

(see [11.5])

z

(~-z’)t-~}~(kl)~(zt)dt

=z{k~~+l(kz)~“(z~)-z~(~~)~“+l(~~) 11.3.16

s zecK,(t)~t=ze”[K,(z~+~~(Z)l-~

11.3.17 m e’t-‘K,(t)& sI

11.3.30

S

ap-“---l

=G

p=o,

ag”,v-l(z)=~“~“(d

11.3.19

ag-,,,+,(z)=--z-“Z”(z)

11.3.22

s0

W>3) 11.3.31

J

E t~J”~l(t)dt=a”e7”(2)

S

z t-*J,+,(t)&= 0

r z t’+p+‘~p(t)~(t)dt =2(;y-;;l)

(B>O) l

2vr(v+ij

-PJ,

(2)

-;

=;

2 n z

StJ?-l(t)&=2 *

,$+l

m-

w--1)J2,-,b)

I

11.3.24

0

11.3.34

2

0

-;

&I kJ2k (2)

11.3.35

[I-WI-&:(t) SzJn(tv.,l(t)dt=~ 0

z t’Y._,(t)dt=z.P.(z)+~ 0

(~x-0

(%>O> Szt[J:-l(t>-~J:+l(t)ldt=2vJt(z) St~~(t)dl=~[J:(z)+J:(z)]

@>O)

J,(t)&

-Jl(Z)

S

PO (vS2&c+&)

11.3.33

1V*,-,(d

11.3.23 Jzn+l(Qdt=SD

{~(z)~(z)+~+l(z)~+l(z)l

11.3.32

0

2n- ‘J2nwt=I t s0

(2nfl)l

%+*(~>~+1(W

/A=fv

11.3.18

11.3.21

c

[K,(z)+K-,@)I

2:

11.3.20

*

- (/.J--v)%?~(kz)9”(zz)

0

Case

1

(gv>o) *seepageII.

INTEGRALS

OF

BESSEL

11.3.36

485

FUNCTIONS

2. There must exist numbers not zero) so that for all n

k, and k, (both

(P+v)szt-‘~(t)W(t)dt 11.4.4

S

--(/h+v+w

Convolution

Type

= 0

In connection with these formulae, see 11.3.29. If o=O, the above is valid provided B=O. This case is covered by the following result.

n-1

=~(z)~(z)+~+.(Z)~+,(2)+2~l~+R(2)~+n(2)

Integrals

11.4.5

11.3.37

go(-)nJr+“+2k+l(2) StJ,(cY,t)J,(a,t)dt=O 1

S

* J,w”(z--w=2

0

kA,%C+l (La> --k,%@na)

zt-%$+.(t)L%+.(t)dt

(m#n,

0

v>-1)

=3[J:hJ12

(9%>-1,9v>-l>

11.3.38

(m=n,

S

zJ”(t)J~-~(2-t)dt=Jo(2)-cos

2

b=O, v>-1)

(-1<9?v
0

(m=n,

11.3.39 2:

S I:t-‘J,(t)J” (z-t)d&$ S J,(t)J-,(z--t)dt=sin

2

zeros of %, . . . are the positive aJV(z) +brJ~(~) = 0, where a and b are real constants.

al,

w4<1>

0

11.3.40

11.4.6

S t-‘J m

0

(9&>0,9?v>-1)

“+*n+l(t)J”+Sm+l(t)~t=O

0

11.3.41

Cm #n) 1 =2(2n+v+l)

c J,(t)J”(Z-t)dt=(~+v)J~+Y(2)

S

(m=n)(v+n+m>-1)

lzvz

t(z-t>

0

(9&4>0,

11.4. Definite Orthogonality

Properties

be a cylinder let

11.4.1

%:(z>

-AJ,(z)

c-?v>O)

Definite

function

Integrals

Over

a Finite

Range

*

Integrals of Bessel

Let $32) In particular,

b#O, v>-1)

11.4.7 of order

S t) S t) S sin

’ Jzn(22

t)dt=i

J",(z)

0

Functions

v.

11.4.8

sin

* Jo(22

cos Zntdt=d~(z)

0

+BYAz)

11.4.9

* ’ Yo(22 sin

cos 2ntdi!=~

J,(2)Yn(z)

0

where

A and B are real constants.

Then 11.4.10

11.4.2 4

S

b

S

t~(x,t)%c(xnt)~t=o

cm

+4

J,(z

sin

0

a

t)

sinp+lt

=2'r(v+1) 2Y+l

tdt

cos2Y+’

(&?/A>-1

Jp+"+I(2)

,

S&J>-1)

11.4.11 provided the following two conditions 1. A, is a real zero of 11.4.3

hold :

i

S

J,(z

sin2 t)J,(z =y

h,X%+*(Xb)--hz%qXb)=O

cos2

t)

csc

2tdt

0

JP+“(z)

WP>O,

9’vw

486

INTEGRALS

Infinite Integrals

of the

OF

BESSEL

Integrals Form

FUNCTIONS

11.4.23

me-wz”(t)dt

s

0

11.4.24

mew-‘J”(t)dt=e t”b+d r(p+v) l-(+--p) S0 r(3)2flr(v-P+l)

(S&r<;9 9&+4>0) 11.4.13 m e-ttp-ll,(t)dt=

-

where T,(w) is the Chebyshev first kind (see chapter 22).

polynomial

of the

11.4.25 t-‘em* orJ, (t)dt --m =z

(4)“(1-u~)w,-,(0)(~2<1)

w(,+v)>o) =O(w”>l)

11.4.14

S0

- 00s bt K,(t)

fit=&

(lJbl
(l+bY

where U,(W) is the Chebyshev second kind (see chapter 22). 11.4.26

11.4.15

S0 S0

m sin bt Ko(t)dt=arc

sinh b (l+b2Y

where P,(W) chapter 22).

r(!I$!A) %<;)

S0

- J,(t)&=1

S0 t’

- [l--Jowt=

mtfiY, (t)dt=F

r($)r(+) 2F{ r (ILL))2

o
X sin T (P-Y) ( LC.&)>--1,

S0

11.4622 mta&)dt=2p-l

(see

y

S0

mYo(t)dt=O

Integrals

Form

s

function

o- e- cza*%PZ” (bt) a2

OD S0 e-‘ata t+J,(bt)dt -r(~v+%i~M(l wr cy+i)

i2&)

+I, 2y

2 pv

+1

6p)

’ -4a2

(9 (P+v)>o, aa2>o> (1~4<1>

where the notation M(a, b, z) stands for the confluent hypergeometric function (see chapter 13).

m S 11.4.29 e

r (E!$!2)r(!!)

of the

gamma

11.4.28

r (p!!)r(EE)

- Y,(t)&=-tan

polynomial

11.4.27

where ~(a, z) is the incomplete (see chapter 6).

11.4.19

S0

is the Legendre

@v>-1)

11.4.18

11.4.21

m t-fe-“uzJn++(t)dt=(--i)n(2n)*P,(w)(w2<1) -m =O(d>l)

- tu,(t)dt=

11.4.17

S

(I ybl
(m+4>-1,

11.4.20

of the

S

w+4w-d u3Pw-~+1) (%<$

11.4.16

polynomial

(D

S0

S0

S -m

11.4.12

-a2%+1J~(bt)dt

0

b= =@-&

e-5

@v>-1,9w>0>

INTEGRALS

OF

BESSEL

487

FUNCTION8

11.4.37

S

- J&t)

cos bt dt=--p-----

0

--ar sin y =(b2-u2)f[b+(b2-a2)f]F

11.4.31 b’J

OD e -“ll.(bt)dt=~

S

@>a>@

esI+,

(z2)

0

(a>-1,

a%z~>O)

11.4.32

SJ,(d) m

S

e -azt’Ko(bt)dt=~

sin bt dt=

0

b=

OD es

K.

($)

arcosy

M’a”>O)

0

Weher-Schafheitlln

Type

=(b2-a2)f[b+(b2-a2)f]p

Integrals

11.4.33

( S-J,(ut)J.(bt)dt= bYr

ta

0

@>a>@ 2

>

11.4.39

S

meratJo (at)dt =

0

r--vy+y

v-p--x+1. 2

b2 a2>

, v+1;

93>-l, O
2

ta

0

p--y;x+l;

(%“(r+v--x+1) rel="nofollow">% For 2F1, see chapter Cases

-1 =(b2_a2)t

>

2Abr-~+lr(p+l)r

x2Fl (Pfv;‘+‘,

of the

Xln

p+l; $)

9?‘x>-l, O
15.

Discontinuous Integral

(O
b . ; arc sm

2a +2-bZ)f

0

arr lJ+v--x+1 J,,(at)J,(bt)dt= ( S

Special

(D S

m

sin bt dt t

0

b- (b2-a2)* a {

S

t’-Y+lJ,,(at)J,(bt)dt=O

(O
0

2r-v+lar(bLa2)v-r-l bYr(v-p)

Weber-Schafheitlin

@ rel="nofollow">a>@

1 =;sin[rorcsini]

(OlbO)

0

mJp(at)Jr-l(bt)dt=~

(WP>---1)

11.4.36

S

S

t

=;

1

cos [parcsin

ar CO8=; =p[b+(b2-a2)t]r

(O
0

=r[b+(b2-a2 rel="nofollow">+]~

cos bt dt

(o

11.4.41 m =

ar sin 3

- J,,(d)

(05 b<4

2i + ,+-&)t

11.4.35

S J,,(d)

(05 b
11.4.40

eib’Yo (at)dt=

11.4.34

’ (a”- b2)+

i =(b2_a2)1

(

W’(cc+v-X+1) rel="nofollow">%

@‘cc>---2)

P+~--x+l

pa’-A+lr(v+l)r

X& ( p+u--x+1, 2

WC----l)

11.4.38

;]

1 =zT;

(O
=o

@>a>01

(O
@P rel="nofollow">O>

11.4.43

S

= ‘+

{l-J,(bt)}dt=O

(O
0

=ln

; (b 2a)O)

488

INTEGRALS

OF

BESSEL

11.4.44 v+‘J,(at)dt s Hankel-Nicholson

Type

aw-r =2’r(p+l)

0 (t2+22)r+l

(a>o,

Integrals

I

FUNCTIONS

11.4.47

7? ODK(at)dt cos~ s 0 tV(t2+22)=4ZY+1

IL,(az)

wa4

-Yhz)l

Wa>O, Wz>O, %<3) 11.4.48

Wz>o+au
11.4.45

a s

(a rel="nofollow">O, Wz>O,

J,(at)dt

=z [Iv(az) - L,(az)] 2zy+l

0 tqt2+2$

(a>O,

11.4.49

hVz>O, L%>-i) OD J,(at)dt

11.4.46 0

K0(ad

Numerical 11.5. Use and Extension

of the Tables

Y,V)dt ,,l IoWt,jy=

KOW

For moderate

values of x, use 11.1.2 and 11.1.P For x sufficiently large, use the asymptotic expansions or the polynomial approximations 11.1.11-11.1.18. 11.1.10 as appropriate.

a.05

Example

1.

$

“r(v+l)

Compute

J,(t)dt

to 5D.

in Tables

9.1 and

s

Using 11.1.2 and interpolating 9.2, we have

(a>O, Wz>O, %>--4)

Methods This value is readily checked using ~~3.1 and h=--.05. Now /Jo(x) 121 for all x and IJn(x)> <2-t, nk 1 for all 2. In Table 11.1, we can always choose Ih( 5.05. Thus if all terms of O(h4) and higher are neglected, then a bound for the absolute error is 2*h4/48<.2.10b6 for all x if (hl 5.05. Similarly, the absolute error for quadratic interpolation does not exceed ha(2++2)/24<.2.10-‘. z Example 3. Interpolation of J,(t)dt using 0 Simpson’s rule. We have

S

so=+* Jo(t)dt=so=

a.oa 09+.31783 69+.04611 52 +. 00283 19+ .00009 72+ .OOOOO211 = 1.37415

Jo(t)dt+S’+*

&,(t)dt=2[.32019

s0

l+” JoWt=$

2.

Compute

to 5D by

J,(t)dt s 0

interpolation We have

of Table 11.1 using Taylor’s

Z+A S S 0

Jo(t)dt=

+;

’ Jo(t)dt +hJo(z) 0

tJP(z)-Job)l+~

-;

JP(e),

x
Now

formula.

JP)(x)=;

[J~(x)-~J&)+~Jo(~$I

Jl(x) and with lhl5.05,

[3J1(~~-Jd~)l+

Then with x=3.0 and h=.O5,

...

it follows that IRl<.s.

Thus if s=3.0

1o-‘o

and h=.05

a.os

a.os J,(t)&=1.387567+(.05)(--260052) 0 - (.00125) (.339059)

S

S

+(.000010)(.746143)

Jo(t)dt

2

CJ,(x)+4Jo(x+;)+Jo(z+h)]+R

-&=,

R=

3.0s

Example

L(~adK~(bxz)

r(2v+l)

(a>O, WzbO)

2

t2+z2

1 Jo(t)dt,l

0

s 0 tqt2+z2p+*=

ODYo(at)dt -=--

S

B!v>-1)

0

=1.37415

J,(t)dt=1.38756

72520+9

[-.26005

19549

+4(- .26841 13883)- .27653 495991 = 1.37414 86481

INTEGRALS

OF

which is correct to 10D. The above procedure gives high accuracy though it may be necessary to interpolate

twice in Jo(x) to compute

Jo ST+; ( >

and Jo(a;+h) . A similar technique based on the trapezoidal rule is less accurate, but at most only one interpolation of Jo(x) is required. aJ,,(t)& and 3 Y,(t)& s s0 to 5D using the representation in terms o\ Struve functions and the tables in chapters 9 and 12. For x=3, from Tables 9.1 and 12.1 Example

Jo= -. IL,=

260052

J1=

.339059

.376850

Yl=

.324674

.574306

H~=1.020110

Using 11.1.7, we have s0

The recurrence formula

+F

[(.574306)(.339059)

js=1.20909

= 1.38757 3 Y,(t)&=

’ lo(t)& s0

6.

and r=O(1)6.

l)!x-‘+!j,(x)

27, g,=1.06701

58, J,,.O4347

and

g,=.96867

9.8 and 12.1, one can

36, g,=.94114

For tables of 2-%(x),

’ K,(t)&. s0

13

of J,,(z)

rj,+l(x>=xj,(x)-(r-ll)j~-l(~)+~jfr--2(x)

gives

12, g,=.90474

64

see [11.16]. Integrals

of Ko(x)

l)Ki,-,(x)

Kio(x)==Ko(x),

and the functions Thus for x=2, K,=.11389

+ (r-

+xKL(x)

Ki,(x)=~mKo(t)dt

on this last line are tibulated

39, K,=.13986

59, Kil=.09712

06

and Ki,=-2Ki,+2&=.08549

06

Similarly,

f-1(2)=-Jl(x),jo(.)=Jo(z),j~(~)=~

Jo@)dt

and the terms on this last line are tabulated. for x=2, .57672 48,jo=.22389

= -xKi,(x)

Ki-,(x)=&(x),

For moderate values of x and r, use 11.2.4. If r=l, see Example 1. For moderate values of x, use the recurrence formula 11.2.5. If x is large and x>r, see the discussion below. Example 5. Compute jl, o(x)=j,(x) to 5D for x=2 and r=O(1)5 using 11.2.6. We have

49

For moderate values of x, use the recurrence formula 11.2.14 for all r. Example ‘7. Compute K&(x) to 5D for x=2 and r=O(1)5. We have rKi,+l(x)

11.1.2~11.1.31. Integrals

86, g,=.98827

and the forward recurrence formula

.19766

For moderate values of 5, use 11.1.19-11.1.23. For x sufficiently large, use the asymptotic expansions or the polynomial approximations

j-1=-

to use the auxil-

Compute g,(x) to 5D for x=10 We have for x=10,

Repeated

Repeated

17

Thus

11.1.8 and Tables

compute

73, js=.25448

This satisfies the recurrence relation

g2=1.02353 s0

10

it is convenient

g&2) = (r-

Similarly,

Using

66, j4=.62451

When x>>r, iary function

Jo=-.24593

-(1.020110)(-.260052)]

gives

j2=2(j,+j-,)=1.69809

Similarly,

Example

a J&)&=3(-.260052)

489

FUNCTIONS

Compute

4.

Yo=

BESSEL

08,j,=1.4257703

Thus

K&=.07696

36, K&=.07043

17, K&=.06525

22

If x/r is not large the formula can still be used provided that the starting values are sufficiently accurate to offset the growth of rounding error. For tables of Ki,(x), see [ll.ll].

INTEGRALS

490

OF

FUNCTIONS

Apart from round-off error, the value of r needed to achieve a stated accuracy for given x and m can be determined a priori. Let

jm(x)=x-‘“S=t~Ko(t)dt 0

Now

BESSEL

Then In

the latter following from 11.3.27 with v=l. 11.3.5, put a=l, b= -1, p=O and v=O. p=m. Then

--Cm- 1)~0(41/~

(G-1)

Using tabular values off0 and fi, one can compute in succession fa, f3, . . . provided that m/x is not large. Example 8. Computef&) to 5D for x=5 and m=0(1)6. We have, retaining two additional decimals K,= Thus

jo=l.

.00369 11

K,=.

00404 46

56738 74

f1=.

19595 54

ja=.O5791 27, j,=.O1458

93, ja=.00685

. . . (r-2k+1)2

~,I[~K~(x)+x(T--~)Ko(~)I/(T--~)~

1>%-2(d--;F2~1(d

M4=[@-

“-““=(~-1)~(~-3)~

Let

since for x fixed, fi(x) is positive increases.

and decreases as T

Example 9.

Computef,(x) We have

m=0(2)10.

K,= .03473 95

to 5D for x=3 and K,=.04015

64

If r=16, e1&.86* 1o-2

fiO<1.4’ 1o-6

Taking

g16=0, we compute the following values Also recorded of Q14, g1a, * * *, go by recurrence. are the required values off,,, to 5D.

36

Similarly starting with fi, we can compute f3 and f5. If m>x, employ the recurrence formula in backward form and write fm-~(x~=~~~~(x)+~~l(x)+~(m-ll)Ko(;c)ll(m-ll)a In the latter expression, replacef, Take r>m and assume g,=O. g,-4, etc. Then lim g+&x)=f,,,(x),

r-f-

by g,,,. Fix x. Compute g,+ -

m=r-2k

-

For tables off,,,(x), see [11.21]. References

Texts

[ll.l]

[11.2] [11.3] [11.4] Ill.51 [11.6]

H. Bateman and R. C. Archibald, A guide to tables of Bessel functions, Math. Tables Aids Comp. 1, 247-252 (1943). See also Supplements I, II, IV, same journal, 1,403-404 (1943) ; 2,59 (1946) ; 2, 190 (1946), respectively. A. Erdelyi et al., Higher transcendental functions, vol. 2, ch. 7 (McGraw-Hill Book Co., Inc., New York, N.Y., 1953). A. Erdelyi et al., Tables of integral transforms, ~01s. 1, 2 (McGraw-Hill Book Co., Inc., New York, N.Y., 1954). W. Griibner and N. Hofreiter, Integraltafel, II Teil (Springer-Verlag, Wien and Innsbruck, Austria, 1949-1950). L. V. King, On the convection of heat from small cylinders in a stream of fluid, Trans. Roy. Soo. London 214A, 373-432 (1914). Y. L. Luke, Some notes on integrals involving Bessel functions, J. Math. Phys. 29, 27-30 (1950).

[11.7] Y. L. Luke, An associated Bessel function, J. Math. Phys. 31, 131-138 (1952). [ll.S] F. Oberhettinger, On some expansions for Bessel integral functions, J. Research NBS 59, 197-201 (1957)

RP

2786.

[11.9] G. Petiau, La thQrie des fonctions de Bessel (Centre National de la Recherche Scientifique, Paris, France, 1955). [ll.lO] G. N. Watson, A treatise on the theory of Bessel functions, 2d ed. (Cambridge Univ. Press, Cambridge, England, 1958). Tables

[ll.ll]

W. G. Bickley and J. Nayler, A short table of the functions Kin(z), from n=l to n=16. Philos. Mag. 7,20,343-347 Ki.(z)

=

(1935).

m K&,-l(t)&, J(.1)2, 3, “9D.

Kir(z)=l-K&)&

n=1(1)16,

~=0(.05).2

INTEGRALS

OF BESSEL

[11.12] V. R. Bursian and V. Fock, Table of the functions

Akad.

Nauk,

Trudy

(Travaux)

Leningrad,

Inst.

2, 6-10

(1931).

Fiz.

Mat.,

jzmKo(t)dt,

7D; 7D; e* z =io(t)dt, 2=0(.1)16, I ‘lo(t)&, 2=0(.1)6, 7D; e-= ’ zo(t)dt, x= s0 s O’(.l) 16, 7D. [11.13] E. A. Chistova, Tablitsy funktsii Besselya ot nogo argumenta i integralov ot deistvitel’ nikh (Izdat. Akad. Nauk SSSR., MOSCOW, U.S.S.R., J,,(X),

Y,,(X), lrn +

dt, 5”

y

&

n=o, 1; s=o(.o01)15(.01)100, 7D. Also tabulated are auxiliary expressions to facilitate interpolation near the origin. [11.14] A. J. M. Hitchcock, Polynomial approximations to Bessel functions of order zero and one and to related functions, Math. Tables Aids Comp. approximations 11, 86-88 (1957). Polynomial 2 -Ko(t)dt. for Jo(t)dt and s sz [11.15] C. W. ‘Horton, A short table’ of Struve functions and of some integrals involving Bessel and Struve functions, J. Math. Phys. 29, 56-58 (1950). 4D;

C.(z)=~t”J,(t)dt,n=l(l)4,z=O(.l)lO,

D,(z) =l

z=O(.l)lO,

4D, where E,(z) is Struve’s function; see chapter 12. [11.16] J. C. Jaeger, Repeated integrals of Bessel functions and the theory of transients in filter circuits, J. Math. Phys. 27, 210-219 (1948). fi(z)= s”

0

Jo(t)dt,

z=O(1)24,8D.

f,(z)

=lz Also

f,-l(t)dk

2-‘f,(z),

r=1(1)7,

o.(z)=~mJo[2(zt)“]J.(t)dt,

a,,(x), a;(x), n=l(l)7, z=O(l;24, 4D. [11.17] L. N. Karmazina and E. A. Chistova, Tablitsy funktsii Besselya ot mnimogo argumenta i integralov ot nikh (Izdat. Akad. Nauk SSSR., MOSCOW, U.S.S.R., 1958). e-zZ&), e-*11(z), ezKo(z),

Denmark,

1953). 5D.

eZZG(x), e=, e-2 = Zo(t)dt, es s - Ko(t)dt, s0 2=0(.001)5(.005)15(.Ol)lOO, 7D exlept for ea which is 75. Also tabulated are auxiliary expressions to facilitate interpolation near the origin.

n=0(1)8,

= J,(t)dt,

s0

Also

s J,(t)e*dt,

a=t,

z=

a=%-t.

s [11.19] Y. L. Luke and D. Ufforod, Tables of the function i?o(z)

=

s

Math.

o=Ko(t)dt.

UMT

129.

A,(z),

A&).

Tables

Aids Comp.

r&r)=-[[y+ln (z/2)lAl(z)+A&), s=O(.O1).5(.05)1, SD.

[11.20] C. Mack and M. Castle, Tables of

s

oaZo(z)dz and

mK&c)dz, Roy. Sot. Unpublished Math. Table s File No. 6. a=O(.O2)2(.1)4, 9D. [11.21] G. M. Muller, Table of the function Kj.(z)

=x-n

zu*Ko(u)du, s Office of Technical Se&es, U.S. Department of Commerce, Washington, D.C. (1954). n=0(1)31, z=O(.O1)2(.02)5, 8s. [11.22] National Bureau of Standards, Tables of functions and zeros of functions, Applied Math. Series 37 (U.S. Government Printing Office, Washington, D.C., 1954).

(1) pp. 21-31: ff

z=O(.ol)lo,

10D.

2=0(.1)10(1)22,

n=0(1)4,

t”E,(t)dt,

[11.18] H. L. Knudsen, Bidrag til teorien for antennesystemer med he1 eller delvis rotations-symmetri. I (Kommission Has Teknisk Forlag, Copenhagen,

O(.Ol)lO,

2=0(.1)12,

1958).

491

FUNCTIONS

Jo(t)dt,

(2) pp. 33-39: 1oD;

F(z)

l sz

=sm

Yo(t)dt,

m Jo(t)dt/t, Jo(t)dt/t

+in (z/2), 2=0(.1)3, 10D; F(“(i)/n!, z= 10(1)22, n=0(1)13, 12D. [11.23] National Physical Laboratory, Integrals of Bessel functions, Roy. Sot. Unpublished Math. Table File No. 17. 1oD. [11.24] M. Rothman,

s,’

Jo(t)&

s,

Yo(t)dt,

2=0(.5)50,

Table of OZ~(x)dz for 0(.1)20(1)25, s Quart. J. Mech. Appl.‘Math. 2, 212-21.7 (1949). 8S-9s. [11.25] P. W. Schmidt, Tables of ‘Jo(t)dt for large z, s J. Math. Phys. 34, 169-172 yl955). z=lO(.2)40, 6D. [11.26] G. N. Watson, A treatise on the theory of Bessel functions, 2d ed. (Cambridge Univ. Press, Cambridge, England, 1958). Table VIII, p. 752: 1 = x=0(.02)1, 7D, with 5 Jo(t) dt, f j- =Y,(t)dt, s the” first 16 m:xima and minima of the integrals to 7D.

492

INTEGRALS

Table

11.1

OF

BESSEL

INTEGRALS s

0

FUNCTIONS

OF BESSEL

LYo (t)dt

FUNCTIONS

e-z

s ,li 10 (t) dt 0.00000 00

ez

s

- Ko (t) dt

0.00000

00000

0.09991 0.19933 0.29775 0.39469

66979 43325 75802 85653

-0.21743 -0.34570 -0.43928 -0.50952

05666 88380 31758 48283

0.09055 0.16429 0.22391 0.27172

92 28 79 46

1.;7079 1.35784 1.25032 1.17280 1.11171

63 82 54 09 28

0.48968 0.58224 0.67193 0.75834 0.84106

05066 12719 68094 44308 59149

-0.56179 -0.59927 -0.62409 -0.63786 -0.64184

54559 15570 96341 88991 01770

0.30964 0.33929 0.36206 0.37910 0.39137

29 99 71 05 42

1.06127 1.01836 0.98109 0.94821 0.91885

17 48 70 80 56

0.91973 0.99399 1.06355 1.12813 1.18750

04101 71082 76711 83885 20495

-0.63706 -0.62447 -0.60490 -0.57911 -0.54783

93766 91607 26964 12548 19295

0.39970 0.40479 0.40721 0.40745 0.40593

88

52 52 78 39

0.89237 0.86829 0.84626 0.82596 0.80719

52 97 10 89 04

1.24144 1.28982 1.33249 1.36939 1.40048

95144 09734 68829 85727 85208

-0.51175 -0.47156 -0.42788 -0.38136 -0.33260

90340 13039 62338 24134 04453

0.40298 0.39891 0.39394 0.38828 0.38211

85 09 29 68 11

0.78973 0.77344 0.75819 0.74386 0.73037

57 80 62 97 44

1.42577 1.44528 1.45912 1.46740 1.47029

02932 81525 63387 80303 39949

-0.28219 -0.23071 -0.17871 -0.12672 -0.07526

28501 32490 50399 97284 50420

0.37555 0.36873 0.36174 0.35467 0.34757

57 67 98 38 29

0.71762 0.70556 0.69412 0.68324 0.67288

95 50 02 16 26

1.46798 1.46069 1.44871 1.43231 1.41181

09446 96081 25408 16899 57386

-0.02480 +0.02420 0.07132 0.11617 0.15839

29261 24953 69288 78353 62206

0.34049 0.33349 0.32659 0.31981 0.31319

93 48 30 99 59

0.66300 0.65356 0.64452 0.63587 0.62757

15 16 98 68 60

1.38756 1.35992 1.32928 1.29602 1.26056

72520 96508 40386 59125 17835

0.19765 0.23367 0.26620 0.29502 0.31996

82565 66986 20748 36222 99576

0.30673 0.30045 0.29435 0.28843 0.28271

62 18 04 67 31

0.61960 0.61193 0.60455 0.59744 0.59059

34 74 84 84 11

1.22330 1.18467 1.14509 1.10496 1.06471

57382 59706 13136 78009 52877

0.34090 0.35775 0.37044 0.37896 0.38335

94657 03989 06831 74266 61369

0.27718 0.27183 0.26668 0.26170 0.25691

02 70 11 94 78

0.58397 0.57757 0.57139 0.56540 0.55961

14 57 13 66 09

1.02473 0.98541 0.94712 0.91021 0.87502

41595 21560 13375 52175 60866

0.38366 0.38000 0.37250 0.36131 0.34665

96479 67672 06552 69475 16398

0.25230 0.24785 0.24357 0.23945 0.23548

18 61 56 46 74

0.55399 0.54854 0.54326 0.53812 0.53314

42 72 15 91 27

E 4: a 4. 9

0.84186 0.81100 0.78271 0.75721 0.73468

25481 72858 50802 10902 94106

0.32872 0.30779 0.28413 0.25802 0.22977

87513 77892 10351 06786 58227

0.23166 0.22799 0.22445 0.22104 0.21775

83 15 13 21 83

0.52829 0.52358 0.51899 0.51452 0.51017

52 03 19 43 24

5. 0

0.71531

19178

0.19971

93876

0.21459

46

0.00000

3. 0 3. 1 ;*3 3: 4 E 3: 7 33:; 4. 0 4.1 4. 2 t:; 4. 5

00000

INTEGRALS

INTEGRALS

,I,Jo (t)dt

OF

BESSEL

OF BESSEL

493

FUNCTIONS

FUNCTIONS

(t)dt so=Yo

e-=

s

Table 11.1

’ l,,(t)dt

ez zp Ko(t)dt s

0.71531 0.69920 0.68647 0.67716 0.67131

19178 74098 10457 40870 39407

0.19971 0.16818 0.13551 0.10205 0.06814

93876 49405 34784 01932 12463

0.210459 0.21154 0.20860 0.20577 0.20303

46 58 68 28 89

0.50593 0.50179 0.49776 0.49382 0.48998

10 55 16 50 19

0.66891 0.66992 0.67427 0.68187 0.69257

44989 67724 98068 18713 19078

0.03413 +0.00035 -0.03284 -0.06517 -0.09630

05806 67983 98697 04775 01348

0.20040 0.19785 0.19539 0.19301 0.19072

08 40 44 81 13

0.48622 0.48256 0.47897 0.47547 0.47204

86 16 75 34 60

0.70622 0.72263 0.74160 0.76290 0.78628

12236 54100 64692 51256 33012

-0.12595 -0.15385 -0.17975 -0.20344 -0.22470

06129 27646 87372 39625 89068

0.18850 0.18635 0.18427 0.18225 0.18030

02 16 20 84 78

0.46869 0.46541 0.46219 0.45905 0.45596

29 11 83 20 99

0.81147 0.83820 0.86618 0.89512 0.92470

67291 76824 77897 09137 60635

-0.24338 -0.25931 -0.27239 -0.28252 -0.28966

05692 37161 18447 78684 45218

0.17841 0.17658 0.17480 0.17308 0.17140

74 44 64 09 55

0.45294 0.44998 0.44708 0.44424 0.44144

98 97 76 15 97

0.95464 0.98462 1.01435 1.04354 1.07190

03155 17153 21344 00558 32638

-0.29377 -0.29486 -0.29295 -0.28811 -0.28043

44843 02239 35658 49927 26862

0.16977 0.16819 0.16665 0.16516 0.16370

82 68 93 39 89

0.43871 0.43602 0.43338 0.43079 0.42824

05 22 34 23 76

1.09917 1.12508 1.14941 1.17192 1.19243

14142 84628 49299 99830 33198

-0.27002 -0.25702 -0.24159 -0.22392 -0.20421

13202 06208 37080 52368 93575

0.16229 0.16091 0.15956 0.15825 0.15698

24 30 91 93 21

0:42574 0.42329 0.42087 0.41850 0.41617

81 20 86 63 40

1.21074 1.22671 1.24021 1.25112 1.25939

68348 60587 13565 88778 12520

-0.18269 -0.15959 -0.13516 -0.10966 -0.08335

75150 61109 40494 01934 07540

0.15573 0.15452 0.15333 0.15217 0.15104

64 08 42 55 36

0.41388 0.41162 0.40940 0.40722 0.40507

07 52 65 37 56

1.26494 1.26777 1.26787 1.26528 1.26005

80240 58297 83120 57796 46162

-0.05650 -0.02940 -0.00230 +0.02451 0.05078

66385 07834 54965 01664 29664

0.14993 0.14885 0.14779 0.14676 0.14575

74 61 88 44 23

0.40296 0.40088 0.39883 0.39681 0.39482

15 04 15 40 69

1.25226 1.24202 1.22946 1.21473 1.19799

64460 70675 51666 08237 38314

0.07625 0.10069 0.12385 0.14552 0.16550

79635 08937 04194 02334 09969

0.14476 0.14379 0.14284 0.14191 0.14099

16 16 16 08 87

0.39286 0.39094 0.38904 0.38716 0.38532

97 15 17 95 41

;:;

1.17944 1.15927 1.13772 1.11499 1.09134

18392 83464 05614 71504 * 58985

0.18361 0.19969 0.21360 0.22523 0.23448

20962 32017 56169 34059 42919

0.14010 0.13922 0.13836 0.13752 0.13669

46 78 79 43 65

0.38350 0.38171 0.37994 0.37820 0.37648

53 20 39 03 06

10. 0

1.06701

13040

0.24129

03183

0.13588

40

0.37478

43

5. 5 267 55:: 2; 6:2 66:; 22 6:7 66:: 7. 0 3:: ::i ::2 7'8 7: 9 8. 0 8"-:. 8: 3 8.4 E xl 8:9 ;*i 9:2 9':: 9.5 9':;

494

INTEGRALS

Table X

OF

BESSEL

INTEGRALS

11.2

s t z 1 --Jo@)

FUNCTIONS

OF BESSEL

FUNCTIONS

& Jz

L

o.“ooooo

000

0.00124 0.00499 0.01121 0.01990

961 375 841 030

-1.3;?38 -0.43423 -0.05107 +0.15238

382 067 832 037

0.00000 0.00113 0.00409 0.00835 0.01347

000 140 877 768 363

0.000000 0.368126 0.460111 0.506394 0.532910

::"9

0.03100 0.04449 0.06032 0.07841 0.09872

699 711 057 882 519

0.26968 0.33839 0.37689 0.39543 0.40022

854 213 807 866 301

0.01910 0.02497 0.03088 0.03667 0.04222

285 622 584 383 295

0.548819 0.558366 0.563828 0.566545 0.567355

z! 1:2 1. 3 1. 4

0.12116 0.14565 0.17211 0.20043 0.23052

525 721 240 570 610

0.39527 0.38332 0.36633 0.34572 0.32256

290 909 694 398 701

0.04744 0.05229 0.05672 0.06070 0.06425

889 376 080 995 420

0.566811 0.565291 0.563058 0.560302 0.557163

0.26227 0.29557 0.33031 0.36636 0.40360

724 796 288 308 666

0.29769 0.27176 0.24529 0.21871 0.19235

696 713 896 360 409

0.06735 0.07002 0.07228 0.07414 0.07563

663 797 458 688 806

0.553745 0.550126 0.546364 0.542506 0.538587

0.44191 0.48117 0.52124 0.56200 0.60333

940 541 775 913 248

0.16650 0.14138 0.11719 0.09408 0.07218

135 594 681 798 365

0.07678 0.07760 0.07813 0.07839 0.07841

298 744 746 884 674

0.534635 0.530670 0.526711 0.522768 0.518854

0.64509 0.68716 0.72942 0.77174 0.81402

164 194 081 836 795

0.05158 0.03235 +0.01457 -0.00174 -0.01655

229 987 248 144 931

0.07821 0.07781 0.07724 0.07652 0.07566

544 809 664 168 245

0.514976 0.511139 0.507350 0.503610 0.499924

0.85614 0.89799 0.93947 0.98047 1.02091

669 596 188 571 428

-0.02987 -0.04168 -0.05201 -0.06088 -0.06833

272 613 554 740 756

0.07468 0.07361 0.07245 0.07121 0.06993

681 124 090 963 006

0.496292 0.492717 0.489198 0.,485736 0.482332

1.06070 1.09975 1.13799 1.17536 1.21179

032 277 707 536 667

-0.07441 -0.07915 -0.08263 -0.08491 -0.08605

025 722 683 323 553

0.06859 0.06722 0.06582 0.06440 0.06297

360 060 033 109 029

0.478984 0.475694 0.472459 0.469280 0.466155

1.24723 1.28163 1.31496 1.34718 1.37826

707 975 504 044 060

-0.08613 -0.08523 -0.08342 -0.08079 -0.07742

706 459 762 769 769

0.06153 0.06009 0.05867 0.05725 0.05584

450 952 042 166 708

0.463085 0.460067 0.457100 0.454185 0.451320

1::

1.40818 1.43694 1.46454 1.49096 1.51622

716 870 052 446 864

-0.07340 -0.06880 -0.06371 -0.05821 -0.05239

123 199 317 690 371

0.05446 0.05309 0.05174 O.(r5042 0.04913

000 325 921 989 691

0.448503 0.445734 0.443012 0.440335 0.437703

5. 0

1.54034

722

-0.04632

205

0.04787

161

0.435114

0.0. 2: 0”:;

0. 5 ::76

::2 i-i 1:9 ::1" 22'32 2: 4 5-z 2:7 ::i 3. 0 ;*; 3:3 3.4 33:: z 3:9 f-1" 4:2 44:43 4. 5 t:;

12.

Struve

Functions MILTON

and Related

Functions

ABRAMOWITZ~

Contents

Pege

Mathematical Properties .................... 12.1. Struve Function H,(z) ................. 12.2. Modified Struve Function L,(z) .............

12.3. Anger and Weber Functions

(2/r) l

t-‘E&)&t,

12.2. Struve

Table

Ho(x)-Y,(x), Io(~>-L&>,

s

499

............

499

..........................

12.1. Struve Functioqs

Table

496 498 498

..............

Numerical Methods ...................... 12.4. Use and Extension of the Tables References

496

(OjzSa~)

x=o(.1)5,5D

Functions

Hlw--l(4, Id+-h(x),

zm [II&)-Y,(t)]t-‘&,

500

.............

501

to 7D

for Large Arguments

. . . . . . . . .

sb; t&w-~cl(~)l~+w7d Jo2 [L,(t)--I,(t)ldt-(2/lr)

502

ln x ln x

x-l= .2(- .Ol)O, 6D

The author acknowledgesthe assistanceof Bertha H. Walter in the preparation and checkingof the tables.

1National Bureau of Standards. (Deceased.) 495

12. Struve

Functions Mathematical

12.1. Differential

Struve

Function

Equation

and

Solution

Functions

Properties Ii&

12.1.11

H,(s)

General

and Related

12.1.12

(2/7r)--HI

-$ (2,H,)=z”H,4

12.1.1

22 dazu+2 dw+(22-va)w= dz2 dz The general solution 12.1.2

where

4(w+’ J;;wJ+ 3)

$

(2-,H”)=&

2,;(v+))-2-,H.,1

is (a$, constants)

w=aJ,(2)+bY,(2)+Hv(2)

Z-“H”(2)

12.1.13

is an entire function Power

Series

of 2.

Expansion

12.1.3

FIGURE 12.1. Integral

H.(z), n=0(1)3

Representations

If 9v>

Struve junctions.

-4,

12.1.6

5 i S

o1(l-ty%in

12.1.7

%]a 2)” =&r(v+f)

12.1.8

=Y. (2)

6

sin

(zt) dt .6

cos 0) sinaVBdO

(2

--I’ (l+tn)‘-3d,

Recurrence

12.1.9

E,-1+H.+,=~

Relations

l3,+

($4” h w+#)

-.6

-.6

FIGURE 12.2. 12.1.10

H,-1--H,+I=BH;-

(a2)” 6 rb+io

Struve junctions.

H,(z), -T&=1(1)3

STRUVE

FUNCTIONS

AND RELATED

497

FUNCTIONS Struve*a

Integral

12.1.25 4 * t-.H,(t)dt=~al(z)+~sm r L -s 12.1.26 2 t-lE&)&=l-f *-s 1

t-1 l.&(t)&

I

2 [ -15.32.j

zb

OI = st~-v-‘H,(t)dt +1%3’.5’.5-

1

***

12.1.27

r(j~)P-“-’ tan (3~~0 w--~P+l>

0

(I9Pl9k’--9)

If j,(z)=

FIGURE

0SH,(t)t*dt

12.1.28 f”+l= (2v+ l)f,< Z)--Z’+W”(Z)

Struve functions.

12.3.

s

++f + (Y+i)2’+lr(~pyY+g)

E,(z), 2=3, 5 Special

12.1.14

Proper&x

Asymptotic

(z>O and u> 3)

H,(z) 2 0

H-(,+),(z)=

H&)=(-g

12.1.16

(n an integer>O)

(-l)nJn+i(z)

for

Large

1x1

12.1.29

H.(z)-Y.(z)=;

12.1.15

Expansions

(wy>-43)

zol

r(k+3)

r(v+3--k) (i)

*-d-R,

(larg 4<4 (1-cos

2)

where R,,,=O(IZI~-*~-~). If Y is real, z positive * and m+ )- v 20, the remainder after m terms is of the same sign and numerically less than the first term neglected.

12.1.17

12.1.30 12.1.18 H,(zemr*)=e”‘n(r+*)ri

H,(z) (m an integer)

12.1.19

x&(z)--Yl(z)-f H,(z)=

!!z’2)‘+1

IFS

dr rG+*>

3

3

1; z+~, &

z2

a0 L(t)&=; st-1

Integrala

12.1.22

(See

chapter 11)

. .] (larg 4<*)

12.1.31

12.1.20 12.1.21

&(z)-yo(z)~~[~-~+q.&$~+~ [1+$-y+y-

. . .]

(larg 4<4 12.1.32 s0

s[Ho(t)--YoWW -5 [In cw+r1

0

12.1.23

z4+A6- **-1 s8Bb(t)dt=~ [g-m

where *i=. 57721 56649 . . . is Euler’s

constant.

12.1.33

0

12.1.24

s0

* t-’ H.+l(w=2”&+p)

- z-‘H,(Z)

*setpage II.

(lw 4<4

498 Asymptotic

Expansions

STRWE

FUNCTIONS

for

Orders

Large

AND

12.1.34

2(32)” H”(Z)-Y”

(z)-&r(y+f)

FUNCTIONS Asymptotic

12.2.6

2 k!b, k=O

RELATED

bz=6(~/2)~-&,

Large

)a i

Lv(z)-I-p(z)

k- 32

(-l’k+lr(k~v+l

(jargzI<+a)

a kPor(Y++--k) (i)

b,=20(~/~)~-4(p/z’) Integrals

12.1.35

2(W -&++$)

=” (4 +iJ”(z)

-jy!!s 4 (Iyl>lzl) k-o 2+1

12.2.7

S

’ LoWt=f

0

12.2. Modified Power

Struve Function Series

L,(5)

12.2.8

Expansion

(VT

12.2.1

for

2+1

(larg 4
Expansion

$+12

.

[

“3: .

4+12

SozM~~---Ldt)ldt-~

.

6+---1

32ze 52 . .

[In @d+rl

L”(z)=--ie-TH”(iz) ($a*”

=(4z)“+’k&r(k+4j)r(k+vt-3) Integral

12.2.2

? S

2($3” Lv(z)=Gr(,+3)

12.2.4 12.2.5

S

’ L&)dt=L,(z)-f

wv>--3)

Relation

to Modified

12.2.10 L-(,++)

S

Recurrence

Relations

msin (tz)(l+t*)l-+ dt 0 my<3, c-0)

L”+

L”-l+L”+1=2L;-

Spherical

Bessel

(2)=1(,++)(z)

Function

(n an integer10)

12.3. Anger and Weber Functions

2(z’2)” J;;w+3>

L”wI-L”+I=;

z

0

o sinh (z cos 0) sin*v e&

12.2.3 I-,(s)-L,(z)=

12.2.9

Representations

(2/2)’ 6 w+i8

Anger’s

12.3.1 12.3.2

J.(z)=-I

S

* cos (vO- z sin 0) de n 0 (n an integer)

J.(4=Jn(z)

(2/2)’ & w+*)

L,(X)

Weher’s

12.3.3 Relations

Function

E.(z)=ai Between

n

Function

S

Tsin (d-

z sin 6) de

0

Anger’s

and

Weher’s

Function

12.3.4

sin (VT) JV(z)=cos (VT)EY(~)-E-Y(~)

12.3.5

sin (VT) E,(z)=J-,(z)-cos

Relations

Between

Weher’s Function Function

(VT) J,(z) and

Struve’s

If n is a positive integer or zero,

12.3.7 FIGURE

12.4. L(z),

Modijied Struve junctions. fn=C(1)5

E-,,(z)=+

“+l[~lr(n-k-t)(aZ)-.+~+l C k=O w+a

--H-,(z)

*

STRUVE

12.3.8

E,,(z)=--Ho(z)

12.3.9

El(z)=:-H,(z)

FUNCTIONS

AND

of the Tables

Example 1. Compute L,(2) to 6D. From Table12.1 I,(2)-L,(2)=.342152; from Table 9.11 we have 1,,(2)=2.279585 so that L,(2)=1.937433. Example 2. Compute H,(lO) to 6D. From Table 12.2 for s-‘=.l. H,,(lO)-~Y,(10)=.063072; from Table 9.1 we have Y,(10)=.055671. Thus, H,,(10)=.118743. Example 5D.

3.

Using

Compute

Tables

‘Ho(t)& for 2=6 to s 12.2, il.1 and 4.2, we have

YdWt

J)ut)dt=Jy

+f In 6+fi (6)

= 1.83148 Example 4. Compute H,(x) for x=4, -n= O(l)8 to 6s. From Table 12.1 we have H,,(4)= .1350146, HI(4)=1.0697267. Using 12.1.9 we find H-,(4) H-,(4) H-3(4) H-4(4)

= - .433107 = .240694 = .152624 = - .439789

H-,(4) H-,(4) H-,(4) H-,(4)

= .689652 = - 1.21906 = 2.82066 = - 8.24933

Example 5. Compute H,(s) for x=4, n= O(l)10 to 7s. Starting with the values of H,(4) and H,(4) and using 12.1.9 with forward recurrence, we get H,(4)= H,(4) = H,(4) = H,(4)= H,(4)= H,(4)=

.13501 1.06972 1.24867 .85800 .42637 .16719

46 67 51 95 41 87

Ha(4) H,(4) H,(4) HQ(4) H,,,(4)

=.05433 =.01510 =.00367 =.00080 = .00018

E&)=g-H&)

Methods We note that for n>6 there is a rapid loss of significant figures. On the other hand using 12.1.3 for x=4 we find H,(4)=.0007935729, H,,(4) = .00015447630 and backward recurrence with 12.1.9 gives H,(4) H,(4) H,(4) H,(4) H,(4)

= .00367 =.01510 = .05433 =.16719 = .42637

54 37 SS Od %5

1495 315 519 87 43

H,(4)= .85800 H,(4)=1.24867 H,(4) = 1.06972 K(4)= .13501

94 6 7 4

Example 6. Compute L,(.5) for n=0(1)5 to 8s. From 12.2.1 we find L,(.5) =9.6307462X lo-‘, L,(.5) =2.1212342X 10m5. Then, with 12.2.4 we get L,(.5)=3.82465 L,(.5)=5.36867

=-.125951+(.636620)(1.791759) + .816764

499

FUNCTIONS

12.3.10

Numerical 12.4. Use and Extension

RELATED

03X10-’ 34X1O-3

L1(.5)=.05394 2181 Lo(.5) =.32724 068

Example 7. Compute L,(.5) for -n=0(1)5 to 6s. From Tables 12.1 and 9.8 we find L,(.5) = .327240, L,(.5) = .053942. Then employing 12.2.4 with backward recurrence we get L-,(.5) = .690562 L-z(.5)=-l.16177 Lb3(.5) = 7.43824

L-,(.5) = -75.1418 L-,(.5) = 1056.92

Example 8. Compute L,(s) for x=6 and -n=0(1)6 to 8s. From Tables 12.2 and 9.8 we fhd L,,(6)=67.124454, L,(6)=60.725011. IJsing 12.2.4 we get L-,(6)=61.361631 L-*(6) =46.776680 L-a(6)=30.159494

L-4(6)=16.626028 Lm5(6)= 7.984089 L-,(6) = 3.32780

We note that there is no essential loss of accuracy until n= -6. However, if furt.her values were necessary the recurrence procedure becomes unstable. To avoid the instability use the methods described in Examples 5 and 6.

STRUVE

500

FUNCTIONS

AND

RELATED

FVNCTIONS

References Texts

[12.1] R. K. Cook, Some properties of Struve functions, J. Washington Acad. Sci. 47, 11, 365-363 (1957). [12.2] A. Erdelyi et al., Higher transcendental functions, vol. 2, ch. 7 (McGraw-Hill Book Co., Inc., New York, N.Y., 1953). [12.3] A. Gray and G. B. Mathews, A treatise on Bessel functions, ch. 14 (The Macmillan Co., New York, N.Y., 1931). [12.4] N. W. McLachlan, Bessel functions for engineers, 2d ed. ch. 4 (Clarendon Press, Oxford, England, 1955). [12.5] F. Oberhettinger, On some expansions for Bessel integral functions, J. Research NBS 59 (1957) RP2786. [12.6] G. Petiau, La theorie des fonctions de Bessel, ch. 10 (Centre National de la Recherche Scientifique, Paris, France, 1955). 112.71 G. N. Watson, A treatise on the theory of Bessel functions, ch. 10 (Cambridge Univ. Press, Cambridge, England, 1958).

Tables

[12.8] M. Abramowitz, Tables of integrals of Struve functions, J. Math. Phys. 29, 49-51 (1950). [12.9] C. W. Horton, On the extension of some Lommel integrals to Struve functions with an application to acoustic radiation, J. Math. Phys. 29, 31-37 (1950). [12.10] C. W. Horton, A short table of Struve functions and of some integrals involving Bessel and Struvefunctions, J. Math. Phys.29,56-58 (1950). [12.11] Mathematical Tables Project, Table of the Struve functions L.(z) and H,(z), J. Math. Phys. 25, 252-259 (1946).

STRTJVE

FUNCTIONS

AND

STRUVE

Ho(x)

X

HI(X)

s;Ho(W

RELATED

501

FUNCTIONS

Table

FUNCTIONS II(X) -L1 (x)

Io(+-b(4

“fo(4 0.00000

12.1

“j-” H$& * 2

0.00000

00

0.00000

00

0.06359 0.12675 0.18908 0.25014

13 90 29 97

0.00212 0.00846 0.01898 0.03359

07 57 43 25

0.003181 0.012704 0.028505 0.050479

1.000000 0.938769 0.882134 0.829724 0.781198

0.000000 0.047939 0.091990 0.132480 0.169710

E.

0.30955 0.36691 0.42184 0.47399 0.52303

59 14 24 44 50

0.05217 0.07457 0.10063 0.13012 0.16281

37 97 17 25 75

0.078480 0.112322 0.151781 0.196597 0.246476

0.736243 0.694573 0.655927 0.620063 0.586763

0.203952 0.235457 0.264454 0.291151 0.315740

0.42982 0.50134 0.56884 0.63262 0.69294

0.799223 0.760044 0.721389 0.683341 0.645976

1. 0 1.1 1. 2 1. 3 1. 4

0.56865 0.61057 0.64855 0.68235 0.71179

66 87 00 03 25

0.19845 0.23675 0.27742 0.32012 0.36452

73 97 18 31 80

0.301090

0.555823 0.527058 0.500300 0.475391 0.452188

0.338395 is ::z 0:396290 0.412679

0.75005 0.80418 0.85553 0.90430 0.95066

0.609371 0.573596 0.538719 0.504803 0.471907

1.5

0.73672 0.75702 0.77261 0.78345 0.78952

35 55 68 23 36

0.41028 0.45704 0.50444 0.55210 0.59966

85 72 07 21 45

0.631863 0.706590 0.783111 0.860954 0.939643

0.430561 0.410388 0.391558 is. :::z

0.427810 0.441783 0.454694 0.466629 0.477666

0.99479 1.03682 1.07691 1.11518 1.15174

0.440D86 0.409388 0;379057 0.351533 0.324450

0.79085 0.78752 0.77961 0.76726 0.75064

88 22 35 65 85

0.64676 0.69304 0.73814 0.78174 0.82351

37 18 96 98 98

1.018701 1.097659 1.176053 1.253434 1.329364

0.342152 0.327756 0.314270 0.301627 0.289765

0.487877 0.497329 0.506083 0.514194 0.521712

1.18672 1.22020 1.25230 1.28309 1.31265

0.298634 0.274109 0.250891 0.228992 0.208417

0.72995 0.70542 0.67729 0.64586 0.61142

77 23 77 46 64

0.86315 0.90036 0.93489 0.96649 0.99496

42 74 57 98 63

1.403427 1.475227 1.544392 1.610577 1.673465

0.278627 0.268162 0.258319 0.249056 0.240332

0.528685 0.535156 0.541164 0.546746 0.551933

1.34106 1.36840 1.39472 1.42008 1.44455

0.189168 0.171238 0.154618 0.139293 0.125242

3: 3 3.4

0.57430 0.53484 0.49339 0.45032 0.40600

61 44 57 57 80

1.02010 1.04177 1.05983 1.07418 1.08477

96 30 03 63 74

1.732773 1.788248 1.839675 1.886873 1.929699

0.232107 0.224348 0.217022 0.210099 0.203553

0.556757 0.561246 0.565426 0.569319 0.572948

1.46816 1.49098 1.51305 1.53440 1.55508

t ZEi o:oa1212 0.073071

3. 5 3.6 3. 7 3.8 3. 9

0.36082 0.31514 0.26935 0.22382 0.17893

08 40 59 98 12

1.09157 1.09457 1.09380 1.08934 1.08127

23 16 77 44 62

1.968046 2.001847 2.031071 2.055726 2.075858

0.197357 0.191488 0.185924 0.180646 0.175634

0.576333 0.579492 0.582442 0.585199 0.587776

1.57512 1.59456 1.61343 1.63176 1.64957

0.065992 0.059928 0.054829 0.050642 0.047311

4. 0 ::4'

0.13501 0.09242 0.05147 +0.01247 -0.02427

46 08 40 93 98

1.06972 1.05484 1.03681 1.01584 0.99214

67 79 86 22 51

2.091545 2.102905 2.110084 2.113265 2.112655

0.170872 0.166343 0.162032 0.157926 0.154012

0.590187 0.592445 0.594560 0.596542 0.598402

1.66689 1.68375 1.70017 1.71616 1.73176

0.044781 0.042994 0.041891 0.041414 0.041502

is 417 4. a 4. 9

-0.05854 -0.09007 -0.11867 -0.14415 -0.16637

33 71 42 67 66

0.96597 0.93759 0.90729 0.87535 0.84208

44 56 01 28 90

2.108492 2.101037 2.090574 2.077406 2.061852

0.150279 0.146714 0.143309 0.140053 0.136938

0.600147 0.601787 0.603328 0.604777 0.606142

1.74697 1.76182 1.77632 1.79049 1.80434

0.042096 0.043139 0.044571 0.046335 0.048376

5. 0

-0.18521

68

0.80781

19

2.044244

0.133955

0.607426

1.81788 (97 [ I

0.050640

K

0: 2 0":: 0. 5 2;

:*; 1: a 1. 9 2; :*: 2:4 22:: z 2:9 3:;

i::

[(-y]

[c-y]

0.000000

i* Ez: 0:489655 0.559399

(-;I6 I:

I

0.09690 0.18791 0.27347 0.35398

1.000000

0.959487 0.919063 0.878819 0.838843

0.112439

(412 1. I

Ho(x), HI(X), LO(Z), L1 (z), compiled from Mathematical Tables Project, Table of the Struve functions L,(x) and H,(x), J. Math. Phys. 25,252-259, 1946(with permission). s;Ho(l)dt,

s;[r,(t)-b(t)]&

is=- H+

dt, compiledfrom M. Abramowitz, Tablesof integralsof Struve

functions, J. Math. Phys. 29,49-51,195O (with permission).

502

STRUVE FUNCTIONS AND RELATED FUNCTIONS Table 12.2 X-’

STRUVE

Hoc@ - Y,(x)

H,(x)- Y,(x)

0.123301 0.117449 0.111556 0.105625 0.099655

0.659949 0.657819 0.655774 0.653818 0.651952

0.15 0.14 0.13 0. 12 0.11

FUNCTIONS

FOR LARGE

I,(x)-L,(x)

fdx)

0.607426 0.610467 0: 618598

0.793280 0.794902 0.796448 0.797910 0.799279

0.815341 0.814541 0.813785 0.813074 0.812411

0.098241 0.091318 0.084474 0.077706 0.071010

0.620955 0.623129 0.625119 0.626927 0.628558

0.800551 0.801721 0.802787 0.803750 0.804611

0.094843 0.088593

0.811796 0.811232 0.810722 0.810266 0.809866

0.064379 0.057805 0.051279 0.044793 0.038340

0.630018 0.631315 0.632457 0.633450 0.634302

0.805374 0.806047 0.806634 0.807140 0.807572

0.063460 0.057147

10 11

i* E%24 0: 038152

:z 17

0.809525 0.809244 0.809023 0.808865 0.808770

0.031912 0.025506 0.019116 0.012738 0.006367

0.635016 0.635596 0.636045 0.636365 0.636556

0.807933 0.808225 0.808450 0.808611 0.808706

0.031805 0.025451 0.019093 0.012731 0.006366

20 25 33

fdx)

W)

-Lo(x)

0.819924 0.818935 0.817981 0.817062 0.816182

0.133955 0.126683 0.119468 0.112319 0.105242

0.093647 0.087602 0.081521 0.075404 0.069254

0.650180 0.648504 0.646927 0.645452 0.644081

0.10 0. 09 0.08 0.07 0. 06

0.063072 0.056860 0.050620 0.044354 0.038064

0.642817

0.05 0. 04 0.03 0. 02 0. 01

0.031753 0.025425 0.019082 0.012727 0.006366

0.638200

0.00

0.000000 (-!)5

0.636620 92

0.20

0.19 0.18 0.17 0.16

k T%5:

0: 639696 0.638888

;- 66::%! 0: 636874 0.636683

ARGUMENTS

i- 66:z

fA4 i* :21zi

0: 113505

00. . :Kz

:

6 i

i* %EB 0: 069761

0.000000 0.636620 0.808738 0.000000 c-y c-g)1 ‘-;I1 92 1 1 c 1 [ 1 c 1 L- 1 [ 1 [(-;I2 1 0.808738

<x>

1% CG

s

:[H,(1)-Y,(t)lat=~Inz+fi(x)

1

Ho(t)- Ye(t) dtzfo(x) t <x > = nearestintegerto 2. Startingwith H,(z) andH,(x), recurrenceformula12.1.9may be usedto generateH,(z) for nx/2, forwardrecurrenceis unstable. To avold the instability, choosen> >x, computeH,(x) and HI+l(x) with 12.1.3, andthen usebackwardrecurrencewith 12.1.9. If n>O, L,(z) mustbegeneratedby backwardrecurrence.If n < 0, L,(X) may be generatedby backwardrecurrence forwardrecurrenceshouldbe used. as long asL,(x) increases.If n
Examples

4-8.

13. Confluent

Hypergeometric

Functions

LUCY JOAN SLATER l Contents Page

Mathematical Properties . . i . . . . . . . . . . . . , . . . 13.1. Definitions of Kummer and Whittaker Functions . . . . .

13.2. 13.3. 13.4. 13.5.

Integral Representations . . . . Connections With Bessel Functions Recurrence Relations and Differential Asymptotic Expansions and Limiting 13.6. Special Cases . . . . . . . . . 13.7. Zeros and Turning Values . . . .

Numerical Methods . . . . . . . . . . 13.8. Use and Extension of the Tables 13.9. Calculation of Zeros and Turning 13.10. Graphing M(a, b, 2) . . . . . . References

. . . . . . . . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

510 511 511 513 513

. . . . . . . . . . . . . . . . . . . .. . . . . . .

514

2=.1(.1)1(1)10,

Hypergeometric a=-l(.l)l,

Table 13.2. Zeros of M(a,

b=.l(.l)l,

. . . . . . . . . . . .

509

. . . . . . . . . . . . Points . . . . . . . . . . . . . . . . . . . . .

Table 13.1. Confluent

a=--l(l)-.I,

. . . . . . . . . . Properties Forms . . . . . . . . . . .

504 504 505 506 506 508

Function

. . . . .

516

b, 2) . . . . . . . . . . . . . . . . . . 7D

535

b=.l(.l)l,

M(a, b, s)

8s

The tables were calculated by the author on the electronic calculator EDSACI in the Mathematical Laboratory of Cambridge University, by kind permission of its director, Dr. M. V. Wilkes. The table of M(a, b, z) was recomputed by Alfred E. Beam for uniformity to eight significant figures.

1 University Mathematical National Bureau of Standards.)

Laboratory,

Cambridge.

(Prepared

under contract with the

503

13. Confluent

Hypergeometric

Mathematical 13.1. Definitions

of Kummer Functions

Kummer’s

13.1.1

Logarithmic

Kummer’s

U(v+L and an irregular

z=O

[

(4 &J . . . +(b),n!+

...

where

2) In z

+goc$$‘r, -WST)-W +n+r) I] I *!tGz+79 z-“M(a-n

, l-n

J 2) n

13.1.7

U(a, l--n,

l+n,

2)=2”U(a+n,

2)

As 9’z~a,

(a)),=a(a+l)(a+2)

. . . (a+n-l),

(ah=l,

13.1.8

and

U(a, b,

2)=~-~[l+O(lzJ-~)]

Analytic

Continuation

13.1.9

13.1.3 bJ z)~~

lr

b, U(a, b, .~e*~‘) ==. em-L I MP-a, sm ab I‘(i+a--b)r(b)

Mb, 4 2) ’ r(l+a-b)r(b)

-,,-eM(1+a--b,2--b, Parameters n positive integers)

b#--n

a+-m

bie-n

a=-m

b= -n b= -n m>n

a#-m a=-m,

M(a,

b, 4

a convergent series for all values of a, 1, and z a polynomial of degree m in 2

4

e*‘*(l-b’zl-bM(l -a, 2-b, r(a)I’(2-bb)

211

l?(a)r(2-b) (m,

M(a,n+l,

for n=O,‘l, 2, . . ., where the last function is the sum to n terms. It is to be interpreted as zero when n=O, and $(a)=I”(a)/I’(a).

13.1.2

‘la’

(-1)n+l Z)=n,r(amn)

+(n-l)! r(a)

Function

a2 @ Mb, h 2)=‘+7;+(b)$!+

Solution

13.1.6

g-au=0

It has a regular singularity at singularity at CD. Independent solutions are

Properties U(a, b, z) is a many-valued function, Its principal branch is given by - r< arg z 5 CT.

and Whittaker

Equation

2 %+(6-z)

Functions

2)

I

where either upper or lower signs are to be taken throughout. 13.1.10

U(a, b, ze2”i”)=[I-e-2~tb”]

rfi:ii\)

a simple pole at b= -n

Mb,

b, d

+e-2ribnU(a, Alternative

b, z)

Notations

b=-n a=-m, undefined m,
$‘,(a; b; z) or *(a; b; 2) for M(a, b, z) P2Fo(a, 1 +a-b; ;- l/z) or *(a; b; 2) for U(a, b, z)

13.1.4

13.1.11

Mu,

6, z)=$j

e’~~-*[l+U((z(-~)~

Complete

(wz>o)

y==AWa,

Ma,

b, 4+BU(a,

where A and B are arbitrary Eight

and 13.1.5 4 d =& 504

(-4-41+0(14-‘)1

(92<0)

Solution

4 4

constants, b ti -n.

Solutionr

b, z)

13.1.12

y,=M(a,

13.1.13

y~=zt-eM(l+a-b,

13.1.14

y8=egM(b-a,

2-b, b,

-2)

z)

iZONFL1TENT HYPERGEOMETRIC

13.1.1.5

y,=zl-bezM(l-cz,

13.1.16

ya=U(a,

13.1.17

y~=z’-“U(l+a--b,

13.1.18

y,=e’U(b--a,

13.1.19

yg=21-berU(1-a,

505

FUNCTIONS

13.1.34

2--b, -2)

b, z)

r(-2r) wK,fi(2)= r(+g-K)

2-b,

2)

M,,,(z)

General

+

Confluent

r(Ly:J-K)

M% -/d2)

Equation

b, -2) 2-b:

-2)

Wronskians

If W(m, n}=y,yi-yy,yk and t=sgn (92)=1 if Jz>O, =-1 if J&O 13.1.20 W{l,

2}=W{3,

4}=W{l,

4}=-V’{2,

Solutions: 3} = (1 --b)zwbea

13.1.21 W{l,

3}=W{2,4}=W{5,

13.1.22

6)=W{7,

13.1.36

Z-Ae-f(z)it4(a,

b, h(Z))

13.1.37

Z-Ae-‘(z)U(a,

b, h(Z))

8)=0

13.2. Integral

W{ 1, 5}=-F(6)zmbe’/I’(a)

Representations

.%b>Wa>O 13.2.1

13.1.23

W{ 1, 7} =,(b)ee**bz-bezlr(b-a)

13.1.24

W{2, 5}= -r(2--b)2-be~/r(l+a-5)

13.1.25

W{ 2, 7) = -r(2-b)2-be~/r(l-a)

r(b;;y(a)

M(a, b, 2) le~t~.-l(l-~)b-.-l&

e:

13.1.26

W{ 5, 7} =e’“f(b-a’z-be* Kummer

s 0

M(a, b, z)=e’M(b-a,

13.1.27

13.2.2

Tranaformations

b,

=21-bet’

-2)

13.1.28

13.2.3

2’-bM(l+a-b,

13.1.29

l t1 e-t~~(l+t)b-‘-l(l--t)~-ldt

s -1

2-4

2)=2’-bezit4(1--a,

U(a, b, 2)=21-aU(l+a-b,

2--b,

-2)

=21-befz s0

2-4

*e-t’ conesin”-9

cotb-% (if?) u!0

2)

13.2.4

13.1.30 e’U(b-a,

b, -2)=e(*1(1-b’e*21-bU(1--a, Whittaker’s

2---b,

=e -A.? ’ ez,(,-A),-l(B--t)b-“-ldt sA

-2)

(A=B-1)

L?Za>O, W2>0

Equation

13.2.5 !g+

13.1.31

[-;+;+Gp]

WC0 r (a)U(a, b, 2) =

Solutions:

13.2.6 Whittaker’s

13.1.32 13.1.33

Functions

MI,r(2)=e-tE2ffrM(4+C(-K,

1+2p,

=e*s

2)

me-art”-‘(l+l)b-.-ldt

s0

m

e-~‘(~-~)“-l~b-“-l&

1

13.2.7 1+2p, 2)

W~.r(2)=e-*z2t+rU(3+~-K,

(--n<arg

257r,

~=*b-a,

p=#b-3)

e-t2 Cosh@sinhbwle cothb-” (@)&

*

,

506 13.2.8

CONFLUENT

r(a)

HYPERGEOMETRIC

FUNCTIONS

13.3.7

U(a, b, z)

Ma, b, 4 =e+z(+bz-uz)~-*b OD I-(b) =eAr SA e-“(l-A)a-l(t+B)b-a-l~~ @=1---B) . neo A,(3z)tn(b-2a)-1nJb-1+,(J(2zb-4za))

Similar integrals for MI,,,(z) and W,,,(z) can be deduced with the help of 13.1.32 and 13.1.33. Barnes-type

Contour

where Ao=l, A1=o, A,=$b,

Integrale

(n+l)A,,+l=(n+b-l)A,-,+(2a-b)A,-P,

13.2.9

r$M(a, b,z)=&

c+fm I-(-s)J?(a+d (-z)“js _ r(b+d Sc 40

for (arg (-z)l<+7r, a, b#O, -1, -2, . . . . The contour must separate the poles of r(--s) from those of r(a+s); c is finite.

(a real) 13.3.8 Ma,

b, 4 r(b) (D C n-0

=eh’

13.2.10

CO=l,

S

a#O,

-2,

-1,

Connections

With

. . ., b-afl,

Bessel

Functions

as Limiting

13.3.2

0-w

a+=

Ma,

Cn+l(a, b>=2aC&+l, Recurrence

=d-fb~,-~(2,@

b,-+)/r(b)}

=d-tbJb-I(2d;i)

(b-u)M(u-1,

b(b-l)M(u, b, -z/u)) = -7rie~*bz+~bH~~:1,(2fi)

Relations Properties

and

b)

Differential

Expaneiona

in

b-l,

z)+b(l--b-z)M(u, +z(b-u)M(u,

b, z) b+l, z)=O

(l+u-b)M(a,

b, z)-aM(u+l, b, z) +(b-l)M(u,

b-l,

z)=O

13.4.4

Series

13.3.6

M(u, b, z)=eW (b-u-$)(+)a-b++ . 5 (2b-2a-l),(b-%),(b--a-i+ * n=O n!(b). (-1)’ I’,-a--f+“(+) (b#O,-l,-2,.

b, z)+(2u-b++)M(u, b, z) -aM(u+l, b, z)=O

13.4.3

(Yz>O)

13.3.5

*see page II.

b+l)/b-C.-l(a,

13.4.2

U(a, b, z/u)} =2d-*b&,-~(2~

13.3.4

lim( I’(l+a-b)U(a, a-+-

b)In(z)

13.4.1

13.3.3

lim {r(i+a-b) a+0

b, 4=n~oG(a,

(h real)

Co= 1, Cl(u, b) =2u/b,

Cases

13.4.

lim( M(u, b, z/u)/I’(b)} lim{M(a,

13.3.9

l)aC,-l

where

Functions

If b and z are fixed, 13.3.1

C,=-$(2h-l)a+$b(b+l)h*,

+[(l-2h)a--h(h-l)(b+n-l)]C,-1 --h(h-

2,

(see chapters 9 and 10) Be-1

4=-b&

(n+1)Cn+I=[(l-2h)n-bh]C,,

3 The contour must separate the poles of r)(--6) from those of r(a+s> and r(1 +u-b+s). 13.3.

b-l+n@d--a&

where

r(a)r(l+a-b)zV(o, b, Z) C+11 r(--s)r(a+a)r(l+a--b+s)Pds =3z c-,o for larg zI<$

CnZ”(-az)i(l--b--n)J

bM(u, b, z)-bbM(u-1,

b, z)-.&(a,

b+l,

z)=O

13.4.5

n)

b(a+z)M(u, . .)

b, z)+z(u-b)M(u,

b+l, z) -abM(u+l, b, z)=O

CONFLUENT

HYPERGEOMETRIC

507

FUNCTIONS

13.4.19

13.4.6 (a-l$z)M(a,

b, z>+@--aMa-1,

(u+z)U(u,

4 2)

t-(1--b)M(a,

b-l,

b, z)--zU(u,

b+l,

2)

+u(b-u-1)U(u+1,

z)=O

b, z)=O

13.4.20

13.4.7

(u+z-l)U(u,

b(l--b+&t&,

b, z)+b(b-l)M(a-1, -azA4(u+l,

13.4.8

k!‘(u, b, z)=+(u+l,

b+l,

b-l,

z)

b-tl,

z)=O

-g

13.4.10

{it&,

ukqu+1,

b, 2)) =&

n

13.4.21

b+l,

z)

KJ(% b, 4 1=(-l>“(u),U(a+~,

b+% 4

13.4.23

b, z)=uM(u,b,

z)+zM’(u,

b, 2)

u(l+u--b)U(u+l,

b, z)=uU(u,

b, 2) fzU’(u,

b, 2)

13.4.24 b, z)=(b--a---z)M(u,

b, 2)

(l+u--b)U(u,

$zM’(u,

z)=bM(u,

z)=(l-b)U(u,

b, 2) --ZU’(u,

13.4.25 bS1,

b-1,

b, 2)

13.4.12 (b-u)M(u,

U’(u, b, z)=-uU(u+l,

z)=O

b+n, 2)

13.4.11 (b-u)A4(u-1,

b-l,

13.4.22

z)

M(u+n,

6, 2)

+(l+u--b)U(u,

$ 13.4.9

b, z)-U(u-1,

b, 2)-4ikP(a,

U(u, b+l,

z)=U(u,

6, z)-U’(u,

b, 2)

b, z)

b, 2)

13.4.26 13.4.13 (b-

l)M(u,

U(u-1, b- 1, 2) = (b-

l)M(u,

b, 2)

b, z)=(u-b+z)U(u,

b, Z)-zU’(u,

13.4.27

+zM’b,

4 4

U(u-1,

b-1,

z)=(l--bfz)U(u,

b-1,

z)=(b-1-z)M(u,

b, 2)

13.4.28

+zM’(a,

13.4.29

b, z)+(a-2u-z)U(u,

(1+2~+fK)M,+*,1r(Z)-(1+2E1-2K)Mr-1,p(Z)

b, 2)

=2(2K-Z)M,,,(Z)

+u(l+u-b)U(ufl,

b, z)=O

13.4.16 (b-a-l)U(fz,

13.4.30 W~+t,p(z)-Z*W.,r+f(%)+(K+I1)Wr--f,p(%)=O

b-1,

2)+(1--~-2)U(u,

b, 2)

+dqu,

63-1,

z)=O

13.4.31 (2K--2)~~,ll(Z)+W,+1,r(Z)

13.4.17 U(u, b, z)-uU(uf1,

b, z)-U(u,

b-l,

=(c(-K+a)(~+K-~)Wr-l,lc(Z)

z)=O

13.4.32

13.4.18 (b-u)U(u,

2~M.-t,,-t(z)-z~M,,,(z)=2~C1M~+t,r-t(~)

4 4

13.4.15

U(u-1,

b, 2) --ZU’(u,

13.4.14 (b-l)M(u-1,

b, z>+U(u-1,

b, 2) -zU(u,

b, 2)

bfl,

z)=O

13.4.33

ZW:,,(z)=(aZ-K)w,,,(z)--~r+l,~(Z)

b, 2)

508

CONFLUENT

13.5. Asymptotic

Expansions Forms

For

Iz\ large,

HYPERGEOMETRIC

and Limiting

r(1-b) Ub, 4 2)=r(l+u-b)

13.5.10

+o((2(l-

9”)

(O
(a, b fixed)

13.5.1 wb

FUNCTIONS

13.5.11

=&+Wlz

In 4)

13.5.12

=,fi~;~~)+“(iz,~

@=O)

h 2) = r(b) Ezr&g;

(ayy).

(WblO,

(-z)-n+o(,Z,-R)} For

large

a (b,

5

bf0)

fixed)

13.5.13

the upper sign being taken if-$r
z<#T,

the

U(u, b, 2) =2-a{Rs

(“)*(l;a-b)n

+WfR) Converging

Factors

I+It

(-z)-”

n-0

1k-i%<arg z<W

for

the

P +fXlW--al -“ rel="nofollow">I

O(,2,-R)=(u)~(1+a-b)~

b- a 1‘and

+ w4-2>1

2-.q

13.5.15

13.5.16

where the R’th and S’th terms are the smallest in the expansions 13.5.1 and 13.5.2.

U(\cG,b, z)=r(3b--a+t)?r-,etz~t-tb cos (d(2bz--4uz)

z (a, b fIxed)

As 1++0, M(u, b, O)=l,

as a+--ol

bf--n

z‘-“+o(,2,9e-2) (9?b22,

=qg

z’-“+o(,ln

a, b, x

so that z>2b-u>l,

13.5.17

sin (a*) exp [(b-%)(4 sinh 2&6+cosh* e)] [(b-2u) cash fp+[?r(3b-u) sinh 201-t

&“+o(l) (l<Wb<2)

=-&

real

A4(u, b, Z) =r(b)

2,)

(b=2) 2g

large

If cosh2 8=z/(2b-4u) bf2)

-$b?r+ur+ts) P+Wl3b-4-W

for b bounded, CCreal. For

ret-l) 13.5.6 U(u, b, z)=r(a)

2 real.

where u is defined in 13.5.13. [#-b+2u+z-S+O(lzl-‘)I

small

for b bounded,

U(u, b, z)= r(~b-u+#e+‘z+-+b[cos (u+Jb-l(,Q2bz-4uz)) -sin (ar)Yb--1(d(2bz-4uz))] [l+O((+b-ul-g)]

13.5.4

For

O_
13.5.14

as a+--cx,

and

o(14-s)= %--aMl-4s 8,

$-#p),

U+Wb--al-*)1

L3+(b+tb-h+t2-ifR) 2

13.5.9

(l--p,

(-z)-R

R!

13.5.8

a=min

M(u, b, z)=r(b)ee(3bz--az)t-tblr-1 COB(d(2bz--4ux)-$bn++r)

Remainders

13.5.3

13.5.7

(~(2bz-dzz))

where

13.5.2

13.5.5

M(u, b, z) = r (b)etz(3bz-uz)t-tbJ~-,

b z+#(4l+O(lzh 21) (b=l)

13.5.18

U(U, b, z)=exp

[1+0(13b--4-91 [(b--t&)($ sinh 28--e+cosh* e)] [(b-2u) cash e]l-b[($b-u) sinh 201-t ~1+Wb-4-31

CONFLUENT

If Z= (2b--4~)[l+t/(b--2a)~],

HYPERGEOMETRIC

FUNCTIONS

509

13.5.21

so that

M(a, b, x) = r(b) exp { (b--u)

xm2b-4a 13.5.19

[(b-2a)

M(a, b, x)=etz(b-2u)f-bI’(b)[Ai(t)

+Bi(t)

cos (ar)

[sin (mr)+sin 13.5.22 U(a, b, z)=exp

V(a, b, ~)=et~+“-~“r(;)a-t~-*

[(b-2a)

{ (+b-a)(2&sin

ae) +t~}

(2B-sin

= I

13.6. Special

Wa, b, 4

cos* e][(b-2u)

cos e]l-”

sin 28]-t{sin

[(*b-a)

so that 2b--4a>x>O,

a

sin 281-4

+o(13b-a(-1)]

{ h-tr(~)(bx-2ux)-f3f?r-~+O(l~b--a~-*)~

= I

cos e]‘-“[r(+b-a)

sin (ar)+O((+b--a(-*)]

13.5.20

If ~0s*e=x/(2b--4u)

cos* e}

[($b-a)

2e)+)T]+O(J)b-aJ-*))

Casee =

Relation

Function

b .-

r(l +v)ei~&)-~J.(z)

v+t

2v+1

13.6.2

-v-k,

-2v+1

IV -v)eis(+z)

13.6.3

v+t

2v+1

IYl+v)e~OzPZ.(z)

13.6.4

n+l

2n+2

r(~+n)6i*(~t)--~J.+t(z>

I

Spherical Bessel

13.6.5

-n

-2n

r(t-n)e’I(fz)“+tJ-.-t(z)

, I

Spherical

Bessel

13.6.6

n+l

2n+2

r(q+n)eg(tz)-“-tZ.+,(t)

I

Spherical

Bessel

r(l+n)e-*rr(fiz*)-n(ber.

13.6.1

Bessel

+ms (vr)J,(z)

--sin (M) Y,(z)]

Bes8el Modified

*

n-l-3

2n-k1

13.6.8

L+l-is

2L+2

eizF&j, z)z-~-~/CL(S)

Coulomb

13.6.9

-n

a+1

&

Laguerm

13.6.10

a

a+1

-@da,

13.6.11

-n

1+v--n

(nl)V (1 +v-n).

13.6.12

a

a

13.6.13

1

2

y

13.6.14

1

2

r sinh t

13.6.15

-iv

3

2+ exp ( +za) Ez”) (2) exp2jfr’)

13.6.7

z+i

bei. z)

Bessel

Kelvin

I) L’:‘(z)

Wave

Incomplete

2)

Gamma

Poieaon-Charlier

Pn(v’ ‘)

Exponential ,-is

sin 2

13.6.16

3-b

B

13.6.17

-n

4

&-j

13.6.18

-n

r)

&

13.6.19

+

t

13.6.20

tm+t

1+n

- erf 2 22 &.-zn+n-I

*see

page

II.

Trigonometric Hyperbolic.

,;1,

Weber

(#I

Pa%bolic Hermite

(-H-*fh&)

C-W”

$ Hea,+lb)

*

Hermite

rt

r(bm+t)

Cylinder

Error Integral

er’T(m, n, r)

*

Toronto

510

CONFLUENT

HYPERGEOMETRIC

FUNCTIONS

13.6. Special Cases-Continued

-

Uh

b, 4

Relation

b

a

2

--

-13.6.21

2v+1

v+t

13.6.22

2v+1

u+3

Function

20

r-fe*(22)-‘K,(z)

-2it

&,1,iMYS3)

Hankel

Modified

-~l(2z)-.~p(z)*

Bessel

EIankel

13.6.23

p+i

2v+1

2iz

3*te-i[n(“St)-Zl(2z)-.HI*‘(zjL

13.6.24

n+l

2n+2

2t

r-ter(2z)-“-tk,++(z)

Spherical

13.6.25

s

0

#z3’l

&Z-I exp ($~3/~)2-2/~3~/@ Ai (z)

Airy

13.6.26

n+i

Zn+l

he

i~~-tev5*(2&)-“[ker,

Kelvin

13.6.27

-n

a+1

2

(- l)%!LIp’

Incomplete

Gamma

Exponential

Integral

z+i

kei,, z]

(2)

Bessel

Laguerre

13.62%

l-a

l-a

2

eT(a, x)

13.6.29

1

1

-X

-e-z

13.6.30

1

1

2

e=E~(x)

Exponential

Integral

13.6.31

1

1

-1nx

-2li

Logarithmic

Integral

13.6.32

*m-n

l+m

2

r(l+n--m)e=-r4(1m-n)w.,

13.6.33

-b

0

2x

r(l+fv)e”k.(z)

13.6.34

1

1

ix

eiz[-+*i+i

13.6.35

1

1

4X

e+l[fri-i

13.6.36

-b

t

3z1

2+es214D.(z)

13.6.37

f-b

B

w

2t+e*‘l’D.

13.6.38

t-h

pr

29

2-*H&c)/x

3

21

13.6.39

f

13.7.

Zeros

(2)

13.7.2

and

Si (2)-Ci Si (2) -Ci

Bateman (z)] (z)]

Sine and Cosine Integral Sine and Cosine Integral Weber

(2)/z

’

Pa$bolic

Cylinder

Hermite

*

Error Integral

Turning

Values

For the derivative, 13.7.4

M’(a, b, X,) =

l/(2&4a.>+O(l/(~b-a)*))

xo-

A*(?-+ fb- 3)” 2b-4a

A closer approximation 13.7.3 XY,=Xo--M(u, *see page II.

for z>o

Cunningham

-

If jb--l,r is the r’th positive zero ofJbel(x), then a first approximation X0 to the r’th positive zero of M(a, b, 2) is 13.7.1 X,=j;-,.,{

,(x)

& exp (9) erfcz

-

-

Ei (2)

is given by b, Xo)/A4’(a, b, Xo)

If XA is the first approximation to a turning value f of M(a, b, r), that is, to a zero of M’(a, b, 2) then a better approximation is

CONFLUENT

The self-adjoint written

equation

&be-z

13.7.6

13.1.1

HYPERGEOMETRIC

can also be

$$z&‘-‘e-‘w

511

FUNCTIONS

is an increasing or decreasing function of z, that is, they form an increasing sequence for M(a, ZJ,z) if a>O, xb--3, and a decreasing sequence if a>0 and x>b--3 or if a<0 and x
The

Sonine-Polya

The turning

Theorem

The maxima and minima of IwI form an increasing or decreasing sequence according as

values of /WI lie near the curves

13.7.7 w= f ~(b)?r-1~2e2~2(~bx-ax)~~~b~

1-x/(2b-4a)}-1’4

-,&b-le-2z

Numerical 13.8. Use and Extension Calculation

In this way 13.4.1-13.4.7 can be used together with 13.1.27 to extend Table 13.1 to the range

of the Tables

of M(a,

Kummer’s

Methods

b, 2)

-lO
Transformation

Example 1. Compute M(.3, .2, - .l) to 7s. Using 13.1.27 and Tables 4.4 and 13.1 we have a=.3, b=.2 so that

M(.3,

.2, -.l)=e-.‘M(-.l, .2, .l rel="nofollow"> = .85784 90. Thus 13.1.27 can be used to extend Table 13.1 to transformation negative values of 2. Kummer’s should also be used when a and b are large and nearly equal, for x large or small. Example 2. Compute M(17, 16, 1) to 7s. Here a=17, b=16, and M(17,

16, l)=e’M(-1, =2.71828 =2.88817 Recurrence

Example 3. Compute Using 13.4.1 and Table b=.2 so that

16, -1) 18X1.06250

00

M(-

1.3, 1.2, .l) to 7s. 13.1 we have a= -.3,

This extension of ten units in any direction is possible with the loss of about 1s. All the recurrence relations are stable except i) if albl, x>O, or ii) blbl, x
M(-1,

b, z)=l-zfor

.l)]

tx.

b,x)=l

for all x, when a=b. .2, .l)-.3M(.7,.2,

= .35821 23.

ati x. ButM(b,

b,x)=e”

M(b,

b, x)=8.

Heocek$l

In the first case b+- 1 along the line a= - 1, and in the second case b-+-l along the line a=b. Derivatives

By 13.4.5 when a= - 1.3 and b= .2, M(-1.3,1.2,.1)=[.26 M(-.3, .2, .l) -.24 M(-1.3, = .89241 08. Similarly

-lOsx
Example 4. At the point (- 1, - 1, x), M(a, b, x)

44.

Relations

M(-1.3,.2,.1)=2[.7M(-.3,

-lOlb
Example 5.

.2, .1)]/.15

7s.

To evaluate M’(-.7, -.6, .5) to By 13.4.8, when a= -.7 and b= -.6, we have M’(-.7,

-.6,

.5)=+4(.3,

.4, .5)

when a= - .3 and b=.2 M(--3,

1.2, .1)=.97459

= 1.724128.

52. Asymptotic

Check, by 13.4.6, M(-1.3, 1.2, .1)=[.2

&Q-.3, .2, .l) +1.2 M(-.3, = .89241 08.

1.2, .1)]/1.5

Formulas

For x2 10, a and b small, M(a, b, 2) should be evaluated by 13.5.1 using converging factors 13.5.3 and 13.5.4 to improve the accuracy if necessary.

CONFLUENT

512 Example 6. using 13.5.1. wg,

Calculate

M(.9,

.l,

HYPERQEOMETRIC

10) to 7S,

FUNCTIONB

Hence U(.l,

61, 10)

=- W) Q-*8)

r(.l)

&.101().8

cN Wn(1.8)n

,.erq()-.o

.2, 1)=5.344799(.371765--194486) = .94752.

Similarly

n-0 n!(-lo)n

U(5

+r(.9)

n!lO”

n-0

Hence by 13.4.15

=-.198(.869)+1237253(.99190

285) -tom

U(1.1, *2, l)=[U(.l, .2, l)-U(-.s, = .38664.

=1227235.23-.17+0(l) = 1227235+ O(1) Check, from Table 13.1, M(.9, .I, 10)=1227235. To evaluate M(a, 5, 2) with a large, z small and b small or large 13.5.13-14 should be used. Example 7. Compute M(-52.5, .l, 1) to 3S, using 13.5.14. M(-52.5,

.l, 1)=I’(.l)e*6(.05+52.5)*a6’*06

By direct application of a recurrence relation, M(-52.5, .l, 1) has been calculated as -16.447. To evaluate M(a, 5, 2) with z, a and/or b large, 13.5.17,19 or 21 should be tried. Example 8. Compute M(-52.5, .l, 1) using 13.5.21 to 3S, cos 0=&210.2. M-(-52.5, J, 1) =r(,l)e106.1 oode L105.1

co0

Example 10. To compute 5s. By 13.4.21 U/(-.9,

11.

=.Ol{

For -lO
But M(.9,

1.8, 1)=.8[M(.9,

M(.9,1.8,1) - r(.l)r(l.8)

.8, 1)-iV(-.l,

= 1.72329, using 13.4.4.

2go

l-.019+.000551-.000021 +o(lo-e)

53.

]*

.8, l)]

-2, .Ol).

1, For

r(.8) =,o+o((.01)~8) =1.09

to 3S, by 13.5.10.

To evaluate U(a, 5, 2) with a large, z small and b small or large 13.5.15 or 16 should be used. To evaluate U(a, 5, z) with 2, a and/or b large 13.5.18, 20 or 22 should be tried. In all these cases the size of the remainder term is the guide to the number of significant figures obtainable. Calculation

.2; l)= ~GiifSj~

U(1, .l, 100) to 5s.

*2, .Ol)= r(l,lr(1--.2) q+“((.ol)l-‘a)

A full range of asymptotic formulas to cover all possible cases is not yet known.

M(.1,.2,1) r(.9)r(.2)

Formular

l-go+;

=.00981

U(.l,

b, x)

1) to

-~jjjjj~+OwOb 1.9 2.9 3.9

+0((52.55)-‘)I=-16.47+0(.02)

of U(a,

.9, -.8,

1)=.9U(.l, .2, 1) =(.9)(.94752) = .85276.

To compute

U(1, .l, loo)=&{

.2, 1)]/.09

U’(-

Example 12. To evaluate U(.l, 2 small, 13.5.6-12 should be used.

e]+s1.5641

52.55-i sin 2tI-*[sin (-52.5~) +sin { 52.55(2e--sin 2e) +ir>

Calculation

-.8,

Asymptotic

Example By 13.5.2

.5642 COB[(.2-4(-52.5))*“-.05~+.25s] [1+O((.05+52.5)-*6)]=-16.34+0(.2)

U(.l,

.9, *2, 1) = .91272.

(-*%w”+*(l(yN)

of the

Whittaker

Functionr

Example 13. Compute M,-.,(l) and W.,,,-.,(l) to 5s. By formulas 13.1.32 and 13.1.33 and Tables 13.1, 4.4 M.o,-.,(l)=e-~6M(.1, W.o,-.,(1)=e-~6U(.l,

.2, 1)=1.10622, .2, 1)=.5’7469.

CONFLUENT

HYPERGEOMETRIC

Thus the values of M,,,(x) and W,,,(Z) can always be found if the values of M(a, 5, 2) and U(a, b, 2) are known. 13.9. Calculation Example

zero of M( -4, Table

13.2.

of Zeros

and Turning

Points

Compute the smallest positive .6,~). This is outside the range of Using 13.7.2 we have, as a first

14.

513

FUNCTIONS

xi=xA =X;

M’(-3, [l--3M(-3,

.6,X;) .6,X;)]

[l-&Q-2,1.6,

X;)/.6M(-3,

=.9715X1.0163=.9873

.6,X;)]

to 4s.

This process can be repeated to give as many significant figures as are required.

approximation x0- (-55a)a -17.2=-

174

*

Using 13.7.3 we have X1=X0--M(-4, But,

-6, X,)/M’(-4,

-6, X0).

by 13.4.8, iW(-4,

.6, X0)=-(.15)-‘M(-3,

1.6, XJ

Hence X,=X,+.15M(-4,

.6,X,)/M(-3,1.6,X,,),

=.174+ (.15) (.030004) = .17850 as a second approximation. If we repeat this calculation, x,=x,+.00002 Calculation

99=.17852 of Maxima

and

99 to 7s. Minima

Example 15. Compute the value of 2 at which M(-1.8, - .2, Z) has a turning value. Using 13.4.8 and Table 13.2, we find that M’(-1.8, --.2,x) = 9M(--.8, .8, x) =0 when %=.94291 59. Also M”(-1.8, -.2, x)=9M’(-.8, .8, x)= -9M(.2, 1.8, Z) and M(.2, 1.8, .94291 59)>0. Hence M(-1.8, - -2, a~)has a maximum in z when x= .94291 59. Example 16. Compute the smallest positive value of z for which M( -3, .6, Z) has a turning value, Xi. This is outside the range of Table 13.2. Using 13.4.8 we have

M’(-3,

.6, z)=-3M(-2,

By 13.7.2 for M(-2,

b

we find that

1.6, x)/.6.

1.6, ;c),

This is a first approximation to XA for M( -3, .6, x). Using 13.7.5 and 13.4.8 we find a second approximation

FIQURB

13.1.

Figure 13.1 shows the curves on which M(o, 5, Z) =0 in the cc, b plane when x=1. The function is positive in the unshaded areas, and negative in the shaded areas. The number in each square gives the number of real positive zeros of M(a;, b, Z) as a function of 2 in that square. The vertical boundaries to the left are to be included in each square. 13.10. Graphing

M(u, b, x)

Example 17. Sketch M(-4.5, 1, 2). Firstly, from Figure 13.1 we see that the function has five, real positive zeros. From 13.5.1, we find that it&+-=, M’+--co as x++m and that M++-, M’++a as x+-w. By 13.7.2 we have &s first approximations to the zeros, .3,1.5,3.7, 6.9, 10.6, and by 13.7.2 and 13.4.8 we find as first approximations to the turning values .9, 2.8, 5.8, 9.9. From 13.7.7, we see that these must lie near the curves

y=fe+“(5r))-~(l-r/ll)%r-+. From these facts we can form a rough graph of the behavior of the function, Figure 13.2.

CONFLUENT

60

HYPERGEOMETRIC

FUNCTIONS

-

6 5

i4

FIGURE 13.2. (From

M(-4.5,

1, z).

ipergeometrioh~ confluenti, F. G. Tricomt, Funzionl, cremonese, Rome, Italy, 1954, with permission.)

Edlzionf.

M 4

IO

b=.5

-1.5

6

0

I

FIGURE 13.4. .5

I

1.5

2

2.5

M

(From

.5, 2). Publlcatlons,

References

IO

b=l

Texts

6

[13.1] H.

6

[13.2] [13.3]

[13.4]

FIGURE 13.3. (From

M(a,

E. Jahke and F. Emde, Tables of functions, Dover Inc., New York, N.Y., 1945, with permiaslon.)

M(a,

[13.5]

1, z).

E. Jahnke and F. Emde, Tables of functions, Dover Publicatlon~, Inc., New York, N.Y., 1945, with permission.) .

Buchholz, Die konfluente hypergeometrische Funktion (Springer-Verlag, Berlin, Germany, 1953). On Whittaker functions, with a large bibliography. A. Erdelyi et al., Higher transcendental functions, vol. 1, ch. 6 (McGraw-Hill Book Co., Inc., New York, N.Y., 1953). On Kummer functions. H. Jeffreys and B. S. Jeffreys, Methods of mathematical physics, ch. 23 (Cambridge Univ. Press, On Kummer Cambridge, England, 1950). functions. J. C. P. Miller, Note on the general solutions of the confluent hypergeometric equation, Math. Table6 Aids Comp. 9,97-99 (1957). L. J. Slater, On the evaluation of the confluent hypergeometric function, Proc. Cambridge Philos. Sot. 49, 612-622 (1953).

CONFLUENT

HXPERGEOMETRIC

[13.6] L. J. Slater, The evaluation of the basic confluent hypergeometric function, Proc. Cambridge Philos. Sot. 50, 404-413 (1954). 113.71 L. J. Slater, The real zeros of the confluent hypergeometric function, Proc. Cambridge Philos. Sot. 52, 626-635 (1956). [13.8] C. A. Swanson and A. ErdBlyi, Asymptotic forms of confluent hypergeometric functions, Memoir 25, Amer. Math. Sot. (1957). [13.9] F. G. Tricomi, Funeioni ipergeometriche confluenti (Edizioni Cremonese, Rome, Italy, 1954). On Kummer functions. [13.10] E. T. Whittaker and G. N. Watson, A course of modern analysis, ch. 16, 4th ed. (Cambridge Univ. Press, Cambridge, England, 1952). On Whittaker functions. Tables

[13.11] J. R. Airey, The confluent hypergeometric function, British Association Reports, Oxford, 276-294 (1926), and Leeds, 220-244 (1927). M(a, b, z), a=-4(.5)4, b=), 1, 8, 2, 3, 4, ~=.1(.1)2(.2)3 (.5)8, 5D.

FUNCTIONS

515

[13.12] J. R. Airey and H. A. Webb, The practical importance of the confluent hypergeometric function, Phil. Msg. 36, 129-141 (1918). M(a, 6, CT), a=-3(.5)4, b=1(1)7, ~=1(1)6(2)10, 45. [13.13] E. Jahnke and F. Emde, Tables of functions, ch. 10, 4th ed. (Dover Publications, Inc., New York, N.Y., 1945). Graphs of M(a, b, 5) based on the tables of [13.11]. [13.14] P. Nath, Confluent hypergeometric functions, Sankhya J. Indian Statist. Sot. 11, 153-166 (1951). M(a, b, z), a=1(1)40, b=3, 2=.02(.02) .1(.1)1(1)10(10)50, 100, 200, 6D. [13.15] S. Rushton and E. D. Lang, Tables of the confluent hypergeometric function, Sankhya J, Indian Statist. Sot. 13, 369-411 (1954). M(a, b, z), a=.5(.5)40, b=.5(.5)3.5, ~=.02(.02).1(.1)1(1)10(10)50, 100, 200, 7s. [13.16] L. J. Slater, Confluent hypergeometric functions (Cambridge Univ. Press, Cambridge, England, M(a, 6, z), a=-l(.l)l, b=.l(.l)l, 1960). s=.l(.l)lO, 8s; M(a, b, l), a=-11(.2)2, b= -4(.2)1, 8s; and smallest positive values of I z for which Mfa, b, z)=O, a= -4(.1)-.l, 6=.1(.1)2.5, 8s. I

516

CONFLUENT

Table

CONFLUENT

13.1

HYPERGEOMETRIC

HYPERGEOMETRIC

FUNCTIONS FUNCTION

M(a,

b, x)

z =O.l

a/b

0.2

0.1

-1.0 -0.9 -0.8 -0.7 -0.6

0.00000

00

-0.5 -0.4 -0.3 -0.2 -0.1

i -1 / 4.88360 -1 5.88807 -1 6.90191 (-1 7.92514 (-1 8.95782

25 94 26 70 77

0.0

( 0) 1.00000

00

0 80 0 0 0

1.64549 1.75647 1.86845 1.98142 2.09538

5.00000

( 0) 1.00000

0.3 00

00

(-1)

6.66666

( 0) 1.00000

0.4

0.5

67

00

( 0) 1.00000

00

07 99 49 05 12

0.6

0.7

8.33333 8.49524 8.65820 8.82221 8.98727

33 54 31 06 18

(-1 (-1 (-1 (-1 (-1

0.0

( 0) 1.00000

00

( 0) 1.00000

0.1

( 0) 1.01725

53

0.5

( 0) 1.08736

79

0.8 -1) -1 -1 II-1 -1

00

00 35 39 38 56

( 0) 1.00000

00

I 0j 0 0 0 ( 0

17 56 21 13 35

1.01289 1.02585 1.03889 1.05200 1.06518

( 0) 1.00000

8.00000 8.19391 8.38915 8.58575 8.78369

00 07 99 33 61

II -1 -1 -1 -1 -1

8.98299 9.18365 9.38567 9.58907 9.79384

40 22 64 21 48

( 0) 1.00000

00

II 0

77 34 83 22 66

1.21388 1.19185 1.16996 1.14822 1.12662

1.0

0.9

8.75000 8.87183 8.99436 9.11759 9.24152

II(-1 -1 -1 -1 -1

00

( 0) 1.00000

00

For Olzl 1, linear interpolation in a, b or z provides 3-4s. Lagrange four-point interpolation gives 7s in a! b or 5 over most of the table, but the Lagrange six-point formula is needed over the range 1 <s< 10. Any interpolation formula can be reapplied to give two dimensional interpolates in a and b, a and 2: or b and z. This calculation can be checked by being repeated in a different order.

CONFLUENT CONFLUENT

HYPERGEOMETRIC

HYPERCEOMETRIC

517

FUNCTIONS

FUNCTION

M(a,

b, ix)

Table

13.1

x=0.2

a\b

0.2

0.1

-1.0 -0.9 -0.8 -0.7 -0.6

-l)-6.30239 -l)-4.39817 -l)-2.45653

72 97 39

-0.5 -0.4 -0.3 -0.2 -0.1

-2)-4.77093 -1)+1.54050 -1) 3.59664 -1 5.69168 -1 7.82601

96 87 50 81 37

0.0

( 0) 1.00000 00

-2) -1) (-1 (-1

0.00000 9.22415 1.86164 2.81785 3.79118

0.3 00 48 63 03 64

( 0) 1.00000 00

0.4 (-1)

( 0) 1.00000 00

0.5

5.00000 00

( 0) 1.00000 00

( 0) 1.00000 00

0":: 0:5 E 0.6 0.7 E l:o

a\b

0.6

0.7

0.8

1.0

0.9

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0:4 E 0.5 0.6 0.7 8.: l:o

II -1 -1 -1 -1 -1 ( 0) 1.00000 00

( 0) 1.00000 00

( 0) 1.00000 00

( 0) 1.00000 00

8.97413 9.17511 9.37817 9.58333 9.79060

99 81 91 69 58

( 0) 1.00000 00

518

CONFLUENT Table

13.1

CONFLUENT

HYPERGEOMETRIC

HYPERGEOMETRIC

FUNCTIONS FUNCTION

M(a,

b, 2)

x=0.3 0.1

a\b -1.0 -0.9 -0.8 -0.7 -0.6

0.2

II-1O)-2.00000 0 -1.73884 -1.19153 00 -1.46940 -9.05127 94 09 81 36

(-2)

( 0) 1.00000 00

II

0) 0 1.17274 1.71742 49 1.53139 1.34985 1.90800 78 94 88 56

a\b

0":: i-: 0:5

0.5

0.00000 8.90939 00 59

II -1) -1 -1 -1 -1

0.6

5.00000 5.45594 5.92137 6.39639 6.88112

0.7

( 0) 1.00000 00

( 0) 1.00000 00

( 0) 1.00000 00

II 0 1.11393 1.34985 1.23054 1.59674 77 1.47191 26 88 56

0.8

0.9

1.0

00 63 29 42 54

-0.5 -0.4 -0.3 -0.2 -0.1 0.0

0.4

I -1 I 4.54351 5.66711 25 5.09916 4.00000 03 00 51 (-1) 6.24750 17

( 0) 1.00000 00

-1.0 -0.9 -0.8 -0.7 -0.6

0.3

( 0) 1.00000 00

( 0) 1.00000 00

( 0) 1.00000 00

II

( 0) 1.00000 00 0 1.03645 1.07349 27 08 0 1.14937 1.11113 16 40 0 1.18822 61

( 0) 1.00000 00

CONFLUENT CONFLUENT

HYPERGEOMETRIC

HYPERGEOMETRIC

519

FUNCTIONS FUNCTION

M(a,

b, x)

Table

13.1

x=0.4

a\b

0.2

0.1

O)-1.00000 00 -1 I -8.32139 43

-1.0

-0.9 -0.8 -0.7 -0.6

0.3

-1

( 0)-1.59134

63

I 0)-1.20063

19

-6.57495

-1 -4.75937 -1 -2.87331

96

91 90

-1 -3.33333 I -1 j -2.19718 -1 -1.01932 -2 +2.01024 (-1 1.46463

0.4 33 27 12 24 65

-2 1-l I -1 -1

0.00000 8.63057 1.75514 2.67677 3.62847

00 33 40 48 08

-0.5 -0.4 -0.3 -0.2 -0.1

-2)+6.90415 20 (-1) 5.25850 66

0.1 :::

( 0) 1.49182 47 0 2.52986 2.00166 27 43

II 0 1.15892 34 0 1.32283 1.49182 47 59

II( 0 1.11772 81 0 1.36358 1.23890 28 21

0"::

I 0I 3.07676 3.64273 82 38

0 1.66597 1.84538 84 67

0 1.49182 1.62369 47 00

-1)

7.66207 59

0 0 0 0 0

i-76 0:8 !:;1

a\b -1.0 -0.9 -0.8 -0.7 -0.6

3.33333 3.92050 4.52459 5.14587 5.78462

00 03 02 39 63

1.0

0.7

0.6 -1 -1 -1 -1 -1

2.03014 2.22033 2.41605 2.61739 2.82445

33 85 74 62 40

-1) -1 I -1 -1 -1

6.00000 6136214 6.73238 7.11085 7.49764

00 28 89 21 78

-0.5 -0.4 -0.3 -0.2 -0.1 0.0

( 0) 1.00000 00

0.1 8:;

j 0I 1.07691 20 0 1.23671 1.15580 59 28

( 0) 1.00000 00

( 0) 1.00000 00

( 0) 1.00000 00

( 0) 1.00000 00

0 1.31966 37 0) 1.40469 04 0.6 EG 0:9 1.0

II 0

1.41655 50

0 1.56883 1.49182 47 03 0 1.64759 75 0 1.72815 18

0) 1.28326 80 01 1.33388 28 0 0 0

520

CONFLUENT

Table

13.1

CONFLUENT

HIYPERGEOMETRIC

FUNCTIONB

HYPERGEOMETRIC

FUNCTION

M(a, b, x)

r=0.5 a\b

-1.0 -0.9 -0.8 -0.7 -0.6

0.2

0.1

( 0 0 II 0 0 0

-4.00000 -3.61201 -3.20079 -2.76573 -2.30622

0.3

00 86 89 85 47

( 0) 1.00000 00

0.5 -2 II -1 -1 -1

-0.5 -0.4 -0.3 -0.2 -0.1 0.0

0.4

( 0) 1.00000 00

( 0) 1.00000 00

II

( 0) 1.00000 00

0.00000 8.38114 1.71019 2.61697 3.55920

00 43 66 96 78

( 0) 1.00000 00

0 1.31281 1.15358 36 87

i-z a:5

0 1.64872 1.47782 42 13 0 1.82563 24

1.0

$9”

-0:s -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1

0.1 ::3 Fl::

-1

4.44444 44

-1

5.00000 00

II -1 -1 -1

4.92975 21 5.42992 27 5.94522 72 6.47594 62

II -1 -1 -1

5.89284 21 5.44007 39 6.35854 17 6.83739 50

CONFLUENT CONFLUENT

HYPERGEOMETRIC

HYPERCEOMETRIC

521

FUNCTIONS

FUNCTION

i)Z(a, b,x)

Table 13.1

z=O.6

a\b

-0.6

0.1

0.2

0.3

II( 0 -1.77497 -1.00575 51 -1.27832 -1.53457 -2.00000 00 65 83 96

( O)-1.00000 00

( 0) 1.00000 00

( 0) 1.000b0 00

0.4

(

0.5

( 0)-3.05183 34

( 0) 1.00000 00

( 0) 1.00000 00

( 0) 1.40083 55

0.1

II 01 7.75149 1.14187 48 1.01404 6.63788 8.91853 04 08 45 76

a\b -1.0 -0.9 -0.8 -0.7 -0.6

85 l:o 8-98

0.6

n.7

0.8

0.9

1.0

0.00000 00

II

0 1.69207 1.82211 88 45 0 1.95661 34 0 2.09565 2.23934 57 48

II 0 1.54938 1.46364 36 57 0 1.63767 83 0 22 0 1.72857 1.82211 88

522

CONFLUENT Table

CONFLUENT

13.1

HYPERGEOMETRIC

HYPERGEOMETRIC

FUNCTIONS FUNCTION

M(a,

b, x)

x=0.7 a/b

0.2

0.1

0.3

0.4

0.5

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1

Ii 0)-3.16446 0 -2.44543 0 -1.67144 -1 -8.40541 -2 +4.92624

06 68 46 00 47

0.0

( 0) 1.00000 00

0.1 0.2

II( 0 2.01375 27 0 3.09264 92

ki 0:5

0 4.23886 5.45463 64 06 0 6.74221 79

II-1 -1 -1 -1 -1 ( 0) 1.00000 00

0.6 0.7 E

II 0 4.41274 94 0 5.09565 95 0 6.56853 5.81389 43 76

l:o

0 7.36066 31

a/b -1.0 -0.9 -0.8 -0.7 -0.6

0.6

0.7

0.8

( 0) 1.00000 00

0.9

00 61 44 66 23

( 0) 1.00000 00

1.0

0.00000 00 t-1) (-1)

1.22710 86 2.30054 51

(-1)

3.43855 96

-0.5 -0.4 -0.3 -0.2 -0.1

-1) 3.43109 52 -1 4.62042 36 -1 5.87022 82 -1 7.18224 16 -1 8.55823 13

0.0

( 0) 1.00000 00

( 0) 1.00000 00

0.1

I 0) 1.15093 86

( 0) 1.12744 17

0.6

( 0) 1.00000 00

2.05299 3.48181 4.98858 6.57561 8.24528

Ii II 0 0 0 0 0

2.01375 2.20933 2.41306 2.62516 2.84585

27 17 50 74 75

0 0 0 0 0

1.85078 2.01375 2.18318 2.35926 2.54213

59 27 94 09 50

( 0) 1.00000 00

( 0) 1.00000 00

( 0) 1.00000 00

CONFLUENT CONFLUENT

HYPERGEOMETRIC

HYPERGEOMETRIC

523

FUNCTIONS

FUNCTION

M(a, b, .x)

Table

13.1

z =0.8

d’\b

-1.0 -0.9 -0.8 -0.7 -0.6

0.1 (II 0 0 0 0 0

-7.00000 -6.50401 -5.94785 -5.32888 -4.64439

0.2 II( 0 0 0 0 -1

00 48 78 96 77

0.4

0.3 -1.66666 -1.48461 -1.28563 -1.06906 -8.34197

67 68 99 32 05

-0.5 -0.4 -0.3 -0.2 -0.1 0.0

( 0)+1.00000 00

( 0) 1.00000 00

0.1 8:;

( 0) 1.00000 00

II

( 0) 1.00000 00

( 0) 1.00000 00

( 0) 1.00000 00

( 0) 1.00000 00

0 1.38374 2.22554 79 1.79197 2.68533 25 39 09

0"::

( 0) 3.17225 39

8:; it: l:o

0.6

n\b -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

( 0) 1.00000 00

0.1 00:; 8:; 2; E l:o

II

( 0) 2.22554 09 0 2.46668 2.98352 2.71923 3.25992 24 90 56 11

( 0) 1.00000 00

( 0) 1.00000 00

524

CONFLUENT Table

a\b

CONFLUENT

13.1

HYPERGEOMETRIC

HYPERGEOMETRIC

FUNCTIONS FUNCTION

M(a,

0.2

0.1

t-11-9.35972

0.6 0.7 0.8

I 1 I 1.16728 1 1.39370 1 1.63551

93 17 72

ki:;

( 1) 1 189334 2:16782

87 94

a\b -1.0 -0.9 -0.8 -0.7 -0.6

b, 2)

0.6

-1 -8.20518 -1 -5.12058 (-1 -1.76920 -1 I +1.86021 t -1) 5.77931

02 10 97 91 14

0 0j 0 0 i 01

1.45345 1.93955 2.45960 3.01492 3.60688

52 77 31 28 44

I 01 4.23689

27

0.7

0.8

II-l)-5.00000 -1 -3.93506 -1 -2.78312 -1 -1.54071 -2 -2.04284

00 44 29 44 74

II -1 -1 -2 -2 -1

-2.85714 -1.92058 -9.13906 +1.65565 1.32057

29 43 92 38 89

( 0) 1.00000

00

( 0) 1.00000

00

I

27

0.9

1.0

0.00000 7.59274 1.56725 2.42566 3.33625

00 35 54 24 68

( 0) 1.00000

00

-2 I -1 -1 -1

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 00.: 0:4 0.5 0:8 i-b7 1"::

( 0) 1.00000 0

I

00

1.15190

18

I 0 1.31197 1.48048 0 1.65771 ( 0) 1.84394

31 24 19 34

( 0) 1.00000 0

I

00

1.11790

61

j 0 1.37111 1.24155 0 1.50677 0) 1.64871

02 10 14 85

CONFLUENT CONFLUENT

-0.6

( O)-6.38931

44

HYPERGEOMETRIC

HYPERGEOMETRIC

( O)-2.62968

FUNCTIONS

FUNCTION

M(a,

525 b, x)

Table

13.1

42

II-1 -3 +7.71680 -2.72739 -5.29840 6.42974 30 3.12589 94 92 36 46 ( 0) 1.00000 00

0.6

( 0) 1.00000 00

( 0) 1.00000 00

0.7

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

II-1

II

6.28763 92 8.08383 4.60681 3.03694 1.57371 81 41 99 08

-1

( 0) 1.00000 00

if: 0:3

8’76 II 8::

0 2.71828 3.06961 97 18

0:8 0.9 1.0

0 3.44142 89 0 3.83447 12 0 4.24952 89

0 0I 0 0 0

2.21650 2.46087 2.71828 2.98919 3.27406

01 06 18 01 39

II 0 0 0 0 0

2.05491 2.26515 2.48615 2.71828 2.96190

39 76 84 18 29

5.27314 4.25195 8.71734 45 7.50355 6.35625 01 07 70 83

526

CONFLUENT Table

13.1

CONFLUENT

HYPERGEOMETRIC

FUNCTIONS

HYPERGEOMETRIC

FUNCTION

M(a,

b, x)

x=2.0

a\b

0.1

0.2

0.3

0.4

0.5

-1.0 -0.8 -0.7 -0.6 -0.9

I

-0.5 -0.4 -0.3 -0.2 -0.1 0.0

( 0)+1.00000

00

( 0)+1.00000

00

( 0)+1.00000

00

( 0)+1.00000

00

( 0) 1.00000 00

II

0 3.07855 7.38905 71 5.77622 4.34381 1.96790 63 17 61 05

a\b

0.6

0.7

0.8

0.9

1.0 0 -1 -1 -1 -1

-1.00000 -9.19616 -8.18288 -6.94107 -5.45057

II ( 0) 1.00000 00

II

0 1.52511 4.35023 2.78211 2.11745 3.52448 72 19 88 92 69

( 0) 1.00000 00

00 98 30 82 11

CONFLUENT CONFLUENT

HYPERGEOMETRIC

HYPERGEOMETRIC

FUNCTIONS

FUNCTION

M(a,

527 b, x)

Table

13.1

x=3.0

a\b

0.2

0.1

0.3

0.4

0.5

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

1 -1.78256 05 1 -1.65079 47 1 -1.41549 22 1 -1.05876 41 0)-5.60854 66 ( 0)+1.00000

00

( 0)+1.00000

00

( 0)+1.0000

00

( 0)+1.00000

00

( 0)+1.00000

00

-1)+2.46564

64

2: 82 0:5 0.6

2 2.05059 14

K

2 2.65765 3.36670 56 66

.i:'o

22) 4.18932 5.13805 19 80

a\b -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

0.6 O)-4.00000

0.7

0.8

00

Ii 0 0 0 0 -1

-3.95879 09 -3.47899 58 -2.77784 38 -1.82229 72 -5.76188 60

( 0)+1.00000

00

( 0)+1.00000

00

1) 3.63241 26

( 0)+1.00000

00

( 0) 1.00000 00

( 0) 1.00000 00

528

CONFLUENT Table

13.1

CONFLUENT

HYPERGEOMETRIC

FUNCTIONS

HYPERGEOMETRIC

FUNCTION

M(a,

b, 2)

2=4.0 a\b

0.1

0.2

0.3

0.4

-1.0 -0.9 -0.8 -0.7 -0.6

0.5

II O)-9.00000 00 1 -1.10723 65 1 -1.28958 24 1 -1.43486 25 1 -1.52885 30

-0.5 -0.4 -0.3 -0.2 -0.1 0.0

( 0)+1.00000

00

( 0)+1.00000

00

( 0)+1.00000

00

( 0)+1.00000

00

( 0)+1.00000

00

II 12 3.20473 1.17799 42 1.44217 8.28815 5.45981 65 11 50 35

0.6 ii:87 Zl

a\b

2 2 3I 3 3

6.58320 8.74427 1.13401 1.44322 1.80888

17 45 20 61 49

0.7

0.6

0.8

-1.0 -0.9 -0.8 -0.7 -0.6

1.0

0.9

Oj-5:38234

50

( 0)+1.00000

00

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0":: i::

( 0)+1.00000

00

( 0)+1.00000

00

( 0)+1.00000

00

( 0)+1.00000

00

CONFLUENT CONFLUENT

HYPlCRGEOMETRIC

HYPERGEOMETRIC

529

FUNCTIONS FUNCTION

M(a,

Table

b, z)

13.1

x=5.0 a\b

0.2

0.1

0.3

0.4

$9" -0:s -0.7 -0.6

II 1 1 1 1

-1.56666 -2.41382 -3.23511 -3.98065 -4.58862

67 36 34 33 62

-0.5 -0.4 -0.3 -0.2 -0.1

II 1 1 1 1 1

-4.98353 -5.07426 -4.75193 -3.88754 -2.32934

39 08 11 12 93

( 0)+1.00000

00

0.0

( 0)+1.00000 00

( 0)+1.00000 1

oo*: 0:s

00

0.5

II 1 1 1 1 1 ( 0)+1.00000

00

-2.33084 -2.33646 -2.15579 -1.73399 -1.00692

19 31 45 46 28

( 0)+1.00000

00

6.28624 01

II 2 2.62678 1.48413 16 96

8::

2 6.01287 4.11434 26 11

a\b

0.7

0.6

0.8

0.9

1.0

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

( 0)+1.00000

1.0

ppE&

0.6 t;: 0:9 1.0

00

Ii

2 1.48413 16 2 1.98603 96

2 3.33018 2.59579 43 07 2 4.20801 74

( 0)+1.00000

pps&

00

( 0)+1.00000

pfiztp

00

( 0)+1.00000

00

pz&

II

1 6.77444 40 1 8.98511 69 2 1.48413 1.16513 78 16 2 1.86309 66

( 0)+1.00000

Poe&

00

530

CONFLUENT

Table

CONFLUENT

13.1

HYF'ERGEOMETRIC

HYPERGEOMETRIC

FUNCTIONS

FUNCTION

M(a,

b, 2)

x=6.0 a\b

0.1

0.2

-1.0 -0.9 -0.8 -0.7 -0.6

1 1 2 Ii 2 2

0.3

-2.90000 -6.43961 -1.01116 -1.37008 -1.69209

00 14 95 05 38

( 0)+1.00000

00

0.4

0.5

-0.5 -0.4 -0.3 -0.2 -0.1 0.0

( 0)+1.00000

00

8::

II 2

00*43 0:5 0.6

( 0)+1.00000

00

4.03428 07 1.66280 79

il 21 2.23669 9.26969 34 33

32 1.16700 7.30095 48 13 3 1.73835 48

2 6.43121 4.03428 79 54 2 9.55746 91

II 3 6.08625 44 43

1.12757 14 8.38957 36

3 1.86253 2.49428 97 70

1":;

4 1.48541 1.92506 80 91

3 4.23039 3.27475 26 92

0.6

( 0)+1.00000 00

II 3 1.35639 99

00::

a\b

( 0)+1.00000 00

0.7

0.8

II

-1.0 -0.9 -0.8 -0.7 -0.6

0.9

1.0

10 -5.66666 -1.56045 67 -1.35713 -1.11025 -8.41150 64 26 62 68

-0.5 -0.4 -0.3 -0.2 -0.1 0.0

( 0)+1.00000

00

( 0)+1.00000

00

( 0)+1.00000

00

( 0)+1.00000

00

0.1

0.6 2: 0:9 1.0

00

II 1 4.32726 9.80333 2.48291 13 1.10148 6.73053 40 09 68 56

8:; i::

( 0)+1.00000

II

2 2.34847 4.03428 1.73291 5.15728 70 3.10736 26 33 89 79

CONFLUENT CONFLUENT

HYPERGEOMETRIC

HYPERGfEOMETRIC

FUNCTIONS

FUNCTION

it&z,

531 b, x)

Table

13.1

x=7.0

a/b

0.2

0.1

0.3

0.4

0.5

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

( 0)+1.00000

00

( 0)+1.00000

00

( 0)+1.00000

00

( 0)+1.00000

00

( 0)+1.00000

00

2: 8'34 0:5

a\b

0.6

0.7

-1.0 -0.9 -0.8 -0.7 -0.6

0.8

j 1l)-4.98346 0 I -9.00000 -4.02257 -2.96917 -1.90770

41 00 88 93 95

( 0)+1.00000

001

0.9

1.0

-0.5 -0.4 -0.3 -0.2 -0.1 0.0

0.6 kz 0:9 1.0

( 0)+1.00000

00

( 0)+1.00000

00

( 0)+1.00000

00

( 0)+1.00000

00

532

CONFLUENT Table

HYPERGEOMETRIC

HYPERGEOMETRIC

FUNCTIONS

13.1

CONFLUENT

FUNCTION

M(a,

b, x)

( 0)+1.00000

00

( 0)+1.00000

00

( 0)+1.00000

00

( 0)+1.00000 00

( 0)+1.00000 00

( 0)+1.00000

00

( 0)+1.00000

00

( 0)+1.00000

00

( 0)+1.jl0000

( 0)+1.00000 00

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 82 0:3 8::

-0.5 -0.4 -0.3 -0.2 -0.1 0.0

2; i-98 l:o

II

3 2.98095 2.12243 80 36

3 4.08075 5.47370 48 63 3 7.22067 87

00

CONFLUENT CONFLUENT

HYPERGEOMETRIC

HYPERG EOMETRIC

FUNCTIONS

FUNCTION

M(a,

533 Table 13.1

b, 2)

z =9.0

0.1

a\b

0.2

0.3

0.4

0.5

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

( 0)+1.00000

00

( 0)+1.00000

00

( 0)+1.00000

00

( 0)+1.00000

00

( 0)+1.00000

00

0":: 8-i 0:5

oo*sB

ii

l:o

4) 5.90279 86 4 8.44810 69 5 1.60777 1.17771 16 47 5 2.15743 14

a/b

0.6

0.8

0.7

0.9

1.0

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

( 0)+1.00000

00

( 0)+1.00000

00

( 0)+1.00000

00

( 0)+1.00000

00

( 0)+1.00000

0.1 00:;

8.10308 1.15389 1.60085 2.17532 2.90602

5.69778 8.10308 1.12277 1.52385 2.03337

4.12286 5.85547 8.10308 1.09842 1.46399

z%E! 5:98502 8.10308 1.07870

2.30549 3.26534 4.50694 6.09425 8.10308

00

534

CONFLUENT HYF'ERGEOMETRIC FUNCTIONS Table

13.1

CONFLUENT

HYPERGEOMETRIC

FUNCTION

M(a, b, x)

x =lO.O

a\b

0.2

0.1

-1.0 -0.9 -0.8 -0.7 -0.6

l)-4.90000 3 -1.04774 II 3 -2.26606 3 -3.65315 3 -5.11412

0.3

00.: 0:3

3 3 3 3 II 3 ( 0)+1.00000 00 4) 4 51 5 5

2.20264 66 5.69563 19 1.09330 93 1.84869 24 2.90713 00

( 0)+1.00000 00 3 41 4) 4 5

0.5

00 98 51 21 18

-0.5 -0.4 -0.3 -0.2 -0.1 0.0

0.4

( 0)+1.00000 00

-2.06370 -2.39329 -2.51877 -2.29844 -1.54205

40 23 45 83 59

( 0)+1.00000 00

( 0)+1.00000 00

8.52983 30 2.20264 66 4.22272 41 7.13160 87 1.12016 64

::t

4 3.28620 4.73642 65 75 4 6.64873 73

l:o E

II 4 5 9.13874 1.23458 32 19

a\b -1.0 -0.9 -0.8 -0.7 -0.6

0.6

0.7

0.8

0.9

1.0

CONFLUENT

IIYPERGEOMETRIC ZEROS

a\b

0.2

OF

M(a,

FUNCTIONS

535

b, r)

Table

13.2

-1.0 -0.9 -0. 8 -0.7 -0.6

0.1 0.10000 0.11054 0112357 0.14010 0.16173

00 47 83 11 42

0.20000 0.22012 0.24477 0.27567 0.31555

00 64 52 24 72

0.3 0.30000 0.32894 0.36411 0.40779 0.46354

00 15 44 72 99

0.40000 0.43713 0.48196 0.53721 0.60707

00 15 35 21 04

0.50000

00

0.54480 0.59858 0.66443 0.74705

16 98 91 02

-0.5 -0.4 -0.3 -0.2 -0.1

0.19128 0.23411 0.30182 0.42537 0.72703

98 73 31 31 16

0.36906 0.44470 0.56019 0.75993 1.20342

09 78 88 80 40

0.53728 0.63961 0.79200 1.04632 1.58016

03 58 44 32 05

0.69839 0.82334 1.00591 1.30289 1.90320

96 00 69 37 51

0.85403 0.99868 1.20695 1.53918 2.19258

26 55 84 36 90

00 19 38 07 45

0.7 0.70000 0.75888 0.82892 0.91376 1.01887

00 50 89 55 44

0.8 0.80000 0.86541 0.94291 1.03637 1.15158

00 05 59 62 21

00 85 10 85 70

1.0 1.00000 1.07763 1.16901 1.27838 1.41205

00 19 22 33 79

1.29771 1.49044 1.75960 2.17271 2.94434

21 27 56 84 51

1.57995 1.79887 2.10045 2.55566 3.38779

68 13 49 24 57

a\b -1.0 -0.9 -0. 8 -0.7 -0.6 -0. 5 -0.4 -0.3 -0.2 -0.1

0.6 0.60000 0.65203 0.71419 0.78986 0.88415

0.4

0.5

0.9 0.90000 0.97164 1.05625 1.15786 1.28256

Table 13.2 gives the smallest zeros in z of Jf(a, b, z), near a = b =O, that is, the smallest positive roots in z of the equation M(a, b, z) =O. Linear interpolation gives 3-4s. Interpolation by the Lagrange six-point formula in two dimensions gives 75.

14. Coulomb Wave Functions MILTON

ABRAMOWITZ

l

Contents

Page

Mathematical Properties .................... 14.1. Differential Equation, Series Expansions ......... .......... 14.2. Recurrence and Wronskian Relations

14.3. 14.4. 14.5. 14.6.

Integral Representations ................ Bessel Function Expansions ............... Asymptotic Expansions ................ Special Values and Asymptotic Behavior

Numerical Methods ...................... 14.7. Use and Extension of the Tables References

538 538 539 539 539 540 542

.........

543 543

............

544

..........................

Table 14.1. Coulomb

lIp120)

Wave Functions of Order Zero (.5 1~520, . . . . . . . . . . . * . . . . . . . . . . * . . .

q=.5(.5)20,

p= l( 1)20,

Table 14.2. Co(q)=e-*“~\I’(l+iq)I q=O(.O5)3, 6s

The author wishes to acknowledge formulas and tables.

1 National

546

5s . . . . . . . . . . . . . . . . I

the assistance of David 9. Liepman

554

in checking the

Bureau of StEbndards (deceased).

537

14. Coulomb Mathematical 14.1. Differential

Equation, Dikl’erential

Series

Expansions

Wave Functions Properties 14.1.12

FL=;

F'!TI,

14.1.13

#h,

P,=~$,,,

P)=CL(~I)P’W~~,P)

Equation

14.1.1

(P>O,

-

O3
L a non-negative

OJ,

The Coulomb wave equation has a regular singularity at p=O with indices L+l and -L; it has an irregular singularity at p= a. General

Irregular

integer)

14.1.2

GL(T,

14.1.16

where FL(v, p) is the regular Coulomb wave function and a(~, p) is the irregular (logarithmic) Coulomb wave function.

14.1.17

P)+&%(T,

Regular

P>

Coulomb

Wave

Function

FL((),

p)

Wave

Function

P)=&

Fdv,

0 flLh,

PNn

~P-I-@-@ pL(q)l+~Lh,

P)=DL(dP-L#LL(tl,

PI

P)

OD tiLbI,P)==~.

-L

4x11>

Pk+L

14.1.18 atL=l,

2L+2,2ip)

p)

DLb)CL(d’&

14.1.3

F,(q, p)=CL(q)pL+le--f~M(L+l-iq,

GLO),

14.1.14

(C,, Cz constants)

w=GFds,

Coulomb

14.1.15

Solution

k&(dpk+’

(k-L-1)

4+1=0,

(k+L)a:=2qu&-a:-,-

(2k-1) pL(q)Af

14.1.4 =cLh)

14.1.5

P”+wrl,

a(~,

14.1.19

P>

P) =,=Z+,

AE(d

PLO&L-if

P’+-]

PLC4

a-1 s2+q2

2L ‘1s-1

s

14.1.6 A:+I=l

, AL.+2==& (See Table 6.8.)

(k+L) (k-L-

1)A: =2qA;-,-A;-*

(k>L+2) 14.1.20

14.1.7

CL(O)=

2Le-y Il?(L+l+itl)l r w+q

rdd=w 91i&f2$~~’

(See chapter 6.) 14.1.8

+2”(iq-L)(iq-L+l)+ (2L-1)

~(7])=27r~(ez*~-l)-1

***

(2!)

+2”(iq-L)(iq-L+1)

. . . (iq+L-1) (2L)!

14.1.9 14.1.10

pL(rl) 14.1.11 -= 211 538

(L2+v21f

cL(77)=L(2L+1)

1

14.1.21

c5-167)

(1+q2)(4+t12) * * * (L2+q2P

w+1)[w)u2

,Qc&-- 2q L dp GM

l+p-lFL(q, +a%

P> 1

PI

COULOMB

14.1.22

S;=$

f&(7, P) =&L(v)

P+%(v,

WAVE

14.2.

P>

539

FUNCTIONS

Recurrence

and

Recurrence

14.1.23

If

c=F’h,

P> or

WI,

Wronskian

Relations

Relations P>,

L ~~=(L2+112)t~~-,-(~+q)~~

14.2.1

14.2.2

GL.FL

~=[~+,1u,-[(L+1)‘+121111,,,

(L+l) 14.2.3

L[w+1)2+v21f~ .+,=(2L+1)[,+L(LP+1)l% -(L+1)[L2+~21k.1 Wronskian

Relations

F;GL- F,G;=

14.2.4

1

FL--GL-FhGL--I=L(L2+q2)-*

14.2.5

14.3.

Integral

Representations

14.3.1

14.3.2 FIGURE 14.1. &(rl, P), &(?I, P>. v=l,

FL-iGL=

p=lO e-*q

L+l

(2L+1~IcL(q) FL, F;

GL;G; .

S-yfm e-tP”(l-t)L-fl(l+t)L+i(lcEt

14.3.3

- 24 - 22 I.1 -

- 20

LO -

- 16

m

( (1- tanh2 QL+’ exp [-i(p s

.9 -

- I6

.6 -

- I4

.7 -

- 12

.6 -

- IO

.s -

-6

.4-

-6 -4 -2 -0

FIGURE

14.2.

FL, FL, (IL and (2;.

?f= 10, p=20

tanh t-2~7t)]

0

+i(l+t2)L 14.4. Expansion

14.4.1

Bessel in Terms

exp [--pt+2r] Function

arctan t]}dt

Expansions

of Bessel-Clifford

Functions

540

COULOMB

WAVE

FUNCTIONS

j1= U/5)$

14.4.3 b2~+1=1, 4q2(k-2L)

hL+z=o,

b,+l+kb,-l+

j2=f

@f'+S)

@x-=+2)

b&-2=0

(84?+1480~*+232Oz)

j3=&

14.4.4

rn=-W5)~ g2=&

(See chapter Expansion

9.)

g3=ko

in Terms

of Spherical

Bessel

(See chapter . C=+1)

,&ds)

(1056x6-11602-2240)

Functions

14.4.5 FL(% P) = 1.3.5..

(7s6--309)

k%L I b, 4;

10.)

Jk+tb)

14.4.6 b&=1, bL+,=2z

1

14.5. Asymptotic

W+l) “=k(k+l)--L(L+l)

Asymptotic

(k--1)(k--2)--L&+1) 2k--3

I’hbn-,-

bke2) (k>L+l)

14.4.7 FL(q, p)=1*3.5

Expansion sJ=

b -(Ic-1) ‘+I (2k--1)

in Terms

@7FP)/h>“”

of Airy I.c=

14.5.2

GL=j

cos e,--g sin eL

14.5.3

FL=g* G;=j*

cos eL+j*

cos eL-g* eL=p-q

sin 0,

sin e,, gj*-jg*=l

In 2p-L

uL=arg

II-+uL 2

l?(L+ l+iq)

uL+,=uL+arctan

14.5.7

Functions

(See Tables

tt>>o

&

p.14, 6.7.)

b--2Kh

14.4.9

14.5.8

j-&k,

g-2

sx,j*+&

where

+Ai’ (4 [r+~+ fl f2 ’ ’ ‘I } Bi’(~) 14.4.10 FXrl, G,(q,

of p

(See 6.1.27, 6.1.44.)

bksl

(21)2’3,

Values

cos flL+j sin eL

14.5.6 b:=--- (k+2) (2k+3)

Large

FL=g

14.5.5

14.4.8

for

14.5.1

14.5.4

. . . (2Lfl)PCdf)

Expansion

Expansions

P) =--at(2q)-tIAW Bib) +d+rfi+

go=o, jo*=O, go*=l-q/p

fk+l=anfn-bngr gk+l=akgk+bkfk

,d+?h

P)

jo=l,

P . . .]+Ai’!d

f:+l=akf:-bkgg:-fk+l/P Bi,

(2)

11 +q

&+l=ak$+bkf:-gk+dP

P2

+i$e.&+ I

. . .]}

,=(zk+l)rl,

(2k+2)p

bk=

L(L+l)--k(k+l)+v2 @k+2) p

g*-&

s:

COULOMB

Asymptotic

14.5.10

r(1/3)8"(lT

F&d 4&>/d3”

245

FUNCTIONS

WAVE

Expansion

for

541

L=O, p=2q>>O

2 r(2/3) 1 32 leT 92672 r(2/3) 1 35 r(1/3) p4 8100 B 7371.104 r(1/3) a"-

* . *I

14.5.11 3’: (271) Gcw/d3

-r(2/3){*1, wa3~

1 rm 15 r(2/3)

1I 8 l+ 11488 W3) 1 I . , *) 82 56700 I36 18711.103 r(2/3) j3B a==(2vj/3)%, r(1/3)=2.6789

38534. . . , r(2/3)=1.3541

17939 . . .

14.5.12 .70633 26373 04959 570165 .00888 88888 89 rv -GAW ( 1.22340 4016 l qw I 1 ~. q% D2 JO245 51991 81 .00091 061 F08958 P q4 L.T.00025

F&d

34684 115 -. 71 'H

. .I

73966 427 +. P

* .I

14.5.13

KOd E&W

.40869 57323 -.70788 17734}@{lf - '

.17282 60369

.00031 74603 174

qs

q2

-+

L.CJ0358 12148 50+.00031 -L

FIGURE

14.3.

9=0, 1, 5,

FIGURE

F,(q, p). 10,

14.4.

r)=o, 1, 5,

7%

P/2

Fi(q, p). 10,

P/2

.

17824 680k.00090 q4

FIGURE 14.5. v=o, 1, 5,

FIGURE q=o,

14.6.

G,,(q, p). 10,

P/2

Gi(q, p).

1, 5, 10, p/2

542

COULOMB

14.6. Special

Values

14.6.1

and Asymptotic

Behavior

WAVE

FUNCTIONS

14.6.10

L>O, p=o F,=O, FL=0 GL=w, (&--cm

14.6.2

L=O, F,=O,

L=O,

Fo-i

Bea; Fi-

Go-Be-y

p=o

F”,=Co(q)

h>>P

(s++&

Gi-

t-“P]

{ -a++&

t-*p4} Go

t=P/$

Go= 1/C,,(t), G;= --oo 14.6.3

a=%{[t(l-t)]++arcsin

L+= J?L-%(~P~+~,

14.6.4

GL’v~&I)P-~

L=O, q=o F,=sin p, F,‘=cos Q=cos

p

L=O,

P>>%

Fo=cu sin 8; Fi=-

p, GA=-sin

i!+-&f}

8= {t/(1-t)}* 14.6.11

t2(bFo-aGo)

p Go=a cos /3; Gi= - t*(aF,+bG,J

14.6.5

P+=

G,+iF,-exp

i[p-q L>O,

14.6.6 FL=

GYP?

CL=

(-

A,++(P) J-w)

L 20,

FL-

P

q=o

lY(3r~Y

14.6.7

t=?!

In 2p-$+nJ

(P>

a= P(l-

h>>P

t)‘, b=[8q(l-

14.6.12

(2L+1)!cdd cwL+’

rl>>%

t)]-1 217-p

@?P)+~12L+1wrlP)fl

2 WL (27lPwGL+1rw7lP)~l

c”~(2LSl)!C,(q) 14.6.8

L=O,

%>>P

Fo-e-“?(?rp>~~l,[2(2qp)tl

x=(PL-P)[-&+

1 W$l)]l,, p;

F&e-“?(2aq)fI,,[2(2qp)*] 14.6.13

Go-2ecq G’ 0-

0

5

kwtlP)tl

[GoSiFoI wr1/2(2q)116[Bi(z)+iAi(x)]

0

14.6.9

[G~+iF~]~-d/2(2q)-1~6[Bi’(~)+iAi’(z)]

L=O,

%>>P

Fomk Bea; FL-a Go-@-a;

29-p

z= (2q- p) (2q) -I’3

??!’ a e”~~0WrlpN

-2

s>>o,

14.6.14

/?-lea

(Jim -b-‘e-a

q>>o PL=l+[112+uL+1)1”2

FL(PL)

r (l/3)

G&L)/&-T cr=2$i&-?rq FZPL) 8=

(P/w*

G~PL>/&

~(213) 2&

F.

E -1’6 03 1

COULOlkfB

14.6.15

WAVE

1 14.6.17

P’h>>o

r(1/3)

FO G,,/&-2J;;

3 WE 03

L+=

I

to 8S for 7>3.)

14.7. Use and Extension

of the Tables

In general the tables as presented are not si.mply interpolable. However, values for L>O may be obtained with the help of the recurrence relations. The values of GL(q, p) may be obtained by applying the recurrence relations in increasing order of L. Forward recurrence may be used for FL(v, p) as long as the instability does not produce errors in excess of the accuracy needed. In this case the backwards recurrence scheme (see Example 1) should be used. Example 1. Compute FL(q p) and F;(q, ,p) for q=2, p=5, L=O(1)5. Starting with Fg=l, Ffi=O, where FZ=CF~, we compute from 14.2.3 in decreasing order of L:

F,/F,*=

2LL!

v--l)1

Numerical

L 11 10 9 8 7 6 5 4 3 2 1 0

constant)

(2L+l)!

14.6.18

Co(T)) ‘v (2mj)%-*“,

(Equality

(r=Euler’s

--VI

C&) -

1l+Q) dd-ii+sOn

rl-*o dd

r (213) Fi -Gv&-2&(2~/3)“” 14.6.16

543

F’UNCTIONS

(1) FZ 0. 1. 4.49284 17.5225 61.3603 191.238 523.472 1238.53 2486.72 4158.46 5727.97 6591.81

(2) FL

(3) FL

(4)

,F;

Methods 2. Compute L=1(1)5.

Example v=2, p=5,

Using 14.2.2 and G,,(2, 5)= .79445, G,‘= - .67049 from Table 14.1 we find G1(2, 5) = 1.0815. Then by forward recurrence using 14.2.3 we find: L

.091 .215 .4313 .72125 .99347 1. 1433

1043 : 2030 .32:05 .3952 3709 : 29380

1.7344~ lo-‘=c-‘.

The values in the second column are obtained from those in the first by multiplying by the normalization constant, F,/F,* where F. ie; the known value obtained from Table 14.1. Repetition starting with F&=1 and F&=0 yields the same results. In column 3, the results have been given when 14.2.3 is used in increasing order of L. Ft. (column 4) follows from 14.2.2.

-CL

GL

*

I

-I i

1. 0815 4969

3 .t

2.0487 5.6298 3.0941

..56619 60286 4.5493 1: 81?9

The values of GL are obtained with 14.2.1. Example 3. Compute G,,(l, p) for ~=2, p=2.5. From Table 14.1, G,(2, 2)=3.5124, $(2, 2) = -2.5554. Successive differentiation of 14.1.1 for L=O gives

Taylor’s .090791 .21481 43130 172124 99346 1: 1433

GL(q, p) and GL(q, p) for

expansion

+@P)’ Twq....

we get:

is w(p-/-Ap)

=w(p)

+ (Ap)w’

With w=G,,(v, p) and Ap=.5 d&Go dpl 3.5124 -2.5554 3.5124 -6.0678 12. 136 -29.540 83.352 -268.26

--(API

k dkGo

k! dpk 3.51i4 - 1.2777 .43905 -. 12641 .03160 -. 00769 .00181 -. 00042

Go(2, 2.5)=2.

5726

As a check the result is obtained with TJ=~, p=3, Ap=-.5. The derivative Gi(q, p) may be obtained using Taylor’s formula with w=GA(q, p). ‘See pageII.

544

COULOMB

WAVE

FUNCTIONS

References Texta

[14.1] M. Abramowitr and H. A. Antosiewicz, Coulomb wave functions in the transition region. Phys. Rev. 96, 75-77 (1954). [14.2] Biedenharn, Gluckstern, Hull, and Breit, Coulomb functions for large charges and small velocities. Phys. Rev. 97, 542 (1955). [14.3] I. Bloch et al., Coulomb functions for reactions of protons and alpha-particles with the lighter nuclei. Rev. Mod. Phys. 23, 147-182 (1951). [14.4] Carl-Erik Froberg, Numerical treatment of Coulomb wave functions. Rev, Mod. Phys. 27, 399-411 (1955). [14.5] I. A. Stegun and M. Abramowit,z, Generation of Coulomb wave functions from their recurrence relations. Phys. Rev. 98, 1851 (1955).

Tables

[14.6] M. Abramowitz and P. Rabinowitz, Evaluation of Coulomb wave functions along the transition line. Phys. Rev. 96, 77-79 (1954). Tabulates F,, F& Go, Go for ~=2z=O(.5)20(2)50, 8s. [14.7] National Bureau of Standards, Tables of Coulomb wave functions, vol. I, Applied Math. Series 17 (US. Government Printing Office, Washington, D.C., 1952). Tabulates

’ (-f$’ Mr], P>and JJ 9=-5(1)5,

‘) for p=O(.2)5,

L=O(1)5, 10, 11,20,21,

7D.

[14.8] Numerical Computation Bureau, Tables of Whittaker functions (Wave functions in a Coulomb field). Report No. 9, Japan (1956). t14.9) A. Tuhis, Tables of non-relativistic Coulomb wave functions, Los Alamos Scientific Laboratory La-2150, Los Alamos, N. Mex. (1958). Values of Fo, Fb, Go,G;, P= 0(.2)40; z=O(.O5)12, 55.

546

COULOMB

Table

14.1

COULOMB

WAVE

4. 5 5. 0

- 4) 2.8622

2.: 6: 5 7.0 5 E 9: 5 10.0

10. 5 11. 0 11. 5 12. 0 12.5 13.0 13. 5 14. 0 14. 5 15.0 15. 5 16. 0 16. 5 17. 0 17.5 18.0 18. 5 19. 0 19. 5 20.0

0. 5 1. 0 i:; ::i :*z 5: 0

10.5 K ;;: ; 13:o 13. 5 14. 0 14. 5 15. 0 15. 5 16. 0 16. 5 17.0 17.5 18. 0 18. 5 19. 0 19.5 20.0 l-3.

FUNC!CIONS

FUNCTIONS

Fo(?,P)

rllP 0.5 ::5” $2 i-5” 4: 0

For use of this table ee Examples

WAVE

OF ORDER

ZERO

COULOMB

COULOMB

WAVE

WAVE

547

FUNC.l’IONE)

FUNCTIONS

OF ORDER

ZERO

Go(T,P)

x Ii5 12.0 :z 13:5 14.0 :::i 15.5 2: ;;:

;

1s:o 18.5 19.0 19.5 20.0

:: k-E 16 I 117969

5 -2.0878 5 -6.3080 6 I -1.9295 10.5 :z 12:o 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0 16.5 x 1e:o 18.5 19.0 ::::

*see page Il.

14 -4.66lO 15 -1.15132 15 I -6.61'?4

11 12 12 13 14 I 14 15 15 16 16 I

-7.7763 -2.6230 -8.8973 -3.0340 -1.0399 -3.5813 -1.2392 -4.3069 -1.5033 -5.2691

Talde

14.1

548

COULOMB

Table

14.1

8

COULOMB

WAVE

WAVE

FUNCTIONS

FUNCTIONS

OF ORDER

ZERO

6 (

0i - 1 - 1 0

9

-1.0286 -1.fJ71a +a.7682 it2850

(-

i)+a.aaoz

- 21 3.5181 - 2 1.5740

10. 11. 11. 12. 12. 13. 13. 14. 14. 15.

5 0 5 0 5 0 5 0 5 0

-

5 6 6 7 7

1.5930 5.9782 2.2113 a.0697 2.9081

- 7 - 7

15.5 16. 0 16.5 17. 0 17.5 18. 0 18.5 19. 0 19. 5 20. 0

-10 I 5.0935 -10 1.7129 -11 5.7147 -ii 1.8924 -12 I 6.2217 -12 2.0316

- 1 +6.5317 - 1 -4.9515 - 1I -a.7151

I

I : g-g; - 1

- 5 - 5 10.5 11. 0 11. 5 12. 0 12.5 13.0 13. 5 14. 0 14. 5 15.0 15. 5 16.0 16.5 17. 0 17.5 18.0 18. 5 19. 0 19. 5 20. 0

2:9346

3.8880 1.4803

I - 76I 5.5384 7.3981 2.0392

- 5 I 2.3388 - 6 9.0675 - 6 3.4579

- 9) 2.7940 - 8) 2.6629

5.3814 2.0569

COULOMB

COULOMB

6

WAVE

7

WAVE

549

FUNCTIONS

FUNCTIONS

Go(v,P)

OF ORDER

ZERO

8

5.5 6. 0 6. 5 7.0 .i:'o 2'0 1::: 10.5 11.0 11.5 12.0 12. 5 13.0 13.5 14.0 14.5 15. 0

3 3I 4 4

15.5 16. 0 16.5 17. 0 17.5 18. 0 18.5 19. 0 19.5 20.0

10.5 11.0 11.5 12.0 12.5 13. 0 13. 5 14. 0 14.5 15.0 15.5 16. 0 16.5 17.0 17. 5 18.0 18.5 19.0 19.5 20.0

3I 4 4 5 5

-1.4717 -1.9033 -4.9246 -1.2929 -3.4407

8 -1.5573 6 -4.4670 9 -1.2923 9 -3.7692 10 i -1.1079 10 I -3.2807 10 -9.7640 11 -2.9371 11 -8.8779 12 -2.6998

1.9070 4.4437 1.0570 2.5623

Table

14.1

550

COULOMB

Table

q\P 0. 5

14.1

COULOMB

WAVE

WAVE

FUNCTIONS

FUNCTIONS

OF ORDER

ZERO

11 - l)-6.9792

::i z 3: 0 2-z 4:5 5.0

- 2 4.3132 - 2 I 2.2096 - 2 1.0980 10. 5 11.0 11. 5 12. 0 12.5 13.0 13.5 22 15: 0 15. 5 16. 0 16.5 17. 0 17.5 18.0 18. 5 19.0 19.5 20.0

10. 5 11. 0 11.5 12. 0 12.5 13.0 13. 5 14. 0 14.5 15.0 15. 5 16.0 16.5 17. 0 17. 5 18.0 18. 5 19.0 19. 5 20.0

1.2422 4.9601 1.9580 7.6449 2.9542 1.1303 4.2845 1.6095 5.9943 2.2143

- 9) 1.0052

- 5 - 5

7.9271 3.5765

COULOMB

COULOMB

WAWE 12

WAVE

551

FUNCTIONS

FUNCTIONS

Go(w)

OF ORDER

ZERO

Table

13

15 -

II

- 01 -1.0783 -8.5560 -1.2510

10.5 11. 0 11. 5 12. 0 12. 5 13.0 13.5 14. 0 14.5 15. 0

2) 2I 3 3 3

2.5735 5.4370 1.1780 2.6115 5.9114

2I 2 2 2

1.0506 2.1519 4.5?09 9.6(54

4 7.6408 5 1.8544 5 I 4.5606

4 4I 4 5

1.0421 2.35'53 5.55~78 1.3286

15. 5 16.0 16.5 17.0 17.5 18.0 18.5 19. 0 19. 5 20.0

- 1I -5.0324 - 1 -5.8597 - 1 -9.1132 0I -1.6356 0 -3.0877 0 -5.9776 1I -1.1842 1 -2.4038 1 -5.0022 2 -1.0663 10.5 11. 0 11.5 12.0 12.5 13.0 13.5 14.0 14. 5 15. 0 15. 5 16. 0 16.5 17. 0 17.5 18. 0 18. 5 19. 0 19.5 20.0

1 1 2

I

1 +6.6972 1I -7.2341 1 -7.2415 1 +2.8479

II

- 1 -5.56,63 -5.7431 -4.9764

1I 2 2 2

4 I -9.2211 5 -2.3041 5 -5.8301 6 6I 7 7 7 8 8I 9 9 10 I

-1.4929 -3.8658 -1.0118 -2.6753 -7.1420 -1.9243 -5.2302 -1.4335 -3.9609 -1.1028

1 1 1 1 1 0 - 1 - 1 0 0

+8.9435 +3.4046 -9.7085 -6.2172 +6.1593 +1.1292 +5.4881 -4.6254 -1.1612 -1.2413

I I I

5.0429 9.6258 1.8964

I I I

4) 9.7988

-

14.1

4 4 4 5 5 5 6) 6 6 7

- 1 +9.7040 - 2 +5.5060 - 1 -9.1975

II

I

- 1 -6.5243 - 1 -5.4972 - 1 -4.9245

-8.8802 -1.8956 -4.1335 -9.1940

1.1531 2.5494 5.7251 1.3047 3.0146 7.0570 1.6726 4.0107 9.7253 2.3833

- l)-4.6958

- 1 -1.4460 - 1 -5.3907 (-

l)-6.8002

I

- 1 -5.5683 - 1 -6.7414 - 1 -6.2956

O)-7.7837

3 1-8.4644 4 -1.9742 4)-4.6712

I

5 -1.1203 5 -2.7217 5 -6.6925

4 -1.1531 4 -2.6329 4 -6.0946 5 -1.4291 5 -3.3924 5 -8.1473 6 1 -1.9785 6 -4.8557 7 1-1.2038 7)-3.0133

552

COULOMB

Table

COULOMB

14.1

WAVE 17

(-

1) 1.8899

(-

l)-7.4641

(-

1) 3.2134

WAVE

FUNCTIONS

FUNCTIONS

Fe(w)

(-

1) 5.0960

E 12: 0 12.5 13. 0 13. 5 14. 0 14. 5 15. 0 15. 5 :s 17:o 17. 5 18. 0 18. 5 19. 0 19.5 20.0

10.5 11. 0 11.5 12. 0 12. 5 13.0 13. 5 14.0 14.5 15. 0 15. 5 16.0 16. 5 17. 0 17.5 18. 0 18. 5 19.0 19.5 20.0

ZERO

18

10. 5

- l)-7.0977

OF ORDER

- 2)+1.8327 - l)-5.3380

(-

1) 7.5308

COULOMB

COULOMB 16

10. 5 11.0 11. 5 12. 0 12.5 13. 0 13.5 14.0 14. 5 15. 0 15. 5 16. 0 :Ei 17: 5 18.0 18.5 19. 0 :E .

WAVE

WAVE

553

FUNCTIONS

FUNCTIONS

OF ORDER

ZERO

Table

14.1

554

COULOMB

Table 9

WAVE

CO(~) =e-i”q

14.2

FUNCTIONS

IW+hY

co(4

co(9)

D

0.05 0.10 0.15 0.20

1.000000 0.922568 0.847659 0.775700 0.707063

1.00 1.05 1.10 1.15 1.20

0.25 0.30 0.35 0.40 0.45

0.642052 0.580895 0.523742 0.470665 0.421667

1.25 1.30 1.35 1.40 1.45

0.50 0.55 0. 60 0.65 0.70

0.376686 0.335605 0.298267 0.264478 0.234025

1.50 1.55 1. 60 1.65 1.70

2.50 2.55 2.60 2.65 2.70

0.75 0.80 0.85 0.90 0.95

0.206680 0.182206 0.160370 0.140940 0.123694

1.75 1.80 1.85 1.90 1.95

2.75 2.80 2.85 2.90 2.95

1. 00

0.108423 (-4)5

2.00

0. 00

II 1

For In J?(l+iy),

see Table

6.7.

(-2)6.33205 (-2)5.52279 (-2)4.81320

(-3)6.61992

co(1)

2.00 2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45

3.00

j -3 I 2.76623 2.06392 3.20136 2.38968 -3)1.78218

(-4)3.50366

15. Hypergeometric

Functions

FRITZ OBERHETTINGER

1

Contents Mathematical Properties . . . . . . . . . . . . . . . . . . . . 15.1. Gauss Series, Special Elementary Cases, Special Values of the Argument . . . . . . . . . . . . . . . . . . . . 15.2. Differentiation Formulas and Gauss’ Relations for Contiguous Functions. . . , . . . . . . . . . . . . . . . . . . 15.3. Integral Representations and Transformation Formulas . . . 15.4. Special Cases of F(a, b; c; z), Polynomials and Legendre Functions. . . . . . . . . . . . . . . . . . . . . . 15.5. The Hypergeometric Differential Equation . . . . . . . . 15.6. Riemann’s Differential Equation . . . . . . . . . . . . 15.7. Asymptotic Expansions . . . . . . . . . . . . . . . . References

1 National

. . . . . . . . . . . . . . . . . . . . . . . . . .

Bureauof Standards.

(Presently,

Oregon State University,

Page 556 556 557 558 561 562 564 565 565

Corvallis, Oregon.) 555

15. Hypergeometric Mathematical 15.1. Gauss Series, Special Elementary Special Values of the Argument Gauss

Functions Properties 15.1.7

Cases,

F(i,

3; Q; -z2)=(1+z2)+F(1,

The circle of convergence geometric series

=2-l F(a, b; b; z)=(l-~)-~

15.1.8 15.1.9

F(a, ++a; 3; z2)=~[(l+z)-2”$(1-z)-2a]

15.1.10

b; c; 2)

F(a, $+a; 4; z2)= &z-‘(1-2a)-’ I-(C)

a r(a+n)r(b+n)

=r(a)r(b>

%Fo

In [z+(l+z2)*]

of the Gauss hyper-

15.1.1 F(a, b;c; 2) =zF1(a,

1; Q; -.z2)

Series

r(C+12)

[(l+~)‘-~“-(l-z)~-~~]

2”

15.1.11

ii!

F(-a,a;~;-z2)=${[(1+z2)f+z]2a+[(1+z2)*-z]2a}

q&q

$mc,

13\i++J

is the unit circle ]z( =l. The behavior of this series on its circle of convergence is: (a) Divergence when 9 (c-a--b) 5 -1. (b) Absolute convergence when 9Z (c-a--b)>O. (c) Conditional convergence when - 1
15.1.12

15.1.2

15.1.15

F (a, l-a;

sin [(2a- 1) z] Q; sin2 z)=~--1)~

15.1.16

F(a, 2-a;

sin [(2a-2) z] Q; sin2 z>= (a-l) sin (22)

15.1.17

F(-a,

-!-

2k

r(c)

(a+m+l,

Elementary

(For cases involving F(l,

15.1.3

Cases

b+m-kl;

of

Gauss

m+2;

Series

higher functions

1; 2; z)=-z-1

15.1.4

F(+, 1; Q; ,z?)=&-’

15.1.5

F(+, 1; Q; -z~)=z-~

2)

In (l-z)

15.1.13

F(a, ++a; 1+2a; ~.)=2~“[1+(1--2)~]-~” =(l--z)+F(l+a, $+a; 1+2a;

F(a, $+a; 2a; z)=22a-1(1-z)-~[l+(1-z)~]1-2a

a; 3; sin2 z)=cos (2~)

15.1.18

F(a, 1 -a; 4; sin2 z) =

15.1.19

F(a, $+a; 3; -tan2 Special

Values

of

co9 [@a-l)z] cos z

z)=coP the

z cos (2a.e)

Argument

15.1.20

arct.an z F(a, b; c; l)=

F(i, ‘See

4; Q; z’)=(l-z2)*F(1, page II.

556

1; Q; .z~)=z-~ arcsin z

z)

15.1.14

In (E)

15.1.6

0 c,,

~(1+2’)-~{[(1+22)~+2]2~-~+[(1+~2)~-2]2~-~}

see 15.4.) *

‘i, \\

+; -z2)=

F(a, b; c; z)=

(&+1(b) n+l p+lF (m+l>! Special

F(a, l-a;

( [y (,!i.i

r(c)r(c--a---b) r(c-a)r(c--b) (c#O, -1, -2,

. . .) 9?(c-a--b)>O)

WPEROE~METRE

15.2.

15.1.21

(l+a--b#O,

-1,

-2,

Differentiation Relations

;

15.2.1

F(u, b; a-b+2;-1)=2-“7r”2(b-l)-‘r(a--b-+2)

1

-1,

(a--b+zzo,

-2,

Formulas for Contiguous

and Gauss’ Functions

Differentiation

Formulas

F(u, b; c; z)=f

F(uf1,

b+l; c+l;

z)

F(u+n,

b+n; c+n;

z)

. . .)

15.1.22

1 1 r(3a>r(g+3a-b)-r(3+6a>r(l+~~--2;j [

557

mcmo~s

15.2.2

-$ F(u, b; c; z)=w

. . .) 15.2.3

F(l, a; a+l;

15.1.23

-1)=3a[lC1(~+3u)-~(~u)] .$

[Za+n-l F(u, b; c; z) I= (a)&-‘F(ufn,

b; c; 2)

15.2.4

(+u++p+~#O,

-1,

-2,.

. .)

15.1.25 F(u,

-$ [z+F(u,

b; c; z>l= (c-n),z”-“-‘F(u,

b; c-n;

z)

15.2.5

3)=2~(a--)-‘r(l+~u+~ib)

b; +u+g+i;

g

{[r(6~>r(B+3b)l-~-[r(3+4u)r(3b)i-~:I (~(u+b)+l#O,

-1,

[Zc-atn-l

-2, . . .)

15.1.26

(l-z)“+“-“F(a,

b; c; z)]

= (~-u),&-=-~(1-,z)“+~-~-“F(a-n,

b; c; z)

15.2.6

F(u, l-u;

b; *>=

2?&(b)

[r(;a+$b)

g

r (~+++~a)]-1 (b#O, -1,

a+b-cF(u, b; c; z)]

[(l--z)

-2, . . .)

-“-“;;;~-““’

(l-z)=+--“F(q

b; c+n; z)

15.1.27

I--

FU, 1; a+l; 3>=4lt(3+%4-e41 (a#-1,

15.2.7

-2,

-3,.

. .)

g

[ (1- z)=+“-~F(u, b; c; z)]

25.1.28 ,.-

-(-l>“(Mc-b>n (c)n

e, a; a+l; 3>=~-‘a[~(3+~ua>-~(3a>l (a#-1,

-2,

-3,.

. .)

b; cfn;

z)

b;c-n;

z)

c-n;

z)

15.2.8

15.1.29 F (a, t-ta;

(l-z)=+F(u+n,

g-%;-+>=(#-2a

-$ [z”-‘(l-z)~-“+“F(u,

;g;g; (g-2uf0,

-1,

-2,.

. .>

15.1.30

6;~; z)]

=(c---~),,~~-~-~(l-~)~-~F(u-n, 15.2.9

g (g+gu#o,

-1,

-2,

. . .>

[~~-‘(l--z)=+~-~F(u,

=(c-n),zc-“-yl-,z)

b; c; z)] a+b-c-nF(u-n,

b-n;

15.1.31

F (a, &a+&; #.~a+#; ef*i3)

Gauss’

(+a# -S,--%--+f, . . J

Relations

for

Contiguous

Functions

The six functions F(u f 1, b; c; z), F(u, f3f 1; c; z), F(u, b; cf 1; z) are called contiguous to F(a, b; c; z). Relations between F(a, b; c; z) and

558

HYPERGEOMETRIC

any two contiguous functions have been given by of these relaGauss. By repeated application tions the function F(a+m, b+n; c+l; z) with integral 171,n, Z(c+Z#O, -1, -2, . . .) can be expressed as a linear combination of F(a, b; c; z) and one of its contiguous functions with coefficients which are rational functions of a, b, C, Z.

FUNCTIONS

15.2.21

[a-l-(c-b-l)z]F(u,

+(c-u)F(u-1,

b; c; 2)

-(C-l)(l-z)F(a,

b; c-l;

(c--a)&-1,

b; c; z)+(2a--c--az+bz)F(a, +a(z-l)F(ufl,

b; c; 2) zq c; z)=O

15.2.11

c; z)+(2!J-c-bz+uz>F(u, b; c; 2) +b(z-l)F(u, bS1; c; z)=O

c(c-l)(z-l)F(u, b; c-l; 2) Sc[c-l-(2c--a--b-l)z]F(u, +(c-u)(c-b)zF(u,

l3; c; 2) 6; c+1; z)=O

6; c; z)---bc(l-z)F(u,

+u(l-z)F(u+1,

b; c; 2.)=o

15.2.14

b; c; z)+bF(u, b+l; -(C-l)F(u,

c; z)+u(l-z)F(u+1,

c; z)=O

b; c; 2)

-(c-b)F(u,

c; 2) b; c-l;

z)=O

15.2.25

c(l-z)F(u,

b; c; 2)-c&,

k-1;

c; 2)

+ (c-a)xF(u,

6; c+1;

z)=O

[b-1-(c--a-l)z]F(u, b; c; 2) +(C-b)F(u, b-1; c; 2) -(C-l)(l-Z)F(U,

b; c; 2) -bF(u, b+l;

15.2.15

(c-u4)F(u,.b;

z)=O

15.2.26

b; c; 2)

-(c-u)F(u-1, 6; c; z)+uF(u+l,

6; c+1;

15.2.24

* 6; c; 2)

c; z)=O

?J+1; c; 2)

+(c-u)(c-b)zF(u,

15.2.13

[C-2u-((b-u)z]F(u,

b-l;

15.2.23

(c--b-l)F(u,

15.2.12

c; 2)

-(C-b)F(u, c[b-(c-u)z]F(u,

(c--b)F(u, b-l;

z)=O

15.2.22 [c--2b+(b--a)z]F(u, 6; c; 2) +a(l-z)F(u, b+l;

15.2.10

(b-u)F(u,

b; c; 2)

b-l;

b; c-l;

z)=O

15.2.27

c[c- 1 - (2c--a--& l)z]F(u, +(c-u)(c-6)zF(u,

6; c; 2) 6; cfl;

-C(C-l)(l-z)F(u,

c; z)=O

2) b; c-l;

z)=O

15.2.16

c[u-(c-b)z]F(u,

6; c; z)-uc(l--z)F(uj-1, +(c-u)(c-b)zF(u,

b; c; 2) 6; c+1;

z)=O

15.3. Integral

15.2.17

(c--a-l)F(u,

b; c; z)+uF(u+l, -(C-l)F(u,

Integral

b; c; 2) b; c-l;

z)=O

15.2.18

(c--a--b)F(u,

6; c; z)-(c-u)F(u-1,

b; c; 2) b+l;

c; z)=O

15.2.19

+(C--6)F(u, 6; c; z)--F(u-1,

b; c; 2) b-l;

c; z)=O

b; c; 2)

+(c-b>zF(u, *seepagrn.

Transfor-

Representations

15.3.1

w Jxb)r(c-bb)

1 s cl

t~-‘(l---~)~-“-‘(l-Lz)-“dt

(9c>g’b>o) b; c; z)-(c-u)F(u-1,

15.2.20

a---)F(a,

and

F(u, 6; c; z)=

+b(l--z)F(u, (b-u)(l-z)F(u,

Representations mation Formulas

6; c+1;

z)=O

The integral represents a one valued analytic function in the z-plane cut along the real axis from 1 to 09 and hence 15.3.1 gives the analytic continuation of 15.1.1, F(u, b; c; z). Another integral representation is in the form of a Mellin-Barnes integral

HYPERGEOMETRIC

559

FUNCTIONS

r(c) +

r(a)r(b)

Here

-r<arg(-z)
r(b+s)

Linear

Transformation

Formulas

From 15.3.1 and 15.3.2 a number of transformation 15.3.3

F(u, b; c; z)=(l--~)~-=-~F(c-u,

for F(a, b; c; z) can bederived.

c-b;, c; z)

15.3.4

=(I-z)-‘F(a,

c-b; c; ;>)

15.3.5

=(I-z)-PF(b,

c-u; c; -;-.&)

r(c)r(c-a-b)

15.3.6

formulas

F(a, b; a+b-c+l;

?(c---a)r(c-b)

1-z)

r(c)r(a+b--c) r (4 r(b)

+(1-$-“-b

F (c-u, c-b; c-a-b+l;

1-z) (larg Cl--z)l
=;;;;;I;;

15.3.7

(--z)-T(u,

l-cfa;

+$&s =(1--z)+

15.3.8

l-b+a;

(-z)-bF(b,

;;;;;-;

F(u, c-b; a-b+l; .

br (c) r (a- b) +(1--z)- r’(a)r(C-b) F r(c)r(c-a-b)

15.3.9

=r(c--@r&-b)

z-” F

a u-c+l.

a+b

;) l-c+b;

I---a+b;

;)

+-)

b, c-a; c+l.

’ ) ) +r(C)r(a:+b-c) ______ (l--~)~-=-~z=--~ rb)r@>

b--a+l;

&

(kg

‘5*3*‘0

WSb) 2 Tk&(b). ‘(a, b;a+b; 2)=r(a)r(b) n=O (nv)2

(m=O,

Cl---z)l
1 F(c-a,

l--a;

c--a-b+l;

(larg zl<~,

Each term of 15.3.6 has a pole when c=a+b:km,

(larg (--4I

1-i)

larg Cl-z>l
1, 2, . . .); this case is covered by

[~(n+l)-~(u+n)-~(b+n)--In

(l--z)l(l-~)~

(lariz W-.4<*, P--21<1) Furthermore 15.3.11

for m=l,

F(% b; a+b+m;

2, 3, . . . rh>W+b+m) Z)=r(a+m)r(b+nE-j

m-l (Mb), go n!(l-m),

-IL(n+m+l).trl(a+n+m)+~(b+n+m)l

(1-z)“

(larg Cl--z)l
HYPERGEOMETRIC

560 15.3.12

F(a, 6; a+ b- m; z)= r(m>r(a+b-ml W4M4

FUNCTIONS

(l--z)-m

(--l)‘Ua+b-mm) - r(a-m)r(b-m)

?iC (a-mL(b-m), n!(l-mm), n-0

2 (aL@)* n10 n!(n+m)!

(l-.#@

(1-z)” (l--z)-#(n+l)

-rL(n+m+l)+~(a+n)+rL(b+n)l (Ia% o--z)l<*,ll-zl
each term of 15.3.7 has a pole when 5 =a f m or b -a=

15.3.13

F(a, a; c; z) = rca)r,(~&

(-2)-a

2

ca)n’~$+a)n

f m and the case is covered by

P[ln

(-2)+2$(n+l>+(a+n)+(c-a-n)]

(lax (-z>Il, (c-a)+4 fl, f2,. . .I The case b-a=m, 15.3.14 F(a, a+m;

(m=l,

2, 3, . . .) is covered by

c; z) = F(a+m,

a; c; z)

r(C>(-2)-a-m -g (4n+m(l-c+aL+m 2--))[In

=r(a+m)r(c-a)

n!(n+m)!

n4

(-z>+N+m+n)+$(l+n)

r(c) m-l r (m-n)(a), -rl(a+m+n)-~(c-a-m-n)l+(-z>-’ r(a+m) ~onlr(e-a-nn) 2-a (larg(--z)l<~Izl>l,(c--a)ZO,~1, f2,...)

The case c-a=O, -1, -2, . . . becomes elementary, 15.3.3, and the case c-a=l, obtained from 15.3.14, by a limiting process (see [15.2]). Quadratic

Transformation

2, 3, . . . can be

Formular

If, and only if the numbers *(l-c), &ta-b), f (a+b-c) are such, that two of them are equal The basic formulas or one of them is equal to 3, then there exists a quadratic transformation. are due to Kummer [15.7] and a complete list is due to Goursat [15.3]. See also [15.2]. 15.3.15 F(a,l;2b;z)=(l-z)-*F($a,b-$a; 15.3.16

=(l-+z)-“F($a,

15.3.17

-(#+3&i)-“F

15.3.18

=(l--z)-4°F

b+$;4&) $+$a; b++; ~~(2-2)~~) a,a-b+3; a, 2b-a;

b+&

bS3; -

(1--111’--2)2

( 15.3.19

15.3.20

F(a, a+3; c;z>=(++&‘i%>-“F =(lf&)-lOF

2a, 2cc-c+l; (2a,c-3;

15.3.23

25; a+b+$;

=(++&/i=i)-”

c;

)

l-;-G 1+&-z

2c-1; =ks>)

2a, 2c-2a-1;

15.3.21 15.3.22 F(a,b; a+b+i;z)=F@a,

4Ji--z

c;

G-1 26

>

i-i=)

F(%, a--b++; .,b+:;E;:)

HYPEROEOMETRIC

15.3.24

F(a, b; a+b-4;

z)=(l--z)-)F(2a--l,2b--l;a+b--3;3-~J1--z) =(1-2)-f

15.3.25 15.3.26

F(a, b; a--b+l;

(&+gilyT(2a-l,

z)=(l+z)-“F($u,

15.3.27

=(1*qz)-“F(

15.3.28

=(l-z)-“F($a,

15.3.29

F (a, b; &a++b++;

=F(&z, F(a, l--a;

a;, a--b+*;

a+b--:;g+J

2

42(1+~)-~)

2u--2b+l;

*4Jz(lf$)-2)

~~-b+4;a--b+1;-42(1-2)-~) -“F (h,

&+$;

$b; ~~t+~b+~;

c; 2)=(1-2>~-‘F(~c-&7a:,

ha+@++;

B)

42-423

~c++-~;

=(1-2)~-‘(1-22)“-‘CF

15.3.32

a--d++;

$a++; a--b+l;

z)= (l-22)

15.3.30 15.3.31

561

FTJ-NCTIONS

c; 42-4z2)

($c-$a,

&-+a+&

c; (4,s42)(1-22)-y

Cubic transformations are listed in [15.:2] and [15.3]. In the formulas above, the square roots are defined so that their value is real and positive O<,z
when

Cases of F(a, b; c; 2)

Polynomials

When a or b is equal to a negative integer, then 15.4.1

b; c; z)=C m F,,,.,.c n

F(--m,

$

n-0

This formula 1.5.4.2

is also valid when c=-m-Z;

F(-m,

b;-m-l;

Some particular

m, Z=O, 1, 2, . . .

z)=~$ :smm)lt-(lb::’5

cases are

15.4.3

F(-n,

n; 3; z)=T,(l-22)

15.4.4

F(-n,

n+1;

15.4.5

F -22, n+2a; a+f; 2 =$&

15.4.6

F(-n,

1; z)=Pn(l--25)

>

cr+l+@+n;

a+l;

z)=&-

Cfp)(:l-2x)

n

P$fl)(l-22)

Here T,,, P,, CF’, Pg*@j d enote Chebyshev, Legendre’s, spectively

(see chapter

Gegenbauer’s

and Jacobi’s polynomials

Legendre

Functions

Legendre functions are connected with those special cases of the hypergeometric a quadretic transformation exists (see 15.3). 15.4.7 15.4.8

F(a, b;2b; z)=22b-‘I’(&+b)z+-b (l-z)+cb-u-*)P:-:-t [(1-;)(l-d-q =22bT-#

re-

22).

r(3+b)

r(2b---a)

15.4.9 F(a,b;2b;-~)=2*~7r -4 !$fd

Z-b(l-_Z)f(b-a)e~‘(a-b)

f’f~t

,-b(]~+Z)W-a)e -**(a-b)P;zf

function

for which

562

HYPERGEOMETRIC

15.4.10

F(a, a+3;c;

FUNCTIONS

z)=2”-‘r(c)z’-~~(l-z)~~-=-~~~~~~[(l-z)-~] (larg zl
15.4.11

F(u,u+~;c;2)=2”-‘r(c)(--z)~-~~(l--z)~~-”-’P:,4,[(1--2)-~]

15.4.12

F(a, b; &+b+*;

z)=20 +“-+r(~+u+6)

z not between 0 and - OD) (-- O3
(-z)‘(‘-“-“‘~-t-~[(l--)l] (larg (-z>l
15.4.13

z not between 0 and 1)

2>=a+“-tr(~+a+b)zt’t-“-b)~~~~~[(l-Z)f]

P(&, b; a+b+i;

(O<x

15.4.14 F(&,~;a--b+i;d=r(a-b+i)2~b-r(i-2)-b~~~=

(kg O-41<

1~~z not between 0 and - W)

15.4.15 F(r,b;a--b+i;x)=r(a--b+i)(i-2)-b(-~)~b-~~b;~

(--- co<x
F(Q, 1--a;~; z)=I’(c)(--z)‘-~(1--z)~c-~~p’=,“(l-2z)

15.4.16

(larg (-z)I<1p,

15.4.17

F(u,L-u;c;2)=

15.4.18

F(u, b; $B+#+$;

15.4.19

F(~,b;$a+#b+);

15.4.20 F(a, Q;a+&$;

15.4.22

F(a,b;&

z not between 0 and 1)

r(c)~~-t~(i-~)tc-'P~~~(i-22)

(O<x
z)=r(3+~+3b)[z(z-l)]t”-a-b)~~~~~~~~(l-2~) (larg z I
Iarg (z-l)l<~,

z not between 0 and 1)

z)=r(~+3a+)b)(2-xz)t(1--.-b)~Pt~~~~:~(1-22)

(O<x
2)=2” +b-tr(u+6-~)(-z)t’t-“-b’(i-Z)-t~b~~~-bf[(l-Z))]

(I arg (-z)l
larg (l-z)l
I arg(l--z)l
.%?[(l-z)+]>O,

z not between 0 and 1)

z)=20+b-tr(u+b-~)zr(i-a-b’(1-z)-tP1-oa-~[(l--z)l]

(O<x
2)=?r-12a+b-fr()+a)r()+b)(2-i)I(t-o-b)[Pt=~~~~(Z))+~~~~~~(-2))1 (larg zl
15.4.23 F(a,b;~;~)=ct~+~-T()+a)r(3+b)(i--z)

z not between 0 and 1)

'('-"-"'[P~~~-t(x')+Pta--Ob--~(-x~)]

15.4.24 F(e, b; $;-z)=r-W-b-1r($+a)r(~-lJ)(2+1)-@-+be*i~

(O<x
(*+) {~:;;-,[~t(l+~)-t]

+z%;-l[--z’(l

+zr+1j

( f according as Y,zSO, z not between 0 and OJ)

*15.4.25 F(u, 6; 3;-x)=~-)20-b-1r(~+a)r(i--b)(l+s)-~-~b{~:;;_l[~~(l+x)-f]+~:;:-,[-X1~2(1+2)-t]} CO<%
F(a,b;4;z)=--?r-t2”+b-~r(a-~)r(b--3)x-l(l-x)t(;-a-b)

15.5. The Hypergeometric

The hypergeometric 15.5.1

z(l-2)

*see page Il.

differential

$$+[C-(a+6Sl)zl

equation $zbw=O

{~~--:-~(~1)--~a-t-~(-~t)

Differential

Equation

}

(O<x
HYPERGEOMETRIC

FUNCTIONS

563

has three (regular) singular points z=O, 1, 03. The pairs of exponents at these points are 15.5.2

p;pi=o, l-c,

respectively. the following

p$=O,

c-a-b,

~&)=a,

The general theory of differential cases.

b

equations

of the Fuchsian type distinguishes

between

a-b is equal to an integer. Then two linearly independent of the singular points 0, 1, m are respectively

solutions

A. None of the numbers c, c-a-b;

of 15.5.1 in the neighborhood

b; c; z)=(l-z)C-a-bF(c--a,

15.5.3

wl(,,)=F(u,

15.5.4

~~~~~=z~-~F(u--e+l,

15.5.5

wIC1)=F(a, b; a+b+l-c;

15.5.6

wZ~l~=(l-z)C-a-bF(c-b,c-a;c-a--b+l;

15.5.7

?blCII))=~-BF(u,

15.5.8

toa

b-c+l;

c-b;

c; z)

z)=zl-c(l-z)C-“-bF(l-a,

2-c;

l-z)=&cF(l+b-c,

l-b;

l+a-c;

2-c;

a+b+l-c;

z) l-2)

l-z)=zl-C(l-z)c-“-“F(l-a,

1-b;c-a-b+l;

a-c+l;

a-b+l;

z--l)-lzb-C(z-l)C-“-bF(l-b,

c-b;

a-b+l;

z-‘)

=z- “F(b, b-c+l;

b-a+l;

z--l)=za-c(z-l)c-a-b~(l-a,

c-u;

b-&l;

z-l)

l-2)

The second set of the above expressions is obtained by applying 15.3.3 to the first set. Another set of representations is obtained by applying 15.3.4 to J5.5.3 through 15.5.8. gives 15.5.9-15.5.14. 15.5.9

a, c-b; c; -)=(l-,z)-bF(b, zf1

~~~,,~=(l-z)-“F

15.5.10

w~~~~=z~-~(~--z)~-~-~F

15.5.11

WluI=PF(a,

a-c+l,

a-c+l;

1-b;

a+b-c+l;

c-a; 2-c;

5

l-z-l)=z-bF(b,

>

This

c; --&)

=z l-c(l--z)c-b-lF b-c+l;

b-c+l,

l-a;

a+b-c+l;

2--e;A -

>

1-z-l)

15.5.12 wm=2

15.5.13

=-c(,-z)-bF(c-u,

l-u;

w lC~~=(z-l)-‘F(a,

c-b;

l-z-l)=zb-C(l-z)c-a-bF(c-b,

c-u-b-t-1;

a-b-l-1;;

&)=(z-l)-bF(b,

c-a;

l-b;

b-a+l;

c-a-b+l;

l-z-‘)

$--)

15.5.14 w4(a.)=2 l-c(~-l)C-a-lF

(a-+1,

1-b; a--1+1;

$--)=zl-C(z-l)C-b-lZi’

(b-c+l,

l-a;

b-a+l;

$-)

15.5.3 to 15.5.14 constitute Kummer’s 24 solutions of the hypergeometric equation. The analytic continuation of wl ,z(O,(z) can then be obtained ‘by means of 15.3.3 to 15.3.9. Then one of the hypergeometric series for B. One of the numbers a, b, c-a, c-b is an kteger. instance wl ,2(0), 15.5.3, 15.5.4 terminates and the corresponding solution is of the form 15.5.15

w=zQ(l-z)@p&)

where p,(z) is a polynomial in z of degree n. This case is referred to as the degenerate case of the hypergeometric differential equation and its solutions are listed and discussed in great detail in [15.2]. C. The number c-a-b is an integer, c nonintegral. Then 15.3.10 to 15.3.12 give the analytic continuation of wr ,2Co,into the neighborhood of z=: 1. Similarly 15.3.13 and 15.3.14 give the analytic continuis an integer but not c, subject of ation of wl ,2(0) into the neighborhood of z= 03 in case a-b (For a detailed discussion of all possible course to the further restrictions c-a=O, fl, f2 . . . cases, see [l&2]). Then 15.5.3, 15.5.4 are replaced by D. The number c=l. 15.5.16

wlto,=F(a,

b; 1; z)

564

HYPERGEOMETRIC

15.5.17

E.

wz(O)=F(a,

The number

b; 1; 2) In 2+g1

c=m+l,

-

m=l,

15.5.18

wIcO)=F(a,

b; m-t-1; z>

15.5.19

wZcOj=JYa,

b; m+l;

wmn

~@,)a z”[~(a+n)-9(u)+~(b+n)-~(~)--2~(n+l)+2~(2)1 2, 3, . . . . A fundamental

2) In z+gI

-

The number c= 1 -m,

15.5.20

wl(,,)=zmF(u+m,

system is

~~~~$--“d,: z-”

m= 1, 2, 3, . . . . A fundamental b+m;

l-km;

(l4<1)

GMb)n cl+mj ~nl. z”[#(u+n)-Ku)++(b+n)+(b)-+(m+l+n)

+Km+l)-~(11+1)+~(1)1-~~ I’:

FUNCTIONS

(lzl<

1 and a,b#O,

1, 2, . . , (m-l))

system is

z)

15.5.21 w2~0~=z~F(u+m,

b+m;

l+m;

2) In z+z~$~z”

cu~~)~j+J)n

n *

[G(u+m+n)-1L(u+m)+~(b+m+n)

(lz/
Differential

Equation

The hypergeometric differential equation 15.5.1 with the (regular) singular points 0, 1, Q) is a special case of Riemann’s differential equation with three (regular) singular points a, b, c

Special

and a, b#O, -1, -2, Cases

of

(a) The generalized

Riemann’s

hypergeometric

(b) The hypergeometric 15.6.5

function

0 OJ 1

w=P

0

11-c

P!(z),

co

1

3P

0

I%++ -3P

t

0

cr+a’+B+B’+r+Y’=l

-4V

r

(d) The confluent

21

J

&:(z)

(1-22)-1 1

J

hypergeometric

function

15.6.7 0

15.6.3

F(u, b; c; z)

15.6.6 w=P

The complete set of solutions of 15.6.1 is denoted by the symbol

function

1 0 c-u-b

a b

(c) The Legendre functions

15.6.2

P Function

w=p{;, ; “,.z}

15.6.4

15.6.1

The pairs of the exponents with respect to the singular points a; b; c are (IL,cr’; 6, B’; y, y’ respectively subject to the condition

. . . -(m-l))

w=P

+ku 3-U

provided

lim c+= .

Q)

c c-k

-c 0

k

z

HYPERGEOMETRIC Transformation

a z-a z-b (>(

15.6.8

k

z-c

-PiY z-b I{

’

a’

Formulas

565

FUNCTIONS for

P Function

Riemann’s

b

c

a

b

B

y

a+k

8-k-l

8’

7’

d-j-k

8,-k--l

and A, B, C, D are arbitrary constants such that AD-BC#O. Riemann’s P function reduced to the hypergeometric function is

The P function on the right hand side is G.auss’hypergeometric function (see 15.6.5). If it is replaced by Kummer’s 24 solutions 15.5.3 to 15.5.14 the complete set of 24 solutions for Riemann’s differential equation 15.6.1 is obtained. The first of these solutions is for instance by 15.5.3 and 15.6.5

15.7.

Asymptotic

15.7.2

Expansions

The behavior of F(a, b; c; z) for large (z] is described by the transformation formulas of 15.3.

F(a, b; c; z)=e-sm[r(c)/I’(c-a>]

+[r(c>lr(a)l

(bz)-“[l+O(lbz(-‘)]

e**(bz)O-“[l+O(lbzl-‘)I

For fixed a, b, z and large ]c( one has [15.8]

(-$<

w (bz)<+r)

15.7.3 15.7.1

?‘(a, b; C;

z)=d@[r(c)/r(c--a)]

(bz)-a[l+O(lbz(-l)l

+~r~~>lr~~>l~bz~b~~-c~~+O~l~~l-‘>l

F(a, b; c; z) =Cn~c~~+o(~cl-‘-‘)

C-+~< For fixed a, c, z, (c#O, -1, and large lb1 one has [l&2]

-2,

. . . , O
arg b+Gb)

For the case when more than one of the parameters are large consult [15.2].

References 115.11 P. Appell and J. Kampe de F&et, Fonctions hypergeometriques et hyperspheriques (GauthiersVillars, Paris, France, 1926). [15.2] A. Erdelyi et al., Higher transcendental functions, vol. 1 (McGraw-Hill Book Co., Inc., New York, N.Y., 1953). [15.3] E.

Goursat, Ann. 3-142(1881).

Sci. Ecole

Norm.

Sup(2)19,

[15.4] E. Goursat, Propri&& generales de l’equation d’Euler et de Gauss (Actualit& scientifiques et industrielles 333, Paris, France, 1936). [15.5] J. Kampe de Feriet, La fonction hypergeom&rique (Gauthiers-Villars, Paris, France, 1937). [15.6] F. Klein, Vorlesungen iiber die hypergeometrische Funktion (B. G. Teubner, Berlin, Germany, 1933).

566

HYPERGEOMETRIC

[15.7] E. E. Kummer, Uber die hypergeometrische Reihe, J. Reine Angew. Math. 15, 39-83, 127-172(1836). [15.8] T. M. MacRobert, Proc. Edinburgh Math. Sot. 42, 84-88(1923). [15.9] T. M. MacRobert, Functions of a complex variable, 4th ed. (Macmillan and Co., Ltd., London, England, 1954). [15.10] E. G. C. Poole, Introduction to the theory of linear differential equations (Clarendon Press, Oxford, England, 1936).

FTJNCI’IONS [15.11] C. Snow, The hypergeometric and Legendre functions with applications to integral equations of potential theory, Applied Math. Series 19 (U.S. Government Printing Office, Washington, D.C., 1952). [15.12] E. T. Whittaker and G. N. Watson, A course of modern analysis, 4th ed. (Cambridge Univ. Press, Cambridge, England, 1952).

16. Jacobian

Elliptic Functions Functions L. M. MILNE-THOMSON

and Theta

1

Contents Mathematical Properties .................... 16.1. Introduction ..................... 16.2. Classifiication of the Twelve Jacobian Elliptic Functions . . 16.3. Relation of the Jacobian Functions to the Copdlar Trio . . 16.4. Calculation of the Jacobian Functions by Use of the Arith16.5.

16.6. 16.7.

16.8. 16.9. 16.10. 16.11. 16.12. 16.13. 16.14. 16.15. 16.16. 16.17. 16.18. 16.19.

16.20. 16.21.

16.22. 16.23. 16.24. 16.25. 16.26. 16.27. 16.28. 16.29. 16.30. 16.31.

16.32.

Page 569 569

........... metic-Geometric Mean (A.G.M.) Special. Arguments .................. Jacobian Functions when m=O or 1 ........... Principal Terms ................... ................. Change of Argument ..... Relations Between the Squares of the Functions Change of Parameter .................. ... Reciprocal Parameter (Jacobi’s Real Transformation) Descending Landen Transformation (Gauss’ Transformation) ....................... Approximation in Terms of Circular Functions ...... ........... Ascendling Landen Transformation .... Approximation in Terms of Hyperbolic Functions ...................... Derivatives Addition Theorems .................. .................. Double Arguments Half Arguments ................... Jacobi’s Imaginary Transformation, ........... ................. Complex Arguments Leading Terms of the Series in Ascending Powers of u. .. Series IExpansion in Terms of the Nome p ........ ... Integrals of the Twelve Jacobian Elliptic Functions. Notation for the Integrals of the Squares of the Twelve Jacobian Elliptic Functions ............. Integrals in Terms of the Elliptic Integral of the Second Kind1 ....................... Theta .Functions; Expansions in Terms of the Nome p . . Relations Between the Squares of the Theta Functions . . Logarithmic Derivatives of the Theta Functions ..... . . Logarithms of Theta Functions of Sum and Difference Jacobi’s Notation for Theta Functions ......... Calculation of Jacobi’s Theta Function O(ulm) by Use of the Arithmetic-Geometric Mean ...........

1 University Standards.)

of .Arizona.

(Prepared

under

contract

with

the National

Bureau 567

570 570 571 571 571 572 572 573 573 573 573 573 573 574 574 574 574 574 574 575 575 575 575 576 576 576 576 576 577 577 577 of

568

JACOBIAN

ELLIPTIC

FUNCTIONS

AND

THETA

FUNCTIONS

Page

16.33. Addition of Quarter-Periods to Jacobi’s Eta and Theta Functions . . . . . . . . . . . . . . . . . . . . . 16.34. Relation of -Jacobi’s Zeta Function to the Theta Functions. 16.35. Calculation of Jacobi’s Zeta Function Z(ulm) by Use of the Arithmetic-Geometric Mean . . . . . . . . . . . . . 16.36. Neville’s Notation for Theta Functions . . . . . . . . . 16.37. Expression as Infinite Products. . . . . . . . ‘. . . . . 16.38, Expression as Infinite Series . . . . . . . . . . . . . .

579 579

References

. . . . . . . . . . . . . . . . . . . . . . . . .

.

581

Theta Functions . . . . . . . . . . . . . . . . . . .

582

?Y,(rO\cYO), &iz

. . . . . . . . . . . . . . . .

579 579

Use and Extension of the Tables . . . . . . . . . . . .

16.1.

. . . . . .

578 578

Numerical 16.39.

Table

Methods

577 578

s,(Eydq

tYn(eO\aO),Jseccy tY,(E;\op) ~t=O~(5~)85~, e, e1=00(50)900, 9-10D Table

16.2.

Logarithmic

Derivatives

of Theta Functions . . . . . .

584

& In 8.(u) =f(~~\oP) $ In GC(u)= -f(e\c~‘) &In

Sn(~)=g(to\aP)

& In ad(u) = -g(ET\OP) c~=O~(5~)85~,t, el=00(50)900,

5-6D

The author wishes to acknowledge his great indebtedness to his friend, the late Professor E. H. Neville, for invaluable assistance in reading and criticizing the manuscript. Professor Neville generously supplied material from his own work and was responsible for many improvements in matter and arrangement. The author’s best thanks are also due to David S. Liepman and Ruth Zucker for the preparation and checking of the tables and graphs.

16. Jacobian

Elliptic

Functions

Mathematical Jacobian

Elliptic

Properties

Functions

16.1. Introduction

A doubly periodic meromorphic called an elliptic function. Let m, ml be numbers such that

function

is

We call m the parameter, ml the complenientary parameter. In what follows we shall assume that the parameter m is a real number. Without loss of generality we can then suppose that O_<m
K and iK’ by

16.1.1

de iK’(m) =iK’=lis0*/a (l-mlsim so that K and K’ are real numbers. K is called the real, iK’ the imaginary quarter-period. We note that 16.1.2

K(m)=K’(mI)=K’(l-m).

the pattern sides.

The Jacobian elliptic function pq u is defined by the following three properties. (i) pq u has a simple zero at p and a simple pole at q. (ii) The step from p to q is a half-period of pq u. Those of the numbers K, iK’, K+iK’ which differ from this step are only quarter-periods. (iii) The coefficient of the leading term in the expansion of pq u in ascending powers of u about u=O is unity. With regard to (iii) the leading term is u, l/u, 1 according as u=O is a zero, a pole, or an ordinary point. Thus the functions with a pole or zero at the origin (i.e., the functions in which one letter is s) are odd, and the others are even. Should we wish to call explicit attention to the value of the parameter, we write pq (ulm) instead of pq u. The Jacobian elliptic functions can also be defined with respect to certain integrals. Thus if 16.1.3

We also note that if any one of the numbers m, ml, K(m), K’(m), K’(m)/K(m) is given, all the rest are determined. Thus K and K’ can not both be chosen arbitrarily. In the Argand diagram denote the points 0, K, K+iK’, iK’ by s, c, d, n respectively. These points are at the vertices of a rectangle. The translations of this rectangle by M, giK’, where x , ccare given all integral values positive or negative, will lead to the lattice .S

Let p, q be any two of the letters s, c, d, n. Then p, q determine in the lattice a minimum rectangle whose sides are of length K and K’ and whose vertices s, c, d, n are in counterclockwise order. Definition

m+ml=l.

We define quarter-periods

and Theta Functions

.S

.n

Zi

.n

.S

.C

.S

:: .C

.n

.d

.n

.d

being

repeated

indefinitely

on all

u=

the angle cpis called the amplitude 16.1.4

p=am

u

and we define 16.1.5

sn u=sin

ip, cn u=cos

‘p, dn u=(l-m

sin2 ‘p>1/2=A((p),

Similarly all the functions pq u can be expressed in terms of cp. This second set of definitions, although seemingly different, is mathematically equivalent to the definition previously given in terms of a lattice. For further explanation of notations, including the interpretation, of such expressions as sn (V\(Y), cn (ulm), dn (u, k), see 17.2. 569

JACOBIAN

570

ELLIPTIC

16.2. Classification

FUNCTIONS

of the. Twelve According

Pole iK’

to Poles

Pole K

Pole K+iK’

AND

THETA

Jacobian and

FUNCTIONS

Elliptic

Functions

Half-Periods

Pole 0

-Half period iK’ Half period K+iK’ Half period K

cd u

sn u -cn u

sd u

dn u

nd u

dc u -nc u -SCu

ns u

Periods 2iK’,

4K+4iK’,

4K

ds u

Periods 4iK’, 2K+2iK’,

4K

cs u

Periods 4iK’,

2K

4K+4iK’,

-

The three functions in a vertical column are copolar. The four functions in a horizontal line are coperiodic. Of the periods quoted in the last line of each row only two are independent.

FIGURE

FIQURE 16.1. Jacobian

16.3.

elliptic junctions

Jacobian elliptic junetiom SCu, es u, cd u, dc u

sn u, cn u, dn u m=l

16.3.

2

The curve for on (~13) is the boundary between those which have an inflexion and those which have not.

0

Relation of the Jacobian Functions Copolar Trio sn u, cn u, dn u

16.3.1

cd u=E

& u=dn” cn u

16.3.2

sd u=E

nc u=-

16.3.3

nd u=-&

snu sc u=cn

r"

K

ZK

-.5 -

3K

1

cnu

I-ISu=gn

to the

1

& uzdnu sn u cnu ca U=sn

4K

And generally if p, q, r are any three of the letters s, c, 4 n, FIGURE

16.2.

Jacobian elliptic junctions

ns u, nc u, nd u

16.3.4

provided that when two letters are the same, e.g., pp u, the corresponding function is put equal to unity.

JACOBIAN

ELLIPTIC

FUNCTIONS

16.4. Calculation of the Jacobian Functions by Use of the Arithmetic-Geometric Mean (A.G.M.)

For the A.G.M. scale see 17.6. To calculate sn (ulm), cn (ulm), form the A.G.M. scale starting with 16.4.1

and dn (ulm)

AND THETA

571

and then compute successively p+,, lpi, p. from the recurrence relation 16.4.3

sin (24+,-&=~

pNm2, . . .,

sin q*,

Then 16.4.4

sn (ujm)=sin

ao=l, bo=Jm1, co=&,

terminating at the step N when cN is negligible to Find lpN in degrees where the accuracy required. (oN=2NaNuIso0 r

16.4.2

FUNCTIONS

(potcn (ulm)=cos

‘OS PO cos (Cpl-PoPg)’ From these all the other functions mined.

dn (ulm)=

can be deter-

16.5. Special Art guments u

*

cn u

Fm u

dn u

16.5.1

0

0

1

1

16.5.2

+K

1 (l+m:l’)“’

mP (l+mll’*)l’a

m$14

16.5.3

K

1

0

rn+l¶

16.5.4

$K')

im-‘I’

(1 + m1’*)1/2 mll’

(l+ml’*)L/*

16.5.5

$K+~K!)

2-‘/z7+/‘[(l

2 l”(l-i) ( >

(

+,121/1)1/* +.~(l--ml”)l”]

16.5.6

K+;(iKt)

16.5.7

iK'

co

16.5.8

f~+iKt

(1

16.5.9

K+iK'

m-m

-i

m-1”

m,1’2)

-i

-1’2

(!-.IlE$)“’

(j-+$)“’

-

0

when m=O or 1 =

16.6. Jacobian Functions r

m=O

m=l

.16.6.1 16.6.2 16.6.3

‘See page II.

If&l “‘[(I + m,l/2)1/* > -i(l-mmlUZ)l/2]

co

-

a

sin u co9 IL 1

tanh u sech u sech u

:t % 16:6:6

cd (ulm) sd (ulm) nd (ulm)

COBu sin 2~ 1

1 sinh u cash u

16.6.7 16.6.8 16.6.9

do (u m) no (u m) 80 (ulm)

set u set u tan u

1 cash u sinh u

16.6.10 16.6.11 16.6.12

:ns (u m) (ds (u Im) (cablm>

csc u csc u cot u

,20th u ,csch u ,osch u

16.6.13

i~1x3 (ulm)

IL

Ited u

im,l/4

n s3-

n SP n sun asn 3u n opn pun Ps n pa-n upn u3 n us-

n s3-

n P~~IW+n ~s~1wW.2 n PAVES

n spn su n asn aun 3~

n UP~I~-W n uw+~~-W n usyltu-

n sa,,,~hu+

n pun psn ~3 n upn us-

1 SPzlr-%lI-wn suzl,-ULn 3szl,hu~ n 3uzll-~zl,k+-n

n us

3pg~l-m

n up+-n nwcn wow

n PUT n ~w+

n P+~U n 3s? n cm,,,-uq n ~P~I~-w

n fpzn wl-vn suzI,--u1

n s3-n SP n su

n s3

n 63

n spn su-

n spn su-

n asn x5-n 3p-

n 3s n aun 3p-

n 38 n aun 3p-

n PU n PS n p3-

n PU n psn pa-

n PU n psn p3-

n UP n u3n us

n UP n us--n us-

n UP n mn us-

n 3sz11bu n 3uz,,h4d

n 3szI,Qun 3uzllbk-

n 3p-

n 3~ n s3zll--Iau

n ~3~,~--1tu -

n fwr-~~

n spzil-l~

n su

n UP~I~-~~ n uIozIl--Itu n us

n su

n UP~I~-~UJ

I a u311,1~~zu-

n us

n dude n PW~

n 3~ n ~3~,~~hu n spz~I-r~

n UP~I,-~~ n ~3~~~--1au n usn PU~-J

n PW~

n psd~--

n p3 -

n ~3

-

n su-

n PUPIL

n spn su-

n 63 n sp n su

n asn 311 n 3~

G su

21’8’91 11’8’91 07’8’91

n 3s n 3u n 3~

38 3U

3P

6’8’91 8’8’91 L’S’91

n PU n @sn ~3

n Pu n PS n ~3

PU ;:

9’8’91 S’S’91 V’S’91

n UP n 013 n us -

n UP n u3 n us

UP

6’8’91

U3 us

Z’S’91 1’8’91

n SD-

n x3zl,~~n 3uz11bu

n ~3

-,szz +

az+-n

m+n+n

I -

n2zfn

-

n-nz

,n!+n

az-n

az+n

n-n

n--n

n-

n+n

=

n

--

n 53

n n n n

SP su 38 311

ZT’t’91 wL’9 t OI’L’91 6’L’9T 8’L’9T

nap

L’L’91

npu n PS n ~3 nup n u3 n us

9’t’91 S’L’91 p’L.91 s-L.91 Z’L’91. I’t’9T.

-= *pagddns aq 03 svq (“2-n) JOS!MP aq? c~eq$smam + pus pa!lddns aq 09 s’~q (“X--n) .ro~m3 aql ~‘t3q$smam X aJaqhi ‘sur.103asaqq sangI gsg &n~o~~o3 aq& -n bd 30 alod ‘13JO ‘oJaz B ‘gu!od Lmup~o 11’13 s: Lx S’B Z?uyp.~oom(‘x-n)-+3 ‘(‘x-n) x& ‘8 sm.103 aql 30 auo ssq pus ma2 pd!mvd aq$ panw st uo!susdxa aqq 30 rnsa%ys.~y aq$ ‘,xz+x ‘,x! ‘;y $0 30 auo s! “x alaqM ‘&-n) 30 maMod Bmpuaoss uy papusdxa so n bd uo~~oun3Di$dma aq? uaq& smra& ~dpupd

‘t-91

JACOBIAN

16.9. Relations

ELLIPTIC

Between the Functions

16.9.1

--dn2u+ml=-m

16.9.2

-mind2u+ml=-mmlsd2u=m

:FUNCTIONS

Squares o:f the

cn2u=m

sn2u-m

AND THETA

16.12.1

573

FUNCTIONS

/quay>

a’&*’

then 16.12.2

sn (ul m) _ (1+c(‘/~) sn (4~)

16.12.3

cn (ul m) =

In using the above results remember that m+ml=l. If pq u, rt u are any two of the twelve functions, one entry expresses tq2u in terms of pq2u and another expresses qt2u in terms of rt%. Since tq% . qt2u= 1, we can obtain from the table the bilinear relation between pq2u and rt2u. Thus for the functions cd u, sn u we have

16.12.4

* t”l m)= (1fpi/2)

16.93

When the parameter m is so small that we may neglect m2 and higher powers, we have the approximations

16.9.3 16.9.4

d2u-m

mlsc2u+ml=m1nc2u=dc2u-m cs2u+ml=ds2u=ns2u-m

nd2u=‘-

m ca2U, dn2u=l-m ml

sn2u

and therefore 16.9.6

Ca2U)(i-m

(l-m

sn*u) =ml.

If m is a positive number,

16.10.2

,=A 1+

Pl=l+mf

v=;;;i

Note that successive applications can be made conveniently to find sn (ujm) in terms of sn (01~) and dn (ulm) in terms of dn ($A), but that the calculation of cn (ulm) requires all three functions. 16.13. Approximation in Terms of Circular Functions

m(u-sin

u co9 u) co9 u

cn (ulm) = cos u+a m(u-sin

u co8 u) sin u

(O
(a[~)

cn (uJ-m)=d

(VIP)

16.10.4

dn (u(-m)=nd

(O/P).

16.11. Reciprocal Parameter Transformation)

p=m-l,

(Jacobi’s Real

v=um1f,2

16.11.2

sn (ujm) =p1/2sn (VIP)

16.11.3

cn (ulm>=dn (4~)

16.11.4

dn (ujm) =cn (VIP)

Here if m>l then m-l=p
dn (ulm) a l-f

16.13.3 16.13.4

am (ulm) =u-a

m(u-sin

u co9 u).

16.14. Ascending Landen Transformation

To in&ease the parameter, 16.14.1 r=(l$::1)2~~l= 16.14.2 sn (ulm) = (l+~~“‘)

cn (ul m) =y

16.12. Descending Landen Transformation (Gauss’ Transformation)

let

m sin2 u

One way of calculating the Jacobian functions is to use Laden’s descending transformation to reduce the parameter sufficiently for the above See also 16.14. formulae to become applicable.

‘16.14.3

To decrease the parameter,

*

U

sn (u(-m)=w+sd

m>O,

_ an2(ulpj

16.13.2

16.10.3

16.11.1

1+ j.P* SIP (81/A)

any~~p~-(i-p~~~~

sn (ulm) =sin u-a

let

1 m’

cn @Id dn (4~)

16.13.1

16.10. Change of Parameter Negative Parameter

16.10.1

- 1+p1/2sn2 (VIP)

16.14.4

dn

(~1

i+j$/2

l-j&l”* m) = cc

let

1+ (qz>:

v= j-&G

sn C4r>cn 04~4 an (VIP) dn 2(~Ijb)-~,1~2

an (44 dn* (OJP)+Prt’*

anbl~)

574

JACOBIAN

ELLIPTIC

FUNCTIONS

Note that, when successive applications are to be made, it is simplest to calculate dn (ulm) since this is expressed always in terms of the same fnnction. The calculation of cn (u/m) leads to that of dn (4.4. The calculation of sn (u/m) necessitates the evaluation of all three functions. 16.15.

Approximation

in Terms Functions

THETA

FUNCTIONS

16.17. 16.17.1

16.17.2

(sinh u cash u-u)

sech2 u

16.15.2

cn (ulm) =sech u -- i ml (sinh u cash u-u)

tanh u sechu

cn(u+v) = cnu.cnv-snu.dnu.snv.dnv 1 - m sn2u ‘sn3v dn(u+v)

16.18. 16.18.1

1 4

+- ml (sinh u cash u+u)

tanh u sech u 16.18.2

16.15.4 am (ujm) =gd u+i

ml (sinh u cash u-u)

16.16.

16.18.3

16.18.4

16.18.5

Derivative

sn 2u = 2snu.cnu.dnu=2snu.cnu.dnu cn2u+sn2u. 1 - msn4u

:z%

16:16:3

cn u dn u --8n u an u -m sn u cn u

Pole n

16.19.1

16.16.4 16.16.5 16.16.6

cd u sd u nd u

-ml sd u nd u cd u nd u m sd u cd u

Pole d

16.19.2

16.16.7 16.16.8 16.16.9

dc u nc u

ml sc u nc u SC u dc u dc u nc u

Pole c

16.19.3

16.16.10 ns u 16.16.11 ds u 16.16.12 csu

-ds u cs u -cs u ns u -na u da u

Note that the derivative is proportional product of the two copolar functions.

16.20. Pole 8

to the

dn2u

cn 2u dn2u dn2u

dn 2u

1 -cn l+cn

2u sn2u . dn% 2u= cn2u

1 - dn 2u msn2u acn2u l+dn 2u= dn2u 16.19.

sn u cn u dn u

SCu

cnv

Arguments

=dn2u-msn2u.cn2u=dn2u+cn2u(dn%-1) 1 - msn4u dn2u-cn2u(dn2u-

Derivatives

Function

Double

= cn2u-sn2u. dn% cn2u-sn2u. l-msn4u = cn2u+sn2u.

sech u.

Another way of calculating the Jacobian functions is to use Landen’s ascending transformation to increase the parameter sufficiently for the above formulae to become applicable. See also 16.13.

dnu+dnv-msnu.cnu.snv. I- msn2u . sn2v

=

Addition theorems are derivable one from another and are expressible in a great variety of forms. Thusns(u+v) comes from l/sn(u+v) in the form (l-msn2usn2v)/(snucnvdnv+snvcnudnu) from 16J7.1. Alternatively ns(u + v) =m112sn { (iK’ - u) -211 which againfronr16.17.1 yields the form (ns u cs v dsu --nsvcsudsv)/(ns2u-ns2v). The function pq(u+v) is a rational function of the four functions pq u, pq v, pq’u, pq’v.

16.15.3

dn (ulm) =sech u

Theorems

u-en v.dn v+sn vacn uedn u 1 -m sn%.sn%

=sn

16.17.3

16.15.1

Addition

sn(u+~)

of Hyperbolic

When the parameter m is so close to unity that ml2 and higher powers of ml can be neglected we have the approximations

sn (ulm) =tanh u+$m,

AND

dn21yu= Jacobi’s

Half

Arguments

m,+dn

u-kmcn lfdn u

Imaginary

u

Transformation

16.20.1

sn(iulm)=isc(ulmJ

1620.2

cn(iujm)=nc(uJ

ml)

16.20.3

dn(iulm)=dc(ul

m,)

1)

JACOBIAN

ELLIPTIC

FUNCTIONS

16.23.6

16.21. Complex Arguments

nd @lm>=sK+

With the abbreviations 16.21.1

s=sn(zIm),

c=cn(z~m),

d=dn(zlm),

sn(z+iylm)

=

16.21.3

cn(z+iyj

m) =

16.21.4

dn(z+iy)m)=

dc (ulm)=&sec +g 16.23.8

c. cl--is. d +51. dl cf + ms2 . st

nc (ul m) =2---& 1

F. (-1)”

2nv

&

CO8 (2nfl)v

set v

-m+&

C-1)” $q

cos @n+l)v

1623.9

16.22. Leading Terms of the Series in Ascending Powers of u 16.22.1

$+(l+14m+m2)

g1 (-l)y&.eos

v

s . dl+ic. de 51. cl c:+ms2 a9.:

d . cl. d,--ims . c . s1 cT+ ms2 ast

sn(ulm)=u-(lfm)

SKl

16.23.7

sl=sn(yIml),

cl=cn(ylml), dl=ddylmd 16.21.2

575

AND THETA FUNCTIONS

$

SC (u Im) =2+K

tan v 1

+

sKmcl

(-1)”

&sin

2nv

1623.10

-(1+135m+135m2+ma)

$+

...

2 Q2n+l 11s(4m>=& cscv-x2u n=O l-p2n+l

sin (2n+l)v

16.23.11

16.22.2 cn(ujm)=l

-$+(1+4m)

v-- g go $g--

ds (ulm)=&csc

$

sin (2n+l)v

16.23.12

-(1+44m+16m2)

$i-

...

16.22.3

16.24. Integrals

dn(ulm)=l-m

$I-m(4+m)

$

- m(16+44m-k m2) $t

No formulae are known for the general cients in these series.

...

sn (ulm)=-

27r 2

pn+1/2 sin (27bfl)v

rn1i2K n=O I-

cn (ul m) = - 2*

16.23.3

dn (ul m) =&+z

p2n+1

5 p.n+1/2 cos (2n.+l)v m1/2K n=O1 + p2n+l 2lrm

n

g1 1 +pq2,, cos 2nv

16.23.4

cd (ul m)=x

(-l)“p+l/2

2 m1/2K n=O 1-p2”+l

cos (2nfl)v

16.23.5

sd (ulm)=(m~~~,2~~~(-l)n~~*2~+~s~

n+r/z

sin 2nv

of the Twelve Jacobian Elliptic Functions

16.24.1

Jsn u du=m-lt2 In (dn u-m’f2cn

16.24.2

$ cn u du=m-1J2 arccos (dn u)

(2n4-1)v

Jdn

16.24.3

u)

u du=arcsin

(sn u)

16.24.4

$ cd u du=m-‘12 In (nd u+m112sd u)

16.24.5

Jsd u du= (mmJ-1’2 arcsin (-m1’2cd

16.24.6

Jnd

16.24.7 16.23.2

gl +ft+

coeffi-

16.23. Series Expansions in Terms of the Nome q=t?‘--*K”K and the Argument v=?ru/(2hr) 16.23.1

cs blm> =z2K cot v-- g

u)

u du=mI-112 arccos (cd u)

Jdc u du=ln

16.24.8

Jnc

16.24.9

Jsc u du=m;li2

u

du=m;“2

(nc u+sc u) In (dc u+m:bc

U)

In (dc ufm:t2nc

u)

16.24.10

$ ns u du=ln (ds u-cs

u)

16.24.11

$ ds u du=ln (ns u-cs

u)

16.24.12

f

u)

cs u du=ln

(ns u-ds

In numerical use of the above table certain restrictions must be put on u in order to keep the arguments of the logarithms positive and to avoid

576

JACOBIAN

trouble tions.

with many-valued

ELLIPTIC

FUNCTIONS

inverse circular

func-

16.25. Notation for the Integrals of the Squares of the Twelve Jacobian Elliptic Functions 16.25.1

Pqu=

16.25.2

Ps u=s,’

s0

Upq2t dt when q#s dt-t

(pq2t-$) Examples

Cdu=

S

’ cd2t dt, Ns u=l(ns2t-$)dt-t

0

16.26. Integrals in Terms of the Elliptic gral of the Second Kind (see 17.4)

mSn u= -E(u)

16.26.1 16.26.2

mCn u=E(u)

+u Pole n

-mlu

mCd u= -E(u)

+u+msn

16.26.6 16.26.7

m,Nd

-mlu-msn u=E(u)

DC u= -E(u)

u cd u

Pole d

-msn

u cd u

+u+sn

u dc u

+miu+sn

16.26.9

m$c u= -E(u)

16.26.10

Ns u=-E(u)+u-cn

Pole c

u dc u

&(z,

FUNCTIONS

a)=&(z)

+sn u dc u

16.27.4 tJ4(2,

*)=tY,(z)=l+2

Ds u=-E(u)+mlu-cn

Pole 8

u ds u

Cs u=-E(u)-cn

u ds u

All the above may be expressed in terms of Jacobi’s ieta function (see 17.4.27). -gu,

where E=E(K)

Functions; Expansions of the Nome p

where K and iK’ are the quarter periods. Since q=q(m) is d et ermined when the parameter m is given, we can also regard the theta functions as dependent upon m and then we write 2, 3, 4

but when no ambiguity is to be feared, we write 9,( 2) simply. The above notations are those given in Modern Analysis [16.6]. There is a bewildering variety of notations, for example the function G,(z) above is sometimes denoted by 8,(z) or b(z); see the table given in Modern Analysis [16.6]. Further the argument u=2Kz/~ is frequently used so that in consulting books caution should be exercised.

b,(z, *)=~,(z)=2p1~‘~~(-l)“q”‘“c”

Squares

~~(z)s:(O)=s:(z)~:(O)-s:(z>st(O>

16.28.2

S:(z)S:(O)=~:(z)9:(0)--61(z)9~(0)

16.28.3

S:(z)r9:(0)=191(~)(93(0)-d:(z)9:(0)

16.28.4

9:(+9:(O)

=S:(z)ti:(O)

f%(O) +fm

of the

-19;(z)@(O) =WO)

Note also the important sin (2nfl)z

(%+l)z

relation

ti;(O)=&(O)&(O)&(O)

Or

6;=t9,6&

16.29. Logarithmic

Derivatives Functions

of ’ the

16.29.1

u+4

sin 2~

16.27.2 cos

Between the Theta Functions

16.28.1

16.28.6

*)=92(2)=29”4~~p.(“+l)

a=l,

in Terms

16.27.1

8,(z,

cos 2ti.2

Theta functions are important because every one of the Jacobian elliptic functions can be expressed as the ratio of two theta functions. See 16.36. The notation shows these functions as depending on the variable z and the nome p, 1~1
16.28.5 16.27. Theta

(-l)“qn2

2 n=1

16.28. Relations

Z(u) =E(u)

$, qn2 cos 2nz

u ds u

16.26.11

16.26.12

=1+2

f%(z, a>=Wlm>,

16.26.8

mlNc u= -E(u)

16.27.3

u cd u

16.26.5

mm,Sd u=E(u)

THETA

P=e--*K’IK 7

Dn u=E(u)

16.26.3

16.26.4

Inte-

AND

$$=cot

1

$

I

&

Theta

JACOBIAN

ELLIPTIC

l?UNCTIONS

AND

THETA

577

FUNCTIONS

16.31. Jacobi’s Notation

1629.2 8; (4 -=-tan fJ*(u> 16.29.3

u-j-4 ‘$$=4

n$l (-1)”

2

(-1)”

sin 2nu

i-C+

sin

ha

16.31.2

2nU

16.31.4

16.29.4

ln &(a+P)-ln ts,-

H(ulm)=H(u)=t$(v) H,(u~m)=H,(u)=&(v)=H(u+K)

16.32. Calculation of Jacobi’s Theta Function @(ulm) by Use of the Arithmetic-Geometric Mean

of !%.un

Form the A.G.M.

sin (a+@)

with

16.32.1

cos bfB) cos (a--/3)

terminating with the Nth step when cN is negligible to the accuracy required. Find VN in degrees, where 180’ 16.32.2 QN= 2”aNu T

+4 5 -iq (-‘)” n-1 n

u()=l,

16.32.3 -

(-1)”

+I

n

p” 1-p

b,=Jm,,

co= &ii

and then compute SUcceSSiVely Q~, Q. from the recurrence relation

@” sin 272~~sin 2nfi

16.30.3

ln$3=4C--

scale starting

sin ((Y-P)

16.30.2 In sab+a)-1, KGB--

~,(u~m)=0~(~)=83(v)=@(u+K)

16.31.3

16.30. Logarithms of Theta Functions and Difference 16.30.1

+

16.31.1 ~(u~m)=O(u)=29&),

It=1

3

for Theta Functions

sin 2 na! sin 2nf3

Sin

(&9.-1-Q,,)=~

QN-1,

Sin

. . .,

QN-~,

Qn.

Then 16.32.4

16.30.4

+i ln Set

The corresponding expressions when /3=i:r are easily deduced by use of the formulae 4.3.55 and 4.3.56. 16.33. Addition U

of Quarter-Periods u+2K

u+K

--u

(2Qo-Ql)

i$

ln

SeC

(2Q1-(02)

+

1 + 2N+ - 1 In set to Jacobi’s Eta and Theta =

u+iK'

uf2iK'

. . .

C&N-I-QN)

Functions

u+K+iK'

u+2K+2iK'

-16.33.1 H(u)

--B(u)

HI (4

-H(u)

iM(u) O(u)

-N(u)Wu)

MW01(4

N(u) H(u)

16.33.2 HI(U)

H,(u)

--H(u)

--H,(u)

M(u) @l(U)

N(u) HI (4

-iM(u)O(u)

- N(u) HI (4

@l(U)

O(u)

@l(U)

M(u)HI(u)

NC.4 O,(u)

iM(u)H(u)

N(u)

O(u)

O,(u)

O(u)

iM(u)H(u)

- N(u)

fif(U)Hl(U)

-N(u)

16.33.3 @l(U) 16.33.4 O(u)

e(u)

@l(U) O(u)

-

where M(u) =[ exp (-$)I

r-z-*,

N(u)=[exp

p-1

(-$)I

1 H(u) and H,(u) have the &(u) have the period 2K. 2iK’ is a quasi-period that is to say, increase of multiplies the function by

period 4K.

O(u) and

for all four functions, the argument by SK’ a factor.

JACOBIAN

578 16.34. Relation

ELLIPTIC

FUNCTIONS

of Jacobi’s Zeta Function Theta Functions

Z(u) =&

to the

AND

THETA

FUNCTIONS

.6.36.2

If A, P are any integers positive, negative, or sero the points uJ-2xK-l-2piK’ are said to be

In O(u)

mngruent to ti. 16.34.1

Z(u)=&

?Tu -

8,(u) 6,(u) 8,(u) 8,(u)

cnudnu sn u

” (6)2K

has zeros at the points congruent to 0 has zeros at the points congruent to K has zeros at the points congruent to iK’ has zeros at the points congruent to

K+iK’

Thus the suffix secures that the function 8,(u) has zeros at the points marked p in the introductory diagram in 16.1.2, and the constant by which Jacobi’s function is divided secures that the leading coefficient of 8,(u) at the origin is unity. Therefore the functions have the fundamentally important property that if p, q are any two of the Letters s, c, n, d, the Jacobian elliptic function pq u is given by

16.34.2

sn u cn u dn u

16.34.3

16.36.3

16.34.4

These functions 16.35. Calculation of Jacobi’s Zeta Function Z(ulm) by Use of the Arithmetic-Geometric Mean

Form

the A.G.M.

16.35.1

h=l,

scale 17.6 starting

b,=JG,

with

c,,=&

terminating at the Nth step when CN is negligible to the accuracy required. Find PN in degrees where

16.36.4

m;“4cP,(K-u)=8,(~)

16.36.5

m;“48~(K-U)=8n(~),

for complementary arguments u and K--u. In terms of the theta functions defined in 16.27, let v=n-u/(X), then 16.36.6

16.35.3

sin (Z(p,-,-(p.)=z

2Kt9, Cd

tYJu)=

16.36.7

and then compute successively (p&l, (pl, (p,, from the recurrence relation

also have the property

b;(o)

sd(u)=s,o’

63 (VI

92 (4 ’ m’s,(o)

~Au>=---.

fJ4(v) 194(O)

$?V+Z,. . .,

sin opt.

Then 16.35.4

Z(ulm)=cl

sin a+c2 sin (pz+ . . . +cnr sin (ON.

16.36. Neville’s

Notation

for Theta

Functions

These functions are defined in terms of Jacobi’s theta functions of 16.31 by 16.36.1

s.(lL)=g$$

S,(u)=

H&SK) H(K)

FIQURE

16.4. fi,(u),

Neuiue’s thetajunctions G,(u),

@d(u),

m=-

1

2

&&)

JACOBIAN 1.0

r

ELLIPTIC

FUNCTIONS

AND

\

\.

THETA

FUNCTIONS

16.38.

579

Expression

as Infinite

Series

Let v=4(2K) 16.38.1

8,(u) =[m~~$~K]l’z

16.5.

FIGURE

Logarithm@ junctions

derivatives

of

theta

2

(--l)fQp(n+l) sin (2ni+l)v

16.38.2

tpC(u)=

[ 1

16.38.3

&(u)=

c1

‘$2

“’ n$O qncn+l) cos (2n+l)v

l/2

&

{ 1+2 gI qn2cos 2nvj

16.38.4

11+Gg (-1Pqn2cos 2?b} 1’2

h(u)=[eK] 16.37.

Expression

as Infinite

a=abO,

Products

sin v nt, (l-2qzn

8*(u)=(~)1’6

cos 2v+q4”)

16.37.2

(2K’/?r)1’2=1f2p1+2~+2q;+

. . . =&(O,

il>

cos 2vfq49 (2m1~2K/7r)1~2=2q1~4(1+q2+q6+q12+q20+. . .)

16.37.3

(1+2p-’

cos 2v+p4”-2)

16.37.4

=w4

n>

16.38.8 l/12

( > m lW4

m

l-l (l--2q+l

co9 2v+q4”-3

r=~

Numerical Use and

Extension

table of principal terms nc u=-m;1f2/(u-K)+ nc (KS .OOlI.64) =-(:yo)l-“‘+

...

=-s+ **. ..

and since the next term is of order .OOl this value -1667 is correct to at least 4s. Example 2. Use the descending Landen transformation to calculate dn (.20(.19) to 6D. Here m=.19, m:12=.9 and so from 16.12.1 1 2 cc=1+p==$? 19 ’

0

. . =fi4(0/,

q).

Methods =lO-‘X7.67 which is negligible. From 16.12.4 dn2 [.I9

. . .

10000

=-1667+.

(2m~‘2K/7r)1’2=1-2p+2p4-2q9+.

of the Tables

Example 1. Calculate nc (1.996501.64) to 4s. From Table 17.1,1.99650=K+.OOl. From the

Also

. . . =11)3(0, q)

16.38.7

co9 v “,= (1+2p

16.39.

(2K/7r)“2=1+2p+2q4+2qe+

16.38.6

16.37.1

r&(u)=

16.38.5

v=7r74@K)

I($)]-(1-i)

dn(*20’-1g) =(l++jnf-

,

-19 1(hyl

Now from 16.13.3 dn [.19 1(A)]=.999951 whence dn (.201.19)=.996253. Example 3. Use the ascending Landen transformation to calculate dn (.20).81) to 5D. From 16.14.1 2 4(.9) 360 C(=(1.9)2=gg

Pl’

w=.19. & is negligible to 4D.

Thus

580

JACOBIAN

1g

ELLIPTIC

dnz(.19)

FUNCTIONS

AND

dn (.20] .Sl) =5X ’

From J-6.15.3 dn

(’

19 3*

1361)

=sech (.lg)+tX&

tanh .19 sech .19

calculation when

[sinh .19 cosh.19+.19] =.982218+:X&

FUNCTIONS

Example 4. Use the ascending Landen transformation to calculate cn (.20).81) to 6D. Using 16.14.4, we calculate dn (.20(.81) and deduce cn (.20].81) from 16.14.3 settling the sign from Figure 16.1. As in-the preceding example, we reduce the

$$)+A

dn(.l9IE)

THETA

of dn (.20).81) to that of dn dn (.19/%)=.982267

(.187746)(.982218) [(.191145)(1.01810)$.19]

= .982218+;X&

dn (.20].81)=.984056

(.184408)[ .384605]

cn (.20].81) = .980278.

= .982218+ .000049=.982267. Thus dn (.20] 31) = .98406. 5.

Example

Use the A.G.M.

scale to compute dc (.672].36) to 4D.

From 16.9.6 we have dc2(.672].36) =.36+l-snZ(~72, method given in 16.4. ?l

Form the A.G.M.

a,

bn

-_______ 0 I 2 3

We now calculate

36)*

sn(.672].36)

by the

scale C”

G

I.

a,

(Pn

sin (Pi

S%” 1: 2. 4117 4. 8234

. 60952 . 93452 . 66679 -. 99384

sin

C&-1

cd

2+%-I--Pn

. 10383 00207

. 10402 00207

-

_______ 8 : 89443 ii;;: :

1 9 89721 . 89721 vn = 2na,u

6 : 1 00279 0’

.6 . 11111 00311 0’

cpa=23(.89721)(.672)

0’

0’

=4.8234

continuing until c,=O to 5D. Then complete as indicated in 16.4 to find q. and so sn u and hence dc u, cpo= .65546 sn u= .60952 dc u=1.1740. Example 6. Use the A.G.M. scale to compute @(.6(.36) to 5D. We use the method explained in 16.32 with a~= 1, b,,=.8, co=.6. Computing the A.G.M. as explained in 17.6, we find (For values of a,, b,, c,, see Example 71

‘pn

sin cp,

0 a

58803 2. 1: 0780 1533

3

4. 3066

. 55472 83509 88101 . -. 91879

sin

ztnd then complete the calculation

5.)

C&h.-1- 4

.

h?k-I-- cp,

00260 09789 0’

outlined

00260 09805 0’

in 16.32 to give

In @(+I)=-.05734+.02935+.00120 = - .02679 O(ulm> =.97357.

The series expansion for 0 is preferable.

set

C&-1-

A

1. 0048 1. 1.

&lnsec

G~-~-(PI)

. 00120 i

JACOBIAN

ELLIPTIC

FUNCTIONS

Example 7. Use the q-series to compute cs (.53601 62l.09). Here we use the series16.23.12, K= 1.60804 862, p.00589

414, v=g=%dians

Since d is negligible

AND THETA

FUNCTIONS

581

Ip,(30°\450)=.59128 (set 45°)~9,(300\450) = 1.02796.

Therefore

or 30’. SC(.61802j.5)=#

to 8D, we have to 7D

cs (.53601 621.09)

(set 45’)$

= .68402.

2

=-r cot 300-2”K -ltq2sin 60’ 2K ) { = (.97683 3852) (1.73205 081)

Example 10. Find sn (.75342(.7) by inverse interpolation in Table 17.5. This method is explained in chapter 17, Example

-3.90733 541[(.00003 4740)(.86602 5404)]

7. Example 11. Find u, given that cs (u1.5)=.75. From 16.9.4 we have

=1.69180 83. Example 8. Use theta functions sn (.61802/.5) to 5D.

to compute

1 sn2u=1 -I+cs

2u

Thus

Here K($)=1.85407

sn2 (u/.5)=.64 and sn (u1.5)=.8. 9, (30°\450) sn L618021.5) z9 (300\450) n .59128 == .56458 1.04729 from Table 16.1. Example 9. Use theta functions to compute

SC (.61802/.5) to 5D. As in the preceding so that

We have therefore replaced the problem by that of finding u given sn (u/m), where m is known. If (o=am u sin cp=sn u and so p=.9272952

radians or’53.13010’.

From Table 17.5,

example

s0=300 , p=45o

Alternatively, starting with the above value of (owe can use the A.G.M. scale to calculate F(~\(Y) as explained in 17.6. This method is to be preferred if more figures are required, or if CYdiffers from a tabular value in Table 17.5.

We use Table 16.1 to give

References Texts

[16.1] A. Erdelyi et al., Higher transcendental funetions, vol. 2 (McGraw-Hill Book Co., Inc., New York, N.Y., 1953). [16.2] L. V. King, On the direct numerical calculation of elliptic functions and integrals (Cambridge Univ. Press, Cambridge, England, 1924). [16.3] W. Magnus and F. Oberhettinger, Formulas and theorems for the special functions of mathematical physics (Chelsea Publishing Co., New York, N.Y., 1949). [16.4] E. H. Neville, Jacobian elliptic functions, :2d ed. (Oxford Univ. Press, London, England, 1951). [16.5] F. Tricomi, Elliptische Funktionen (Akademische Verlagsgesellschaft, Leipzig, Germany, 1948). 116.61 E. T. Whittaker and G. N. Watson, .4 course of modern analysis, chs. 20, 21, 22, 4th ed. ((Cambridge Univ. Press, Cambridge, England, 1952).

Tables

[16.7] E. P. hdams and R. L. Hippisley, Smithsonian mathematical formulae and tables of elliptic functions, 3d reprint (The Smithsonian Institution, Washington, D.C., 1957). [16.8] J. Hoiiel, Recueil de formules et de tables numeriques (Gauthier-Villars, Paris, France, 1901). [16.9] E. Jahnke and F. Emde, Tables of functions, 4th ed. (Dover Publications, Inc., New York, N.Y., 1945). [16.10] L. M. Milne-Thomson, Die elliptischen Funktionen von Jacobi (Julius Springer, Berlin, Germany, 1931). [16.11] L. M. Milne-Thomson, Jacobian elliptic function tables (Dover Publications, Inc., New York, N.Y., 1956). [16.12] G. W. and R. M. Spenceley, Smithsonian elliptic function tables, Smithsonian Miscellaneous COIlection, vol. 109 (Washington, D.C., 1947).

582

JACOBIAN

0

25 :; 40 45

0.00000 0.08715 0.17364 0.25881 0.34202

0000 5743 8178 9045 0143

0.42261 0.50000 0.57357

8262

0.64278

0.70710

0.76604 0.81915 0.86602 0.90630 0.93969

ELLIPTIC

5 0.00000 0000

0.08732 0.17397 0.25931 0.34267

FUNCTIONS

10 0.00000 0000

15 0.00000 0000 3070

2476

0.08782 0.17497 0.26080

0.08867

4191

0.34464

3695

0.17667 0.26332 0.34797

1584 6099 7361

4343 3708

0446 6358 7994 1085 4820

0.42998 0.50871 0.58357 0.65399 0.71944

1306 3952 6134 8067 3681

1966 9362

2677

0000 6436 7610 6781

5688

4443 2044 5404 7787 2621

0.76750 0;82071 0.86767 0.90803 0.94148

5843 4821 7668 6964 5546

0.77192 0.82544 0.87267 0.91326 0.94691

5893 2256 6562 9273 1395

0.77941 0.83345 0.88115 0.92214 0.95611

4712 4505 1505 2410 4956

5826

0.96776

8848

0.97334 0.99237 1.00384 1.00768

6839 4367 9133 3786

0.98281 1.00202 1.01361 1.01748

0311 5068 2807 5224

E\QI 0'

30 0.00000 0000

7753 4698 0000

0;98668 0.99809 1.00190

0526

3768

6836 5528 8098

36 0.00000 0000

40 0.00000 0000 0.09914 0.19754 0.29449 0.38924

0.09353 0.18636 0.27778 0.36710

4894 3367 4006 5393

0.09606

0073

0.19139 0.28530

9811 3629

0.37706

5455

0.45365 0.53676 0.61581 0.69019 0.75934

1078 4494 3814 6708 4980

0.46599 0.55141 0.63268 0.70917 0.78030

3521 5176 1725 3264 3503

0.65339 0.73250 0.80611

0.82272 0.87986 0.93030 0.97366 1.00961

9031 2121 4365

6431 2870

0.84552 0.90433 0.95626 1.00092 1.03795

4503 1298 6326 3589 2481

1.03786 1.05820 1.07047 1.07456

5044 3585 0366 9932

1.06706 1.08801 1.10066 1.10488

1179 9556

:: 20

60 0.00000 0.11968 0.23861 0.35604 0.47120

0000 1778 4577 4091 6153

25 30 35 40 45

0.58332 0.69160 0.79525 0.89344 0.98538

3727 6043 0355 6594 4972

50

1.07026 1;14731 1.21579 1.27502 1.32438 1.36335 1;39150 1.40851 1.41421

:; 60 65 70 75 s850 90 ta\ 0" 5

65: 65 70 75 2 90

4152 9967

0.42586 0.50383 0.57797 0.64772 0.71253

0.96592 0.98480 0.99619 1.00000

::

THETA

0.42342 0.50095 0.57467 0.64401 0.70845

75 80 85 90

1:

AND

45 0.00000 0000

FUNCTIONS

20 0.00000 0000

25 0.00000 0000

0.08988 0.17909 0.26693 0.35274

7414 1708 4892 9211

0.09149 0.18229 0.27171 0.35907

5034 6223 4833 2325

0.43588 0.51570 0.59159 0.72935

2163 1435 9683 9145 6053

0.44370 0.52497 0.60225 0.67495 0.74253

5382 0857 0597 6130 3161

0.79016 0.84496 0.89332 0.93489

4790 1783 9083 7610

5863 0899 llbb 9199 0216 4761 0908 6011 7974

0.66299

0.96935

0025

0.80446 0.86028 0.90955 0;951as 0.98700

0.99642

3213 0350 2527 9925

1.01458 1.03444 1.04641 1.05041

1.01591 1.02766 1.03158

50 0.00000 0000

55 0.00000 0000

2353 9961 2321 7478

0.10287 0.20501 0.30564 0.40405

9331 0420 8349 4995

0.10740 0.21405 0.31918 0.42204

5819 3194 5434 9614

0;11291 0.22506 0.33569 0.44403

2907 4618 3043

0.48110

6437

0.56937

7735

2178 7761 4729

0.49950 0.59127 0.67868 0.76106 0.83776

2749 8602 8658 3101 1607

0.52189 0.61799 0.70961 0.79606 0.87664

9092 6720 8904 0581 1114

0.54932 0.65080 0.74770 0.83928 0.92480

5515 1843 4387 2749 2089

0.87364 0.93455 0.98837 1.03467 1.07308

0739 6042 8598 8996 5074

0.90817 0.97175 1.02796 1.07635 1.11651

9128 1955 3895 2410 4503

0.95071 1.01765 1.07692 1.12798 1.17041

1025 9399 1759 8100 0792

1.00355 1.07485 1.13807 1.19262 1.23801

1297 2509 1621 9342 2299

1511 6686

1.10328 1.12503 1.13815 1.14254

blO0 6391 8265 4218

1.14811 1.17087 1.18461 1.18920

2152 7087 4727 7115

1.20381 1.22789 1.24242 1.24728

2008 0346 6337 6586

1.27378 1.29959 1.31518 1.32039

3626 2533 2322 6454

0.00000 0.12814 0.25558 0.38160 0.50544

0000 8474 9564 3032 4270

70" 0.00000 0.13904 0.27741 0.41467 0.54994

0000 1489 6571 2740 7578

0.62633

0.96294 1.06350

5361 9784 1570 9380 5669

6403 5349 4546 0900 1718

1.15670 1.24161 1.31733 1.38304 1.43795

0747 9855 3549 3601

1.27329

7730

1.36953

6895

1.45580 1.53099 1.59408

7011 8883 7380

0417 0813 9209 3562

1.48140 1.51284 1.53187 1.53824

2159 3876 4716 6269

1.64417 1.68050 1.70253 1.70991

0149 3336 2036 3565

65

0.74345 0.85596

Ob87

0.68254

0.81164 0.93630

1.05553 1.16824

9331 3704 8263

5305 3466

75 0.00000 0000

80 0.00000 0000

4769

85”

0.15372 0.30706 0.45960 0.61082

0475 5715 9511 7702

0.17522

3596

0.35063 0.52633 0.70219

9262 5260 9693

0.00000 0.21321 0.42844 0.64743 0.87146

0000 7690 3440 4941 4767

0.76005 0.90647 1.04907 1.18666 1.31788

8920 6281 2506 0037 6740

0.87783 1.05251 ii22511 1.39412 1.55769

8622 4778 1680 6403 2334

1.10111 1.33612 1.57526 1.81633 2.05616

6239 3616 8297 9939 7815

1.44126 1.55522 1.65814 1.74846

6644 4175 9352

2.72469

1.82467

1283 2258 2358 3523 7685

2.29072 2.51529

0610 1332

1.71363 1.85953 1.99285 2.11103 2.21162

2.91357 3.07668

3417 0558 4161 4159 6743

1.88545 1.92971 1.95660 1.96563

5864 0721 6998 0511

2.29242 2;35155 2.38762 2.39974

2061 6149 2438 3837

3.20921 3.30704 3.36705 3.38728

2227 7313 9918 7004

In calculating elliptic functions from theta functions, when the modular angle exceeds about 60”, use the descending Landen transformation 16.12 to induce dependence on a smaller modular angle. Compiled from E. P. Adams and R. L. Hippisley, Smithsonian mathematical formulae and tables of elliptic functions, 3d reprint (The Smithsonian Institution,

Washington, D.C., 1957) (with permission).

65 60 55 50 45

15 10 5 0 45 90" :50 75 70 65 EE 50 45

JACOBIAN

ELLIPTIC

FUNCTIONS TIIETA

AND THETA

FTJNCTIONS

583 Table

FUNCTIONS a&\4

;

5 1.00000 1.00001 1.00005 1.00012 1.00022

00000 44942 75362 78184 32051

1.00000 1.00005 1;00023 1.00051 1.00089

00000 83670 16945 47160 88322

1.00034 1.00047 1.00062 1.00078 1.00095

07982 70246 77451 83803 40492

1.00137 1.00192 1.00252 1.00317 1.00384

1.00111 1.00128 1.00143 1.00156 1.00168

97181 03532 10738 73002 48932

1.00178 1.00185 1.00189 1.00190

lo”

15 20 1.0000000000 1.00000 00000 1.00013 1.00052 1;00117 1.00204

28199 72438 12875 53820

1.00023 1.00095 1.00211 1.00369

99605 25510 61200 53131

25 1.00000 1.00038 1.00152 1.00337 1.00589

00000 29783 02770 73404 77438

23717 09464 78880 47551 18928

1.00312 1.00437 1.00575 1.00722 1.00874

29684 13049 24612 44718 26104

1.00564 1.00789 1.01039 1.01305 1.01579

21475 74700 27539 21815 49474

1.00900 1.01260 1.01658 1.02083 1.02520

49074 44231 69227 14013 88930

1.00450 1.00515 1.00576 1.00631 1.00678

90305 58975 28392 14139 49535

1.01026 1.01173 1.01311 1.01436 1.01543

07491 27599 39167 22536 98405

1.01853 1.02119 1.02369 1.02594 1.02789

77143 71444 24323 77596 45992

1.02958 1.03383 1.03781 1.04141 1.04452

63905 08852 34098 29561 01522

02800 05621 36042 80984

1.00716 1.00745 1.00762 1.00768

90696 20912 54187 37857

1.01631 1.01695 1.01735 1.01748

39354 79795 24037 52237

1.02947 1.03063 1.03134 1.03158

37972 73701 99632 99246

1.04704 1.04889 1.05003 1.05041

05862 76746 49895 79735

30" 1.00000 1.00056 1.00224 1.00499 1.00872

00000 64294 85079 51300 28461

35 1.00000 1.00079 l.00316 1.00702 1.01226

00000 66833 25308 56701 87413

40 1.00000 1.00108 1.00429 1.00954 1.01667

00000 26253 76203 73402 23379

45" 1.00000 1.00143 1.00570 1.01267 1.02212

00000 67802 35065 06562 67193

50" 1.00000 1.00187 1.00745 1.01655 1.02891

00000 71775 17850 47635 00179

55 1.00000 1.00243 1.00964 1.02143 1.03743

00000 05914 88003 61311 56974

1.01331 1.01864 1.02453 1.03081 1.03728

83978 21583 23743 00797 45330

1.01873 1.02622 1.03450 1.04333 1.05244

24599 04548 52308 50787 17208

1.02545 1.03563 1.04689 1.05889 1.07126

62012 21191 09786 07481 68617

1.03378 1.04729 1.06223 1.07816 1.09458

46028 03271 37524 10137 82886

1.04414 1.06179 1.08131 1.10213 1.12360

27466 07561 84270 29153 21058

1.05716 1.08002 1.10531 1.13227 1.16009

29130 00285 40947 78297 27802

1.04375 1.05003 1.05592 1.06125 1.06584

90125 67930 71242 10260 67280

1.06154 1.07037 1.07866 1.08615 1.09261

84606 85902 37978 23221 66042

1.08364 1.09564 1.10690 1.11708 1.12586

32917 39724 42279 18582 75438

1.11101 1.12694 1.14189 1.15540 1.16706

64844 63970 38846 45920 77783

1.14507 1.16589 1.18543 1.20309 1.21834

37802 54205 40490 54999 25328

1.18791 1.21489 1.24021 1.26310 1.28287

40899 61356 82552 97835 36204

1.06957 1.07232 1.07400 1.07456

45853 13226 34764 99318

1.09786 1.10172 1.10408 1.10488

02047 37756 99048 66859

1.13299 1.13824 1.14146 1.14254

42539 53698 12760 42177

1.17652 1.18350 1.18776 1.18920

88244 00363 94140 71150

1.23071 1.23982 1.24540 1.24728

12287 51648 69243 65857

1.29890 1.31072 1.31795 1.32039

75994 29838 95033 64540

60 1.00000 1.00313 1.01245 1.02768 1.04834

00000 85295 94672 16504 57003

65" 1.00000 1.00406 1.01615 1.03589 1.06269

00000 92257 50083 51569 75825

1.ooooc~ 1.00534~ 1.02121 1.047151 1.0823Ei

00000 44028 95717 56657 38086

75 1.00000 1.00720 1.02862 1.06363 1.11122

00000 88997 79374 90673 86903

1.00000 1.01026 1.04076 1.09068 1.15864

00000 06485 43440 07598 11101

85 1.00000 1.01663 1.06618 1.14751 1.25875

00000 88247 38299 59063 62174

1.07382 1.10335 1.13604 1.17088 1.20684

76019 71989 11010 93642 51910

1.09575 1.13408 1.17651 1.22176 1.26848

73598 00433 06705 77148 10938

1.12585 71388 1.17627 97795 1.23214. 31946 1.291761 91861 1.35335' 85717

1.17001 1.23826 1.31398 1.39491 1.47863

24008 96285 80140 71251 07744

1.24276 1.34068 1.44960 1.56636 1.68752

19421 05139 33094 90138 66770

1.39725 1.55957 1.74151 1.93815 2.14389

25218 26706 57980 19599 95792

1.24281 1.27771 1.31046 1.34007 1.36565

67937 04815 39783 a9457 16965

1.56259 1.64425 1.72108 1.79070 1.85094

67789 25175 41609 70015 39670

1.80942 1.92833 2.04054 2.14249 2.23090

88493 82823 54606 29245 12139

2.35264 2.55792 2.75309 2.93165 3.08742

71220 12198 84351 25995 47870

11169 28947 92570 35624

31927 17261 31647 53793 81348 00916 10514 83232 62687

1.41504 43413 1.47494 78592 1.53123 64694 1.58218s 06891 1.62620 90720

1.38640 1.40169 1.41105 1.41421

1.31523 1.36060 1.40320 1.44173 1.47501 1.50203 1.52194 1.53413 1.53824

1.66195 1.68832 1.70447 1.70991

1.89989 1.93602 1.95816 1.96563

92030 35909 92561 05108

2.30289 2.35609 2.38873 2.39974

04563 12550 86793 38370

3.21489 3.30946 3.36764 3.38728

91220 52989 82512 70037

70

87940 00831 27784 35651

80”

In calculatingelliptic functionsfrom theta functions,whenthe modularangleexceeds about 60’, usethe descending Landentransformation16.12 to inducedependence on a smallermodularangle.

16.1

584

JACOBIAN Tuhle

ELLIPTIC

LOGARITHMIC

16.2

FUNCTIONS DERIVATIVES

AND OF

THETA THETA

FUNCTIONS FUNCTIONS

$&lnWu)=f(t\a) ‘\a O0 1;

0 11.4m3005

25 ;50 40 45 50 2: 65 70 75 80 98: c\a O0 5

1.00000

2.10787 1.70248 1.40378 1.17143 0.98296

2.07952 1.67962 1.38497 1.15577 0.96985

2.04325

1.71888 1.41729 1.18270 0.99240

0.83910

0.83750

0.36397

oo*% 0:46542 0.36328

0.83273 0.69489 0.57297 0.46277 0.36121

0.82481 0.68830 0.56754 0.45839 0.35779

0.81383 0.67915 0.56001 0.45232 0.35306

0.79987 0.66754 0.55047 0.44464 0.34708

0.26592 0.17499 0.08683 0.00000

0.26340 0.17334 0.08600 0.00000

0.25992 0.17105 0.08487 0.00000

0.25553

KZ 0:08732 0.00000 35

40

45"

50

55

9.;7764 4.60585

a.;657 4.38332

%%E .

;z!. 1.66695 1.35001 1.11647 0.93462 0.78679

65

40

0.70021 0.57735 0.46631

0.26795 0.17633 0.08749 0.00000

30 10.6m5083 5.28496 2.56090

10.?7113 5.14645 3.38730 2.49430

1.61498 1.33189 1.11167 0.93301

:%t;3 1:29791 1.08352 0.90958

0.78307

0.76355

0.65359

:i 20 25

0.53902 0.43543 0.33992

Lx;93 ;:M; .

0.25028 0.16471 0.08173 0.00000

0.24424

a.;941 4.13843 2.72935 2.01530

;.f;;;; p;; . E%

1.57876 1.28047

2.12820

1.65041

1.36096 '0%: .

KKi o:ooooo

80

75 70 66: 55 50 45

15 10 i

1.88828 1.52607 1.25919

1.82172 1.47292 1.21591

1.74793

xz.

E595: .

0.82139

0.74151 0.61923 0.51093 0.41292 0.32248

0.71714 :z3:28 0:39991 0.31242

0.69066 0.57749 0.47705 0.38595 0.30168

0.66232 0.55441 0.45846 i%:: .

:i 25 20

0.23151 0.15634 0.07759 0.00000

0.23017 0.15155 0.07522 0.00000

0.22235 0.14645 0.07270 0.00000

0.21419 0.14114

15 10

%oooo~ .

z

70” ;A;: 2:36323 1.75208

0.71131

0.63242

0.60125 0.50526 0.41932 0.34063 0.26719

0.20584 0.13572 0.06742 0.00000

0.19749 8%:~~ o:ooooo

2.66414

9.;479 4.80696 3.16502 2.33179

1:00096 0.84142

0.53023 0.43911 0.35605 0.27885

3.61876

4.98711 3.28290 2.41789

lO.om4914

Xf 0193737 0.79086 0.67101

50

i50 90

0.07977 0.00000 65"

;50 40 45

75

O.lb076

60

1.06066 0.88940 0.75000

2; 65 70

10.8~811 5.40253 3.55536 2.61756

2.14043 1.72875 1.42543 1.18949 0.99810

1.99919

ta \ 00 5

5.49902

ll.iil275

2.14451 1.73205 1.42815 1.19175

25 30

ii 90

11.2:449 5.57427 3.66823 2.70051

a/4 0 li

:%z 2174225

3.47816

75

11.;306

25

3.73205 2.74748

:i 20

iz 45

ll.iia29

20

15

5.62812 3.70365 2.72658

5.67128

2105

10”

5

75 6.4009756 3.24056 2.15026 1.60057 1.26603

1.41419 1.16828 0.97687

80" 5.7:041 2.85790 YEf . 1.13996 %27E 0:68225

85" 4.7?263 2.37760 1.60605 1.22261 0.99169 0.83453

65 2

tzs5 p;'9"4: .

0.58682

:*x 0:54358

0.56918 0.47987 0.39943 0.32532 0.25574

0.53662 0.45454 0.37992 0.31054 0.24484

0.50411 0.42988 0.3b140 0.29684 0.23497

0.47247 0.40690 0.34488 0.28513 0.22685

0.18935

0.18170 0.12026 0.05988 0.00000

0.17490

0.16949 0.11272 0.05628 0.00000

Oo.Ef o:ooooo

0.11601 0.05784 0.00000

565" 50 45

5";

In calculatingelliptic functionsfrom theta functions,whenthe modularangleexceedsabout 60”,usethedescending Landentransformation16.12 to inducedependence on a smallermodular angle.

JACOBIAN

ELLIPTIC

LOGARITHMIC

DERIVATIVES

5” 0.000000

0.000331

0 i 0

30" 0.000000

0.012059 0.023711 0.034569 0.044277

FUNCTIONS

:to" o.oc~oooo

OF

AND THETA THETA

585

FUNCTIONS

FUNCTIONS

20"

Table

15” 0.000000

0.000000

0.002984 0.005875 0.008583 0.011024

0.005318 0.010466 0.015283 0.019616

0.000000

0.036462 0.041075

25"

K%z 0:001224

O.OCll324 O.OCl2607 O.OC13811 O.OCl4897

0.001458 0.001649 0.001788 0.001874 0.001903

"o.mm;;: o:oc17147 O.OC17486 O.OCf7596

0.013124 0.014819 0.016057 0.016804 0.017037

0.023332 0.026318 0.028487 0.029776 0.030154

E%4: 0:046846

0.001873 0.001787 0.001647 0.001457 0.001222

O.OCl7476 O.OCl7129 O.OCt6566 O.OC~5805 O.OCl4868

0.016753 0.015962 0.014691 0.012979 0.010879

0.029616 0.028185 0.025912 0.022871 0.019154

0.045938 0.043654 0.040077 0.035328 0.029556

0.000951 0.000650 0.000330 0.000000

0.003786 O.OCi2589

0.008455 0.005780 0.002933 0.000000

0.014877 0.010165 0.005157 0.000000

0.022935 0.015661 0.007942 0.000000

35 0.000000

0.016511 0.032444 0.047248 0.060427

oo%E . 4OO"

0.000000 0.021734 0.042671 0.06'2057 0.079221

45

0.000000 0.027787 0.054498 0.079124 0.100783

0.008337 0.016401 0.023933 0.030690

0.034760 0.068087 0.098650 0.125308

8-E%

ii: 80 75 70

0:083685 0.120939 0.153099

0.065561 0.062183 0.056989 0.050157 0.041905

0.142097 0.133678 0.121592 0.106302 0.088310

0.172615 0.161784 0.146658 0.127835 0.105932

0.205102 0.191402 0.172831 0.150136 0.124058

0.032483 0.022163 0.011235 0.000000

0.043344 0.029545 0.014968 0.000000

0.055309 0.037660 0.019067 0.000000

0.068143 0.046339

0.081578 0.055395 0.028000 0.000000

0.095321 0.064622 0.032631 0.000000

0.052098 0.101680 0.146471 0.184635

0.063034 0.122704 0.176024 0.220691

rr(0 0 0.000000 0.076222 0.147856 0.210938 0.262588

0.092860 0.179233 0.253725 0.312762

0.214885 0.236514 0.249349 0.253651 0.250000

0.255225 0.278976 0.292010 0.294931 0.288691

0.301193 0.326329 0.338517 0.338908 0.325990

0.239181 E;:",: 01172751 0.142285

0.274426 0.253326 0.226549 0.195171 0.160167

rl.109049 0.073794 0.037222 0.000000

0.122405 0.082664 0.041645 0.000000

60"

65" 0.000000

75"

0.000000

2 Q/c,

0.179081 0.198206 0.210188 0.215082 0.213212

0.000000

15 10

55"

0.147169 0.163627 0.174358 0.179298 0.178606

Ki%o3;fo"3 .

sbs 50 45

50"

0.118758 0.132533 0.141791 0.146411 0.146447

0.071558

65

0.000000

0.093605 0.104784 00'00~8~4028 0.112477 0.116544 0:089827 0.116978 0.090424 0.113888 0.088287 0.107483 0.083549 0.098051 0.076408 0.085943 0.067122 0.071553 0.055989

0.052528 0.059074 0.063730 0.066384 0.066987

16.2

80" 0.000000

40 2 25 20

85

0.000000

EE .

0.153481 0.289421 0.395712 0.467893

0.354775 0.379918 0.389553 0.385698 0.370590

0.420046 0.442452 0.446532 0.435687 0.413176

0.507818 0.520777 0.512966 0.490013 0.456422

65 60

0.3113353 0.28,4538 0.25.2950 0.215820 0.17'7204

0.346389 0.315020 0.278119 0.237026 0.192823

0.381811 0.343874 0.301140 0.254956 0.206331

0.415539 0.369741 0.320668 0.269431 0.216780

40

0.13,1996 0.090960 0.045763 0.000000

0.146375 0.098382

0.156015 0.104574 0.052449 0.000000

0.163217 0.109083 0.054618 0.000000

15 10 5 0

K%o3 .

0.115687 0.221544

0

z50 45 5: ;5

In calculatingelliptic functionsfrom theta functions,whenthe modularangleexceedsabout 60”,usethe descending Landentransformation16.12 to inducedependence on a smallermodular angle.

17. Elliptic

Integrals

L. M. MILNE-THOMSON

1

Contents Mathematical Properties .................... 17.1. Definition of Elliptic Integrals

............. 17.2. Canonical Forms ................... 17.3. Complete Elliptic Integrals of the First and Second Kinds . . 17.4. Incomplete Elliptic Integrals of the First and Second Kinds . 17.5. Landen’s Transformation ................ 17.6. The Process of the Arithmetic-Geometric Mean ...... 17.7. Elliptic Integrals of the Third Kind ...........

Numerical Methodls ....................... 17.8. Use and IExtension of the Tables References

Page 589 589 589 590 592 597 598 599

600 600

............

..........................

606

Table 17.1. Complete

Elliptic Integrals of the First and Second Kinds and the Nome p With Argument the Parameter m ......... K(m),

K’(m),

15D; n(m), nl(m>,

15D; E(m), E’(m),

608

9D

m=O(.Ol)l Table 17.2. Complete

Elliptic Integrals of the First and Second Kinds and the Nome p With Argument the Modular Angle (Y ...... WY), K’(a),

ab>, PIN),

W4,

E’b),

610

151~

(r=o”(10)900 Table 17.3. Parameter

m With

Argument

K’(m)/K(m)

.......

612

10D

K’(m)/K(m)=.3(.02)3,

Table 17.4. Auxiliary Functions Parameter m .........................

for Computation

of the Nome p and the 612

15D

Qh>=alh>h, L(m)=-K(m),tKT

l,(s),

10D

ml=0(.01).15 Table 17.5. Elliptic a=o”(20)900, Table 17.6. Elliptic

a=O”(20)900,

1 University Standards.)

Integral

of the First Kind

5’3(100)850, o=o”(50)900,

Integral

(p=oo(50)900,

(Prepared

under

. . . . . . .

613

. . . . . .

616

8D

of the Second Kind

5”(10°)850,

of Arizona.

F((P\cz)

E((P\~)

8D

contract

with

the

National

Bureau 587

of

588

ELLIPTIC

INTEGRALS

Page Table

17.7.

Jacobian

Zeta Function

Z(~\CY) . . . . . . . . . . . .

619

Values of K(cr)Z( ~\a) cY=o”(20)900, 5°(100)850, (p=o”(50)900, Table

17.8.

Heuman’s

Lambda

A0 (rp\or)=F(~\go0-a)+2K(a) K’(a) 7r a=0°(20)900, Table

17.9.

n=o(.l)l,

Elliptic

Function

6D

A,,(q\a) . . . . . . . . . .

2(,\90°-a),

622

6D

5°(100)850, (p=O“(5°)900

Integral

of the Third Kind lT(n; q\(y) . . . . . .

cp,a=0°(150)900,

625

5D

The author acknowledges with thanks the assistance of Ruth Zucker in the computation of the examples, Ruth E. Capuano for Table 17.3, David S. Liepman for Table 17.4, and Andreas Schopf for Table 17.9.

i89

17. Elliptic Mathematical 17.1. Definition

of Elliptic

S

Properties 17.1.5

Integrals

If R(x, y) is a rational function of x anld y, where y* is equal to a cubic or quartic polyno:mial in z, the integral 17.1.1

Integrals

(2--s)boJ,-s+3b1(3--2s)J,-z+bz(l--s)J,-l +$b3(1-2~)Js-~b4Js+l=y(x-~)-S

(s=l,

R(x,y)dx

is called an elliptic integral. The elliptic integral just defined can not’, in general, be expressed in terms of elementary functions. Exceptions to this are (i) when R(x, y) contains no odd powers of y. (ii) when the polynomial yy2has a repeated factor. We therefore exclude these cases. By substituting for yz and denoting polynomial in x we get 2

By means of these reduction formulae and certain transformations (see Examples 1 and 2) every elliptic integral can be brought to depend on the integral of a rational function and on three canonical forms for elliptic integrals. 17.2. Canonical

by p&c) a

Definitions

m=sin2

a!; m is the parameter, a! is the modular x=sin

(0=sn u

17.2.4

+YPdx) =R (x)+Iljp!dd 1 YPh) Y

(1-m

sin2 v)*=dn

u=A(cp),

where RI(x) and R2(x) are rational functions of x. Hence, by expressing R,(x) as the sum of a polynomial and partial fractions

17.2.5

q=arcsin

(sn u)=am

JR(x,

17.2.6

y)dx=SR,(x)dx+Z,A,Sx”yU1:

Reduction

S

[ (x-c)‘y]-‘dx

Elliptic

Integral

the delta amplitude u, the amplitude

of the

First

Kind

F(~\~)=F(alm)=~‘(1--sinlcr

17.2.7

=

Formulae

s0

sin28)-+d0

‘[(l-t”)(l-mt2)]-+dt

=St4

Let 17.1.2

dw=u

0

y2=aG’+a1$+a2z8+a3x+a4

(la01+ Iad +O>

=b,(~-C)4+b!(x-C)3+b&-C)2+b&-c)+t’4 1’7.1.3 I,=

s

xsy-‘dx, J,=

S

(lbol+ hl$0)

Elliptic

17.2.8

of the

Second

Kind

E(~\a)=E(a~m)=~z(1-t2)~~(l-mt2)~dt 0

and

17.1.4

17.2.9

=

17.2.10

=

s0

y1-

sin2 cysin2 O)ide

S

Udn2wdu,

0

(s+2)a0~~+3+3a,(2s+3)I,+2+a2(s+1)1,+,

a See [17.7] 22.72.

Integral

[y(x-c)s]-‘dx

By integrating the derivatives of yx” y(~-c)-~ we get the reduction formulae

+3~(2s+1)Is+sa4111_l=x~y

angle

cos cp=cn u

17.2.3

JPl(4+YP2bN km-YPh>lY { [P3(X)12-Y2[P4(X) I’1 Y

+z,B,

Forms

17.2.1

17.2.2

=PdX)

2, 3, . . .)

(s=O, 1, 2, . . .) 17.2.11

=m++m

u cn2 w dw

S 0

590

ELLIPTIC

In computation the modulus is portance, since it is the parameter ment which arise naturally. The the modular angle will be employed to the exclusion of the modulus.

u

17.2.12

E((P\a)=u-

s1-2w dw

m s 0

17.2.13

=-

T 8:(42K)+E(m)u 2K(m) &b@K) K(m)

(For theta functions, Elliptic

see chapter 16.)

Integral

of the

Third

The

Kind

17.2.14 Cl- n sin2 6)-l [l -sin2

n(n;P\4=Jo~

(Ysin2 B]-1’2dB

If 2=sn (ulm), 17.2.15

n(n;ulm)=

S S

’ (l-nt2)-1[(l-t2)(l-mt2)]-1/2dt

=

o” (l- n sn2 (wlm))-ldw The

17.2.17

Amplitude

cp=am u=arcsin

can be calculated

from Tables The

z

17.5 and 4.14.

Complete

Elliptic Integrals and Second Kinds

of the

First

17.3.1

m

Parameter

11

Referred to the canonical forms of 17.2, the elliptic integrals are said to be complete when the amplitude is +n and so z=1. These complete integrals are designated as follows

p

(sn u) =arcsin

Characteristic

of minimal imand its compleparameter and in this c’hapter

The elliptic integral of the third kind depends on three variables namely (i) the parameter, (ii) the amplitude, (iii) the characteristic n. When real, the characteristic may be any number in the interval (- 03, a). The properties of the integral depend upon the location of the characteristic in this interval, see 17.7. 17.3.

0

17.2.16

INTEGRALS

[K(m)]=K=~1[(l-t2)(l-mt2)]-1~2dt

Dependence on the parameter m is denoted by a vertical stroke preceding the parameter, e.g., Fblm>. Together with the parameter we define the complementary parameter ml by

=S *I2

(l-m

sin2 O)-1/2dB

0

17.3.2

K=F($rlm)=F(%r\cr)

17.3.3 m+ml=

17.2.18

1

E[K(m)]=E=~01(l-t2)_‘Il(l_mt2)112dt

When the parameter is real, it can always be arranged, see 17.4, that 0
=S

*I2

(I- m sin2

0

The

Modular

Angle

cc

Dependence on the modular angle (Y, defined in terms of the parameter by 17.2.1, is denoted by a backward stroke \ preceding the modular angle, thus E((P\(Y). The complementary modular angle is r/2--a or 90°--a! according to the unit and thus ml=sin2 (90°-a)=cos2 LY. The

Modulus

k

17.3.4

In terms of Jacobian elliptic functions (chapter the modulus k and the complementary modulus are defined by

E=E[K(m)]=E(m)=E(&\F\ol)

We also define 17.3.5 rt !

K’=K(m,)=K(l-mm)=

S

(l-ml

K’=F(&rjm,)=F($r\$7r--m)

17.3.7

E’=E(m,)=E(l-mm)=

*I2

S

(1 -ml

0

They

k=ns (K+iK’),

are related

k’=dn

K.

t,o the parameter

by k2=m,

k’2=m

Depkndence on the modulus is denoted comma preceding it, thus II&; u, k).

by a

sin2 8)-1/2d8

0

17.3.6

16),

17.2.19

e)‘i2d8

17.3.8

sin2

e)lj2d0

E’=EIK(m,)]=E(mI)=E($r\h-4

K and iK’ are the (‘real” and “imaginary” quurter-periods of the corresponding Jacobian

elliptic

functions

(see chapter 16).

ELLIPTIC Relation

to the

Hypergeometric (see chapter 15)

17.3.9

K=$

17.3.10

E=*aF(-*,

Function

17.3.22

K=f

Series

17.3.24 am u=v+g

[l+c>’

rn+(Ey

m2

+(e6>”

(Iml
m3+ . . .]

y-(&g

Legendre’s

lii

17.3.26

52 [K-4

17.3.27

lim m-‘(K-E)=lim

17.3.28

lim p/m=lim

Auxiliary

L(m)=

m=

1- 16 exp

17.3.16

m=

16 exp

m-‘(E-mm,K)=r/4 m-10

m-10

m-90

Alternative

ql/m,=1/16

ml+1

hvaluations

of

K and

E (see also 17.5)

E(m)=(l+mj~2)E([(l-m~~2)/(i+m~~2)]2) -2m~~2(1+m~~2)-1K([(1-m~~2)/(i+m~~2)]~)

[--r(K(m) + L(m))/K’(m)]

17.3.31

K(cu)=2F(arctan

17.3.32 The function

In (16/m,)]=O

17.3.30

In $-K(m)

17.3.15.

=o

K(m)=2[1+m~‘2]-1K[(l-m~~2)/(i+m~~2)]2

Function

KG

K’(E-K)

17.3.29

Relation

cEK’+E’K--K.K’=;a

17.3.13

17.3.14

(I4-G)

. . .]

Values

17.3.25

f

-(F6)y--

2’ sin 2sv where v=ru/(2K)

.9=1 SC1+a’“>

Limiting

17.3.12 [l-(;y

A?.!-

a=1 l+qzs

3; 1; m)

Infinite

E(m)=&

,r+~2~ 2

17.3.23

rF($, 3; 1; m)

17.3.11

K(m)=&

INTEGRALS

[--*(K’(m)

+ L(ml))/K(m)]

L(m) is tabulated

in Table 17.4.

q-Seriee

The Nome q and the Complementary 17.3.17

q=q(m) =exp [-rK’/Kj

17.3.18

ql=q(m,)=exp

Nome p1

I%(Q) =2E(arctan Polynomial

(sec1/2 &)\a) (sec’12 CX)\CX) -1 fcos

Approximations

3 (0 5

Q

m< 1)

17.3.33

K(m)=[~+alm,+azm:l+[bo+blm, +b2mfl In OlmJ+c(m> le(m)j13X10-6 ao= I. .38629 al= .11197

[-rK/K’]

44 23

az= .07252 96

b,,= .5 bl=.12134

78

bz= .02887 29

17.3.34

17.3.19

K(m)-[ao+arml+ 17.3.20

. . . +a4m~l+[bo+bIm,+

...

+b4mtl ln Wml>+4m> le(m)l12XlO-*

log,, i log,, i=

(?rlog,, e)a= 1.86152

28349

to 10D

17.3.21

q=exp

[--lrK’/K]=E+8

(Gy+y +992

0

g

(zy

4+ . . .

(Id
ao= 1.38629

436112

al=

.09666 344259

a~=

.03590

a3= a(=

.03742 563713 .01451 196212

092383

bo= b,= b2= b3= b4=

.5 .12498

593597

.06880 248576 .03328 355346 .00441 787012

8 The approximations 17.3.33-17.3.36 are from C. Hastings, Jr., Approximations for Digital Computers, Princeton Univ. Press, Princeton, N. J. (with permission).

ELLIPTIC

592

INTEGRALS

17.3.35

~(m)=[l+a~m~+azm~l+[blm,+bzm~l In (lh> f&d Jr(m)J<4X10S6 b1= .24527 27 b2=.04124 96

al= .46301 51 a,=.10778 12 17.3.36

. . . +a4mf]+[blmI+

E(m)=[l+aIm,+

. ..

+b4m:] In (l/m,) +6(m) Ie(m)1<2X10-g al= .44325 a3= .06260 a3= .04757 al=.01736 17.4.

Incomplete and

I 00

FIGURE

I I 100

I 20’

17.1.

I

I I 300

I I 400

I t 500

I 600

Complete elliptic

I

I 700

1

1 1 1 600 90’

17.4.1

R--h4

17.4.2

EC-

integral of the first

kind.

of the

Negative

.Q

i

17.4.4

1.6

17.4.5

c

.04069 697526 .00526 449639 of the

First

Tables

Amplitude

= --F(colm)

cplm) = --E(h)

Amplitude

17.4.3

.24998 368310 .09200 180037

Elliptic Integrals Second Kinds

Extension ,,SL

bl= bz= ba= bd=

141463 601220 383546 506451

of Any

Magnitude

cp[m) =2sKfF(cp(m)

F(s?rf

E(u+2K) E(uf2X’)

=E(u)+2E

=E(u) +2i(K’--E’)

17.4.6

E(u+2mK+2niK’) 17.4.7

=E(u)

E(K-u)

+2mE+2ni(K’-E’)

=E-E(u)+msn Imaginary

u cd u

Amplitude

If tan @=sinh cp 17.4.8

F(icp\cY) =iF(B\&r-a)

17.4.9

E(+\a)

= -iE(e\+-cz)

+iF(e\+-a) +i tan

Jacobi’s

Imaginary

e(i -cos2 ar sin2 e>i

Transformation

17.4.10

E(iulm) FIWRE

17.2.

Complete elliptic second kind.

=&+dn(ulmI)sc(ulml) Complex

ini%grd of the 17.4.11

F(q+i$lm)

Amplitude

=F(Xlm)+iF(~ImI)

-E(ulmdl

ELLIPTIC

593

INTEGRAL&

where cot2 X is the positive root of the equation x2-[cot2 cp+m sinhv csc2p--mJx-ml cot:“(a=O and m tan2 PI= tan2p cot2X- 1.

Parameter

17.4.15

Than

F(cp(m)=m-+F(Olm-l),

Unity

sin O=m* sin cp - (m- 1)~

17.4.16 E(ulm) =mtE(um~lm-l)

17.4.12 E((P+i#\cY)

=E(X\a)

by which a parameter greater than unity replaced by a parameter less than unity.

-iE(p\90°-a) +iF(p\90°-a)+qp

Negative

where

can be

Parameter

17.4.17

bl=sin2 bz=(l

Greater

a sin X cos X sin2 ~(l-sin2 -sin2

(Ysin2 A)*

a! sin2 A) (1 - cos2 LYsin2 ~)4 sin j.6mcosP

b3=cos2 p+sin2 Amplitude

(11sin2 X sin2 P Near

to s/2

(see

-(l+m)+F

G--PI

m(l+m)+)

17.4.18

also 17.5)

If cos LYtan (0 tan $= 1 17.4.13

F((P\a)

+F(l/\a)

=F(?r/2\cY)

=K

cd(u(l +m)*lm(l +m)-l)}

17.4.14 E((p\a)

+@#\a)

=E(?T/~\cY) +sin2a

sincp sin+

Values when cpis near to lr/2 and m is near to unity can be calculated by these formulae.

whereby computations can be made for negative parameters, and therefore for pure imaginary modulus.

I.0.6 -

406

.6 30”

.42o” .2

-

IO’ 0 I, I I I I I I I I I I I I I I <---,a 3o” 4o” so” 60° 7o” 60° 9’f 0o IO’ 20°

FIGURE

17.3.

Incomplete elliptic integral first kind. F(v\d,

p constant

qf

the

00,00 FIGURE

100

200

17.4.

,o*

4OQ

300

600

70*

SO’

SO’

Incomplete elliptic integral of the .first kind. F(y~\4,

a constant

ELLIPTIC

594

INTEGRALS

2 2

I.2 -

60' 7.5.

2

1.0 -

.

.6 I

.6 -

I

.4 -

I I 0'

IO'

20'

30'

FIGURE 17.7.

40°

50'

60'

70'

Incomplete elliptic second kind.

60'

integral

90'

'

of the

aconstant

E(9\4, goe.ELQp-Q. t

Q

FIGURE 17A.

p-90’

F((P\~ 7~

a! constant.

1.6 -

LO .6 -

500 10~

3CP

200

FIGURE 17.8.

400

SO0

m'

6Q

E((o\4 90’ 7-q’

60'

(Yconstant.

17.4.21 F((p\90°) =ln (set pftan FIGURE

17.6.

Incomplete elliptic second kind. 9constant E(v\a),

17.4.20

of the

F(icp\SOO) =i arctan (sinh (p>

17.4.23

Eb\O) = cp

17.4.24

= cp

17.4.25

F(icp\O) =icp

17.4.26

F(v\O)

.p) =ln tan

17.4.22

Casee

Special

17.4.19

integral

E@o\O) =ilp E(v\900) E(i(p\90’)

SO0

=sin

lp

=i sinh cp

ELLIPTIC Jacobi’s

l-7.4.27

Zeta

Heuman’s

Function

Z(,\CY) =E(cp\cu) -E(cr)F(+o\dK(c.~)

17.4.28

Z(+-L)=Z(u)=E(u)-uuE(m)/K(m)

17.4.29

Z( -u) = -Z(u)

17.4.30

Z(u + 2K) = Z(u)

Lambda

Function

17.4.39 Aob\4

=

F(qY\90°-a) 2 +; K(a) Zb\90° -a) K’(a)

17.4.44I

=~{K(a)&Y\90°-cY)

-[K&l --E(4 lQ\90°--a) 1

Z(K-u)=-z(K+u)

17.4.31 17.4.32

595

INTEGRALS

r0fcpa)

Z(u)=Z(u--K)--msn(u--K)cd(u--IQ

t

Special

17.4.33

Values

*

Z(u(0) =o

Z(uIl)=tanh

17.4.34

Addition

u

Theorem

17.4.35

Z(u+v)=Z(u)+Z(v)--msn Jacobi’s

u sn 2, sn(u+E)

Imaginary

Transformation

17.4.36

iZ(iu~m)=Z(zllm,)+~,-dn(ulml)sc(zllm,) Relation

17.4.37

to Jacobi’s

Theta

Z(u)=@‘(u)/C3(u)=~ln

.2 .I -

Function

5’

O(u) FIGURE 17.10.

Heuman’s

lambda junction

A,(p\a).

q-Series

Numerical

17.4.38

27r (ID

Z(u) =K8s

f(l-

p**)-* sin (?rsu/K>

KlalZlp\oJ

FIGURE 17.9.

*seepageII.

Jacobian

zeta junction

K(ar)Z(+~\a).

Evaluation First

of Incomplete and Second Kinds

Integrals

of

the

For the numerical evaluation of an elliptic integral the quartic (or cubic 4, under the radical should first be expressed in terms of t*, see Examples 1 and 2. In the resulting quartic there are only six possible sign patterns or combinations of the factors namely (t*+a*)(t*+b*), (a*-t*)(t*-b*), (a*-t*)(ZP-t*), (t*-a*)(t*-b*), (t*+a”>(t*-b*), (t*+a*)(b*-P). The list which follows is then exhaustive for integrals which reduce to F(p\a) or E(p\a). The value of the elliptic integral of the first kind is also expressed as an inverse Jacobian Here, for example, the notation elliptic function. u=sn-I2 means that z=sn u. The column headed “t substitution” gives the Jacobian elliptic function substitution which is appropriate to reduce every elliptic integral which contains the given quartic. 4 For an alternate 17.4.70.

treatment

of cubits see 17.4.61 and

Equivalent Inverse Jacobian Elliptic Function

&~\a)

yo

t Substitution

Wv\a) -

cos a = b/a a>b m= (aa-P)/aa

17.4.41 Ia Z dt 0 [(F+d)(ta+ba)]1’2 IJ < 17.4.42

SC-* ($y)

tan q=i

l=b

SC v

a s .m[(P+a2)fj+P)]*12

m-1 ($y)

tan +T=%

l=a

cs v

17.4.43 E dt a s b [(aa-ta)(P-bz)]*‘4

nd-l(ily)

l=a

dn v

dn-1 (ZIy) I dl s [(a2-P)(bl--ta)]*'a 17.4.46

a0

sin (I = b/a a>b m = Plaa

(a

at-x2 aa--ba

sin c=? b cd-1

(I >

&-I

" !! (I)a a2

17.4.47

? b” b a2

ns-*(Ixb2 -a-aa > cota=- ab 17.4.50 m=as/(as+bl) I

sin2 cp=-

t=bsnv

I &la~=bqa2Lx2) ~a2(P-x2) l=b sin2 (p=-

x2-ae 39-F

Pdt -1 a a s = [(az-F)(F-P)]l’z 1 = (aa - t*) dl ii s 0 [(aa-ta)(ba-t2)]1’2

cd v

t=a dc v

sin +J=:

t=a ns v

aJzrn(

b

t=b nc v

ba z ta+a2 (aa+b2)*” s b ta [(P+a2)(EP)]l/*

cos ‘p=i sin* cp=-d+bn a2+xa

t=(d+P)*f*dsv

xa(aa+ P) sina p= p(a2+xs)

t=(aa+P)Ila ab sdv

CO8 $2

t=bcnv

b

7)

[(P--a2);lLP)]*~”

(a2+@‘qm& 1 aa(aa+W1’z s o’(f2+a*)

dt [(p+aa)(&2-~)]l/’ dt [(c+aZ)@2-@)]1/2

ELLIPTIC Some Important

~597

INTEGRALS Special

Cases

=

Z!Z

fF(9\4

008

a

9

&(9\4

co9

17.4.53 -

dt

s + (1Sl”P

45O

l-29 1+x1

45O

=

dt s 0 u+w

L s-

17.4.55 dt * (P-l)+

2-l-h x-1+&

dt

=

&+1-x &1+x

dt

s 1 (ta-i)+ 17.4.59 ’

45O

X

17.4.56 1-2 111 1 dt s 0 (l---t’)4

-

s z (ta-i)* 17.4.58

1

=

2 “1

17.4.57

X2-1 x2+1

17.4.34

&-1+x &+1-x

dt

s I (1-W 17.4.60

1-43-x 1+&x

dt

45O

X

(p

_-

_-

s d-(1--P)+

-

Reduction polynomial 0-h)

of dt@ where P=P(t) is a cubic s with three real factors P=

(t-&> 0-h)

where A>A>h

Write

Reduction of s&/p when P=P(t)=t3fa,t2 +a2t+e is a cubic polynomial with only ones real root t=@. We form the first and second dehivatives P’(t), P”(t) with respect to t and then jvrite

17.4.61

x=;(81-83)1’2,

17.4.70

m=sin2 a=-,62-Sa

ml=ccps2 ~=k& A-83

17.4.62

sina--x-83

= dt

A

X2=[P’(/3)]i/2, m=sin2 a=$-; =

17.4.71 = dt x s u JF

co*

F(9\4

(x-8) CO8 9=(x-p)+p

~

‘=p+(,+)

17.4.72

17.4.63 x

K-(x-8)

F(v\4

9-B2-83

s4

s2

-AZ

(81-/%)(x-A) cos8 9=(h-83)(81-4

h dt s JF

17.4.64 * dt

A

sin’

F(v\4

s 81 JF

9=-

x--81

z-82

17.4.63 A

F(v\a)

17.4.74 u dt A -s z J(-P)

X-81 COB% (p=x-83

17.5. Landen’s Transformation

17.4.66

Descending

F(v\

(90” -a”))

F(v\

(90” - ~‘1)

17.968 = dt Xs 8s d-P

cod

c-81-2

on-63

s- =

F(9\

(90” -a”))

sin’

dl CP

Transformation

17.5.1

b--Ba)@--8s) 9=(8*-/9*)(x-883j

cosla

-- Z-81 ‘-&-&

17.5.2

6

angles such thpt

(1 +siIl (Y&I) (1 +cos CYJ=2

and let Q,,, Q,,+~ be two corresponding such that

17.4.69 h

Landen

Let CY*,CY,,+~be two modular

17.4.67

x

F(cD\W’-4)

tan (Q,,+~- an>=cos (Y, tan pn

(%‘+I $a”) amplithdes (vn+l>cp.)

5 The emphasis here is on the modular angle sine is an argument of the Tables. All formulae Landen’s transformation may also be expressed in of the modulus k=mi=sin a and its complement k’ =cos (1.

598

ELLIPTIC

INTEGRALS

With

Thus the step from n to n+ 1 decreases the modular angle but increases the amplitude. By iterating the process we can descend from a given modular angle to one whose magnitude is negligible, when 17.4.19 becomes applicable. With LY~=(Ywe have

17.5.13 17.5.14 17.5.15

17.5.3 F(cp\cr)=(l+cos

4-1F(P1\LYl) =Hl+sin

17.5.16

+Y(P~\~

F(cp\ru) =2(l+sin

F(Cp\cY) = 8i,(1 +cos CYJF((On\(Y,)

I%\4

= Cpjl

(1 +sin

a,>

K=F(b\a)=$r

8tI (l+sin

F&\a)

17.5.8

a,)

=F((p\a)

1

-tF sin a1 sin K+

[ 1 -i

of a Right

17.6. The Process

=27r-‘Ka

sin2 (Y(1 +f sin cyl

. . .)]+sinct![a

Angle

(see also 17.4.13)

of the Arithmetic-Geometric Mean

Starting with a given number triple (a,, b,,, c,) we proceed to determine number triples (al, h, cd, (a~, b2, cd, . . . , (aN, bN, CN) according to the following scheme of arithmetic and geometric means

17.5.9

I&\,)

*=lim (pn n+m

When both (p and (Y are near to a right angle, interpolation in the table F(cp\a) is difficult. Either Landen’s transformation can then be used with advantage to increase the modular angle and decrease the amplitude or vice-versa.

17.5.6 17.5.7

csc or8Qsin (r8]$In tan ($?r+*@)

F(p\a)=[

Neighhorhood

17.5.5

ff)-lF&\~~~) (1 +sin a,)-lF((p,\(y,)

F(cp\ol) =2n 2:

17.5.17

F(p\~4=2-“~~~(l+sinol,)F(q,\a,)

17.5.4

~,=cr we have

(sinar,)1/2sin~,

1

+i2 (sin (Yesin 1y2)l’* sin pz+ . . , 17.5.10 E=K

[

l-f

sin* (Y l+f (

sin a,-$

sin a1 sin a2 ev=Hkl+bN-l)

1 +Psinculsina2sinaQ+. Ascending

Landen

(l-+-sin cu,)(l+cos

sin (2cp,+,--,J

cl=3

(ao--bo)

c2=3

(al-h)

angles such that

cy,+J =2

and let (on, vn+l be two corresponding such that 17.5.12

)I

co

..

Transformation

Let (Y,, anfl be two modular 17.5.11

bN=(aN-lbN-l)f

=sin ~~,sin (Pi

(ar,+I>or,) amplitudes (v~+,<&

Thus the step from n to n+l increases the modular angle but decreases the amplitude. By iterating the process we can ascend from a given modular angle to one whose difference from a right angle is so small that 17.4.21 becomes applicable.

We stop at the Nth step when aN=bN, i.e., when cN=O to the degree of accuracy to which the numbers are required. To determine the complete elliptic integrals K(a), E(a) we start with 17.6.2

~=l,

b,,=cos LY,c,,=sin (Y

whence 17.6.3

K(a) =& N

ELLIPTIC 1,

.

6 4 .

w-4

-m4

=z 1 [c”o+26+2*c;+

. . . +-2Ncz,]

K(a)

To determine

K’(a),

E’(a)

aA=l, bi=sin

17.6.5

599

INTEGRALS

17.7.4 s*(v+a> -=!2 + It1 S*(v+3)

we start with

LY,ch=cos a!

5*31 a-‘@(1 - q2S)-1 sin 2sv sin 2s/3

17.7.5

&(8) ---Cot

whence

fl+4 5

$1 (P)

17.6.6

K’(a)=&

In the above we can also use Neville’s functions 16.36.

17.6.7

=z1 [c;~+2c:*+2%;*+

K’(a)-,??(a)

K’ (4

Q2S(1-2q28 cos 2fi+q*“)-l

. . . $2%&2,21

17.7.6

(ii)

Hyperbolic

Case

n> 1

To calculate F(cp\cr), E(cp\cy) start from 17.5.2 which corresponds to the descending Landen transformation and determine cpl, cpz, . . , q,V successively from the relation

The case n>l can be reduced O
17.6.8

17.7.8

tan G,

CPO=(O

Then to the prescribed accuracy 17.6.9

F(v\4

of the Third

@

17.7.9

&\a> Case

Kind

e=arcsin

17.7.1

II(n; (p\a)=

s0

17.7.2

p (l- n sin2 0) -‘(l -sin*

(i)

Hyperbolic

Case

(iii)

=KW Circular

sin* ff
Case

I

O
P=!m sin2 a

O
(n/sin* cr)‘,

-WV\4

[(l --n)/cos* (u]’

17.7.10 v=$~F(cp\a)/K(a),

B=arcsin

17.2.4.

j3=$aF(t\90°---a)/K(a)

(Ysin2 0) %13

II (n; 4 ~\a) = II (n\a) Case

(~\a, +F(v\d

where A(q) is the delta amplitude,

=cl sin (Pl+c2 sin qn+ . . . +cN’sin Integrals

cw)]’

+&In1 [(A(v) -I-& tan cp> (A(v)--P, tan (PI-Y

= E(cp\4 - u-m7 F(cp\4

17.7. Elliptic

to the case

sin2 cr, pl=[(n-l)(I-nn-1sin2

N=n-’

*

17.6.10 a\4 *

17.7.7

H(n; (~\a, = -n(N;

=v%/(2%nr)

theta

n(n\~)=K(ar)+~lK(cu)Z(e\cr) Case

tan (cpn+l-cpJ=(bn/~n>

sin 2j3

a=1

016<$A

17.7.11

H(7L;

b2=[n(1-n)-1(n-sin2 (P\(Y)

cu)-l])

=6,(ii--4~V)

17.7.12

!I= d4

X=arcta.n

v = 3~G\OI)PW, 61=[n(l-n)-1(siny

(tanh ,5 tan v)

+2 2 (-l)~-1~--lp2s(l-p2a)-1 .?=l CY--~)-~]+

17.7.13

I[

C2 .a@*sinh 2sfi

17.7.3

p=

Ww\4=~1

17.7.14 p*ge

II.

1 -1

2s/3

a-1

n(n\a)=K((y)+3as2[1-~o(~\a)]

where & is Heuman’s *see

1+2 2 q”” cash

a=1

I-3 ln [&(v+@/~&-i31

sin 2su sinh 2s/3

Lambda

function,

17.4.39.

ELLIPTIC

600

INTEGRALS Special

Cases

n=O

17.7.18

no% (0\4 =Fb\4 17.7.19

n=O, a=0 NO;

cp\O) =cp

17.7.20

cY=o

n.l.y=45’

n(n; cp\O)=(l---n)-+

1.0

arctan

[(l-n)4

tan cp],

n- O.p-45.

* n
= (n- 1)-t arctanh [(n- 1)” tan cp], n>l FIGURE

17.11.

Elliptic integral NW cp\a).

Case

(iv)

Circular

=tan cp

of the third kind 17.7.21

The case n
(Y’S/2

II(n; (p\r/2) = (1 -n)-‘[ln

n
Case

to the

case

17.7.15

d In (l+n+

-f

ar; cp\cx)--F(cp\cr)}

17.7.23

cw-n)]g

n=l

[(lTsin

&cos (Y

n=sin2 cx

17.7.24

II(sin2 cu;~\a)=sec2~E(~\~)-(tan2 a-n)-l

II(N\~)

+sin2 ar(sin2CY--n)+K(cr)

Numerical

Example

Use and 1.

Extension

of the

Tables

Reduce to canonical form

where

s

Q1=3x2-10x+9, First

Q~-XQ2=(3+X)~2-(10+8X)~+9+10x *see page n.

Q2=-22+8x-10

sec2~~E((p\a)+sec~ (Ytan cpA(cp)

Methods i.e., if X=-i

is a per-

or f

and then Q2=; (z--1)2, Q,-f

Q2=; +2)2

Solving for Q1and Qz we get Ql=(2-1)2+2(2-2)2,

Q2=2(z--1)2-3(2-2)2

The substitution t = (z- 1)/(x-2)

Method

fect square if the discriminant

~(l;cp\ol)=~(cp\or)-

Q,+;

By inspection or by solving an equation of the fourth degree we find that

CYsin2p)/(2A((p))

n=l

17.7.25

(10+8X)2-4(3+X)(9+10X)=0; y-l&c,

y2=-3x4+3423-1192+1722-90

y2=QlQ2where

cp

7 (17 cos a)F((P\a)

&p2 sin 2p/A((p)]

17.7.17

17.8.

CY)tan +LY/A(c,o)]

a; cp\or)=&$ln[(l+tan

sin2 CY)]~II(N; cp\a)

II(~\CY) = (-n cos2a) (1 -n)-*(sin2

n#l

-A(cp>)(l-tan ~A((P))-~l-ki In [(A(v) +cos cr. tan q)(A(cp>-cos CYtan cp>-‘]

sin2 cy)]+II(n; (P\CY)

-I-p;’ sin2&‘((~\a) +arctan

sin cp)-‘1

(1Tsin (~){2II(fsin

=arctan

17.7.16

=[(l-N)(l-N-l

sin cp)(l-n) n= fsin (Y

2 cos cJI(lfcos

[(1--n)(l--n-l

(tan p+sec cp)

17.7.22

N= (sin2 a-n) (1 -TX)-’ p2=[-n(l--n)-‘(sin2

n=l

~y-1dz=i-~[(t2+2)(2t2-3)]-+dt

then gives

EL:LIPTIC

If the quartic y2=0 has four real roots in z (or in the case of a cubic all three roots are real), we must so combine the factors that no root of C&=0 lies between the roots of Q2=0 and no root Provided of Q2=0 lies between the roots of Q,=O. this condition is observed the method jus,t described will always lead to real values of X. These values may, however, be irrational. Second

601

INTEGRALS

The discriminant is

4 w= (2wf 1)2-4w=4w2+

and if we write

be

4T2=(8t2+10)2-4(t2+3)(10t2+9) (2t2-1)

Then y

dx= f

T-‘dt=

f

[(3t2+2)

(2t2-1)1-f&

The first method of Example 1 fails with the above values of Q, and Q2 since the root of Ql=O lies between the roots of Q2=0, and we get imaginary values of X. The method succeeds, however, if we take Q1=x, Q2=(2-l)(2-2), for then the roots of Ql =0 do not lie between those of Q2=0. Example 3. Find K(80/81).

Table

17.1.

Method

Q1 3x2--10x+9

4W=4(3w+2)

of Q2w-Ql

be

(2w- 1) =4(AW2+Bw+

c:)

Then if z2=W/w and Z2=(B-z2)2-4AC=(22-1)2-t48 $y-ldz=

iSZ-‘dz

However, in this case the factors of 2 are complex and the method fails. Of the second and third methods one will always succeed where the other fails, and if the coeffic.ients of the given quartic are rational numbers, the factors of T2 or 22, as the case may be, will be rational. Example 2. Reduce to canonical form ,y-lds s where y2=2(2- 1) (z-2). We use the third method of Example 1 taking &1=(x-l), Q~=z(z-2) and writing

Q, x-1 w=a;=-

Method

Table 17.4 giving L(m) is useful for computing K(m) when m is near unity or K’(m) when m is near zero.

w=~=--s2+8x-10 and let the discriminant

Method

Use 17.3.29 with m=80/81, ml=1/81, m:l’=1/9. Since [(l-mm:la) (1 +m:‘“)-‘J2=.64, K(80/81) = 1.8 K(.64)=3.59154 500 to 8D, takingK(.64) from Second

Write

C=i

z2= W/w and

Fkst

This method will succeed if, as here, T2 as a function of t2 has real factors. If the coefficients of the given quartic are rational numbers, the factors of T2 wiIl likewise be rational. Third

B=O,

J1/-‘dx=+z2-l)(z2+l)J-“Pda

Q2 -x2+8x-10

S-l s s

where A=l,

W=Au++Bw+C

t2-Q~- 3x2-10x+9

=4(3t2+2)

1

so that

Write

of Q#-Q1

(2wf 1)x+ 1

Z2=(B-z2)2-4AC=(z2)2-1=(z2-1)(z2+1),

Method

and let the discriminant

of Q2w- Q1=Zw-

K(80/81)=:

K’(80/81)

In (16X81)--L(80/81).

By interpolation in Tables 80/81=.98765 43210,

17.1 and 17.4, since

K’(80/81) =1.57567 8423 L(80/81) =.00311 16543 K(80/81) =~r-‘(1.57567 8423)(7.16703

7877)

-.00311 =3.59154

5000 to 9D. Third

Method

The polynomial approximation 8D K(80/81) =3.59154 Fourth

Method,

Here sin2 a=80/81 uo=l, giving

a,=;,

16543

17.3.34 gives to 501

Arithinetic-Geometric

Mean

and we start with co= m

2.99380

79900

602

ELLIPTIC

INTEGRALS

=

n

a,

0 1 2 3

d

b,

I

1.00000 .55555 .44444 .43738 .43735 .43735

00000 55555 44444 79636 95008 95003

.lllll .33333 .43033 .43733 .43735 .43735

-_ 11111 33333 14829 10380 94999 95003

The computation could also be made using common logarithms with the aid of 17.3.20. The point of this procedure is that it enables us to calculate p1 without the loss of significant figures which would result from direct interpolation in Table 17.1. By this means In (l/al) can be found without loss of accuracy. Example 6. Find m to 10D when K’/K=.25 and when K’/K=3.5. From 17.3.15 with K’/K=.25 we can write the iteration formula

c, .99380 .44444 .11111 .00705 .00002

79900 44444 11111 64808 84628

0

-

1

Thus K(80/81)=; Example

4.

?ru,-l=3.59154

510(Il.

Find E(80/81). First

m(“+“=

Method

Use 17.3.30 which gives, with m=80/81 E(80/81)=;

E(.64)-5 =1.01910

Then by iteration

K(.64)

0

17.1. i 3

Method

Method

1. .99994 .99994 .99994

]

&a)

n

0 .(3)26841 .(3)26837 .(3)26837

i 3

25043 65 65

[1.43249 712981

= .71624 85649. Using the value of K(80/81) fourth method, we have

found in Example

E(80/81) = 1.01910 6048 to 9D. 5. Find p when m=.9995. Here ml = .0005 and so from Table 17.4

Example

563013 =.00003

3,

Thus m= .00026 83765. The above methods in conjunction with the auxiliary Table 17.4 of L(m) enable us to extend Table 17.3 for K’/K>3, and for K’lKC.3. Example 7. Calculate to 5D the Jacobian elliptic function sn (.753421.7) using Table 17.5. Here

m=sin’ 12578 15.

From 17.3.19

ln (i)=dln

we can write the

m(“+‘)= 16e--3.br exp [ -rL(m,‘“))/K(m’“))

0

Q(m) =.06251 ql=mlQ(m)

and 17.4

42025 42041 42041

Thus m= .99994 42041. From 17.3.16 with K’/K=3.5 iteration formula,

Arithmetic-geometric mean, 17.6. The numbers were calculated in Example 3, fourth method, and we have

=;

17.1

&I’

n

Polynomial approximation, 17.3.36 gives E(80/81)=1.01910 6060. The last two figures must be dropped tokeep within the limit of accuracy of the method. Third

using Tables

6047

taking P(.64) and K(.64) from Table Second

1- 16em4” exp [ --~rL(m(“‘)/K’(m’“‘)].

a=.7,

Thus, sn (.753421.7)=sin from

a=‘56.789089’. Q where (p is determined

F(cp\56.789089’)= @=4/10.37324

= .95144 84701 q=.38618 125.

.75342.

1132

Inspection of Table between 40” and 45’. of F(9\d

17.5 shows that 9 lies We have from the table

ELLIPTIC

603

INTEGRALS Second

a

56’

58’

% 0

: 63803 73914

. 63945

. ;;;;;

45’ 50’

.84450 .95479

: E::: .95974

: 85122 .96465

v

\

60’

Method,

From this we form the table of F((p\56.789089’) F

P

A

A3

I35’

. 63859

4o”

. 74003

Quadrature

Simpson’s formula with 11 ordinates and interval .l gives .141117. Example 9. Evaluate

s2

-I

Numerical

First

Method,

4 [ (t”-2)

Reduction Bivariate

(t2-4)]-tdt. to Standard Interpolation

Form

and

that

a2=4,

A3

Here we can use 17.4.48 noting b2=2, and that

I10144 437 10581

45’

.84584

50’

.95674

72 509

s2

4 [(P-2)

(P-4)]-+&=

11090

s

I-

-

s

=f [F(q~~\45’) “F&45’) =f

A rough estimate now sEows that Qlies between 40’ and 41’. We therefore form the following table of F(q\56.789089’) by direct interpolation in the foregoing table

where

]

[1.854075-.535623]=.659226

2 2 2 sin (o~=--I sin p2=-7 sin2 a=--’ 2 4 4

Thus

F

4o:oo 40.5O 41.0°

.74003 .75040 .76082

Second

whence by linear inverse interpolation $0=40.5Of.5O

1

.75342- .75040 =40 644go .76082- .75040 ’

and so sin cp=.65137=sn (.75342].7). This method of bivariate interpolation is given merely as an illustration. Other more dfiect methods such as that of the arithmetic-geomletric mean described in 17.6 and illustrated for the Jacobian functions in chapter 16 are less laborious. Example 8. Evaluate

Method,

Numerical

Integration

If we wish to use numerical integration we must observe that the integrand has a singularity at t=2 where it behaves like [8(t--2)1-t. We remove the singularity at t=2, by writing

where f(t)=[(t2-2)(t2-4)]-f-[8(t-2)]-*. If we definef(2)=0,

S‘f (W 2

3 [(2Pfl) s2 , First

Method,

(P-2)]-“W.

Bivariate

Interpolation

From 17.4.50 we have

can be calculated by numerical quadrature.

S 2

4 [8(t-2)]-*dt=[$

Also

(t-z)*]‘+

and thus we calculate the integral as l+

S4f

(t)dt=l-.340773=.659227.

2

where 1 4 cos (p2=-Jz sin2 (Y=-I co9 (oi=-~ 5 3 2 Thus (r=26.56505 12’, (pi=61.87449 43’, lp2=45’, F((~~\~~)=1.115921 and F(a\a)=.800380 and therefore the integral is equal to .141114.

Example 10. Evaluate u=

S

- (z3-7z+6)-*ddz.

17

2--7x+6=(x-l)(z--2)(z+3) with &=2, p2=1, p3=-3,

and we use 17.4.65

ELLIPTIC

604 ~=-&2,C0S2

TYi=Sit12 &=4/s,

Thus cr=63.434949’,

Q=30’

Q=3/4.

the integral=F(90°\300)

and

=2(5)-fi(.543604)=.486214

Second

from Table 17.5.

n

We have

co9alI ; 3.

There is only one real zero and we therefore use 17.4.74 with P(t)=t?-22t2+12t-24, fl=2 so that P’(2)=16, P”(2)=8, X=2 and therefore m=sh2

,=I,

4

a=30°

2

,:,=;

Transformation

I

:02943 33333 333 725 00021 673

sin (2~~-90°)=sin

30°,

sin Q* .99956 904 .94280 663 .99999 998 Qpl=60°

sin

(2Q2-Q1)=Sin.

al sin (01,

Sin

(2Qa-&=Sin

a2

Sin

(2Qp-Q~)=Si.tl

cU3 Sin

Sin

cr,)

e,

lp2=57.367805' Q~= 57.348426'

Qa,

~,=57.348425'=@.

From 17.5.16

Therefore the given integral’is s-s0

Amending

1+cos CY,+~=2/(l+sin

dt.

(24-12t+2t2-ta)-1'2

0

Method,

to 5D.

We use 17.5.11 to give

The above integral is of the Weierstrass type and in fact 17= @(&A; 28, -24) (see chapter 18). Example 11. Evaluate 20

(1.071797) (1.001292)

=1.68575

u=2(5)-+F(30”\63.434949°)

S

=i

F(go0\300)

[F(Q,\~Q~)--F(Q~\~Q~)~

=A . 1 .942:O 904 1.999:6 663

where CO8 Ql=--1

1 3

~,=70.52877

CO9 Q2=-,

1 2

~2=6O'

93’

and the integral=3[1.510344-1.212597]=.148874. Example 12. Use Landen’s transformation evaluate

=1.37288

050 In tan 73.674213’

=1.37288

050(1.22789 30)

F(90°\300) = 1.68575 to 5D. Example 13. Find the value of F(89.5’\89.5’). First

to

This is a case where interpolation in Table 17.5 is not possible. We use 17.4.13 which gives F(89.5°\89.50)

First

Method,

Descending

Transformation

We use 17.5.1 to give 1 +sin cxl=

2 .=1.071797 1+cos 30

cos ,,=[(l-sin

CYJ(1 +sin CXJ]~/~= .997419

Method

= F(90’\89.5’)

where cot $=sin (.5O) cot (.5O)=cos (.5O) $=45.00109 084O and F($\89.5’) = .881390 from Table 17.5. F(90’\89.5’)

=K(sin2

89.5’) =K(.99992

=6.12777 1 +sin (Ye=

2 = 1.001292 ; co9 CY2=.999999 1 +cos al

Thus F(89.5q89.5’)

Landen’s gives

38476)

88

z5.246389. Second

1 +sin cys= 1 +cos 2 (Y2= 1 .oooooo Thus from 17.5.7,

- F(#\89.5O)

ascending

Method

transformation,

17.5.11,

ELlUPTIC

CO8 Cq=(l-sin

89.5°)/(l+sin

sin q=[(l-cos

CYJ(l+cos

89.5O) q)]*=.99999

99997

605

INTEGRALS

This is case (i) of integrals O
CO8 as=0 sin rYz=l.

n=$

89.5Osin 89.5’ =.99992

2+q-89.5°=89.29290490,

38476

q,=89.39645

a1 sin fi,

sin (2+9-tpcp,)=sin sin(2$o~-~)=sin~,

n=89.39645 Qs=fi=@.

hind,

(0=45~,ar=30~,

c=arcsin (n/sin2 0r)*=30°, /3= $rF(30°\300)/K(300) = .49332 60 V=&rIq45°\300)/K(300)=.74951 51,

17.5.12 ‘then gives

sin (2~--89.5°)=sin

of the third

61= (16/45)*

24L5' 602'

and so from 17.7.3 rl (A; 450\30°) =

Thus 17.5.16 gives (16/45)* F(89.5’\89.5’)

-3 In m+$$

C}

=

1

t 1 In (tan 89.69822 801°)=5.24640. .99996 19231 > Example

Evaluate

14.

q= .01797 24. Using the q-series, 16.27, for the d functions II (A; 45’\30’)

S

* [(9-P)(16+ta)3]-fdt 1

= (16/45)+ { - .02995 89 +(1.86096

to5D. Table

From 17.4.51 the given integral

=ssa- ,l=f W(Ql\4 -mQ2\4 0

21)(.74951 51)) =.813845.

17.9 gives .81385 with 4 point Lagrangian

interpolation. Example

Evaluate

16.

rI &\30°) a=36.86990’

sin QlD1=4fi, 5 s*

II (&\30”)

to 6D.

Table

p2=m’

=K(30°)

+ (16/45)“2K(~)Z(e\300)

e=arcsin(n/sin2@=300.

where lp2=23.84264'. 17.6 we

=;[.80904-.41192]=.00496. Simpson’s rule with 3 ordinates gives

= 1.743055.

II (g-; 450\30°) r/4

=

S

(1-g sin2 0)-l (1-a

0

15.

Evaluate

r/4

0

(1-f

sin”e)-yl-~

sin2

ey2 &I

to 6D.

lT($45°\300)=

S

using

Table 17.9 gives 1.74302 with 5 point Lagrangian interpolation. Example 17. Evaluate

~[.00504+.01975+.005]=.00496. Example

Thus

17.7

l-l (&\30”> in Table

elliptic

From 17.7.6 we have

pl=48.18968°

By bivariate interpolation find that the given integral

the complete

integral

where sin cy=&

we get

sinae)-+de

to 6D.

This is case (iii) of integrals sir? ff
Q=45’,

a=30°

of the third hind,

606

ELLIPTIC

e=arcsin [(1-n)/cos2 ~$=45~ ,9=~~rF(45"\60~)/K(30")=.70317 v=$rF(45°\300)/K(300)=

INTEGRALS

Here n=i, 74

(p=45’, (r=30° and since the character-

istic is greater than unity we use 17.7.7

.74951 51

N=n-’

sin2 (~=.2, p,=(1/5)* II (2; 450\30°) = -H( 2; 45’\30’) + F (450\30°)

62= (40/g)* q= .01797 24

and so from 17.7.11

+(‘I@

(7/w+ (l/5)1 In (T/8)+- (l/5)+

II ($; 450\30°)= (40/9)“yx-4jLv) =2.10818 51{ .55248 32-4(.03854

=--83612t.80437

26)

(.74951 51)) =.921129. Table

17.9

gives .92113 with 4 point

Lagrangian

interpolation. Example

=1.13214. Evaluate

18.

the complete elliptic

Numerical

integral

Example

II (9\30’) to 5D.

From 17.7.14

IL (-*;

we have

quadrature gives the same result. Evaluate

20.

450\30°)

=S */4

II (~\30°)=K(300)+;~~ where e=arcsin Table

to 5D.

[(~--~)/cos~cx]~~~=~~~. Thus using

17.8

Table 17.9 gives 2.80126 by 6 point Lagrangian interpolation. The discrepancy results from interpolation with respect to n for (o=90° in Table

sin2 @-‘(l-t

is negative

with n=-i,

N=(l--n)-‘(sin2

a-n)=.4,

+$5/2)*F(45°\300) *I4

sin2 c?)-‘(1-a

and we there-

sin2 cx=i

45’\30’) =(9/4O)*II

H (f ; 450\30°)

=S (l-9

sin2 0)-k&

~~~=fi

and therefore (5/2)* II (-;;

Evaluate

19.

Here the characteristic fore use 17.7.15

l-I ($\3OO) =2.80099.

17.9. Example

(l++

0

[1-~~(e\30~)]

Using Tables

sin2 19)~“~d0

4.14,

17.5,

(6; 45’\30°)

farctan

(35)-t

and 17.9 we get

0

to 5D.

II (-#;

45’\30’) = .76987

References Texts

[17.1] A. Cayley, An elementary treatise on elliptic functions (Dover Publications, Inc., New York, N.Y., 1956). [17.2] A. Erdelyi et al., Higher transcendental functions, vol. 2, ch. 23 (McGraw-Hill Book Co., Inc., New York, N.Y., 1953). [17.3] L. V. King, On the direct numerical calculation of elliptic functions and integrals (Cambridge Univ. Press, Cambridge, England, 1924). [17.4] E. H. Neville, Jacobian elliptic functions, 2d ed. (Oxford Univ. Press, London, England, 1951). [17.5] F. Oberhettinger and W. Magnus, Anwendung der elliptischen Funktionen in Physik und Technik (Springer-Verlag, Berlin, Germany, 1949).

[17.6] F. Tricomi, Elliptische Funktionen (Akademische Verlagsgesellschaft, Leipzig, Germany, 1948). [17.7] E. T. Whittaker and G. N. Watson, A course of modern analysis, chs. 20, 21, 22, 4th ed. (Cambridge Univ. Press, Cambridge, England, 1952). Tables

[17.8] P. F. Byrd and M. D. Friedman, Handbook of elliptic integrals for engineers and physicists (Springer-Verlag, Berlin, Germany, 1954). [17.9] C. Heuman, Tables of complete elliptic integrals, J. Math. Phys. 20, 127-206 (1941). [17.10] J. Hoiiel, Becueil de formules et de tables numbriques (Gauthier-Villars, Paris, France, 1901).

ELLIPTIC [17.11] E. Jahnke and F. Emde, Tables of functions, 4th ed. (Dover Publications, Inc., New York, N.Y., 1945). [17.12] L. M. Milne-Thomson, Jacobian elliptic function tables (Dover Publications, Inc., New York, N.Y., 1956). (17.131 L. M. Milne-Thomson, Ten-figure table of the complete elliptic integrals K, K’, E, E’ and a 1 table of 8&, sm, Proc. London IMath. sot. 2, 33 (1931).

INTEGRALS [17.14] L.

607

M. Milne-Thomson, The Zeta function of Jacobi, Proc. Roy. Sot. Edinburgh 52 (1931). [17.15] L. M. Milne-Thomson, Die elliptischen Funktionen von Jacobi (Julius Springer, Berlin, Germany, 1931). [17.16] K. Pearson, Tables of the complete and incomplete elliptic integrals (Cambridge Univ. Press, Cambridge, England, 1934). [17.17] G. W. and R. M. Spenceley, Smithsonian elliptic function tables, Smithsonian Miscellaneous Collection, vol. 109 (Washington, D.C., 1947).

608

ELLIPTIC T&k

17.1

COMPLETE AND

ELLIPTIC INTEGRALS THE NOME p WITH

INTEGRALS OF THE ARGUMENT

AND SECOND PARAMETER

K(m)=Jj (1-msin2B)-‘cl0

K’(m)=K(m,)

E(m)=Jj (l-m sin*O)‘dfl

E’(m)=E(n/])

g(m)=exp [-sK’(m)/K(m)] 1,L

FIRST THE

0.03 0. 04

1.57079 1.57474 1.57873 1.58278 1.58686

K(m) 63267 55615 99120 03424 78474

94897 17356 07773 06373 54166

3.69563 3.35414 3.15587 3.01611

73629 14456 49478 24924

0.05 0.06 0.07 0. 08 0.09

1.59100 1.59518 1.59942 1.60370 1.60804

34537 90792 82213 21610 32446 58510 96546 39253 86199 30513

2.90833 2.82075 2.74707 2.68355 2.62777

0.10 0.11 0.12 0.13 0.14

1.61244 1.61688 1.62139 1.62595 1.63057

13487 90905 31379 48290 55488

20219 05203 80658 38433 81754

0.15 0.16 0.17 0.18 0.19

1.63525 1.63999 1.64480 1.64967 1.65461

67322 98658 64907 82052 66675

0.20 0.21 0.22 0.23 0.24

1.65962 1.66470 1.66985 1.67507 1.68037

0.25 0.26 0.27 0.28 0.29

KINDS m

ql(m)==9(w)

K’(m)

0.00000

9(1”) 00000

00000

89875 99160 91841 77648

0.00062 0.00126 0.00190 0.00255

81456 26665 36912 13525

60383 23204 69025 13689

1.00 0.99 0.98 0.97 0.96

72484 24967 30040 14063 33320

44552 55872 24667 15229 84344

0.00320 0.00386 0.00453 0.00521 0.00589

57869 71356 55438 11618 41444

70686 22010 98018 66885 34269

0.95 0.94 0.93 0.92 0.91

2.57809 2.53333 2.49263 2.45533 2.42093

21133 45460 53232 80283 29603

48173 02200 39716 21380 44303

0.00658 0.00728 0.00798 0.00870 0.00942

46515 53858 28484 49518 89058 49815 30002 35762 53141 02678

0.90 0. a9 0.88 0. a7 0.86

64580 64511 98881 94514 22527

2.38901 2.35926 2.33140 2.30523 2.28054

64863 35547 85677 17368 91384

25580 45007 50251 77189 22770

0.01015 0.01089 0.01164 0.01240 0.01316

60362 53620 34936 06407 70202

37153 10173 87540 58856 86392

0. a5 0.84 0. a3 0.82 0.81

35986 07858 00860 34293 28228

10528 45692 83368 77219 48361

2.25720 2.23506 2.21402 2.19397 2.17482

53268 77552 24978 09253 70902

20854 60349 46332 19189 46414

0.01394 0.01472 0.01552 0.01632 0.01714

28572 83850 38457 94906 55806

75318 66891 56320 37206 74605

0.80 0.79 0. 78 0.77 0.76

1.68575 1.69120 1.69674 1.70237 1.70808

03548 81991 86201 39774 67311

12596 86631 96168 10990 34606

2.15651 2.13897 2.12213 2.10594 2.09037

56414 01837 18631 83200 27465

99643 52114 57396 52758 52360

0.01797 0.01881 0.01965 0.02051 0.02139

23870 01914 92872 99793 25853

08967 93399 66940 66788 82708

0.15 0.74 0. 73 0.72 0. 71

0.30 0. 31 0.32 0.33 0.34

1.71388 1.71978 1.72577 1.73186 1.73805

94481 78791 48080 56405 56096 29320 47782 52098 53734 56358

2.07536 2.06088 2.04689 2.03336 2.02027

31352 16467 40772 94091 94286

92469 30131 10577 52233 03592

0.02227 0.02317 0.02408 0.02500 0.02594

74361 57154 48765 35013 52661 67250 89803 73177 64110 66576

0.70 0. 69 0.68 0. 67 0.66

0.35 0.36 0.31 0. 38 0.39

1.74435 1.75075 1.75726 1.76389 1.77064

05972 25613 38029 15753 85048 82456 83888 a3731 73233.33534

2.00759 1.99530 1.98337 1.97178 1.96052

83984 27776 09795 31617 10441

24376 64729 27821 25656 65830

0.02689 0.02786 0.02884 0.02984 0.03085

79677 40785 51915 17757 43225

51443 93729 76181 44138 51033

0. 65 0.64

0.40 0. 41 0.42 0.43 0.44

1.77751 1.78451 1.79165 1.79891 1.80632

93714 88046 01166 80391 75591

91253 81873 52966 87685 07699

1.94956 1.93890 1.92852 1.91841 1.90854

77498 76652 63181 02691 70162

06026 34220 14418 09912 81211

0.03188 0.03292 0.03399 0.03507 0.03617

33473 13363 93907 86003 30208 70043 48344 66773 54594 93133

0. 60 0.59 0.58 0. 57 0.56

0.45 0.46 0.47 0.48 0.49

1.81388 1.82159 1.82945 1.83749 1.84569

39368 16983 27265 56821 97985 64730 13633 55796 39983 74724

1.89892 1.88953 1.88036 1.87140 1.86264

49102 30788 13596 02398 08023

71554 53096 22178 11034 32739

0.03729 0.03843 0.03959 0.04077 0.04198

55570 58239 69950 98463 51981

75822 43468 38753 75263 67183

0. 55 0.54 0.53 0.52 0. 51

0. 50 ml

1.85407

46773 K’(m)

1.85407

46773 K(nd

01372

0.04321

39182 91(m)

63772

0.50 m

0. 00 0. 01 0. 02

01372

ml

kz 0: 61

r.(73 1 &XZ Examples

3-4.

E(m) and E’(m) from L. M. Milne-Thomson, Ten-figure table of the complete elliptic integrals 1 1 K, K’A E’ and a table of -p s;(olv) 7’ Ji (o,v) Proc. London Math. Soc.(2)33,1931(with permission).

ELLIPTIC COMPLETE ELLIPTIC INTEGRALS AND THE NOME q WITH

X(m=Jj

INTEGRALS

OF THE FIRST AND SECOND ARGUMENT THE PARAMETER

(l-m sin2O)-IdO

E(m)=Jj (l-m sin28)‘dO q(m)=exp [-rK’(m) /K(m)] 7n 0. 00

41 (nl)

E(m)

KINDS m

Table 17.1

K’(m)=K(m,) E’(m)=E(q) CIl (m)=cJ(w) E’(m)

ill,

0. 01 0.02 0. 03 0.04

1.00000 0000000000 0.26219 6267917709 0.22793 4574067492 0.20687 9810847842 0.19149 6308209940

1.57079 6327 1.56686 1942 1.56291 2645 1.55894 8244 1.55496 8546

1.00000 0000 1.01599 3546 1.02859 4520 1.03994 6861 1.05050 2227

1.00 0.99 0. 98 0.97 0.96

0.05 0.06 0.07 0.08 0.09

0.17931 6006955723 0.16920 7531146133 0.16055 4201073011 0.15298 1481009741 0.14624 4269473236

1.55097 3352 1.54696 2456 1.54293 5653 1.53889 2730 1.53483 3465

1.06047 3728 1.06998 6130 1.07912 1407 1.08793 7503 1.09647 7517

0.95 0.94 0.93 0.92 0.91

0. 10 0.11 0.12 0.13 0.14

0.14017 3126954262 0.13464 5884592091 0.12957 1469520553 0.12488 0122352049 0.12051 7195728729

1.53075 7637 1.52666 5017 1.52255 5369 1.51842 8454 1.51428 4027

1.10477 4733 1.11285 5607 1.12074 1661 1.12845 0735 1.13599 7843

0.90 0.89 0.88 0.87 0.86

0.15 0.16 0.17 0.18 0.19

0.11643 90607 17472 0.11261 0316423363 0.10900 1833023834 0.10558 9345798477 0.10235 2423513544

1.51012 1831 1.50594 1612 1.50174 3101 1.49752 6026 1.49329 0109

1.14339 5792 1.15065 5629 1.15778 6979 1.16479 8293 1.17169 7053

0.85 0.84 0.83 0.82 0.81

0.20 0.21 0.22 0.23 0.24

0.09927 3697338825 0.09633 8274965990 0.09353 3288880648 0.09084 75434 60707 0.08827 12359 87862

1.48903 5058 1.48476 0581 1.48046 6375 1.47615 2126 1.47181 7514

1.17848 9924 1.18518 2883 1.19178 1311 1.19829 0087 1.20471 3641

0.80 0.79 0.78 0.77 0.76

0.25 0.26 0.27 0.28 0.29

0.08579 5733702195 0.08341 3393883117 0.08111 7417341165 0.07890 1728126084 0.07676 0874004317

1.46746 2209 1.46308 5873 1.45868 8155 1.45426 8698 1.44982 7128

1.21105 6028 1.21732 0955 1;22351 1839 1.22963 1828 1.23568 3836

0.75 0.74 0.73 0.72 0.71

0.30 0. 31 0. 32 0.33 0.34

0.07468 9943537179 0.07268 4496537110 0.07074 0505387511 0.06885 4305247167 0.06702 25515 69108

1.44536 3064 1.44087 6115 1.43636 5871 1.43183 1919 1.42727 3,821

1.24167 0567 1.24759 4538 1.25345 8093 1.25926 3421 1.26501 2576

0. 70 0.69 0. 68 0.67 0.66

0.35 0. 36 i-z 0: 39

0.06524 21836 78738 0.06351 0393400746 0.06182 45979 15898 0.06018 2416179938 0.05858 1648356838

1.42269 1133 1.41808 3394 1.41345 0127 1.40879 0839 1.40410 5019

1.27070 7480 1.27634 9943 1.28194 1668 1.28748 4262 1.29297 9239

0.65 0.64 0.63 0.62 0. 61

0.40 0.41 0.42 0. 43 0. 44

0.05702 0257814610 0.05549 6355309081 0.05400 8185043499 0.05255 4112342653 0.05113 2612721764

1.39939 2139 1.39465 1652 1.38988 2992 1.38508 5568 1.38025 8774

1.29842 8034 1.30383 2008 1.30919 2448 1.31451 0576 1.31978 7557

0.60 0.59 0.58 0.57 0.56

0.45 0.46 0.47 0.48 0.49

0.04974 2262164574 0.04838 1728453289 0.04704 9763416424 0.04574 5195980149 0.04446 6925925028

1.37540 1972 1.37051 4505 1.36559 5691 1.36064 4814 1.35566 1135

1.32502 4498 1.33022 2453 1.33538 2430 1.34050 5388 1.34559 2245

0.55 0.54 0.53 0.52 0. 51

0.50 ml

0.04321 3918263772 ?(4

1.35064 3881

1.35064 3881 E(m)

0.50

E’(m) (94 II 3

nc

610

ELLIPTIC

Table 17.2

COMPLETE AND

ELLIPTIC THE NOME

R(.)=Jj

INTEGRALS

INTEGRALS OF THE q WITH ARGUMENT

(l-sin2 01sin2 8)-‘d0

FIRST AND SECOND KINDS THE MODULAR ANGLE a

K’(a)=K(g~-c2) E’(a)=E(90%) ql(“)=q(9V-a)

K’ (4

Kc4 1.57079 63267 94897 1.57091 59581 27243 1.57127 49523 72225 1.57187 36105 14009 1.57271 24349 95227

Q(Q) 0.00000 00000 00000

5.43490 4.74271 4.33865 4.05275

9m8296 25564 72652 78886 39759 99725 81695 49437

0.00001 0.00007 0.00017 0.00030

90395 61698 14256 48651

55387 24680 42257 48814

19997 59694 24991 80266 29421

84146 78752 71838 68445 43555

0.00047 0.00068 0.00093 0.00122 0.00154

65699 66451 52197 24470 85045

16867 27305 97816 64294 16579

1.58054

09338 95721

65776

88648

3.83174 3.65185 3.50042 3.36986 3.25530

1.58284 1.58539 1.58819 1.59125 1.59456

28043 41637 72125 43820 83409

38351 75538 27520 13687 31825

3.15338 3.06172 2.97856 2190256 2.83267

52518 86120 89511 49406 25829

87839 38789 81384 70027 18100

0.00191 0.00231 0.00276 0.00324 OiOk76

35945 79450 18093 54674 $2262

90170 15821 29252 43525 86978

1.59814 1.60197 1.60608 1.61045 1.61510

20021 85300 13494 41537 09160

12540 86952 10364 89663 67722

2.76806 31453 2.70806 76145 2.65213 80046 2.59981 97300 2.55073 14496

68768 90486 30204 61099 27254

0.00433 0.00493 0.00558 0.00627 0.00700

34205 84132 45970 23946 22602

09983 64213 58517 95994 97383

1.62002 1.62523 1.63072 1.63651 1.64260

58991 36677 91016 74093 41437

24204 58843 30788 35819 12491

2.50455 00790 2.46099 94583 2.41984 16537 2.38087 01906 2.34390 47244

01634 04126 39137 04429 46913

0.00777 0.00859 0.00944 0.01035 0.01130

46804 01752 92999 26461 08432

16442 53626 75082 44729 78049

1.64899 52184 1.65569 69263 1.66271 59584 1.67005 94262 1.67773 48840

78530 10344 91370 69580 80745

2.30878 2.27537 2.24354 2.21319 2.18421

67981 64296 93416 46949 32169

67196 11676 98626 79374 49248

0.01229 0.01333 0.01442 0.01555 0.01673

45605 45085 14412 61584 95077

27181 07947 80638 97708 33023

1.68575 03548 12596 1.69411 43573 05914 1.70283 59363 12341 1.71192 46951 55678 1.72139 08313 74249

2.15651 2.13002 2.10465 2.08035 2.05706

56474 14383 76584 80666 23227

99643 99325 91159 91578 97365

0.01797 0.01925 0.02059 0.02197 0.02341

23870 57475 05967 80013 90910

08967 39635 10437 16901 88188

1.73124 51756 1.74149 92344 1.75216 52364 1.76325 61840 1.77478 59091

57058 26774 68845 59342 05608

2.03471 2.01326 1.99266 1.97288 1.95386

53121 65652 97557 22662 48092

85791 05468 34209 74650 51663

0.02491 0.02646 0.02807 0.02974 0.03147

50625 71830 67957 53239 42771

23981 76961 17219 19583 20286

1.78676 1.79922 1.81215 1.82560 1.83956

91348 15440 98536 18981 67210

85021 49811 62126 35889 93652

1.93558 1.91799 1.90108 1.88480 1.86914

10960 75464 30334 86573 75460

04722 36423 63664 80404 26462

0.03326 52566 95577 0.03511 99625 22096 0;03704 02001 87133 0.03902 78889 26607 0.04108 50703 79885

1.85407

46773 01372 K'(a)

1.85407

46773 01372 KM

1.57379 21309 24768 1.57511 36077 77251 1.57667 79815 92838 1.57848

0.04321

39182 63772 9lH

cI C-i’9

Compiled from G. W. and R. M. Spenceley, Smithsonian elliptic function tables, Smithsonian Miscellaneous Collection, vol. 109, Washington, D.C., 1947 (with permission).

ELLIPTIC

611

INTEGRALS

COMPLETE ELLIPTIC INTEGRALS OF THE FIRST AND SECOND KINDS AND THE NOME q WITH ARGUMENT THE MODULAR ANGLE a K(a)=Jj

(l-&n2

a sin2

Table 17.2

O)-Bdfl 1

E(a)=f

4(4=exp

(l-

km2 * 01sin2 O)“dO

[-rK’@)/K(4]

E'(a)=E(gOL) q1 (a) =q(90"-a)

-w

4164

E’(a)

1.00000 0.40330 0.35316 0.32040 0.29548

00000 00000 93063 38378 56482 96037 03371 34866 83855 58691

1.15707963267 94897 l.!j7067 67091 27960 l.fi7031 79198 97448 l.!i6972 01504 23979 l.!i6888 37196 07763

1.00000 1.00075 1.00258 1.00525 1.00864

00000 00000 15777 01834 40855 27552 85872 09152 79569 07096

0.27517 0.25794 0.24291 0.22956 0.21754

98048 73563 01957 66337 29743 06665 71598 81194 89496 99726

1.15678090739 77622 l.!i6649 67877 60132 l.!i6494 75629 69419 1.56316 22295 18261 l.!j6114 17453 51334

1.01266 1.01723 1.02231 1.02784 1.03378

35062 34396 69183 41019 25881 67584 36197 40833 94623 90754

0.20660 0.19656 0.18728 0.17865 0.17059

97552 00965 76611 43642 51836 10217 56628 04653 45383 49477

l.!i5688 l.!i5639 1.55368 1.55073 l.!i4755

71966 01596 97977 70947 08919 36509 19509 84013 45758 69993

1.04011 1.04678 1.05377 1.06105 1.06860

43957 06010 64993 44049 69204 07046 93337 53857 95329 78401

0.16303 0.15591 0.14919 0.14283 0.13680

35348 21581 66592 65792 73690 67429 65198 36280 08474 28619

l.!i4415 1.54052 1.53666 1.53259 1.52830

04969 14673 15741 27631 97975 68556 72877 45636 62960 54359

1.07640 1.08442 1.09265 1.10106 1.10964

51130 76403 52193 72543 03455 37715 21687 57941 34135 42761

0.13106 0.12559 0.12037 0.11539 0.11062

18244 99858 47852 09819 82455 07894 33684 49987 35386 78854

1.52379 1.51907 1.51414 1.50900 1.50366

92052 59774 85300 25531 69174 93342 71479 16775 21353 53715

1.11837 1.12724 1.13624 1.14534 1.15454

77379 69864 96377 57702 43646 84239 78566 80849 66775 24465

0.10605 0.10167 0.09746 0.09342 0.08953

40201 85996 16783 93444 47524 70352 26672 88483 58769 52553

1.49811 49284 22116 1.4.9236 87111 24151 1.4!8642 68037 44253 l.d.8029 26638 27039 1.4.7396 98872 41625

1.16382 1.17317 1.18258 1.19204 1.20153

79644 93139 93826 83722 90849 45384 56765 79886 81841 13662

0.08579 0.08219 0.07872 0.07537 0.07215

57337 02195 43773 66408 46415 92073 99738 58803 43668 98737

1.4.6746 22093 39427 1.4.6077 35062 13127 1.4.5390 77960 65210 1.4,4686 92406 95183 1.4,3966 21471 15459

1.21105 1.22058 1.23012 1.23966 1.24918

60275 68459 89957 54247 72241 85949 11752 88672 16206 07472

0.06904 0.06603 0.06313 0.06033 0.05763

22996 09032 86859 10861 88302 96461 83890 33716 33361 79494

1.43229 09693 06756 1.4.2476 03101 24890 1.4,1707 49233 71952 1.4,0923 97160 46096 1.40125 97507 85523

1.25867 1.26814 1.27757 1.28695 1.29627

96247 79997 65310 65206 39482 50391 37387 83001 80079 94134

0.05501 0.05249 0.05005 0.04769 0.04541

99336 98829 47051 04844 44121 29953 60340 17056 67490 83529

1.39314 02485 23812 1.2~848865913 75413 1.3'7650 43257 72082 1.26799 91658 73159 1.35937 69972 75008

1.30553 li31472 1.32384 1.33286 1.34180

90942 97794 95602 64623 21844 81263 99541 17179 60581 29911

0.04321 39182 63772 d")

1.35064 38810 47676

1.35064 38810 47676 E(4

E'(a) C-95)3

[I 1

612

ELLIPTIC Table

17.3

K’ K 0. 30 0.32 0.34

PARAMETER

INTEGRALS

m WITH A-’ K 1.20 1.22 1.24 1.26 1.28

E

0.99954 0.99912 0.99844 0.99740 0.99590

69976 85258 79307 80762 01861

0. 40 0. 42 0. 44 0.46 0.48

0.99380 0.99101 0.98739 0.98284 0.97726

79974 23521 58502 72586 54540

0.50 0.52 0.54 0.56 0.58

0.97056 0.96266 0.95352 0.94310 0.93138

27485 75125 60602 38029 57063

0. 60 0. 62 0.64 0. 66 0. 68

0.91837 0.90409 0.88859 0.87191 0.85413

61134 80105 18214 38254 42916

0. 70 0.72 0.74 0.76 0.78

0.83533 0.81560 0.79505

54217 91841 51193

1.64

0.77377 0.75188

81814 66711

0.80 0.82 0. a4 0. 86 0. 88

0.72949 0.70669 0.68361 0.66035 0.63700

0.90 0.92 0. 94 0.96 0.98

ARGUMENT

K’(m)/K(m) K’ K

m

,,a 0.02158

74007

0.02028 0.01906 0.01791 0.01683

61803 26278 21974 05990

0.30866 0.29292 0.27782 0.26335 0.24949

25998 52811 39170 17107 94512

2.10 2.12 2.14 2.16 2.18

0.23625 0.22360 0.21154 0.20003 0.18908

58558 78874 10467 96393 70181

2.20 2.22 2.24 2.26 2.28

0.01395 0.01311 0.01232

94517 49385

0.17866 0.16875 0.15934 .0.15040 0.14193

58032 80773 55603 97635 21249

2.30 2.32 2.34 2.36 2.38

0.01157 0.01087 0.01021 0.00959 0.00901

52117 41433 62118 44574

0.13389 0.12627 0.11906 0.11223

0.10577

41273 73987 38004 54993

50300

2.40 2.42 2.44 2.46 2.48

0.00846 0.00795 0.00747 0.00701 0.00659

78199 41974 lb117 82011 22140

1.68

0.09966 0.09388 0.08843 0.08327 0.07841

53447 98538 24583 75739 01486

2.50 2.52 2.54 2.56 2.58

0.00619 0.00581 0.00546 0.00513 0.00481

20026 60167 27984 09763 92610

03078 84707 86358 50204 74395

1.70

0.07381

0.06948

56747

1.72 1.74 1.76

0.06539 0.06153 0.05789

03054 31533 64327

0.00452 0.00425 0.00399 0.00375 0.00352

64398 13725 29873

1.78

2.60 2.62 2.64 2.66 2. b8

0.61367 0.59043 0.56737 0.54457 0.52209

03730 22404 48621 30994 46531

1.80 1.82 1.84 1.86 1.88

0.05446 0.05123 0.04819 0.04532 0;04262

83767 77481 38272 63995 57408

2.70 2.72 2.74 2.76 2,78

0.00330 0.00310 0.00291 0.00274 0.00257

81448 69966 80blO 05988 39151

1. 00 1 02 1.04 1.06 1.08

0.50000 0.47834 0.45716 0.43651 0.41642

00000 24497 83054 71048 19278

1.90 1.92 1.94 1.96 1.98

0.04008 0.03768 0.03543 0.03331 0.03131

26022 81947 41720 26147 60134

2.80 2. a2 2.84 2.86 2.88

0.00241 0;00227 0.00213 0.00200 0.00188

73568 03103 21990 24811 06475

1.10 1.12 1.14 1.16 1.18

0.39690 0.37800 0.35971 0.34205 0.32503

97552 18621 42366

0.02943 0.02766 0.02600 0.02444 0.02297

72515 95892 66464 23873 11038

2.90 2.92 2.94 2.96

0.00176

2.98

0.00155 0.00146 0.00137

62198 87487

98919

2.00 2.02 2.04 2.06 2.08

78119 30127 39785

1.20

0.30866

25998

2.10

0.02158

74007

3.00

0.00129

03591

1.30

1.32 1.34 1.36

1.38 1.40

1.42 1.44 1.46 1.48 1.50 1.52

1.54 1.56 1.58

1.60 1.62 1.66

80100

01950

0.01581 0.01485

0.00165

37845 79356

11967

53165

02764

22924

c 1 (-32

For

Table

K’ K >3.0,

K’ K < 0.3, see Example

6.

17.4

AUXILIARY

FUNCTIONS

FOR

COMPUTATION

OF THE

NOME

q AND

THE

PARAMETER

m

y(m)2$ 9 0. 00 0. 01

0.06250

00000

00000

0.00000

00000

0.06281

45660

38302

0.00251

65276

0.02 0.03 0.04

0.06313 0.06345 0.06378

33261 63756 38128

60188 34180 42217

0.00506 0.00765

0;06411

57394

13714

0.01027 OiO1292

66040 09870

04595 58301

0.06445 0.06479

22603 34842

66828 57396

0.01561 0.01834

79344 76360

Q (4

See Examples

L(m)

[5716 1

3, 5 and 6.

ml

0.08 0. 09 0.10 0.11 0.12 0.13

0.14 0.15

Q(4 0.06513 0.06549 0.06584 0.06620 0.06657 0.06694 0.06732 0.06770

95233 36060 04937 14101 b5155 38584 77131 77434 42154 15123 61556 59704 36721 61983 69082 47689

’

L(4 0.02111 0.02392 0.02677 0.02966 0.03259 0.03556 0.03858 0.04165

58281 34345 14110 07472 24678

76342 73466 27452

c(-;I6 3

ELLIPTIC ELLIPTIC

INTEGBlAL

613

INTEGRALS

OF

THE

FIRST

KIND

F(v\a)

Table

17.5

F(c\Q)=J~’ (l-sin2 a sin2@)+dO 5”

15”

10”

20”

25”

30”

0.08726 0.08726 0.08726 0.08726 0.08726

646 660 700 767 860

0.17453 0.17453 0.17453 0.17454 0.17454

293 4011 72:l 25!j 999

0.26179 0.26180 0.26181 0.26183 0.26185

939 298 374 163 656

0.34906 0.34907 0.34909 0.34914 0.34919

585 428 952 148 998

0.43633 0.43634 0.43639 0.43647 0.43659

231 855 719 806 086

0.52359 0.52362 0.52370 0.52384 0.52403

878 636 903 653 839

0.08726 0.08727 0.08727 0.08727 0.08727

980 124 294 407 703

0.17455 0.17457 0.17458 0.17459 0.17461

94') lo;! 451. 99:. 714

0.26188 0.26192 0.26197 0.26202 0.26208

842 707 234 402 189

0.34927 479 0.34936 558 0.34947 200 0.34959 358 0.34972 983

0.43673 0.43691 0.43711 0.43735 0.43761

518 046 606 119 496

0.52428 0.52458 0.52493 0.52533 0.52578

402 259 314 449 529

0.08727 0.08728 0.08728 0.08728 0.08729

940 199 477 773 086

0.17463 0.17465 0.17467 0.17470 0.17472

611. 67!i 89!i 261 76i!

0.26214 0.26221 0.26228 0.26236 0.26245

568 511 985 958 392

0.34988 0.35004 0.35022 0.35040 0.35060

016 395 048 901 870

0.43790 0.43822 0.43856 0.43893 0.43932

635 422 733 430 365

0.52628 0.52682 0.52741 0.52804 0.52872

399 887 799 924 029

0.08729 0.08729 0.08730 0.08730 0.08730

413 755 108 472 844

0.17475 0.17478 0.17480 0.17483 0.17486

386 110 950 0611 840

0.26254 0.26263 0.26273 0.26282 0.26293

249 487 064 934 052

0.35081 0.35103 0.35126 0.35150 0.35174

868 803 576 083 218

0.43973 0.44016 0.44060 0.44107 0.44154

377 296 939 115 622

0.52942 0.53017 0.53094 0.53174 0.53257

863 153 608 916 745

0.08731 0.08731 0.08731 0.08732 0.08732

222 606 992 379 765

0.17489 0.17492 0.17496 0.17499 0.17502

88i' 96i' 07:; 185' 30C1

0.26303 0.26313 0.26324 0.26335 0.26345

369 836 404 019 633

0.35198 0.35223 0.35249 0.35274 0.35300

869 920 254 748 280

0.44203 0.44252 0.44302 0.44353 0.44404

247 769 960 584 397

0.53342 745 0.53429 546 0.53517 761 0.53606 986 0.53696 798

0.08733 0.08733 0.08733 0.08734 0.08734

149 528 901 265 620

0.17505 0.17508 0.17511 0.17514 0.17517

392' 44E1 455~ 397' 26Cl

0.26356 191 0.26366 643 0.26376 936 0.26387 020 0.26396 842

0.35325 0.35350 0.35375 0.35400 0.35424

724 955 845 269 101

0.44455 0.44505 0.44555 0.44604 0.44652

151 593 469 519 487

0.53786 0.53876 0.53965 0.54053 0.54139

765 438 358 059 069

0.08734 0.08735 0.00735 0.08735 0.08736

962 291 605 902 182

0.17520 0.17522 0.17525 0.17527 0.17529

024' 69C1 232 64C' 90:

0.26406 0.26415 0.26424 0.26432 0.26440

355 509 258 556 362

0.35447 0.35469 0.35490 0.35511 0.35530

217 497 823 081 160

0.44699 0.44744 0.44787 0.44828 0.44867

117 153 348 459 252

0.54222 0.54304 0.54382 0.54456 0.54527

911 111 197 704 182

0.08736 0.08736 0.08736 0.08737 0.08737

442 681 898 092 262

0.1 532 0.1 ! 533 0.17535 0.17537 0.17538

010 949 712 289 672

0.26447 0.26454 0.26460 0.26465 0.26470

634 334 420 883 671

0.35547 0.35564 0.35579 0.35592 0.35604

959 377 326 721 488

0.44903 0.44936 0.44967 0.44994 0.45019

502 997 538 944 046

0.54593 0.54654 0.54710 0.54760 0.54804

192 316 162 364 587

0.08737 408 0.08737 528 0.00737 622 0.08737 689 0.08737 730

0.17539 0.17540 0.17541 0.17542 0.17542

a54 830 594 143 473

0.26474 0.26478 0.26480 0.26482 0.26483

766 147 795 697 842

0.35614 0.35622 0.35629 0.35634 0.35636

560 881 402 086 908

0.45039 0.45056 0.45070 0.45079 0.45085

699 775 168 795 596

0.54842 0.54873 0.54898 0.54916 0.54927

535 947 608 348 042

0.08737

0.17542

583

0.26484

225

0.35637

851

0.45087

533

0.54930

614

744

[ (7)3] 0.08726 0.08727 0.08728 0.08730 0.08732 0.08734 0.08735 0.08736 0.08737 _ .

730 387 623 289 185 084 756 998 659 .

[ (-l)3] 0.17453 0.17459 0.17469 0.17482 0.17497 0.17512 0.17526 0.17536 0.17541

962 198 061 397 630 935 454 525 895

[ (-f)l] 0.26182 0.26199 0.26232 0.26277 0.26329 0.26382 0.26428 0.26463 0.26481

[ (-!)2] 180 739 912 965 709 007 466 238 840

0.34911 0.34953 0.35031 0.35138 0.35261 0.35388 0.35501 0.35586 0.35631

[ (-f)5] 842 092 330 244 989 123 092 223 976

0.43643 0.43722 0.43874 0.44083 0.44328 0.44580 0.44808 0.44981 0.45075

[ (-yq 361 998 792 848 233 113 179 645 457

0.52377 095 0.52512 754 0.52772 849 0.53134 425 0.53562 273 0.54009 391 0.54419 926 0.54735 991 0.54908 352

The table can also be usedinversely to find cp=amu where u=F(lp\a) and so the Jacobian elliptic functions, for example sn u=sin (D, cn u=cos IP, dn u= (l-sin2 a sin2 P) 112. See Examples 7-11. Compiled from K. Pearson, Tables of the complete and incomplete elliptic integrals, Cambridge Univ. Press,Cambridge,England, 1934(with permission). Known errorshave beencorrected.

614

ELLIPTIC Table

ELLIPTIC

17.5

INTEGRALS

INTEGRAL

OF THE

FIRST

KIND

F(v\a)

F(e\a)=K (l-sin2 01 sin20)-*&I ace \ 0 2" 4 : 10 12 :"b 18 20 2': :i 30

:: :i 40 t% ft 50 z24 2s" :; :i 68 :2" :z 78 :2" 8": 88 90

60" 1.04719 755 0.61086 524 0.61090 819 0.69819 0.69813 170 436 0.78548 0.78539 509 816 0.87266 0.87278 463 045 0.96008 0.95993 109 037 1.04738 465 0.61103 691 0.69838220 0.78574 574 0.87312 784 0.96052 821 1.04794 603 1.04888 194 0.78617 644 974 0.87451 0.87370 649 593 0.96231 0.96127450 911 1.05019 278 0.61125 108 0.61155 010 0.69913 0.69869 161 484 0.78678 35”

40”

45”

50”

55”

0.61193 318 0.61239927 0.61294 707 504 0.61357 0.61428140

0.69969 159 0.70037 358 0.70117 730 608 0.70209 0.70313 511

0.78756 494 0.78851 403 0.78963 221 768 0.79091 0.79236 827

0.87555 545 0.87682 412 0.87832 076 0.88004 389 0.88199 174

0.96366 180 0.96530 224 0.96723 438 998 0.96947 0.97200 462

1.05187 911 1.05394 160 1.05638 099 1.05919 813 1.06239 384

0.61506 406 0.61592871 071 0.61684 0.61784 515 682 0.61890

0.70428 706 0.70555 037 0.70692 183 0.70839 451 788 0.70997

0.79398 143 0.79575 422 0.79768 324 0.79976 461 0.80199 389

0.88416 214 0.88655 254 0.88915 992 0.89198 071 0.89501 076

0.97482 960 0.97794 790 0.98135 773 0.98505 681 0.98904 227

1.06596 891 1.06992 405 1.07425 976 1.07897 628 1.08407 347

0.62003 018 0.71164 728 0.80436 610 0.89824 524 0.99331 059 1.08955 067 1.09540 656 0.62244138 0.62121 622 0.71341 0.71526 124 098 0.80687 0.80951 558 599 0.90167 0.90530852 415 0.99785 1.00267 743 749 1.10163 899 0.62373 019 840 0.71719 052 0.81228 024 039 0.90911 465 148 1.00776 438 1.10824 474 0.62505 0.71919 335 0.81516 0.91310 1.01311 039 1.11521 933 0.62642 563 0.62782 630 0.62925 446 0.63070 385 0.63216 783

0.72126 235 0.72338 982 0.72556 741 0.72778 615 640 0.73003

0.81814 765 0.82123 227 0.82440 346 0.82764 941 0.83095 712

0.91725 487 0.92156 370 0.92601 535 0.93059 558 0.93528 835

1.01870 633 1.02454 127 1.03060 230 1.03687 427 1.04333 948

1.12255 667 1.13024 880 1.13828 546 1.14665 369 1.15533 731

0.63363 947 0.63511 150 0.63657 639 0.63802 636 0.63945 343

0.73230 789 0.73458 970 0.73687 028 0.73913 751 0.74137 870

0.83431 247 0.83770 010 0.84110 344 0.84450 468 0.84788 483

0.94007 568 0.94493 756 0.94985 177 0.95479 381 0.95973 682

1.04997 735 1.05676 412 1.06367 248 1.07067 128 1.07772 516

1.16431 637 1.17356 652 1.18305 833 1.19275 650 1.20261 907

0.64084944 0.64220 613 0.64351 521 0.64476 839 0.64595 751

0.74358 071 0.74572 998 0.74781 266 471 0.74981 0.75172 208

0.85122 375 0.85450 024 0.85769 220 0.86077 677 0.86373 057

0.96465 156 0.96950 647 0.97426 773 0.97889 946 0.98336 406

l..O8479 434 1.09183 436 1.09879 601 1.10562 535 1.11226 392

1.21259 661 1.22263 139 1.23265 660 1.24259 576 1.25236 238

0.64707458 189 0.64811 0.64906 209 829 0.64991 0.65067 415

0.75352 078 0.75519 716 0.75673 800 076 0.75813 0.75936 376

0.86652 996 0.86915 135 0.87157 159 0.87376 830 0.87572 037

0.98762 253 0.99163 507 0.99536166 0.99876 287 1.00180 067

1.11864 920 1.12471 530 1.13039 401 1.13561 610 1.14031 304

1.26185 988 1.27098 218 1.27961 482 1.28763 696 1.29492 436

0.65132 394 0.65186 270 0.65228 622 0.65259 116 0.65277 510

0.76042 640 0.76130 931 0.76200 457 582 0.76250 0.76280 846

0.87740 833 0.87881481 0.87992 495 0.88072 675 0.88121 143

1.00443 942 1.00664.678 1.00839 470 1.00966 028 1.01042 658

1.14441 892 1.14787 262 1.15062 010 1.15261 652 1.15382 828

1.30135 321 1.30680 495 l.31117 166 1.31436 170 1.31630 510

0.65283658 0.76290 965 0.88137 359 1.01068 319 1.15423 455 1.31695 790 c-y [‘-y] [(-i)3] [‘-;‘“I [‘-p-1 [‘-;“I

[ 1

0.61113 335 0.61325114 0.61733 857 0.62308 236 0.62997 691 0.63730 374 0.64414 930 0.64950 235 0.65245 368

0.69852 295 0.70162198 0.70764702 0.71621 617 0.72661 222 0.73800 634 0.74882 464 0.75745 364 0.76227 978

0.78594 111 0.79025 416 0.79870 514 0.81088 311 0.82601 788 0.84280 548 0.85924 936 0.87269 924 0.88036 502

0.87338828 0.87915 412 0.89054 388 0.90718 679 0.92829 036 0.95232 094 0.97660 210 0.99710 535 1.00908 899

0.96086 405 0.96832 014 0.98317 128 1.00518 803 1.03371 296 1.06716 268 1.10223 077 1.13306 645 1.15171 457

1.04836 715 1.05774 229 1.07657 042 1.10489 545 1.14242 906 1.18788 407 1.23764 210 1.28370 993 1.31291 870

ELLIPTIC ELLIPTIC

INTEGRAL

615

INTEGRALS OF THE FIRST

KIND

F(+e\a)

Table

17.5

P(l.\a)=~P(l-sina~sin2e)-‘de 65”

70”

75”

80”

85”

90”

1.13446 401 1.13469 294 1.13537 994 1.13652 576 1.13813 158

1.22173 048 1.22200 477 1.22282 810 1.22420 180 1.22612 810

1.30899 694 1.30931 959 X.31028 822 1.31190 491 I..31417 314

1.39626 340 1.39663 672 1.39775 763 1.39962 909 1.40225 598

1.48352 986 1.48395 543 1.48523 342 1.48736 769 1.49036 470

1.57079 633 1.57127 495 1.57271 244 1.57511 361 1.57848 658

:z 18

1.14019 906 1.14273 032 1.14572 789 1.14919 471 1.15313 409

1.22861 010 1.23165 180 1.23525 808 1.23943 470 1.24418 827

I..31709 778 1.32068 514 I..32494 296 1.32988 047 I..33550 840

1.40564 522 1.40980 577 1.41474 871 1.42048 728 1.42703 700

1.49423 361 1.49898 627 1.50463 742 1.51120 474 1.51870 904

1.58284 280 1.58819 721 1.59456 834 1.60197 853 1.61045 415

$2” 228

1.15754 967 1.16244 535 1.16782 525 1.17369 362 1.18005 472

1.24952 627 1.25545 700 1.26198 957 1.26913 385 1.27690 045

1.34183 901 I..34888 616 I.35666 531 1.36519 359 1.37448 981

1.43441 578 1.44264 399 1.4$174 466 1.41174 360 1.4 266 958

1.52717 445 1.53662 865 1.54710 309 1.55863 334 1.57125 942

1.62002 590 1.63072 910 1.64260 414 1.65569 693 1.67005 943

1.18691 274 1.19427 162 1.20213 489 1.21050 542 1.21938 520

1.28530 059 1.29434 605 1.30404 906 1.31442 210 1.32547 772

1.38457 455 1.39547 013 I..40720 064 1.41979 198 1.43327 179

1.48455 455 1 49743384 1.51134 1 644 1,52633 523 1.54244 734

1.58502 624 1.59998 406 1.61618 906 1.63370 398 1.65259 894

1.68575 035 1.70283 594 1.72139 083 1.74149 923 1.76325 618

1.22877 499 1.23867 392 1.24907 904 1.25998 475 1.27138 210

1.33722 824 1.34968 545 1.36286 013 1.37676 148 1.39139 640

1.44766 938 1.46301 $65 1.47934 287 1.49668 437 1.51507 416

~1.55973441 1.57825 301 1.59806 493 1.61923 762 1.64184 453

1.67295 226 1.69485 156 1.71839 498 1.74369 264 1.77086 836

1.78676 913 1.81215 985 1.83956 672 1.86914 755 1.90108 303

1.28325 798 1.29559 414 1.30836 604 1.32154 149 1.33507 910

1.40676 855 1.42287 717 1.43971 560 1.45726 935 1.47551 372

1.53454 619 1.55513 354 1.57686 709 1.59977 378 1.62387 409

1.66596 542 1.69168 665 1.71910 125 1.74830 880 1.77941 482

1.80006 176 1.83143 068 1.86515 414 1.90143 591 1.94050 873

1.93558 110 1.97288 227 2.01326 657 2.05706 232 2.10465 766

1.34892 643 1.36301 803 1.37727 323 1.39159 384 1.40586 195

1.49441 087 1.51390 609 1.53392 332 1.55435 972 1.57507 940

1.64917 867 1.67568 359 1.70336 398 1.73216 516 1.76199 085

1.81252 953 1.84776 547 1.88523 335 1.92503 509 1.96725 237

1.98263 957 2.02813 570 2.07735 219 2.13070 052 2.18865 839

2.15651 565 2.21319 470 2.27537 643 2.34390 472 2.41984 165

1.41993 796 1.43365 925 1.44684 001 1.45927 266 1.47073 163

1.59590 624 1.61661 644 1.63693 134 1.65651 218 1.67495 873

1.79268 736 1.82402 292 1.85566 175 1.88713 308 1.91779 814

2.01192 798 2.05903 582 2.10843 282 2.15978 295 2.21243 977

2.25177 995 2.32070 416 2.39615 610 2.47892 739 2.56980 281

2.50455 008 2.59981 973 2.70806 762 2.83267 258 2.97856 895

1.48098 006 1.48977 975 1.49690 410 1.50215 336 1.50537 033

1.69181 489 1.70658 456 1.71876 033 1.72786 543 1.73350 464

1.94682 231 1.97316 666 1.99562 118 2.01290 452 2.02384 126

2.26527 326 2.31643 897 2.36313 736 2.40153 358 2.42718 003

2.66935 045 2.77736 748 2.89146 664 3.00370 926 3.09448 898

3.15338 525 3.36986 803 3.65185 597 4.05275 817 4.74271 727

50 52

1.50645 424 1.73541 516 2.02758 942 2.43624 605 3.13130 133 [ (-;‘“]

5 :: i: 2: 75 85

13589544 II* 14740244 1: 17069811 1. 20625 660 1. 25446 980 31490567 :: 38443225 45316 359 :: 49977412

[c-:)5]

22344604 :: 23727471 26548460 :: 30915104 36971948 :: 44840433 54409676 :* 64683711 1: 72372395

[ ‘;yJ]

1.31101 537 1.32732 612 1.36083 467 1.41338 702 1.48788 472 1.58817 233 1.71762 935 1.87145 396 2.00498 776

[ ‘;$2]

1.39859 928 1.41751 762 1.45663 012 1.51870 347 1.60847 673 1.73347 444 1.90483 674 2.13389 514 2.38364 709

m

[ (-3Y]

1.48619 317 1.50780 533 1.55273 384 1.62477 858 1.73081 713 1.88296 142 2.10348 169 2.43657 614 2.94868 876

1.57379 213 1.59814 200 1.64899 522 1.73124 518 1.85407 468 2.03471 531 2.30878 680 2.76806 315 3.83174 200

ELLIPTIC

616 Table

17.6

ELLIF’TIC

INTEGRAL

INTEGRALS OF THE

SECOND

KIND

E((o\a)

E(~\Ix)=J~’ (l-sin2asin28)‘tls 5”

10”

15”

20”

0.08726 646 0.08726 633 0.08726 592 0.08726 525 0.08726 432

0.17453 293 0.17453 185 0.17452 864 0.17452 330 0.17451 587

0.26179 939 0.26179 579 0.26178 503 0.26176 715 0.26174 224

0.34906 585 0.34905 742 0.34903 218 0.34899 025 0.34893 181

0.43633 231 0.43631 608 0.43626 745 0.43618 665 0.43607 403

0.52359 878 0.52357 119 0.52348 a56 0.52335 123 0.52315 981

0.08726 313 0.08726 168 0.08725 999 0.08725 806 0.08725 590

0.17450 636 0.17449 485 0.17448 137 0.17446 599 0.17444 879

0.26171 041 0.26167 182 0.26162 664 0.26157 510 0.26151 743

0.34885 714 0.34876 657 0.34866 055 0.34853 954 0.34840 412

0.43593 011 0.43575 552 0.43555 106 0.43531 765 0.43505 633

0.52291 511 0.52261 a21 0.52227 039 0.52187 317 0.52142 828

0.08725 352 0.08725 094 0.08724 al6 0.08724 521 0.08724 208

0.17442 985 0.17440 926 0.17438 712 0.17436 353 0.17433 a62

0.26145 391 0.26138 485 0.26131 056 0.26123 141 0.26114 778

0.34825 492 0.34809 262 0.34791 800 0.34773 la7 0.34753 510

0.43476 831 0.43445 488 0.43411 749 0.43375 767 0.43337 709

0.52093 770 0.52040 357 0.51982 a27 0.51921 436 0.51856 461

0.08723 881 0.08723 540 0.08723 187 0.08722 824 0.08722 453

0.17431 250 0.17428 529 0.17425 714 0.17422 al7 0.17419 852

0.26106 005 0.26096 a67 0.26087 405 0.26077 666 0.26067 697

0.34732 a63 0.34711 342 0.34689 050 0.34666 093 0.34642 580

0.43297 749 0.43256 075 0.43212 880 0.43168 368 0.43122 748

0.51788 193 0.51716 944 0.51643 040 0.51566 820 0.51488 638

0.08722 075 0.08721 692 0.08721 307 0.08720 920 0.08720 535

0.17416 835 0.17413 779 0.17410 700 0.17407 613 0.17404 531

0.26057 545 0.26047 261 0.26036 a93 0.26026 492 0.26016 110

0.34618 625 0.34594 343 0.34569 a50 0.34545 266 0.34520 710

0.43076 236 0.43029 055 0.42981 431 0.42933 594 0.42885 776

0.51408 a62 0.51327 866 0.51246 037 0.51163 767 0.51081 454

0.08720 152 0.08719 774 0.08719 402 0.08719 039 0.08718 686

0.17401 472 0.17398 449 0.17395 477 0.17392 571 0.17389 745

0.26005 795 0.25995 600 0.25985 574 0.25975 765 0.25966 224

0.34496 302 0.34472 162 0.34448 409 0.34425 159 0.34402 529

0.42838 212 0.42791 134 0.42744 775 0.42699 368 0.42655 138

0.50999 501 0.50918 310 -0.50838287 0.50759 a31 0.50683 341

0.08718 345 0.08718 017 0.08717 704 0.08717 408 0.08717 130

0.17387 013 0.17384 388 0.17381 a83 0.17379 511 0.17377 283

0.25956 996 0.25948 126 0.25939 660 0.25931 640 0.25924 104

0.34380 631 0.34359 575 0.34339 465 0.34320 404 0.34302 487

0.42612 308 0.42571 097 0.42531 712 0.42494 358 0.42459 224

0.50609 207 0.50537 all 0.50469 523 0.50404 700 0.50343 686

0.08716 a71 0.08716 633 0.08716 416 0.08716 223 0.08716 053

0.17375 210 0.17373 302 0.17371 568 0.17370 018 0.17368 659

0.25917 090 0.25910 634 0.25904 767 0.25899 519 0.25894 917

0.34285 805 0.34270 443 0.34256 478 0.34243 984 0.34233 022

0.42426 495 0.42396 339 0.42368 913 0.42344 363 0.42322 817

0.50286 804 0.50234 359 0.50186 633 0.50143 a86 0.50106 351

0.08715 909 0.08715 789 0.08715 695 0.08715 628 0.08715 588

0.17367 498 0.17366 539 0.17365 789 0.17365 250 0.17364 926

0.25890 983 0.25887 737 0.25885 195 0.25883 370 0.25882 271

0.34223 650 0.34215 915 0.34209 857 0.34205 507 0.34202 889

0.42304 389 0.42289 175 0.42277 258 0.42268 700 0.42263 547

0.50074 232 0.50047 707 0.50026 923 0.50011 993 0.50003 003

0.08715 574 0.17364 ala

[ 1

See Example

(-i)4

[ (-l)3]

0.08726 562 0.08725 905 0.08724 671 0.08723 006 0.08721 113 0.08719 220 0.08717 554 0.08716 317 0.08715 659

0.17452 624 0.17447 391 0.17437 550 0.17424 275 0.17409 157 0.17394 015 0.17380 680 0.17370 770 0.17365 493

25”

30”

0.25881 905 0.34202 014 0.42261 826 0.50000 000

[‘-p”] 0.26177 698 0.26160 165 0.26127 157 0.26082 567 0.26031 693 0.25980 639 0.25935 592 0.25902 064 0.25884 192

[‘f’“] 0.34901 329 0.34860 188 0.34782 632 0.34677 648 0.34557 562 0.34436 714 0.34329 797 0.34250 043 0.34207 467

[ (-;)“I 0.43623 105 0.43543 791 0.43394 028 0.43190 776 0.42957 525 0.42721 938 0.42512 769 0.42356 271 0.42272 556

[c-y1

0.52342 670 0.52207 785 0.51952 597 0.51605 197 0.51204 932 0.50798 838 0.50436 656 0.50164 622 0.50018 720

14.

Compiled from K. Pearson, Tables of the complete and incomplete elliptic integrals, Cambridge Univ. Press, Cambridge, England, 1934 (with permission). Known errors have been corrected.

ELLIPTIC ELLIPTIC

INTEGRAL

617

INTEGRALS OF THE

J:(v\a)=Jl(l-sin2

SECOND

KIND

E(p\ol)

Table

17.6

n sin2 S) ‘do

a(P \

35”

40”

45”

50”

55”

60”

4

a

0.61086 524 0.61082 230 0.61069 365 0.61047 983 0.61018 171

0.69813 170 0.69806 905 0.69788 136 0.69756 935 0.69713 427

0.78539 al6 0.78531 125 0.78505 085 0.78461 792 0.78401 409

0.87266 463 0.87254 a83 0.87220 la3 0.87162 487 0.87081 998

0.95993 109 0.95978 la4 0.95933 459 0.95859 083 0.95755 301

1.04719 755 1.04701 051 1.04644 996 1.04551 764 1.04421 646

:2” :“6 la

0.60980 055 0.60933 793 0.60879 577 0.60817 636 0.60748 229

0.69657 784 0.69590 226 0.69511 023 0.69420 492 0.69318 999

0.78324 162 0.78230 343 3.78120 308 D.77994473 3.77853323

0.86979 001 0.86853 863 0.86707 031 0.86539 034 0.86350 481

0.95622 460 0.95461 005 0.95271 478 0.95054 522 0.94810 a78

1.04255 047 1.04052 491 1.03814 615 1.03542 177 1.03236 049

20 22

0.60671 652 0.60588 229 0.60498 319 0.60402 308 0.60300 616

0.69206954 0.69084 al4 0.68953 083 0.68812 308 0.68663 077

il.77697 402 10.77527316 0.77343 735 0.77147 387 0.76939 059

0.86142 062 0.85914 545 0.85668 781 0.85405695 0.85126 295

0.94541 386 0.94246 984 0.93928 709 0.93587 699 0.93225 la6

1.02897 221 1.02526 804 1.02126 023 1.01696 224 1.01238 a73

0.60193 687 0.60081 994 0.59966 035 0.59846 332 0.59723 431

0.68506 023 0.68341 al7 Oi68171170 0.67994 830 0.67813 578

0.76719 599 Cl.76489908 Cl.76250947 Cm.76003 726 k75749 309

0.84831 663 0.84522 958 0.84201 414 0.83868 340 0.83525 115

0.92842 504 0.92441 083 0.92022 452 0.91588 234 0.91140 150

1.00755 556 1.00247 977 0.99717 966 0.99167 469 0.98598 560

0.59597 a97 0.59470312 0.59341278 0.59211 406 0.59081 324

0.67628 229 0.67439 630 0.67248 651 0.67056 191 0.66863 167

0.75488 a09 0.75223 383 0.74954 234 0.74682 605 0.74409 773

0.83173 189 0.82814 080 0.82449 369 0.82080 700 0.81709 775

0.90680 017 0.90209 742 0.89731 325 0.89246 a58 0.88758 513

0.98013 430

0.58951 664 0.58823 065 0.58696 171 0.58571 622 0.58450 056

0.66670 515 0.66479 la3 0.66290 130 0.66104 317 0.65922 707

0.74137 047 0.73865 766 0.,73597286 O-73332979 0.,73074229

0.81338 346 0.80968 217 0.80601 230 0.80239 262 0.79884 217

0.88268 551 0.87779 305 0.87293 la4 0.86812 660 0.86340261

0.94929 830 0.94300 285 0.93673 272 0.93051 931 0.92439 505

0.58332 103 0.58218 382 0.58109 497 0.58006 032 0.57908 549

0.65746 255 0.65575 905 0.65412 585 0.65257 197 0.65110 612

0.72822 416 0.72578 915 0.72345 085 0.72122 260 0.71911 737

0.79538 015 0.79202 582 0.78879 a39 0.78571 685 0.78279 987

0.85878 561 0.85430 169 0.84997 709 0.84583all 0.84191 082

0.91839 329 0.91254 a21 0.90689 460 0.90146 778 0.89630 323

0.57817 584 0.57733 641 0.57657 la9 0.57588 663 0.57528 450

0.64973 667 0.64847 154 0.64731 al2 0.64628 328 0.64537 322

0.71714 767 0.71532 545 0.71366196 0.71216 766 0.71085210

0.78006 562 0.77753 157 0.77521 434 0.77312 952 0.77129 143

0.83822 090 0.83479 335 0.83165223 0.82882 031 0.82631 a79

0.89143 642 0.88690 237 0.88273 530 0.87896 al0 0.87563 la5

0 2” 6

:z 28

“8: a4

0.97414

397

0.96803 a99 0.96184 497 0.95558 a73

it

0.57476 897 0.57434 302 0.64459 0.64394 347 879 0.70972 0.70879 381 019 0.76840 0.76971 298 544 0.82416 0.82238 694 177 0.87275 0.87036 520 381 0.57400 912 0.64344 316 0.70805 745 0.76737 830 0.82097 770 0.86847 970 0.57376 921 0.57362 470 0.64307 0.64286 973 075 0.70721 0.70753 289 050 0.76663 0.76619 912 339 0.81996 0.81935 631 604 0.86712 0.86629 990 068

90

0.57357 644 0.64278 761 0.70710 678 0.76604 444 0.81915 204 0.86602 540 [ (-;)l]

5 :: 35

5”: ;:a5

0.61059 734 0.60849 557 0.60451 051 0.59906 618 0.59276 408 0.58633 563 0.58057 051 0.57621 910 0.57387 732

[ Qw]

0.69774 083 0.69467 152 0.68883 790 0.68083 664 0.67152 549 0.66196 758 0.65333 a44 0.64678 548 0.64324 351

[ (-i)3]

0.78485 586 0.78059 337 0.77247 109 0.76128 304 0.74818 650 0.73464 525 0.72232 215 0.71289 304 0.70776 799

[ (-l)4]

0.87194 199 0.86625 642 0.85539 342 0.84036 234 0.82265 424 0.80419 500 0.78723 a20 0.77414 195 0.76697 232

[ (-i)5]

0.95899 964 0.95166 385 0.93760 971 0.91'807la6 0.89489 714 0.87052 066 0.84788 276 0.83019 625 0.82042 232

[(-(y]

1.04603 012 1.03682 664 1.01914 662 0.99445 152 0.96495 146 0.93361 692 0.90415 063 0.88079 972 0.86773 361

618

ELLIPTIC Table

ELLIPTIC

17.6

INTEGRAL qP\4=[“p

INTEGRALS

OF THE (l75”

SECOND

KIND

E(a\a)

65”

70”

1.13446 401 1.13423 517 1.13354 929 1.13240 a37 1.13081 573

1.22173 048 1.22145 628 1.22063 443 1.21926 717 1.21735 a20

1.30899 694 1.39626 340 1.30867 442 1.39589 024 1.30770 767 1.39477 lb5 1.30609 91b 1.39291 030 1.30385 297 1.39031 062

1.48352 986 1.48310 448 1.48182 929 1.47970 717 1.47674 288

1.57079 633 1.57031 792 1.56888 372 1.56649 679 1.56316 223

1.12877 602 1.12629 522 1.12338 066 1.12004 099 1.11628 624

1.21491 274 1.21193 748 1.20844 065 1.20443 195 1.19992 262

1.30097 484 1.29747 215 1.29335 393 1.28863 089 1.28331 541

1.38697 886 1.38292 302 1.37815 292 1.37268 017 1.36651 a23

1.47294 312 1.46831 652 1.46287 363 1.45662 693 1.44959 085

1.55888 720 1.55368 089 1.54755 458 1.54052 157 1.53259 729

1.11212 778 1.10757 a34 1.10265 204 1.09736 439 1.09173 228

1.19492 542 1.18945 465 1.18352 618 1.17715 743 1.17036 745

1.27742 153 1.27096 502 1.26396 337 1.25643 578 1.24840 326

1.35968 233 1.35218 961 1.34405 903 1.33531 146 1.32596 967

1.44178 179 1.43321 al3 1.42392 023 1.41391 049 1.40321 335

1.52379 921 1.51414 692 1.50366 214 1.49236 a71 1.48029 266

1.08577404 1.07950 942 1.07295 961 1.06614 728 1.05909 660

1.16317 686 1.15560 796 1.14768 469 1.13943 273 1.13087 946

1.23988 a58 1.23091 635 1.22151 305 1.21170 705 1.20152 a70

1.31605 a41 1.30560 436 1.29463 629 1.28318 499 1.27128 343

1.39185 532 1.37986 503 1.36727 328 1.35411 306 1.34041 965

1.46746 221 1.45390 780 1.43966 215 1.42476 031 1.40923 972

1.05183 322 1.12205 408 1.04438 435 1.11298 760 1.03677 875 1.10371 291 1.02904 677 1.09426 484 1.02122 034 1.08468 023

1.19101 036 l.laola 648 1.16909 366 1.15777 077 1.14625 a99

1.25896 675 1.24627 240 1.23324 019 1.21991 241 1.20633 398

1.32623 Obb 1.39314 025 1.31158 614 1.37650 433 1.29652 865 1.35937 700 1.28110 340 1.34180 606 1.26535 a37 1.32384 218

1.01333 305 1.07499 796 1.13460 200 1.00542 010 1.06525 908 1.12284 604 0.99751 a35 1.05550 682 1.11104 010 0.98966 632 1.04578 671 1.09923 b04 0.98190414 1.03614 663 1.08748 883

1.19255 255 1.17861 a73 1.16458 621 1.15051 210 1.13645 710

1.24934 449 1.23311 580 1.21672 971 1.20024 724 1.18373 339

1.30553 909 1.28695 374 1.26814 653 1.24918 lb2 1.23012 722

0.97427 354 0.96681 780 0.95958 158 0.95261 084 0.94595 256

1.12248

1.10866 752 1.09507 580 1.08178 986 1.06889 476

1.16725 747 1.15089 364 1.13472 145 1.11882 b5a 1.10330 172

1.21105 603 1.19204 568 1.17317 938 1.15454 bb8 1.13624 437

1.02171 634 1.05648 221 1.01220 781 1.04465 133 1.00333 091 0.92833 088 1.03350 951 0.92340 024 0.96208 074 0.99517 606 1.02317 331 776 0.98783 b70 1.01376 904 0.91901 a02 0.95638

1.08824 773 1.07377 505 l.ObOOO556 1.04707 504 1.03513 640

1.11837 774 1.10106 217 1.08442 522

0.91522 691 0.95143 a47 0.98140 781 0.91206588 0.94729297 0.97598 331 0.90956 905 0.94400 544 0.97165 228 0.90776 445 0.94162 171 0.96849 392 0.90667 305 0.94017 677 0.96657 142

1.00543 295 0.99831 000 0.99255 019 0.98830 025 0.98568915

1.02436 393 1.01495 896 1.00715 650 1.00123 026 0.99748 392

1.04011 440 1.02784 362 1.01723 692 1.00864 796 1.00258 409

0.90630 ['-;'"]

0.98480

0.93965 0.93376

447 462

779

1.13303 553 1.12176 337 1.10005 236 1.06958 479 1.03292 bb0 0.99358 365 0.95606 011 0.92579 978 0.90857 a73

1.02663 689 1.01731 023 1.00822 192 0.99942 966 0.99099

354

1.07585 669 1.06440 132 1.05318 al4 1.04228 653 1.03176 998

85”

590

0.98297 0.97544

583 068 0.96845 360

0.93969 ['-$11

262

1.22001 a78 1.20649 962 1.18039 569 1.14359 al3 1.09900 a29 1.05063 981 1.00378 508 0.96518 626 0.94269 al3

0.96592

583

[qw] 1.30698 342 1.29106 728 1.26026 405 1.21665 a53 1.16345 846 1.10513 448 1.04769 389 0.99915 744 0.96992

775

[‘-y]

212

1.39393 358 1.37550 358 1.33976 099 1.28896 903 1.22661 050 1.15755 065 1.08838 943 1.02823 305 0.99022 779

1.06860

953

1.05377 692

0.99619 470 1.00000 000

[

C-i)3

1 [ 1

1.48087 384 1.45984 990 1.41900 286 1.36076 208 1.28885 906 1.20849 656 1.12673 373 1.05342 b32 1.00394 027

($4

1.56780 907 1.54415 050 1.49811 493 1.43229 097 1.35064 388 1.25867 963 1.16382 796 1.07640 511 1.01266 351

ELLIPTIC JACOBIAN

ZETA

619

INTEGRALS FUNCTION

+-\a)

K(,)z(~\,)=K(,)E(,\o)-E(~)F(~\\n) K(gO”)2(~\a)=K~~90’)Z(~~~l)=K(9~) tanh IL=- for 5” 10” 20” 15” 0.000000 0.000000 0.000000 0.000083 0.000164 :- K% 0.000308 0.000332 0.000655 0:000957 0.001231 0.000748 0.001474 0.002155 0.002770 0.001331 0.002621 0.003832 0.004928

&I? Example

0.002080 0.002997 0.004082 0.005337 0.006761

0.004098 0.005905 0.008043

Table

17.7

all ?L 25” 0.000000 0.000367 0.001467 0.003302 0.005875

30” 0.000000 0.000414 0.001658 0.003734 0.006644

it. EE!

t "OK;;: 0:011765 is. E:E

0.007706 0.011107 0.015136 0.019796 0.025094

0.009188 0.013246 0.018055 0.023621 0.029951

0.010393 0.014987 0.020433 0.026740 0.033919

0.008357 0.010125 0.012067 0.014186 0.016483

0.016470 0.019958 0.02379-I 0.027972 0.032508

0.024105 0.029216 0.034834 0.040968 0.047624

0.031035 0.037627 0.044878 0.052799 0.061401

0.037055 0.044942 0.053626 0.063119 0.073438

0.041981 0.050941 0.060814 0.071617 0.083373

0.018962 0.021625 0.024476 0.027520 0.030761

0.037403 0.042664 0.048298 0.054315 0.060725

0.054811 0.062540 0.070823 0.079674 0.089108

0.070696 0.080700 0.091430 0.102905 0.115148

0.084599 0.096624 0.109534 0.123356 0.138120

0.096103 0.109834

0.034205 0.037860 0.041734 0.045835 0.050177

0.067540 0.074774 0.082444 0.090569 0.099172

0.099145 0.109807 0.121118 0.133109 0.145813

0.128185 0.142046 0.156765 0.172383 0.188947

0.153860 0.170614 0.188428 0.207353 0.227450

00.:z; 0:215197 0.237025 0.260240

0.054771 0.059634 0.064786

0.159273 0.173536 0.188661 0.204716 0.221785

0.206513 0.225145 0.244921 0.265933 0.288294

0.248789 0.271452 0.295538

0.284929 0.311193 0.339150

: . EZ

0.108280 0.117925 0.128146 0.138989 0.150510

"0. . ::zl:

i*. z%

0.082227 0.088818 0.095876 0.103468 0.111676

0.162776 0.175872 0.189901 0.204994 0.221320

0.239971

t :::'b:i 0:364981 i*. :z;:

t 34707sK 0:442321 0.478462 0.517644

0.434726 0.471170 0.510371 0.552710 0.598675

0.120612 0.130420 0.141301 0.153537 0.167542

0.239097 0.258615 0.280272 0.304631 0.332519

0.353322 0.382351 0.414575 0.450832 0.492356

0.461145 0.499384 0.541857 0.589673 0.644462

0.560402 0.607444 0.659739 0.718657 0.786214

0.648900 0.704225 0.765737 0.835238 0.914934

0.183967 0.203902 0.229402 0.265091 0.325753 m

0.365230 0.404937 0.455734

0.541075 0.600229 0.675918 0.781873 0.962000 00

0.708771 0.786884 0.886859 1.026844 1.264856 m

0.865556 0.961976

0.000519 0.004688 0.013105 0.025973 0.043755 0.067477 0.099601 0.147228 0.245478

0.001023 0.009238 0.025838 0.051258 0.086448

0.001496 0.013513

0.001923 0.017387 0.048754 0.097073 0.164459 0.255266 0.379430 0.565011 0.949910

2. E: m

is E8O: 0:292070 0.487761

i* z;; 0:302637 0.326895

t E'7i 0:127026 0.196567 0.291216 0.432134 0.723644

:* E%! 1:552420 al

is :GE; 0:157347

:- E56E 11268462 1.472953 1.820811 m 0.002592 0.023479 0.066098 0.132373 i* :::E: 0:531121 !*. :E+:

16.

Compiled from P.F. Byrd and M.D. Friedman, Handbook of elliptic integrals for engineers and physicists, Springer-Verlag, Berlin, Germany, 1954 (with permission).

620

ELLIPTIC

Table

JACOBIAN

17.7

INTEGRALS

ZETA

FUNCTION

Z(P\=)

K(~)z(~\~>=K(or)E((p\u)-E(a)F(~\p\a)

35” 0.000000

R(SV)Z(v\P\(r)=K(900)Z(ull)=K(900) tanh u=m for all u 40” 45” 50” 55” 0.000000 0.000000 0.000000 0.000000 0.000471 o.ooiaa6 0.004248 0.007561

0.000479 0.001916 0.004314 0.007681

0.000471 0.001887 0.004250 0.007567

0.000450 0.001800 0.004056 0.007224

60” 0.000000

o.oli284 0.016276 0.022197 0.029060 0.036876

o.olia33 0.017073 0.023293 0.030505 0.038728

0.012023 0.017353 0.023683 0.031029 0.039411

0.011849 0.017106 0.023354 0.030610 0.038897

0.011313 0.016337 0.022312 0.029257 0.037194

0.010433 0.015070 0.020588 0.027006 0.034347

0.045662 0.055435 0.066216 0.078026 0.090893

0.047979 0.058279 0.069655 0.082132 0.095744

0.048850 0.059372 0.071005 0.083783 0.097742

0.048238 0.058663 0.070203 0.082895 0.096782

0.046150 0.056156 0.067246 0.079461 0.092844

0.042639 0.051912 0.062203 0.073551 0.086003

0.104844 0.119914 0.136138 0.153557 0.172220

0.110525 0.126515 0.143758 0.162305 0.182211

0.112924 0.129375 0.147147 0.166300 0.186898

0.111909 0.128330 0.146103 0.165296 0.185983

0.107447 0.123327 0.140549 0.159186 0.179319

0.099613 0.114438 0.130548 0.14aoia 0.166934

0.192178 0.213492 0.236228 0.260466 0.286295

0.203541 0.226365 0.250764 0.276831 0.304671

0.209016 0.232738 0.258158 0.285383 0.314535

0.208248 Fl-ZE 01285531 0.315196

0.201042 0.224459 0.249691 0.276871 0.306156

0.187395 0.209512 0.233413 0.259243 0.287169

0.313816 0.343151 0.374438 0.407844 0.443565

0.334405 0.366173 0.400138 0.436490 0.475457

0.345755 0.379203 0.415067 0.453565 0.494956

0.347064 0.381317 0.418166 0.457861 0.500691

0.337723 0.371776 0.408552 0.448328 0.491428

0.317383 0.350108 0.385601 0.424167 0.466161

0.481836 0.522947 0.567251 0.615191 0.667330

0.517310 0.562378 0.611064 0.663870 0.721434

0.539547 0.587709 0.639896 0.696670 0.758741

0.547003 0.597211 0.651822 0.711460 0.776910

0.538238 0.589220 0.644933 0.706068 0.773487

0.512007 0.562214 0.617399 0.678320 0.745922

0.724397 0.787359 0.857536 0.936789 1.027859

0.784577 0.854390 0.932355 1.020563 1.122089

0.827024 0.902728 0.987491 1.083621 1.194508

0.849178 0.929590 1.019938 1.122735 1.241670

0.848294 0.931931 1.026343 1.134246 1.259612

0.8214il 0.906356 1.002860 1.113848 1.243568

1.135017 1.265447 1.432669 1.667113 2.066078

1.241721 1.387516 1.574623 1.837147 2.284127

1.325428 1.485245 1.690632 1.979107 2.470622

1.382470 1.554749 1.776579 2.088611 2.620801

1.408589 1.591484 1.827639 2.160541 2.729164

1.398577 1.589820 1.837791 2.188502 2.788909

m

m

al

m

m

co

0.002813 0.025510 0.071991 0.144695 0.248154 0.390865 0.590735 0.895883 1.538234

0.002948 0.026774 0.075754 0.152865 0.263583 0.418002 0.636916 0.975016 1.692810

0.002994 0.027228 0.077249 0.156547 0.271538 0.433972 0.667669 1.033955 1.820471

0.002949 0.026855 0.076403 0.155518 0.271473 0.437641 0.680968 1.069585 1.916972

0.002815 0.025662 0.073210 0.149686 0.263028 0.428046 0.674774 1.078397 1.977347

0.002594 0.023683 0.067742 0.139108

0.000450 0.001800 0.004052 0.007212

0.000415 0.001659 0.003739 0.006660

8* 24% 0:647089 1.056317 1.995386

ELLIPTIC

JACOBIAN

621

INTEGRALS

ZETA FUNCTION

Z(v\a)

‘Table 17.7

K(+(,~\~)=K(+E(v\~)-E(~)F(~\~)

R(9OO)Z(~\a)=K(900)Z(u~l)=K(90”) tanh u=- for all u 65” 70 75” 80" 85” 0.000000 0.000000 0.000000 0.000000 0.000000

90”

0.000367 0.001468 0.003308 0.005893

0.000308 0.001232 0.002776 0.004946

0.000239 0.000958 0.002160 0.003849

it 8E~ 0:001477 0.002633

0.000083 0.000333 0.000750 0.001337

t w: 0:018231 0.023922 0.030438

0.007751 0.011202 0.015312 0.020098 0.025581

0.006032 0.008718 0.011920 0.015649 0.019924

0.004127 0.005966 0.008158 0.010713 0.013642

0.002096 0.003030 0.004143 0.005442 0.006930

0.037803 0.046047 0.055206 0.065319 0.076431

0.031783 0.038732 0.046459 0.055000 0.064397

0.024763 0.030188 0.036225 0.042905 0.050260

0.016959 0.020680 0.024823 0.029411 0.034466

0.008617 0.010509 0.012617 0.014952 0.017526

0.088594 0.101867 0.116315 0.132015 0.149053

t 87sE16 01098224 0.111585 0.126114

0.058332 0.067164 0.076808 0.087324 0.098779

0.040018 0.046099 0.052747 0.060004 0.067920

0.020354 0.023454 0.026845 0.030550 0.034595

0.167527 0.187551 0.209254 0.232785 0.258315

0.141905 0.159064 0.177713 0.197996 0.220078

0.111254 0.124839 0.139641 0.155784 0.173414

0.076554 0.085973 0.096255 0.107493 0.119798

0.039011 0.043833 0.049104 0.054874 0.061201

0.286045 0.316206 0.349070 0.384960 0.424255

0.244154 0.270454 0.299246 0.330854 0.365664

0.192704 0.213858 0.237121 0.262789 0.291220

0.133299 0.148154 0.164550 0.182720 0.202947

0.068157 0.075826 0.084312 0.093745 0.104281

0.467411 0.514976 0.567621 0.626169 0.691653

0.404143 0.446860 0.494517 0.547987 0.608372

0.322854 0.358236

0.116121 0.129521 0.144812 0.162430 0.182965

0

: tE58 0:494668

0.225584 0.251076 0.279993 0.313069 0.351277

0.765385 0.849072 0.944993 1.056298 1.187535

0.677086 0.755975 0.847508 0.955095 1.083634

0.554038 0.623195 0.704762 0.802400 0.921408

0.395917 0.448779 0.512376 0.590350 0.688163

0.207230 0.236382 0.272114 0.317015 0.375226

0

1.345674 1.542281 1.798909 2.163806 2.790834

1.240571 1.438150 1.698985 2.073357 2.721008

1.069839 1.260828 1.518315 1.894760 2.555104

0.814374 0.983236 1.220780 1.583040 2.241393 0

0.453784 0.565578 0.736684 1.028059 1.628299 m

0.001025 0.009390 0.027060 0.056296 0.101748 0.173397 0.295957 0.549278 1.380465

0.000520 0.004769 0.013755 0.028657 0.051923 0.088901 0.153297 0.293208 0.860811

m 0.002295 0.020975 0.060141 0.124003 0.220781 0.366615 0.596098 0.998480 1.962673

on 0.001926 i- oa157066:: 0:104764 0.187640 0.314676 0.520463 0.899033 1.866624

0.001498 0.013718 0.039483 0.081953 t y;w; 0:419877 0.751288 1.686113

8 8 0

00 8 0

8 8 0 : 00 0 0 0 i 0

8 8 0

i 0 0

i 00

Ii 8 0 L-a

8 0 0 00 00 0

622

ELLIPTIC

Table

AO(P\4

a%+ \0 F! 4

86

HEUMAN’S

17.8

$2 28 30 32 ;"b 38 40 It 46 48

5” 0.087156

15”

20”

25”

30”

0.258819

0.342020 0.341916 0.341604 0.341084 0.340359

0.422618 0.422490 0.422104 0.421462 0.420566

0.500000 0.499848 0.499391 0.498633

0.339430 0.338299 0.336969 0.335445 0.333729

0.419419 0.418024 0.416385 0.414506 0.412394

0.331827 0.329743 0.327483

0.410054

0.485184

0.407492 0.404717

0.482176 0.478920 0.475428

0.173595 0.173437

0.086917

0.173173

0.258504 0.258111

0.086732

0.172804

0.257562

0.086495 0.086206 0.085866

0.172332 0.171757

0.256858 0.256001 0.254994 0.253838 0.252536

0.084549 0.084013 0.083432 0.082806 0.082136 0.081425 0.080674 0.079884

0.079058

0.078198

0.171080 0.170303 0.169429 0.168458 0.167393

0.166236 0.164991 0.163661 0.162247

0.160755 0.159187 0.157548 0.155842

0.258740

0.251092 0.249509 0.247790

0.245941 0.243966 0.241870 0.239657 0.237335

0.124009 0.122121 0.120307

0.118583

0.296727

0.367209 0.362720

0.430127

0.358145

0.424860

0.353500

0.419519 0.414121 0.408685 0.403228

0.224292 0.221447

0.218543

0.197331

0.194307 0.191324 0.188396

0.185540 0.182774

0.180119 0.177596

0.293022 0.289242 0.285399 0.281505 0.277573 0.273616

0.269648 0.265684 0.261739 0.257832 0.253979

0.250200

0.380037

0.348799 0.344057

0.339290 0.334516 0.329751 0.325015 0.320328 0.315710 0.311185

0.376331

0.371186

0.230436

0.287571

0.344410 0.341017 0.338088 0.335718 0.334046

2 88

0.173054 0.171099 0.169410 0.168043

0.227922 0.225750

0.223992

0.284573 0.281983 0.279887

0.167078

0.222751

0.278408

90

0.055556

0.290914

0.111111

0.166667

0.222222

0.341370

00.z::2" 0: 356706

[ 1 [(-y-1 [c-y1 [c-y1 [c-y1 0.277778

0.333333 c-p [1 I

0.086990

0.173318

0.085677

0.170704

0.083124

0.165625 0.158377

0.421815 0.415475 0.403252 0.386013

0.499050 0.491565

0.079476 0.074953

0.258327 0.254434 0.246882 0.236134

0.477203 0.457086

0.149408 0.139334 0.128968 0.119433 0.112490

0.208034 0.192809 0.178839 0.168682

0.364976 0.341676

0.432729 0.405958

C-55)2

2;

0.069861 0.064614

75

0.059779

a5

0.392328 0.386926 0.381586

0.352309 0.348194

0.115479 0.114143 0.112988 0.112053 0.111392

E 35 45

0.397769

0.294547

0.302515 0.298427

0.057095 0.056508 0.056034 0.055698

5

0.435306

0.306778

0.175231

0.057773

0.445330

0.246517 0.242952 0.239531 0.236282 0.233238

0.116967

8820

0.463642 0.459316 0.454813 0.450147 0.440378

0.200380

0.062100 0.061143 0.060223 0.059348 0.058528

0.467777

0.371600

0.134126 0.132049 0.129989 0.127955 0.125958

0.066175 0.065131 0.064100 0.063088

0.471710

0.375880

0.215587 0.212589 0.209558 0.206506 0.203443

0.067226

0.395191 0.391645 0.387930 0.384057

0.487937

0.303869 0.300346

0.144464 0.142428 0.140370 0.138295 0.136211

0.070385

0.316806

0.492638 0.490424

0.229767 0.227068

0.072455

0.150367

0.319707

0.401736 0.398558

0.496219 0.494572

0.234908 0.232383

0.154073

0.148439 0.146470

0.325052 0.322458

0.497574

0.313764 0.310587 0.307286

0.077307 0.076385 0.075436 0.074463 0. 073469

0.152246

~&\a)

{K(rr)E(v\90”-a)-[K(a)-E(~)]qco\SOo-+

0.087050

0.069336 0.068281

:i 78

10” 0.173648

FUNCTION

0.087129

0.071426

7720

LAMBDA

=- F(p’go”-~)+~K(~)Z(p\90”-~)=H K’(ol)

0.085476 0.085037

20 22

INTEGRALS

0.056256

0.222878

0.336231 0.326288 0.312192 0.294884 0.275597

0.255897 0.237883

0.224814

0.318009 0.296459 0.280867

0.378946 0.354475

0.336826

Compiled from C. Heuman, Tables of complete elliptic integrals, J. Math. Phys. 20, 127-206, 1941 (with permission).

ELLIPTIC HEUMAN’S

623

INTEGRALS

LAMBDA

35” 0.573576 0.573402 0.572878 0.572009 0.570795

40” 0.642788 0.642592 0.642006 0.641032 0.639674

0.569244 0.567360 0.565150

0.701786

t . %%

0.637940 0.635836 0.633373 0.630561 0.627412

0.556657 0.553238 0.549546 0.545591 0.541389 0.536953

FUNCTION

+\a)

Table

17.8

50” 0.766044 0.765811 0.765113 0.763956 0.762347

55” 0.819152 0.818903 0.818157 0.816922 0.815210

60” 0.866025 0.865762 0.864975 0.863674 0.861876

is E% 0:693729 0.690306

0.760298 0.757822 0.754937 0.751660 0.748011

0.813034 0.810416 0.807375 0.803935 0.800123

0.859602 0.856877 0.853731 0.850194 0.846297

0.623939 0.620157 0.616080 0.611725 0.607107

0.686540 0.682450 0.678054 0.673372 0.668422

0.744012 0.739683 0.735049 0.730130 0.724951

0.795963 0.791483 0.786709 0.781667 0.776384

0.842073 0.837553 0.832766 0.827743 0.822510

is :;:z 0:522388 0.517165

0.602244 0.597153 0.591851 0.586356 0.580687

0.663225 0.657801 0.652170 0.646351 0.640365

0.719533 0.713900 0.708073 0.702074 0.695923

0.770883 0.765190 0.759326 0.753314 0.747177

0.817093 0.811517 0.805804 0.799976 0.794052

0.511786 0.506266 0.500622 0.494873 0.489034

0.574862 0.568898 0.562815 0.556632 0.550366

0.634231 0.627970 0.621600 0.615142 0.608615

0.689642 0.683251 0.676769 0.670217 0.663613

0.740932 0.734602 0.728203 0.721756 0.715277

0.788051 0.781992 0.775891 0.769764 0.763627

0.483125 0.477164 0.471170 0.465163 0.459163

0.544038 0.537668 0.531275 0.524879 0.518502

0.602038 0.595432 0.588817 0.582212 0.575640

0.656976 0.650326 0.643682 0.637064 0.630491

0.708785 0.702298 0.695832 0.689405 0.683037

0.757496 0.75-1385 0.745310 0.739286 0.733329

0.453192 0.447272 0.441428 0.435683 0.430065

0.512167 0.505895 0.499711 0.493642 0.487715

0.569122 0.562680 0.556339 0.550124 0.544062

0.623985 0.617567 0.611258 0.605085 0.599072

0.676745 0.670549 0.664469 0.658528 0.652749

0.727455 0.721680 0.716024 0.710504 0.705142

0.424604 0.419332 0.414284 0.409500 0.405026

0.481959 0.476408 0.471098 0.466070 0.461371

0.538183 0.532519 0.527106 0.521985 0.517202

0.593247 0.587641 0.582290 0.577231 0.572511

0.647159 0.641784 0.636659 0.631818 0.627303

0.699961 0.694985 0.690244 0.685770 0.681601

0.400915 0.397229 0.394049 0.391477 0.389662

0.457055 0.453189 0.449853 0.447157 0.445255

0.512813 0.508883 0.505494 0.502754 0.500823

0.568181 0.564307 0.560967 0.558268 0.556366

0.623166 0.619464 0.616276 0.613700 0.611884

0.677782 0.674368 0.671427 0.669053 0.667379

0.444444 (-;I1 c 1 [(-j)l1

0.500000 c-y [ I

0.555556

0.611111 (-$)l [ I

0.666647

13.705765 13.695307 11.675748 0.649283 11.618381 0.585512 11.553214 0.524506 0.504034

0.764592 0.753346 0.732623 0.705094 0.673501 0.640369 0.608153 0.579721 0.559529

0.817600 0.805703 0.784220 0.756337 0.724985 0.692612 0.661480 0.634200 0.614903

0.864388 0.852010 0.830282 0.802903 0.772830 0.742291 0.713246

0.388889

0.572487 0.563926 0.547600 0.524935 0.497760 0.468167 0.438541 0.411857 0.392679

0.641567 0.632010 0.613936 0.589127 0.559735 0.528076 0.496661 0.468546 0.448417

0.707107 0.706891 0.706247 0.705177 0.703687

[(-p1

[c-y1

624

ELLIPTIC

Table

17.8

HEUMAN’S

65

70"

INTEGRALS

LAMBDA

FUNCTION

75”

~,+\ol)

80

85"

0.906308 0.906032 0.905210 0.903857 0.901997

0.939693 0.939407 0.938559 0.937172 0.935282

0.965926 0.965633 0.964769 0.963376 0.961512

0.984808 0.984511 0.983652 0.982315 0.980599

0.996195 0.995903 0.995130 0.994063 0.992833

0.899660 0.896881 0.893699 0. a90152 0. a86280

0.932934 0.930177 0.927061 0.923634 0.919940

0.959244 0.956638 0.953755 0.950646 0.947355

0.978597 0.976384 0.974016 0.971534 0.968969

0.991511 0.990135 0.988727 0.987299 0.985858

0.882119 0. a77704 0.873068 0. a68240 0.863249

0.916018 0. 91.1904 0.907630 0.903221 0.898703

0.943918 0.940364 0.936718 0.933000 0.929226

0.966343 0.963671 0.960968 0.958241 0.955500

0.984410

0. 0. 0. 0. 0.

a58117 a52869 a47523 a42100 a36615

0. a94095 0. a89416 0.884681 : . t:E::

0.925409 0.921563 0.917695 0.913817 0.909935

0.952751 0.949998 0.947247 0.944502 0.941766

0.977159 0.975719 0.974286 0.972861 0.971445

0.831085 0. a25524 0.819946 0.814365 0.808792

0. a70277 0.865449 0.860625 0. a55814 0.851026

0.906056 0.902188 0.898337 0. a94508 0.890708

0.939042 0.936335 0.933647 0.930981 0.928341

0.970039 0.968644 0.967262 0.965894 0.964540

0.803241 0.797724 0.792252 0.786839 0.781496

0.846269 0. a41553 0. a36887 0.832280 0. a27742

0.886942 0.883216 0. a79537 0. a75911 0. a72345

0.925731 0.923152 0.920610 0.918108 0.915649

0.963204 0.961885 0.960586 0.959309 0.958055

0.776237 0.771077 0.766029 0.761110 0.756338

0.823283 0. ala913 0.814645 0.810490 0.806464

0. a68846 0.865421 0. a62080 0. a58831 0.855685

is 99E: 0: 908588 0.906357 0.904198

0.956826 0.955626 0.954457 0.953321 0.952223

0.751731 0.747312 0.743104 0.739137 0.735442

0.802581 0.798860 0.795319 0.791983 0.788877

0.852654 0. a49751 0.846990 0. a44390 0. a41 972

0.902119 0.900129 0.898237 0.896456 0.894800

0.951166 0.950154 0.949193 0.948288 0.947446

0.732059 0.729036 0.726434 0.724333 0.722852

0.786036 0.783497 0.781312 0.779549 0.778307

0. a39759 0.837783 0. a36083 0. a34711 0. a33745

0.893286 0. a91933 0. a90770 0.889831 0.889170

0.946677 0.945990 0.945400 0.944923 0.944587

90” : 11 1 1 : : :

i- K”o: 0: 980054 0.978604

[(-p1 [(-y 1 [(-i)7 1 c(-!)5 1 [ 1

0.722222

0.777778

0. a33333

0.888889

0.944444 t-y

0.904599 0.891969 0.870676 0.844820 0.817155 0.789537 0.763552 0.741089 0.725315

0.937930 0.925384 0.905441 0.882297 0. a58217 0.834576 0.812552 0.793624 0.780373

0.964135 0.952226 0.934867 0.915757 0.896419 0. a77717 0.860443 0.845669 0.835352

0.983037 0.972787 0.959607 0.945873 0.932311 0.919353 0.907464 0. a97332 0.890270

0.994624 0.988015 0.980779 0.973573 0.966576 0.959944 0.953885 0.948733 0.945145

: 1 : : 1 : 1 : i : 1 : 1’ 1 i : 1 1’ : 1 1

: 1 : 1 : 1

ELIJPTIC

ELLIPTIC

INTEGRAL

625

INTEGRALS

OF THE

THIRD

an;v\+sop(l- 72sin2 @-'[l--sin2

15-20.

n (n; +‘\a)

Table 17.9

LYsin2 0]+ds

15" 0.26180 0.26200 0.26254 0.26330 0.26406 0.26463 0.26484

30" 0.52360 0.52513 0.52943 0. 53562 0 542413 0: 54736 0. 54991

45" 0.78540 0.79025 0.80437 0.82602 0.85122 0.87270 0.88137

60" 1.04720 1.05774 1.08955 1.14243 1.21260 1.28371 1.31696

75" 1.30900 1.32733 1.38457 1.48788 1.64918 1.87145 2.02759

1.57080 1.59814 1.68575 1.85407 2.15651 2.76806 06

0.26239 0.26259 0.26314 0.26390 0.26467 0.26524 0.26545

0.52820 0.52975 0.53412 0.54041 0.54712 0.55234 0.55431

0.80013 0.80514 0.81972 0.84210 0.86817 0.89040 0.89939

1.07949 1.09058 1.12405 1.17980 1.25393 1.32926 1.36454

1.36560 1.38520 1.44649 1.55739 1.73121 1.97204 2.14201

1.65576 1.68536 1.78030 1.96326 2.29355 2.966101 co

0.26249 0.26319 0.26374 0.26450 0.26527 0.26585 0.26606

0.53294 0. 5345.2 0.53896 0. 545315 0.55217 !*. zt13

0.81586 0.82104 0.83612 0.85928 0.88629 0.90934 0.91867

1.11534 1.12705 1.16241 1.22139 1.30003 1.38016 1.41777

1.43078 1.45187 1.51792 1.63775 1.82643 2.08942 2.27604

1.75620 1.78850 1.89229 2.09296 2.45715 3.20448 m

0.26359 0.26379 0.26434 0.26511 0.26588 0.26646 0.26667

0.53784 0.53945 0.54396 0.55046 0.55739 0.56278 0.56483'

0.83271 0.83808 0.85370 0.87771 0.90574 0.92969 0.93938

1.15551 1.16791 1.20543 1.26812 1.35193 1.43759 1.47789

1.50701 1.52988 1.60161 1.73217 1.93879 2.22876 2.43581

1.87746 1.91309 2.02779 2.25038 2.65684 3.49853 al

0.26420 0.26440 0.26495 0.26572 0.26650 0.26708 0.26729

0.54291 0.54454 0.54912 0.55573 0.56278 0.56827 0.57035

0.85084 0.85641 0.87262 0.89756 0.92670 0.95162 0.96171

1.20098 1.21419 1.25419 1.32117 1.41098 1.50309 1.54653

1.59794 1.62298 1.70165 1.84537 2.07413 2.39775 2.63052

2.02789 2.06774 2.19629 2.44683 2.90761 3.87214 m

0.26481 0.26501 0.26557 0.26634 0.26712 0.26770 0.26792

0.54814 0.54980 0.55447 0.56119 0.56837 0.57394 0.57606

0.87042 0.87621 0.89307 0.91902 0.94939 0.97538 0.98591

1.25310 1.26726 1.31017 1.38218 1.47906 1.57881 1.62599

1.70919 1.73695 1.82433 1.98464 2.24155 2.60846 2.87468

2.22144 2.26685 2.41367 2.70129 3.23477 4.36620 00

0.26543 0.26563 0.26619 0.26696 0.26775 0.26833 0.26855 C-f)5

0.55357 0.55525 0.56000 0.56684 0,57414 0.57982 0.58198 c-:)4

0.89167 0.89770 0.91527 0.94235 0.97406 1.00123 1.01225

1.31379 1.32907 1.37544 1.45347 1.55884 1.66780 1.71951

1.85002 1.88131 1.98005 2.16210 2.45623 2.88113 3.19278

2.48365 2.53677 2.70905 3.04862 3.68509 5.05734 00

[

$3~ Examples

KIND

1 c 1 [

(-72

1 1 [C-f)7

90

626 Table

ELLIPTIC

17.9

ELLIPTIC

INTEGRAL

INTEGRALS

OF THE

THIRD

Nn; v\a,=JJ (l- n sin2 e)-'[l-sin2 n 0.7

ff\P O0

i::

15 30 45 60

i::

;o'

0"::

8-Z 0:9 0.9 iE 0: 9 1.0 ::: i-i 1: 0 1. 0

KIND

n(n;

~\a)

a sin2 e]-*dfI

0.26605 0.26625 0.26681 0.26759 0.26838 0.26897 0.26918

30" 0.55918 0.56090 0.56573 0.57270 0.58014 0.58592 0.58812

45" 0.91487 0.92116 0.93952 0.96784 1.00104 1.02954 1.04110

60" 1.38587 1.40251 1.45309 1.53846 1.65425 1.77459 1.83192

75" 2.03720 2.07333 2.18765 2.39973 2.74586 3.25315 3.63042

2.86787 2.93263 3.14339 3.56210 4.35751 6.11030 03

0.26668 0.26688 0.26745 0.26823 0.26902 0.26961 0.26982

0.56503. 0.56676 0.57168 0.57877 0.58635 0.59225 0.59449

0.94034 0.94694 0.96618 0.99588 1.03076 1.06073 1.07290

1.47370 1.49205 1.54790 1.64250 1.77145 1.90629 1.97080

2.30538 2.34868 2.48618 2.74328 3.16844 3.80370 4.28518

3.51240 3.59733 3.87507 4.43274 5.51206 7.96669 w

0.26731 0.26752 0.26808 0.26887 0.26966 0.27025 0.27047

0.57106 0.57284 0.57785 0.58508 0.59281 0.59882 0.60110

0.96853 0.97547 0.99569 1.02695 1.06372 1.09535 1.10821

1.58459 1.60515 1.66788 1.77453 1.92081 2.07487 2.14899

2.74439 2.79990 2.97710 3.31210 3.87661 4.74432 5.42125

4.96729 5.09958 5.53551 6.42557 8.20086 12.46407 m

0.26795 0.26816 0.26872 0.26951 0.27031 0.27090 0.27112 Q5)5

0.57735 0.57916 0.58428 0.59165 0.59953 0.60566 0.60799 C-$5

1.00000 1.00731 1.02866 1.06170 1.10060 1.13414 1.14779

1.73205 1.75565 1.82781 1.95114 2.12160 2.30276 2.39053

3.73205 3.81655 4.08864 4.61280 5.52554 7.00372 8.22356

15”

[

1 [

1

[(-f)l1

90”

a, 00 00 00 co 00 co

18. Weierstrass

Elliptic Functions

and Related

THOMAS H. SOUTHARD ’

Contents Page Mathematical 18.1. 18.2. 18.3. 18.4.

18.5. 18.6. 18.7. 18.8. 18.9. 18.10. 18.11. 18.12. 18.13. 18.14. 18.15.

Properties

............

+ .......

Definitions, Symbolism, Restrictions and Conventions . . Homogeneity Relations, Reduction Formulas and Processes. Special Values and Relations ............. ......... Addition and Multiplication Formulas Series Expansions. .................. Derivatives and Differential Equations ......... Integrals, ...................... Conformal Mapping ................. Relations with Complete Elliptic Integrals K and K’ and Their Parameter m and with Jacobi’s Elliptic Functions , Relations with Theta Functions ............ Expressing any Elliptic Function in Terms of Q.3 and @’ , . Case A=0 ...................... Equianharmonic Case (g2=0, g3= 1) ........... Lemniscatic Case (g,=l, g3=O). ............ Pseudodemniscatic Case (g,= - 1, g3=O) .......

Numerical Methods ...................... 18.16. Use and Extension of the Tables References

1 University at Hayward.

629 629

631 633 635 635 640 641 642 649 650 651 651 652 658 662 663 663

............

..........................

670

of California, National Bureau of Standards, and California State College 627

628

WEIERSTRASS

ELLIPTIC

AND

RELATED

FUNCTIONS

Table for Obtaining Periods for Invariants g2 and g3 G2=g2g3-2fi). . . . . . . . . . . . . . . . . . . . . . . .

Table

18.1.

Non-Negative

Discriminant

wgY, f$f+g

Page 673

(3 I z2 5 03)

ln(J2-3);

T2=3(.05)3.4,

7D

wg;le, w’g:“/i; ;2=3.4(.1)5(.2)10, 7D wg;‘eg:‘4, w’g:‘“ij:“/i; ?;I=.1 (-.Ol)O, 7D Non-Positive

Discriminant (- a0 5J2 5 3) w’zg:‘eljj$‘4/i; J;‘=O(-.01)--.2,

w2gY61&p4,

wzg;“, w;g:‘6/i; J;‘=-.2(-.05)-l, w2gi”, +

g:‘“+$

18.2. Table Real Half-Period-

Table

Positive

h(3-y,);

7D

7D

J2=-1(.2)3,

7D

for Obtaining @, @’ and 5 on Oz and Oy (Unit Period Ratio a). . . . . . . . . . . . . . .

Discriminant

674

(0 5s 5 1, 0
z2 p(z), 2 @j’(z), zf(z),u=l, 1.05, 1.1, 1.2, 1.4, 2, 4 2=0(.05)1, y=O(.O5) 1.1, 1.2 (.2) a, 6--8D Negative

Discrimiuant

(0 5x 5 1, 0 -

2 g(z), 2 @‘(z rel="nofollow">, z{(z),a=l, 1.05,1.15,1.3, x=0(.05)1, y=O(.O5)1 (.l)b(bla/2), 7D 18.3. Invariants Real Half-Period).

and Values at Half-Periods

Table

a=i(.o2)i.s(.d5)‘z.~(.i)‘q,‘oD,

Ndn-Negative s2, g3,

s3,

(1 <~<m)

(Unit

S--&b ’ ’ * * ’ ’ ’ * ’ * ’ * ’

680

Discriminant

el= @ (0,

Non-Positive 92,

1.5, 2, 4

e3=

@ (4,

v=SO>,

tl’/~=lbW,

4),

4W,

&J

Discriminant el,

a=W>,

d/~=W)/~,

u(l),

u(w#,

u(w’)

The author gratefully acknowledges the assistance and encouragement of the personnel of Numerical Analysis Research, UCLA (especially Dr. C. B. Tompkins for generating the author’s interest in the project, and Mrs. H. 0. Rosay for programming and computing, hand calculation and formula checking) and the personnel of the Computation Laboratory (especially R. Capuano, E. Godefroy, D. Liepman, B. Walter and R. Zucker for the preparation and checking of the tables and maps).

629 18. Weierstrass

Elliptic

Mathematical 18.1.

Definitions,

Symbolism, Conventions

Restrictions

and

Related

Functions

Properties Invarlants

and

An elliptic function is a single-valued doubly periodic function of a single complex variable which is analytic except at poles and whose only singularities in the finite plane are poles. If w and w’ are a pair of (primitive) half-periods of such a function f(z), then f(z+2il4~+2Nw’)=f(z), M and N being integers. Thus the study of any such function can be reduced to consideration of its behavior in a fundamental period paradelogram (FPP). An elliptic function has a finite number of poles (and the same number of zeros) in a FPP; the number of such poles (zeros) (an irreducible set) is the order of the function (poles and zeros are counted according to their multiplicity). All other poles (zeros) are called congruent to the irreducible set. The simplest (nontrivial) elliptic functions are of order two. One may choose as the standard function of order two either a function with two simple poles (Jacobi’s choice) or one double pole (Weiemtrass’ choice) in a FPP. Wekrstrass’ @-Function. Let W, w’ denote a pair of complex numbers with Jf(w’/w)>O. Then p(z)= @ (zlw, w’) is an elliptic function of order two with periods 20, 2w’ and having a double pole at z=O, whose principal part is zwa; @ (z) - .z-’ is analytic in a neighborhood of the origin and ‘vanishes at z=O. Weierstrass’ ~-Function j(z) = f(zIw, w’) satisfies the condition l’(z) = - @ (z) ; further, c(z) has a simple pole at z=O whose principal part is 2-I; {(2)-z-’ vanishes at z=O and is analytic in a neighborhood of the origin. t(z) is NOT an elliptic function, since it is not periodic. However, it is quasi-periodic (see “period” relations):, so reduction to FPP is possible. Weierstrass’ u-Function u(z) = u(zIw, w’) satisfies the condition u’(z)/u(z)=~(z); further, u(z) is an entire function which vanishes at the origin. Like t, it is NOT an elliptic function, since it is not periodic. However, it is quasi-periodic (see “period’, relations), so reduction to FPP is possible.

~8 and

Let W=2Mw+2Nw‘, Then ga=6OZ’W-’

18.1.1.

g3

M and N being integers. and g3=1402’Wse

are the INVARIANTS, summation pairs M, N except M=N=O. Alternate

Symbolism

18.1.2 18.1.3 18.1.4 18.1.5

being over all

Emphasising

Invariants

@ (4 = @ (2; $72,93) @‘(4=P’(z; 92, Sal e9=T(z; 92, Sal “M=&; g2, gal

Fundamental

Differential Related

18.1.6

9

“k+=4

Equation, Quantities

Discrim,inant

@

~“(4-92

and

(4-93

18.1.7 =4(

P (4---ed(

@

Cd--e2)(

P (4-e3)

18.1.8

A=&-27&=16(e2-e3)2(e3-e1)2(e1-e2)2 18.1.9 ga=--4(elea+ele3+e2e3)=2(et+e:+e~)

18.1.10

g3=4ele2e3=#(ef+e:+e~)

18.1.11

el+e2+e3=0

18.1.12

eHf#-ei=gV3 4et-gsel-g3=0(i=l,

18.1.13 Agreement

ahout

Values

of Invariants inant)

2, 3) (and

Discrim-

We shall consider, in this chapter, only recal g2 and g3 (this seems to cover most applications)hence A is real. We shall dichotomize most of what follows (either A>0 or A. Homogeneity relations 18.2.1-18.2.15 enable a further restriction to non-negative g3 (except for one case when A=O). Note

on Symbolism for Roots of Complex Numbers for Conjugate Complex Numbers

and

In this chapter, elln _ (n a positive integer) is used to denote the principal nth root of z, as in chapter 3; 5 is used to denote the complex conjugate af z.

WEIERSTRASS

630

ELLIPTIC

FPP’s,

Symbols

AND RELATED FUNCTIONS for

Periods,

etc.

Y

A>0

A<0

202

2w’

J

R3

R2 co' R,=

R4

+FPP

r

r w

0

WX 2w

FIGURE 18.1

RECTANGLE

RHOMBUS

w1=w w=wfw’

w&w’-w

w=w’

w REAL w’ PURE

wa REAL w; PURE

IMAG.

IMAG.

Iw:I 2 02, since ga2 0

1W’ 12 w, since g3 Z 0 Fundamental

Rectangles

Study of all four functions (,f$ , @ ‘t 1, u) can be reduced to consideration of their values in a Fundamental Rectangle including the origin (see 18.2 on homogeneity relations, reduction formulas and processes). A<0 A>0 Fundamental

Rectangle

is i FPP, which has ver-

Fundamental

Rectangle

has vertices 0, w2, w&$

tices 0, w, waand w’

UP FUNDAMENTAL RECTANGLE (+

FPP)

-x FIGURE 18.2

There is a point on the right boundary

of Fundamental

Rectangle

where @ =O.

Denote it by 4.

WEIERSTRASS

18.2.

Note

18.2.1 18.2.2 18.2.3 18.2.4

Period

@‘(tzlto,

(Suppose

Ratio

Thus the case g3<0 can be reduced to one where sa>o*

0’)

“Period”

e,(tw, td)=te2et(w,

A&o, tw’) =t-“A(w, H,(tw, t~‘)=t-~H~(w,

4

(See 18.10)

m(tw, tw’)=m(w,

18.2.12

@‘(tz;

18.2.13

@ (tz; t-‘g2, t-Oga)=t-“@

18.2.14

f&z; t-4gs, t-“gJ =t-‘{(z;

18.2.15

a(&; t-‘ga, t+ga) =tu(z;

=(-1)

= p (2)

=@)

+2Mq+2N$

M+N+MNu(~) exp [(z+Mw+Nw’)(2Mq

18.2.21

(z; 92, ga)

where q={(w), “tin

ga, ga)

jugate”

q’={(w’) Values

j(Z)=J
ga, ga) to ‘/

FPP

(See

Figure

(Z denotes conjugate

18.1) Ah<0

of s)

R,

~4 in

JI.qI,)=-p’(2C@--z4)

a’=---$mBw) g (4 = 3 @w- 24)

Lw4)=PcG=a --

18.2.24

rw

18.2.25

u(zJ=u(2w-2,)

= --J@w-4

--

~(2,)=-~(2w2-223+2(~+~‘)

+a

exp [2q(z,--w)]

~(23 =$GZj Point

@‘&a)=-

9 6%)= @ Gk?- za)

18.2.28

l(za> = --1er-%za)+~h+B’~ ~(4 =u@w-4

5a in

@‘me-4

18.2.27

@ (4 =B G=zo’) -s%) =fh--2w’) +%I’ -u(zg)= -u(z+2w’) exp [2T&?3-w’)]

(a--41

--@‘&r~~

@ @a) = @

ch-

4

ea)=--~(2~-zJ+2(q+q’) u(za)=u(2@2-22) exp P(ll+ll’)(%---o2)1

exp Ph+r13 b-w31

@‘(z$) =~$pizo’)

exp Dh+rl’)

Ra

@‘(za)=

Point

18.2.33

(M,N

+24w1

Point

18.2.32

FPP

u(z+2Mw+2Nw’)

t-4g2, t-“g3) =t+ @‘(z; gz, ga)

Reduction

18.2.31

the

18.2.20

(See 18.9)

w’)

A>0

18.2.30

to

p(z)

@ (z+2Mo+2Nw’)

c(z+2Mw+2Nw’)

w’) 2, 3 (See 18.3)

18.2.11

18~%~

Reduction integers)

18.2.19

2, 3

w’), i=l,

!I@% iw=!&,

18.2.26

and

p(Z+2&+2Nw’)~

18.2.18

4

w’), i=l,

18.2.10

18.2.23

Relations

18.2.17

to’) =f4g2(w, w’)

g&w tf.4 = f-%b,

18.2.22

@(z; 92, gJ = - @‘(iz; ga, -ga)

u(tzlkd, ta’) =tu(zIw, 0’)

18.2.6

18.2.9

18.2.16

(z/w, w’)

g*(tw,

18.2.8

ga
Put t=i and obtain, e.g.,

0’)

{(tzltw, tw’) =t-‘{(zlw,

Case

t # 0)

h’)=t-3@‘(ZIq

@ (tzltw, tw’) =t-‘$p

631

FUNCTIONS The

is preserved.

18.2.5

18.2.7

RELATED

Relations,

Relations

that

AND

Reduction and Processea

Homogeneity Formulas Homogeneity

ELLIPTIC

53

in

RI

Gw2)=m2) mz1)=PGl m-J =m 4%) =x2)

632

WEIERSTRASS

ELLIPTIC

!W’

AND

RELATED

FUNCTIONS

Reduction

from

l/r FPFa;

T;u
Rectangle

We need only be concerned with the case when z is in triangle A2 (therefore 2w’--z is in triangle AI>.

1 JP + $

18.2.34

@(z)=&D(2d-2)

18.2.35

@p’(z)=--@‘(20’-2)

18.2.36

Z(z)=2?f---5(26+-z)

18.2.37

-x 2

u(z)=u(~w’-2)

exp [24(2-a’)]

FJGURE 18.3 Reduction

to

Case

where

Real

Half-Period

is

Unity

(preserving period ratio)

00

A<0 (w*=w+w’)

18.2.38 18.2.39 18.2.40 18.2.41 18.2.42 18.243

.g3(0, w’)=w-6g3

18.2.44 (i=1,2,3) 18.2.45

in

(i=1,2,3) A(w, w’)=w-l*A

NOTE:

New real half-period

is

WEIERSTRASS

ELLIPTIC

AND

RELATED

633

FUNCTIONS

18.3. Special Values and Relations Values @‘,

@‘9

and { are infinite,

at

Periods

u is zero at 2.=2w1, i=1,2,3

and at 2oi(A
00

A<0 Half-Periods

18.3.1

$I) (wl)=eI(i=1,2,3)

18.3.2

@‘(wJ=O(i=1,2,3)

18.3.3

vf=s(4(i=l,

18.3.4

tl1=:7,

18.3.5

E=2e3+e,ek

2,3)

m=rl++,

(i, j, k=l,

l)a=rl’

2,3; i#j,

i#k,

j#k)

= (et- e,) (er-en) =2et+4-j=3e+$ f

18.3.6 18.3.7

e: real

18.3.8

el>O2 (equality

ea real and non-negative (es=0 when g,=(I)

e2>ea

el=-fx+ig,

when ga=O)

e3=Z1

where C-Z>0, /3>0 (equality 18.3.9

~>0

18.3.10

$l$O

18.3.11

(o’l/~~1.91014

18.3.12

Hl>O,

18.3.13

Hg = im

r/4<arg(HJ

18.3.14

u(o)=erUIS/Hf/’

~(0~) = eW21z/P2fa

18.3.15

~(w’)=ie”‘~“~/@~‘~

b(&) =~eq~w~‘a/~l~

18.3.16

Qe(w2)=e~2W2/(-H2)

ti(w’)=eq’@‘/(-Ha)

18.3.17

arg

when ga=O)

d=l(4=1’-11 if &i$O

050 (approx.)

Ha>0

[u(wJ]=+- tfw

if jw#$3.81915

447 (approx.)

Ha>0

7r

I?r/2 (equality

if ga=O) ; H,=a

arg[u(w’)]=4i iwa+” 2-2 1 ar g( eg+H$-et)

2 Quarter

PerioQ

18.3.18

@ (w/2)=el+H~>e~

@

18.3.19

@‘(w/2)=--2H&&&

QJ ‘(wJ2) = -2Hjd2H,+3e,

18.3.20

z(w/2)=4h+d%i34

Zba/2)=3hr+@zTa

b@)=es+H,>el

634

WEIERSTRASS

ELLIPTIC

AND

RELATED

FTJWCTIONS A<0

A>0 er291~

ew18

18.3.21

a(w/2)=21/4~/8(2H1+3e1)l/8

18.3.22

@ W/2)

=e3-&<e3<0

18.3.23

@‘(w’/2)

= -2H,i4-3

18.3.24

[(0’/2)

~(~2~2)=21,4~/8(2~2+3,2)1/8

@ bib3 =e2--H2= @ (w2+4/2> <es<0 @‘(w;/2)=-2H,i,/~=@‘(w,+w;/2)

=~[q’---iJm

Hw&‘)

=H&d2H2--3ezl=

-T(wa+~P)+W

~&fI~

18.3.25

u(w’~2)=21/4H~“(2H3-3e3)1/8 =dw2+W;/2)

6 (w’/2)=e3-Ha

18.3.26

@ (w2/2)=e2-Ha

18.3.27

@‘(~/2)

18.3.28

T(oz/2)=3[r12-i(2H2-3e2)11

18.3.29

u(w2’2)=[~(2H2-&?2)]1’8

,(P’(w’/2) = -2iH,(2H,-3e#

= -2Hsi(2Hz-33e,)+

r(w’/2)

=$[q’--i(2H,--3e#]

e920~la&d4

One-Third

At z=2w,/3(i=l, equivalently

Period

Relations

2, 3) or 20;/3,@“~=12@@‘~;

:

4g@4--24g2@2-48g3@ -d=O

18.3.30

A-0

00 18.3.31

{(2w/3) =2+[

@(y’3)]

18.3.32

*(2w~,3)=!!!&[

@(,‘j3)]

18.3.33

@w2/3)

@ (F’3)]1

18.3.34

u(2w/3) =

18.3.35

u(2w’/3)

18.3.36

u(2w2/3)

=%+[

{(2w2/3)=F+[

--exp Pvd91

4243)

W(243) -exp [2q’w’/9] = [ @ ‘(2w’/3)]1/3e2*1’3

u(2w;/3) =

-exp P72wJ91 = [ @‘(2w2/3)]1’3e2rf13

u(2w’/3) Legendre’e

18.3.37

I]w’-

=

=

Relation

lfw=ui/2

q2w;-lj;wz=ti

(also valid for A
18.3.38 18.3.39

Among

E+z+G=Ql2/4

WW-~P3+~~=0

the

Hi

@(,,,,]t

--exp

P72w2/91

7 @ ‘@~~/3) -exp [@

P&d/91

‘(2w~/3)]‘13e2~*‘”

-exp [2q’w’/9] [ @ ‘(2w’/3)]1/3e2Tg13

exp [--rl’~~l

WEIERSTRASS

ELLIPTIC

AND

RELATED

LT;H;fl=

18.3.40

--A/l 6

16H:-l:!gzH:+A=O(i=l,

18.3.41

18.4. Addition

FUNCTIONS

2, 3)

and Multiplication

Addition

Formulasa

(51#

Formulas 52)

18.4.1 18.4.2 18.4.3 18.4.4

~(~I+~2)~(~1-22)~=-~2(Zl)u2(Z2)[lP Duplication

L-

Note that@“=6@2(z)-$j

and

(ZJ-@ Triplication

Formulas

@‘“(z)=4@,“(z)-g2@(z)-g

3 and @“‘(z)=lZ$B(z)g’(z)]

18.4.5

18.4.6

p ‘(22)=

18.4.7

m4 =X(z) 1

18.4.8

+ @“W/2

u(22) == -

lP’c4

gJ’(z)B(z)

4 09 ‘3(2>

18.4.9

~(32)=3Kz)i-~,(,)~,,,(~)-

18.4.10

u(32)= - p”“(z)u”(z)[@

18.5.1 18.5.2

g.+z-2+&

where

ca=g2/20,

Series ckza-2

ca=g3/28

and 18.5.3 18.5.4 18.5.5 18.5.6 * Formulas

for { and u are nof true algebraic

gj”“&.)

(22) - P(z)]

18.5. !3eries Expansions ILaurent

(z2)l

add.ition formulas.

635

WEIERSTRASS

636

AND RELATED

where a,,=1

18.5.7 18.5.8

ELLIPTIC

a~,,=3(m+l)a~+,,.-I+~

FUNCTIONS

and (2m+3~-1)(4m+6n-1)u,,+~,.,

(n+l)a,_S..+l-;

it being understood that a,,,= 0 if either subscript is negative. (The radius of convergence of the above series for @--z-‘, @ ‘+2zm3 and l-z-’ smallest of 12~1,12w’I and (2wf2w’l; series for u converges for all 2.) Values

of Coefficients3

18.5.9

CL in Terms

of ca and

cs=3c2c3/11

18.5.11

cs=

18.5.12

[2$+3c;]/39

c,=2&3/33

18.5.13

cs=5c~(11&+36c~)/7293

18.5.14

c,=c,(29c;+llc:)/2717

18.5.15

clo= (242$+1455&)/240669

18.5.16

c,,=14c,c~(389c;+369c;)/3187041 c~2=(114950c~+1080000c;cc:+166617c~)/891678645

18.5.18 18.5.19

c3

c*=cy3

18.5.10

18.5.17

is equal to the

c,,=10c~c3(297c;+530c~)/l1685817 2c2(528770c!:+7164675c;c;+2989602cj) c14=

(306735)

(215441)

18.5.20

4c3(62921815cs,+179865450c;c~+14051367c’,) c15= (179685) (38920531)

18.5.21

cl~=cj(58957855c~+1086511320c~~+875341836c~) (5909761)(5132565)

18.5.22

c,,,czc~(30171955c:+126138075c~c~+28151739c:) (920205)(6678671)

18.5.23 18.5.24

c1o=2c~c3(3365544215c~+429852433.45c~cf+8527743477c~) (91100295)(113537407)

J NOTES: 1. c,-clp were computed and checked independently by D. H. Lehmer; these were double-checked by substituting ga=20 CZ, g3=28 cz in values given in [18.10]. 2. clrcls were derived from values in [18.10] by the same substitution. These were checked (numerically) for particular values of gz, ga.

3. cl0 is given incorrectly in [18.12] (factor 13 is missing in denominator of third term of bracket) ; this value was computed independently. 4. No factors of any of the above integers with more than ten digits are known to the author. This is not necessarily true of smaller integers, which have, in many instances, been arranged for convenient use with a desk calculator.

=

=

Value

=

=

* of Coefficients

a,, n

=

=

1 I,

-2.3.

6.69

3 .107893773

7

ii t

6

I 10 -2.3. 6.23

-2.3.

.257.13049

.107896773

I

1,

6.69

6 II

8 10

-2:&.~

-2.3.

2.57

131.1699

~41.6947

.13049

2803

,4922497

2:3%

&:f#.~

.0:03

.*-n,?on

6.7

li 10 -2.3. b ,nc*r\.nn Nulyuy

-2.3.

-2$’

6.7

6.7

In .m,l .“il.l,O J42ql

II,

2.3.6

2:3:6.9103

4

-2!&.7 13.37.41

.31 IO 2.3.6.31

2f3f23

: zf35.17

a

-2.3’

2f3i3

-2!3:6.11.31

-2.3.

33609

313.190337

263.43489b3

6.7.2.3 .17b2630144977

-&&.,j3

-2!3:b.b03

-2:3:3$

-2!3:6.7.11.29

-2!3$.7.613

.109

3911

166217

,316939669

a33.1129.9bbl

.176052%081

: !:3:b.b3

i3s.37

-2.3:5X

-2f3h.151

-213’

-2f3k17.63

-2.3yb.71117

;a!&;

167

3037

,853

2337260103

.2967.41X39

R7.19b6blObQ

35866647631801

!!3f311

l~b.‘XJN7

-2.3:11

-3f17

-2f3t7.13

-3’

-2.3:6.7:193

%Q

2609

.1678267

.2742.%7

13679.274973

39bbbew341079

sJ7.23.37

33313.503

-3f7

-347.23.14337

-3f71

-3f7.11.23’

sf7124733

\bOj ;fi,

y i, Id? ’ ’ (_ t

a5973

40763

~1763ca760639

a3.739.13539

19392278bsll

2

zI59

-2.3’

1 10 t

-2f3fb.691

1.107

-1 \ 1

?,? / ” I 4

5

3,fll.37 267981 7

6

8

9

10

11

12

+m

‘Value3

de&

calculators;

Of a..

.I in unfactored form for 4m+‘3n+113b primality of large factors was established

are given in k3.251, p. 7; of (ar, with the aid of SWAC (National

form n)3-m in factored Bureau of Standards

in [13.15], Western

Vol. 4, p. 89 for 4m+6n+l<25. Automatic Computer).

Additional

values

were computed

and

cheeked

on

638

WEIERSTRASS Reversed

Series6

for

Large

ELLIPTIC

1 [

ru2 U9f,- km

2u+c,u~+c,u7+3

RELATED

[@ 1

18.5.25 zy

AND

u 11

FUNCTIONS Reversed

Series

(3*~+5a:)u1~+Q3u”+~

+2

(a;+7a:)U’g+f+j

+35W3 92

(12&+7a;)u”

(3a:+10a:)d1

(9a;+4&

z=Alu+A5~5+A,~7+Ag~9+

18.5.30

where u=( @‘1/3)--lefr’3

A

18.5.33

(lla;+loa~)

u2’

+g9

(143a;+1155&~~+21Oa;)

+g

(143a~+220a;a~+6aol:)u31

A13=%

18.5.37 u29

18.5.38 18.5.39

18.5.4Q + s; . . (195a~+456&;+42a:)U3~

A = 17

14a2AT -51 (d+lW)

where &=g2/6, a3=g3/6

18.5.42

&=---6&I

18.5.43

A,= -4J7

18.5.44

Ag= &i/7

(1615a~+7140a~+2520a$&-24a~)~43

18.5.47

1

18.5.48

+ o(u4y

18.5.28

f

as=gt& u=(

p -‘)i

b In this and other series a choice of the value of the root has been made so that z will be in the Fundamental Rectangle(Figure 18.2). whenever the value of the given function is appropriate.

for

Large

1f 1

. .

18.5.41 where u= {-’

f$$$

18.5.27

Series

z=ufAguSfA,u’+A,us+.

&=36,6,/11

A Is’&

18.5.4.6

a2=g2/8

(az”+S&)

AIS= -96&a3/175

18.5.45

where

7

All=8a2a3A~/11

Reversed

18.5.26

=-+a+% -I-

Ag=O

18.5.36 +?$

...

18.5.32

18.5.35

7 +zoo (33~~+lS0~~~~+10~~)Uz6

( @ ‘(

4=2113

18.5.34

uB

Large

18.5.29

18.5.31

+&

for

(--86:+7%)

Alb= -41&J&/91 82 (13496;-41166;) A17=9163 (1154316;--2256861,)

18.5.49

A,,=&

18.5.50

where 62=g2/12

18.5.51

63=g3/20

WEIERSTRASS Other

Seriesnear

z.

Series

Involving

ELLL-PTIC

AND

RELATED

18.5.57

@

where u= (z-

[@(zo)=Ol

Other

18.5.52

Series near

~~~;u[1--3e2u4~-4e,u'+~2L8+~u~lo

1 [

+702+-5& 13

un 4@3c:~3~14 +u2 -5cy--14c3u2 33 84c;- 10~; u8, 1363~;~~~‘~ +5c:u4+33czc3ue+ 3

18.5.53

where u=(z-z,),@o)=

1jr*..

Involving

30c:us+

264~~4~~’

+(84Oca,-1OOci) u8 54524~~ u,l 3 11 +70q(55&2316d) u13 143

1

+ @

;[ -

15C3U4-

28C3Ue+

3oC;U3+

1 14C3C3d”

@‘(zo)=i& +7(12+5~)u’~-~u’Ll+.

u= @,l[v+av”+2u2v”+

q+5a3) d+f (3@$ ( +15g3@~2+70a3)2P+22(2@~4+7g3@~2+21a3)z+ !J3@‘0

+( ++

{ d+20a3}

+15a (*

“+ { !f+fjas}

@i4+15a2g3@i2+132a8)a’ @;a+?

18.5.59

where u= (z--zo)

(z--zo)=A-bA3-

pi2

‘9

A4+3(ca+b2)As

+ lot’@ a’--3[36c3-3@

6f

-3P

@;4+2002a”5ss @i2+57ke) ~6*+llg3~9~a3+&

{ya3+d}Pi4

..

and b = 4g3/g2 wi

@ ‘=2(3e+5c2)(u+4(10c2ei+21e3)(r3+6(7ti2ei

wi

+21c3e,+5~~)~sS24(6c3e~t10c~e,

18.5.56 (@ -ei)=(3e?-&&t

(loC2eit21C3)U2+

22f$ei+92c2c3ei+105C

(7C3t!:

+21c3ef+5ci)u3+(18cae:+30c~ei +3h&)U4+

+ F

A”+.

18.5.63

18.5.55 where v= @ /(Q.J 6)” and a=g2/4

-y

>

where A= (,f$ ‘- @ 6)/(- 10~2)

Series near

+ 16016a6g3@ A”+ 19448 a”) VI”]+ . . . near

i2b+132b4

@3

18.5.62 +2002a3

(285Q2c2

~9

18.5.61 +a (3Wa3+i73

;+4b3]A7

c2+21 b2)As+$

+100+279@ + { 154a3+33d}

..

18.5.60

(;g.y+15g3@;~

dJ+$f

Series

@ ’

lOc2u-56c3u3+

18.5.54

+F)

Series

wJ2

z,

18.5.58 (P w=[-

33

5c~(55c;-2316c2,)u12 143

+

639

FUNCTIONS

>

(

22c~~+92c2c3ei+105~

Us+ ‘z (

2c3)

-~)u’+24(~~2~3e:+~~e, +42c$e,+E

dc3) a?'+70 (g

de:

w~e~+~4e,+84&

u”+(gde:+J$

+gc$e:+T

c$eset+E

c24+&

4)

al3

c$ej t . .. ,

>

d+

. . .,

18.5.64

where a= (z-w<).

WEIERSTRASS

640 Other

Series

Involving

ELLIPTIC

AND

RELATED

18.6.

r

and

Derivatives

Ordinary

Series near a [ @ (20) = 01

FTJNCTIONS

Differential

(c2=gd20,

ca=gJ28)

18.6.1

s’W=-

9 (4

18.6.2

a’ W&>

=I(d

Equations

18.6.3 +(5c;-1%;) 26 7c3u” +2-F--

I[

u,4+61&3u’6 66 5&L’

+ 5s~~ 3

llczc& --3-+

~““(2)=4~3(z)-g*~(z)-g~=4(~3-~c*~-7c3)

(loc;-84c%) 33

1 ...

1363&a ,,+&316+-55~~) + 429 429 18.5.66

where u= (Z-G),

18.5.67

~o=eo)

ul*

utg +

18.6.4

QY’(z)=6~2(z)-~g~=6@2-10cz

18.6.5

@“‘(Z)

= 12

@Q’

18.6.6 @(4)(2)=12(

QQ”,

@‘@‘>

=5!

@3--3c@

18.6.7 gP(z)=12(@~“‘+2@‘~“+

Series near wi

-y

[ @”

@“>

=3.5!@‘[p-cz] 18.6.8 @‘“‘(~)=12(~@(~)+3~‘@“‘+3~“~” +@,“‘@‘I)

_ (7cae:+21c3e,+5c~)d 7 -(6c~e~+10c~e~+llc~&ze 3

(

18.6.9

==7![ ~4-4c*p--4c3@+5c~/7]

18.6.10

@‘7~(z)=4~7!~‘[~a-2c~~-c~]

18.6.11

22c&$+92cflae1+105c&j

ci

>

~(‘“‘(2)=9![~6-5C2~a-5ca~~

o?I

+(lOc:~+~lc*ca>/3l

18.6.12

11

~(‘“‘(2)=5.9!~‘[~‘-33~z~~-2ca~

+2c1,/3]

18.6.13 ~~“0’(z)=ll![~“-6c~~4-6c~@8+7~~a

+x681 c%3el+~9150 wi+~

5 4 > &-

--.,

18.5.69 where Reversed

a= (zSeries

for

WJ Small

18.6.14

+ (342cgQ

lOc,8)/33]

~‘“‘(z)=6~11!~‘[~*-4c~~8-3cg~2 + (774 L? +57cd331

18.6.15 1~1

+84+

p’yZ)=13![&07-7c~p-7ca@4+35C$p/3 +210c~a~2/11+(84c+35c~))/13-1363&~/4291

18.5.70

18.6.16 ~‘~~~(z)=7.13!~‘[~~-5q~4-4c*~~ +-

18.5.71 18.5.72 For reversion [18.18].

1%2%

u 11 +3fW+W+

55

6006

ula+

. . . ,

where -ya= gal48 Ya=gdl20 of Maclaurin

series, see 3.6.25 and

18.6.17

+5~~2+60c~c~~/11+(12+55Q)/13]

~(14)(z)=15![~8-8c&P-8c~$iP+52~~4/3 +32&m @ “Ill + (444&---3284) @ *I39 -488&s@ /33+s(55+2316eo,)/429] 18.6.18 ~‘1s~((z)=8~15!~‘[~7-6c&‘6-5ca~4+26e’,~’/3 f123~q @a/11+(111t+82c$@

/39-61&/33]

WEIERSTRASS Partial

Derivatives

with

Respect

ELILIPTIC

AND

RELATED

y”=

to Invariants

641

FUNCTIONS

[a @ (2) +a]y

(LamtYs equation&see

118.81,

2.26

18.6.19

For other (more specialized) equations (of orders l-3) involving p(z), see [18.8], nos. 1.49, 2.28, 2.72-3, 2.439-440, 3.9-12. For the use of @ (z) in solving differential equations of the form y’“+A(z,y)=O, where A(a,y) is a polynomial in y of degree 2m, with coefficients which are analytic functions of z, see [18.7], p. 312ff.

A~=~‘(3g2l-~g~z)+6g2~‘-9$~~~-$~ 18.6.20

A~=~‘(-;g&+$)-9g.lP’+$+;gzg~ 18.6.21

18.7. Integrals hfk$nite

d”--33.($2P Adg,-

+&)

18.7.1 +;

2 (%I@

+;,,>-;

s

@P(z)dz=;

@ W+~

$22

$2P

18.6.22

x7.2

A~=;,(sss@+$)

(formulas for higher powers may be derived by integration of formulas for @ (a) (2) ) For f @“(z)dz, n any positive integer, see [18.15] vol. 4, pp. 108-9.

-ij

A E=;

18.6.23

$22

g2uf’+;

($

$2 P

+i

$3)+:

!?a P

garr+; g;z2,-;

’

gtza’

s

@ 3(2)d2=&o

gas(z) +b

gaz

If @ ‘(a)#0

18.6.24

18.7.3

1 A bg,= -4 gau” --4 gg-- 1”8asaz2u+; 9

bu

$W @ ‘@) j-9

(

here ’ denotes =&

Differential

y’“=yyy-a)’

u(z-a)-ln

DeJnite

@’ ( g; 0, -g

A>0

>

18.6.26

A<0

18.7.4

Y’a=w-3aY2+3Y)2

Y”,-3

@ T(2; 0, ga)’

4:-3a2

$a =-- 27

18.7.5

w’=i

18.6.27

(y+a)“(y+b)”

u(z+a)

For Jdz/[ @ (z)- Ip (a)]“, ( @ ‘(a) #O) n any positive integer, see [18.15], vol. 4, pp. 109-110.

solution y=;+g

(a) =22f(a)+ln

Equations

Equation

(2)f%

>

18.6.25

Y ‘,=y

@ “‘(z)-&

Y=~@~(z;

$2, W-b, s2=-,

2

(a-b)

18.7.6 18.7.7

s #3

-Jqj

- dt

where t is real and s(t)=4P-g2t-$2

WEIERSTRASS

64‘2

ELLIPTIC

AND

RELATED

18.8 Conformal

FUNCTIONS

Mapping

w=@ (a) maps the Fundamental Rectangle onto the half-plane vI0; if Iw’l=&7~=0), the isoscelestriangle Owwzis mapped onto ~20, vIO.

w=@(z) maps th e Fundamental Rectangle onto the half-plane ~50; if jw:)=wz(g3=O), the isosceles triangle OQW’ is mapped onto ~10, VlO.

w=@‘(z) maps the Fundamental Rectangle onto the w-plane less quadrant III; if lw’l=u, the triangle Owwzis mapped onto v 20, a 2~.

w=@ ‘(2) maps the Fundamental Rectangle onto most of the w-plane less quadrant III; if jwil =w2, the triangle Owzw’is mapped onto v 20, v2u.

(a = period

ratio)

w={(z) maps the Fundamental Rectangle onto the half-plane ~20. If a 11.9 (approx.), ~50; otherwise the image extends into quadrant I. For very large a, the image has a large area in quadrant I.

w={(z) maps the Fundamental Rectangle onto the half-plane u>O. The image is mostly in quadrant IV for small a, entirely so for (approx.) For very large a, the image has a 1.3 la I3.8. large area in quadrant I.

w=a(z) maps the Fundamental Rectangle onto quadrant I if a < 1.9 (approx.), onto quadrants I and II if t.9
w=a(z) maps the Fundamental Rectangle onto quadrant I if a<3.8 (approx.), onto quadrants I and II if 3.8
a, w&d41

a, arg[r

=z;

consequently the image winds

(az+$)]-$

consequently the image

around the origin for large a.

winds around the origin for large a.

Other maps are described in [18.23] arts. 13.7 (square on circle), 13.11 (ring on plane with 2 slits in line) and in [18.24], p. 35 (double half equilateral triangle on half-plane).

Other maps are described in [18.23] arts, 13.8 (equilateral triangle on half-plane) and 13.9 (isoscelestriangle on half-plane).

Obtaining

Fundamental

@ ’ from

@ ‘2

Rectangle

Fundamental

A rel="nofollow">O FUNDAMENTAL

FUNDAMENTAL

RECTANGLE

I

I

0

.4

I

w=1

RECTANGLE

X

-X FIGURE

In region A LZ?(@‘) 20 if ~2.4 andxS.5;

Rectangle

A<0

18.4

In region A 9(@‘)

20 elsewhere

(1) If ~~11.05, use criterion for region A for A>O. (2) If 1 $z<1.05: a(@‘) 20 if ~2.4 and x1.4, -1~/4<arg (@‘)<3*/4 if .4< ~1.5 and .4
WEXERSTRASS ELLLPTIC AND RELATED FUNCTIONS In region B

In region B

Use the criterion

The sign (indeed, perhaps one or more significant digits) of @ ’ is obtainable from the first term, -21.9, of the Laurent series for @‘. (Precisely similar A>0 Map:

643

criteria apply when the real half-periodf

for region B for A>O.

1)

w=l

@(z)=u+iv

Near zero : @ (2) =$+c,

w'=1*4i

-4

* -3

-2

-I

0

I

2

3

-_ _ --_ -----/I[[ _._

4

1.5

”

- - -

--

-

1.0 /.3

’ ,‘.3i

Ltd-~-i .5- M

--

- -

-

-

-

r &_

_ - -

- i - - L c-- -t --

0’=2.Oi FIGURE 18.5

l--y’ 0

I

1.0

I

644

WEIERSTRASS

ELLIPTIC

AND

RELATED

A<0 Map:

FUNCTIONS

wa=l

@ (z)=u+iv

wj=1.5i

-2 -

-5-

lE,l5 .07 lC215 .03

4=2.oi

Iv

-6-

FIGURE

18.6

WEIERSTRASS

ELLIPTIC

AND

00

RELATED

w=l

Map: c(z)=u+iv

FIGURE

18.7

FUNCTIONS

645

646

WEIERSTRASS

ELLIPTIC

AND

RELATED

F’TJNCTIONB

Map: Near zero:

Y strq--y -1 -_____ ###L 0

.5

1.0

x.

If, IS.04 IQ .0002

w&i

"

.5

0

I.0 ’

I.5 .9 H

-.5

,, (1’ 1 ;

4si!F

_-

,.’ &

,!5i

-.

-2.0

j 1.41

\

,: /’

:’

.,

__---3.5

1 , I I

.2

I’

b *-

/’

-

0;=2.oi -3.5

_-

-

1 _

kik 0

\

,/

0 -3.0

#’

_

.5

_ I

\

’

.75

t-----G

FIGURE

1 _

\

: ,’

;=1.5i

I.31

’

------

.li ‘.

\

_--*

-2.5

‘-, ’

.3

,I

Y

xi2i

,

”

._

I’

’

I “&

‘I -4.5

,I.

3.5

i--or---

‘\ \

“!7i

.5

-.

\

I

.75i,

-1.0

6

-. \\

I I (I

3.0

-_2&L2.5

.7_

18.8

_ _ _-

__

_-_ _--

-. __ .5

_

-i.o

x

WEIERSTRASS

ELLWTIC

AND

RELATED

w=l

00

Map:

“(2) =u+iu

kY 1.0 lI--~l ---_--__--. ---_-- -_ _ --_ .5---------_--_ --------. - ---_--

I.0

.5

w’=2

647

FUNCTIONS

.

.5

0

,\J,’ .

I.0

jy

,1.2i

1.4 - ---1.0 ---

-i-d---/.81

--

--

-

0.5-

-#I

-

--

i

I

_ -

-

_

--

-

--

--

-

---

/ill!!l~ 0 -

i

-

-__

--

. - a !4i -.2i

--_ - --

--

--

2.Oi ’ 1.8i r, I\I ’

x

‘6i

2.0

Ai

.5

-

--

co

IY ------

--~

-_--.

-----,-se ----

-.. --

-._--.-..-

----

-_.-

--.-

--

--

1.5 ---- --_--..

-._

.---

.--------

-..-

_

-- -

_ -.-

-----.--.v_

w’=2.Oi 1.0 FIGURE

u 18.9

1.0 -:‘.” -------.---_-- ----.-5 ---.------_. --.-mm.---. -.----0

_-._ -

--

-

----- - -.---..--.----_.--_ --.__ ----- - .----.- -- -- - -- .5 1.0

x

648

WEIERSTRASS

ELLIPTIC

AND

A<0

RELATED

FUNCTIONS

t&=1

Map: u(z) =u+iv

#’’

\

It-1 1.liiiiit Y

.5i

*5

.ai

-

--_ ----_ _ -- _ --0-- ---

-

.5

1.0

1.2

0

0

.2

.4

.6

.8

.5

1.0

.5

Y - ----- -

BB~ -

--

0

U .4

.6

.u

FIGURE 18.10

1.0

----

-

---- - - --

_--

*2

1.0

U

I.0

V

0

x

7.0

-

.5-

-

I.0

x

WEIERSTRASS

18.9. Relations

with

Complete

and with

(Here K(m)

and K’(m)

Jacobi’s

=K(l-mm)

ELLIPTIC

AND

Ellipltic

Integrals

Elliptic

are complete

RELATED

K and

Functions

elliptic

e1=(2

18.9.2

e2=

integrals

18

. . 9

5

18.9.6

(2m-l)IC(m)

e2=

3wa

see chapter

2(1--2m)Kym) 34 m-1)-6iJm-m2 3w2 a

30’

- JWd

34

g,,8(2m-1)(32mz-32m-l)IP(m)

g3=4(m-2)(2m-l)(m+l)~(m) 27~’

274

A~16ma(m-1)a~2(m)

A =-256(m-m2)K12(m)

WI2

Wp

iK’(m)w K(m)

w’=

iK’( m)w2 K(m)

18.9.8

w=K(m)/(eI-eJ1/2

e=K(m)/a12

18.9.9

m=

1 3e2 m=Z-4H,

(e2-e3Y(el-e3)

18.9.10 18.9.11

of the 1st kind;

gl=4(16mz-16m+l)K4(m)

gl,4w-m+l~K4(m)

w’=

16)

-mW(m) 3w2

e3=(2

18.9.7

m

A<0

18.9.3

18.9.4

K' and Their Parameter

(see chapter

A>0 18.9.1

649

FUNCTIONS

D<m I*, f&e g3201 ~(z>=e3+(el-e3>/sn”(z*lm)

:+z[z:/ti -

@Yz)=e2+&

18.9.12 @‘(z)=-2(el-e3)3/2

~cn(z*~m)dn(z*~m)/sn3(z*~n)

where

-4IWsn( z’ Im)dn( z’ Im) [l-cn(z’(m)]2

P(z)= where

z*=(el-e3)b 18.9.13 18.9.14 [E(m)

~=t(w)=-

Kiz’

.z~=22Hp

W(m)+(m--2Mm)l

r2=rcw*,=K*

[6E(m)+(4m--5)K(m)l

q’=l(w’)=SW’;W

is a complete

elliptic

integral

of the 2d kind (see chapter l?).]

17.)

650

WEIERSTRASS

ELLIPTIC

AND

RELATED

FIJWCTIONS

18.10. Relations with Theta Functions (chirpter 16) The formal definitions of the four 9 functions are given by the series 16.27.1-16.27.4 which converge for all complex z and all p defined below. (Some authors use ?FZ,instead of z, as the independent variable.) These functions depend on z and on a parameter p, which is usually suppressed. Note that 8:(O) =S2(0)8,(O)84(O),

where 6,(O) =Gr(O, q).

A>0 18.10.1

r=w‘/w p,eir+,e-rK‘lK

18.10.2 18.10.3

q is real and since g320(]o’]20),

18.10.4

O
(v=7rz/2w)

P & q-;e-$2@av andsinceg~2O(Iw~I2wl), P(v=rz/2wJ

j=l,2,3

18.10.7 18.10.8

c(z)=?

NV> exp -w2 ( 20 > qq

18.10.9 12w2eI=+[t9:(0) +0:(O)]

12w:e1=?r2[6:(0)-89:(O)

18.10.10 12w2e2=lr”[8~(0)-8:(O)] 18.10.11 12w2e3=-+[9:(0) +6$(O)]

12wie2=79[9:(0) +8:(O)] 12w:ea=--Ir”[tP:(0)

18.10.12 (e2-e#=-;(e,-e2)*=&@(0)

(e2-es)f=i(e,-e2)+=22:(0)

18.10.13 (e~-et)+=-i(es-eI)+==~~~(o)

h-

18.10.14 (eI-e2)+=-i(e2-e,)+=f$$))

(e2-e#=-i(el-e2)*=2:(0)

18.10.16 g3=4eIe2e3 18.10.17 Ai=-&2(0)

(-A)i=&8;2(0)e-(r/4 2

7&?:"(o) 18.10.18 m4=-12wg;(0) 18.10.19 +f(~')="~';~

+r9pb(O)]

e#=i(e,-e,)i=a:(0)

18.10.15 g2=;(@':(0)+@(0)+~:(0)]

&;“(o) Ir2=r(~2~=-12y&(o)

]

WEIERSTRASS

ELLIPTIC

AND

RELATED

651

FUNCYl’IONSr

Series

18.10.20

81(O)

18.10.21

ll)a(o)=zq~[l+q~‘2+q2’3+d~4+

=o

. . . +q*(“+‘)+

tPg(0)=l+:l[q+$+qQ+

18.10.22

~4(o)=1+2[--p+$--q9+

18.10.23

. . . +qG+

. . .]

. . . +(-l)“p~2+

Attainable

. . .]

Accuracy A-0

A>0

Note: a,(O)>O,

Note: &(O) =Aefr/8,

j=2,3,4

9z&(O)>O; 85

(0) :

j=2,3,4

. . .]

A>6; 84(0)=s3o

2 terms give at least 5s

2 terms give at least 3s

3 terms give at least 11s

3 terms give at least 5s

4 terms give at least 21s

4 terms give at least 10s

18.11

Expressing

If f(z) is any elliptic

any Elliptic

function

and P(Z)

Function

in Terms

has same periods, write

f(z>=3[f(z)+f(-:?)l+3[(f(z)-f(-z)l

18.11.1

of @ and @ ’

{JY’(4~-‘l@‘W.

Since both brackets represent even elliptic functions, we ask how to express an even elliptic function g(z) (of order 2k) in terms of @ (2). Because of the evenness, an irreducible set of zeros can be denoted by a, (i=l, 2, . . . , k) and the set of points congruent to -a2 (i=l, 2, . . . , k); correspondingly in connection with the poles we consider the points fbi, i= 1,2, . . . , k. Then 18.11.2

3 where A is

a constant. If any a, or b, is congruent to the origin, the corresponding factor is omitted from the product. Factors corresponding to multiple poles (zeros) are repeated according to the multiphcity. 18.12.

Case A=O(c>O)

18.12.9

Subcase I

.18.12.10

18.12.1

g2>0, g9
18.12.2

Hl=Hs=o,

e3= -2~) 18.12.11

H,=3c 18.12.12

18.12.3

@(z; 12dL,-gc3)=c+3c{sinh

[(~c)+z]}-~

18.12.4

18.12.13 18.12.14

l(z; 12c2,-8~+)=-~+(3c)+

coth [(3c)iz]

18.12.5

18.12.15 18.12.16

a(~; 12c2,--88) =(3c)-*

sinh [(3c)+z]e-“‘l” 18.12.17

18.12.6

co==, w’=(12c)-+lri

18.12.7

~=~(w)=--o>

18.12.8

q’=f(w’)

= -co’

18.12.18 18.12.19

!I=19 u(o)=0

m=l

652

WEIERSTRASS

ELLIPTIC

a(w’/Z)=

RELATED

FUNCTIONS

18.12.46

&,*2/96~

18.12.20

AND

~‘bm=O

T 18.12.47

18.12.21

g (42)=c

18.12.22

@‘(~*/2)=0

18.12.23

l(w*/Z)=-

18.12.24

18.12.48

Subcase II 18.12.25

18.12.26

&=3c,

(el=2c,

@(z;

18.12.49

g2=0, g3=0(el=e2=e3=O)

18.12.50

@(z; 0, O)=z-’

18.12.51

t(z; 0, 0) =2-l

18.12.52

u(z; 0, O)=r

18.12.53

UC -&‘=a0

e2=e3= -c)

H*=H,=O

12c*, 8c3)=-c+3c{sin

128, 8c3)=cz+(3c)*

cot [(3c)*z]

18.12.29 u(z; 128,

18.12.30

8~s) = (3c)-1

sin [(3c)*z]eCP’*

w= (12c)-+lr, Tj=[(u)

=cw

18.12.32

q’=(+,J)

&xl

18.12.33

If g2=0 and g3>0, homogeneity relations allow us to reduce our considerations of @ to@ (z; 0, 1) (@ ‘, r and u are handled similarly). Thus @(z;0,g3)=g:/3QJ(zg~‘o;0, 1). The case g2=0, g3=l is called the EQUIANHARMONIC case. 1 FPP; Reduction to Funo!wnentd Triangle

w’=&

18.12.31

Case (g,=O, g,=l)

[(3c)*z]}-*

18.12.28 {(z;

m

u(w,/2) =o

18.13. Equianharmonic 18.12.27

cw 2

Subcase III

ca co’ -2

a(w,/Z)=O

g2>0, g3>O:

+,/z)=-+i

A1=AOo2z,, is the Fundamental

Let Bdenote etTJ3throughout wzl.5299

54037

05719

28749

Triangle 18.13. 13194

17231’

m=O

!2=0,

‘Joer2/24

18.12.34

u(co)=y

18.12.35

u(d)

=o

18.12.36

u(q)

=o

18.12.37

g (w/Z) = 5c

18.12.38 18.12.39 18.12.40 18.12.41 18.12.42

u(w/Z)=

e**/%Jz 7r

@(d/2)=-c @‘(d/2)=0

18.12.43

{(d/2) = +im

18.12.44

u&‘/2) =o

18.12.45

@ 6.d~) = --c

FIGURE 18.11 6 This value was computed precision on a desk calculator

30s.

and checked by multiple and is believed correct to

WEIERSTRASS

Reductionsfor ,Q in A$ zl=t& 18.13.1 @ (4 = @P (4 18.13.2

ELLIPTIC

AND

RELATED

Special

is in AI.

and

Formulas

A= -27,

H,=&4-1’3)Z, H2=&(4-“3),

H3=,h(4-1/3),

~-lf(~l)

18.13.10

u(zJ =t&>

18.13.11

Q2(0) = Aeir18

18.13.12

0,(O) = Aeir/z4

18.13.13

G4(0) = Ae-i*124

z(zz>

18.13.4

Values

653

18.13.9

g ‘(22)= - ml,

18.13.3

FUNCTIONS

=

is in Al

Reduction.for z3 in A3: zl=e-'(2w'-~) 18.13.5 @(z3>=@Lw1)

m=sin2 15°=2*j

q=ie

-*ti/j/z

18.13.14 18.13.6

@‘(z3)=

18.13.7 {(Za) = -t-‘{(ZJ 18.13.8

&3)

P’W

exp

=d%)

where A= (~~/l~)'/2~'~3~'~~1.0086

+2$,

‘I’= CC@‘)

cw)l

[k3-4

18.13.15 Values at Half-periods

.-

= I __

.- --

i-

18.13.16 W=Wl

0

18.13.17 w

0

18.13.18 W"6.Q

0

18.13.19 Y'

0

-

-

=

=

_-

_-

values ’ along (0, wJ = --

.P’

--

18.13.N - p[ GG-GF i/cos--

20*/9

+ 4/cos 407 2/‘m 4o”

18.13.21 l/(21”-

WI3

1)

- p(21”+

1)/(2’!‘-

1)

18.13.22 - P[ 2/cos

402/9

D-

18.13.23 WI2

a+&

18.13.24 1

ad3 18.13.25 8wl9 7

Value& at 2w2/9, 4w1/9 and 8ws/9 from 118.141.

+ Qcos so01 %:os 80”

u2=K(W1/3-r3W3) 3114

4n

67

WEIERSTRASS

654

18.13.26

2012

-21/ae2

3i

l a(ez - HZ)

i(3”‘)

0

i

ELLIPTIC

AND

RELATED

Values

along

(0, zO)

F’UNCTIONS

18.13.27

32014

18.13.28 20

Duplication

18.13.29 18.13.30

&qi

where tan 4=@‘(z), choose C$in intervals

Formulas

@(2@$$7~

0<2<20~

and we must

< (-iE)969$)9($7$)toget

2~6(2)-lo~3(2)-1 @‘(22)= t @‘(z>P

@ (g), @ (g+?), 18.13.31

@ (:+2),

respectively.

rczz,=zrc,,+~ Complex

a(22)=-

18.13.32

Trisection

~‘(2)u4(2) Formulas

(x real)

Multiplication

18.13.35

gB(EZ)=e--2@(2)

18.13.36

JD’(ez)=-p’(z>

18.13.37

l(a) = c-‘&g

18.13.38

u(ez) = CU(2)

In the above, E denotes (as it does throughout section 18.13), ei*fi. The above equations are useful as follows, e.g. : If z is real, EZ is on 00’ (Figure 18.11); if tz were purely imaginary, z would be on Oz, (Figure 18.11).

Conformal -60’

Equianharmonic

.2

.4

Maps .6

.6

Case

Map: f(z)=u+iw

4

i

I\

+-----‘\,

‘..

1.0

1.2

I.4

WEIERSTRASS

ELLIPTIC

.2

0

AND

.4

RELATED

.6

.6

655

FUNCTION&

1.0

1.2

s-(z) Near zero: {(z)=,+q

1

-.4 x -.6 1.5, I 5 .007

-3

(.$*I 5

I x10-5

t” v 1.0

1.6

.8

I

FIGURE

18.12

656

WEIERSTRASS

ELLIPTIC

Coefficients

for Laurent (cm=0

EXACT

k

RELATED

FUNCTIONS

Series for @, @ ’ and { for mf3k)

cab

:\~~3.2s) = l/10192 l/(13.19.28”) = l/5422144 3/(5.13’.19.28’)=234375/(7709611X10*) 4/(5.13*.19.31.28’)=78125/(16729 (7.43)/(13”~19*.31.37.283 (6.431)/(5.13”.19’.31.37.43.28’) (3.7.313)/(5’.13’.19’.31.37.43.28*) (4.1201)/(5*.134.1~.31.37.43.280) (2’.3.41.1823)/(5.13’.19*.31’.37.43.61.28i”) (3.79.733)/(5.13’.196.31’.37.43.61.67.28”) 3.1153.13963.29059 53.136.194.31s.37s.43.61.67.73281’ 2’.3*.7.11.2647111 5’.13’.19’.31s.37s.61.67.73.79.28’6

APPROXIMATE

85587X

First 5 approximate values determined from cak/c3k+ using at least double precision arithmetic use of the recursion relation; cx -CH are believed c3k<

AND

108)

cxs

3. 9.8116 5714 1.8442 3.0400 4.6697 6.8662 9.7990 1.3685 1.8800 2.5497 3.4222

28571 16954 88901 36650 95161 18676 31742 06574 72610 66946 48599

42857 47409 21693 35758 83961 79393 57961 79360 01329 68202 51463

733iZx %88X lo-6 55885 78983X10-’ 61350 20301X lo-10 00384 33643X10-1’ 36788 98X10-‘@ 41839 66X10-” 13026 87X10-n 79236 40X10-x’ 63683X10+ 05316X lo-80

4.5541

38864

99184

3O391X1O-J”

6.0171

15776

98241

99591

X 10-s’

exact values of cxk; subsequent values determined by using exact ratios with a desk calculator. All approximate c’s were checked with the correct to at least 215; C~O-C~O are believed correct to 205.

Ca 13k-1

.

2@-1t k=2, 3, 4, . . .

Other Series Involving Reversed

Series

for

Large

18.13.46

@ I@

1

18.13.39

18.13.47 where w =( @ -&)/3ea Other Series Involving 18.13.40 where u=&B-“/S and z is in the Fundainental Triangle (Figure 18.11) if @ has an appro-

priate value. Series

near

~0

Reversed

Series

for

Large

@’ 1@ ’ (

18.13.48 z=21/a( @‘1/8)-l&/3

1-g

(@‘)-2

18.13.41

18.13.42

z being in the Fundamental Triangle if @ ’ has an appropriate value.

18.13.43 where u=(z-z,J

18.13.49

Series

Series

near

w2

(Lo’--i)==x[2-ir+A

X3

18.13.50 where z= (z-z,J3

18.13.44 (@ --e2)=&?& [1+X+2’+;

+; x4+;x6+$ x~+o(xq, 18.13.6 where u=( z-w)~,

z=e2u

18.13.51

x=2a!

l-k-;

near

(Figure 18.11)

SO

x2+$

a++o(xq]

13s cu2+- 7 +o(u4)

18.13.52 where U= (@ ‘--i)/(-4)

1 ’

WEIERSTRASS Series

near

ELLIPTIC

AND

RELATED

FUNCTIONS

w2

+

18.13.53 @‘=6ei(z--w2)

1+2u+3d+~

+F

d

v4+7

27.36.52.31 28*3vi.9103 25! 226+ 31!

v6+2$

- 214.310.5*23.257.18049 243 43!

v~+o(v7)],

215*312*5*59*107895773 24g+‘o( P) 49!

18.13.55 @‘/6e;)

[1-2~+9~+0

18.13.66

d 212058 - 13 we+O(W’)

+330w4-2268w’+

18.13.56 where w= @’ ‘2/9 Other

Series Involving

Reversed

Series

for

[

Large

I

41d3 13*337$9 27.32.52.11.13+210.34.5a.ll.l7.ig

t

31.101u” +2’5.35.5.112.17.23 +#0($7 Economized

IfI

18.13.67

near

[-~+~-~]+~~-~]+0(2117),

18.13.60 where u= (z-z,,) near

~,J?“+c(z)

a,=(-1)9.99999

96

a4= - (-9)2.20892

aI= (-2)3.57143

20

at,=(-10)1.74915

a2= (-5)9.80689

93

aB= - (- 12)4.46863

Ua= (-7)2.00835

02

47 $5

50

18.13.59

Series

(0 <x < 1.53)

le(z)~<2XlO-’

y= p-y20 Series

Polynomials

cc2@(z)=$

1-?+17y2 , 143 -4g+o(rq],

18.13.58

(l-{o)=i

1

a7 2=a+2a.3.5.7

t

18.13.57 z=p

zal

- 212.3Q-5-229.2683 237 37!

18.13.54 where v=h(z--w2)’

(z-w2)=(

657

18.13.68

w2

2”@ ‘(z)=$

18.13.61

93

a,cP+e(s)

le(z)j<4X10-7

c-02) 1+0+; [

(l-72)=-2(2-

+g

18.13.62

?Y+g

ua,;

va+; v4 a(=

ao= - 2 .OOOOO00 ti+g

v7+o(v))],

v=e2(z-cdJ2

-(-9)2.12719

aI=(-1)1.42857

22

us=(-

a2= (-4)9.81018

03

a(=--(-11)1.70510

aa=(-6)3.00511

93

66

10)6.53654 $7 78

18.13.63 12w2 l-w+5-35+5-

(r-92)

(z--02)=-

-2

267d

139~’

18.13.69

a$(~)=$

a,$!*+&>

~c(z)~<3xlo-~

18.13.64

w= Series

*=(-1)9.99999

(t-722)2/e2

Involving

v

18.13.65 u=z-y-

2.3

&,7

26~3~.23 1g 13! z13+2 191

98

a4= (- 10)6.12486 14

aI=-(-3)7.14285

86

a,=(-11)4.66919

$5

al=-(-6)8.91165

65

a,=(-12)1.25014

65

tZa=-((-8)1.44381

84

2a*3a

658

WEIERSTRASS

18.14. Lemniscatic

ELLIPTIC

AND

RELATED

FUNCTIONS

tFPP; Reduction to Fundamental Triangle A1= AOwwzis the Fundamental Triangle

Case

(g2=1, ga=O)

w ~1.8540 74677 30137 1928 Reducticm for zz in 4: zl=a& is in A1

If ga>O and g3=0, homogeneity relations allow us to reduce our consideration of @ to @ (z; 1, 0) (@ ‘, j- and u are handled similarly). Thus QJ (z; g2, O)=g2’@(zg2~; 1, 0). The case ga=l, g3=0 is called the LEMNISCATIC case.

18.14.1

m%)=-ml)

18.14.2

g ‘(22) =@@d

18.14.3

ea) = -m &)=iQ(zJ

18.14.4

Special Values and Formulae

18.14.5 A=l,

w’ (=

Hl=Ha=2-4,

H,=i/2, m=f&P 45O=3, q=e--

18.14.6

-

&(O)=&(O)=(w&+;

18.14.7 w=K(sin2

&(O)=(~W/?T)

45O)=p=k \*

Jz

where

3~2.62205 75542 92119 81046 48395 89891 11941 36827 54951 43162 is the Lemniscate constant

FIGURE 18.13

correct

to 18s.

Values at Half-periods

18.14.8 w-0,

a=3

0

y=*/4w

e*b(21/4)

18.14.9 w-a

a-0

0

7+4

erl4(~&4*/4

18.14.10 co’=01

s--t

0

‘)’ = - *i/40

&/8(21/4)

Valuesalong(O,w)

18.14.11 WI4

&$&+

2’14)

(1 + 21’4)

18.14.12

42

aI2

--Q

,m

-G55* &

,r1”‘(21/16)

l+Q 8w

af

2&i

18.14.13 2013 18.14.14 3014

$(&-2$(1+2f) a=1+fi

edla(31/.9) (2+

&)l’l’

WEIERSTRASS

ELLIPTIC

Values

AND

RELATED

along

FUNCTIONS

(0, zO)

P 18.14.15

659

r

-

%I4 18.14.16

-&++Jz,)

42 18.14.17

-i/2

22013

2

dsec

30”-

@

1

(2.zo/3) ‘la 3 I

18.14.18

-- ; (a-&q

32014

Duplication

18.14.19

18.14.24

Formulas

@ (22)

u

=[@“(d+tl”/{

(~)=~‘(2)~[2~(2)+fli~(r)-?

Q%9[4P(d--lU

-[2@(~)--31JP(~>f3

18.14.20

-2@'"/"(z,)

~‘(2~)=(P+l)(P~-68+1)/[32~‘~(~)1,

0=4@“(z)

[Use - on O<xlw,

18.14.21

Complex

18.14.22

18.14.25

a(22)=-@'(z)d(z) Bisection

Formulas

18.14.26

(O
18.14.23

[Use + on O<x
Case

Map: f(z)=u+iu P(z) Near zero:

@(z)=-$+c,

md=$+g+~2,

lzl
Near z,,: ~(~)=-(‘h’~)~+~~, I---zol<&

~(~)~-(~-~o)2+(z-zo)6+6, 4

80

+ on ~52<2f.0] Multiplication

@(iz)=-P,(Z) ~‘(iz)=i@‘(z)

18.14.27

I(k)=-i{(z)

18.14.28

a(iz)=iu(z)

The above equations could be used as follows, e.g.: If z were real, iz would be purely imaginary.

- on o<x<2w] Conformal

Lemniscatie

(See [18.13].)

Maps

660

WEIERSTRASS

ELLIPTIC

AND

RELATED

FUNCTIONS

Near z,,: g’(z)=

Near zero: @ ‘( z)=$+Q

P’(z)=

l(z) 1

Near zero: c(z)=--+Q

-.2

-.4 (z-zo)3 Near

20:

T(z)=T~+T+Q,

-.6

12-r

&.)=ro

; (.-zo)3

12

I

(z--20)’

560

‘I er

-.6 t

”

FIGURE

18.14

--(z-z32

+Q

-(2-2cJ+3(2-20)6+c,

2

40

WEIERSTRASS

ELLIPTIC

Coefficients

for

AND

RELATED

L.aurent

Series

Cc,.=0

for m odd)

for

1661

FUNCTIONS

!?,@‘,andI

Z-Z

EXACT

k

APPROXIMATE

c2k

Czk

.-

.05 .8333

l/20 l/(3.202) = l/1200 2/(3.13.203) = l/156000 5/(3.13.17.20’) = l/21216000 2/(3z.13.17.205) = 1/(31824X 105) 10/(33.13*.17.206)=1/(4964544X 105) 4/@132.17.29.2d’) =‘ij(7998432X 167) 2453/(3’.11.132.172.29.2o8)=9582O3125/(1262OO2599X 2.5.7.61/(33.13a.172.29.37.2O~)=833984375/(18:194643943X

z

c&+9

k=l,

10’6) 10’7)

. . . x10-3 , .~.

.641025 .47134 .31422 .20142 .12502 .75927 .45338

23831 641025 82554 83688 45048 19109 43533

10-T ;8::X 99296X10-Q 32882% lo-11 37651X10+3 59917X 10-l” 06092X 10-I”

07bi8’ 04725 49183 02941 76468 93461

2, . . .

-

Other Series Involving Reversed

,/

Series

for

Large

,f$

Other Series Involving Reversed

1@

for

Large

I@

’1

18.14.38

18.14.29

3wK 231~’ +7+x+

18.14.39 A=2113,

429~~ -195~~ 464 + 128

1215520~ 4618920’~ +o(w”) + 10496

bp/

Series

.@ ’

1 )

18.14.30 w= g-“/S, and z is in the Fundamental Triangle (Figure 18.13) if @ has an appropriate Series

18.14.31 2 @=

near

Series

w=(z--zo)4/m

18.14.42

-5+;-~+&+0(~+‘)~

(z-zo)=2@’ 18.14.32

z=(z-20)2/2

18.14.43

d 7wh w+,+~++$+

18.14.33 x=-

Series

near

a0

(2-20) [ -1+3w-~+~+o(w~)],

18.14.41

~0

near

18.14.4 Q&f

value.

v=Au4/6, and z is in the (Figure 18.13) if @’ has

Fundamental Triangle an appropriate value.

O(w”)

1

[l+~+y+8g+O(u4)],

u=4@‘4 Series

near

w

18.14.44

w

18.14.34

18.14.45

5=(2--w)

18.14.46 18.14.35

v=(z-co)%

18.14.36

f)=y

+

1-y+6yz-@ [

5

5

1 172Y4

18.14.37

!I=(@-ed

819@‘l’ 8 +O(@‘13)

Other Series Involving

75

52@ 1064g -ij+ 195 +wi7>

SO@” ___3

Reversed

1 ,

Series

18.14.47 z=c-’ 18.14.48

v=t-4112

for

Large

{ [rj

662

WEIERSTRASS Series

near

ELLIPTIC

AND

RELATED

FUNCTIONS

Economized

a9

Polynomials

(0 5% 5 1.86)

18.14.49 18.14.5’7

(S-i-o)=a(z-zJ3[;-;+g-&+o(vq, 18.14.50

2@(x)=$z.x4”+c(x) (e(x)C)(<2X10-7

v= (z-%J)4/20 Series

near

0

18.14.51

98

a4=

(-8)4.81438

20

al= (-2)4.99999

62

as= (- 10)2.29729

21

a2= (-4)8.33352

77

ae=(-12)4.94511

45

a,=(-6)6.49412

86

112’3 -qo(xq, 31200 9750

---- 2"

825

x= (z-w)

18.14.52

ao= (- 1)9.99999

18.14.58

SC”@‘(x) =~a,x4n+a(x)

18.14.53 Ie(x):)1<4x10-7

x=w-p+~-~+~-‘+g~~~~~~3

891

ao= -2.00000 + O(d5)

18.14.54

w= -2( c-7) Series

Involving

(r

00

a,=(-7)6.58947

52 49

al=(-l)1.00000

02

a,=(-9)5.59262

a2=(-3)4.99995

38

as=(-11)5.54177

u3= (--5)6.41145

59

69

18.14.55 25 3%+’ -3.23~~~ 3~107~‘~+3~~7~23~37z~~ ~ a=z--2.5!-22.91+ 23.13! + 24.17! 25.21!

18.14.59

x{(x) =&7.x4’+E(x)

+ (e(x)[<3X10-8 18.14.56

f&=(-1)9.99999

18.15. Pseudo-Lemniscatic (gz=---1,

a4= - (-9)2.57492

99

aI=-(-2)1.66666

74

a,=-(-11)5.67008

a2=-(-4)1.19036

70 a,=(-13)9.70015

a,=-(-7)5.86451

63

Case

g,=O)

If g2<0 and g3=0, homogeneity relations allow us to reduce our consideration of @ to @ (z; -1, 0). Thus 18.15.1 @(z; gz, 0) = Ig$‘“@ (zlgzf1’4;-1, 0) [ @ ‘, { and u a,re handled similarly]. Because of its similarity to the lemniscatic case, we refer to the case g2= - 1, g3=0 as the pseudo-lemniscatic case. It plays the same role (period ratio unity) for A< 0 as does the lemniscatic case for A>O. u,=&x (real h aIf -period for lemniscatic case) =i; (the Lemniscate Constant-see 18.14.7)

2w FIGURE

18.15

62 00 80

WEIERSTRASS

ELLIPTIC Special

A=-1,

18.15.2

g2=-1,

AND

Values

RELATED

and

g,=O

FUNCTIONS

663

Relations

18.15.4

&(O) = R21/4ei*/8,6,(O) = Rein’*, a,(O) = Re-*~/8,

18.15.3

HI= --;/,I$

m=$, q=&-‘2

Hz=+, H,=i@,

Values

18.15.6 18.15.7 18.15.8 18.15.9

w = 01 w2 CO’ = 03 02’

18.15.10

@I (z; -1,

18.15.11

@‘(z;

at Half-Periods

i/2

0

0 8 0

0

0)=i@(zefX’4;

with

Numerical Use and

d2w2

f(B2+$2’)

112’ = --ill,

Lemniscatic

Values

18.15.12

1, 0)

-1, 0)=e3nf’4@‘((ze”W/4; 1, 0)

18.16.

fh2-?2’)

92 =

-i/2

Relations

where R= ~&Z/T

18.15.5

I 18.15.13

{(z;

-1, O)=eiH’4[(zei*/4; 1, 0)

a(~; -1,

0)=e-fr’4u(ze2r’4; 1, 0)

Methods Extension

of the

Tables

1. Lemniscat.ic Case (a) Given z=z+iy in the Fundamental Triangle, find @(@‘,{,u) more accurately than can be done with the maps. u-Use Maclaurin series throughout the Fundamental Triangle. Five terms give at least six significant figures, six terms at least ten. @, {-Use Laurent’s series directly ‘(near” 0, (if IdI< 1, four terms give at least eight significant figures for @, nine for {; five terms at least ten significant figures for @, eleven for 0. Use Taylor’s series directl,y “near” z,,. Elsewhere (unless approximately seven or eight significant figures are insufficient) use economized polynomials to obtain @ (x), P’(Z) and/or [s(z) as appropriate. To get @(;y), @‘(;y) and/or {(iy), use Laurent’s series for “small” y, otherwise use economized polynomials to compute @ (y), ,fp ‘(y) and/or l(y), then use complex multiplication to obtain @ (iy), @ ‘(iy) and/or [(iy) . Finally, use appropriate addition formula to get @ (z) and/or c(z). @‘-Use Laurent’s series directly “near” 0 (if lzl< 1, four terms give at least six significant figures, five terms at least eight significant figures). Ekewhere, either use economized polynomials and addition formula as for @ and {, or get @“2=4@3@ and extract appropriate square root (g@ ‘2 0). (b) Given P (@I ‘, l, u> corresponding to a point in the Fundamental Triangle, compute z more accurately than can be done with the maps. Only a few significant figures are obtainable from the use of any of the given (truncated) reversed series, except in a small neighborhood of the center of the series. For greater accuracy, use inverse interpolation procedures. Example

Equianharmonic Case (a) Given z=z+iy in the Fundamental Triangle, find @ (@‘, [, U) more accurat,ely than can be done with the maps. Four terms give at least eleven u-Use Maclaurin series throughout the Fundamental Triangle. significant figures, five terms at least twenty one. @J--Use Laurent’s series directly “near” 0 (if lzj< 1, four terms give at least 10s for @, 11s for {; five terms at least 13s for @, 14s for l). Elsewhere (unless approximately seven or eight significant figures are insufficient) use economized polynomials to obtain $9 (z), P’(Z) and/or T(z), as appropriate. To get @ (iy), @ ‘(iy) and/or I, use Laurent’s series. Then use appropriate addition formula to get @ Cd and/or W. Example

2.

664

WEIERSTRASS

ELLIPTIC

AND

RELATED

FUNCTIONS

@‘-Use Laurent’s series directly “near” 0 (if Izi< 1, four terms give at least 8S, five terms at least 11s). Elsewhere, either proceed as for @ and {, or get @‘2=4@3-1 and extract appropriate square root (g@‘ZO>. (b) Given @ (@‘,l,a) corresponding to a point in the Fundamental Triangle, compute z more accurately than can be done with the maps. Only a few significant figures are obtainable from the use of any of the given (truncated) reversed series, except in a small neighborhood of the center of the series. For greater accuracy, use inverse interpolation procedures. Example 3. Given period ratio a, find parameters m (of elliptic integrals and Jacobi’s functions of chapter 16) and p (of 6 functions). m-In both the cases A>0 and A3, use the method of Example 6 in chapter 17. An alternative method is to use Table 18.3 to obtain the necessary entries, thence use

m= (e2-e3)/(el-e3)

in case A>O,

m = 4 - 3e2/4Hz in case A< 0 a

q----In both the cases A>0 and AO, p=ie+j2 if A
of Values

at Half-Periods,

Invariants

and

Related

Quantities

from

Given

A>0 Given w and u’, form ~‘/iw and enter Table

18.3.

4.

Given w=lO,

18.3)

Given w2 and w2’, form w2’/zIw2and enter Table 18.3. Multiply the results obtained by the appropriate power of w2 (see footnotes of Table 18.3) to obtain value desired. Example

w’=lli,

(Table

A<0

Multiply the results obtained by the appropriate power of w (see footnotes of Table 18.3) to obtain value desired. Example

Periods

find ef, gt, and A.

4.

Given w2=10,

Here w’/iw= 1.1, so that direct reading of Table 18.3 gives

find ef, gc and A.

w2’=lli,

Here w3w2= 1.I, so that direct reading of Table 18.3 gives

el(l)=1.6843 e2(l)=-.2166

041

cl(l) = -.2166

258 (=-el--e3)

e2(1)=.4332

2576+3.0842 5152=--2LS(eJ

e3(l) = - 1.4676 783

e3(l) =&Cl>

92(l) = 10.0757 7364

g2(1) = -37.4874

ga(1) =2.1420

g3(1) = 16.5668 099.

Multiplying obtain

1000.

by appropriate e1=.01684

powers of w=lO

we

Multiplying we obtain

3041

e3= - .01467 6783

e3=Zl

gz=1.0075

77364 x 1O-3

g2=-3.7487

g,=2.1420

1ouo x 10-B 3191 x lo-lo

powers of w2=10

el= - .00216 62576+ .03084 25893 e2=.00433

A=8.9902

912

by appropriate

e2= - .00216 6258

whence

589i

g3=1.6566

25152 4912 x lo-$ 8099 x 1O-6

whence A=-6.0092

019 x 1O-8

WEIERSTRASS

ELLIPTIC

AND

Example 5. (A>O) Given w=lO, w’=55i, find q, q’, u(w), u(:w’) and u(wJ. Forming w’/iw=5.5 and entering Table 18.3 we obtain q=.82246704, u(w) = .96045 40. Using Legendre’s relation we find q’=qw’-k/2=2.9527 723i. Since interpolation for ~(0’) and u(w+w’) is difficult, use is made of 18.3.15-18.3.17 together with 18.3.4 and 18.3.6. Values of g2, g3 and el can be read directly to eight significant figures and e3 to about five significant figures giving gz=8.1L74 243, g3=4.4508 759, e,=1.6449 341, and e3 =-.82247. Use of 18.3.6 ields H,=.OO174 69 and Hz=.00174 69i. App %‘cation of 18.3.S 18.3.17 yields u(w’)/i=.OO71177 and u(wJ = -.002016-.01055i. Multiplying the results obtained by the appropriate powers of w we obtain q = .08224 6704., q’= .29527 7233, u(w) =9.6045 40, u(w’)=.O71177% and a(~~)=-.02016-.1055i. Determination

Given g2>0 and (if g3=0, Jw’J= w ; &=g2gr2”. From and w’g3lt8, thence Example

of Perk&

from

Given

6.

Invariants

Given g2=8, g3=4, find w and w’. With J2=g2g3-2/a=3.1748 02104, from Table 18.1 wg31’B= 1.2718 310 and w’g,“6=1.8702 425i whence w=1.009453 and w’=1.484413i. @I,

or f for

18.1.)

6.

Given 2= -10, g3=2, find Q and w;l. With g2z-g2g3-2L - 1O/1.5874 0105=-6.2996 053, from Table 18.1 w2g31/6= 1.5741 349 and wkg31’6=1.7J24 396i whence w2=1.40239 48 and wj=1.52561 02i. Example

7.

(Table

5.

A<0 Given g2 and g3>0 such that A=d--27d
of @,

065

(A
g3>0 such that A=d-27gi’>O see lemniscatic case), compute Table 18.1, determine wg;/e w and w’.

Computation

FUNCTIONS

Example

Given g2=10, g3=2, find w and w’. With g2= g2g3-2/a=6.2996 05249, from Table 18.1 wg3’16= 1.1267 806 and w’g3‘16=1.2324 295i whence w= 1.003847 and w’= 1.09797Oi. Example

RELATED

7.

Given g2=7 s=6, find w2 and 0;. With &=g2g3-2 9 =7/3.3019 2725=2.119974, from Table 18.1 w2g31’6=1.3423 442 and w2’gJ’6=3.1441 141i whenoew2=.99579 976 and wi=2.33241 83ti. Given

z and

arbitrary

g2, ga

(or arbitrary periods from which g2 and g3 can be computedin any case, periods must be known, at least approximately) First reduce the problem (if necessary) to computation for a point z in the Fundamental by use of appropriate results from 18.2.

FIGTJEE 18.16

Rectangle

666

WEIERSTRASS

ELLIPTIC

AND

RELATED

FUNCTIONS

1 (as accurate as desired) If both 2 and y are “small,” (point in double-cross hatched region) use Laurent’s series in z directly. If either z or y is ‘L1arge,” use Laurent’s series on Ox, then on Oy and finally use an addition formula. (For 9 j an alternative is to get @, then compute the appropriate root of @‘z=4 p3-g2@ -ga; see18.8.) Method

A<0

A>0 Method

Method

2 (for @ or @ ’ only)

2 (for @ or @ ’ only)

Compute et(i= 1,2,3) (if only 92, g3 are given use Table 18.1 to get the periods, then get ei in Table 18.3; if periods are also given, use Table 18.3 directly). In any case, obtain m(=[ez-es]/ [el- e3]), thence Jacobi’s functions sn(z*jm), cn(z*lm), dn(z*)m), from 16.4 and 16.21 and @ or @,’ from 18.9.11-18.9.12.

Compute e2 and Hz (if only g2,g3 are given, use Table 18.1 to get the periods, then get e( in Table 18.3; if periods are also given use Table 18.3 directly). In any case, obtain m(=$--3e2/4H2) thence Jacobi’s functions sn(z’)m), cn(z’lm),

Method 3 (accuracy limited by Table 4.16 of eenz and by the method of getting periods). Obtain periods, their ratio a, then q=e-” from Table 4.16. Hence get St(O), i=2,3,4 from truncated series 18.10.21-23. Compute appropriate d functions for z=x and for z=iy, whence and/or JW), Q (in), @ ‘(iy> get P(x), @‘(4 and/or [(iy), then use an addition formula (if either x or y is “small”, it is probably easier to use Laurent’s series).

Method

Example

8.

Given 2=.07+.li,

g2=10, g3=2,

find @. Using Laurent’s series directly with cz=.5 c,=.O7142 85714 cq= .08333 33333 c5= .00974 02597 2-2= -22.97193 820-63.06022 25i + c2z2= .00255 000+ .00700 OOi +c3z4=.OOOOl214.OOOOl02i fc4z6= f .OOOOO 024.OOOOO Oli

dn(z’jm),

from 16.4 and 16.21 and @ or @ ’ from

18.9.11-18.9.12. 3 (accuracy limited

as in the case A>O).

Obtain periods, their ratio a, thence q2=e+J2 from Table 4.16. Then proceed as in the case A>O, using corresponding formulas.

Example 8. Given z=.1+.03i, g2=-10, find@. Using Laurent’s series directly with

g3=2,

C2=-.5

~a=.07142 85714 ch= .80333 33333 z.-2=76.59287 938-50.50079 960i c&= - .00455 ooo.00300 oooi c3z4= $ .OOOOO 334 + .OOOOO 780i c4z6= - .OOOOO 002 + .OOOOO 01 li @ (z)=76.58833

270-50.50379

169i.

@ (z) = -22.97450 010-63.05323 28i. Example 9. Given z=15+73i, g2=8, g3=4, find @. From Example 7, w=1.009453, c,+= 1.484413i. From Table 18.3, e,=1.61803 37, e3= -.99999 96, whence m=.14589 79. From 18.2.18 with M=7 and N=24, 9(.867658+ 1.748176i)=@(l5+73i). Since z lies in Rz, by 18.2.31 @(15+73i)=F(.867658+1.220653). From 16.4 with z*=1.40390+1.97505i, sn(z*lm) =2.46550+1.96527i. Using 18.9.11, ,f$(15+733) = - .57743t .067797i.

Example

find @.

9.

Given 2=1.75-t-3.6(

g2=7,

g3=6, 98, &=

From Example 7, o2=.99579 2.33241 83i. Using 18.2.18 with M= 1, N= 1, @(1.75+3.6i)=@(-.24159 96-l.O64836i)= @(.24159 96+1.0648 36i). WithA
WEIERSTRASS

ELLIPTIC

AND

RELATED

A>0

A<0

Given w=lO, w’=2Oi, find Example 10. {(9+19i) by use of theta functions, 18.10 and addition formulas. For the period ratio a=w’/wi=2 with the aid of Table 4.16, p=e-2”=.00186 74427. Using the truncated approximations 18.10.2118.10.23 we compute the theta functions for argument zero. Using 16.27.1-16.27.4 we compute the theta functions for arguments v where Then, with 18.10.5-18.10.7 Z=X and z=iy. together with 18.10.9 and 18.10.18 we obtain l(9) =.09889 5484, s(19i) = -.00120 0 155i, @ (9) = .01706 9647, p’(9) = - .00125 3460, @ (19i) = -.00861 2615, @‘(19i) = -.00003 :757i. Using the addition formula 18.4.3, we obtain [(9 + 19i) = .07439 49- .00046 8%. Use of Table

667

FUNCTIONS

18.2 in Computing

Example 10. Given wZ=5, wi=7i find @‘(3+2i) by use of theta functions, 18.10 and addition formulas. With the use of Table 4.16 and 18.10.2, q=ie-.‘r =.11090 12784i. The theta functions are computed for argument zero using 18.10.21-18.10.23 and the theta functions for arguments vl and vz corresponding to z=zlfzz using 16.27.1-16.27.4. Using 18.l.O.518.10.6 together with 18 10.10, we find @ (3) = .10576 946, @ (2i) = -.24497 773, @’ (3) = - .07474140, Q ‘(Zi) = - .25576 007i. The addition formula 18.4.1 yields @(3+2i)=.O1763 210 -.07769 187i, and 18.4.2 yields @‘(3+21)= -.00069 182 t.04771 305i. @,

@‘,

r for

Special

Period

Ratios

If the problem is reduced to computing @ , @ ‘, { in the Fundamental Rectangle for the case when the real half-period is unity and pure imaginary half-period is ia, for certain values of a Table 18.2 may be used. Consider @ as an example. If Iz( is “small”, then use Laurent’s series directly for @ (z) [invariants for use in the series are given in Table 118.31. If 2 is “large” and y “small” use Table 18.2 to obtain x2@ (2) and S?@‘(X), thence @ (x) and g’(x) ; use Laurent’s series to obtain @ (iy) and @‘(iy) ; finally, use addition formula 18.4.1. For z “small” and y “large”, reverse the procedure. For both 2 and y “large,” use Table 18.!2 to obtain JB (x), P’(X), @ (iy) and @‘(iy), thence use addition formula 18.4.1. Similar procedures apply to @ ’ or [. For @ ‘, one can also first obtain $J5, then compute @ I2 =4&?‘--g,@ -g3 and extract the appropriate square root (see 18.8 re choice of sign for @‘).

A>0

A<0

Example 11. Compute @(.8+i) when a==1.2. Using Table 18.2 or Laurent’s series 18.5.1-4 with g2=9.15782 851 and g,=3.23761 717 from Table 18.3, @(.8)=1.92442 11, @‘(A) = -2.76522 05, @ (i) = - 1.40258 06 and p’(i)=-1.19575 5%. Using the addition formula 18.4.1 @(.8+i)=-.3 81433-.149361i.

Example 11. Compute @(.9+.li) for a=1.05. Using Table 18.2 or Laurent’s series 18.5.1-4 #with g2=-42.41653 54 and g3=9.92766 62 from Table 18.3, @ (.9) = .34080 33, @‘(.9) = -2.164801, @ (.li) = -99.97876, With the addition for@‘(.li) = -2000.4255i. mula 18.4.1 @(.9+.1i)=.231859-.215149i.

Example 12. Compute ((.02+3i) for (2~4. Using Table 18.2 or Laurent’s series 18.5.1-5 with g2=8.11742 426 g3=4.45087 587 from Table 18.3, [(.02) =49.99999 89, @ (.02) =2500.00016, @‘(.02) = -249999.98376, p(3i) = .89635 173i, @(3i)=-32326 511, @ ‘(3i) = - .00249 829i. Applying the addition formula 18.4.3, {(.02+3i)=.O16465+.89635i.

Example 12. Compute @‘(.4+.9i) for o=2. Using Table 18.2 or Laurent’s series 18.5J-4, with g,=4.54009 85, g,=8.38537 94 from Table 18.3, @ (.4) =6.29407 07, @‘(.4) = -30.99041, @ (.9i) = - 1.225548, @,‘(.9i) = -3.19127 03i. Using the addition formulas 18.4.1-2, 9’(.4+.9i)=1.10519 76-.56489 OOi.

668

WEIERSTRASS

ELLIPTIC

Computation

of s for

AND Given

RELATED z and

(or periods from which g2 and g3 can be computed-in approximately)

FUNCTIONS

Arbitrary

g, and

g3

any case, periods must be known, at least

First reduce the problem (if necessary) to computation for a point z in the Fundamental Rectangle (see 18.2). After final reduction let z denote the point obtained. A<0 A>0 If $?z>w,@ or If $?a>~/2 or, $z>w:/4, use duplication formula as in case A>O. Otherwise, use Maclaurin series for u Nz>w’/2, use duplication formula directly. obtaining u(P@) by use of Maclaurin series for u and @‘(z/2) by method explained above. Otherwise, simply use Maclaurin series for u directly. An alternate method is to use theta functions 18.10 first computing A>0 Example 13. Compute u(.4+1.3i) for g2=8, g3=4. From Example 7, w=1.009453 and w’ =1.484413i. Since Yz>w’/2, the Maclaurin series 18.5.6 is used to obtain “(z/2) =a(.2+.65i) = .19543 86+ .64947 28i, the Laurent series 18.5.4 to obtain @‘(.2+.65i)=5.02253 80-3.56066 93i. The duplication formula 18.4.8 gives u(.4+1.3i) = .278080+ 1.272785i.

p and S<(O), i=2,3,4. A<0 Compute

Example 13. u(.8+.4i) for ga=7, g3=6. From Example 7, w,=.99579 976, w: =2.33241 83i. Since k%‘z>w2/2, the Maclaurin series 18.5.6 is used to obtain u(z@)=u(.4+.2i) =.40038 019-j-.19962 017i, the Laurent series 18.5.4 to obtain @‘(.4+ .2i) = -3.70986 70 +22.2185444. The duplication formula 18.4.8 gives u(.8+.4$=.81465 765-k.38819 473i.

Rectangle, as well as g2 and g3 or Given Q(@, P’, C> corresponding to a point in the Fundamental the equivalent, find z. Only a few significant figures are obtainable from the use of any of the given (truncated) reversed series, except in a small neighborhood of the center of the series. For greater accuracy, use inverse interpolation procedures. If t,he given function does not correspond to a value of z in the Fundamental Rectangle (see Conformal Maps) the problem can always be reduced to this case by the use of appropriate reduction formulas in 18.2. This process is relatively simple for g(z), more difficult for the other functions (e.g. if A>0 and @=a+ib, where h>O, simply consider &a--ib and find z1 in RI [Figure 18.11; then compute z2=ZL+2w’, the point in R2 corresponding to the given @).

A>0

A<0

Given @=1-i, g2=10, g3=2, find z. Using the first three terms of the reversed series 18.5.25 z1= .727+.423i. The Laurent series

Given J? = 1 -t-i, g2= - 10, g3=2, find z. From Example 6, w,=1.40239 48 and 0:=1.52561 02i. Since b>O, z exists in R2 and z is computed with <E. Using 18.5.25 with a2= 1.25, a3=.25, u=[(@)-‘]‘/~ and the coefficients c, from Example 8

Example

18.5.1

gives

and

@(z,)=

14.

@‘(.727+.423i)=.825-.895i

@ (22)= @ (.697 + .393&)= .938- 1.038i.

Inverse interpolation gives (I)= .707 + .38Oi. Repeated applications of the above procedure yield 2= .706231+.379893i.

21

Example

14.

2u= 1.55377 3973 f .64359 424933 c2u5= .08044 9281-.19422

c3u7=-.01961 dUS -=-* 3

9359f.00812

17466i 660473

10115 7160- so419006673i

WEIERSTRASS

ELLIPTIC

AND

RELATED

669

FUNCTIONS

A<0

A>0

Stopping with the term in u7, z1 ~.81+~23i. Assuming AZ= -.03-.Oli, using 18.5.1, @(.81-l.23i)=.91410 95-.86824 37i, @(.78+.2&)= 1.03191 60- .91795 22i; with inverse interpolation zi” =. 7725 + .2404i. Repeated applicationg of inverse interpolation yield z=. 772247-.239$58i. Example 15. Given a=.4+.li, g2=7, g$=6, find z. Using the reversed series 18.5.70 with y,=.14583, y3=.05

Example 15. Given {=lO-15i, g2=8, 9:$=4, find Z. Using the reversed series 18.5.40 with A5= -.I3333 A,= u=

333,

u= +.40000

.03076 923076+.04615

38461%

A&=

- .OOOOO001402 + .OOOOO00686Oi

A&=

- .OOOoO 000004z=

000+.10000

oooi

$=+.OOOll

783+.00032

696i

Yd T=-.OOOOO

208+.00001

432i

- .02857 14286,

.OOOOO000003i

.03076 921670+.04615

3-&J9 -= 14

391472&.

w2Y3~11

55

- .OOOOO093 + .OOOOO126i = -.OOOOO 013+.00000

z=.40011

469+.10034

006i 260i

Methods of Computation of @ (@‘, f or U) for Given a and Given gz, g3 (or the equivalent), with the Use of Automatic Digital Computing Machinery

(a) Integration of Differential Equation @ and @’ may be generated for any z close enough to a “known point” z*( @ (z*) and @“(z*) being given) by integrating @“=6 p2-g2/2. A program to do this on SWAC, via a modification of the Hammer-Hollingsworth method (MTAC, July 1955, pp. 92-96) due to Dr. P. Henrici, exists at Numerical Analysis Research, UCLA (code number 00600, written by W. L. Wilson, Jr.). The program has been tested numerically in the equianharmonic case, using integration steps of various sizes. For example, if one starts with z*=02, using an “integration step” (h,k), w h ere h and k are respectively the horizontal and vertical components of a step, with (h,k) having one of the six values (&2&O), (fh,,fk,J, &=w2/2000, ko=~w~~/2000, one can expect almost 8S in @ and 7S in @,’ after 1000 steps, unless 2 is too near a pole. (b) Use of Series The process of reducing the computation problem to one in which z is in the Fundamental Rectangle can obviously be mechanized. Inside the Fundamental Rectangle the direct use of Laurent’s series is appropriate when the period

ratio a is not too large. However, if aZ&A>O) or a 22$(AO) and at z=ia/2 if uh4(AO). Use of Maps If the problem (of computing @, @‘, J‘ or u for given z) is reduced to the case where the real bdfperiod is unity and imaginary half-period is one of those used in the maps in 18.8 inspection of the

WEIERSTRASS

670

ELLIPTIC

appropriate figure will give the value of P(Z) If @’ is wanted instead, [c(z) or u(z)] to 2-35. get @, use 18.6.3 to obtain @ I2 and select sign (s) of @ ’ appropriately. (See Conformal Mapping (18.8) for choice of sign of square root of @ “>.

Computation

AND RELATED

approximation to (zol by Graeffe’s process, we may use the fact that z,,=wi-iyo(A>O), zo=w2 +ivo(A0 and

of z.

[lz

Given gz, g3 (or equivalent)

y,,/w>i

arccosh fi( x .7297)].

yo/wz is a monotonic increasing function of a for A<0 and

Since z$@ (zo) =O, the Laurent’s series gives O=1+C&&L3+C~U4+

FUNCTIONS

. . .

[0 5 yo/wz
where u= .zt. We may solve this equation [by Graeffe’s (root-squaring) process or otherwise] for its absolutely smallest root [having found an

Further data is available from Table 18.2 or from Conformal Maps defined by .Cp(2).

References Texts

and

Articles

[18.1] P. Appell and E. Lacour, Principes de la theorie des fonctions elliptiques et applications (GauthierVillars, Paris, France, 1897). [18.2] A. Erdelyi et al., Higher transcendental functions, vol. 2, ch. 13 (McGraw-Hill Book Co., Inc., New York, N.Y., 1953). [18.3] E. Gmeser, Einfiihrung in die Theorie der elliptischen Funktionen und deren Anwendungen (R. Oldenbourg, Munich, Germany, 1950). [18.4j G. H. Halphen, Trait6 des fonctions elliptiques et de leurs applications, 1 (Gauthier-Villars, Paris, France, 1886). [18.5] H. Hancock, Lectures on the theory of elliptic functions, vol. 1 (John Wiley & Sons, Inc., New York, N.Y., 1910, reprinted, Dover Publicat,ions, Inc., New York, N.Y., 1958). [18.6] A. Hurwite and R. Courant, Vorlesungen iiber allgemeine Funktionentheorie und elliptische Funktionen, 3d ed. (Springer, Berlin, Germany, 1929). [18.7] E. L. Ince, Ordinary differential equations (Dover Publications, Inc., New York, N.Y., 1944). j18.81 E. Ksmke, Differentialgleichungen, LBsungsmethoden und Liisungen, vol. 1, 2d ed. (Akademische Verlagsgesellschaft, Leipzig, Germany 1943). (18.91 D. H. Lehmer, The lemniscate constant, Math. Tables Aids Comp. 3, 550-551 (1948-49). [18.10] S. C. Mitra, On the expansion of the Weierstrassian and Jacobian elliptic functions in powers of the argument, Bu!l. Calcutta Math. Sot. 17, 159-172 (1926). [lS.ll] F. Oberhettinger and W. Magnus, Anwendung der elliptischen Funktionen in Physik und Technik (Springer, Berlin, Germany, 1949). (18.12) G. Prasad, An introduction to the theory of ellipt,ic functions and higher transcendentals (Univ. of Calcutta, India, 1928). (18.13) U. Richard, Osservazioni sulla bisesione delle funzioni ellittiche di Weierstrass, Boll. Un. Mat. Ital. 3, 4, 395-397 (1949).

[18.14] E. S. Selmer, A simple trisection formula for the elliptic function of Weierstrass in the equianharmonic case, Norske Vid. Selsk. Forh. Trondheim 19, 29, 116-119 (1947). [18.15] J. Tannery and J. Molk, Elements de la theorie des fonctions elliptiques, 4 ~01s. (Gauthier-Villars, Paris, France, 1893-1902). [18.16] F. Tricomi, Elliptische Funkt.ionen (Akademische Verlagsgesellschaft, Leipzig, Germany, 1948). [18.17] F. Tricomi, Funzioni ellittiche, 2d ed. (Bologna, Italy, 1951). [l&18] C. E. Van Orstrand, Reversion of power series, Phil. Mag. (6) 19, 366-376 (Jan.-June 1910). [l&19] E. T. Whittaker and G. N. Watson, A course of modern analysis, ch. 20, 4th ed. (Cambridge Univ. Press, Cambridge, England, 1952).

Guides,

Collections

of Formulas,

etc.

[18.20] P. F. Byrd and M. D. Friedman, Handbook of elliptic integrals for engineers and physicists, Appendix, sec. 1030 (Springer-Verlag, Berlin, Germany, 1954). [18.21] -4. Fletcher, Guide to tables of elliptic functionst Math. Tables Aids Comp. 3, 247-249 (1948-49). [18.22] S. Fliigge, Handbuch der Physik, vol. 1, pp. 126 146 (Springer-Verlag, Berlin, Germany, 1956). j18.231 H. Kober, Dictionary of conformal representations (Dover Publications, Inc., New York, N.Y., 1952). [18.24] L. M. Milne-Thomson, Jacobian elliptic function tables (Dover Publications, Inc., New York, N.Y., 1950). [18.25] K. Weierstrass and H. A. Schwarz, Formeln und Lehrsiitze zum Gebrauche der elliptischen Functionen. Nach Vorlesungen und Aufzeichnungen des Herrn K. Weierstrass bearbertet und herausgegeben von H. A. Schwarz, 2d ed.

(Springer,Berlin, Germany, 1893).

WEIERSTRASS

ELLIPTIC

Tables [18.26]

[18.27]

$

mostly

(4, 4D.

RELATED [18.28]

Chih-Bing Ling, Evaluation at half-periods of Weierstrass’ elliptic function with rectangular primitive period-parallelogram, hlath. Comp. 149 69, 67-70 (19GO). Values of ei (i-1, 2, 3) to 15D for various period ratios in case A>O. E. Jahnke and F. Emde, Tables of functions, 4th ed., pp. loo-106 (Dover Publications, Inc., New Equianharmonic case, real York, N.Y., 1945). argument,.P(u),P

AND

T(u),u

(u),u=O

(go)

T.

H. Southard, Approximation Weierstrass @J function in case for real argument, Math. 11,

58, 99-100

to 7D . with 1~=0(.1).8(.05)1.55. [18.29]

671

FUNCTIONS

D.

(Apr.

1957).

modified

and table of the the equianhatmonic Tables Aids IComp. f(u)=@(u)-$ central

differences,

A. Strayhorne, A study of an elliptic function (Thesis, Chicago, Ill., 1946). Air Documents Division T-2, AMC, Wright Field, Microfilm No. Rc-734F15000. @,(a; 37, -42), 4D, z= .04i(.O4i) 1.36i.

WEIERSTRASS

ELLIPTIC

AND

RELATED

673

FUNCTIONS Table

TABLE

FOR

OBTAININC:

PERIODS @2-=&G

Non-Negative

3.25 3.30 3.35 3.40

::: i:: ::2 2; 4:9 5. 0 ::: ::i 2; t-2 6: 8

Non-Positive

1.28254 1.27944 1.21637 1.27333 1.27031

98 73 43 03 49

1.52168 1.51892 1.51685 1.51505 1.51342

83 22 48 45 84

1.26732 1.26436 1.26143 1.25853

80 90 77 38

1.51193 1.51053 1.50923 1.50799

A-

g, AND

18.1 g,

Discriminant

-0.00 -0. 01 -0.02 -0.03 -0.04

2.62205 2.62025 2.61693 2.61258 2.60737

76 54 53 67 43

2.62205 2.62304 2.62710 2.63126 2.63611

76 98 11 10 20

-

18 84 08 63

-0.05 -0.06 -0.07 -0.08 -0.09

2.60137 2.59464 2.58720 2.57909 2.57032

48 00 37 05 09

2.64151 2.64735 2.65355 2.66002 2.66669

34 75 47 55 74

-

38 64 42 45 47 23

' 6. WiT 3/z 1.69503 33 1.64719 87 1.60789 93 1.57451 65 1.54548 31 1.51978 54

-0.10 -0.11 -0.12 -0.13 -0.14

2.56091 2.55088 2.54025 2.52905 2.51729

33 61 86 23 09

2.67350 2.68037 2.68725 2.69409 2.70081

25 66 00 09 77

- 10 I ; I

7"

70 03 26 31 52 70

;

94 06 57 10 63

2.70738 2.71375 2.71986 2.72568 2.73117 2.73630

1

1.49672 1.47581 1.45668 1.43905 1.42269

11 23 70 01 90 29

7

47 95 44 69 50

2.50500 2.49221 2.47895 2.46527 2.45118 2.43675

-

1.22569 1.22055 1.21551 1.21055 1.20568

-0.15 -0.16 -0.17 -0.18 -0.19 -0.20

I -

; 5

1.20089 1.19618 1.19156 1.18700 1.18253

62 86 00 83 18

1.40744 1.39316 1.37973 1.36706 1.35506

84 72 79 51 88

1.17812 1.16953 1.16120 1.15314 1.14532

83 35 96 34 23

1.34368 1.32250 1.30316 1.28537 1.26889

10 70 60 08 69

49 74 94 71 14

Wig t Ii 1.82987 88 1.94863 05 2.04569 84 2.12452 34 2.18836 07

1.13773 1.13036 1.12321 1.11626 1.10950

46 91 55 38 49

1.25356 1.23923 1.22577 1.21310 1.20113

57 29 96 78 41

4

1.10293 1.09653 11.09030 1.08423 1.07831

00 11 03 04 46

1.18978 1.17901 1.16874 1.15895 1.14959

83 03 82 67 65

1.07254 1.06691 1.06142 1.05606 1.05083

63 95 83 74 15

1.14063 1.13203 1.12377 1.11583 1.10817

29 51 59 09 84

1.04571 1.04071 1.63582 1.03104 1.02636 1.02178

58 56 65 44 52 54

1.10079 1.09367 1.08678 1.08012 1.07367 1.06742

87 40 83 69 66 51

0.10 0.09 0.08 0.07 0. 06

%!FF$ 1.81>01 1.82207 1.82696 1.83165 1.83611

99 90 90 87 17

1.8981E"01 1.89119 06 1.80476 56 1.87888 68 1.87354 40

0.05 0.04 0.03 0.02 0. 01 0.00

1.84028 1.84412 1.84756 1.85050 1.85280 1.85407

47 45 35 78 73 47

1.86873 1.86447

53 02

1.85769 1.86077 1.85534 1.85407

72 37 90 47

.&-I

INvARIANTS

3

Discriminant

wg3 1.25853 1.25280 1.24718 1.24166 1.23624 1.23092 4. 0

FOR

w’g$g

[‘;;I-11 1s=0.20412

b/i

o

-P;, -0:25 -0.30 -0.35 -0.40

:i :: 17

:z 100 -

OPg3 1.62955 1.66926 1.68880 1.69574 1.69529

?J

-0.45 -0.50 -0.55 -0.60 -0.65

1.69080 1.68433 1.67705 1.66962 1.66240

53 20 44 98 65

2.24023 2.28267 2.31773 2.34701 2.37174

31 03 31 74 42

-0.70 -0.75 -0.80 -0.85 -0.90

1.65555 1.64914 1.64320 1.63771 1.63264

57 98 64 44 84

2.39284 2.41102 2.42683 2.44070 2.45294

34 56 68 05 80

-0.95 -1.00

1.62797 1.62366

70 67

2.46384 2.47359

40 62

g2 -1.0 -0.8 -0.6 -0.4 -0.2

cg,>

'

w2g t

w4$

1.62366 1.60646 1.58820 1.56918 1.54967

67 93 63 06 81

3.03954 3.05518 3.06892 3.08070 3.09053

85 40 24 50 50

1.52995 1.51022 1.49067 1.47143 1.45262

40 67 44 75 13

3.09846 3.10458 3.10899 3.11182 3.11318

47 18 55 48 95

1.43430 1.41652 1.39933 1.38273 1.36672

15 88 41 24 71

3.11320 3.11196 3.10955 3.10604 3.10147

22 36 78 84 38

1.35131 1.33647 1.32220 1.30847 1.29526 1.28254 (,3)3

24 63 24 11 10 90

3.09584 3.08910 3.08116 3.07175 3.06025 3.04337

00 74 35 37 10 67

i: $‘=0.40824

20 17 14 13 11

<&?> I

:

1

:

-

3

1 -

2' 2

I

:

I

:

I : -1 1

:

In (3-g,)

[ 1

4145

cc

-100 - 50 - 33 - 25

829

a-

0

,

674

WEIERSTRASS Table

z=r\,,

18.2

ELLIPTIC

TABLE

1.00

1.05

FOR

OBTAINING

AND

RELATED

@, P’

AND

(Positive Discriminant--Unit zV(z) 1.1 1.2

FUNCTIONS I ON Ox AND

Oy

Real Half-Period) 1.4

2.0

4.0

0.00 0.05 0.10 0.15 0.20

1.00000 1.00000 1.00005 1.00029 1.00094

00 37 91 91 57

1.00000 1.00000 1.00005 1.00027 1.00086

00 34 41 41 77

1.00000 1.00000 1.00005 1.00025 1.00081

00 32 05 59 12

1.00000 1.00000 1.00004 1.00023 1.00074

00 29 59 31 02

1.00000 1.00000 1.00004 1.00021 1.00068

00 26 22 46 25

1.00000 1.00000 1.00004 1.00020 1.00066

00 25 08 75 02

1.00000 1.00000 1.00004 1.00020 1.00065

00 25 07 73 97

0.25 0. 30 0.35 0.40 0.45

1.00230 1.00479 1.00889 1.01520 1.02442

98 35 27 23 50

1.00212 1.00441 1.00821 1.01408 1.02269

32 61 33 14 65

1.00198 1.00414 1.00772 1.01326 1.02144

79 21 00 70 00

1.00181 1.00379 1.00709 1.01224 1.01985

79 79 99 31 94

1.00167 1.00351 1.00659 1.01140 1.01857

98 80 56 98 24

1.00162 1.00340 1.00640 1.01108 1.01807

64 97 03 69 36

1.00162 1.00340 1.00639 1.01107 1.01806

51 71 57 93 19

0.50 0.55 0.60 0.65 0.70

1.03738 1.05504 1.07855 1.10923 1.14872

54 92 23 99 15

1.03486 1.05152 1.07381 1.10307 1.14092

08 36 21 22 35

1.03302 1.04895 1.07036 1.09857 1.13524

47 81 11 95 09

1.03071 1.04572 1.06601 1.09291 1.12807

36 73 29 64 45

1.02883 1.04309 1.06246 1.08829 1.12222

08 40 70 58 46

1.02810 1.04207 1.06109 1.08650 1.11995

10 28 15 29 41

1.02808 1.04204 1.06105 1.08646 1.11990

38 87 91 07 05

0.75 0.80 0.85 0.90 0.95 1.00

1.19894 1.26229 1.34171 1.44091 1.56460 1.71879

38 01 37 81 22 62

1.18933 1.25071 1.32807 1.42515 1.54671 1.69885

40 86 28 17 40 59

1.18232 1.24227 1.31812 1.41364 1.53366 1.68430

81 98 18 80 04 41

1.17348 1.23162 1.30556 1.39912 1.51717 1.66592

94 95 03 31 65 77

1.16627 1.22292 1.29529 1.38725 1.50370 1.65090

18 96 60 23 31 68

1.16346 1.21955 1.29130 1.38264 1.49846 1.64507

98 14 97 14 94 17

1.16340 1.21947 1.29121 1.38253 1.49834 1.64493

37 17 57 27 59 41

[(-;‘“I CTi-!/\,l

[ (-i’4]

1.00

[(-i’“]

[(-;‘“I

1.1

1.05

[(-i’4]

1.2

[(-i’4]

1.4

[(-;‘“I

2.0

4.0

0.00 0. 05 0.10 0.15 0.20

1.00000 1.00000 1.00005 1.00029 1.00094

00 37 91 91 57

1.00000 1.00000 1.00005 1.00027 1.00086

00 34 40 31 20

1.00000 1.00000 1.00005 1.00025 1.00080

00 31 03 42 14

1.00000 1.00000 1.00004 1.00023 1.00072

00 29 57 05 54

1.00000 1.00000 1.00004 1.00021 1.00066

00 26 19 13 38

1.00000 1.00000 1.00004 1.00020 1.00063

00 25 05 39 99

1.00000 1.00000 1.00004 1.00020 1.00063

00 25 04 37 94

0.25 0.30 0.35 0.40 0.45

1.00230 1.00479 1.00889 1.01520 1.02442

98 35 27 23 50

1.00210 1.00435 1.00804 1.01371 1.02194

14 08 86 37 93

1.00195 1.00403 1.00743 1.01263 1.02016

05 04 81 dl 25

1.00176 1.00362 1.00667 1.01129 1.01792

15 91 40 28 92

1.00160 1.00330 1.00605 1.01020 1.01612

81 38 50 38 33

1.00154 1.00317 1.00581 1.00978 1.01542

88 81 59 33 64

1.00154 1.00317 1.00581 1.00977 1.01540

75 52 03 34 99

0.50 0.55 0.60 0.65 0.70

1.03738 1.05504 1.07855 1.10923 1.14872

54 92 23 99 15

1.03345 1.04901 1.06955 1.09614 1.13001

04 44 87 60 89

1.03061 1.04466 1.06309 1.08675 l.11663

34 92 37 16 04

1.02707 1.03925 1.05504 1.07507 1.10003

18 21 64 92 09

1.02421 1.03488 1.04856 1.06569 1.08671

09 20 45 47 44

1.02310 1.03319 1.04606 1.06208 1.08160

77 83 96 70 18

1.02308 1.03315 1.04601 1.06200 1.08148

17 85 09 18 16

0.75 0.80 0.85 0.90 0.95

1.19894 1.26229 1.34171 1.44091 1.56460

38 01 37 81 22

1.17264 1.22578 1.29157 1.37264 1.47224

63 78 86 39 79

1.15387 1.19980 1.25602 1.32443 1.40736

03 68 53 52 61

1.13065 1.16777 1.21233 1.26544 1.32835

03 18 97 15 02

1.11207 1.14221 1.17761 1.21873 1.26610

03 52 18 89 10

1.10494 1.13243 1.16435 1.20095 1.24247

84 76 46 66 14

1.10478 1.13220 1.16404 1.20053 1.24191

09 79 34 95 74

1.00 1.05 1.10

1.71879

62

1.59449 1.74462

89 36

1.50769 1.62902 1.77589

66 39 10

1.40258

06

1.32024

17

1.28909

73

1.28836

81

[(-;‘4] Z.I iL!/\rr

:*i 1:4 1.6 1. 8 2.0 :*:

2: 6 2.8 3.0

z’ i ;: ; 4: 0

[(-:‘3] 1.05

1.00 1.71879

[(-;‘3]

62

1.59449

[‘-$11

1.1 89

1.50769

[‘-:‘“I

1.2 66

1.85616 1.40258

[(-;‘“I

1.4 29 06

1.32024 1.61789 2.09401

[(-;‘6]

2.0 95 17 44

4.0

1.52970 1.28909 1.86127 2.28676 2.80921

17 73 05 23 52

1.527649 1.288368 1.855916 2.273495 2.777516

3.43759

29

3.363868 4.028426

If the real half-period ~1, see 18.2 Homogeneity Relations. Interpolation with respect to II will, in general, be difficult because of the non-uniform subintervals involved. Aitken’s interpolation may be used in this ease. As few as 3s may be obtained. For the computation of P’, 9’ or c at z=.r+i!l, an addition formula may be used (18.4 and Examplea 11-12).

z: :$~~~ ‘* 459856 7.409386 98. $$~~~ lo:660867 11.877621 13.160574

WEIERSTRASS TABLE

ELLIPTIC FOR

AND RELATED

OBTAINING

9, Pi’ AND

(Positive Discriminant-Unit _.W(:) 1.1 1.2 -2.oooao 00 -2.00000 00 -1.99999 37 -1.99999 43 -1.99989 a9 -1.99990 80 -1.99948 63 -1.99953 10 -1.99836 70 -1.99850 41 -1.99598 17 -1.99630 33 -1.99158 17 -1.99221 67 -1.98420 07 -1.98530 95 -1.97260 99 -1.97437 35 -1.95525 47 -1.95785 77

;=s\fl 0.00' 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

1.00 -2.00000 -1.99999 -1.99988 -1.99940 -1.99810 -1.99537 -1.99038 -1.98210 -1.96928 -1.95036

00 26 18 16 75 33 23 95 90 :3

1.05 -2.00000 -1.99999 -1.99989 -1.99945 -1.99825 -1.99572 -1.99107 -1.98332 -1.97121 -1.95319

00 32 17 07 79 57 69 00 06 16

0.50 0.55 0.60 0.65 0.70

-1.92339 -1.88593 -1.83488 -1.76619 -1.67451

01 a3 99 53 43

-1.92730 -1.89106 -1.84127 -1.77376 -1.68307

50 43 27 97 45

-1.93016 -1.89480 -1.84594 -1.77931 -1.68934

21 97 09 45 72

-1.93377 -1.89954 -1.85184 -1.78633 -1.69729

0.75 0.80 0.85 0.90 0.95 1.00

-1.55271 -1.39118 -1.17683 -0.89169 -0.51095 0.00000

74 65 20 al a7 00

-1.56189 -1.40041 -1.18536 -0.89858 -0.51505 0.00000

13 70 53 la 33 00

-1.56861 -1.40713 -1.19163 -0.90361 -0.51806 0.0000'3

96 15 25 00 28 00

-1.57715 -1.41579 -1.19959 -0.91006 -0.52188 0.00000

[C-32]

[‘-321

1.05 -2.00000 -1.99999 -1.99989 -1.99945 -1.99828

[(72]

1.1 -2.00000 -1.99999 -1.99989 -1.99940 -1.99840

675

FUNCTIONS r ON Ox AND

0,

Table

18.2

Real Half-Period) 1.4 -2.00000 -1.99999 -1.99991 -1.99956 -1.99861

00 47 53 73 55

2.0 -2.00000 -1.99999 -1.99991 -1.99958 -1.99865

00 49 a1 14 86

4.0 -2.00000 -1.99999 -1.99991 -1.99958 -1.99865

00 49 a2 17 97

-1.99656 -1.99273 -1.98621 -1.97581 -1.95998

50 38 31 22 33

-1.99666 -1.99293 -1.98656 -1.97637 -1.96080

63 42 35 02 a2

-1.99666 -1.99293 -1.98657 -1.97638 -1.96082

08 89 17 34 78

03 33 a2 a9 96

-1.93671 -1.90341 -1.85668 -1.79209 -1.70382

95 73 71 a0 60

-1.93786 -1.90492 -1.85856 -1.79433 -1.70636

53 32 93 95 76

-1.93789 -1.90495 -1.85861 -1.79439 -1.70642

23 86 37 25 75

61 29 24 69 70 00

-1.58416 -1.42286 -1.20613 -0.91535 -0.52503 0.00000

75 23 88 50 45 00

-1.58689 -1.42561 -1.20869 -0.91741 -0.52626 0.00000

93 79 13 70 26 00

-1.58696 -1.42568 -1.20875 -0.91746 -0.52629 0.00000

39 30 17 57 14 00

[!-;‘“I

[C-j’“]

[(-;‘“I

[‘3’2]

00 37 95 33 62

1.2 -2.00000 -1.99999 -1.99990 -1.99954 -1.99856

00 43 a9 15 33

1.4 -2.00000 -1.99999 -1.99991 -1.99958 -1.99869

00 48 65 07 07

2.0 -2.00000 -1.99999 -1.99991 -1.99959 -1.99873

00 49 94 59 99

4.0 -2.00000 -1.99999 -1.99991 -1.99959 -1.99874

00 49 95 62 11

0.25 -1.99537 33 -1.99581 31 -1.99613 0.30 -1.99038 23 -1.99133 a2 -1.9920;! 0.35 -1.98210 95 -1.98398 06 -1.98533 0.40 -1.96928 90 -1.97268 69 -1.9751:; 0.45 -1.95036 13 -1.95619 80 -1.96030

14 a9 03 44 48

-1.99652 -1.99289 -1.98701 -1.97818 -1.96561

94 25 63 68 82

-1.99685 -1.99359 -1.98837 -1.98065 -1.96982

19 12 91 01 60

-1.99697 -1.99386 -1.98890 -1.98159 -1.97144

66 12 48 94 57

-1.99697 -1.99386 -1.98891 -1.98162 -1.97148

95 76 71 la 38

0.50 0.55 0.60 0.65 0.70 0.75 0.80 0;85 0.90 0;95

10 25 39 48 64

-1.94845 -1.92574 -1.89643 -1.85930 -1.81290

17 23 16 08 09

-1.95533 -1.93661 -1.91313 -1.88437 -1.84984

26 23 16 77 78

-1.95797 -1.94078 -1.91952 -1.89395 -1.86392

74 35 74 96 68

-1.95803 -1.94088 -1.91967 -1.89418 -1.86425

95 17 77 46 71

62 80 26 95 84

-1.75545 -1.68471 -1.59780 -1.49093 -1.35912

41 79 32 la 08

-1.80902 -1.76134 -1.70615 -1.64263 -1.56972

61 96 96 75 20

-1.82937 -1.79034 -1.74698 -1.69950 -1.64818

52 89 46 14 82

-1.82985 -1.79102 -1.74793 -1.70082 -1.65001

21 80 96 95 75

z/i=?/\u

0.00 0.05 0.10 0.15 0.20

1.00

-2.00000 -1.99999 -1.99988 -1.99940 -1.99810

-1.92339 -1.88593 -1.83488 -1.76619 -1.67451 -1.55271 -1.39118 -1117683 -0.89169 -0.51095

00 26 18 16 75

01 a3 99 53 43 74 65 20 al a7

-1.93299 -1.90123 -1.85861 -1.80221 -1.72827 -1.63184 -1.50639 -1.34312 -1.13018 -0.85145

00 32 21 48 08

a4 75 50 44 05 71 22 50 63 23

-1.93989 -1.91218 -1.87553 -l.a278(1 -1.76629 -1.6875? -1.5869E -1.45865 -1.29452 -1.08387

1.00 1.05 1.10

0.00000 00 -0.48485 79 -0.81220 52 -1.19575 58 -1.48600 58 -1.59338 a5 -1.59588 68 o.uoooo 00 -0.45984 59 0.00000 00 [(-;I”] [‘-y] [!-p] [(-i’4] [‘-;“I [(-i!4] [‘-$)“]

z/i=y\a

1. 0 1.2 -_1.4 ::i

1.4 1.00 1.2 2.0 4.0 1.05 1.1 0.00000 00 -0.48485 79 -0.81220 52 -1.19575 58 -1.48600 58 -1.59338 85 -1.59588 0.00000 00 -0.99449 51 -1.34717 40 -1.35527 0;OOOOO00 -iii7521 03 -1;09935 -0.78786 76 -0.85550 -0.46104 27 -0.64191

68 93 a3 88 20

:*64 21 a

0.00000 00 -0.46669 -0.33022 -0.22828 -0.15467 -0.10296

27 92 a9 43 79

-0.06745 -0.04346 -0.02734 -0.01629 -0.00795 0.00000

48 22 75 07 66 00

3.0 3.2 3.4

676

WEIERSTRASS Table

TABLE

18.2

ELLIPTIC

AND

RELATED

FUNCTIONS

FOR OBTAINING @, P AND r ON Or AND 0, (Positive Diswiminant-Unit Real Half-Period) S(i) 1.1 1.2 1.4 2.0 1.00000 000 1.00000 000 1.00000 000 1.00000 0.99999 895 0.99999 905 0.99999 912 0.99999 0.99998 319 0.99998 471 0.99998 595 0.99998 0.99991 481 0.99992 246 0.99992 868 0.99993 0.99973 030 0.99975 429 0.99977 377 0.99978

:=LY\, 0.00 0.05 0.10 0.15 0.20

1.00 1.00000 0.99999 0.99998 0.99990 0.99968

000 876 031 029 483

1.05 1.00000 0.99999 0.99998 0.99990 0.99971

000 887 198 871 119

0.25 0.30 0.35 0.40 0.45

0.99923 0.99840 0.99704 0.99494 0.99189

041 360 076 715 577

0.99929 0.99853 0.99727 0.99534 0.99251

399 355 741 298 583

0.99934 0.99862 0.99744 0.99563 0.99296

010 782 912 028 602

0.99939 0.99874 0.99766 0.99599 0.99353

799 617 478 122 179

0.99944 0.99884 0.99784 0.99628 0.99399

501 235 008 469 196

0.50 0.55 0.60 0.65 0.70

0.98762 0.98183 0.97419 0.96430 0.95174

541 783 386 782 028

0.98854 0.98315 0.97599 0.96671 0.95486

726 105 894 478 674

0.98921 0.98410 0.97731 0.96846 0.95714

683 521 096 489 079

0.99005 0.98530 0.97896 0.97066 0.96000

855 511 146 726 343

0.99074 0.98628 0.98030 0.97246 0.96233

0. 75 0. 80 0. 85 0.90 0.95 1.00

0.93598 0.91647 0.89251 0.86334 0.82800 0.78539 ['-;'"I

819 208 910 108 562 822

0.93995 0.92140 0.89855 0.87059 0.83659 0.79543 ['-j'"]

720 960 136 177 307 267

0.94284 0;92500 0.90294 0.87587 0.84284 0.80274 ['-;I"]

503 321 299 177 790 283

0.94648 0;92952 0.90847 0.88252 0.85073 0.81195 ['-;'"]

146 973 617 588 222 906

0.94944 0;93322 0.91298 0.88795 0.85716 0.81947 [c-y]

z/i=y\a 0.00 0.05 0.10 0.15 0.20

1.00 1.00000 0.99999 0.99998 0.99990 0.99968

000 876 031 029 483

1.05 1.00000 0.99999 0.99998 0.99990 0.99971

000 887 200 891 234

1.1 1.00000 0.99999 0.99998 0.99991 0.99973

000 895 322 516 226

1.2 1.00000 0.99999 0.99998 0.99992 0.99975

000 905 476 299 725

0.25 0.30 0.35 0.40 0.45

0.99923 0.99840 0.99704 0.99494 0.99189

041 360 076 715 577

0.99929 0.99854 0.99731 0.99541 0.99266

836 660 033 639 485

0.99934 0.99865 0.99750 0.99575 0.99322

758 014 544 586 092

0.99940 0.99877 0.99774 0.99618 0.99391

0.50 0.55 0.60 0.65 0.70

0.98762 0.98183 0.97419 0.96430 0.95174

541 783 386 782 028

0.98882 0.98364 0.97684 0.96808 0.95701

817 988 238 373 320

0.98969 0.98495 0.97875 0.97080 0.96080

725 820 291 464 810

0.75 0.80 0.85 0. 90 0.95

0.93598 0.91647 0.89251 0.86334 0.82800

819 208 910 108 562

0.94322 0.92626 0.90559 0.88063 0.85068

518 102 833 688 069

0.94842 0.93328 0.91496 0.89299 0.86683

1.00 1.05 1.10

0.78539

822

0.81491 0.77237

420 164

0.83587 0.79939 0.75655 [(-;)3]

[‘-;‘“] zli=y\a 1. 0 1.2 1.4 ::"8 2. 0 2; 2 2. 4

1.00 0.78539 822

[‘-i’s] 1.05 0.81491 420

000 915 643 109 130

4.0 1.00000 0.99999 0.99998 0.99993 0.99978

000 915 644 115 148

0.99946 0.99887 0.99790 0.99639 0.99417

321 957 793 831 016

0.99946 0.99888 0.99790 0.99640 0.99417

364 045 954 099 438

340 174 531 106 582

0.99100 0.98666 0.98082 0.97315 0.96324

867 012 605 633 002

0.99101 0.98666 0.98083 0.97317 0.96326

490 904 833 272 132

525 007 848 364 486 977

0.95059 0.93465 0.91473 0.89005 0.85966 0.82239

446 128 876 936 076 820

0.95062 0.93468 0.91478 0.89010 0.85971 0.82246

155 503 003 902 964 703

1.4 1.00000 0.99999 0.99998 0.99992 0.99977

000 912 601 935 752

2.0 1.00000 0.99999 0.99998 0.99993 0.99978

000 915 649 181 537

4.0 1.00000 0.99999 0.99998 0.99993 0.99978

000 916 650 187 555

928 991 989 100 695

0.99945 0.99888 0.99794 0.99652 0.99448

935 517 811 557 077

0.99947 0.99892 0.99802 0.99665 0.99469

871 586 472 871 855

0.99947 0.99892 0.99802 0.99666 0.99470

917 682 653 184 368

0.99078 0.98659 0.98113 0.97419 0.96553

438 357 896 926 710

0.99166 0.98791 0.98306 0.97694 0.96935

445 646 740 003 061

0.99200 0.98842 0.98381 0.97799 0.97081

425 700 123 651 949

0.99201 0.98843 0.98382 0.97802 0.97085

225 902 874 138 406

600 385 295 175 386

0.95489 0.94200 0.92657 0.90827 0.88676

807 908 574 878 908

0.96010 0.94902 0.93589 0.92051 0.90268

986 381 412 815 849

0.96211 0.95172 0.93947 0.92521 0.90878

557 061 230 144 307

0.96216 0.95178 0.93955 0.92532 0.90892

276 405 644 176 628

315 419 714

0.86166

128

0.88219

209

0.89003

731

0.89022

154

1.1 0.83587 315

[(-941 1.2 0.86166 128 0.71573 454

[(-;)3] 1.4 0.88219 209 0.76897 769 0.59293 450

[ 1

[(-jP1

(93

(Vi)3

2.0 0.89003 0.78909 0.64073 0.43846 +0.17708

731 505 496 099 802

-0.14800

012

22:: 3. 0

:2 3:6 20” [(-3’s]

r.

I

4.0 0.89022 0.78956 0.64184 0.44095 to.18250

15 60 73 77 43

-0.13652 -0.51809 -0.96348 -1.47349 -2.04858

01 61 97 03 lb

-2.68905 -3.39508 -4.16677 -5.00417 -5.90734 -6.87630 [(j;)g]

52 38 17 86 21 32

WEIERSTRASS TABLE

z=r\c,

1.00

1.05

ELLIPTIC FOR

OBTAINING,

AND RELATED L?‘, @’ AND

(Negative Discriminant-unit A~(:) 1.15 1.3

677

FUNCTIONS I ON Ox AND

Table

Oy

18.2

Real Half-Period)

0.00 0.05 0.10 0.15 0.20

1.00000 0.99998 0.99976 0.99880 0.99622

00 52 37 40 33

1.00000 0.99998 0.99978 0.99893 0.99663

00 68 83 08 32

1.00000 0.99998 0.99983 0.99918 0.99743

00 98 74 15 55

1.00000 0.99999 0.99990 0.99950 0.99845

00 38 10 43 77

1.5 1.00000 0.99999 0.99996 0.99980 0.99940

00 75 06 51 30

2.0 1.00000 1.00000 1.00002 1.00011 1.00038

00 14 30 a3 24

4.0 1.00000 1.00000 1.00004 1.00020 1.00065

00 25 07 71 92

0.25 0.30 0.35 0.40 0.45

0.99079 0.98097 0.96495 0.94070 0.90617

63 a2 11 57 03

0.99182 0.98317 0.96915 0.94811 0.91839

47 67 65 25 70

0.99381 0.98736 0.97703 0.96174 0.94051

16 11 14 61 05

0.99631 0.99255 0.98664 0.97810 0.96656

17 06 20 01 45

0.99860 0.99725 0.99525 0.99255 0.98928

26 51 02 94 71

1.00096 1.00205 1.00396 1.00705 1.01183

01 83 14 13 11

1.00162 1.00340 1.00639 1.01107 1.01805

38 46 11 17 02

0.50 0.55 0.60 0.65 0.70

0.85939 0.79882 0.72356 0.63382 0.53123

83 11 52 07 69

0.87853 0.82744 0.76469 0.69080 0.60756

56 45 39 48 14

0.91254 0.877,14 0.83537 0.78725 0.734'35

55 80 63 05 90

0.95189 0.93426 0.91429 0.89316 0.87276

16 12 23 80 38

0.98573 0.98244 0.98031 0.98063 0.98521

01 30 24 64 20

1.01895 1.02925 1.04381 1.06395 1.09136

42 89 01 05 32

1.02806 1.04202 1.06102 1.08641 1.11984

66 47 67 83 70

0. 75 0.80 0.85 0.90 0.95 1.00

0.41930 0.30366 0.19233 0.09574 0.02666 0.00000

23 33 10 08 27 00

0.51830 0.42820 0.34438 0.27605 0.23446 0.23286

a4 16 12 07 42 11

0.681!,5 0.63143 0.59046 0.56611 0.56753 0.6051>3

50 16 32 51 12 48

0.85577 0.84585 0.84771 0.86731 0.91197 0.99060

68 35 96 78 25 83

0.99643 1.01739 1.05201 1.10523 1.18314 1.29335

13 07 81 21 77 96

1.12815 1.17693 1.24098 1.32440 1.43234 1.57134

05 44 76 72 85 70

1.16333 1.21939 1.29112 1.38242 1.49822 1.64479

76 20 16 38 24 64

[ C-l)“3

zji=?j\o

[C-i)51

1.00

[ ‘-;;)5]

1.05

[C-83)4]

1.15

1.3

[C-83)4]

1.5

[C-i)41

[(-R3)4]

2.0

4.0

0.00 0.05 0.10 0.15 0.20

1.00000 0.99998 0.99976 0.99880 0.99622

00 52 37 40 33

1.00000 0.99998 0.99978 0.99892 0.99658

00 67 76 27 78

1.000(10 0.99998 0.99903 0.99916 0.99734

00 98 59 47 10

1.00000 0.99999 0.99989 0.99948 0.99834

00 37 93 51 96

1.00000 0.99999 0.99995 0.99978 0.99931

00 75 93 96 61

1.00000 1.00000 1.00002 1.00011 1.00034

00 14 24 15 41

1.00000 1.00000 1.00004 1.00020 1.00063

00 32 04 35 88

0.25 0.30 0.35 0.40 0.45

0.99079 0.98097 0.96495 0.94070 0.90617

63 a2 11 57 03

0.99165 0.98266 0.96786 0.94525 0.91264

20 22 42 04 56

0.9934.5 0.98628 0.97423 0.92846

16 83 43 47 67

0.99589 0.99132 0.98354 0.97122 0.95268

95 10 71 41 27

0.99827 0.99626 0.99275 0.98701 0.97806

12 60 81 30 19

1.00081 1.00162 1.00285 1.00459 1.00684

39 14 94 41 49

1.00154 1.00317 1.00580 1.00976 1.01539

61 22 47 35 36

0.50 0. 55 0.60 0.65 0. 70

0.85939

83

0.86784 0.80881

46 13

0.89009 0.83817 0.77024

57 66 24

0.92592 0.88861 0.83812 0.77163

17 10 71 28

0.96465 0.94522 0.91784 0.88019 0.82955

71 83 50 00 45

1.00955 1.01258 1.01563 1.01827 1.01983

92 51 95 41 61

1.02305 1.03311 1.04595 1.06191 1.08136

58 90 22 71 14

0.76286

31

1.01942 1.01585 1.00758 0.99269 0.96882 0.93312 ['-;'I]

61 25 28 39 29 29

1.10461 1.13197 1.16373 1.20012 1.24136 1.28763 [(-yq

36 83 23 24 39 91

0.95576

0.75 0.80 0.85 0.90 0.95 1.00 [‘-;I”]

[p”]

[C-i’“3

[‘-921

[‘-923

4.0

z/i=y\u

1.1

, 3

::1 1.' 1 1.5

I. 6 $z ;: i

2.0

If the real half-period ~1, see 18.2 Homogeneity Relations. Interpolation with respect to ‘6 will, in -general, be diflicult because of the non-uniform subintervals involved. Aitken’s interpolation may be used in this case. As few as 3s may be obtained. For the computation of @, Ip’ or I at z-.r+,!/, an addition formula may be used (18.4 and E&plea 11-12).

_1.39585 ____^ 1.67719 1.85056 2.04521

80 ^^ 97 87 26

1.5Z55Y

tlu

2.26025

62

678

WEIERSTRASS

Table

ELLIPTIC

TABLE

18.2

FOR

AND

OBTAINING

RELATED

P, 9

FUNCTIONS

AND

(Negative Discriminant-Unit

r ON

Ox AND

Oy

Real Half-Period)

3cP(r)

r=r\,, 0.00 0.05 0.10 0.15 0.20

-2.00000 -2.00002 -2.00047 -2.00239 -2.00753

00 95 25 01 43

-2.00000 -2.00002 -2.00042 -2.00212 -2.00667

00 65 27 89 30

-2.00000 -2.00002 -2.00032 -2.00161 -2.00502

00 04 37 92 56

-2.00000 -2.00001 -2.00019 -2.00097 -2.00297

00 24 63 17 32

-2.00000 -2.00000 -2.00007 -2.00037 -2.00110

00 50 74 44 66

2.0 -2.00000 -1.99999 -1.99995 -1.99975 -1.99919

00 71 34 65 66

4.0 -2.00000 -1.99999 -1.99991 -1.99958 -1.99866

00 49 83 21 07

0.25 0.30 0.35 0.40 0.45

-2.01829 -2.03755 -2.06843 -2.11379 -2.17550

41 78 88 74 18

-2.01608 -2.03274 -2.05907 -2.09713 -2.14789

73 55 94 03 87

-2.01196 -2;02197 -2.04247 -2.06835 -2.10148

38 $9 95 37 48

-2.00694 -2.01358 -2.02334 -2.03614 -2.05106

49 73 71 78 10

-2.00246 -2.00448 -2.00696 -2.00922 -2.00992

05 84 68 15 37

-1.99793 -1.99544 -1.99095 -1.98338 -1.97120

23 16 74 63 64

-1.99667 -1.99294 -1.98657 -1.97639 -1.96084

11 36 99 65 72

0.50 0.55 0.60 0.65 0.70

-2.25339 -2.34395 -2.43881 -2.52318 -2.57463

16 53 27 49 40

-2.21047 -2.28098 -2.35140 -2.40840 -2.43241

72 85 73 49 27

-2.14013 -2.18023 -2.21466 -2.23248 -2.21839

46 97 43 50 89

-2.06592 -2.07692 -2.07815 -2.06116 -2.01460

49 41 03 83 73

-2.00685 -1.99665 -1.97452 -1.93392 -1.86620

64 49 31 01 81

-1.95234 -1.92399 -1.88246 -1.82286 -1.73878

05 70 83 83 53

-1.93791 -1.90499 -1.85865 -1.79444 -1.70648

93 42 81 54 76

0.75 0.80 0.85 0.90 0.95 1.00

-2.56240 -2.44770 -2.18496 -1.72414 -1.01321 0.00000

86 16 84 78 01 00

-2.39712 -2.26959 -2.01105 -1.57813 -0.92423 0.00000

18 69 50 99 16 00

-2.15233 -2.00933 -1.75959 -1.36864 -0.79716 0.00000

79 39 77 82 03 00

-1.92378 -1.77031 -1.53168 -1.18057 -0.68374 0.00000

08 11 32 88 39 00

-1.76023 -1;6017a -1.37288 -1.05066 -0.60580 0.00000

25 75 13 42 78 00

-1.62181 -1;46089 -1.24141 -0.94387 -0.54202 0.00000

13 21 08 76 52 00

-1.58702 -1;42574 -1.20881 -0.91751 -0.52632 0.00000

84 81 20 44 04 00

1.00

1.05

1.15

1.5

1.3

[c-y] i/i=?/\cl

1.00

1.05

1.15

1.3

[ (-3

[c-y]

0.00 0.05 0.10 0.15 0.20

-2.00000 -2.00002 -2.00047 -2.00239 -2.00753

00 95 25 01 43

-2.00000 -2;00002 -2.00042 -i:iO216 -2.00685

00 65 55 ii 42

-2.00000 -2.00002 -2.00032 -2.00168 -2.00540

00 05 97 65 32

-2.00000 -2.00001 -2.00020 -2.00104 -2.00340

00 25 30 a7 55

1.5 -2.00000 -2.00000 -2.00008 -2.00043 -2.00145

00 50 28 62 41

2.0 -2.00000 -1.99999 -1.99995 -1.99978 -1.99935

00 72 58 38 00

4.0 -2.00000 -1.99999 -1.99991 -1.99959 -1.99874

00 49 95 6t 22

0.25 0.30 0.35 0.40 0.45

-2.01829 -2.03755 -2.06843 -2.11379 -2.17550

41 78 88 74 18

-2.01677 -2.03479 -2.06420 -2.10841 -2.17036

67 40 40 06 66

-2.01340 -2.02825 -2.05319 -2.09200 -2.14879

12 59 59 85 02

-2.00859 -2.01849 -2.03567 -2.06346 -2.10597

22 50 60 12 25

-2.00378 -2.00844 -2.01691 -2;03134 -2.05462

54 10 87 51 43

-1.99851 -1.99718 -1.99536 -ii99123 -1.99120

75 99 97 08 21

-1.99698 -1.99387 -1.98892 -ii98164 -1.97152

24 40 95 4i 19

0.50 0.55 0.60 0.65 0.70

-2.25339

16

-2.25173 -2.35170

01 68

-2.22747 -2.33108 -2.46061

67 42 76

-2.16805 -2.25504 -2.37230 -2.52442

61 79 39 19

-2.09057 -2.14403 -2.22089 -2.32798 -2.47283

56 61 13 29 02

-1.99006 -1.99107 -1.99605 -2.00760 -2.02919

63 16 96 83 12

-1.95810 -1.94097 -1.91982 -1.89440 -1.86458

18 97 80 95 73

-2.66308

69

-2.06534 -2;12ii7 -2.20596 -2.32643 -2.49375 -2.72008

90 04 83 60 12 43

-1.83032

90

4.0 -1.48398 -1.36337 -1.24144 -1.12345 -1.01509

95 47 17 13 75

-0.92286 -0.85472 -0.82134 -0.83783 -0.92645

21 55 27 54 86

0.75 0; 80 0. a5 0.90 0.95 1.00 [ c-p3

zii=q\n

1.1

::‘3 ::“5 1.6 1. 7 1.8 ::i

[C-i)31

[‘-;‘4]

IIC-i)61

WEIERSTRASS TABLE

z=r\n 0.00 0.05 0.10 0.15 0.20 0.25 0.30

1.00 1.00000 00 1.00000 49 1.00007

88

1.00039 88 1.00125 98 1.00307 33 1.00636

38

0.40 0.45

1.01999

45

0.50 0.55 0.60 0.65

1.04821 1.06990 1.09776

0.35

0.70 0.75 0.80 0. 85 0. 90 0.95

1.00

1.01176 23

ELLIPTIC FOR

OBTAINING

AND

RELATED

9,

P

AND

1.00274 1.00566

09 06

1.02805

07

1.01043 07 1.01767 00

1.00048 81 1.00098 1.00175 1.00285

15 16 61

0.99968 0.99934 0.99875 0.99781 0.99639

98 32 38 57 49

1.00619 1.00840

68 79

0.99432

31

0.97457

57

91

39 26 64 76

1.02543 1.03459 1.04547 1.05796

63 22 13 45

1.22444 09 1.28188 76 1.34648 26

1.13570

79

1.07181

59

1.41726 1.49272

1.23'329

81 88 58

1.16765 25 1.20.248 62

20 42

22

1.271579 52 1.31332 66

1.57079 62

1.08659 1.10165 1.11613 1.12887 1.13842 (-p4

33 80 35 36 65

1.3 1.00000 1.00000 1.00003 1.00017 1.00054 1.00134 1.00281 1.00529 1.00917 1.01496 1.02322 1.03466 1.05006 1.07029

00 21 34 04 31 04 53 28 72 03 84 71 29 97

[ 1

zli=v\n

1 nn

0: oo- 'l.Oooii, 00 0.05 1.00000 49 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

0.50

1.00007

88

1.00039 88 1.00125 98 1.00307 33 38 1.01176 23

1.00636 1.01999

45

I.. 03186 18

1.04821 35

0.55 0.60 0.65 0. 70 0.75 0; 80 0.85 0.90 0. 95 1.00

1.05 1.00000 1.00000 1.00007 1.00035 1.00113

00 44 08 86 51

1.15 1.00000 1.00000 1.00005 1.00027 1.00088

00 34 46 73 05

1.00277

55

1.00216

14

1.00576 1.01069 1.01824 1.02921 1.04444 1.06483

38 02 62 31 39 58

1.00451 03 1.001341 42 1.01445 97 1.02333 32 l.O3!i81 72 1.05;!77 97 1.07515 67

1.00433 47

1.01087 1.01343 1.01581 1.01765 1.01845

54 17 69 94 50

1.01754 1.01408

41 58 73 76

1.00702 0.99506

[‘-;‘“]

[ I

[‘-;‘“I

[‘--;I’]

[‘-;‘“]

0.99139 16 0.98734 37 0.98186 55 0.96501 30 0.95262 09

1.5 1.00000 1.00000 1.00001 1.00006 1.00022

00 08 35 91 22

1.00055

4.0 1.00000 00 0.99999 0.99998 0.99993 0;99978

92 65 12 17

0.99946 0.99888

41 13

0.99640

37

0.99102 0.98667 0.98085 0.97318 0.96328

12 79 06 91 27

0.99791 11 0.99417 86

0.95064

87

0.93672

94

0.93471 88 0.91482 13

0.89105 0.85912

46 29

0.89015 0.85977 0.82253

86 85 59

0.91653 15

43

94 03 67

2.0 1.00000 0.99999 0.99999 0.99996 0.99988

00 95 25 24 28

0.99971 0.99943

90 06

1.00117 1.00225 1.00396 1.00658 1.01042 1.01588 1.02344

41 39 73

0.99897 0.99830 0.99739 0.99619 0.99471 0.99295

1.03369

45

0.99095

77 58

0.98656 0.98447 0.98273 0.98166 0.98165

79 25 54 56 63

42

1.04730 93

[(-;‘“I

41 68 10 89 80

0.98878 64

0.98319 64 [(-i’2]

4.0 1.00000 00 0.99999 0.99998 0.99993 0.99978

92 65 19 57

0.99947 0.99892 0.99802 0.99666 0.99470

96 78 83 50 88

0.99202 0.98845 0.98384 0.97804 0.97088

03 10 63 63 86

0.96221 00 0.95184 0.93964 0.92543 0.90906 0.89040

1:5

::; i-98 2:o

75 06 21 94 57

c(-$)3 I

ZJi=Y\U

::: :*i

18.2

[C-i'"3

1.06508 51

C-216

51

1.00125 79

1.04478 1.06180 1.08258 1.10724

1.08493 1.11454 1.15021

00 95 24 10

1.00256

54

27 77 02 52

1.04227 15 1.06102 21

Table

Oy

1.00208 94

1.00467 1.00779 1.01217 1.01799

1.13248 70 1.17462 06

0.x AND

1.00429

32 74 50 19

35 78 14

f ON

(Negative Discriminant-Unit Real Half-Period) :r(:) 1.05 1.15 1.3 1.5 2.0 1.00000 00 1.00000 00 1.00000 00 1.00000 00 1.00000 1;00000 44 1;00000 34 1;00000 21 1.00000 08 0.99999 1.00003 31 1.00001 32 0.99999 1.00007 06 1.00005 43 1.00035 70 1.00027 40 1.00016 65 1.00006 60 0.99996 1.00112 60 1.00086 16 1.00052 15 1.00020 48 0.99987 1.00787 1.01325 1.02090 1.03127

1.03186 18

679

FUNCTIONS

4.0 0.84561 0.79003 0.72274 0.64295 0.55003

98 67 36 89 38

0.44345 0.32282 0.18790 +0.03858 -0.12508

14 70 92 90 40

WJGIERSTRASS

680 ‘I’ulh! n=d/i

18.3 I/?

ELLIPTIC

AND

FUNCTIONS

RELATED

1NVARlANTS AND VALllES AT HALF-PERIODS (Non-Negative Discriminant-Unit Real Half-Period) ‘:(=[P(w’) q=r(l) (‘1= P(l) II3 0.00000 000 1.71079 64 -1.71079 64 0.70539 0.55310 992 1.71005 96 -1.66130 15 0.70979 1.03405 699 1.70235 77 -1.60703 69 0.79367 1.45404 521 1.69556 79 -1.55707 59 0.79700 1.02151 090 1.60950 10 -1.51123 63 0.00009 2.14201 000 1.60430 41 -1.46767 03 0.00274 2.42241 937 1.67965 00 -1.42690 19 0.00507 2.66790 153 1.67554 00 -1.30094 40 0.00713 2.00320 000 1.67193 04 -1.35330 12 0.00095 3.07195 910 1.66074 05 -1.32011 96 0.01054

n’/i=r(u’)/i -0.70539 -0.76520 -0.74537 -0.72500 -0.70669

02 32 75 50 61

-0.60777 -0.66910 -0.65066 -0.63241 -0.61434

92 00 09 30 79

906 160 717 299 453

-0.59644 -0.57069 -0; 56106 -0.54356 -0.52616

54 03 78 50 97

0.01606 0.01752 0.01011 0.01062 0.01907 0.01947 0.01903 0.02014 0.02041 0.02066 0.02007 0.02106 0.02122 0.02137 0.02150

533 732 103 572 950 977 269 309 031 031 370 i$i 707 423 329

-0.50007 -0.49166 -0.47452 -0.45746 -0.44046 -0.42352 -0.40663 -0.30970 -0.37290 -0.35621

14 03 75 53 65 46 39 91 56 91

0.02161 0.02104 0.02201 0.02213 0.02222

711 620 364 509 516

-0.33940 -0.32270 -0.30610 -0.20945 -0.27202 -0.25620 -0.21475 -0.17337 -0.13205 -0.09079

50 22 4 55 11 90 00 32 05 10

0.02229 0.02233 0.02237 0.02239 0.02241 0.02243 0.02244 0.02244 0.02245 0.02245 0.02246 0.02246 0.02246 0.02246 0.02246 0.02246 0.02246 0.02246 0.02246 0.02246 0.02246 0.02246 0.02246 0.02246 0.02246 0.02246 0.02246 0.02246 [c-y]

030 000 201 020 676 032 022 745 274 659 146 406 546 619 659 600 691 690 701 702 703 703 703 704 704 704 704 704

-0.04955 -0.00035 +O. 03203 0.07400 0.11515

91 41 07 01 00

0.15630 0.19745 0.23050 0.27972 0.32005

73 01 01 23 30

0.40311 0.40536 0.56761 0.64906 0.73211 0.01435 0.09660 0.97005 1.06109 1.14334 1.22559 1.30703 1.39000 1.47233 1.55457 1.63602 1.71907

12 30 39 24 01 74 44 13 01 40 16 03 50 17 04 51 10

1. 00 1.02 1.04 1. 06 1. 00

11.01704 11.37372 10.90419 10.64177 10.34065

500 304 107

1.10 1.12 1.14 1.16 1.10 1.20 1.22 1.24 1.26 1.20 1.30 1.32 1.34 1.36 1.30 1.40 1.42 1.44 1.46 1.40 1.50 1.52 1. 54 1.56 1. 50

10.07577 9.04269 9.63754 9.45693 9.29709

364 105 049 072 413

9.15702 9.03445 0.92575 0.02999 0.74560

051 117 043 055 130

3.23761 3.30300 3.51000 3.62320 3.72197

717 317 223 977 756

1.66592 1.66344 1.66126 1.65933 1.65763

77 74 03 17 09

-1.20900 -1.25900 -1.23262 -1.20710 -1.10320

20 23 55 65 95

0.01195 0.01320 0.01429 0.01526 0.01611

0.67123 0.60560 0.54791 0.49690 0.45209

169 620 374 090 746

3.00005 3.00529 3.95256 4.01170 4.06392

265 056 351 462 070

1.65613 1.65400 1.65364 1.65261 1.65170

11 06 22 37 67

-1.16002 -1.13905 -1.12021 -1.10100 -1.00454

70 91 33 31 05

0.41252 0.37763 0.34607 0.31975 0.29503 0.27475 0.25616 0.23977 0.22531 0.21257

263 305 203 220 997 500 404 191 604 036

4.10905 4.15029 4.10593 4.21732 4.24490 4.26936 4.29004 4.30970 4.32647 4.34119

014 019 045 430 720 502 965 602 752 120

1.65090 1.65020 1.64957 1.64903 1.64054 1.64012 1.64774 1.64741 1.64711 1.64606

60 13 92 06 60 02 39 20 94 13

-1.06037 -1.05321 -1.03099 -1.02566 -1.01316 -1.00144 -0.99044 -0.90012 -0.97045 -0.96137

47 20 50 55 45 04 37 04 19 37

1.60 1.65 1.70 1.75 1.00 1.05 1.90 1.95 2.00 2. 05 2.10 2.15 2.20 2.25 2.30

0.20133 0.17070 0.16217 0.15011 0.14129 0.13406 0.13016 0.12672 0.12421 0.12230 0.12104 0.12007 0.11935 0.11003 0.11045 0.11797 0.11771 0.11750 0.11750 0.11746 0.11744 0.11743 0.11743 0.11742 0.11742 0.11742 0.11742 0.11742 0.11742 0.11742 0.11742 0.11742 0.11742

033 300 907 147 012 127 001 634 044 671 003 164 791 660 503 459 705 007 702 004 004 694 103 707 619 529 401 455 441 434 430 426 426

4.35416 4.30026 4.39931 4.41322 4.42337 4.43079 4.43620 4.44016 4.44305 4.44516 4.44670 4.44702 4.44064 4.44924 4.44960 4.45024 4.45053 4.45069 4.45077 4.45002 4.45004 4.45006 4.45006 4.45007 4.45007 4.45007 4.45007 4.45007 4.45007 4.45007

210 291 441 294 010 360 096 375 205 152 219 746 934 963 000 222 705 555 969 457 052 130 011 174 360 472 520 556 572 501

1.64663 1.64617 1.64504 1.64559 1.64541 1.64520 1.64519 1.64512 1.64507 1.64503 1.64500 1.64490 1.64497 1.64496 1.64495 1.64494 1.64494 1.64493 1.64493 1.64493

30 54 00 63 70 73 21 25 17 45 74 76 32 26 49 51 00 71 57 49

1.64493 1.64493 1.64493 1.64493 1.64493

45 43 42 41 41

-0.95205 64 -0.93379 17 -0.91752 00 -0.90365 10 -0.09100'0? -0.00169 76 -0.07306 52 -0.06569 37 -0.05939 02 -0.05402 10 -0.04942 70 -0.04550 41 -0.04215 20 -0.03920 00 -0; 03604 11 -0.03296 37 -0.03013 20 -0.02006 54 -0; 02655 50 -0.02545 33 -0.02464 01 -0.02406 01 -0.02363 06 -0.02331 60 -0.02300 70

1.64493 1.64493 1.64493 1.64493 1.64493 1.64493 1.64493

41 41 41 41 41 41 41

-0.02292 -0.02279 -0.02270 -0.02264 -0.02259 -0.02256 -0.02253

,j2=d,

q3=U,

q=&‘/Z,

2; 2: 6 22:; :-z 3: 1 ::3'

Forn=l:

347

794

4.45007 505 4.45007 507 4.45007 590

cc-p ,- 1

04 02 09 37 61 13 59

1.64493 41 -0.02246 70 [(-j"] [c-y] ^ . I?:,=-02/2,~=*‘4, n’jr=-r/4.

016 710 192 535 279 203 017 637 045 949

[95 I

For a=-: 92-r4/12, 9s-(i/216, c, =r2/6, ‘;I = -+/12, v&/12, v’/;= m. (w= 1.85407 4677 is the real half-period in the Lemniscatic case 18.14.) -, .to obtain V’ use Legendre’s relation q’ = W’ -+/2. -For 4
WEIERSTRASS

ELtLIPTIC

INVARIANT!5

AND

AND RELATED VALUES

1. 02 1. 04 1. 06 1.08 1.10 1.12 1.14 1.16 1.18 1.20 1.22 1. 24 1.26 1.28

(Non-Negative Discriminant-Unit o(w’), i 0.949i9 88 0.949899 0.95114 80 0.967481 0.95224 92 0.984884 0.95321 98 1.002097 0.95407 54 1.019107 0.95482 97 1.035904 0.95549 47 1.052476 0.95608 10 1.068811 0.95659 79 1.084899 0.95705 36 1.100727 0.95745 55 1.116285 0.95780 98 1.131562 0.95812 22 1.146546 0.95839 77 1.161227 0.95864 07 1.175594

1.30 1.32 1.34 1.36 1.38 1.40 1.42 1.44 1.46 1.48

0.95885 0.95904 0.95921 0.95935 0.95948 0.95960 0.95970 0.95979 0.95986 0.95993

49 38 04 73 68 10 18 06 89 80

1.189636 1.203344 1.216707 1.229716 1.242361

1.50 :* :: 1: 56 1.58

90 27 01 19 87 13 67 45 94 49 35 71 70 43 96 35 63 84 99 10 24 31 35 37 38 39 40 40 40 40 40 40 40 40 40 40 40

1.310087 1.319941 1.329364 :: EE 1.354990 1.373224 1.388539 1.400869 1.410170 1.416408 1.419573 1.419665 1.416707 1.410733 1.401800 1.389977 1.375349 1.358018 1.338098 1.291016 1.235264 1.172151 1.103091 1.029557

::"5 33'; 31 a 3.9 4.0

0.95999 0.96005 0.96010 0.96014 0.96017 0.96021 0.96027 0.96032 0.96035 0.96038 0.96040 0.96041 0.96042 0.96043 0.96043 0.96044 0.96044 0.96044 0.96044 0.96045 0.96045 0.96045 0.96045 0.96045 0.96045 0.96045 0.96045 0.96045 0.96045 0.96045 0.96045 0.96045 0.96045 0.96045 0.96045 0.96045 0.96045

2=*-o

0.96045 40

0.000000

1.50

:-66; 1: 70 1.75 1.80 1.85 1.90 1.95 2. 00 2. 05 2.10 2.15 2.20 2. 25 2.30 ::z z 2: a 2.9 ::1" :::

1.254633 1.266522 1.278021 1.289120 1.299811

0.953025 0.874937 0.796655 0.719428 0.644360 0.572395 0.504299 0.440663 0.381903 0.328268 0.279851 0.236623

681

FUNCTIONS

AT HALF-PERIODS

Real Half-Period) m+Jz) 1.182951 1.170397 1.157316 1.143695 1.129522 1.114782 1.099457 1.083531 1.066989 1.049814 1.031991 1.013507 0.994349 0.974506 0.953970 0.932733 0.910790 0.888138 0.864776 0.840704 0.815927 0.790449 0.764278 0.737425 0.709900 0.681719 0.652896 0.623452 0.593404 0.562777 0.531593 0.451372 0.368286 0.282840 0.195588 0.107125 +o. 018074 -0.070918 -0.159199 -0.246114 -0.331019 -0.413290 -0.492330 -0.567579 -0.638522 -0.765682 -0.870782 -0.951807 -1.007808 -1.038896 -1.046157 -1.031530 -0.997636 -0.947586 -0.884775 -0.812687 -0.734720 -0.654024 -0.573398 -0.495196 -0.421291 -0.353075 0.000000 (-i)3

[ 1

Table 18.3 y44 1.182951 1.218650 1.253864 1.288619 1.322935 1.356827 1.390301 1.423362 1.456007 1.488231 1.520022 1.551369 1.582254 1.612657 1.642557 1.671930 1.700750 1.728989 1.756618 1.783607 1.809925 1.835542 1.860425 1.884541 1.907860 1.930348 1.951974 1.972707 1.992515 2.011370 2.029242 2.069439 2.102914 2.129313 2.148344 2.159783 _~~ .~. 2.163478 2.159353 2.147412 2.127732 2.100473 2.065864 2.024211 1.975882 1.921308 1.795415 1.650936 1.492779 1.326086 1.155967 0.987255 0.824296 0.670787 0.529666 0.403050 0.292246 0.197780 0.119493 0.056643 +o. 008033 -0.027857 -0.052740 0.000000

cc-y 1

w2=l+o', ez=(P (l+J')=-(e,+e$, 12=I(l+w')=1+1'. For a= 1: u(1) =eR/821/4,$ ~(a’) =i+l), +J =$@$#‘4/,. For a= oD. #(I) = !&~*‘*4+r, U(J) = 0, u(~*) =o (~=1.854& 4677 is the real half-period in-the Lemniscatic case 18.14.) To obtain the corresponding values of tabulated quantities when the real half-period 0~1, multiply g by w

WEIERSTRASS

682 Table

INVARIANTS

18.3

a=o;/i

Q2

1.08 1.10 1.12 1.14 1.16 1.18

-37.48749 -35.54027 -33.62168 -31.73930 -29.89938

12 17 02 91 64

1.20 1.22 1.24

45 62

1.26

-28.10693 -26.36591 -24.67936 -23.04950

1.28

-21.47786

60

-19.96535 -18.51237 -17.11886 -15.78441 -14.50828

52

:*'3; 1:34 1.36 1.38

lb 71 82 67

1.40 1.42 1.44 1.46 1.48

-13.28947 -12.12676 -11.01876

27 19 70

-

9.96396 8.96072

40 32

21.80880 21.28756 20.74000 20.17372 19.59530

1.50 1.52 1.54 1.56

-

8.b0733 7.10204 6.24304 5.42853

71 36 63 20

19 30 84 53

58 83

1.58

- 4.65668

53

1.60 1.65

- 3.92570 - 2.26537

12 64

1.70

- 0.82241

58

1.75 1.80

+ 0.42844

48

1.51045

44

2.44471 3.25015

18 81

2.00 2.05

3.94365 4.54009 5.05259

25 85 79

2.10 2.15 2.20 2.25 2.30

5.49261 5.87014 6.19388 6.47134 6.70905

76 05 49 42

2.4

7.08692 7.36377 7.56643 7.71470 7.82312

59 30 61 39 83

7.90239 7.96032 8.00265 8.03358

07

8.05618

01

1.85 1.90 1.95

::2 ::; ;:: :: 3:s

AND

0.00000 00

00

1.06

AND RELATED VALUES

AT

FUNCTIONS

HALF-PERIOD>

(Non-Positive Discriminant-Unit Real Half-Period) .RPe,= <(If?,= g+;) E?(;-;) 112=<(l) 93

-47.26818 -45.35272 -43.40071 -41.42954 -39.45420

1. 00

1.02 1.04

ELLIPTIC

57

11 32 32

0.00000 -0.04867 -0.09452

000

1.57079 1.53091

63

1.49282

30

39

1.45647

40

1.42184

87 01

1.38885

99

29 69 68

00 58 28

-0.13769

202

3.22711

14.25448

26

-0.17834

547

3.15578

16.56680 18.47603

99 08

-0.21662

576 894

3.08425 3.01273

89 84

315 915

2.94140 2.87040

17 90

-0.25266

22.20294

45

-0.28660 -0.31854 -0.34862

22.90208 23.38397 23.67693

34 82 85

-0.37692 -0.40356 -0.42863

23.80660 23.79610

45 09

23.66620

08

23.43548 23.12052 22.73602 22.29496

95 98

20.02550

17

21.25543

82

29 60

22 31

810

3.43759 3.36827 3.29802

4.41906 8.23156 11.49257

083

086

2.79990

29

571

2.73000

96

-0.45222 -0.47442

512 481 513 139

2.66084 2.59249 2.52505 2.45859

-0.49530 -0.51494 -0.53342 -0.55081

414 941 897 058

2.39318 2.32886

49

2.26569 2.20369

11 72

-0.56715

817

2.14291

32

209

2.08336 2.02506

24 27 64

-0.58253 -0.59698

926

81

-0.61058 -0.62336

339 513

70

-0.63538

226

19.01038 18.42378 17.83959 17.26123 lb.69159

59 52 12 98 27

-0.64667 -0.65730

980 023

-0.66728 -0.67666 -0.68548

357

16.13300 14.79653 13.56033 12.43388 11.41927

57

36

14

-1.57079 -1.58005

-1.58905 -1.59772 -1.60600

63 81 67 52 53

1.35748

74

1.32766 1.29935 1.27247

96 18 81

-1.61384 -1.62120 -1.62804 -1.63434 -1.64006

1.24699 1.22283 1.19995 1.17830 1.15780

24 82 95 09 77

-1.64520 -1.64973 -1.65364 -1.65693 -1.65959

1.13842 1.12010 1.10279

65

52 31

-1.66lb3 -1.66305 -1.66384

1.08644 1.07100

09

-1.66403

10

-1.66361

75 bl 43 14 83

-1.66259 -1.66099 -1.65881 -1.65608 -1.65280

76 36 21

26

1.96802 1.91226 1.85777

13 09

1.05642 1.04267 1.02970 1.01747 1.00593

1.80455 1.75261 1.70192

50 00 94

0.99506 0.98482 0.97517

68 68

93 46 85 18 00 28 36 88 82 38

99 31 13 42 26

85 44

40

751 761

1.65250

41

0.96608

09

1.60432

26

0.95751

90

-1.64899 -1.64466 -1.63982 -1.63450 -1.62871

1.55737 1.44527 1.34049 1.24271 1.15159

16

0.94945 0.93130 0.91571 0.90232 0.89084

69 88 53 74 07

-1.62246 -1.60493 -1.58487 -1.56251 -1.53807

17 31 67 97 94

l.Obb78 0.98792 0.91466

0.88099 0.87254 0.86531

10 91

-1.51175 -1.48374 -1.45422 -1.42334 -1.39126

93 94 51 69 17

-1.35810 -1.32398

23 93

-1.28903 -1.25332

05 31

77 94 28

-0.75360

734 375 198 441 961

10.51370 9.71138 9.00473 8.38537 7.84470

92 21 54 94 38

-0.76358 -0.77212 -0.77942 -0.78567 -0.79101

973 691 883 351 353

7.37428 6.96611 6.61278

09 56 90

-0.79557 -0.79948 -0.80282

957 352 119

6.30752

86

-0.80567

458

23

-0.69377 -0.71238 -0.72831 -0.74194

07 39 44 58

63

$/i=f(w$/i

36

21 21 40 48 73

0.84665

65 46

0.85912

0.78355

46

0.85382

0.72504 0.67080

25 91

0.62056 0.57401

06 95

6.04422

78

-0.80811

383

0.53092

40

5.62231

14

-0.81198

137

0.45410

32

5.31058 5.08099 4.91228 4.78851

54 59 49 39

-0.81480 -0.81687 -0.81837 -0.81948

718 lb7 985 158

0.38831 0.33200 0.28383

56 75 23

0.24262

75

4.69782

05

4.63142

26

636 422 361 725

0.20739 0.17726 0.15151 0.12949

21 58 09 50

67

29 00

0.84928

11

0.84539 0.84207 0.83923 0.83679

69 37 09 93

0.83294 0.83012 0.82805

16 09 92

0.82655 0.82545

25 16

0.82464

72 96

13 08 76 65

-1.21695

43

-1.14253 -1.06629 -0.98863 -0.90990 -0.83032

28 03 87 09 82

4.58284 4.54731

25 53

-0.82028 -0.82087 -0.82130 -0.82161

03

-0.66941 -0.58833

4.52134

25

-0.82184

634

0.11067

62

0.82331 0.82308

67 77

-0.42540

32

93 72 14 62

368

0.09459

0.82259

-0.26179 -0.17984 -0.09781 -0.01572

06 10

65

04 82 89 37 61

33

0.08084 0.06909 0.05904 0.05046

0.82292 0.82279 0.82270 0.82264

-0.34366

590 517 038 800

10 29 25 97

0.82405 0.82363

-0.75011

58

39 07

-0.50697

92

::z

8.07268 8.08474

80 69

2; 3:

a

8.09355 8.09999 8.10469

57 01 00

4.50235 4.48848 4.47835 4.47094 4.46553

65

-0.82201 -0.82213 -0.82222 -0.82229 -0.82233

3.9 4. 0

8.10812 8.11063

30 05

4.46158 4.45869

47 80

-0.82237 -0.82239

279 820

08

0.82256

13

+O. 06639

64

0.03686

13

0.82253

59

+0.14855

08

co

8.11742

43

4.45087

59

-0.82246

703

0.00000

00

0.82246

70

A=0

J

(7)3

Fora=l:q2=-4d,g3=U,

1

[(-;)8] &,=U,

[(-;)4] ^-

0.04313

[‘-;“I

[(-;)“I

[T-:)3]

Ye,=~~,?~=~l/2,?;/i=-rr,2.

g2=?r4/12, c7s=i+/216, G?e,= -7+2/12, Se, =o, 7,=+/12, $/;=-. (m= 1.8540’7 4677 is the real half-period in the Lemniscatic case 18.14.) For 4
91 75

WEIERSTRASS

ELLIPTIC

INVARIANTS (Non-Positive

AND

AND

RELATED

VALUES

AT HALF-PERIODS

Discriminant-Unit

683

FUNCTIONS

Table

18.3

Real Half-Period)

u(cq/i

R+‘)

1.00

1.02 1.04 1.06 1. 08

1.18295 1.17091 1.15940 1.14841 1.13793

13 79 62 45 68

1.182951 1.219157 1.255842 1.292964 1.330480

0.474949 0.475654 0.476433 0.417275 0.478169

Su(w’) 0.474949 0.483826 0.492792 0.501851 0.511006

1.10 1.12 1.14 1.16 1.18

1.12796 1.11848 1.10948 1.10094 1.09285

39 38 26 49 44

1.368342 1.406502 1.444910 1.483513 1.522257

0.479107 0.480078 0.481074 0.482085 0.483104

0.520259 0.529611 0.539064 0.548616 0.558268

1.20 1.22 1.24 1.26 1.28

1.08519 1.07794 1.07109 1.06461 1.05850

40 61 31 72 11

1.561089 1.599952 1.638790 1.677548 1.716167

0.484122 0.485132 0.486126 0.487098 0.488041

0.568019 0.577866 0.587809 0.597843 0.607968

1.30 1.32 1.34 1.36 1.38

1.05272 1.04727 1.04214 1.03729 1.03272

75 97 12 63 96

1.754591 1.792765 1.830630 1.868133 1.905218

0.488949 0.489817 0.490639 0.491410 0.492126

0.618179 0.620474 0.638850 0.649302 0.659828

1.40 1.42 1.44 1.46 1.48

1.02842 1.02437 1.02055 1.01696 1.01357

64 26 48 00 57

1.941832 1.977922 2.013437 2.048327 2.082544

0.492783 0.493376 0.493902 0.494357 0.494739

0.670422 0.681082 0.691804 0.702582 0.713414

1.50 1.52 1.54 1.56 1.58

1.01039 1.00739 1.00457 1.00191 0.99942

05 28 23 88 27

2.116040 2.148771 2.180693 2.211766 2.241950

0.495045 0.495272 0.495418 0.495480 0.495458

0.724295 0.735221 0.746189 0.757192 0.76a229

1.60 1.65 1.70 1.15 1.80

0.99707 0.99179 0.98727 0.98340 0.98008

51 98 79 36 56

2.271208 2.340071 2.402437 2.457895 2.506120

0.495348 0.494687 0.493456 0.491645 0.489246

0.779295 0.807059 0.834917 0.862812 0.890687

1.85 1.90 1.95 2.00 2.05

0.97724 0.97481 0.97273 0.97095 0.96943

49 36 30 31 05

2.546866 2.579972 2.605345 2.622973 2.632902

0.486255 0.482673 0.478503 2 E,"

0.918490 0.946170 0.973680 1.000975 1.028011

2.10 2.15 2.20 2.25 2.30

0.96812 0.96701 0.96606 0.96524 0.96455

82 46 23 80 19

2.635245 2.630169 2.617892 2.598678 2.572828

0.462516 0.456054 0.449041 0.441488 0.433405

1.054750 1.081151 1.107179 1.132799 1.157978

2.4 2.5 2.6

0.96344 0.96264 0.96205 0.96162 0.96130

79 13 18 12 65

2.502604 2.410244 2.299090 2.172666 2.034544

0.415693 0.395997 0.374417 0.351055 0.326022

1.206881 1.253647 1.298044 1.339858 1.378884

0.96107 0.96090 0.96078 0.96069 0.96063

67 89 62 67 12

1.888235 1.737097 1.584242 1.432486 1.284291

0.299435 0.271420 0.242114 0.211664 0.180224

1.414929 1.447812 1.477367 1.503441 1.525899

0.96058 0.96054 0.96052 0.96050 0.96049

34 86 31 44 08

1.141740 1.006520 0.879924 0.762869 0.655914

0.147962 0.115052 0.081678 0.048028 +0.014297

1.544621 1.559512 1.570495 1.57751a 1.580552

0.96048 0.96047

09 37

0.559298 0.472982

-0.019318 -0.052618

1.579595 1.574671

0.96045

40

0.000000 (93

0.000000

0.000000 (95

;:7:

&;+$

, rel="nofollow">:{=,p f+$ +, ( >

g~~i:

.(i)d+,

=:

.(1)=2&4/r,

[

/,,=a’

+;)=b(i),

++o,

(1) = -2Be,,

1 f-f

~(W,)=pXIRp’~/4/21i4~.

[. 1 -(-;I2

(

;+f

)

[

=f (v2+ni).

+‘)=o.

(w=1.85407 4677 is the real half-period in the Lemniscatic case 18.14.) To obtain the corresponding values of tabulated quantities when the real half-period wzfl, multiply c by 02.

1

19.

F’arabolic

Cylinder

J. C. P. MILLER

Functions l

Contents Page Mathematical Properties . . . . . . . . . . . . . . . . . . . . 19.1. The Pare.bolic Cylinder Functions, Introductory . . . . . The

Equation

G-

6i6 686

($z”+a)y-0

19.2 to 19.6.

Power Series, Standard Solutions, Wronskian and Other Relations, Integral Representations, Recurrence Relations . . . . . . . . . . , . . . . . . . . . . . 19.7 to 19.11. Asymptotic Expansions . . . . , . . . . . . . . 19.12 to 19.15. Connections With Other Functions . . . . . . The

Equation

686 689 691

g+(as2---a)y=0

19.16 to 19.19.

Power Series, Standard Solutions, Wronskian and Other Relations, Integral Representations . . . . . . . . 19.20 to 19.24. Asymptotic Expansions . . . . . . . . . . . . 19.25. Connections With Other Functions . . . . . . . . . . . 19.26. Zeros . . . . . . . . . . . . . . . . . . . . . . . . 19.27. Bessel Functions of Order & 2, + $ as Parabolic Cylinder Functions . . . . . . . . . . . . . . . . . . . . Numerical 19.28. References Table

Table

Methods

. . . . . . . . . . . . . . . . . . . . . .

Use and Extension of the Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19.2.

W(a,

&x)

Table

19.3.

Auxiliary

697 697 697

700

. . . . * . * . . . .

702

(0<2<5). . . . . . . . . . . . * . . . x=0(.1)5, 4-5D or S

712

19.1. U(a, 2) and V(a, x) &u=o(.l)l(.5)a; x=0(.1)5,

fa=0(.1)1(1)5;

692 693 695 696

Functions

(O$c15) 5s

. . . . . . . . . . . . , . . . .

~720

The author acknowledges permission from H.M. Stationery Office to draw freely from [19.11] the material in the introduction, and the tabular values of W(a, 5) for a= -51(1)5, +2=0(.1)5. Other tables of W(a, z) and the tables of U(a, 2) and V(a, z) were prepared on EDSAC 2 at the University Mathematical Laboratory, Cambridge, England, using a program prepared by Miss Joan Walsh for solution of general second order linear homogeneous differential equations with quadratic polynomial coefiicients. The auxiliary tables ‘were prepared at the Computation Laboratory of the National Bureau of Standards.

1 The University Mathematical Laboratory, contract with the National Bureau of Standards.)

Cambridge,

England.

(Prepared 685

u/rider

19. Parabolic

Cylinder

Mathematical 19.1.

The

Parabolic

Cylinder

Functions

Functions

Properties 19.2.1

Introductory

These are solutions d* a+y

19.1.1

of the differential

equation =e -L”(1-t(a++)

(ax2+~x+c)Y=o

with two real and distinct

standard forms

d2y &*--

($x”+a)y=o

19.1.3

g+

(4x2-a)

y=o

The functions Y(-a,

ix)

y(-a,

y(--ia,

xefir)

y(--ia,

-xefi*)

Y( ia, -xe-ftw)

y(ia, xe-ii”)

Both variable x and the parameter a may take on general complex values in this section and in many subsequent sections. Practical applications appear to be confined to real solutions of real equations; therefore attention is confined to such solutions, and, in general, formulas are given for the two equations 19.1.2 and 19.1.3 independently. The principal computational consequence of the remarks above is that reflection in the y-axis produces an independent solution in almost all cases (Hermite functions provide an exception), SO that tables may be confined either to positive x or to a single solution of 19.1.2 or 19.1.3. The

Equation 19.2.

Power

Even and odd solutions 686

(u-3) b-9)

$+

. * .}

$+ c.

... 1

$+

... 1

$; +x2)

=e~x2M(-&a++,

4, --$X2)

=e fz2(1+

;+

(a-+)

dA dx2-(i Series

x2+a)

8, +x2)

-ix)

are all solutions either of 19.1.2 or of 19.1.3 if any one is such a solution. Replacement of a by --ia and x by xefir converts 19.1.2 into 19.1.3. If y(a, x) is a solution of 19.1.2, then 19.1.3 has solutions: *

. . .}

=e-fz2,F~(+a+j-;

y,=xe-fz2M(+a+2, Y(U, -2)

19.1.5

$+

19.2.3

19.1.4 2)

(uf8)

19.2.2

19.1.2

da,

$+(a++)

YE0

=e -fz2 x+(a++)$+(a+*)(afG) 1 19.2.4

=xdZ2M(-~a+~,

Q, -4x”)

=e fZ2 z+(a-+g {

;;+(a-@(a-g)

these series being convergent for all values of x (see chapter 13 for M(a, c, 2)). Alternatively, 19.2.5 ?h=1+a

$+(az+i)

$+(as+S

u) $

+(a4+lla3+~)~+(a~+25a3+~

a)$+

19.2.6 .=x+a$+(a3+3g+(a3++);

+(a4+17a’+~)$+(a~+35a3+5+a)~+

in which non-zero connected by

. . .

coefficients

a, of x”/n!

in x

of 19.1.2 are given by

. . .

19.2.7

a,,+z=a

. an+;

n(n-1)

an-2

are

PARABOLIC

19.3.

Standard

CYLINDER

$87

FUNCTIONS

19.4.4

Solutions

These have been chosen to have the asymptlltic behavior exhibited in 19.8. The first is Whittaker’s function [19.8, 19.91 in a more symmetrical notation.

r($-+u)

19.3.1

- r(f-+uj

u(a,2)=D~,~~(2)=COSa(~+~a)~Y~

cos n(++$u) &2&-f

sin n($++u).Y,

yI=2

=U(u, 2) +U(u, -c-x) 19.4.5

sin?r($++a) J;;2w:

y2=2

=U(u,

.Y,

-sina(~+&z)

cos a($++z)-Yz 2) -U(u,

y>

19.4.6

19.3.2

V(a, 5)=&

$zrU(--a,

{sin7r($++u).Y1

2

r(g+u) (e-i”‘@-t’U(u,

scos n(~+$u)*:F~j

in which

iiz)=

*z) +ei”‘+f’U(u,

rix) }

19.4.7

l9 3 3 . .

y

19.3.4

Yz=&

set 7r($+$u)

=’ r(t-34 1 J;; 2wt

?A=fi 21a+tr(f+4u) csc n(;++a)

“;*:2”’

2~a-t~(5+3u)

yz=fi

y1

&TrU(u,

xtz)=

yz

19.3.5

19.5.

19.4.

4% u’(“‘o)=-2*=-,r(a+~u)

r (B-4 e+

S =2nieSBe-Q2(Z+t)a-4d

2@+t sin?r(z-$3)

r($-*u)

r(3-4

19.5.2

V’(u, O)=

2@+t sina(+--$2) -

r($-+u)

In terms of the more familiar D,(X) of Whittaker,

(I ers-lsaSa-td,y I

U(u, Z)=F

19.5.1

19.3.6

19.3.7

Representations

A full treatment is given in [19.11] section 4. Representations are given here for U(u, z) others may be derived by useof the relations

fi U(“F0)=2#a+tr(p+$u)

V(u, o>=

Integral

tza

where (Yand p are the contours shown in Figares 19.1 and 19.2. When a++ is a positive integer these integrals become indeterminate; in this case

U(u, 2)=D-a-*(2)

U(u, z)=&

19.5.3

19.3.8

V(U,X)=~ r(++u){sin 19.4.

Wronskian

19.4.1

W{U,

m

S

e -pa

e-zs-+8zsa-+dsi

0

~u.D_,-~(~)+D_,-~(-:~>}

and

Other

Relations

S

V) =@Yr

t

PLANE

PLAN4

a c

19.4.2

rV(a, 2) = r (++a) {sin TU.U(U,

P

L

S> +U(a,

-2)

} --Z L

19.4.3 I

ua+w(a,

4=

7r see* *a { V(u,

-2)

FIGURE19.1 -sin 7mV(a,

2) ]

-*<wz

a<*

FIGURE19.2 --r<arg (z+O>r

~

PARABOLIC

688 19.5.4

U(a, z)=~-&

efr’

FUNCTIONS

19.5.9

--IJ+~sls-a-~ds

U(a, 4

=iIyf-+a>

S

,ca-ud

=Jz;;;

19.5.5

(e

s

CYLINDER

efz2 (3 ezs+@‘S-a%?S

Sml)

3ze-fzaf(l+t)-f”-f(r-t)~-fdt

2wtr 19.5.10

=iIy*-$a)

19.5.6

21W~

where E, e3 and e4 are shown in Figures

19.3 and

19.4. s

PLANE

S

e-“(tz”+v)-‘“-‘(tz”-v)f”-‘dv

‘11

The contour cl is such that (+z2+v) goes from coevfr to mei* while v= is2 is not encircled ; ($z2-v)+-t has its principal value except possibly in the immediate neighborhood of the branch-point when encirclement is being avoided. Likewise ql is such that (a~“-v) goes from me*, of v= -2~” is simito me -** while encirclement larly avoided. The contours (11) and (sl) may be obtained from cl and qI by use of the substitution v=$z”t.

The expressions 19.5.7 and 19.5.8 become indeterminate when a=~, 8, J+, . . .; for these values FIGURE

19.3 s
FIGURE

-t*<arg

On ea I*<arg On r, --:r<arg

19.5.11

19.4 a<;s a<-**

rl

19.5.7 ;r(a, 2)=

Iyq-+>

2$a+tr;

S(fl)

19.5.12 =W--;;‘S,,

~2e~(~22+v)ta-f(~z2-v)-*a--1dv

U(a, ~)=~(~l+~~)

Barnee-Type

U(a, 2)=x

e-b2 + 2

S

+mf r (4 r @+a--2.4

-mrl

V(a, z) =,/;

2

za-f_+.;

19.6. Recurrence

tGzJ28ds

(larg 4
r #+a>

U’(a,z)+&S7(a,a$+(a+~)U(a+l,z)=O

19.6.2

??(a, z)-$ST(a,

19.6.3

2U’(a,

19.6.4

zU(a, z)-U(a-1,

z)+U(a-1,

r (‘) $--;2s)

(Jzz)**

z)+U(a-1,

Similarly

cos sr ds

19.6.7

Relations

19.6.1

S

e-tr2 ,me-‘s~a-~(Z2+2s)-to-tdg

Integrals

where the contour separates the zerosof r(s) from those of l’(a+$-2s). 19.5.14

0

Again 19.5.9 and 19.5.10 become indeterminate when a=&, #,#, . . .; for these values

eql+t)ia-t(l-t)-wfdt

19.5.8

19.5.13

S

Ub, Z)=r(&a) m-f22 me-~~~o-f(Z2+2S)-ta-tds

2V’(a,2)-v(a+1,2)-(a-~)V(a-1,2)=0 x)=0

z)+(a+~)U(a+l, z)+(a+$)U(a+l,

These are also satisfied by r(+a)V(a,

19.6.8

x)=0 cc)=0 z).

zV(a,2)-V(u+l,z)+(a-~)V(a-l,2)=0 These are also satisfied by U(a, z)/I’($-a)

19.6.5

V’(a,z>-3zV(ap)--(a-+>V(a-l,z)=o

19.6.9

19.6.6

V’(a,z)+&sV(a,z)-V(a+l,s)=o

19.6.10

y;(a,2)+3~yl(a,2)=(a+3)ya(a+1,2)

y;(a,z)-$q/~(a,z)=(a-i)yz(a-1,~)

PARABOLIC

19.6.11

~;(~,~)+3~yz(~,~)=y~(~+1,~)

19.6.12

yxa, xl--bY21/2(% x)=Yl(@--1,x)

Asymptotic

CYLINDER

FUNCTIONS

sense of Watson [19.6], although valid for a wider range of ]arg z] in Poincare’s sense; the second series is completely valid only for x real and posithe.

Expansions

19.9. 19.7.

Expressions

in

Terms

When a is large OIx< m

and

2=2JH[

of Airy

negative,

I 689

Expansions

for

a Large

With

z Mod

Functions

write,

for

(i) a positive When u>>x2,

with p=&

then

t= (4lul)G

19.7.1

19.9.1

Ub,

19.92

U(-)=2*a+‘r($t)a)

Ia) exp (-px+s)

2) =2,“+l&

~

4 2

T=-(gs,y

Jr

exp (px

t v2) I

where 19.7.2

$(4x)” ($2)”-+3x-- : (3x)5 (2P)2 -I-- (2 >” v1J%- %--+md’* ($8W-4(3x)‘+ . . . 04 (2P)5

19.9.3

T=+@w

&=+

S

arccosh 5

l’JsZ-ds=$~,/~-+

Then for 220, a+-

({2 1)

(a++ a>

0~ The upper sign gives the first function, lower sign the second function.

19.7.3

U(a, ~))--2-+@~

($--+a)

‘Ai When -a>>~~,

r (&-a) V(a, 2) -2-*-W

(*-+a)

with p=d?i,

U(u, 2) +ir

(9-a) .V(u, 2)

19.3

Expansions

When z>>

for z Large

and

=

eir(t+ta) r ($-$a)

etpz exp (v,-t&J

2+++

a Moderate

where

]a]

19.9.5 (32>”

19.8.1

2(3x)4 --

9(~x)‘--$8(~x)”

%- + (2p)“+ (2p)4

+(a+))(u+~)(a+~)(u+D_ . . . 2 * 4x4

19.8.2

I

then

19.9.4

‘Bj (t)

gives T as a function of [. See [19.5] for further developments.

19.8.

and the

(ii) a negative

19.7.4

Table

rate

t

(x+-t

vt--

(2p)fl

-*..

g($x)3I 3x+~(3x)“+-$B(fx)“-to7_ 2p (‘2P)5 (2P)3

.;,

>

a>

Further expansions found in [19.11]. 19.10.

(i) a positive,

(x+t m> These expansions form the basis for the choice of standard solutions in 19.3. The former is valid for complex x, with (arg s]
19.10.1

0=4a& (x/2&) =a

of a similar

Darwin’s

type

(a-v-

a)

wi$

be

Expansions

x2+4u large.

Write

x=Jiiqz

’ Xdx=+xX+u S0

=z J&jZ+u

In sf

~ I

arcsinb ,z&

690

PARABOLIC

CYLINDER

19.10.11

(see Table 19.3 for &), then 19.10.2

V(a, s) =

exp

U(a, z)=~%

FUNCTIONS

2

ear sin { +s++7ra+e+v,}

(27r)tJl?(+-a)

U(a, -z)=~%

19.10.3

exp {e+v(a,

-2))

where

where

19.10.12

In Y-$+$&.

vlm-&

..

19.10.4

v(a, x)--

*In

X-kg1

“f-y3

(-1>“d3JX3*

(a>O, x2+4a-++ and d3a is given by 19.10.13. (ii) a negative, x2+4a large and positive.

Write

e=dlal~,(x/2Jiiil)=~S~;,

a Xdx=+xX+a =+xdw+a

ax)

da=% x2-2a

In x&z arccosh 5

2&q

do=ii(

-A0

xg-go

ax7-go

a2x5

+3’ a3x3-19a4x 12 > exp { ++v(a,

U(a, x)=-(~~)~,~

m)

19.10.13

19.3 for &), then

(see Table

d 12=i!!?

x) }

8

x4-186ax2+80a2

See [19.11] for d15, . . ., d24, and [ 19.51 for an alternative form.

19.10.7 V(a, x)=

2

exp

(27r)“4Jm

Ie+v(a,

-2)

1

19.11. Modulus

where again

and Phase

When a is negative and Ixl<2Jlal, the functions U and V are oscillatory and it is sometimes convenient to write

19.10.8

v(a, x)--

(x2+4a-+-

*. .

In each case the coefficients d3r are given by

d3=k ($++

X=Jx2--4la(

19.10.5

19.10.6

a)

d3 -- & yg+

* In X+x

(-l)“d3JX3S a=1

(a
~0)

Write

19.11.1

U(a, x)+ir($-a)V(a,

x)=F(a,

19.11.2

U’(a, x)+iI’($-a)V’(a,

x)=-G(a,

Then, when a<0

x)efxca*“’ x)e*‘@**’

and lal>>x’,

19.11.3 19.10.9

Y=&q=P

e=41aP4(x/Ni$

+W-ta) ga+tJ;7

=S

+ ‘YG%=txY

+ Ial arcsin 2.L. 24

0

(see Table

19.3 for 84=&r-83),

al

eD, ’

x=(~+~>?r+px+vt

where v,, vr are given by 19.9.5 and p=1/--a. Alternatively, with p=Jla(, and again -a>>x2, 19.11.4

then

19.10.10

U(a , x) =2w

(27r)“4

ear co9

+w

++*?ra+e+v, 1

1

-144x2 (4p)B

+

’ * *)

PARABOLIC

p9,i~

-$x6-176z2 (4p)G 19.11.7

(+a -ga+px

9-

*.)

x2 l-gp-.(qp)

,JYS-+a)

19.11.6

2/c*-16 L4P)4

1 ---p-3x2 (4zv { -+x6-y,*(4PY

x- (+a+$T+px

19.11.5

CYLINDER

$54

-...

691

FUNCTIONS

U(a, ~)=2-+“x-‘iW-~,

19.12.2 19.12.3

U(a *x)=dG2-:-w~z , r(f+3u>

U(a, 2)=2-f-:“e-f’*CT(3a+4,+,

l-~*-~Yf$

* *>

Again, when x2+4a is large and negative, Y=d4ja] -x2, then

with

19.13.

Connection

19.11.8

where 13, v, and v$ are given

With Hermite Functions

When n is a non-negative

x=$r+&ra+f?+v, by 19.10.9

and

19.10.12.

from

Polynomials

integer

19.13.1 U(-n-4,

Another form is

4, ix”)

Expressions for V(a, x) may be obtained these by use of 19.4.2.

and

en7

ix’)

=2-?-@xe- tz”U(+a+$,

.

(4PY

(2r) f

M(jYaS$, +,+x2)

19.12.4

1 -+x6+yx*-

&w3

&x2)

x)=e-fz2Hen(x)=2-~“e-~““H,(x/~~)

19.13.2

19.11.9

3

5a

V(n+$,

621

*’ *

l+4y4+j%+32ys+

kc*+4a+-

w)

~)=J~e~“~He~(x)=2-~“e~“~H~(x/&)

in which H,(x) and He,(x) are Hermite nomials (see chapter 22) while

poly-

19.11.10

5

cw

7a

835 ‘--4y4-F-32yB --

He:(x)=e-+“’

d$

19.13.4

H*,(x)=e-“*

&*=

(--i)“He,(ix)

’’ ’

(2*+4a+while $ and x are connected by

&

ez2= (-i)“H,(ix)

w) This gives one elementary solution to 19.1.2 whenever 2a is an odd integer, positive or negative.

19.11.11

*-x--+T-+

19.13.3

(1+$*+gf+gg+

. . .j

19.14.

Connection and Dawson’s

With Probability Integrals Integral (see chapter 7)

If, as in [19.10] Connections 19.12.

With

Connection With metric Functions

Other

Functions

Confluent

Hypergeo-

(see chapter 13)

19.14.1

Hh-,(x) =e+*

19.14.2

Hh, (x) = ~mH~~-l(t)dt=(l/n!)~m(~-x)‘e-i’2~t z

19.12.1

(G20)

then +21-+x-t r(‘+la) 4

2

M-ta.dW

19.14.3

U(n+$, x)==ef’*Hh,,(x)

(n>,-1)

PARABOLIC

692

CYLINDER

FUNCTIONS

19.15.14

Correspondingly

19.15.15

V(+,s) =meiz2

19.14.4

V(-1, v(-2,

x)=(tx>+(Y~+Y$) x)=$($x)“(~Y*+~&-$~)

19.15.16

and

V(-3,

19.14.5 V(-n-3,

19.15.17

x)=e-fz2

2) is closely

related

>

WO)

to Dawson’s

Set2at z

19.15.19

5(3x):(5~~+g9.~~-5~~--~) x)=Ja/~($x)K+

U(-+,

19.15.18

-__- sin ina ~ aW(3n+l) Here V(-a, integral

x)=8.

U(-3,

~)=&$(32)~2K~

U(-8,

x)=&$~($z)~(~K~-K;)

19.15.20

VC+, x>= (3x> (It+I-d

19.15.21

V&, x) = (+r)“(2r*+2I+)

0

These relations give a second solution of 19.1.2 whenever 2a is an odd integer, and a second solution is unobtainable from U(a, 2) by reflection in the y-axis. 19.15. Explicit Formula Functions When

in Terms of 2a Is an Integer

Bessel

Write 19.15.1

I-,--I,=(2/?r)

19.15.2

I-,fI,=cos

where the argument tions is ax2. Then 19.15.3

sinna.

K,

7m * 9, of all modified

U(2, x)=2.

The

v(1,2)=~(42)~(~--~~) v(2,

19.15.8

Equation

$$+(i

+-a)

y=O

Series in x

Even and odd solutions are given 19.2.4 with -ia written for a and the series involves complex quantities imaginary part of the sum vanishes Alternatively,

by 19.2.1 to xeii* for x; in which the identically.

yl=l+a

-&)

$+(a’

$+(a3-fa)

+(a”-lla2+y)

$

$+(a5-25a3+*)

$+.

..

$+.

..

a, of xn/n!

are

19.16.2 X3

X7

X5

g+ W-E9

3+ (a”--!?a)

fl

$. $r-+(&z);(-5Kt+9Ki-5K;+K;)

19.15.6 19.15.7

x) = ($x)“(514+51-+-I;-I-;)

19.16. Power

y2=x+a

U(3, x)=2.

2)=4(32)~(2~*-3~~++~)

+(a”-17a2+q) in which connected

$+(a5-35a3+~~a)

non-zero by

coefficients

V(3, z) =3(32):(5~~-9~~+55~--~) 19.16.3

19.15.9

a. an-tn(72-l)an-2

an+2=

U(0, z)=?r-+(+x)fKt

19.15.10

U(-1,

i7(-2,

19.17.1

W(a,*x)=(co$,“a)’

19.17.2

=2-314

Solutions

(see L19.41) (G~~I+‘%~YJ

d

iz$=?r-~(~z):(2Kt+3K~--K~)

19.15.12 U(-3,

19.17. Standard

z>=d(3x);(Ki+Kt)

19.15.11

19.15.13

Bessel func-

$T-~($x)*(~K~-~K~+K~)

19.15.5

V(;,

19.16.1

U(1, z)=2d($x):(-Kt+Ks)

19.15.4

19.15.22

z)=T-~(+T):(~K~+~K~-~K;-K;) V(0, x)=3(32)*4

(-\iZ

?/,rg

where 19.17.3

G1=II’(4+3i4

G3=Ir
y2)

PARABOLIC CYLINDER

FUNCTIONS

693

19.20.1

At x=0,

T=-(~a,y

19.20.2 Complex

T= +(9w

Solutiom3

19.17.6

E(a, x)=k-)W(a,

19.17.7

E*(a,

x)+ik*W(a,

-2)

2)-iik+W(a,

-2)

2)=/k-~W(a,

&,=i Then for x>O, a++

where 19.17.8

(21)

0~

19.20.3 k= &i?=

l/k.=Jl+e”““t

era

era

In terms of U(a, x) of 19.3, 19.17.9

arccosh 5

W(U, x)~~(4~)-t,-r~~(EZ4i)tBi(-tj 19.20.4

E(a, x) =$Zetra+*i*++fQz

U(ia, xe-*‘“)

~(a, -2) -2&4a)-*e+”

with 19.17.10

&=arg

Table 19.3 gives r as a function [19.5] for further developments.

l?(++iu)

where the branch is defined by &=O and by continuity elsewhere. Also

of t. ; See

when a=0 19.21.

19.17.11

Expansions

for

x Large

and

a Moderate

When x>>kl, xe:t”*)

{ eha-tf*u(-ia,

xe- ~fi)=l?(+ia)

Jz?rU(ia,

19.21.1

+e-)‘a+~f”U(-ia,-x,!fi*)}

E(a, x)=y’2/2 19.18. 19.18.1 19.18.2 19.18.3 19.18.4

Wronskian

and

Other

In x+&+$~)]SQa,

x) I

19.21.2

w{W(a,x),W(a,-x)j=l W(u, ~)=J~{s,(a,

W{E(a, x), E*(a, 5)) =-2i 41 +e’*‘E(a,

x) =e@E*(a, x) +iE*(a,--2)

x)=e-%+f~‘E(-a,

E*(a,

2) cos ($x2-a In x+$7&&)

--~(a, x) sin (ix’--a

W(a,-x)=Ji@ii{s,(a,

ix)

In xftnf~~)

]

19.21.3

2) sin ($x2-a In x+$+3&)

+~,(a, 2) cos ($x2-a In x+$*+16)

19.18.5

JF($TqE*(a, 19.19.

x)=e-fqiygG&q-a, Integral

i.r)

Asymptotic Expressions

Expansions in Terms

of Airy

t= (4a)?7

s(a, 2) =%(a,

]

and

2) +is,ca,

4

19.21.5

19.21.6 Funcl;ions

When a is large and positive, write, for O_<x< w x=2*(

where & is defined by 19.17.10 19.21.4

ReprFsentations

These are covered for 19.1.3 as well as for 19.1.2 in 19.5 (general complex argument).

19.20.

exp {i(ax”-a

Relations

u2

s,(a, x):>-----1!2x2

v4

2!22x4

+i!!L+L!!L-

3!23x6 4!24xs

.).

with b4-

m>

694

PARABOLIC

19.21.7

CYLINDER

FUNCTIONS

(see Table 19.3 for S,), then.

u,+iv,=r(r+~+ia)/r(~+ia)

19.23.2

V’(a, x) = &ke+ cos (&+e+v,)

sin ($n+fI+z+)

19.23.3 W(a, -cc) = meor 19.22. Expansions

When a>>x2,

for a Large With

2 Moderate

where

(i) a positive with p = &, then

19.23.4

19.22.1

W(a, cc)= W(a, 0) exp ( -ps+vl)

19.22.2

W(a,-cc)=

up-g+gg+.

. . . .

W(a, 0) exp (ps+vz)

(x2--4a+

(ii) a positive,

19.22.3

4a-x2>>0

Write %(Bx)3 I (t”)21tr+% 2p

(2PY

$2 Gx)“,

(4x)5 19.23.5

(2P)3

Y=J4a-x2

-$s($x)3++ (6x)’ (2p)5

(2P)4

+ ***

cl2Sz

8=4~6,(~//2&)

arcsin 2

Ydx=+xY+a

2&

0

The upper sign gives the first function, lower sign the second function.

and the

When -a>>x2, 19.22.4 W(a, x)+iW(a,

with p=JTi,

then

19.23.7

0) exp {v,+i(pz+r+vJ

}

W(a, --2)=exp{e+v(a,

-2)}

6 da + In Y+p+y~+p+

do

... a)

and d3, is again given by 19.23.12.

-- (6x)” 2k44 __(zp)“+

9(+x)2+v (&z)B+

(2p)4

Bx+8&Y (2P) 3

(2p)6

(iii) a negative, x2-4a>>O

..’

+wBd3+w’(2P)5

expansions of a similar

Write

. .

19.23.9

(a+in [19.3].

x)}

(x2--4a+-

19.22.5

Further

--8+v(a,

19.23.8 v(u, x)--

where W(a, 0) is given by 19.17.4, and

2P

W(a, x)=exp{

then

where -x)

=JzW(a,

,*,t(M3

(see Table 19.3 for S4=+r-G3), 19.23.6

(ii) a negative

&-

a)

and d3r is given by 19.23.12.

where W(a, 0) is given by 19.17.4, and

Vl,w-f

X d!+d12 xB xI2-.

v,- -i-In

~0)

type will be found

x=J22+4/a/

19.23. Darwin’s Expansions

(i) a positive,

e=4[albl(z/2y’la[)=+~ =+xX-a

x2--4a>>O

Write

=

Xdx

In -x+x 2m

~x~x2+41ul - a arcsinh --?-Niq

19.23.1

(see Table 19.3 for 0,) then 19.23.10 W(a, 2) =J%e”r

=*xX-a = +x4--a

In -x+x

19.23.11

2&

arccosh A!-2&

W(a, -x)=J22e‘+

cos (b+e+v,)

sin (b+e+u,)

where v, and vI are again given by 19.23.4. each case the coefficients d3r are given by

In

PARABOLIC

CYLINDER

19.24.8

19.23.12

d3=-;

~695

FUNCTIONS

-&+(4p)i19.24.9

d6=%xz+ 2a

~x4+16++x6+yx2 (4p)B

$2'

h=$

(

a3x3+19a4x

~~-~~az’+~a~x~+~

&j

i J$xe+152x2 (4p)6 + . * ’

8x4+8

($ax)

lfo”-(4p)4

>

19.24.10

d12=- 153 x4+186ax2+80a2 8

$x6+168x2 +

See [19.11] for d15, . . ., d24, and [19.5] for an alternative form. 19.24. Modulus

(4p)6

-

. . .

19.24.11 $x4--16 (4p)4 +

+-

When a is positive, oscillatory when x< when a is negative, the all x. In such cases it to write

- . i. .

and Phase

the function W(a, x) is 2&i and when x>:!JZ; function is oscillatory for is sometimes convenient

Again, when aO, then 19.24.12

F---\i%?“r

with X=dx2m,

x=$r+e+vi

where 8, v, and vr are given by 19.23.4 and 19.23.9. Another form also when a>O, x2--4a+m is

19.24.1

k-W(a,

x)=Fe””

x)+ikhW(a,-x)=E(a,

(XX)

19.24.13

F-

19.24.2 k-t dWa, dx 4 +ikt my(~~mx)=,f(a,

x)=-Gc!W

19.24.14

(XX)

Then, when x2>>lal,

Gwhile J, and

19.24.3

x

are connected by

19.24.15

F-

J/-X-

19.24.4

X-4x2---a In x+#~++T+,,+

4a2-3

4a3-19a+ 8x4

...

19.25. Connections Connection

Confluent

Hypergeometric

and

Be/ssel

Functions

19.24.5

G-

With

With Other Functionp

J(

19.25.1

zz +y-14a;;663a-

... >

W(a, &x)=2-f

(J

$ H(-$7

+a, +x2)

3

19.24.6

*- $x-j-a In x+$#~~-$r+-

4a2+5+4a3+29a

8x2 where +2 is defined by 19.17.10.

When a
8x4

+ ...

19.25.2

H(m, n, x) =e-fZIFl(m+

lal>>x” F~&W(a,

where

19.25.3

0) eor

where V, is given by 19.22.5 with p=J=o.

AIso

1 --in; 2m+2;

=e- fZM(m+l-iin,

2i;c)~

2m+2, 2ix) 1

19.25.4 W(0, ~x)=2-~~?rz{J-t(4x2)fJ~(i~2)

I

(xqx

696

PARABOLIC

CYLINDER

integer for W(a, x) or its derivative, or an even integer for W(a,-2) or its derivative, the zeros fc, stc’ have expansions

19.25.5

& W(0, ~2)=-2-tz~{J*(t22)fJ_*(a22)} (X20)

19.26. Zeros

Zeros of solutions U(a, s), V(a, cc) of 19.1.2 occur only for ]z]O. Approximations may be obt,ained by reverting the series for # (or x for zeros of derivatives) in 19.11, giving 9 (or x) values that are multiples of +a, odd multiples for V(a, x), even multiples for V(a, 2). Writing a=(gr-+-$r as an approximation

FUNCTIONS

to a zero of the function,

or

19.26.4

c z;-~

19.26.5

c’ =;-”

2d--3a+52~~~-240~~~+315a+ 768OpQ 48~

---

j3 2a”+ 38+52f15+ 280fi3-285/3+ 7680~~ 48~

*”

When x is large and a moderate, we may solve inversely the series 19.24.4 or 19.24.6 with a=+ (m-+a-&), /3=$ (r7r++qi2), r odd or even as above; the presence of the logarithm makes it inconvenient to revert formally. The expansions 19.26.4 and 19.26.5 fail when x is in the neighborhood of 2m. When a is positive, a zero c of W(a,-x) is obtained approximately by solving 19.26.6

as an approximation to a zero of the derivative, we obtain for the corresponding zero c or c’, with -a =p2 t,he expressions c=BJii[ 19.26-J 19.26.2

,,;+!&$+52a5-240a3$315a+

768Op

c’ = P - 2/33+3P+52P5+280/33-285/3+ 7680~~ P+-- 48p5

’ ’’

These expansions, however, are of little value in the neighborhood of the turning point x=2-a. Here first approximations may be obtained by use of the formulas of 19.7. If a, (negative) is a zero of Ai( the corresponding zero c of U(a, 2) is obtained approximately by solving 19.26.3

tY3=+{ arccos f-s-}

=w c=Nm

(a>>01

...

(a<<@

This may be done by inverse use of Table 19.3. For a zero of V(a, x), a, must be replaced by b,, a zero of Bi(t). For further developments see [19.5]. Zeros of solutions W(a, 2), W(a,-z) of 19.1.3 occur for ]x I>2fi when a is positive; the general solution may, however, have a single zero between -2& and +2& If a is negative, zeros are unrestricted in range. Approximations may be obtained by reverting the series .for $ (or x) in 19.24. With -a=p2, cr=($r-t)*, S=($r+i)?r, rl0 being an odd

with the aid of Table 19.3. For a zero of W(a, x) we replace a, by b,. When a is negative we solve, again with the aid of Table 19.3, 19.26.7

zr),=+{[Jl2+1+arcsinh

[} =@$$@ c=wl45

where n=l, 2, 3, . . . for an of W(a, -x), and n=+, Q, 8, . mate zero of W(a, x). Further given in [19.5]. Any of the approximations above may readily be improved Let c be a zero of y, and c’ a is a solution of

(--a>>01

approximate zero . . for an approxidevelopments are to zeros obtained as follows: zero of y’, where y

Y "-Iyd-J

19.26.8

Here I=uf+x2, I’= f&x, I”= &$; the method is general and the following formulae may be used 0. Then if y, y’ are approximawhenever I”‘= tions to the zeros c, c’ and 19.26.9

u=y(r)/y’(r)

with 1=1(r)

or I=l(r’)

v=y’W)l~YW)

respectively,

then

PARABOLIC

CYLINDER

I697

FUNCTIONS

19.26.10

19.27.1

J&2,=$

19.27.2

J&z’)

{W(O,

-2)

rW(0,

z) 1 ~

Cy-u-*IU3+*I’U4 ~($aI”+~P)u5 19.26.11

+&$II’d+

. . .

-

y’(c)-y’(y)

{ l-~Iu2$~I’u3 - (~I”+~I~)u4+&II’u5+

. . .)

19.26.12

19.27.3

. . .

. . .)

19.27.5

The process can be repeated, if necessary, using as many terms at any stage as seems convenient. Note the relations, holding at zeros,

8F

19.26,14

V’(u,

c’) =&$/U(u,

19.26.16

W’(a,c)=-l/W(a,

I-&x2)

c’)

19.27.7

-c)

f

+ I&x2)

=:

2 v (0,x) 4X

I-&x2)+I&x2)=-4

U(0,

d V(O,z) x& dx

=

wa, c’)=U{ -%(a, da: -x) 1 z-c’ =-l/W’(a, Bessel Functions Parabolic Cylinder

Use and

Extension

of Order f 2, f 2 as Functions

of the

-sx

g “‘f,

--c’)

Most applications of these functions refer to cases where parabolic cylinder functions would be more appropriate. We have

19.28.

2)

&(+x2)=I-*(fx2)-I&x2)

19.26.17

19.27.

+&z?J~-~(~z2)+0

K~(~22)=I-~(tx*)-I&x2)=&

19.27.6

U’(u,c>=--@pqv(a,c)

19.26.15

a~“JY+1(4rz)-2~JY(:5’)

19.27.4

{ 1--4pb--gPI’v3 - (+I3I’2--gJ4I”+gI~)v4+

-x)}~

Again

19.26.13

y(c’) -y(g)

(W(O,x)&W(O,

Functions of other orders may be obtaine by use of the recurrence relation 10.1.22, which ere “, becomes

c’-~‘-Iv-~II’v2+(~121”-~II’~-~Iq? + ($JTI”--fJI’“--&J~I’)v4+

= g

$1 I

As before, Bessel functions of other orders ma Jt be obtained by use of the recurrence relation 10.21.23, which here becomes 19.27.8

~221y+l(~x2)+2~I~(~x2)-~x21v-~(~x2)

19.27.9

~xz~,+,(~x2)-2~~v(~x2)-~x2~v~,(~x2)~=0

=fj

Numerical

Methods

Tables

reverse direction, from arbitrary starting va ues (often 1 and 0) for two values of a some hat beyond the last value desired. This is because4 the recurrence relation is a second order homogen ous linear difference equation, and has two i deLoss of accuracy by cant llapendent solutions. tion occurs when the solution desired is diminis i: ing as a varies, while the companion solution is increasing. By reversing the direction of prog ess in a, the roles of the two solutions are in erchanged, and the contribution of the desired s lution now increases, while the unwanted solu ion By st rtdiminishes to the point of negligibility. ing sufkiently beyond the last value of a for w ich the function is desired, we can ensure that the unwanted solution is negligible but, because : the starting values were arbitrary, we have an un-

For U(u, x), V(u, x) and W(u, x), interpolation x-wise may be carried out to 5-figure accuracy almost everywhere by using 5-point or 6-point Lagrangian interpolation. For la] 5 1, com.parable accuracy u-wise may be obtained with 5- or 6-point interpolation. For [aI> 1, U(u, x) and V(u, x) may be obtained by use of recurrence relations from two values, possibly obtained by interpolation, with Ial :I 1; such a procedure is not available for W(u,. &a), lal> 1. In cases where straightforward use of the a-wise recurrence relation results in loss of accuracy by cancellation of leading digits, it may be worth while to remark that greater accuracy is usuSally attainable by use of the recurrence relation in the

698

PARABOLIC

CYLINDER

FUNCTIONS

known multiple of the solution desired. The computation is then carried back until a value of a with (al < 1 is reached, whan the precise multiple that we have of the desired solution may be determined and hence removed throughout. Compare also 9.12, Example 1.

If less accuracy is needed they selations 19.6.1-2. :an be found by use of mean central differences of U(u, s), V(a, x) and also of W(u, x) with the formula

Example 1. Evaluate . . ., using 19.6.4.

using h= .l; this usually gives a 3- or 4-figure value of duldx. If greater accuracy is needed for dW(u, x)/dx it may be obtained by evaluating d2W/d2 with the help of the differential equation satisfied by W and integrating this second derivative numerically. This requires one accurate value of dW/dx to start off the integration; we describe two methods for obtaining this, both making use of the difference between two fairly widely separated values of W, for example, separated by 5 or 10 tabular intervals.

(a++)

U(a+

1,~) +zU(a,

Forward Recurrence

a

3 4 5 ; 8 9 10 11 12 13 14 15 16 17 18 19

(-6) (-7) i-7j

5. 2847’ 9. 172* 1. 5527

[I;; 2. 5609 4. 1885 (-10) 6. 2220 (-lO)+l. 2676 (-11)-O. 1221 (-ll)+l. 2654 (-12)-5.6079 (-12)+3. 2555

U(a, 5) for u=5, 2) - U(a-

Backward Recurrence 12) 11) 10) 9) 9) 8) 7) 6) 0)

1. 59035 2. 76028 4. 67131 7. 72041 1. 24785 1. 97488 3. 06369 4.66352 697082 102444 14789 2111 292 42 5 1+

1,z)

Final

(-6) (-7) (-7) (-8) ( -9) (- 10) (-10) (-11) (-12) (-13) (-14) (-15) (-16)

*From tables. +Starting values. **This value was used to obtain the constant d _(-6)5.2847 =(-18)3.32298 for converting F-(12) 1.59035 vious column into this one.

6, 7,

=0

Values

5.2847** 9. 1724 1. 55227 2.5655 4. 1466 6.5625 1.01806 1. 5497 2. 3164 3.404 4.91 7.01 9.7

du

hu’=h &=&

-~pccs3u+&63u- . . .

(i) Write j,, j& j:’ for W(u, xO+rh) and its first two derivatives, then j: may be found from

hj;-&

(jn-j-.)-gm$

h2 -- 2n I +-&

(n-r)(j:‘-ji’?) s2+*

64- . . .} (f;--jQ

-h2{+&--~~~3+&&+-.

. .} j;’

multiplier the

pre-

The second column shows forward recurrence starting with values at a=3,4 from Table 19.1. Backward recurrence starts with values 0 and 1 at a=19 and 18, containing a multiple kU(u, 5) and a subsequently negligible multiple of the other solution I?($-a) V(u, 5). Rounding errors convert kU(u, 2) into k*U(u, z) without uflecting the values in the lust column. The value of l/k* is identified from the known value of U(3,5), and used to obtain the final column by multiplying throughout by l/k*. The improvement in U(5,5) is evident by comparison with Table 19.1. Derivatives. These are not tabulated here. Since the functions U(u, z), V(u, s) and W(u, z) satisfy differential equations, values of derivatives are often required. For all these functions the equation is second order with first derivative absent, so that secom derivatives may be readily obtained from function values by use of the differential equation. First derivatives can be obtained for CJ(a,x) and V(a, s) by applying the appropriate recurrenct

(ii) Consider a solution y of the differential equation for W(u, x), namely y”= (-&r”+u)y. If we are given values y and y’ at a particular x=x,, and write T,=H”y’*‘/n!, T-l==T-2=0, then we may compute T2, T3, T4, . . . in succession by use of the recurrence relation obtained from the differential equation,

--W2T,-21 These are computed, to a fixed number of decimals until they become negligible, thus giving

y(x,fH)=TofT~+TsfTa+

. . .

This may be applied, with H=rh, h being the tabular interval, and r a small integer, say r=5, to the solutions y=y], y=y2 having Yl (ZJ = W@, ZJ

YI (20)= w*‘(a, x0)

Y2(xo)=O

Y;:(x0) = 1

in which W*‘(u, so) is an approximation to W’(a, x0), not necessarily a good one; it may be

PARABOLIC

obtained from differences, for example. obtain y1(G AH) and yz(~0 f H> . Now suppose W’(u, x0) = w*‘(u,

CYLINDER

We thus

x0) +x

Thus W’(-3, l)=-.74453. This might have an error up to about li units in the last figure but is, in fact, correct to 5 decimals. (ii) Using the second method, with yl(l)=W(-3, y;(l)=-.745

then, for all z Wb,

4 =y1

(4

the following

W’(u, so) = W*‘(u, to a suitable

W( -3,

accuracy. 2. Evaluate

Example

X0)+x

W’(-3,

.5)=-.05857

W(-3,

l)=-.61113

W(-3,

1.5)=-.69502

(i) Using the first method v”(-3,x:

W-3,x)

0. 4

=

= 6”

6

.-

KS

t-0: ;;;“5’: -. 18832 -. 31226 -. 42646

0. 9

-.

:: :

SE’

_- -+131

.

1. 0 1. 1

-.

1.68842

61113

1.98617

-.

67522

2.22991

-. -. -. -. -.

71706 73488 72761 69502 63774

2.40932 2.51513

29775

- 1095

24374

- 1032

17941 ::i :2 1: 6

-

.oooo 677 26

Ts

-

Te

+

279 3 134

+ +

24 2

T, -I54 5 TB f + .4323 y(1.5)-.695223 -.4371 y(.5) -.058363

1) = - .74453 which is correct to

Example 3. Evaluate the positive zero1 of U(-3, x). We use 19.7.3 to obtain a Crst approximatEon, see 19.26.3. The appropriate zero of Ai is at

-k.

T=-(2.338)X(12)-1=-.4461

-9129

+JW’(-3,1)

-- 1 100

+ .24827 2 $ 5680 9 1407 4

t=(4lul)f~=-2.338

The fifth decimal in W”( -3, X) is only a guard figure which is hardly needed. Only the difl’erences needed have been computed. Then

-&- {A

T2 T3 T,

$1 =l/*+bh

At x=1.5 X-. 695223 + .4423X = - .69502 A= .000203/.43213 = .000470 so W/(-3, 1) =-.745fX = - .744530 At x=.5 -.058363-.4371X = - .05857 X= .000207/.4371 = .000474 so W’(-3, I) =-.745 +x = - .744526

whence

; E8~~ 2: 32137

=A(--.69502+.05857)

.oEoo + .5000

Thus W’(-3, 5 decimals.

34081 52722

Tl

T .61113 - .37250

To

1) using r=5.

From Table 1.9.2

X

values result, with H=.5,

~ofH)=Yl(~ofH)+~Y,(~ofH)

The values of W(a, z. f H) may be read from the tables and two independent estimates of X obtained, whence

W(-3,

l)=-.61113 to 5 de&n@ to about 3 decimals

+ XY2 (4

and in particular wa,

1699

F’UNCTIONS

(10.38874)

Hence, from Table - 19.3, [= .3990 and the approximate zero is x=26ju15=1.382. We improve this by using 19.26.10, but takei for convenience, Z= 1.4 as an approximation, so that the value of Scan be read directly from the tables. U’ can be obtained as in the section following Example

1.

We tind (2.29664)-k.

& (.54149) -Go

(-.09260)}

(-.02127)

=-.0636450-.0103887-.0001918-.00022;’2 = - .0744527

U(-3,

1.4) =.02627

U/(--3,

1.4) =2.06317

Then 19.26.9 gives u= U/U’=.012730

I= -2.51 I’=.7

and

I”!=.5

PARABOLIC

700

CYLINDER

FUNCTIONS

y'(c)=2.0637(1+.000203)=2.0641

c=1.4-.012730+,000002=1.38727 which is correct to 5 decimals, while 19.26.11 gives

compared

with the correct value 2.06416.

References Texts

[19.1] H. Buchholz, Die konfluente hypergeometrische Funktion (Springer-Verlag, Berlin, Germany, 1953). [19.2] C. G. Darwin, On Weber’s function, Quart. J. Mech. Appl. Math. 2, 311-320 (1949). [19.3] A. Erdelyi et al., Higher transcendental functions, vol. 2 (McGraw-Hill Book Co., Inc., New York, N.Y., 1953). [19.4] J. C. P. Miller, On the choice of standard solutions to Weber’s equation, Proc. Cambridge Philos. Sot. 48, 428-435 (1952). [19.5] F. W. J. Olver, Uniform asymptotic expansions for Weber parabolic cylinder funcjions of large order, J. Research NBS 63B, 2,131-169 (1959), RP63B2-14. [19.6] G. N. Watson, A theory of asymptotic series, Philos. Trans. Roy. Sot. London, A 211,279-313 (1911). [19.7] H. F. Weber, Ueber die Integration der partiellen Differential-gleichung: &@a?+ @r&1/2+ k*u=O, Math. Ann. 1, l-36 (1869). (19.81 E. T. Whittaker, On the functions associated with the parabolic cylinder in harmonic analysis, Proc. London Math. Sot. 35, 417-427 (1903).

[19.9] E. T. Whittaker and G. N. Watson, A course of modern analysis, 4th ed. (Cambridge Univ. Press, Cambridge, England, 1952). Tables

[19.10] British Association for the Advancement of Science, Mathematical Tables, vol. I, Circular and hyperbolic functions, exponential, sine and cosine integrals, factorial (gamma) and derived functions, integrals of probability integral, 1st ed. (British Association, London, England, 1931; Cambridge Univ. Press, Cambridge, England, 2d ed., 1946, 3d ed., 1951). [19.11] National Physical Laboratory, Tables of Weber parabolic cylinder functions. Computed by Scientific Computing Service Ltd. Mathematical Introduction by J. C. P. Miller. (Her Majesty’s Stationery Office, London, England, 1955). [19.12] National Physical Laboratory Mathematical Tables, vol. 4, Tables of Weber parabolic cylinder functions and other functions for large arguments, by L. Fox (Her Majesty’s Stationery Office, London, England, 1960).

702

PARABOLIC

CYLINDER

FUNCTIONS

Table 19.1

c’(-5.0, x)

T

U(-4.5, z)

U(-4.0, x)

lJ-3.5, .A)

0) 1.5204 0 1.1869 -1 8.0608 -1 I +3.9325 -1 -0.3518

::L” 0”.:

0:4

0)

1.4678

-1 I 8.8615 -1 +2.6550 -1 -3.6676 -1 -9.8321

U(-3.0, r)

0.0000

( 0)-0.8721

2.1235

(-2)

u(-2.5, x)

U(-2.0, x)

U(-1.5, x)

-1 1-4.6224 -1 -8.7118 0 1-1.2462 0 -1.5731 0)-1.8397

0)-3.0982 0 I -3.0617 0 -2.9073 0 -2.6435 0 -2.2824

:*: 314

i

0I 0

4.3739 4.8038 5.1246

5.0

( 0)

1.8800

(-1)

9.2276

For interpolation, see19.28.

(-1)

4.4586

(-1)

9.9802

-2)

4.6331

(-2)

2.1262

(-3)9.6523

PARABOLIC

CYLINDER

: 703

FUNCTIONS

Tab14 19.1 V(p5.0, z)

V(-4.5, .r)

V(-4.0, .r)

TT(-3.5,J)

V(-3.0, cc)

l7-2.0, z) -1 -1 -1 -1 -1

0.0000

0.0000

(-2)-5.8311

V(-2.5, z)

1.8125

1.0

6.7728 -0.3303 -3.9309

:-: 1:3 1.4

-7.3916 +3.2819

(-1) (-1)

i-2

1.5812 1.1580

r;

-

.5385 .9387 .2553

-1 -1 - .4995 -1 ii - .6835

-7.b762

l-1 -1.6465 I'-1 -2.0148 ('-1 i-1 I -2.3214 -1.2266

I

1.5 :*; 1:8

1.9

-l)-1.0927 -2 1 -6.9034 -2 -2.7540 -2 +1.4424 -2) 5.6116

-1 3.4421 -1 1 3.7545 -1) 4.0712

-1) 3.8093 -1) t-78 4:9 5.0

4.1462

-1) 5.5449 (-1)

1.8370

(-1)

3.3533

(-1)

6.2047

( 0) 1.1734

( 0) 2.2757

( 0) 4.5254

( 0) 9.2067

( 1) 1.9107

704 Table x

PARABOLIC

CYLINDER

FUNCTIONS

19.1

U(-1.0,s)

U( - 0.9,5)

U(-O&2)

U(-0.7,x)

U(-0.6,~)

U(-0.5,x)

ix-0.4,s)

0.0 0.1 0.2 il -10 1.0594 9.7698 1.0035 1.0261 1.0448

i::

2.0 2.1 214

(-lj3.7407

-1 -1 -1 -1 -1

I

II II II II

4.9754 4.5701 4.1741 3.7910 3.4238

-1 4.2938 2.8559 3.1876 3.5391 3.9086

-1 2.9165 3.2511 3.6054 3.9779 2.6029

-1 2.3693 2.6647 3.3204 3.6788 2.9820

-1 ;*;;;; 1,' 2.4313 2:7312 2.1538

2.5 f:4

3.0 3.1 3.2 z*z .

II

-1 1.8488 -1 1.6124 -1 1.3985

-1 1.2064 1.0351

3.5 3.6 3.7 3.8 3.9 t*10 4:2 4.3 4.4 4.5 4.6 4.7

-2 4.6771 r; 2:7052 2.2315 33';;;;

II -2 3.4324 1.9411 2.3589 4.1098 2.8525

1; 4.4657 -2 2:5638 ;.:;g 5.3190

ii

-2 3.6903 1.6688 2.0512 2.5079 3.0502

t:9" 5.0

II II Ii

Ii-2 2.4280 1.9923 1.0676 1.3211 1.6265

(-2j1.1618

II

-3 9.3333 4.6914 5.9310 7.4594

(-3)4.3375

(-3)3.6919

-3 I 9.9881 -3 8.0067 -3 6.3856 -3 5.0667 -3 3.9996 (-3)3.1412

II

-2 -3 1.7268 9.1898 1.1397 2.1094 1.4064

-3 6.8657 3.4085 4.3266 8.5831 5.4641

(-3)2.6716

(-3)2.2714

(-3)1.9305

(-3)1.6401

PARABOLIC

CYLINDER

RO5

FUNCTIONS

Table 119.1

V(-1.0,x)

V( - 0.9,x)

V(-0.8,~)

V( - 0.7,x)

V(-0.6,~)

V(-0.5,x) I I0.0000

V(--014,s)

-1 3.1502 0.7972 2.3760 1.5905

iI -1

3.3194 5.0435 4.1939

(-1) (-1)

5.8660 6.6605

(-1) (-1)

:-: 1:4

6.4993

6.0492 -1) 8.5594 -1) 9.2113

I 0) 1.3191

f Oj1.3881

2.3 2.4 ( 0) 1.5902

2.5

t 0) 1.7975 0) 1.9338

22::

22::

( 0) 2.4881 t 0) 2.1558

3.0 3.1 ;:: 3.5 3.6

9.9377 1.1805 1.4113

( 1) 4.0344

( 1) 4.6937

( 1) 5.4639

( 1) 6.3641

( 1) 7.4168

( 1)8.6484

( 2)1.tjO90

706 Table

PARABOLIC

CYLINDER

FUNCTIONS

19.1

U(O.3,x)

U( -0.2,x)

Ii -10 1.0421 9.8431 1.1000 1.1581 1.2163

82 2; 0:9 t-1)6.8072

1.0 :::

(-lj4.9087 1.5 II -1 4.0657 3.6765 4.4769 3.3102 (-1)2.9673

(-1)2.5142

2.5 il -1 -2 1.0248 6.4422 5.4703 7.5534 8.8173

2.9

(-2)6.9114

3.0 ;-21 3:3 3.4

II

-2 9.2134 5.7406 6.7502 7.9031 4.8608

-2 4.0978 1.9799 2.3907 2.8739 3.4393

-2 4.4006 2.5730 5.2146 3.0912 3.6967

t-213.1669

22

t-2)2.7772

t-2)2.4340

32 3:9

t:;

II ii

t-3)2.9336

t-3)2.5122

t-3)2.1504

I

i-87 4:9 4.6

-3 2:9173 2.2914 -3 1.7909 I'I:';.;G;l"

5.0

(-3)1.3929

(-4)8.5136

(-4)7.2201

(-4)6.1210

-3)7.7613 -3 3.1779 4.0011 5.0135 6.2526

II

(-3J3.9954 -3 2.4912 -3' 3.1626 1.5233 1.9528

(-3)1.1825

(-3)1.0035

f-2)1.8659

II

4.0

4.5

II ii II

-1 1.8627 1.0695 1.2363 1.4232 1.6315

(-lj1.3136

-3 4.3184 3.4390 5.3973 6.7143 2.7259

-3 5.8057 2.3371 2.9546 4.6568 3.7179

(-4)5.1875

PARABOLIC

CYLINDER

707

FUNCTIONS Table

Ij

-1 4.5280 5.0724 5.6069 6.1307 (-1 6.6436

-1)5.7994 -1 6.2358 -1 I 6.6661 -1 7.0905 -1 7.5093

t-1)6.1992

t-1)7.1460

-1)7.92:38

(-lj8.4321

(-lj9.0756

Ii -1 8.3353 9.15138 8.7460 (-1)9.5771

II

-1 7.5184 8.1782 7.1901 6.8621 7.8474

(-1)8.9640

0 I 1.0591 0 1.1013 0 1.1490 0 1.2032 0)1.2649

II -10 1.0019 1.1100 1.0553 9.4914 0I 1.25:,3 0 1.3147

II 0 1.2839 0 1.3515

II 0 1.3115 0 1.3848

00 y:!;; 0 1:55:!2

0 1.5142 1.4277 0 1.6130

00 :2:; 0 1:6738

0)1.3353 0 1.4160 0 1.5085 0 1.6150 0 I 1.7379

II 0 1.5886 1.6941 1.8149 2.1153 1.9541 Ii 0 1.8799 2.7195 2.4589 2.2360 2.0446

2; 2:9

0 0 0 0 II 0

2.1614 2.3551 2.5818 2.8478 3.1612

Oj2.4576

14.1

I Oj2.6278

( Oj2.8159

( Oj3.0247

II 0 2.3028 3.0803 2.7785 2.5218 3.4366

I

0 0 0 0 0

1.35 i 1.44 1.54 1.66 1.80,

9 6 9 2 0

I

0 0 0 0 0

3.25 I 2 3.66 7 4.14 5 4.70 4 5.38 0

3.0 3.1 1)1.0158 3.5 3:;

0I 6.7730 0 9.1860 7.8635

II 0 7.5658 10 1.0340 8.8182

2;

1 1.0797 1.2766

1 1.4470 1.2196

0 I 8.4638 0 9 90213 1 1:1653 ; ;.;m; .

t-l 4:2

2; 29" ( 2)1.1778

( 2)1.3756

( 2)1.6073

( 1)1.0637

II II II ii II II II II 1 1 1 1

4.5

5.0

( Oj9.4818

2.2395 2.7041 3.2829 4.0073 4.9179

1 1 1 1

2.5539 3.0927 3.7653 4.6086 5.6708

1 1 1 1

3.5401 2.9150 4.3219 5.3040 6.5433

1 1 1 1

3.33 f 04 4.05 4.96 4 6.10 5 7.55 0

1 6.0680

1 7.0147

1 8.1143

1 9.39 I 1

1 7.5270 9.3866 2 1.4831 1.1768

21 1.0904 8.7230 2 1.3703 1.7309

2 1.0115 1.2674 2 1.5964 2.0211

2 1.47 1.17 60 2 2.36 1.86 81

( 2)1.8791

( 2)2.1979

( 2)2.5720

( 2)3.01$2

708 Table x

0.0 0":: 24'

PARABOLIC

CYLINDER

FUNCTIONS

19.1

U(O.4,x)

U(O.5,x)

U(0.6,~)

U(O.7,x)

U(0.8,~)

U(O.9,x)

0)1.2579 0)1.1672 0)1.0811 -1)9.9946 -1)9.2205

U(l.O,x) 0)1.1627 -1)8.4523 -1)7.5790

(-1)8.2327 (-1)7.5219 t-1)6.8555

(-1)4.8280 (-1)4.3327

-1)4.2896 -1)3.0003 -1)2.6475

(-1)2.9390

1.9

(-1)1.9402

2.0

(-1)1.7003

(-1)2.7238

(-1)2.5204 t-112.2177

(-1)1.6216

(-1)1.4798

(-1)1.3487

-1)2.1487 -1)1.8774 -1 1.6351 -1 11.4193 (-1)1.2278

I

t-21713793 (-2)6.2874 2.5

t-2)8.2754

2:9

i-2)4:3157

-1)1.9797 I -1 I 1.7240 1.2948 1.4965 (-1)1.1165

(-2j5.0508

II

-2 2.5078 2.0830 1.4189 1.7228

(-2)1.1636 3.5 3.6 3.7 3.8 3.9

5.0

-3)9.5009

il-3 8.2868 3.4952 4.3655 5.4288 6.7217

(-4)4.3948

t-314.3344

f-3)3.7425 -3 2.9826 -3 I 2.3663 -3 1.8689 -3 1.4693

(-4)3.7221

(-4)3.1512

(-4)2.6671

(-4)2.2566

(-4)1.9086

(-4)1.6138

PARABOLIC!

CYLINDER

709

FUNCTIONS

Table

5

V(O.4,x)

V(O.5,x)

::9"

0) 1.3024 ( 0)1.3779

:*i 1:9

0 1.5943 1.7281 0 1.8829

-1 7.1733

( 0)1.3158

-1 I 9.7713 0 1.0488 0 1.1309 0 1.2251 0 1.3330

0 1.4784

t Oj2.0622

( 012.1689

II

00' 2.6757 3.3676 2.9943 2.4030

2.5

II 0 3.5166

;:"7

0 3.9749 4.5165

;:t

0 5.1589 5.9235

w.oJ$

V(OB,x)

-1 17.8124 -1)8.5344

II

1.5

2.0

V(O.7,x)

-1)8.4934 -1 I 8.7302 -1 9.0186 -1 9.3633 -1 9.7698

0.5 0.6 0;7

1.6

V(0.6,~)

14.1

(

0)2.1703

0I 1.4949 0 1.6542 0 1.8373 0 2.0484

0)2.2926

Oj2.28116

0)3.6363

0)4.2741

0)2.7481 0I 3.1169 0 3.5483 0 4.0548 0 4.6517

0 4.4944 0 5.1536 0 5.9365 0I 6.8696 0 7.9862 1 1.0378 1 1.2220 111.4455

3.0

I

3.5

1 1.7060 1 2.0373

3:8 3'2 3.9

1 1 2.4452 2.9495 1)3.5756

Ii 1 4.65 6.99i 6 5.69 3.82 3.15' 5

4.3 4.4 4.5 i-7" 4:8 4.9 5.0

( 2)3.5270

( 2)4.1331

( 2)4.8456

( 2)5.6833

( 2)6.6688

( 2)7.8285

( 2)9.19b8

710 Table 2

PARABOLIC

CYLINDER

U(2.5~)

U(3.0,2)

FUNCTIONS

19.1

U(1.5,x)

lJ(2.0,x)

-1 I 1.9302 -1 1.6146 -1 1.3490 -1 1.1256 -2 9.3785

U(3.5,x)

-1 -1 -2 -2 -2

1.2931 1.0674 8.8019 7.2491 5.9624

z.7" 2:8 2.9

II II -2 2.3966

-2 1.3223

-2 1.6441 1.9886 -2 1.3544 -2 1.1116

-2 1.0837 -3 8.8509 -3 7.2040 -3 5.8431

U(4.5,x)

U(5.0,x)

II II II II -2 I 1.7849 -2 1.4503 -2 1.1759 -3 9.5127 -3 7.6780

2.5

U(4.0,x)

-2 -2 -2 -2 -2

8.4374 6.8788 5.6025 4.5579 3.7035

-2 -2 -2 -2 -2

5.3758 4.3316 3.4869 2.8040 2.2523

-2 -2 -2 -2 -2

3.3518 2.6707 2.1262 1.6910 1.3434

II II II I II -2 -3 -3 Ii:

1.0327 8.2953 6.6500 ;.;g .

-3 -3 -3 -3 -3

5.8705 4.6645 3.6991 2.9276 2.3122

-3 -3 -3 -3 -3

3.2833 2.5816 2.0262 1.5873 1.2409

-3 -3 -3 -3 -3

3.3818 2.6869 2.1296 1.6837 1.3277

-3 -3 -3 -4 -4

1.8222 1.4328 1.1240 8.7960 6.8665

-4 i 9.6810 -4 7.5364 -4 5.8538 -4 4.5364 -4)3.5071

-5 -5 -6 II-6 -5

1.4817 1.1039 8.1946 6.0609 1.9818

3.5 3.6

I II II I II -4 -5 -5 -5 -5

5.0

(-5)6.9418

1.2259 9.3061 7.0352 5.2961 3.9701

(-5)2.9634

-5 -5 -5 -5 -5

5.4198 4.0787 3.0571 2.2819 1.6964

(-5)1.2558

-5 -5 -5 -5

7.3727 5.5875 4.2185 3.1726 9.6913

-5 -5 -5 -6 -6

2.3767 1.7736 1.3183 9.7593 7.1961

( -6)5.2847

-5 3.3191 1; :*:;;:

II-5 1:3920 4.4015 -5 -6 -6 -6 -6

1.0342 7.6538 5.6428 4.1440 3.0315

(-6)2.2089

ii -6 4.4663 -6 3.2790 -6 2.3983 -6 1.7475 -6)1.2685 (-7)9.1724

IIII III -6 -6 -6 -6

6.5617 4.8485 3.5701 2.6194 8.8495

-6 -6 -6 -7 -7

1.9150 1.3949 1.0124 7.3205 5.2737

(-7)3.7849

-7 -7 -7 -7 -7

8.1539 5.8942 4.2455 3.0469 2.1788

(-7)1.5523

PARABOLIC

CYLINDER

711

FUNCTIONS Table

V(2.0,x)

V(1.5,x)

2

0.0

V(2.5,~)

V(3.0,x)

V(3.5,x)

0.0000

V(4.0,x)

V(4.5,x)

19.1

V(5.0,s)

0.0000

I -1 I 1.6118 3.3218 2.4481 0.7999 1.0497 7.5647 4.8999 Ii -1 -10 2.4076

0)5.2778

1) 3.0195 1) 3.7699 1 I 4.7150 1 5.9076 1 7.4155

(

0)6.7480 0 I 7.9725 0 9.4452 1 1.1222 1) 1.3374

1) 2.276El

( 1) 3.9709

2 :2 2:9

(

1)3.0364 1) 3.7393 1 j 4.6150 1 5.7092 1) 7.0801

( 1)5.2689

1)6.5656

1)9.3262

1 1 8.1989 2 1.0262 I

2)1.2873

1) 8.8025

I( I I(

1) 9.2982

( 3)2.9574

I i ( i( 1 ( I ( I

1) 5.1442 1) 6.4978 1 8.2198 2 1 1.0415 2) 1.3218

2) 1.6806 2) 2.14108 2 2.73125 2 3.49148 2)4.4794

2) 9.1055

I(

(

3)5.4084

4) 1.1642

I

3) 8.10129 4 1.07’22 4 1 1.42’32 4 1.89’50 4) 2.53113

i

5.0

(

3)2.0666

(

3)4.6909

4) 1.0746

( 4)2.4833

(

( 4)5.7864

( 5)1.3589

4 1 6.7384 4 9.1425 5I 1.2450 5 1.7018 5)2.3348 ( 5)3.2156

I (

5)7.6639

712

PARABOLIC Table

FUNCTIONS

19.2 W(-5.0,X)

W(-4.0,x)

W(-3.0,x)

E i*:

0.47348 0.35697 +0.07727 0.22267

0:4

-0.07200

0.50102 0.39190 0.26715 +0.13172 -0.00899

0.53933 0.43901 0.32555 0.20231 +0.07298

i::

-0.21764 -0.35231 -0.46911 -0.56198 -0.62597

-0.14933 -0.28362 -0.40634 -0.51236 -0.59713

1.0

-0.65752

:::

-0.61732 -0.65470

2: :t 1:7

5

CYLINDER

W(-2.0,x)

W(-5.0,-z)

W(-4.0,-z)

W(-3.0,-x)

W(-2.0,-x)

0.60027 0.51126 0.41203 0.30453 0.19088

0.47348 0.56641 0.63113 0.66435 0.66434

0.50102 0.59017 0.65576 0.69515 0.70666

0.53933 0.62350 0.68900 0.73381 0.75649

0.60027 0.67730 0.74078 0.78939 0.82206

-0.05857 -0.18832 -0.31226 -0.42646 -0.52722

+0.07334 -0.04569 -0.16377 -0.27838 -0.38697

0.63099 0.56583 0.47199 0.35408 0.21799

0.68972 0.64485 0.57370 0.47898 0.36441

0.75622 0.73285 0.68690 0.61955 0.53268

0.83798 0.83665 0.81785 0.78173 0.72875

-0.54700 -0.44716

-0.65688 -0.68881 -0.69121 -0.66357 -0.60670

-0.61113 -0.67522 -0.71706 -0.73488 -0.72761

-0.48704 -0.57617 -0.65204 -0.71255 -0.75583

+0.07061 -0.08044 -0.22724 -0.36189 -0.47700

0.23458 +0.09483 -0.04897 -0.19063 -0.32388

0.42880 0.31103 0.18303 +0.04890 -0.08688

0.65975 0.57594 0.47890 0.37059 0.25333

-0.32290 -0.18077 -0.02851 +0.12535 0.27194

-0.52270 -0.41495 -0.28803 -0.14758 -0.00009

-0.69502 -0.63774 -0.55733 -0.45625 -0.33785

-0.78031 -0.78484 -0.76869 -0.73166 -0.67412

-0.56602 -0.62369 -0.64634 -0.63218 -0.58147

-0.44262 -0.54122 -0.61480 -0.65945 -0.67250

-0.21962 -0.34454 -0.45694 -0.55237 -0.62680

0.12978 +0.00294 -0.12397 -0.24749 -0.36405

0.40253 0.50907 0.58468 0.62416 0.62438

+0.14739 0.28751 0.41299 0.51702 0.59364

-0.20633 -0.06661 +0.07581 0.21503 0.34495

-0.59707 -0.50217 -0.39174 -0.26879 -0.13696

-0.49661 -0.38212 -0.24445 -0.09171 +0.06678

-0.65271 -0.60042 -0.51764 -0.40802 -0.27680

-0.67684 -0.69989 -0.69432 -0.65962 -0.59652

-0.47006 -0.56198 -0.63649 -0.69061 -0.72184

0.58460 0.50668 0.39507 0.25669 +0.10057

0.63810 0.64722 0.61968 0.55625 0.45985

0.45960 0.55333 0.62119 0.65920 0.66463

-0.00046 +0.13603 0.26749 0.38872 0.49459

0.22095 0.36067 0.47637 0.55973 0.60434

-0.13062 +0.02276 0.17482 0.31672 0.43980

-0.50704 -0.39454 -0.26363 -0.12008 +0.02936

-0.72830 -0.70889 -0.66340 -0.59265 -0.49853

-0.06260 -0.22123 -0.36354 -0.47850 -0.55672

0.33555 0.19042 +0.03320 -0.12614 -0.27701

0.63631 0.57472 0.48225 0.36312 0.22333

0.58021 0.64123 0.67411 0.67637 0.64681

0.60627 0.56451 0.48124 0.36184 0.21471

0.53615 0.59915 0.62397 0.60808 0.55155

0.17727 0.31588 0.43747 0.53481 0.60167

-0.38404 -0.25332 -0.11153 +0.03530 0.18042

-0.59128 -0.57849 -0.51836 -0.41490 -0.27601

-0.40886 -0.51196 -0.57820 -0.60177 -0.57982

+0.07050 -0.08654 -0.23816 -0.37452 -0.48622

0.58576 0.49519 0.37883 0.24205 +0.09180

+0.05079 -0.11714 -0.27544 -0.41066 -0.51073

0.45725 0.33088 0.18074 +0.01731 -0.14737

0.63325 0.62663 0.58111 0.49849 0.38313

0.31672 0.43701 0.53447 0.60305 0.63793

:*i! 412

-0.11306 +0.05995 0.22741 0.37359 0.48406

-0.51295 -0.40534 -0.26474 -0.10210 +0.06923

-0.56500 -0.60443 -0.60059 -0.55252 -0.46263

-0.06370 -0.21535 -0.35365 -0.46937 -0.55413

-0.56615 -0.57098 -0.52367 -0.42750 -0.29056

-0.30058 -0.42985 -0.52406 -0.57448 -0.57571

0.24189 +0.08387 -0.08010 -0.23812 -0.37804

0.63597 0.59605 0.51937 0.40960 0.27290

4.5 44:;

0.54726 0.55583 0.50770

::9"

0.26226 0.40664

0.23443 0.37847 0.48758 0.55059 0.56028

-0.33674 -0.18393 -0.01604 +0.15314 0.30893

-0.60118 -0.60601 -0.56693 -0.48549 -0.36666

-0.12531 +0.05237 0.22465 0.37342 0.48233

-0.52643 -0.42982 -0.29363 -0.12977 +0.04660

-0.48847 -0.55975 -0.58492 -0.56059 -0.48753

5.0

[(-y 1 [(-f)7 1 [996 1 [ 1 [(-y 1 [53P31 [53161 [ 1

0.51440

0.43707

-0.21874 ($3)5

21" :*: 214 Ei ?i 2:9 3.0 3.1

Z-8 3:9

0.08936

0.53861

0.21827

-0.37095

+0.11769 -0.04573 -0.20576 -0.35036 -0.46788 -0.54818 (-!)5

Values of W( a,z),for integral values of a are from National Physical Laboratory, Tables of Weber parabolic cylinder functions. Computed by Scientific Computing Service Ltd. Mathematical Introduction by J. C. P. Miller. Her Majesty’s Stationery Office, London, England, 1955 (with permission).

PARABOLI:C

CYLINDER

713

FUNCTIONS Table

x

W(2.0,x)

W(3.0,x)

W(4.0,x)

W(5.0,x)

W(2.0,-2)

W(3.0, -4

i 1 I 1.4686 1.6117

I 1 \ 3.0749 3.5113

( 0)-9.6664

( 1)6.6590

W(4.0,-2)

19.e

W(5.0,~2j

1.8377 1.4984 1.2246 1.0035 a.2455

5.0

(-2)-1.9179

For interpolation,

t-31-3.8449 see 19.28.

(-4)-1.4564

(-4)1.1577

( 2)5.9987

( 3)2.8528i

714

PARABOLIC

Table

CYLINDER

FUNCTIONS

19.2

x

W(-1.0,x)

0.0

0.73148 0.65958 0.58108 0.49671 0.40726

W(-0.8,~)

W(-0.7,x)

W(-0.6,~)

W(-0.5,x)

0.75416 0.68457 0.60881 0.52750 0.44133

0.77982 0.71267 0.63980 0.56175 0.47908

0.80879 0.74421 0.67441 0.59981 0.52089

0.84130 0.77940 0.71281 0.64187 0.56693

0.87718 0.81803 0.75477 0.68766 0.61696

0.91553 0.85912 0.79925 0.73610 0.66984

0.31359 0.21659 0.11723 +0.01657 -0.08429

0.35102 0.25734 0.16111 +0.06324 -0.03529

0.39240 0.30233 0.20958 0.11490 +0.01912

0.43811 0.35200 0.26311 0.17206 +0.07954

0.48837 0.40658 0.32198 8EE .

0.54293 0.46584 0.38601 0.30379 0.21956

0.60064 0.52866 0.45409 0.37715 0.29811

-0.18412 -0.28164 -0.37549 -0.46422 -0.54635

-0.13342 -0.23002 -0.32384 -0.41357 -0.49783

-0.07684 -0.17198 -0.26523 -0.35538 -0.44119

-0.01369 -0.10679 -0.19880 -0.28870 -0.37536

+0.05650 -0.03384 -0.12386 -0.21269 -0.29933

0.13380 +0.04704 -0.04009 -0.12687 -0.21246

0.21727 0.13503 +0.05185 -0.03172 -0.11502

-0.62034 -0.68464 -0.73771 -0.77808 -0.80439

-0.57517 -0.64409 -0.70310 -0.75070 -0.78547

-0.52130 -0.59431 -0.65875 -0.71317 -0.75611

-0.45753 -0.53393 -0.60317 -0.66382 -0.71446

-0.38270 -0.46162 :;*'b;;1:; -0:65854

-0.29594 -0.37627 -0.45231 -0.52280 -0.58645

-0.19728 -0.27764 -0.35510 -0.42857 -0.49684

-0.81541 -0.81014 -0.78787 -0.74822 -0.69124

-0.80610 -0.81144 -0.80054 -0.77279 -0.72790

-0.78618 -0.80212 -0.80282 -0.78741 -0.75531

-0.75365 -0.78003 -0.79238 -0.78960 -0.77089

-0.70628 -0.74273 -0.76654 -0.77649 -0.77153

-0.64186 -0.68765 -0.72243 -0.74486 -0.75373

-0.55864 -0.61261 -0.65738 -0.69156 -0.71385

-0.61743 -0.52785 -0.42412 -0.30847 -0.18374

-0.66601 -0.58777 -0.49436 -0.38753 -0.26968

-0.70633 -0.64071 -0.55918 -0.46303 -0.35416

-0.73570 -0.68391 -0.61582 -0.53224 -0.43455

-0.75086 -0.71398 -0.66079 -0.59164 -0.50739

-0.74799 -0.72686 -0.68984 -0.63684 -0.56821

-0.72301 -0.71801 -0.69802 -0.66256 -0.61149

-0.05335 +0.07873 0.20811 0.33006 0.43974

-0.14378 -0.01339 +0.11741 0.24412 0.36198

-0.23506 -0.10884 +0.02083 0.14977 0.27340

-0.32474 -0.20540 -0.07973 +0.04850 0.17504

-0.40948 -0.29995 -0.18146 -0.05729 +0.06875

-0.48485 -0.38820 -0.28034 -0.16395 -0.04232

-0.54517 -0.46444 -0.37075 -0.26614 -0.15327

E 3:8 3.9

0.53233 0.60334 0.64885 0.66575 0.65207

0.46613 0.55184 0.61476 0.65118 0.65834

0.38695 0.48557 0.56460 0.61986 0.64786

0.29527 0.40440 0.49761 0.57035 0.61858

0.19236 0.30891 0.41360 0.50168 0.56868

+0.08071 0.20083 0.31342 0.41373 0.49706

-0.03541 +0.08365 0.19963 0.30797 0.40397

4.0 4.1 4.2 4.3 4.4

0.60721 0.53214 0.42952 0.30382 0.16115

0.63466 0.58002 0.49593 0.38565 0.25422

0.64616 0.61356 0.55042 0.45874 0.34234

0.63904 0.62958 0.58939 0.51923 0.42158

0.61072 0.62476 0.60892 0.56270 0.48725

0.55906 0.59598 0.60496 0.58437 0.53398

0.48303 0.54088 0.57391 0.57944 0.55599

4.5 4.6 t-i 4:9

+0.00918 -0.14329 -0.28674 -0.41153 -0.50861

+0.10831 -0.04397 -0.19348 -0.33057 -0.44572

0.20677 +0.05918 -0.09193 -0.23720 -0.36694

0.30072 0.16266 +0.01497 -0.13360 -0.27352

0.38544 0.26194 +0.12315 -0.02310 -0.16782

0.45522 0.35129 0.22716 +0.08947 -0.05374

0.50355 0.42375 0.31998 0.19740 +0.06277

5.0

-0.57025

-0.53023

-0.47182

-0.39516

-0.30146

-0.19341

-0.07580

0”:: 8:: :*56 0:7 E :*: 1:2 ::i 1.5 ::; :::

2.0 2.1 5-G 2:4 2.5 $2 22:: 3.0 t: 313 3.4 3.5

[(-/'"I

W(-0.9,x)

[(-;'7

[

(-l)4

1

[(-;)"I

[(-;)4]

W(-0.4,x)

1 1 [ (-i)4

(-l)4

1

PARABOLIC

CYLINDER

715

FUNCTIONS

Table

W(-1,0,-x)

W(-0.9,-x)

W(-0.7,-z)

W(-O&-x)

W(-0.6,-z)

W(-0.5,-z)

19.2

W(-0.4,-z)

0.73148 0.79607 0.85267 0.90067 0.93946

0.754i6 0.81697 0.87241 0.91990 0.95892

0.77982 0.84073 0.89490 0.94182 0.98099

0.80879 0.86771 0.92053 0.96682 1.00612

0.84130 0.89814 0.94958 0.99522 1.03467

0.87718 0.93193 0.98201 1.02707 1.06677

0.91553 0.96827 1.01711 1.06178 1.10197

0.96849 0.98722 0.99521 0.99202 0.97734

0.98892 1.00940 1.01990 1.01997 1.00923

1.01192 1.03413 1.04713 1.05048 1.04374

1.03797 1.06191 1.07745 1.08414 1.08151

1.06749 1.09323 1.11143 1.12160 1.12325

1.10070 1.12843 1.14951 1.16343 1.16966

1.13729 1.16736 1.19170 1.20981 1.22114

0.95092 0.91262 0.86244 0.80055 0.72729

0.98738 0.95418 0.90952 0.85341 0.78603

1.02655 0.99859 0.95962 0.90954 0.84835

1.06912 1.04657 1.01355 0.96978 0.91515

1.11589 1.09904 1.07228 1.03523 0.98760

1.16769 1.15695 1.13693 1.10714 1.06714

1.22511 1.22112 1.20855 1.18680 1.15529

0.64322 0.54911 0.44603 0.33528 0.21849

0.70774 0.61912 0.52099 0.41443 0.30081

0.77623 0.69355 0.60091 0.49914 0.38936

0.84963 0.77341 0.68684 0.59053 0.48532

0.92923 0.86006 0.78025 0.69014 0.59032

1.01659 0.95525 0.88304 0.80004 0.70659

1.11351 1.06102 0.99750 0.92281 0.83697

+0.09757 -0.02528 -0.14758 -0.26660 -0.37941

0.18179 +0.05934 -0.06427 -0.18651 -0.30459

0.27298 0.15171 +0.02758 -0.09709 -0.21967

0.37236 0.25309 0.12930 +0.00305 -0.12323

0.48166 0.36531 0.24278 +0.11588 -0.01322

0.60326 0.49090 0.37070 0.24419 +0.11327

0.74025 0.63319 0.51665 0.39182 0.26028

-0.48297 -0.57415 -0.64990 -0.70733 -0.74387

-0.41552 -0.51623 -0.60356 -0.67449 -0.72615

-0.33731 -0.44698 -0.54551 -0.62975 -0.69663

-0.24685 -0.36487 -0.47416 -0.57149 -0.65363

-0.14203 -0.26774 -0.38730 -0.49748 -0.59492

-0.01983 -0.15248 -0.28178 -0.40451 -0.51729

+0.12398 -0.01472 -0.15309 -0.28802 -0.41615

-0.75737 -0.74633 -0.70996 -0.64841 -0.56281

-0.75605 -0.76219 -0.74323 -0.69863 -0.62881

-0.74331 -0.76738 -0.76692 -0.74077 -0.68862

-0.71748 -0.76019 -0.77937 -0.77320 -0.74065

-0.67629 -0.73841 -0.77841 -0.79386 -0.78300

-0.61660 -0.69897 -0.76108 -0.79994 -0.81309

-0.53384 -0.63739 -0.72310 -0.78743 -0.82721

-0.45542 -0.32961 -0.18992 -0.04191 +0.10799

-0.53525 -0.42059 -0.28860 -0.14423 +0.00657

-0.61114 -0.51016 -0.38867 -0.25086 -0.10208

-0.68160 -0.59701 -0.48899 -0.36092 -0.21739

-0.74490 -0.67961 -0.58833 -0.47349 -0.33883

-0.79874 -0.75603 -0.68515 -0.58750 -0.46582

-0.83985 -0.82349 -0.77725 -0.70141 -0.59756

0.25266 0.38471 0.49679 0.58208 0.63477

0.15702 0.29976 0.42722 0.53205 0.60759

+0.05134 0.20225 0.34303 0.46597 0.56372

-0.06416 +0.09203 0.24366 0.38285 0.50171

-0.18934 -0.03124 +0.12831 0.28140 0.41981

-0.32421 -0.16811 -0.00420 +0.15987 0.31572

-0.46872 -0.31938 -0.15545 +0.01587 0.18634

0.65055 0.62708 0.56440 0.46513 0.33464

0.64841 0.65075 0.61301 0.53614 0.42379

0.62979 0.65910 0.64846 0.59705 0.50672

0.59285 0.64997 0.66833 0.64531 0.58085

0.53543 0.62083 0.66982 0.67800 0.64328

0.45473 0.56851 0.64950 0.69154 0.69050

Ez: 0:60280 0.68125 0.71794

0.18091

0.28240

0.38215 (-l)5

0.47771

0.56635

0.64481 t-z’6

0.70889 C-l’6

[(-i)5 1 c 1 [ (-;)5

1 Ii 1 [ 1 [ (-l)5

(-i)5

1 II 1

716

PARABOLIC

Table

CYLINDER

FUNCTIONS

19.2

W(-0.3,x)

W(-0.2,x)

W(-0.1,x)

we-4 4

W(O.l,x)

W(O.2,s)

W(O.3,x)

0.95411 0.90030 0.84377 0.78461 0.72293

0.98880 0.93725 0.88381 0.82851 0.77137

1.01364 0.96381 0.91299 0.86116 0.80828

1.02277 0.97388 0.92496 0.87595 0.82673

1.01364 0.96480 0.91691 0.86984 0.82344

0.98880 0.93920 0.89145 0.84540 0.80084

0.95411 0.90311 0.85480 0.80896 0.76536

0.65878 0.59225 0.52341 0.45236 0.37924

0.71237 0.65150 0.58875 0.52410 0.45756

0.75426 0.69902 0.64245 0.58445 0.52493

0.77719 0.72716 0.67647 0.62496 0.57244

0.77753 0.73192 0.68637 0.64067 0.59459

0.75757 0.71533 0.67388 0.63296 0.59228

0.72375 0.68386 0.64540 0.60809 0.57163

0.30421 0.22751 0.14946 +0.07042 -0.00912

0.38918 0.31906 0.24734 0.17425 0.10007

0.46383 0.40111 0.33677 0.27090 0.20361

0.51877 0.46381 0.40744 0.34961 0.29032

0.54790 0.50038 0.45186 0.40217 0.35118

0.55160 0.51063 0.46915 0.42691 0.38374

0.53573 0.50010 0.46446 0.42854 0.39209

-0.08857 -0.16725 -0.24435 -0.31894 -0.38999

+0.02522 -0.04982 -0.12443 -0.19788 -0.26933

0.13514 +0.06577 -0.00407 -0.07387 -0.14299

0.22960 0.16760 0.10454 +0.04073 -0.02340

0.29883 0.24510 0.19006 0.13384 0.07667

0.33945 0.29393 0.24713 0.19904 0.14975

0.35491 0.31679 0.27761 0.23725 0.19569

-0.45633 -0.51674 -0.56989 -0.61444 -0.64903

-0.33779 -0.40219 -0.46135 -0.51400 -0.55882

-0.21066 -0.27600 -0.33802 -0.39560 -0.44755

-0.08731 -0.15034 -0.21170 -0.27048 -0.32569

+0.01891 -0.03902 -0.09655 -0.15300 -0.20756

0.09941 +0.04828 -0.00327 -0.05478 -0.10567

0.15296 0.10917 0.06450 +0.01926 -0.02617

-0.67233 -0.68311 -0.68033 -0.66313 -0.63097

-0.59448 -0.61966 -0.63315 -0.63385 -0.62088

-0.49261 -0.52947 -0.55686 -0.57356 -0.57846

-0.37619 -0.42082 -0.45833 -0.48749 -0.50710

-0.25934 -0.30731 -0.35040 -0.38745 -0.41729

-0.15523 -0.20267 -0.24709 -0.28749 -0.32283

-0.07129 -0.11551 -0.15811 -0.19829 -0.23518

-0.58369 -0.52157 -0.44541 -0.35655 -0.25697

-0.59365 -0.55190 -0.49584 -0.42613 -0.34402

-0.57063 -0.54943 -0.51451 -0.46594 -0.40427

-0.51607 -0.51344 -0.49851 -0.47084 -0.43039

-0.43878 -0.45085 -0.45256 -0.44315 -0.42215

-0.35203 -0.37401 -0.38777 -0.39239 -0.38713

-0.26783 -0.29526 -0.31648 -0.33055 -0.33663

-0.14924 -0.03654 +0.07742 0.18846 0.29213

-0.25134 -0.15050 -0.04453 +0.06302 0.16814

-0.33055 -0.24643 -0.15413 -0.05645 +0.04330

-0.37754 -0.31318 -0.23871 -0.15612 -0.06794

-0.38941 -0.34517 -0.29013 -0.22549 -0.15299

-0.37148 -0.34523 -0.30852 -0.26190 -0.20639

-0.33401

0.38382 0.45904 0.51364 0.54413 0.54793

0.26651 0.35370 0.42535 0.47744 0.50658

0.14132 0.23354 0.31572 0.38368 0.43357

+0.02278 0.11257 0.19762 0.27395 0.33764

-0.07486 +0.00615 0.08689 0.16386 0.23342

-0.14349 -0.07518 -0.00389 +0.06754 0.13597

-0.18313 -0.12880 -0.06948 -0.00733 +0.05511

0.52370 0.47151 0.39312 0.29197 0.17327

0.51029 0.48726 0.43762 0.36308 0.26703

0.46212 0.46690 0.44663 0.40138 0.33274

0.38503 0.41300 0.41921 0.40237 0.36248

0.29194 0.33601 0.36270 0.36981 0.35608

0.19809 0.25059 0.29037 0.31476 0.32171

0.11504 0.16948 0.21549 0.25027 0.27144

0.04376 (-$)4

0.15455 (-l)3

0.24393 (-l)3

0.30095 (-;)3

0.32145 (-2)3

0.31009 c-y

0.27719

[ 1 II 1 II 1 [ 1 II 1 [

I"o';;g;; -0:27027 -0.23072

1

PARABOLIC

CYLINDER

717

FUNCTIONS

Table

x

W(-0.3,-z)

W(-0.2,-z)

W(-0.1,--z)

W(O.l,-z)

W(O, -4

19.2

W(O.2,~2)

W(O.3,~2)

0:4

0.95411 1.00506 1.05296 1.09759 1.13868

0.98880 1.03835 1.08581 1.13097 1.17362

1.01364 1.06245 1.11016 1.15665 1.20172

1.02277 1.07165 1.12050 1.16924 1.21771

1.01364 1.06348 1.11435 1.16622 1.21899

0.98880 1.04037 1.09399 1.14968 1.20741

0.95411 1.00797 1.06483 1.12477 1.18782

8-Z 0:7 0.8 0.9

1.17589 1.20884 1.23706 1.26006 1.27725

1.21344 1.25007 1.28307 1.31193 1.33606

1.24510 1.28645 1.32534 1.36129 1.39368

1.26568 1.31285 1.35884 1.40315 1.44521

1.27248 1.32644 1.38053 1.43429 1.48719

1.26706 1.32845 1.39129 1.45520 1.51968

1.25396 1.32307 1.39494 1.46928 1.54567

1.28802 1.29171 1.28761 1.27501 1.25320

1.35480 1.36744 1.37321 1.37129 1.36083

1.42185 1.44504 1.46241 1.47304 1.47598

1,48433 1.51974 1.55054 1.57575 1.59429

1.53855 1.58760 1.63341 1.67498 1.71113

1.58412 1.64775 1.70967 1.76885 1.82408

1.62356 1.70224 1.78087 1.85841 1.93366

1.22150 1.17926 1.12596 1.06115 0.98458

1.34098 1.31091 1.26983 1.21705 1.15200

1.47020 1.45469 1.42841 1.39039 1.33973

1.60502 1.60672 1.59813 1.57800 1.54509

1.74059 1.76201 1.77390 1.77474 1.76299

1.87401 1.91713 1.95181 1.97628 1.98870

2.00522 2.07150 2.13072 2.18093 2.22000

0.89620 0.79618 0.68503 0.56357 0.43300

1.07426 0.98365 0.88026 0.76448 0.63710

1.27565 1.19757 1.10510 0.99819 0.87711

1.49825 1.43644 1.35882 1.26478 1.15405

1.73709 1.69557 1.63706 1.56041 1.46471

1.98714 1.96968 1.93446 1.87972 1.80390

2.24569 2.25565 2.24752 2.21894 2.16770

z*e7 2:9

0.29492 0.15140 +0.00489 -0.14168 -0.28503

0.49932 0.35277 0.19959 +0.04242 -0.11563

0.74256 0.59571 0.43825 0.27241 +0.10100

1.02673 0.88342 0.72523 0.55388 0.37173

1.34942 1.21444 1.06021 0.88776 0.69887

1.70575 1.58440 1.43949 1.27129 1.08078

2.09177 1.98946 1.85956 1.70140 1.51507

z! 3:2 3.3 3.4

-0.42150 -0.54722 -0.65815 -0.75027 -0.81974

-0.27098 -0.41967 -0.55742 -0.67978 -0.78229

-0.07258 -0.24442 -0.41011 -0.56487 -0.70368

+0.18182 -0.01213 -0.20574 -0.39404 -0.57158

0.49606 0.28264 +0.06279 -0.15855 -0.37567

0.86979 0.64105 0.39827 +0.14618 -0.10952

1.30151 1.06267 0.80159 0.52249 +0.23083

-0.86311 -0.87754 -0.86098 -0.81248 -0.73233

-0.86067 -0.91101 -0.93010 -0.91559 -0.86631

-0.82147 -0.91331 -0.97470 -1.00185 -0.99193

-0.73259 -0.87118 -0.98158 -1.05844 -1.09719

-0.58228 -0.77162 -0.93674 -1.07077 -1.16728

-0.36221 -0.60449 -0.82836 -1.02554 -1.18779

-0.06670 -0.36232 -0.64721 -0.91187 -1.14634

-0.62227 -0.48559 -0.32717 -0.15346 +0.02771

-0.78249 -0.66595 -0.52024 -0.35070 -0.16437

-0.94343 -0.85640 -0.73270 -0.57611 -0.39249

-1.09434 -1.04786 -0.95753 -0.82515 -0.65483

-1.22069 -1.22662 -1.18240 -1.08743 -0.94350

-1.30732 -1.37730 -1.39231 -1.34891 -1.24610

-1.34070 -1.48554 -1.57256 -1.59514 -1.54901

0.20739 0.37594 0.52351 0.64069 0.71919

+0.03014 0.22299 0.40359 0.56113 0.68534

-0.18962 +0.02291 0.23414 0.43218 0.60494

-0.45301 -0.22843 +0.00810 0.24408 0.46598

-0.75508 -0.52942 -0.27649 -0.00874 +0.25940

-1.08573 -0.87285 -0.61582 -0.32626 -0.01876

-1.43285 -1.24877 -1.00271 -0.70462 -0.36835

0.75259 5316

0.76721 (-p5

0.74090 (-l)5

0.65996 (-l)5

0.51219 5316

+0.28970

-0.01132

E 00-32

1.0 1.1 1.2 ::: 1.5 1.6 :*a7 119 2.0 2.1 z.3

2:4 22

4.5 4.6 i-87

4:9 5.0

[

1 1 1 [

1 [

1 [

1 c(-[I71 [(-;P1

718 Table

PARABOLIC

CYLINDER

FUNCTIONS

19.2

x

W(O.4,2)

W(O.5,2)

W(O.6,s)

W(O.7,x)

W(0.8,~)

W(O.9,x)

W(l.O,x)

0.0 0.1

0.2 0.3 0.4

0.91553 0.86271 0.81331 0.76709 0.72376

0.87718 0.82232 0.77155 0.72456 0.68104

0.84130 0.78433 0.73205 0.68408 0.64007

0.80879 0.74973 0.69590 0.64687 0.60222

0.77982 0.71874 0.66339 0.61328 0.56794

0.75416 0.69116 0.63436 0.58321 0.53718

0.73148 0.66667 0.60852 0.55639 0.50970

0.5 0.6 3.7 0.8 0.9

0.68304 0.64462 0.60820 0.57347 0.54011

0.64064 0.60305 0.56793 0.53495 0.50380

0.59964 0.56244 0.52810 0.49629 0.46666

0.56155 0.52446 0.49058 0.45952 0.43095

0.52692 0.48979 0.45614 0.42558 0.39774

0.49578 0.45853 0.42499 0.39476 0.36745

0.46791 0.43051 0.39703 0.36704 0.34013

1.0 1.1 1.2 1.3 1.4

0.50782 0.47630 0.44523 0.41435 0.38338

0.47414 0.44567 0.41808 0.39108 0.36438

0.43889 0.41266 0.38765 0.36358 0.34015

0.40452 0.37992 0.35682 0.33494 0.31399

0.37228 0.34888 0.32720 0.30697 0.28790

0.34271 0.32020 0.29960 0.28063 0.26299

0.31594 0.29412 0.27435 0.25634 0.23981

1.5 1.6 1.7 1.8 1.9

0.35206 0.32018 0.28752 0.25395 0.21934

0.33771 0.31084 0.28354 0.25561 0.22689

0.31709 0.29416 0.27111 0.24773 0.22384

0.29370 0.27382 0.25410 0.23433 0.21430

0.26973 0.25219 0.23506 0.21812 0.20115

0.24643 0.23071 0.21559 0.20085 0.18629

0.22451 0.21019 0.19662 0.18361 0.17094

2.0 2.1 2.2 2.3 2.4

0.18363 0.14682 0.10899 0.07029 +0.03094

0.19726 0.16665 0.13504 0.10248 0.06908

0.19927 0.17390 0.14767 0.12054 0.09255

0.19384 0.17280 0.15107 0.12857 0.10528

0.18398 0.16644 0.14841 0.12976 0.11045

0.17173 0.15700 0.14195 0.12647 0.11045

0.15845 0.14595 0.13331 0.12038 0.10707

2.5 2.6 2.7 2.8 2.9

-0.00872 -0.04827 -0.08719 -0.12486 -0.16058

0.03504 +0.00063 -0.03378 -0.06773 -0.10069

0.06378 0.03440 +0.00466 -0.02513 -0.05457

0.08121 0.05645 0.03113 +0.00547 -0.02025

0.09043 0.06972 0.04840 0.02659 +0.00447

0.09385 0.07662 0.05879 0.04042 0.02163

0.09330 0.07900 0.06416 0.04879 0.03296

?Y 3:2 3.3 3.4

-0.19356 -0.22295 -0.24788 -0.26746 -0.28083

-0.13202 -0.16105 -0.18700 -0.20910 -0.22656

-0.08319 -0.11043 -0.13568 -0.15826 -0.17749

-0.04569 -0.07041 -0.09392 -0.11569 -0.13511

-0.01769 -0.03960 -0.06087 -0.08106 -0.09969

+0.00259 -0.01649 -0.03531 -0.05355 -0.07080

0.01677 :;*;g;; -0:03216 -0.04774

3.5 3.6 3.7 3.8 3.9

-0.28722 -0.22598 -0.27664 -0.25895 -0.23299

-0.23861 -0.24455 -0.24381 -0.23596 -0.22079

-0.19265 -0.20307 -0.20814 -0.20735 -0.20033

-0.15158 -0.16446 -0.17317 -0.17718 -0.17604

-0.11623 -0.13014 -0.14088 -0.14793 -0.15084

-0.08664 -0.10061 -0.11222 -0.12101 -0.12652

-0.06242 -0.07581 -0.08750 -0.09707 -0.10411

4.0 4.1 4.2 4.3 4.4

-0.19913 -0.15813 -0.11115 -0.05975 -0.00585

-0.19835 -0.16901 -0.13343 -0.09266 -0.04811

-0.18692 -0.16717 -0.14143 -0.11032 -0.07481

-0.16946 -0.15730 -0.13965 -0.11684 -0.08947

-0.14922 -0.14284 -0.13162 -0.11566 -0.09531

-0.12836 -0.12624 -0.11996 -0.10948 -0.09494

-0.10824 -0.10912 -0.10653 -0.10030 -0.09046

4.5 4.6 4.7 4.8 4.9

+0.04828 0.10016 0.14714 0.18659 0.21607

-0.00149 +0.04518 0.08968 0.12967 0.16286

-0.03614 +0.00411 0.04416 0.08203 0.11567

-0.05843 -0.02485 +0.00985 0.04406 0.07604

-0.07112 -0.04392 -0.01477 +0.01506 0.04414

-0.07669 -0.05525 -0.03141 -0.00614 +0.01943

-0.07716 -0.06075 -0.04174 -0.02086 +0.00100

5.0

0.23350 (-;I2

0.18712

0.14307

0.10399

0.07092

0.04399

0.02281

c 1 [!-;I11 [c-y1 [(-217 1 [t-y 1 IIc-y1 IIt-y 1

PARABOLIC

CYLINDER

P19

FUNCTIONS Table

5

W(O.4,-2)

W(O.5,

-x)

W(0.6,-:e)

W(O.7,

-x)

W(O.8,-2)

W(O.9,

-2)

19.2

W(l.O,-z)

0.91553 0.97201 1.03235 1.09671 1.16520

0.87718 0.93642 1.00031 1.06911 1.14300

0.84130 0.90331 0.97072 1.04386 1.12302

0.80879 0.87352 0.94433 1.02166 1.10591

0.77982 0.84714 0.92122 1.00258 1.09173

0.75416 0.82396 0.90115 0.98636 1.08022

0.73148 0.80361 0.88375 0.97265 1.07106

1.23789 1.31475 1.39567 1.48046 1.56879

1.22215 1.30664 1.39648 1.49158 1.59174

1.20846 1.30040 1.39896 1.50419 1.61602

1.19746 1.29663 1.40371 1.51888 1.64225

1.18917 1.29538 1.41079 1.53574 1.67051

1.18338 1.29644 1.42000 1.55459 1.70068

1.17975 1.29949 1.43106 1.57519 1.73254

1.6602 1.7541 1.8497 1.9460 2.0418

1.6966 1.8057 1.9184 2.0337 2.1506

1.7343 1.8586 1.9884 2.1230 2.2613

1.7738 1.9133 2.0603 2.2144 2.3746

1.8153 1.9700 2.1345 2.3083 2.4909

1.8586 2.0286 2.2107 2.4048 2.6102

1.9037 2.0891 2.2891 2.5037 2.7327

2.1358 2.2263 2.3115 2.3891 2.4570

2.2677 2.3833 2.4956 2.6023 2.7009

2.4020 2.5437 2.6843 2.8216 2.9529

2.5397 2.7083 2.8785 3.0480 3.2141

2.6811 2.8777 3.0788 3.2823 3.4854

2.8264 3.0520 3.2856 3.5249 3.7674

2.9756 3.2316 3.4991 3.7762 4.0605

2.5125 2.5529 2.5754 2.5770 2.5548

2.7886 2.8623 2.9188 2.9546 2.9660

3.0752 3.1853 3.2793 3.3532 3.4030

3.3737 3.5231 3.6583 3.7748 3.8678

3.6849 3.8770 4.0573 4.2209 4.3624

4.0097 4.2479 4.4775 4.6931 4.8889

4.3487 4.6368 4.9201 5.1930 5.4490

2.5061 2.4283 2.3192 2.1772 2.0013

2.9496 2.9018 2.8196 2.7001 2.5413

3.4241 3.4124 3.3634 3.2734 3.1389

3.9321 3.9626 3.9538 3.9007 3.7984

4.4760 4.5555 4.5944 4.5863 4.5251

5.0582 5.1940 5.2887 5.3346 5.3240

5.6811 5.8811 6.0405 6.1502 6.2008

1.7914 1.5484 1.2746 0.9733 0.6496

2.3419 2.1015 1.8213 1.5038 1.1529

2.9573 2.7270 2.4478 2.1206 1.7487

3.6430 3.4312 3.1612 2.8324 2.4466

4.4050 4.2211 3.9697 3.6486 3.2576

5.2495 5.1041 4.8822 4.5794 4.1934

6.1832 6.0883 5.9081 5.6359 5.2669

+0.3098 -0.0381 -0.3848 -0.7198 -1.0317

0.7746 iO.3767 -0.0314 -0.4385 -0.8319

1.3369 0.8923 +0.4244 -0.0553 -0.5332

2.0074 1.5210 0.9962 +0.4445 -0.1199

2.7987 2.2767 1.6994 1.0779 +0.4263

3.7241 3.1746 2.5511 1.8636 1.1259

4.7985 4.2315 3.5700 2.8225 2.0016

-1.3084 -1.5382 -1.7095 -1.8124 -1.8391

-1.1977 -1.5216 -1.7893 -1.9871 -2.1032

-0.9940 -1.4209 -1.7966 -2.1039 -2.3268

-0.6804 -1.2184 -1.7136 -2.1453 -2.4930

-0.2378 -0.8941 -1.5199 -2.0907 -2.5817

+0.3558 -0.4249 -1.1915 -1.9160 -2.5692

1.1251 +0.2152 -0.7013 -1.5936 -2.4280

f-i 4:9

-1.7844 -1.6469 -1.4292 -1.1387 -0.7876

-2.1283 -2.0567 -1.8870 -1.6231 -1.2742

-2.4513 -2.4668 -2.3670 -2.1513 -1.8252

-2.7376 -2.8632 -2.8579 -2.7153 -2.4359

-2.9685 -3.2291 -3.3452 -3.3040 -3.0995

-3.1213 -3.5437 -3.8110 -3.9027 -3.8054

-3.1692 -3.7818 -4.2326 -4.4924 -4.5392

5.0

-0.3927

0.0 0”:: E

0.5 i:; E 1.0 1.1 1.2 ::: 1.5 1.6 287 1:9 2.0 22.: 2:3 2.4 2.5 22.; 218 2.9 3.0 ;*: 3:3 3.4 3.5 33:; ;:: t:l" t *: 4:4 2:

-0.8557 -1.4010 -2.0281 -2.7346 -3.5149 -4.3599 c-p (-;I1 C-f)2 c-y c-y C-f)1 [ 1 [ 1 [ 1 [ 1 [ 1 [ 1 [(-;I3 1

720

PARABOLIC CYLINDER

Table 19.3

AUXILIARY

FUNCTIONS

FUNCTIONS

The functions 8,. 82,83of 19.10 and 19.23 are neededin Darwin’s expansionand also the function Tof 19.7 and 19.20. 93

$1 0.00000

E i-i 0:9

z

81

62

7

0.05008 0.10066 0.15222 0.20521

0.39270 0.34278 0.29337 0.24498 0.19817

-0.70270 -0.64181 -0.57855 -0.51304 -0.44540

6.9519 7.2093 7.4716 7.7388 8.0109

5.5506 5.7981 6.0507 6.3084 6.5712

4.1079 4.2291 4.3511 4.4738 4.5972

0.26006 0.31713 0.37678 0.43929 0.50492

0.15355 0.11182 0.07387 0.04088 0.01468

-0.37574 -0.30415 -0.23071 -0.15549 -0.07857

8.2880 8.5700 8.8569 9.1487 9.4454

6.8391 7.1120 7.3901 7.6732 7.9614

4.7213 4.8461 4.9716 5.0977 5.2246

E

192

81

92

7 5.3521 5.4803 5.6092

0.01513 0.04341 0.08086 0.12617

0.00000 0.08015 0.16185 0.24502 0.32964

9.7471 10.0537 10.3652 10.6817 11.0031

8.2546 8.5530 8.8564 9.1649 9.4784

0.97473 1.06696 1.16344 1.26422 1.36937

0.17866 0.23786 0.30347 0.37527 0.45309

0.41566 0.50304 0.59175 0.68175 0.77300

11.3295 11.6608 11.9970 12.3382 12.6843

9.7970 10.1207 10.4494 10.7832 11.1220

5.9996 6.1310 6.2631 6.3958 6.5290

1.47894 1.59299 1.71155 1.83466 1.96236

0.53679 0.62626 0.72142 0.82220 0.92853

0.86549 0.95917 :- l":tt:: 1:24716

13.0354 13.3914 13.7524 14.1183 14.4892

11.4659 il.8148 12.1688 12.5278 12.8919

6.6629 6.7974 6.9325 7.0682 7.2045

2.09467 2.23163 2.37325 2.51956 2.67058

1.04036 1.15764 1.28034 1.40843 1.54187

1.34539 1.44470 1.54506 1.64646 1.74888

14.8651 15.2459 15.6316 16.0223 16.4180

:t 26% 14:0144 14.3987 14.7880

:* E9' 717555 7.8947

2.82632 2.98681 :- :::o": 3149688

1.68063 1.82470 1.97406 2.12867 2.28853

1.85229 1.95669 2.06206 2.16837 2.27562

16.8186 17.2242 17.6348 18.0503 18.4708

15.1823 15.5817 15.9861 16.3956 16.8101

8.0344 a.1747 a.3155 8.4569 8.5989

3.67648 3.86089 4.05011 4.24416 4.44305

2.45363 2.62394 2.79946 2.98017 3.16606

:* i9":;; 2:60281 2.71365 2.82536

18.8962 19.3266 19.7620 20.2024 20.6477

::- ;:2 la:0838 18.5184 18.9581

a.7413 8.8844 9.0279 9.1720 9.3166

4.64678 4.85537 5.06880 5.28711 5.51028

3.35712 3.55335 3.75474 3.96127 4.17295

2.93791 3.05131 3.16554 3.28058 3.39643

21.0980 21.5532 22.0135 22.4787 22.9488

19.4028 19.8525 20.3073 20.7671 21.2319

9.4617 9.6074 9.7535 9.9002 10.0474

::9"

5.73833 5.97126 6.20908 6.45178 6.69938

44.22 4183875 5.07093 5.30822

:* Z~ 3174872 3.86770 3.98743

23.4240 23.9041 24.3892 24.8792 25.3742

21.7017 22.1766 22.6565 23.1414 23.6314

10.1951 10.3433 10.4920 10.6411 10.7908

5. 0

6.95188

5.55062 [ '-;'"] _

4.10792

25.8742

24.1264

10.9410 (-y [ . .. I

::1" :*: 1:4 ::2 :*i 1:9 21" z 2:4 2.5 ::7" 2: 3. 0 ::: 33:; 3.5 333 ::: 4. 0 ::: 44:: 4. 5 44::

0.57390 0.64640 0.72261 0.80265 0.88666

T

0.00000

[ 5W.3 . .

.

.

:*. 'BEi

7.3414

When interpolating for 92and 83for 6near unity, it is better to interpolate for + and then use ti2 = f r312

or a3 = i (-.)3/Z.

20. Mathieu

Functions

GERTRUDE

BLANCH

1

Contents Page

Mathematical Praperties . . . . . . . . . . . . . . . . . . 20.1. Mathieu’s Equation . . . . . . . . . . . . . . . 20.2. Determination of Characteristic Values . . . . . . . 20.3. Floquet’s Theorem and Its Consequences . . . . . . 20.4. Other Solutions of Mathieu’s Equation . . . . . . . 20.5. Properties of Orthogonality and Normalization . . . . 20.6. Solutions of Mathieu’s Modified Equation for Integral v 20.7. Representations by Integrals and Some Integral Equations 20.8. Other Properties . . . . . . . . . . . . . . . . . 20.9. Asymptotic Representations . . . . . . . . . . . . 20.10. Comparative Notations . . . . . . . . . . . . . . References

. . . . . . . . . . . . .

722 722 722 727 730 732 732 735 738 740 744

. . . . . . . . . . . . . . . . . . . . . . . . . .

745

Table 20.1. Characteristic

Values (O
. .

. . . . . .

Values, Joining Factors, Some Critical . . . . . . . . . . . . . . . . . . . . .

748

Even Solutions

b, d(O, a), se, (ljr *)p 84 ($ cz), (Jn rel="nofollow">f go,4d, (4P)If,, I(a) q=O(5)25, G+RI-

(4~+2) Ji h+2qq-+=.16(-.04)0, ?+=o,

Table 20.2. Coefficients

8D or S (4+-2) 45 8D

1, 2, 5, 10, 15

A, and B, . . . . . . . . . . . . . . . .

p=5, 25; r=O, 1, 2, 5, 10, 15,

1 Aeronautical

Research Laboratories,

Wright-Patterson

750

9D

Air Force Base, Ohio.

721

20. Mathieu Mathematical 20.1. Mathieu’s Canonical

Form

of the

Mathieu’e

Relation Equation

a=

Differential

cash 2u)f =0

Between Mathieu’s equation for the Elliptic Cylinder

The wave equation

in Cartesian

Equation

(v=iu,

y=f)

and

Wave

the

coordinates

is

g;+gp$+k’W=o

20.1.3

A solution W is obtainable by separation of variThus, let ables in elliptical coordinates. x=p cash u cos v; y=p sinh u sin v;

2=2;

p a positive constant ; 20.1.3 becomes 20.1.4 SW 2 *m+ p2 (cash 2u-cos

Assuming a solution

An Algebraic

2v) of the fprm

9-t

(1-t2)

(1-P)

20.1.8

Thus!

Mathieu’s

2v) variables,

it follows

20.1.5

where c is a constant. Again, from the fact that G=c and that u, v are independent variables, one sets 20.1.6

* a=d2f A+ (k2Lc) p2co& 2u 722

Equation

(co9 v= t>

Wave

Equation

$+(c--4@‘)Y=o

equation is a special c=a+2q. of Characteristic

y=mgo (A,,, cos mz+

B,,, sin

* case of

Values

A solution of 20.1.1 with v replaced by period u or 2r is of the form 20.2.1

du2f

‘$-2(btl)t

20.2. Determination

2

U, v are independent

of Mathieu’s

to Spheroidal

20.1.8, with E=-4,

where

2,

Form

$+(a+2*-4*P)y=O

Relation

1 d2cp (oz2+G=o

Since that

2v

cos

20.1.7

and substituting the above into 20.1.4 one obtains, after dividing through by W,

p2 (cash 2u-cos

&”

where a is a constant. The above are equivalent to 20.1.1 and 20.1.2. The constants c and a are often referred to as separation constants, due to the role they play in 20.1.5 and 20.1.6. For some physically important solutions, the function g must be periodic, of period r or 2~. It can be shown that there exists a countably infinite set of churacteristic values a,(q) which yield even periodic solutions of 20.1.1; there is another countably infinite sequence of characteristic values b,(q) which yield odd periodic solutions of 20.1.1. It is known that there exist pei :odic solutions of period kr, where k is any positive integer. In what follows, however, the term character&tic value will be reserved for a value associated with solutions of period r or 2r only. These characteristic values are of basic importance to the general theory of the differential equation for arbitrary parameters a and q.

w= &)f(u)dv)

*G=

d2gs+--.21 (k2-c) $

-

Equation

cos 2v)y=O

Modified

‘z2-(a-2q

20.1.2

Properties

Equation Differential

d2y a+(a-2q

Functions

2,

having

mz)

where B, can be taken as zero. If the above is substituted into 20.1.1 one obtains 20.2.2 $i2

[(a--m2)A,-q(A,-2+A,+2)1

+m$-I

2

[(a-m2)Bm-q(Bm-2+Bm+2)] A+,,, B-,=0

*seepage n.

~09 m2

’

sin mz=O m>o

MATHIEU

Equation 20.2.2 can be reduced to one of four simpler types, given in 20.2.3 nnd 20.2.4 below 20.2.3

yo=~oA2,+,

20.2.4

yl=go

~0s @m+p)z,

p=O or 1 p==O or 1

Bz,+n sin (2m+p)z,

If p=O, the solution is of period S; if p= 1, the solution is of period 2x. Recurrence

Relations

Even solutions 20.2.5

the

Coefficients

(u-m2)A,-q(A,~2+Am+2)

20.2.8 ’

)

I I

UC,,,

Gm+2=Vm-

=

=0

(mL3)

=0

I pl=d=O;

cPr=d=&=l,

(m>3!

Odd solutions of period 2~

Let Go,=B,IB,-2;

G,,,=Ge, or Go,,, when the same operations apply to both, and no ambiguity is likely to arise. Further let

Equations 20.2.14

are equivalent

20.2.22

VI-1

to

Ge2= Vo; Gel= Vz-$

solution

If 20.2.19 is suitably combined with 20.2.13-20. .18 there result four types of continued fractions, the roots of which yield the required character’ 2 tic values 2 1 1 Roots:1 az, 20.2.21 vcl _-__ v,- v,- -ve- . . . =0

(a-m2>/q.

20.2.5-20.2.7

I

associated with even periodic solutions

a = b,, associated with odd periodic

along wit,11 20.2.10 for m > 3.

V,=

if Gn+2=&a+J&a-1

The four choices of the parameters (pl, (p, d correspond to the four types of solutions 20. .320.2.4. Hereafter, it will be convenient to eparate the characteristic values a into two m i jor subsets: u=u,,

(a-l)&+q(B,-Bd=o,

20.2.13

if G,,,+z=B2,/B2,-2 cpo=d=l, if G,,,+~=A2a+JA~r--l ~

a=-1;

of period ?r:

Ge,=A,IA,+

(po=2, if G,,+2=A21/A21-2

q,=d=+q,=O,

* 20.2.10 (a-m2)B,--(B,_,+B,+,)=O

20.2.12

(mtZ3)

. * * vo;+‘pl

where

of period 27r:

(a-4)B,--qB,=O

20.2.11

1

v m--vm-,lVffI-2-

along with 20.2.7 for m 2 3.

20.2.9

R, I

20.2.19

(a-lb%-&%+&=Oj

Odd solutions

of perio$

These three-term recurrence relations amon the coefficients indicate that every G, can be devel ped into two types of continued fractions. 1 hus 20.2.15 is equivalent to

of period ?r:

(a--4)Ar-q(2A,+A,)

Even solutions

20.2.18 K=Goc, for odd solutions along with 20.2.15

20.2.20

aAo-qAz=O

20.2.6 20.2.7

Among

~723

FUNCTIONS

20.2.23

_--- 1 v,V2

----v,-1

1 v,1 v,-

1 Jr,1 vs-

* * * =0 - . . =O

Roots: uC,+1 Roots:~ bz,

2

20.2.15

G,=l/(V,-Gm+p)

20.2.24

Cm2 31,

for even solutions of period ?r. Similarly

VI+1

20.2.16 VI-- 1 =Ge,; for even solutions 21r, along with 20.2.15

of period

20.2.17 VI+ 1= Goog,for odd solutions 2r, along with 20.2.15 *seepageII.

of period

_---v,-1

1 v,-

1 v,-

. * . =o

Roots:

b’,+l f

If a is a root of 20.2.21-20.2.24, then the co responding solution exists and is an entire func { ion of 2, for general complex values of p. If p is real, then the Sturmian theory of second order linear differential equations yields the

724

MATHIEU

following: (a) For a fixed real p, characteristic values a, and b, are real and distinct, if p#O; aoO and dn>, preach r2 as p approaches zero. (b) A solution of 20.1.1 associated with a, or b, has T zeros in the interval 0 5 z
FUNCTIONS Power

Series

for

Characteristic

8

64

1536

Values

20.2.2s

al(++n-92+9-L-h(a)

llq6 __49q6 36864+589824

55q' --m-35389440+

83n" -' *

b,(P)=4-b+~4-7g~2~2~o 21391ps '+458647142400+ 1002401$ 7g626240

a2(*) =J+5$-si+

16690684Olq8 -458647142400+ 32

-

28

-

24

-

/

a3(-*)=g+92-93+13p4 5q6 16 64 20480+16384 b&d 1961 p6 609q' -23592960+104857600+ b,(p)=16+~-~+27~l~o~o~o+

a&)=36+2+

' ' *

..'

...

37qe +891813888+ b&)=36+2+

' ' *

' ' *

70

5861633q' 187q4 43904000-92935987200000+.

**

70

6743617q' 187q4 43904000+92935987200000+

' *'

For ~27, and IpI not too large, a, is approximately equal to b,, and the following approximation may be used 20.2.26 a, br -24L

FIGURE

20.1.

Characteristic

Values

a,, b,

1,=0,1(1)5

="+2(:-1)a+

(59+7) q4 32(~~-1)~(r~-4) (9r4+58++29) +64(+1)"(+4)(+9)+

$ ' ' '

MATHIEU

The above expansion is not limited to integral values of r, and it is a very good approximation for r of the form n++ where n is a.n integer. In case of integral values of r=n, the series holds only up to terms not involving r2-n2 in the denominator. Subsequent terms must be derived specially (as shown by Mathieu). Mulholland and Goldstein [20.38] have computed characteristic values for purely imaginary Q and found that a, and a2 have a common real value for IpI in the neighborhood of 1.468; Bouwkamp [20.5] has computed this number as qo= l i 1.46876852 to 8 decimals. For values of -iq>-iqo, a0 and a2 are conjugate complex numbers. From equation 20.2.25 it follows that the radius of convergence for the series defining a0 is no greater than \qol. It is shown in [20.36], section 2.25 that the radius of convergence for a2,,(q), n22 is greater than 3. Furthermore a,-b,=O(qr/rr-l), Power

in q for the sufficiently

Series

FUNCTIONS

p25

se2(2,q)=sin%--

..

&g+n’(e!.&eg9+.

20.2.28

cdz,n>=cos

se,k d

~1 4(r+l) -cos [(r-22)z-p(~/2)] (rz--p(r/2))

--p

1

cos (r+2) 2-q E [

4(r-1)

I

i

+a” 1

cos[(r+4)~-pW)l+cos [@-4)2--p(+)] 32(r-1) (r-2) Wr+ 1)(r+2) -cos,rz,,2)l[~~~:t!]}~.

with p=O for ce,(z, q), p=l

~

for se,(z, a), r23.

r+ co.

Periodic Functions small 1q I)

(for

20.2.27 ceo(z, q)=2-1

[

1-i

cos 2z+a2

-a” &g-”

(

‘*-&

>

;;; 2z)+. . .]

cel(z, q)=cos z--S cos 32 8

sq(z, q) =sin z-z

8

sin 32

FIGURE

ce2(z, q) =cos 22-q

(s!p)+,

(E.&l9;;;2~)+.

20.2.

..

Even

Periodic

Mathieu q=l.

Functions,

Orders1 O-6

,.

726

MATHIEU

FUNCTIONS

IE

I .4

Ii

IC

.a

.E .4

.2 -.2 C

-.4 -

-.2 :6:4 -6 55

-1.0 -

FIGURE

20.3.

Odd

Periodic

Mathieu

Functions,

Orders

1-5

-.6

q-l. -I 0

20.5.

FIGURE

cr 1.6 e

Odd Periodic

dfathieu q= 10.

Functions,

Orders

1-5

For coefficients associated with above functions

20.2.29 A;(O)=2-~; A;(O)=B;(O)=l, r>O A;,=[(-l)“q”/s! s! 22*-1] A;+ . . ., s>O A:+,, =[(-l)“r! B:+s.

Asymptotic

@/4”(r+s)!

Expansion

Let w=2r+l,

for

s!] c:+ . . . rs>o,C:=A;

Characteristic

p=w*p, (oreal.

Values,

orB;

q>l

Then

20.2.30 a - br+1 --2p+2w&-~----

4 -----~ 212p

w2+1 w?

4 217$/2

where

FIGURE

20.4.

Even

Periodic

Mathieu

q= 10.

Functions,

Orders

O-5

33 410 405

d2=w++-g

d, ---. 4 22bqb/2

~20~2

..

MATHIEU

t727

FUNCTIONS

proportional to F, (-2) ; the second, independent solution of 20.3.1 then has the form 20.3.6

y2=zce,(z, q)+kg

&kfP sin Pk+p)z,

)

associated with ce,(& a) 20.2.31

b,+~-ar~24r+6~gtr+fe-4~/~!,

(given in [20.36] without 20.3.

Floquet’s

Theorem

y”+

(a-2q

20.3.7

proof.)

equation

cos 22)y=O

are periodic functions of Z, it follows from the known theory relating to such equations that there exists a solution of the form 20.3.2

F”(Z) =efV(z),

F,(--z)=e-“““P(-2)

sat.isfies 20.3.1 whenever 20.3.2 does. and FJ-z) have the property

~ ,

From 20.3.8 it follows that, if v is a proper fraciion of 20.3.1 is perio ic, and of period at most 2?rm,. This agrees ith results already noted in 20.2; i.e., both indepen ent solutions are periodic, if one is, provided the pe* iod is different from A and 27r.

ml/mz, then every solution

Method

Both

F”(z)

of Generating

the

Characteristic

Define two linearly independent 20.3.1, for fixed a, 9 by

Expone4t

solutionsi

of

y,(o)=l;y:(o)=o.

y(z+k3f~=Cky(d, y=F,(z) or FA-z), C=e’“” for F”(z), C=e-‘v*

20.3.9 ys(O)=O; yi(O)=l.

for FJ-2)

Solutions having the property 20.3.4 will hereafter be termed Floquet solutions. Whenever F.(z) and Fy(--z) are linearly independent, the general solution of 20.3.1 can be put into the form

If AB#O,

j2k+P cos (2k+p)z,

The coefficients IL+~ and jit+P depend on the presponding coefficients A, and B,, respectiqly, of 20.2, as well as on a and p. See [20.301, set ion (7.50)-(7.51) and [20.58], section V, for det ils. If v is not an integer, then the Floquet solut i ons F”(z) and Fy(-z) are linearly independent. I It is clear that 20.3.2 can be written in the form

20.3.4

20.3.5

q)+g

20.3.8

where v depends on a and q, and P(z) is a periodic function, of the same period as that of the coefficients in 20.3.1, namely A. (Floquet’s theorem; see 120.161 or [20.22] for its more general form,) The constant v is called the characteristic exponent. Similarly 20.3.3

yz=zser(z,

associated with 8eJ4, a)

and Its Consequences

Since the coefficients of Mathieu’s 20.3.1

!Fa

y=AFv(z)+BFv(-z)

the above solution will not be a Floquet solution. It will be seen later, from the method for determining v when a and q are given, that there is some ambiguity in the definition of v; namely, v can be replaced by v+2k, where k is an arbitrary integer. This is as it should be, since the addition of the factor exp (2ikz) in 20.3.2 still leaves a periodic function of period r for the coefficient of exp ivz. It turns out that when a belongs to the set of characteristic values a, and b, of 20.2, then v is zero or an integer. It is convenient to associate V=T with a,(q), and v=--T with b,(q); see [20.36]. In the special case when v is an integer, F,(z) is

Then it can be shown that 20.3.10 20.3.11 Thus

cos w-y/1(7r)=O cos nv-1-2~; v may be obtained

6)

yz (‘;-)=o

;

from

y1 (r) or from a knowledge of both For numerical purposes 20.3.11 may be desirable because of the shorter range of inte tion, and hence the lesser accumulation of rou doff errors. Either V, -v, or f v+2k (k an a Fbitrary integer) can be taken as the solutioni of 20.3.11. Once v has been fixed, the coefficiebts of 20.3.8 can be determined, except for an arbitr ry it multiplier which is independent of z. The characteristic exponent can also be puted from a continued fraction, in a analogous to developments in 20.2, if a sufficie close first approximation to v is available.

728

MATHIEU

systematic tabulation, this method is considerably faster than the method of numerical integration. Thus, when 20.3.8 is substituted into 20.3.1, there result the following recurrence relations:

20.3.12

FUNCTIONS 0 A 4.0

3.5 -

V*nC2n=CZn4+C2n+l

------.-* IJ-H’-

where

3.0 -

20.3.13 Vz,,=[a-

(2n+~)~]/q,

--CD
.I

m20

. . .’

./

‘\

v- m?4-

. .

, .

‘\I.0

m20.

From the form of 20.3.13 and the known properties of continued fractions it is assured that for sufficiently large values of ]mJ both ]Gn] and IH-,,,j converge. Once values of G,,, and H-, are available for some sufficiently large value of m, then the finite number of ratios G,,,+ G,,,-,, . . ., G0 can be computed in turn, if they exist. Similarly for H++*, . . ., Ho. It is easy to show that Y is the correct characteristic exponent, appropriate for the point (a, q), if and only if H,,G,,= 1. An iteration technique can be used to improve the value of v, by the method suggested in [20.3]. One coefficient cj can be assigned arbitrarily; the rest are then completely determined. After all the cj become available, a multiplier (depending on q but not on z) can be found to satisfy a prescribed normalization. 1% is well known that continued fractions can be converted to determinantal form. Equation 20.3.14 can in fact be written as a determinant with an infinite number of rows-a special case of Hill’s determinant. See [20.19], [20.36], [20.15], or [20.30] for details. Although the determinant has actually been used in computations where high-speed computers were available, the direct use of the continued fraction seems much less laborious. Cases

LO,/ .’

.5 -

Special

./

2.0 -

1.0 (

1

H-m=V-,‘_,-

*.I’-

.’

20.3.14 v,,,--

e I.&‘--

,/--

C-

__---

I.5 -

fractions

1 -Gm-vm-

2.5 -

___------

-0--

_/-

When Y is complex, the coefficients V,,, may also be complex. As in 20.2, it is possible to generate the ratios G,,,=c,,Jc,-a and H-m=c+,,-~/c-,,, from the continued

--‘/-

FIQURE 20.6. Characteristic Region8 y=e’*“P(x) where period r.

Exponent-First P(z) is a periodic

Two Stable function of

Definition of v; Zn first etable region, 0 Iv 5 1, Zn second stable region, 15 Y 5 2. (Constructed

from tabular values supplied by T. Tamir, Polytechnic Institute)

Brooklyn

a

(a, q Real)

Corresponding to q=O, yI=cos $iz, y*=sin ,lZz; the Floquet solutions are exp(iaz) and exp(-&z). As a, q vary continuously in the q-a plane, Y describes curves; Y is real when (q, a), q >O lies in the region between a,(q) and b,+,(q) and

FIGURE 20.7. Characteristic Exponent in First Unstable Region. Differential equation: y” + (a- 2q cos 22) 1/ = 0. The Floquet solution y=e’ *v*P(z), where P(z) is a periodic function of period T. In the first unstable region, v=iN; CIis given for a 2 - 5. (Constructed at NBS.)

MATHIEU 2.01

,

.6 ,

,

.? ”

‘729

FUNCTIONS

.0 .9 I \ ,I,\\\\vl\\‘l\\\k\\

1.0 h

I

I.1 I

I

1.2 I 1~ I

1.3 I

I,4 I O., ‘I -

I.5 I

-4i6

1.8

I.6

-- 29 0

FIGIJRE~ 20.8

t

.5

.6

.7

.0

.9

1.0

1.1

1.2

I.3

1.4

l5

1.6

--)

da

2.0

1.6

1.2 m-2q 0

FIGURE

120.9

I.0

t

.6

I 1.7

I.6

1.9

21)

2.1

2.2

2.3

Charts of the Characteristic (From 8. J. Zamdny, An elementary tions, Ballistic Researob Laboratory

2.4

2.5

2.6

2.7

28

2.9

Exponent.

review of the Matbieu-Hill equation of real variable based on numerical soluMemo. Rept. 878, Aberdeen Proving Ground, Md., 1955, with permission.)

s=eii*=constant; in unstable regions - - - - v=constant; in stable regions -.-* - Lines of constant values of -9.

--+

4

730

MATHIEU 3.1

II

3.2

I

3.3

I I

I

3.4

I

FUNCTIONS Expansions

3.5

I I1

for

Small

q ([20.36]

chapter

2)

If v, p are fixed: 20.3.15

c5v2+v q4

z+ a=V2+2(~f-1)

1.6

32(~~-l)~(v~--4) (9v4f5sv2+29)q6

. . . (Vf1J2J3).

+64(v2-1)5(v2-4)(v2-g)+

For the coefficients czj of 20.3.8 20.3.16 ’ (vz+4v+7)a” 128(v+1)3(v+2)(v-l)+

___CZ/Co=4(;-l)

. **

(v#l,2) c4/Co=q2/32(v+1)(v+2)+

. . .

C2*/CO=(-1)8q*r(v+l)/2%!r(v+S+1)+ .6

. . .

20.3.17

F”(z)=co [efcq{$-g-G}]+

...

(v not an integer) 3.1

3.2

3.3

3.4

3.5

20.3.18

---da FIGURE

20.10.

Chart

of the Characteristic

For small values of a

Exponent.

cos v7r=

(From S. J. Zaroodny, An elementary review of the Msthieu-Hill equation of real variable based on numerical solutions, Ballistic Research Laboratory Memo. Rept. X78, Aberdeen Proving Ground, Md., 1955, with permission)

(

a# l-T+=+

a2r4

. . .

>

-q,,,(l-;)+

s = e”‘= constant; in unstable regions Y= constant; in stable regions - . - . - Lines of constant values of -4. ----

. . .] +q(;-g+...)+...

20.4. Other Solutions of Mathieu’s

all solutions of 20.1.1 for real z are therefore bounded (stable); v is complex in regions between b, and a,; in these regions every solution becomes infinite at least once; hence these regions are termed “unstable regions”. The characteristic curves a,, b, separate the regions of stability. For negative q, the stable regions are between bzr+, and b2r+2,az, and a2,+l ; the unstable regions are between a2r+l and b2r+l, a2, and bzr. In some problems solutions are required for real values of 2 only. In such cases a knowledge of the characteristic exponent v and the periodic function P(z) is sufficient for the evaluation of the required functions. For complex values of z, however, the series defining P(z) converges slowly. Other solutions will be determined in the next section; they all have the remarkable property that they depend. on the same coefficients c,,, developed in connection with Floquet’s theorem (except for an arbitrary normalization factor).

Following 20.4.1

Erdelyi

[20.14],

[20.15],

cpn(z)=[ err cos (z-b)/cos

Equation

define

(z+b)]tkJk(j)

where 20.4.2

j=2[q

cos (z-b)

cos (z+b)]!,

and J,Jj) is the Bessel function,of order k; b is a fixed, arbitrary complex number. By using the recurrence relations for Bessel functions the following may be verified: 20.4.3 ~$-2&3

2z)(o,+P(M-2+M+2)+~~~=0.

It follows that a formal solution by 20.4.4

y=

5 9&c--”

C2?&72n+v

of 20.1.1 is given

MATHIEU

where the Floquet’s complex. an integer,

coefficients c,, are those associated with solution. In the above, v may be Except for the special case when v is the following holds:

QZ?k+"--9 -ly-rv

Q-2n+v

Qzn+u

Q-zn+v+z

-4n2 q [cos

where j satisfies 20.4.2. An examination ratios #Z/Zn+v/fi2LZn+v-2 shows that

will be a solution

(n-4

(z-b)]2

I731

FUNCTIONS

[Qz2n+v/Qzb2n+v-z]

[cos (~-b)]~

On the other hand CPn C--2n -q CZn-C-2n+2-GF

(n-p OJ)

20.4.7

JP (cc)= p

It follows that 20.4.4 converges absolutely uniformly in every closed region where

Y, (2) = 2:) (2) ; I-p (ix) = 2:) (2) ; Hf’ (2) = z& (ix)

If z is replaced by --iz in 20.4.5 and 20 .4.6 solutions of 20.1.2 are obtained. Thus 20.4.8

regions:

y:“(z)= (1) ~b-b)>4>0; (II) JJ(z--b)<--d,
(2) ;

and

Ices (z-b)l>&>l. There are two such disjoint

(cos (z+b)l>l.

The above two conditions are necessary iven when v is an integer. Once b is fixed, the reg ions in which the solutions converge can be rer dily established. Following [20.36] let

- - [cos (z-b)]zq/4n2 - -4d/q

[~-2~+~/~-2~+~+21

provided

(cos (z-b)j>l;

If v and n are integers, J-,,+,(f>=(-l)yJzn-,,(f>.

of ~the

(Ices b---b)l>dl>l) (jcos (z-b)j>dl>l)

5 cz,(-l)'%ifn:y(2& It=--m

cash z)

(bosh 4bl> 20.4.9

If v is an integer 20.4.4 converges for all values of 2. Various representations are found by specializing b.

yP(z)=

2 c~,J;$+~(~& 7&3--m

sinh z) (Isinh zl>l,

j=l,

2,

4)

cos 4

The relation between ?/i”(z) and yP(z) ca be determined from the asymptotic properties of ,the It Bessel functions for large values of argument. can be shown that

cos zl-
20.4.10

1

20.4.5

If b=o, yl=e+12 $-

czn(-1w2n+“w$

(1~0s zl>l,

larg 2&

20.4.6

y1(‘)(2)/y6’)(2)=

If a=$ y= 5

n--m

c~,J~~+~(%&~

([sin zl>l,

sin

2)

larg 2J;j sin 2[<7r)

If b-+-i, y reduces to a multiple of the solution 20.3.8. The fact that 20.3.8, 20.4.5, and 20.4.6 are special cases of 20.4.4 explains why it is t’hat these apparently dissimilar expansions involve the same set of coeffiaents czn. Since 20.4.4 results from the recurrence properties of Bessel functions, Jk(f) can be replaced by Hij’(f), j= 1, 2, where Hi” is the Hankel function, at least formally. Thus let &=[d*

cos (z-b)/cos

(z+b)] Wij’(f)

[F”(o)/F”@e’““/2

(9zz

0).

When v is not an integer, the above solut’ons 1 do not vanish identically. See 20.6 for inte ral g values of v. Solutions

Involving

Products

of

Bessel

Functiods

20.4.11

y:‘)(z)=I

2

&(-lyZ(‘)

c2s?&=--m

n+Y+J(~eiZ)J,-,(~e-~i’> I (j=l, 2,3j, 4)

satisfies 20.1.1, where Zy) (u) is defined in 20. .7, the coefficients czn belong to the Floquet solut on, and s is an arbitrary integer, c2,f: 0. The solu I,ion converges over the entire complex z-plane if q Written with z replaced by --iz, one solutions of 20.1.2.

732

MATHIEU

FUNCTIONS

20.5.4

20.4.12

2A”,+A;+

. . . =A;+A:+

...

=B;L+E+ It can be verified from 20.4.8 and 20.4.12 that

. . . =B2,+Ba,+

. . . =l.

20.5.5 2r 2* ce2,(z, q)dz; Ai=: ce,(z, q) cos nzdz 2r s 0 = s0

AZ=- 1 20.4.13

M$-&)=E:(0))

L%%>O)

provided cza$0. If cZJ=O, the coefficient of l/czs in 20.4.11 vanishes identically. For details see [20.43], [20.15], [20.36]. If s is chosen so that Ic2sl is the largest coefficient of the set /I+(, then rapid convergence of 20.4.12 is obtained, when L%?‘z>O. Even then one must be on guard against the possible loss of significant figures in the process of summing the series, especially so when q is large, and ]z] small. (If j#l, then the phase of the logarithmic terms occurring in 20.4.12 must be defined, to make the functions single-valued.) 20.5.

Properties of Orthogonality Normalization

If a(v+2p, 20.3.10 then 20.5.1

s

a), a(v+2s,

and

a) are simple roots of

n#O

se,(z, q) sin nzdz

0

For integral values of v, the functions ce,(z, a) and se,(z, q) form a complete orthogonal set for the interval 0 5 2 5 2~. Each of the four systems cezl(z), ce21+l(z), se2,(z), se21+1(z) is complete in the smaller interval 0 5 z<+?r, and each of the systems cc,(z), se,(z) is complete in O
Bs&,

n>

defmed so that ace,(O, p) = 1;

42, *I=; [F~(z)+Fv(-z)l; se,(z, q)=-i f [F”(z)-FJ-z)]

ceV(z, q), seV(z, a) are thus even and odd functions of z, respectively, for all v (when not identically zero). If v is an integer, then ce,(z, CJ), ae,(z, n) are either Floquet solutions or ident,ically zero. The solutions ce,(z, a) are associated with a,; se,(z, CJ)are associated with b,; T an integer. Normalization

20.5.3

*

or FY+21)(~)FY+21(-~)d~=0, if p#s.

Define 20.5.2

B;=i s2+

Integral

Values

of Y and

Real

q

s,” [ce,(z, q)lzdz=Jozr [se&, q)]%z=?r

For integral values of v the summation in 20.3.8 reduces to the simpler forms 20.2.3-20.2.4; on account of 20.5.3, the coefficients A,,, and B, (for all orders r) have the property

L%(z, *I]

z1o=1

are always possible. This normalization has in fact been used in [20.59], and also in (20.581, where the most extensive tabular material is available. The tabulated entries in [20.58] supply the conversion factors A=l/a, B=l/& along with the coefficients. Thus conversion from one normalization to another is rather easy. In a similar vein, no general normalization will be imposed on the functions defined in 20.4.8. 20.6.

for

[i

Solutions of Mathieu’s Modified Equation 20.1.2 for Integral v (Radial Solutions)

Solutions

of the first kind

20.6.1 Ce2r+r(z,

a)

=ce2t+p(iz,

q>

==po A;;$; (q) cash (2k+p)z

associated with a,

MATHIEU

20.6.2

LS&+~(Z, p)=--i~e~,+~(iz,

writing A$f;

(q)=Azr+l,

q) =gO @;T;(q)

for brevity;

similarly

733

FUNCTIONS

sinh (2k+p)z,

associated with b,

for Bzn+p; p=O, 1,

ce2,

20.6.3

Ce2k

a) =

z.

&,

(-

1)‘LA2,J&fi

cash

I cezr+l Ce2r+l(z, a> = fiAf,+~

20.6.4

=

20.6.5

ce2,+l@,

Se2,(z, a) 7

&-;,+‘-

!m

4

=

““‘$j

go (--~>“+‘&+~z~+ICW? cl)

coth zgO (2k+1)A2k+IJ2t+l(2fi

-

ks (-1>k2kB,sJs(2&

=sd@, a> coth z & 2k&JJ,,(2& !-I&’ 20.6.6

Se2,+h,

2 B2k+I J2k+1(2fi kc,-,

See [20.30] for still other forms. Solutions of the second kind, as well as solutions functions) are obtainable from 20.4.12. ~c,‘f,)(z, d=&

cosh4

sinh z)

cash z)

sinh z)

tanh z k$ (-1)‘(2k+l)Bzk+,J2,+,(2~

a)= =se;r+l(O,@ ,hjB:‘+’

20.6.7

*) p0 A2JJ2,(2,iTj sinh z)

cash 2)

sinh z)

of the third

and fourth

kind (analogous

to Hankel

(-1>‘+“A~;(~>[Jk-~(ul>~~‘~~(Uz)+Jk+~(u~)~~~l,(uz>l/eaA::

where eo=2, t8= 1, for s= 1, 2, . . .; s arbitrary,

associated with a2,

20.6.8

20.6.9

Ms$(z,

q)=&

(-1)k+‘Ba”~(Q)[Jk-,(u,)~~~~!,(U2)-Jk+a(U~)~~~,(U2)]/B~~,

associated with bzr

20.6.10 where ul=fiee-g,

ua=&e’,

Bijl++pP,A::fj’#O,

p=O, 1.

See 20.4.7 for definition of Zg’ (x). Solutions 20.6.7-20.6.10 converge for all values of z, when q#O. If j=2, 3, 4 the logarithmic entering into the Bessel functions Y,(u2) must be defined, to make the functions single-valued. can be accomplished as follows: .DeCne (as in [20.58]) 20.6.11

ln (fie”>=ln

See [20.15] and [20.36], section 2.75 for derivation.

t&)+2

terms This I

734

MATHIEU Other

Expressions

20.6.12

Mc$(z,

20.6.13

Ms$)(z, Ms:;!,,

for

the

Radial

q)=[ce2,(0,

q)l-’

q)=[se:,(O,

q)]-’

FUNCTIONS Functions

&

Over

More

(-1>“+‘&(q).Gf(2~

tanh z g

(2, q) =[se’ zr+l(O, a)]-’

(Valid

Limited

Regions)

co& z>

(-1)“+‘2kB,2;(q)Z~P(2~

cash z)

tanh k k% (-1)“+‘(2k+l)B~;~:(q)Z~I,:,(2~

cash z)

Valid for %z>O, ]cosh zl>l; if j=l, valid for all z. They agree with 20.6.7-20.6.10 functions Y,(2q” cash z) are made single-valued in a suitable way. For example, let y,(u)=~ where

4(u) is single-valued

20.6.14

On u>Jm(u> +4(u)

for all finite values of u.

In (2q’ cash z)=ln

if the Bessel

2q*+z+ln

With

u=2q+ cash z, define

+(l+e-zz)

-g

<arg

$(l+e-““)

5;

(If q is not positive, the phase of In 2qi must also be specified, although this specification will not affect continuity with respect to z. If Y,(u) is defined from some other expression, the definition must be compatible with 20.6.14.)

FIGURE

FIGURE 20.11. Radial Mathieu Function of the First Kind. (From J. C. Wiltse and M. J. King, Values of the Mathieu functions, The Johns Hopkins Univ. Radiation Laboratory Tech. Rept. AF-53, 1959, with permission)

/y

20.13.

Radial

Mathieu

Function

of the

Second

Kind.

(From J. C. Wiltse and hf. J. King, Values of the Mathieu functions, The Johns Hopkins Univ. Radiation Laboratory Tech. Rept. AF-53, 1958, with permission) &

(” (Z.-q) Ms2

$, hld; (z,q) 32I-

-3cl:2 25 Derivative of the Radial Mathieu Function of the First Kind. (From J. C. Wiltse and M. J. King, Derivatives, zeros, and other data per‘ taining to Mathieu functions, The Johns Hopkins Univ. Radiation Laboratory Tech. Rept. AF-57, 1955, with permission) FIGURE

20.12.

FIGURE

20.14.

Radial

Mathieu

Function

of th,e Third

Kind.

(From J. C. Wiltse and M. J. King, Values of the Mathieu functions, The Johns Hopkins Univ. Radiation Laboratory Tech. Rept. AF-53, 1958, with permission)

MATHIEU

If j= 1, A@;:, and Ms!‘!,,, p=O, 1 are solutions of the first kind, proportional to Celr+p and Sezl+P, respectively. Thus ?r ce2, ipz ce240, a> ( > MczY’(z, n> Ce24z, a>= (-l)‘&’

(0, abei,

n>=

3 p ( >

(-l)‘&’ s&+l(Q, dse2,+l

Se2,+l(z, n) =

q>=-Mc:“(z,

(

Ej p >

20.6.20

q) q)

20.6.17 a> ~)+24W+,(z,

Mi?+,(z+in~, n> =(-l)“p[W3!,,(~, a>- 2nM6::&, M!?+,(z+in~, n> = (--l)nPIW:,(~, c-d+‘JnM:?+&,

Properties

ar and

Functions, b,)

and Some

G(u) =$ K(u, t)V(t)dt ‘c

I

be defined for u in a domain U and let the co tour C belong to the region T of the complex t- 4lane, with t=yo as the starting point of the co tour and t=y, as its end-point. The kernel K & , t) and the function V(t) satisfy 20.7.3 and1 the hypotheses in 20.7.2.

n>l

20.7.2 K(u, t) and its first two partial derivqtives with respect to u and t are continuous for t pn C

a>1

and u in U; V and -

dV are continuous in t. dt

20.7.3 n>l

or Ms throughout any of the above of Characteristic (Associated With

a)

See [20.58]. In particular the above equations can be ushd to extend solutions of 20.6.12-20.6.13 when g&l, they do not reprkt the same functions as 20.6.9-20.6.10.

20.7.1

From 20.6.7-20.6.11 and the known properties of Bessel functions one obtains

where M=Mc equations.

q)-2f,,,M$)(z,

20.7. Representations by Integrals Integral Equations Let

functions are

=(-l)"p[ME+,(z,

q>=MsP’(z,

a)

M&)+1(2, a>

Mi3’(z, q)=Mp’(z, q)+iMp’(z, Mt”(z, q)=Mt”(z, a>--iMY’(z, M$“=Mc;” orMs’?‘.

IJ)

fe, ,= --Mcs2’(0, q)/McS”(O, q>

20.6.16

M%,(z+im,

q)-2f,.,McI”(z,

20.6.21

Ms2 (2, n>

(-l)‘&jBy+

The Mathieu-Hankel

Other

Mc:‘(-z,

where

nce&+l Z’P ce2t+1@,a> ( > n>= M&‘+,(z, (-l)‘+l,/‘ijA;‘+l 4,

Sdz,

20.6.19

Ms:‘(-zz,

20.6.15

Cb+l(z,

735

FUNCTIONS

q

Real

C

70

If K satisfies

2. 7 4 . .

Consider

2t)V+o.

~V-d~K]l’=o;~+(a--Zgcos

b2K b2K a+dt2+2&osh

2u-cos 2t)K=O

20.6.18

Xl=-M@(z, X,=MS:~‘(Z,

q)+Mc:‘(-z, ~)-Ms$~)(-z,

q); q)

Since X1 is an even solution it must be proportional to McI”(z, a); for 20.1.2 admits of only one even solution (aside from an arbitrary constant factor). Similarly, X2 is proportional to Msj” (z, q). The proportionality factors can be found by considering values of the functions at z=O. Define, therefore,

then G(u) is a solution of Mathieu’s equaiion 20.1.2. If K(u, t) satisfies 20.7.5 $$+%;+2q(cos

mojlified

2u-cos 2t)K=O

then G(u) is a solution of Mathieu’s 20.1.1, with u replacing v.

equbtion

736

MATHIEU Kernels

20.7.6

Kl(z,

KI(z,

Kz(a, t)

t) and

t)=Z$‘)(u)[M(z,

t)]-““,

@z>O)

where

20.7.7 u=1/2q(cosh 2z+cos 2t) 20.7.8 M(z, t)=cosh (z+it)/cosh (z-a) To make M-i”

single-valued,

define

20.7.9

FUNCTIONS

where F”(t) is the Floquet solution 20.3.8. The path C is chosen so that G(z, t, a) exists, and 20.7.2, 20.7.3 are satisfied. Then it may be verified that K3(z, t, a), considered as a function of z and t, satisfies 20.7.4; also, considered as a function of a and t, K3 satisfies 20.7.5. Consequently G(z, p, a)=Y(z, &/(a, a), where Y and y satisfy 20.1.2 and 20.1.1, respectively. Choice of Path C. Three paths will be defined:

20.7.16 cash (z+&r) =ek cash z cash (z-i?r)=e+cosh

Path C3: from -d,+im

z

-dl<arg

M(z,

0) = 1

[M(z,

7r)]+=e-iv*M(z,

0)

20.7.10 G(z, p)=;f: j-Or Kl(u,t)Fv(t)dt,

[&{ cash (z+b)

-d2<arg

Let (9 z>O)

to d2--ia,

t)]-+Af’, s any integer.

Kz(z, t)=Z’?+zs(u)[M(z,

See 20.4.7 for definition

(same dI, d2 as in 20.7.16)

20.7.18 4; e2’AwF,(t)dt

ql=k

SFci

j=3,4

where M?(z, q) is also given by 20.4.12. 20.7.19

Path C.i: from -d,+ia

q)=

to 2r-dl+im

4”;

F”(a)M?(z,

20.7.12

q)=k

e2’~~wFv(t)dt

See [20.36], section 2.68.

where MY (z, q) is given by 20.4.12, s=O, 1, . . ., cza#O, and F”(t) is the Floquet solution, 20.3.8. Kernel

K&,

t, a)

20.7.13 K&z, t, a)=e2+ 20.7.14 w=cosh z cos a cos t+sinh

If Y is an integer the paths can be simplified; for in that case F,(t) is periodic and the integrals exist when the path is taken from 0 to 2~. Still further simplifications are possible, if z is also real. The following are among the more important integral representations for the periodic functions ce,(z, n), se,(z, n) and for the associated radial solutions. Let r=2s+p,

where

20.7.15

-d,

of Z(!)+,,(u).

From the known expansions for El,)+.,,(u) when 8,s is large and positive it may be verified that

M!“(z,

f 1 }]
20.7.17

F&)Mi(z,

20.7.11

f 1 }I<*-do

[&{ cash (z-k)

Path C4: from d2--ico to 2~+i~ where F,(t) is defined in 20.3.8. It may be verified that K,F, satisfies 20.7.3, K satisfies 20.7.2 and 20.7.4. Hence G is a solution of 20.1.2 (with z replacing u). It can be shown that KI may be replaced by the more general function

dI, dz real

z sin a sin t

p=O or 1

20.7.20 ce,(z,p)=p,~‘,s(2~cos

zcost-p;)

cf+(t, q)o!t

MATHIEU

737

FUNCTIONS

r12

20.7.21

ce,(z, n>= 6

20.7.22

cash (24 sin z sin t)[(l-p)+p

s

a sin 2 sin t se& >

sin . ( 2Jq cos 2 cos t+p

se,(z, ql=irl'2

cos z cos t]ce,(t, q)dt q)dt

*12

20.7.23

S

se&,

n> = Q,

where 20.7.24

2 ~~=;ce~~ p,=t’

R 2’ p I&(q); ( >

R s4,(0,q

if p=O;

)&Es(q),

if

z sin t)[(l-p)

-2

p=Op,=;-

se:, $ p /fiE*(q>; ( >

CT,=: cezs(O,q)/&(q) G=!

sinh (2&sin

cos z cos t+p]se,(t,

q)dt

0

P,=:

cei,+l

se2*+1

(

(

;, p /,iijAF+l

$ p /e+‘(q), >

c,,=4 ce2s+l(0, q)/A:“+‘(q), R

Integrals

Involving

Bessel

for functions

ce,(z, *)

for functions se,(z, q) if p-l;

I uI=; 2 se2a+l(0, q)/&e+‘(q),

p=O;

(q) if p=l,

>

associated with functions

if p=l;

Function

ce,(jz, q)

associated with se,(z, q)

Kernels

Let u= 42q(cosh 2z+cos 27), (9cosh

20.7.25

22>1; if j=l,

valid also when z=O)

20.7.26

Mf$)(z,q)=*

J’ &$j)(u)ce,,(t, *)&;M&)+l(z,*)=(-1)‘8d ‘Osh’ J” zP’(u)‘OSt 032.+~(t,~&& ,A;‘+’ 0

20.7.27

U

0

Ms:!’ k5 !7>=Msil,)+l(z,

(-l)‘+%q

sinh 22 ?rB2,

2

q)=(-1)r84j

t .Q)(u)

s’in 2t sen,(t, p)dt

S S

u2

0

sinh z =B27+1 1

a W(U)

sin t w,+l(t,

ddt

U

0

In the above the j-convention of 20.4.7 applies and the functions MC, Ms are defined in 20/.5.120.5.4. (These solutions are normalized so that they approach the corresponding Bessel-Hankel functions as 9?2+ 0 .) Other

Integrals

for

20.7.28

cos

20.7.31 20.7.32 20.7.33

( 2~5

cash z cos

20.7.29

20.7.30

Mc!“(a,

~~~f)+,(Z, q)=;

Jw+,(z,

a)=-

2

M&‘(z, q)=z 4j ;e;;;‘d,,

and

Msj”(s,

cash z cos t-p

t] cos (2&

q)

E) ce,(t, q>dt

sinh z sin

Z

a>

ST a S S’ ’

q)dt

t)ce,(t,

sin (2rq sinh z sin t)se2,+l(t,

4 J-(-l)’

= s&+l(O,

q)

q)dt

sinh z sin t cos (2Jq cash z cos t)se2,+l(t,

q)dt

0

sin (2Jq cash z cos

0

sm

t)

[sinh z sin

t

ae2, (t,

n>ldt

(2& sinh z sin t)[cosh z cos t ses,(t, q)ldt

738 Further

MATHIEU

with w=cosh

z cos cr cos t+sinh

20.7.34

ce,(a, dMc!“(z,

20.7.35

=,(a, dM4”(5

The above can be differentiated

z sin (Ysin t (-l)“(i)-p 2?r

d=

(-l)“(-i)P 2n

a)=

s0

‘*ezifi %e,(t,

q)dt

S

‘*e2idiwge,(t,

a)dt.

0

with respect to (Y,and we obtain

cet(ct, q)Mc~l)(z,

20.7.36

FUNCTIONS

q)=(-l)‘(~mp+lfi

szre2idw

$

ce,(t, p)dt

0

20.7.37

se:&

q)M+)(z,

(-lp(ip+‘JTi a

q)=

Integrals

With

s0 Infinite

2”e2idw g

se,(t, q)dt

Limits

r=2s+p In 20.7.38-20.7.41 20.7.38

below, z and p are positive. MC?’ (2, a>=-Yr

s0

m sin

(

2 &j cash z cash t+p;)

McS”(t,

q)dt

yT =2ce28

m MsY(z, a> =Yr S

20.7.39

sinh z sinh

0

-yT=

20.7.40

-4se:,

(

McJ2”(2, q)=-f,

Ms?' (2, n>=-rr Y,=-4sei,($u,

Additional

if p=O

s0

m sin

q)&

(

24

Replacing 20.8.1

Solutions

for

Parameters

(

q)dt

g2 p /?~@+l, if p=l > q)dt

if p=l

cash z cash t+p t) sinh z sinh t Ms’:’ (t, q)dt

?rBa2*,if p=6

forms in [20.30], 120.361, [20.15].

Between

-4se2,+1

yr=2ce;s+1(4u, q)/qiTjA?+‘,

~,=4se2,+1(&r,

4-v, q and

z by +a- z in 20.1.1 one obtains

y”+ (a42p cos 2z)y=O Hence if u(z) is a solution of 20.1.1 then u&-z) satisfies 20.8.1. It can be shown that

~)/TB~~‘+~, if p=l

20.8.2

20.8. Other Properties Relations

x=

Ms:“(t,

cos 24 cash z cash t-p ;) Mcl”(t,

q)/rAi8, if p=O

y,= -2ce2,(+r, 20.7.41

24 cash z cash t-p

a0 s(

;I p /@rlP, >

t

-p

n>=4v,

-q)=a(v,

c;,(-a)=~(-l)?&(q), (c2, defined in 20.3.8) normalization ; FJz, -q)=pe-

q), v

v

not an integer

not an integer

and p depending

on the

MATHIEU

739

FUNCTIONS

20.8.3

azr(-q>=a2,(q>; b21(-q>=b2r(a>, forintegraly ~2r+l(--d=bzr+1(q),

+~~+,+1(~d&-,(9-42)1

bzl+l(-d=azr+l(d

20.8.4 ce2&,

-q>=(-l>'ce21(i~-z,

q>

ce2,+l(z,

-q)=(-lI)'se2,+l(~?r--,

a)

se2,+l(z,

-q>=(-l>'ce2,+l(~?r--,

q>

-q)=(-l)‘-‘se2,(+7r-2,

n)

se2,k

For the coefficients associated solutions for integral Y:

with

20.8.10

20.8.5 A;:(-q)=(-l)“-‘A;&(q); A~~~:,‘,(-qP)=(--l)~-‘~~~;t,:(q); ~~~ll(-q)=(-l)“-‘A:~::,‘,(q). For the corresponding

20.8.6

y”-(a+2q

modified

A2s+1

where I,,,(z), K,(z) are the modified Bessel &unctions, ul, u2 are defined below 20.6.10. Sbperscripts are omitted, c,=2, if s=O, es=1 if s#lO. Then for functions of first kind:

the above

E:(-q)=(---l)“-‘Edq)

/

M&)(z,

-q>=(-l>‘h(z,

q>

M&‘(z,

--a)=(-l)‘Ioz,(z,

q)

~

Mc~)+~(z,

-q>=(--l)‘~~e~~+~(~,

a)

Ms:‘!,,(z,

-~)=(-1)‘i1oo2,+,(z,

q)

For the Mathieu-Hankel

’

function of first

kind:

20.8.11

equation

I M&)(z,

-q)=(--‘)‘f1ifKe2r(z,

q) I

cash 2z)y=o

Ms,‘:)@,

20.8.7

-q)=(-1)‘++02,(z,

q)

~ I

ML’)@, - q) = M;” (z+

!I)) Mez’?+,(z,

--4=(-l)

M#+l(z,

-n)=(-l)‘+‘~Ko,,+l(z,

Mij) (z, q) defined in 20.4.12. For integral

values of Y let

20.8.8 lez,(z,

n,=&

For M:“(z, tions

(-l)“+‘Azt[l~-~(ul)Z+~(uz) +I~+s(ut>l~-s(u2>1/Az,e,

IO2&,

*,=g

n)=po

(--

q,=$

-i(M$3)-Mp))

My’@,

-q)=2MI”(z,

; M,=Mc,

or Ms4

or

-q)-MM13’(z,

Ms;for

--p i

real z, p, M?’ (z, -q)l

are in general complex if j=2,4. Zeros

(-1)“+“A2~+~[~~--s(u~)~~+s+~(u2) -In+r+~(u~)~~-,(~z)I/A2a+,

20.8.9

My)=

M=Mc

l)“+*B2,+,[1,-,(2(,)I,+,+,(~,)

+I~~+l(a)l~-,(zL2)1/~2~+~ 102,+1(~,

4, one may use the dbfini-

j=2,

also

(-1)“+“Bzn[l~-~(ul>~~+*(~2) -I~s(~l)~~-*(~2>1/~2,

h2,+l(z,

-q),

a> ~

of the

See [20.36], Zeros

of

ce,(s,

Functions

Xoor

Real

Values

of

section 2.8 for further q)

and

se,(z,

q),

Mc!"(5,

q),

4.~

results. Ms!')@,

q).

In OO. If zo=zo+iy, is any zero of ce,(z, q), se,(qp) in -~
then k?rf q, ka f z,,

are also zeros, k an integer.

740

MATHIEU

In the strip

-;<xO<$

the imaginary

FUNCTIONS

McJ”(o, a)=

zeros of

ce,(z, q), se,(z, q) are the real zeros of Ce,(z, q), Se,(z, q), hence also the real zeros of MC:) (z, q) and MS’,‘) (z, q), respectively. For small p, the large zeros of Ce,(z, q), Se,@, q) approach the zeros of J,(2&cosh z).

Ince zero

[20.56]

Tabulation

of Zeros

tabulates

the first

“non-trivial”

from 0, z, 7r for M,(Z), se,(z), > and for se,(z) to within “10m4, for q=O(l) 10(2)40. He also gives the “turning” points (zeros of the derivative) and also expansions for them for small p. Wiltse and King [20.61,2] tabulate the first two (non-trivial) zeros of A@)(z, q) and Msp)(z, q) and of their derivatives r=O, 1, 2 for 6 or 7 values of p between .25 and 10. The graphs reproduced here indicate their location. Between two real zeros of Mcp)(z, q), Ms~)(z, q) there is a zero of McP’(z, q), Msj”‘(z, q), respectively. No tabulation of such zeros exists yet. Available tables are described in the References. The most comprehensive tabulation of the characteristic values a,, b, (in a somewhat different notation) and of the coefficients proportional to A, and B, as defined in 20.5.4 and 20.5.5 can be found in [20.58]. In addition, the table contains certain important “joining factors”, with the aid of which it is possible to obtain values of Mcji’ (z, q) and Ms:” (z, q). as well as their derivaValues of the functions ce,(z, q) tives, at x=0. and se,(x, q) for orders up to five or six can be found in [20.56]. Tabulations of less extensive character, but important in some aspects, are outlined in the other references cited. In this chapter only representative values of the various functions are given, along with several graphs. ( r=2(1)5

i.e. different

Special

Values

for

Arguments

0 and

MsP(z, a)=-go,r(d 4: The functions j,,, ,, go.,, jc,,, gc,, are tabulated in

[20.58] for q<25. 20.9.

Asymptotic

Representations

The representations given below are applicable to the characteriskic solutions, for real values of p, unless otherwise noted. The Floquet exponent v is defined below, as in [20.36] to be as follows: In solutions associated with a,: v=r In solutions associated with b,: v=-r. For the functions defined in 20.6.7-20.6.10: 20.9.1 ~c?‘(z,

a>

(-l)‘MSf3’(2)

p)

e+fieosh

+T-f)

.&

-&114(cosh z--a)*

D,

m=,,[-4i&(cosh

z-u)],,’

where D-1=D-2=O; DO=l, and the coefficients D,,, are obtainable from the following recurrence formula :

20.9.2

i

20.8.12

+2p-a]

D,,,+( m-i)

[16q(l-~~)-%fi

+4q(2m-3)

urnID,-1

(2m-~)~(l--a2)D,,+~=0

20.9.3 c&+1

se&

(

;I P =(-l)‘+lgc,z,+l(q)A:‘+’ >

(!-I> &I

n2’ q =(-1)‘g0,2r(q)B~‘(p).q ( >

nse2,+1 5’ P =(-1>‘g~.2r+l(dB1 ( >

-\i

Mcl”‘(% a> (-l)‘Ms’:‘(z, q)

e--t[2+m z-++]

;

o

dm

- ?r+q”4(coshz-u) 4 go [4i ,@(cosh z-u)]” 2r+1(!d

&7

d-I=d-z=O;

d,=l,

and

MATHIEU

741

FUNCTIONS

20.9.4

20.9.9

(m+l)dm+l+[(m+;)l+(m+;) +2*-+,+(m-;)

&&a

Fo(Z)-l+-w! S&j Gosh2z

[16p(l-u*)+8i~am]d,-,

__ 1 +2048p [

+4q(2m-3)(2m-l)(l--a2)d,-2=0. In the above -2?r<arg I

+ 16384~~‘~ l cosh4 z

Sz>O,

but u is otherwise arbitrary. If u2=1, 20.9.2 and 20.9.4 become three-term recurrence relations. Formulas 20.9.1 and 20.9.3 are valid for arbitrary a, p, provided v is also known ; they give multiples of 20.4.12, normalized so as to approach the corresponding Hankel functions H?‘(fie”), II?’ (&ez), as 2-m . See [20.36], section 2.63. The formula is especially useful if jcosh z/is large and p is not too large ; thus if u= - 1, the absolute ratio of two successive terms in the expansion is essentially

Jn+” +2 m 4&i If a, p, z, v are real, components of M$‘(z, Mcy’(z, q), respectively; ponen ts of &Ls$~‘(z, q). complex

- (w5+14w3+33w) cosh2 z

i

- (zw5+124w3+1122w)+3w5+290w3+1627p

fi cash zlufl(,

20.9.5

w4+c%6w~~lo5

/(cash z+l)

.

the real and imaginary q) are Mcp)(z, q) and similarly for the comIf the parameters are

Mcs”(z, a)=$ [Mca’(z, q)+Mcy’(z,

,

20.9.10

F,(z) - (w6-47w4+667w2+2835) 12 cosh2 z +(w~+505w*+12139w2+10395J 12 cosh4 z See [20.18] for details and an added erm in -6/2; a correction to the latter is noted in t,[20.58]. P The expansions 20.9.7 are especially usef 1 when p is large and z is bounded away from zer . The order of magnitude of Mc:(O, q) cannot be clbtained from the expansion. The expansion can [also be used, with some success,for z=ix, when p 115large, if lcos xj>>O;

they fail at zc=i ?r. Thus, if q, x are

real, one obtains

q)]

1 .. +

coshe z

I

20.9.11 20.9.6

Mc~)(z, q)=-f

[Mci3’(z, q)-Mcy’(z,

q)]

Replacing c by s in the above will yield corresponding relations among MS? (z, q) . Formulas in which the parameter a does not enter explicitly : Goldstein’s

20.9.7 Mf$‘(z,

q)-iMSp:,(z,

20.9.12

Expansions

q) = [F,(z)--iF~(z)]e’~/~~qt(cosh

z)+

In the above, P,,(Z) and P,(Z) are obtainable from FO(z), F,(z) in 20.9.9-20.9.10 by Ireplacing cash z with cos x and sinh z with sin 12. Thus

P,,(x) = F,(k) ; P,(x) = --iFI where

:

:

20.9.13

20.9.8 t$=2 Jq sinh z-k

(2rSl)

Wl=e 2zliisinz [cos (+x+~n)]“‘+’

arctan sinh z, 9’>0,

q>>l,

w=2rfl

W2=e-2dPshz [sin ($x+$7r)]“‘+‘/

/(COS~X)‘f’ (cO$

x)‘+*

742

MATHIEU

FUNCZ’IONB

20.9.14

20.926

41

(2~+3)J7~+4W-.

..

1024q

64&

See 20.9.23-20.9.24 for expressions relating to ce,(O, q) and se:(O, q). When ]cos z]>dw/p’, 20.9.11-20.9.12 are useful. The approximations become poorer as r increases. Expansions

in Terms

of Parabolic

Cylinder

Functions

(Good for angles close to $s, for large values of p, especially when ]cos s]<2t/q’.) Due to Sips [20.44-20.461.

sin 5,

Let Dk=Dk(a)=(-l)“e’a”s

+ ii&@+j

-?4 1

**

j2=216+5r4-416?-629T1-1162r-476 It should be noted that 20.9.15 is also valid as an approximation for se,+l (2, q), but 20.9.16 may give * slightly better results. See [20.4.] Explicit

Expansions

for

Orders

0, 1, to Terms

in

q-*I*

20.9.21 For r=O:

20.9.16

se,+~(x,n)-fCK&)--Z~(41

r’+2P-121rZ-122r-84 2048q

(9 Lw34

4x, n>-CWX4+%(41

20.9.15

+

a=2pf co9 5.

Z,,- D,,-

e-+u2.

+64p

1

(-$+gp&)+

...

20.9.17

Z,b)-Dr+&j[ -++i (i) Dr-41 +kq

pgp+;;Dr+4+;

(P-l)

(r;)D,-4

20.9.22

++y@*]+

For r=l

:

.. .

20.9.18 Z,(a)-@

1 C

-4 ’

+kq

Dr+2

rq+

-9

Drm2 ($12;-36)

64 +r(r-l)(-fl-27r+lO)

D r_

2-4

+64p

1

1

(-g+;-2&)+.

D,+2 1530, 350, DII 32 +256-2048

1

6 45 0 r D r-6 + . . .

Formulas

20.9.19

Involving

ce.(O,h)

20.9.23

+

r’+213+263?+262r+lOS 2048~~

..

+ ii&p+* j

-54 ceo(O,n> 1 cedh,

n)

-21/Z ee2fi

**

ce2(0,

ce2&,

d ---321$2 n)

e-2fi

l+

and

se,(O,

q)

MATHIEU

20.9.24

sei@, d -4q&iee2&

se1 (ia, n) s&O, sdh,

a> ----64q@t~-~~ !I>

sd(O, d seX$r, a>

(

l21 l- ----+ 16,hj (

17 128q

FUNCTIONS

;743

For higher orders, these ratios are increasi gly more difficult to obtain. One method of esti t ating values at the origin is to evaluate both 2019.11 and 20.9.15 for some x where both expansion$ are I satisfactory, and so to use 20.9.11 as a means to solve for ce,(O, q); similarly for se:(O, q). Other asymptotic expansions, valid over various regions of the complex z-plane, for real values of ... a, p, have been given by Langer 120.251. It i not > to determine the li“n ea,r always easy, however, combinations of Langer’s solutions which coincide with those defined here.

20.10. Comparative

-

Notation8

= Tbie Volume

StrattEi2se-

-Paremet8rs

:omments

t. etc

I ,

in 24X1.1 _-_----

a 9 or R

I I Periodic solutions, of 20.1.1

&aNotel.

Even ___________. _______ Odd ---------_Co~~~knta

- -- -- -- -- -

in Perk&c

80111

Odd.. ___ _- _- _- _- _. - - - v’dz, y Is the Stander lolutlon of 20.1.1. Flcquet’s Solutfom 20.3.lI.. Cbwecterlstic Exponent..N&om&elilons of Floquet’ 80~om&Modi0ed . . .

A: (9)

AVDL

*

A:,

-4:

B:(Q)

B’FL

*

B:

B:

1

(A’)-’

1

1

or (B’)-,

F&l

See Note 1.

h) p=b

Lpeclded

Equa

Ag.. .QJer(c,

cash I) ~r(~,

~,w~, 0

9)

MC ,(I) (2 9 A) Ms”’ I

Jofniug Fectors __.___.______

(I I 9)

Mu .(I’ (z, A)

A4d2)(Z, I

g)

id2; ,

Ad2+z, r

9)

Ma;2’(z,

A)

XmMc

r(I’ (0’9)

&/MC

t(I’ (0 ’ A)

&@[M,” ‘dr

’

(2,9)ld

d Ms ‘I’ (2, q) ;i; (2’9) [gMI’:’ 1r. -MC

,&/$fdL

,I(n (0 g)/ikfc ,I(I’ (0 g)

--Mc~”

I

(z, A)

dz

’

SW Note 2.

(L A)],-o ’

(0, A)/bfe .(” (0, A)

Seme es this volume

NOTE: 1. The conversion factors A’ and Br are tabulated in I20.581 along with the coefficients. 2. The multipliers p, and ar are defined in [20.30], Appendix 1, section 3, equations 3, 4, 5, 6. 3. &e 120.591, sections (5.3) and (5.5). In eq. (316) of (5.5), the !irst term should have a minus sign. *see page II.

&eNotea.

s

MATHIEU

FUNCTIONS

745

References Texts

[20.1] W. G. Bickley, The tabulation of Mathieu functions, Math. Tables Aids Comp. 1, 409-419 (1945). [20.2] W. G. Bickley and N. W. McLachlan, Mathieu functions of integral order and their tabulation, Math. Tables Aids Comp. 2, l-11 (1946). {20.3] G. Blanch, On the computation of Mathieu functions, J. Math. Phys. 25, l-20 (1946). [20.4] G. Blanch, The asymptotic expansions for the odd periodic Mathieu functions, Trans. Amer. Math. Sot. 97, 2, 357-366 (1960). [20.5] C. J. Bouwkamp, A note on Mathieu functions, Kon. Nederl. Akad. Wetensch. Proc. 51, 891-893 (1948). [20.6] C. J. Bouwkamp, On spheroidal wave functions of order zero. J. Math. Phys. 26, 79-92 (1947). [20.7] M. R. Campbell, Sur les solutions de periode 2 su de l’equation de Mathieu associee, C.R. Acad. Sci., Paris, 223, 123-125 (1946). [20.8] M. R. Campbell, Sur une categoric remarquable de solutions de 1’6quation de Mathieu aasociee, C.R. Acad. Sci., Paris, 226, 2114-2116 (1948). [20.9] T. M. Cherry, Uniform asymptotic formulae for functions with transition points, Trans. Amer. Math. Sot. 68, 224-257 (1950). [20.10] S. C. Dhar, Mathieu functions (Calcutta Univ. Press, Calcutta, India, 1928). [20.11] J. Dougall, The solution of Mathieu’s differential equation, Proc. Edinburgh Math. Sot. 34, 176196 (1916). [20.12] J. Dougall, On the solutions of Mathieu’s differential equation, and their asymptotic expansions, Proc. Edinburgh Math. Sot. 41, 26-48 (1923). [20.13] A. Erdelyi, Uber die Integration der Mathieuschen Differentialgleichung durch Lsplacesche Integrale, Math. 2. 41, 653-664 (1936). [20.14] A. Erdelyi, On certain expansions of the solutions of Mathieu’s differential equation, Proc. Cambridge Philos. Sot. 38, 28-33 (1942). [20.15] A. Erdelyi et al., Higher transcendental functions, vol. 3 (McGraw-Hill Book Co., Inc., New York, N.Y., 1955). I20.161 G. Floquet, Sur les equations differentielles lineaires a coefficients periodiques, Ann. Ecole Norm. Sup. 12, 47 (1883). [20.17] S. Goldstein, The second solution of Mathieu’s differential equation, Proc. Cambridge Philos. Sot. 24, 223-230 (1928). [20.18] S. Goldstein, Mathieu functions, Trans. Cambridge Philos. Sot. 23, 303-336 (1927). [20.19] G. W. Hill, On the path of motion of the lunar perigee, Acts Math. 8, 1 (1886). [20.20] E. Hille, On the zeros of the Mathieu functions, Proc. London Math. Sot. 23, 185-237 (1924). [20.21] E. L. Ince, A proof of the impossibility of the coexistence of two Mathieu functions, Proc. Cambridge Philos. Sot. 21, 117-120 (1922).

[20.22] E. L. Ince, Ordinary differential equations (Longmans, Green & Co., 1927, reprinted by Dover Publications, Inc., New York, N.Y., 1944). [20.23] H. Jeffreys, On the modified Mathieu’s equation, Proc. London Math. Sot. 23, 449-454 (1924). [20.24] V. D. Kupradze, Fundamental problems in the mathematical theory of diffraction (1935). Translated from the Russian by Curtis D. Benster, NBS Repor) 200 (Oct. 1952). [20.25] R. E. Langer, The solutions of the Mathieu equation with a complex variable and at least one parameter large, Trans. Amer. Math. Sot. 36, 637-695 (1934). [20.26] S. Lubkin and J. J. Stoker, Stability of columns and strings under periodically varying forces, Quart. Appl. Math. 1, 215-236 (1943). [20.27] 8. Mathieu, M&moire sur le mouvement vibratoire d’une membrane de forme elliptique, J. Math. Pures Appl. 13, 137-203 (1868). [20.28] N. W. McLachlan, Mathieu functions and their classification, J. Math. Phys. 25,209240 (1946). 120.291 N. W. McLachlan,jMathieu functions of fractional order, J. Math. Phys. 26, 29-41 (1947). [20.30] N. W. McLachlan, Theory and application of Mathieu functions (Clarendon Press, Oxford, England, 1947). [20.31] N. W. McLachlan, Application of Mathieu’s equation to stability of non-linear oscillator, Math. Gaa. 35, 105107 (1951). [20.32] 3. Meixner, Uber das asymptotische Verhalten von Funktionen, die durch Reihen nach Zylinderfunktionen dargestellt werden konnen, Math. Nachr. 3, 9-13, Reihenentwicklungen von Produkten zweier Mathieuschen Funktionen nach Produkten von Zylinder und Exponentialfunktionen, 14-19 (1949). 120.331 J. Meixner, Integralbeziehungen zwischen Mathieuschen Funktionen, Math. Nachr. 5, 371-378 (1951). [20.34] J. Meixner, Reihenentwicklungen vom Siegerschen Typus fiir die Sphiiroid Funktionen, Arch. Math. Oberwolfach 1, 432-440 (1949). [20.35] J. Meixner, Asymptotische Entwicklung der Eigenwerte und Eigenfunktionen der Differentialgleichungen der Sphilroidfunktionen und der Mathieuschen Funktionen, Z. Angew. Math. Mech. 28, 304-310 (1948). 120.361 J. Meixner and F. W. Schafke, Mathieusche Funktiomn und Spharoidfunktionen (SpringerVerlag, Berlin, Germany, 1954). l20.371 P. M. Morse and P. J. Rubinstein, The diffraction of waves by ribbons and by slits, Phys. Rev. 54, 89.5898 (1938). [20.38] H. P. Mulholland and S. Goldstein, The characteristic numbers of the Mathieu equation with purely imaginary parameters, Phil. Mag. 8, 834-840 (1929). [20.39] L. Onsager, Solutions of the Mathieu equation of period 4~ and certain related functions (Yale Univ. Dissertation, New Haven, Conn., 1935).

746 [20&l] [29.41] [20.42]

(20.431 120.441 [20.45] [20.46]

[20.47]

[20.48] [20.49]

L20.501

[20.51] [20.52]

MATHIEU

F.

W. Sch&fke, Uber die Stabilitiitskarte der. Mathieuschen Differentialgleichung, Math. Nachr. 4, 175-183 (1950). F. W. Schiifke, Das Additions theorem der Mathieuschen Fusktionen, Math. Z. 58,436-447 (1953). F. W. Schiifke, Eine Methode zur Berechnung des charakteristischen Exponenten einer Hillschen Differentialgleichung, Z. Angew. Math. Mech. 33, 279-280 (1953). B. Sieger, Die Beugung einer ebenen elektrischen Welle .an einem Schirm von elliptischem Querschnitt, Ann. Physik. 4, 27, 626-664 (1908). R. Sips, Representation asymptotique des fonctions de Mathieu et des fonctions d’onde spheroidales, Trans. Amer. Math. Sot. 66, 93-134 (1949). R. Sips, Repr&entation asymptotique des fonctions de Mathieu et des fonctions spheroidales II, Trans. Amer. Math. Sot. 90, 2, 340-368 (1959). R. Sips, Recherches sur les fonctions de Mathieu, Bull. Soo. Roy. Sci. Liege 22, 341355, 374-387, 444-455, 530-540 (1953); 23, 37-47, 90-103 (1954). M. J. 0. Strutt, Die Hiiische Differentialgleichung im komplexen Gebiet, Nieuw. Arch. Wisk. 18 31-55 (1935). M. J. 0. Strutt, Lambche, Mathieusche und verwandte Funktionen in Physik und Technik, Ergeb. Math. Grensgeb. 1, 199323 (1932). M. J. 0. Strutt, On Hiil’s problems with complex parameters and a real periodic function, Proc. Roy. Sot. Edinburgh Sect. A 62,278-296 (1948). E. T. Whittaker, On functions associated with elliptic cylinders in harmonic analysis, Proc. Intl, Congr. Math. Cambr. 1, 366 (1912). E. T. Whittaker, On the general solution of Mathieu’s equation, Proc. Edinburgh Math. Sot. 32, 75-80 (1914). E. T. Whittaker and G. N. Watson, A course of modern analysis, 4th ed. (Cambridge Univ. Press, Cambridge, England, 1952). Tables

L20.531 G. Blanch and I. Rhodes, Table of characteristic values of Mathieu’s equation for large values of the parameter, J. Washington Acad. Sci. 45, 6, 166-196 (1955). Be,(t)=a,(q)+2q-2(2r+l)fi Bo,(t)=&(q)+2q-2(2r-l)fit=1/2Jq, r=O(l) 15, O
FUNCTIONS

[20.56] E. L. Ince, Tables of the elliptic cylinder functions, Proc. Roy. Sot. Edinburgh 52, 355-423; Zeros and turning points, 52,424-433 (1932). Characteristic values aa, al, . . ., as, bl, bl, . . ., be, and coefficients for 0=0(1)10(2)20(4)40; 7D. Aiso ce,(z, 0), se&r, e), @=O(l)lO, z=O”(lo)900; 5D, corresponding to characteristic values in the tables. a,=be,-2q; b,=bo,--29; e=q. [20.57] E. T. Kirkpatrick, Tables of values of the modified Mathieu function, Math. Comp. 14, 70 (1960). r=0(1)5, r=1(1)6; C4u, d, fh(u, d, u=.l(.l)l; q=1(1)20. [20.58] National Bureau of Standards, Tables relating to Mathieu functions (Columbia Univ. Press, New York, N.Y., 1951). Characteristic values be,(s), ho,(s) for 0 5s
c=2qf,

b,=a,+2q,

b:=b,+2q.

120.601 T. Tamir, Characteristic exponents of Mathieu * equations, Math. Comp. 16, 77 (1962). The Floquet exponent Y, of the first three stable regions; namely r=O, 1, 2; q=.1(.1)2.5; a=r (.l)r+l, 5D. [20.61] J. C. Wiltse and M. J. Ring, Values of the Mathieu functions, The Johns Hopkins Univ. Radiation Laboratory Technical Report AF-53, Baltimore, Md. (1958). (Notation of [20.58] used: IX&J, q)/A, se&, q)/B for 12 values of q between .25 and 10 and - from 8 to 14 values of v; dir/2 Mc$(u, q), 442 M&u, q), j=l, 2 for 6 to 8 values of a between .25 and 10 and about 20 values of i, r=O, 1, 2; 48 McP’(1~1, q), d?r/2 Me?(]u], q), r=O, 1, 2 for about 9 values of u and q, 2 to 4 D in all. [20.62] J. C. Wiltse and M. J. Ring, Derivatives, seros, and other data pertaining to Mathieu functions, The Johns Hopkins Univ. Radiation Laboratory Technical Report AF-57, Baltimore, Md. (1958). [20.63] S. J. Zaroodny, An elementary review of the Mathieu-Hill equation of real variable based on numerical solutions, Ballistic Research Laboratory Memorandum Report 878, Aberdeen Proving Ground, Md. (1955). Chart of the characteristic exponent. See also [20.18]. It contains, among other tabulations, values of a,, b, and coefficients for ce,(z, q), se,@, q), q=40(20)100(50)200; 5D, rs2. *See

page 11.

748

MATHIEU

Table

20.1

CHARACTERISTIC

FUNCTIONS

VALUES,

JOINING

FACTORS,

SOME

CRITICAL

VALUES

EVEN SOLUTIONS 7

9 i

10 15 20 25 2

602 996 776 007 955

781 817 757 832 829 018

(-1)

7.07106 1.33484 1.46866 1.55010 1.60989 1.65751

78 87 05 82 09 03

4.00000 7.44910 7.71736 5.07798 1.15428 3.52216

000 974 985 320 288 473

1.00000 (-1 7.35294 (-1 2.45888 (-2 I 7.87928 (-2)2.86489 (-2)1.15128

000 308 349 278 431 663

-1.00000 (-1 -7.24488 C-1 -,';',

00 15 ;;

-1.07529 -1.11627

32 90

100.00000 100.12636 100.50677 101.14520 102.04891 103.23020

000 922 002 345 602 480

1.00000 1.02599 1.05381 1.08410 1.11778 1.15623

000 503 599 631 863 992

a, 1.00000 1.85818 2.39914 8.10110 14.49130 21.31489

000 754 240 513 142 969

cd@ 1.00000 (-1)2.56542 (-2 5.35987 (-2 1 1.50400 (~3 5.05181 (-3 1,1.91105

9) 000 879 478 665 376 151

ce:(fr, q) -1.00000 00 -3.46904 -4.85043 -5.76420 -6.49056 -7.10674

21 83 64 58 15

-5.00000 -5.39248 -5.32127 -5.11914 -5.77867 -7.05988

00 61 65 99 52 45

000

-

5.80004 13.93697 22.51303 31.31339 40.25677

t -

1: :i 25 0 5 10 15 z

7 10

Q 5 10 :z 25

+ -

5

0 5 10 15 20 25

25.00000 25.54997 27.70376 31.95782 36.64498 40.05019

000 175 873 125 973 099

1.00000 1.12480 1.25801 1.19343 (-1)9.36575 (-1)6.10694

000 725 994 223 531 310

15

0

225.00000 225.05581 225.22335 225.50295 225.89515 226.40072

000 248 698 624 341 004

1.00000 1.01129 1.02287 1.03479 1.04708 1.05980

000 373 828 365 434 044

1: :z 25

(4d%, Ad

ce,W, 4)

-1)7.07106 -2)4.48001 (-3)7.62651 (-3)1.93250 -4)6.03743 I -4)2.15863

0.00000

0

10

ce,(O, q)

“F

0

(-1)7.97884 1.97009 2.40237 2.68433 2.90011 3.07743

56 00 95 53 25 91

4.29953 32 1.11858 69 23 I 2.30433 23 2.31909 23 2.36418 23 2.44213 12)1.40118

72 77 54 04

52

w%e, r(q)

@We.,AZ)

1.59576 7.26039 Q1.35943 1)1.91348

2.54647 1.02263 2i 9.72660 2 1.19739 3 1.84066 4)3.33747

91 46 12 95 20 55

8 I 4.80631 8 5.11270 8 6.83327 9 I 1.18373 9 1.85341 9 2.09679

83 71 77 72 57 12

91 84 49 51

-

Compiledfrom National Bureau of Standards,Tables relating to Mathieu functions, Columbia Univ. Press,New York, N.Y., 1951(with permission). q +2qq-?/r

0.16 0.12 0.08 0.04 0.00

1

0

-0.25532 -0.25393 994 098 -0.25257 851 -0.25126 918 -0.25000 000

-1.28658 -1.30027 -1.27371 -1.26154 -1.25000

(4r+2)&

2 212 972 191 161 000

-3.39777 -3.45639 -3.34441 -3.29538 -3.25000

10

5 483 782 938 745 000

-16.92019 -17.84809 -16.25305 -15.70968 -15.25000

225 551 645 373 000

-76.84607 -76.04295 -63.58155 -58.63500 -55.25000

15 855 314 264 546 000

-141.64507 - 80.93485 -162.30500 -132.08298 -120.25000

841 048 052 271 000

2 156 625 03

For ge,~ and f,, ~ see20.8.12. < q> = nearestinteger to q. Compiledfrom G. Blanch and I. Rhodes,Table of characteristicvaluesof Mathieu’s equation for large valuesof the parameter,Jour. Wash.Acad. Sci., 45, 6, 1955(with permission).

MATHIEU

CHARACTERISTlC

VALUES,

749

FTJNCTIONS

JOINING

FACTORS,

SOME

CRITICAL

Table

VALUES

20.1

ODD SOLUTIONS 1

2

i 10 15 20 25

10

5" 10 15 2

045 824 680 325 062

00 22 84 14 78 17

-2.00000 -3.64051 -4.86342 -5.76557 -6.49075 -7.10677

00 79 21 38 22 19

100.00000 000 100.12636 922 100.50676 946 101.14517 229 102.04839 286 103.22568 004

( 1)1.00000 9.73417 9.44040 9.11575 8.75554 8.35267

00 32 54 13 51 84

l)-1.00000 1 -1.02396 11 -1.04539 l)-1.06429

00 46 48 00

898

4(0, 1.00000 (-1 1.74675 -2 4.40225 I -2 i 1.39251 (-3)5.07788 (-3)2.04435

u) 00 40 66 35 49 94

000 605 636 060 133 590

5.00000 4.33957 3.40722 2.41166 1.56889 (-1)9.64071

00 00 68 65 69 62

225.00000 000 225.05581 248 225.22335 698 225.50295 624 225.89515 341 226.40072 004

1 1.50000 I 1 I 1.48287 1 1.46498 1 1.44630 1 1 I 1.42679 ( 1)1.40643

00 89 60 01 46 73

+ -

2.09946 2.38215 8.09934 14.49106 21.31486

1.00000 5.79008 13.93655 22.51300 31.31338

- 40.25671

Li 10 :05 25

15

0 5 10 :i 25

25.00000 25.51081 26.76642 27.96788 28.46822 28.06276

000

000 060 248 350 617

@qP%,r(q)

se;(ta, d

2.00000 (-1)7.33166 -1)2.48822 I -2)9.18197 (-2)3.70277 (-2)1.60562

4.00000

+ 5

40, q)

br

Q

6.38307 1)1.24474 1)1.86133 1)2.42888 1)2.95502 1)3.44997 11)1.51800 11)1.56344 11)1.62453 11)1.70421

65 88 36 57 89 83 43 50 03 18

( (

1)8.14873 1)2.24948 3.91049 (- 1)7.18762 (- 1)1.47260 (- 2)3.33750 23)2.30433 23)2.31909

31 08 85 28 95 27 72 77

i

se,(%,59 1.00000 1.33743 1.46875 1.55011 1.60989 1.65751

00 39 57 51 16 04

1.59576 2.27041 2.63262 2.88561 3.08411 3.24945

91 76 99 87 21 50

1.00000 9.06077 8.46038 8.37949 8.63543 8.99268

00 93 43 34 12 33

3)9.80440 4 1.14793 4j 1.52179 4)2.20680 4 3.27551 4 4.76476

55 21 77 20 12 62

-1.00000 (-1 -9.88960 (-1 -9.78142 -1 -9.67513 I -1 1 -9.57045 (-l)-9.46708

00 70 35 70 25 70

-1) -1) -1) -1) -1)

I 19)3.78055 I(1919)3.73437 3.83604

81 49 43 (19 3.90140 52 19 i 3.97732 29 I 19)4.06462 83

2.54647 91 I-(((-

3)2.21737 2)3*74062 4)2.15798 4)2.82474 6)4.53098

82 88 83 71 74

( ( ( I

8)5.46799 8)4.26215 8)5.27524

57 66 17

(

8)2.94147

89

( 40)2.19249

18

b,+Zq- (4r-2)@ cl-*\? 0.16 0.12 0.08 0.04 0.00

1 -0.25532 -0.25393 -0.25257 -0.25126 -0.25000

5

2 994 098 851 918 000

-1.30027 -1.28658 -1.27371 -1.26154 -1.25000

164 971 191 161 000

-11.53046 -11.12574 -10.78895 -10.50135 -10.25000

10 855 983 146 748 000

-51.32546 -56.10964 -51.15347 -47.72149 -45.25000

For go,Tand f,, r see20.8.12. < q> = nearestinteger to 4.

15 875 961 975 533 000

- 55.93485 112 -108.31442 060 -132.59692 424 -114.76358 461 -105.25000 000

39 69 156 625 m

MATHIEU Table

20.2

FUNCTIONS

COEFFICIENTS

9,

AND

II,,

4 r/=5 7rA7 0 0 +0.54061 2 -0.62711 4 +0.14792 6 -0.01784 8 +0.00128 10 -0.00006 12 +o.ooooo 14 -0.00000 16 +O.OOOOO

10

2 2446 5414 7090 8061 2863 0723 2028 0050 0001

+0.43873 +0.65364 -0.42657 +0.07588 -0.00674 +0.00036 -0.00001 +O.OOOOO -0.00000

7166 0260 8935 5673 1769 4942 3376 0355 0007

:: 22

+O.OOOOO +0.00003 +0.00064 +0.01078 +0.13767 +0.98395 -0.11280 +0.00589 -0.00018

?ll’,T 1679 3619 2987 4807 5121 5640 6780 2962 9166

1 3 5 9' 11 13 15 17

-0.00000 +O.OOOOO 0071 4226 +O.OOOOO 0001

1

co.76246 -0.63159 +0.13968 -0 01491 +0'00094 -0:00003 +O.OOOOO -0.00000 +O.OOOOO

3686 6319 4806 5596 4842 9702 1189 0027 0001

5 +0.07768 +0.30375 +0.92772 -0 20170 +0'01827 -0:OOOSS +0.00003 -LX00000 +O.OOOOO

5798 1030 8396 6148 4579 9038 3457 0839 0016

:;) 23 25

15 0.00000 0000

+o.ooooo +O.OOOOO +O.OOOOO +0.00014 +0.00428 +0.08895 +0.99297 -0.07786 +0.00286 -0.00006 +O.OOOOO -0.00000

0002 0106 4227 8749 1393 2014 4092 7946 6409 6394 1092 0014

q=25 77’ 1 0 0 +0.42974 2 -0.69199 4 +0.36554 6 -0.13057 8 +0.03274 10 -0.00598 12 +0.00082 14 -0.00008 16 +O.OOOOO 18 -0.00000 20 +O.OOOOO 22 -0.00000 24 26 28

1038 9610 4890 5523 5863 3606 3792 7961 7466 0514 0029 0001

2 +0.33086 -0;04661 -0.64770 +0.55239 -0.22557 +0.05685 -0.00984 +0.00124 -0.00012 +O;OOOOO -0.00000 +o.ooooo -0.00000

10 5777 4551 5862 9372 4897 2843 6277 8919 1205 9296 0578 0030 0001

+0.00502 co.02075 +0.07232 +0.23161 +0.55052 +0.63227 -0.46882 +0.13228 -0.02206 +0.00252 -0.00021 +0.00001 -0.00000 +O.OOOOO -0.00000

d,r 1 3 5 7 9 11 13 15 17 19 21 23 25

6361 4891 7761 1726 4391 5658 9197 7155 0893 2374 3672 4078 0746 0032 0001

1 to.39125 -0.74048 +0.50665 -0.19814 +0.05064 -0.00910 +0.00121 -0.00012 +0.00001 -0.00000 +O.OOOOO -0.00000

5 2265 2467 3803 2336 0536 8920 2864 4121 0053 0660 0036 0002

+0.65659 +0.36900 -0.19827 -0.48837 +0.37311 -0.12278 +0.02445 -0.00335 +0.00033 -0.00002 +o.ooooo -0.00000 +o;ooooo

15 0398 8820 8625 4067 2810 1866 3933 1335 9214 6552 1661 0085 0004

31

+O.OOOOO +0.00003 +0.00032 +0.00254 +0.01770 +0.10045 +0.40582 +0.83133 -0.35924 +0.06821 -0.00802 +0.00066 -0.00004 +O.OOOOO -0.00000 +o.ooooo

4658 7337 0026 0806 9603 8755 7402 2650 8831 6074 4550 6432 1930 2090 0085 0003

9=5

10

2 +0.93342 -0.35480 +0.05296 -0.00429 +0.00021 -0.00000 +b.ooooo -0.00000

94'42 3915 3730 5885 9797 7752 0200 0004

+0.00003 +0.00064 +0.01078 to.13767 +0.98395 -0.11280 to.00589 -0.00018 +O.OOOOO -0.00000 +o.ooooo

1

TV\?. 3444 2976 4807 5120 5640 6780 2962 9166 4227 0070 0001

: 5 7 9 :: 15 17

5

+0.94001 -0.33654 +0.05547 -0.00508 +0.00029 +O.OOOOO -0.00001

9024 1963 7529 9553 3879 1602 0332

-0.00000

0007

+0.05038 +0.29736 +0.93156 -0.20219 +0.01830 -0.00096 +0.00003 -0.00000 +o.ooooo

2462 5513 6997 3638 5721 0277 3493 0842 0017

5 +0.30117 +0.62719 +0.17707 -0.60550 +0.33003 -0.09333 +0.01694 -0.00217 +0.00021 -0.0Ou01 +O.OOOOO -0.00000 +o.ooooo

4196 8468 1306 5349 2984 5984 2545 7430 0135 5851 0962 0048 0002

:: ::

15 0.00000 0000

+o.ooooo +O.OOOOO +O;OOOOO +0.00014 tO.00428 +0.08895 +0.99297 -0.07786 +0.00286 -0.00006 +o.ooooo -0.00000

0002 0106 4227 8749 1392 2014 4092 7946 6409 6394 1093 0013

q=25 ?n~s 2 2 8 10 12 :: 18 20 22 24 26

2 +0.65743 -0.66571 +0.33621 -0.10507 to.02236 -0.00344 +0.00040 -0.00003 +O.OOOOO -0.00000 +O.OOOOO

10 9912 9990 0033 3258 2380 2304 0182 6315 2640 0157 0008

+0.01800 +0.07145 to.23131 +0.55054 +0.63250 -0.46893 +0;13230 -0.02206 +0.00252 -0;00021 +0.00001 -0.00000 +o.ooooo

For -1, and II,,, see 20.2.3-20.2.11

1

m/r 3596 6762 0990 4783 8750 3949 9765 3990 2676 3694 4079 0746 0033

: 5 ;

+0.81398 -0.52931 +0.22890 +0.01453 -0.06818

3846 0219 0813 2972 0886

11 13 15

-0.00229 +0.00027 -0.00002

5765 7422 6336

:< ;:

+O.OOOOO -0.00000 0126 2009 +o.ooooo 0007

::

Compiled from National Bureau of Standards, Tables relating to Mathieu Press, New York, N.Y., 1951 (with permission).

15 +o.ooooo +0.00003 +0.00032 +0.00254 +0.01770 +0.10045 +0.40582 +0.83133 -0.35924 +0.06821 -0.00802 +0.00066 -0.00004 +O.OOOOO -0.00000 +o.ooooo

3717 7227 0013 0804 9603 8755 7403 2650 8830 6074 4551 6432 1930 2090 0086 0003

functions, Columbia Univ.

21. Spheroidal

Wave Functions

ARNOLD N. LOWAN

1

Contents Page Mathematical 21.1.

21.2. 21.3.

21.4. 21.5. 21.6.

21.7. 21.8. 21.9. 21.10. 21.11. References Table

Properties.

...................

Definition of Elliptical Coordinates .......... Definition of Prolate Spheroidal Coordinates ...... Definition of Oblate Spheroidal Coordinates. ...... Laplacian in Spheroidal Coordinates .......... Wave Equation iu Prolate and Oblate Spheroidal Coordinates Differential Equations for Radial and Angular Spheroidal Wave Functions .................. Prolate Angular Functions .............. Oblate Angular Functions .............. Radial Spheroidal Wave Functions .......... . Joining Factors for Prolate Spheroidal Wave Functions Notation ......................

Eigenvalues-Prolate m=O(l)Z, n=m(l)mf4 c2=O(1)16, c-‘=.25(--.Ol)O,

and Oblate

21.2. Angular Functions-Prolate m=0(1)2, n=m(1)3, ?j=O(.l)l ~=0”(10”)90”, c=1(1)5, 24D

Table

21.3. Prolate Radial Functions-First m=0(1)2, n=m(1)3 5=1.005, 1.02, 1.044, 1.077, c=1(1)5,

Table

. . . . . . . . . . .

760

4-6D

Table

Table

753 753 756 756 757 758 759

..........................

21.1.

752 752 752 752 752 752

and Oblate

. . . . . . . .

and Second Rinds

. . .

766

768

4s

21.4. Oblate Radial Functions-First m=O, 1, n=m(l)m+2; m, n=2 E=O, .75, c=.2, 3, 23, 1(.5)2.5, 5s

and Second Rinds . . . .

769

Prolate Joining Factors-First 1, n=m(l)m+2; m, n=2 c=1(1)5, 4s

Rind . . . . . . . . . .

769

21.5.

m=O,

1 Yeshiva University. Standards.) (Deceased.)

(Prepared

under

contract

with

the

National

Bureau

of 751

21. Spheroidal

Wave Functions

Mathematical 21.1. Definition

of Elliptical

Coordinates

Properties 21.3.1

21.1.1

Z=T

r1 and rz are the distances to the foci of a family of confocal ellipses and hyperbolas ; 2f is the distance between foci. a=ft,

21.1.2

axis;

Equation

21.1.3

b=semi-minor

of Family

axis;

of Confocal

$-&=f

4; X=T sin 4; 02412*

co8

where 4,~ and 4 are oblate spheroidal Relations

Between

Cartesian Coordinates

qnd

coordinates.

Oblate

Spheroidal

21.3.2

es- f a

b=jm,

a=sen-&major centricity.

e=ec-

21.4. Laplacian

in Spheroidal

Coordinates

21.4.1

Ellipses

$+&=f Equation

21.1.4

of Family

of Ckmfocal

Hyperbolas

(---1
e-L=fa q2

Relations

l--$

Between

21.1.5

Cartesian

x=fb?;

and

Elliptical

Coordinates

Y=.fJ(ta--lw--lla)

21.2. Definition of Prolate Spheroidal Coordinates

Metric

If the system of confocal ellipses and hyberbolas referred to in 21.1.3 and 21.1.4 revolves around the major axis, then $+&=f;

21.2.1 y=r

cos

$-&=f

4; Z=T

sin

4; 0 54 12~

Relations

Between

Cartesian Coordinates

and

Prolate

Prolate

Spheroidal

Coordinates

ht=j$Z

h&s Coefficienta

+jJ(p-l)(l-92) for

Oblate

Spheroidal

* Coordinates

coordinates. Spheroidal

21.5. Wave Equation in Prolate and Oblate Spheroidal Coordinates Wave

c_os4; z=fJ(+l)(l-v2)

sin 4

21.3. Definition of Oblate Spheroidal Coordinates

If the system of confocal ellipses and hyperbolas referred to in 21.1.3 and 21.1.4 revolves around the minor axis, then 752

for

21.4.3

21.2.2 Y=fJG2--lm--lla)

doe5cients

21.42

Metric

where [, 7 and 4 are prolate spheroidal

x=flrl;

’ ’ j5+,,_,=f”;

Equation

in

Prolate

Spheroidal

Coordinates

21.5.1 v*+k+z+

[(P-l)

$I+;

[(l-r13

$1

2- 2 +(~~~1)(:-r7~+c~(E’-13~=0

(c=; *see page II.

jk)

SPHEROIDAL Wave

Equation

in

Oblate

Spheroidal

WAVE

Coordinates

753

FlJNC.?l’IONS

(21.6.3 may be obtained from 21.6.1 by the transformations [*fit, c++ic; 21.6.4 may be obtainedfrom21.6.2 by the transformation c-+~ ic.)

21.5.2

21.7.

Prolate

Angular

Functions

21.7.1 WC,

21.5.2 may be obtained from transformations p++i[, c++c. 21.6. Differential Angular Prolate

Equations Spheroidal

21.5.1

by

d =rg;

dY (4 E+r

=Prolate

the

b>

angular function

of the first kind

21.7.2

for Radial and Wave Functions

=Prolate

angular function

of the second kind

If in 21.5.1 we put

then the “radial “angular solution” equations

solution” R,,(c, 5) and the &,,(c, 7) satisfy the differential

(E?(v) and R‘(d are associated Legendre functions of the first and second hinds respectively. However, for -1Izj1, ~(z)=(l-z2)“‘2d”P,(z)/ dz” (see 8.6.6). The summation is extended over even values or odd values of T.) Recurrence

21.6.1

Relations

Between

the

Coefficients

21.7.3

(2m+k+2)(2m+k+l)c2 ak=(2m+2k+3)(2mf2kf5)

21.6.2

he,=(m+k) (m+k+l) ’ where the separation constants (or eigenvalues) A,,,,, are to be determined so that R&c, 5) and &,,(c, TJ) are finite at ,$=f 1 and q=f 1 respectively. (21.6.1 and 21.6.2 are identical. Radial and angular prolate spheroidal functions satisfy the same differential equation over different ranges of the variable,)

2(m+k)(m+k+l)-2m2-1 (2m+2k-1)(2m+2k+3)

Transcendental

Equation

Equations for Spheroidal

Radial and Functions

21.6.3

E)] $ [ Q”+U$ Gnn(c,

Angular

X,,

21.7.4 u(~,,)=u1(~,)+uz(xmn>=o R-2 nm -

Differential

for

c2

-&-A,,-

* . .

Oblate U&xmn)=-

iT+z Y?+a--X,,-

8” r+4 r7+4--X,.-

- * *

k(k-1)(2m+k)(2m+k-l)c’ ~‘(2m+2k-l)2(2m+2k+1)(2m+2k-3)

Y?= (m+k) (m+k+l) +4cfl-(zm+2k~Y~$+2k+31 (The choice of r in 21.7.4 is arbitrary.)

(k2 2)

0) 1(k>

754

SPHEROIDAL Power

WAVE Series

FUNCTIONS

Expansion

for

X,”

21.7.5

&=n(n+l) 1J2m--1Pm+l) (27+-l) (%+3)

1

I,=-(n--m+l)(n--m+2)(~+m+l)(n+m+2)+(7a--m-l)(n--m)~~+m-l)(n+m) 2(2n+l) (2n+3)3(2n+5) ze= (4d-

(n-m+l)(n-m+2)(n+m+l)(n+m+2)~ (Zn-1) (2n+l) (2n+3)‘(2n+5)

1)

ls=2(4mp-1)2A+~

B+;

C+;

(~-m+1)(7a--m+2)(n+m+l)(~+m+2) (2n-l)2(2n+1)(2n+3)7(2n+5)(2n+7)2

Expansion

for

X,”

Refinement

21.7.6 A,n.(c)=cp+m2-;

-A2

($+5)-s

21.7.7 X,=X$+6x,

-- 1 - 1 (33pb+1594$+5621q) c3 [ 1282

-z

1 [

c

of &,m

I

6X,,= Ul WJ

W6+167P)+~

+u2ou

f&-l-~2

q]

(115*4+1310~+735)+~

(pl+l)]

A2=(N7+2)2

-?%?-

-2

Values

(63$+4940q4+43327tf+22470)

&

-g2 1

of Approximate

If Q, is an approximation to &,, obtained either from 21.7.5 or 21.7.6 then

($+ll-32m2)

[5(q4+26d+21)--384m2(qB+1)l

-&

1

(VP-m-l)(n-m)(n+m-l)(n+m) (2n--5)(2n-3)(2n-l)6(2n+l)(2n+3)

D

(?--m-l)(n-m)(n+m-l>(n+m) A=(2n-5)2(2~-3)(2~-l)7(2n+l)(2n+3)2-

Asymptotic

(2n+7)

2(2n-3)(2n-l)3(2n+l)

A2

(hT+2W+4)2+ #%2@+4

0?‘+2W+4W+J2 /37+2/3%n+a

+...

(527q7+61529pb+1043961$

+2241599(J) -A4 +29s951q)+g

(573iW+l27550$

(355n”+15o5p)-~]+o(c-7 p=2(n-m)+1

(2m+r)(2m+r-l)c’ w=(2m+2r-l)(2m+2r+l)

d, 4-2

(r22)

T(T-1)(2m+T)(2m+r-1)c’ fl~=(2m+2r-l)y2m~+2r+l)(2m.t27--3) (f 22)

SPHEROIDAL Evaluation

Step 1.

WAVE

of Coeflicienta

Calculate

755

FUNCTIONS

21.7.15

NT’s from

21.7.8 Ny+,=+bn.-g

CT221

I

N?=rT-Ln;

(n-m)

Ny=ry’-L,,

= Step 2.

Calculate

ratios $ and & 2r

21.7.9

?(n+m+l)! C-1) ! (“+;+1>

2”-” (y-1)

of Angular SchZke

Functions

S-1l [Sm&, d12&=& C,

T!

(This normalization fwd as v-+1.)

&,(c,O)=~(O)=

Snn(C,

V)

21.7.17 &f&,

9) = 0 -112>‘~mn(c, 7)

um&keJ5m

(c--

>

1-n-m

Wl,&)

cylinder

func-

$$$ and the H,(z) are the Hermite polynomials (see chapter 22). (For tables of &,/hi see [21.4].)

d =(n+m)! ’

(n-m)!

has the effect that S,,,Jc, q)+

Expansion

of S..(c,

7) in Powers

of r)

21.7.18

Flammer

21.7.13

for

Scheme

(T+2m)!

r=o, 1

Expansions

where the D,(z)‘s are the parabolic tions (see chapter 19).

Scheme

Stratton-Morse-Chu-Little-Corbat6

21.7.12

Odd

(The normalization scheme 21.7.13 and 21.7.14 is also used in [21.10].)

and the formula for NT in 21.7.7. The coefficients dy” are determined to within the arbitrary factor do for r even and d, for T odd. The choice of these factors depends on the normalization scheme adopted.

21.7.11

(n-m)

P+l

Asymptotic

Meixner-

!

from

$=(>($)...&)

Normalization

even

21.7.16

Scheme

[21.4]

“--(ll C-l) 2 (n+m)! 2” (?4)!(!!+?9

(n-m)

even

n-m-1

(The derivation for A,,,,,is similar

“(n+m+l)!

21.7.3.)

21.7.14 S;n(c,O)=fy(0)=

(-l)

of the transcendental equation to the derivation of 21.7.4 from

2”(“-3!(“+;+‘)! Expansion

(n-m) The above lead to the following 4 mn

conditions

odd for

of S=“(c,

(I) in Powers

of (l-3)

21.7.19 &&,(c, r)=(1-tl’)“/2~ocb”;(l-b)L

(n-m)

even

756

SPHEROIDAL

WAVE

FUNCTIONS

~n=-2-gp[33p4+114p2+37-2m2(23q2+25)

21.7.20 (n-m)

+ 13m*]

odd ,?y= -2-10[63q6+340q4+239q2+

1

___~

m (2m+2’)!(--T)k(m+T+$@ 2 (a?)!

C’m;‘2mk!(m+k)!

-10m2(10p4+23q2+3)+m4(39q2-18)-2me]

(n-m)

even

~‘“=v(v+m)a,‘+(v+l)(v+m+l)a:’ q=n+l

c2; = m (2m+2ry)! z (2r+l).

2’k!(ml+k)!

Asymptotic

(n-m)

(The @‘s

for (n-m)

odd

* . . (a+k+l)

are the coefficients

even; q=n for (n-m)

Expansion

for

Oblate

Angular

Functions

21.8.3

Smn(-ic, +(1+)~‘2

5 A,mn{e-c~1-~)LSm+![2C(1--l)] *=--Y

in 21.7.1.)

+(-l)n-me-ccl++L$ Prolate.

Angular

Expansion 21.7.2

Functions--Second

ultimately

odd

(For the definition of agr see 21.8.3.)

(-r,,(m+r+;)$y+,

(&=&+1>(~+2)

14

[Zc(l+~j)]]

Kind

where the L$““’(2) are Laguerre polynomials (see chapter 22) and

leads to

21.7.21

f&g

a,f l( m, n>cmk

(Expressions of a&*’ are given in [21.4].) 21.9. Radial

(The coefficients G$‘“”are the same as in 21.7.1; the coefficients d;: are tabulated in [21.4].) 21.8. Power

Oblate Series

21.8.1

Angular Expansion

Spheroidal

Wave

Functions

21.9.1

Functions for

Eigenvalues

q)(z)= Jz-.J?&++(z)

x,.=~(-1)$kc’”

(p=l)

where the lk’s are the same as in 21.7.5. Asymptotic

Expansion

for

Eigenvalues

=J g Yn+&> (p=2)

[21.4]

21.8.2

A,,= -c2+2c(2v+m+

(J,++(z) and Y,++(z) are Bessel functions, order n++, of the first and second kind respectively (see chapter lo).)

1)-2v(v+m+l) -(m+l>+Amfl

v=f (n-m)

for (n-m) v=f

even; (n-m-

1) for (n-m)

odd

21.9.2

R7%(C, s> =E~(c,

21.9.3

R~~(c,I)=R~~((c,~)--iR~~T!,(c,~)

Asymptotic

@‘“=

--2-%q*+

10q2+ 1 -2m2(3q2+

1) +m’]

Behavior

8 +W~(c,

of

s>

R$,((c, E) and R~;((c, C)

21.9.4

C,@, t)c 01$ ~0sI@-Hn+W

21.9.5

R%(C,S)---& i sin [~[-+(n+l)~]

SPHEROIDAL

21.10.

Joining

Factors

for

WAVE

Prolate

757

FUNCTIONS

Spheroidal

Wave

Functions

21.10.1

Kg;(c)

=

(n-m)

even

(n-m)

odd

(n-m)

odd

21.10.2 S~~(c,I)=K~~(C)R~~(C,5‘)

(The expression for joining factors appropriate to the oblate case may be obtained from the above formulas by the transformation c*-k)

21.11.

Ang. coord.

Stratton, Morse, Chu, Little and CorbaM

7

h

E

I

I

I

Independent variable

Rsd. coord.

&z(h,

kd(h,

d

7

Chu and Stratton

I)

Meixner and Schilfke

B

Morse and Feshbath

*=cos

I

Wave I

Red. wave function

Ang. wave function

Functions I

1 Eigenvalue

0

1

AmzV4

Smz(h,

1= Flammer’s Amz= L,,,

&n”(C, 7)

e2(c, .9

hn”(C>

Smn(c, 0) = F?(O) &,,,,(c, 0) = R’ (0)

C

&!k

R%

A

fQ(c, 0) = P-+,(O) S::‘(c, 0) = E:,,(O)

c

uzmw

11)

Remarks

O=Pi”(l)

C

.-

I9

I

Normalisation of angular functions

,mzp ;] ml I

I

Flammer and this chapter

Notation

Notation for Prolate Spheroidal

#

U

ntZ

n

(n - 74 even (n - 74 bdd (I even) (I odd)

I= Flammer’s n - m &z= -A,, n-m,

E=cosh /.I

.Page

tl

vzIn(d

~(l-PP’~~zmn(U1=1

QZWI

E=l

;:-8m Notation

Stratton, Morse, Chu, Little and CorbaM

7

Flammer and this chapter

v

g

j S,dig,

for 7d

Oblate

Spheroidal

1 jemz(ic7,

TVa - ve Functions A

-3

“Z

I 1) =P?(U

$= FkFmer’s ml *n

n

-E

C

&d-k

3

R”)I* ( -

ic 2 it)

LA-ic)

S,,( -ic,O) =e(o) S:,(--ic,

7

t

C

Ass ( - ic, 7))

Rg/ ( - ic,

iE)

B IPSZ

-Meixner and Schiifke

n

I S,z(ig,

O)=P$(O)

(n - m) even (n - m)odd

-Chu and Stratton

I = Flammer’s crzm=Xmn-Ca

B

Y

PC h, - -8

S(f) ” ( - i& i+)

,S’s( -ic, 0) =e+z(0) S$‘( - ic, 0) = P=;z(O)

W-r? s

A

ntZ

Z=Flammer’s n--m - X,, n-,,,

Bz,=

A;(-ya)=X,,(-ic)+cl

-1, UPS% --?)I% 2 =2n+l

--

(I even) (I odd)

(n+m)! (n-m)!

r(1--?2)-~‘2&zGB, dl,1 =[(l -~P’2P;(?)lbl

2 F1,;mer’s ml mn

n

I= Flammer’s azm=XnnfCa

n

-LeEtner and Spence The notation

QZWI

in this chapter closely follows the notation

in 121.41.

-

SPHEROIDAL

WAVE

FUNCTIONS

759

References [21.1] M. Abramowitz, Asymptotic expansion of spheroidal wave functions, J. Math. Phys. 28, 195-199 (1949). [21.2] G. Blanch, On the computation of Mathieu functions, J. Math. Phys. 25, l-20 (1946). [21.3] C. J. Bouwkamp, Theoretische en numerieke behandeling van de buiging door en ronde opening, Diss. Groningen, Groningen-Batavia, (1941). [21.4] C. Flammer, Spheroidal wave functions (Stanford Univ. Press, Stanford, Calif., 1957). [21.5] A. Leitner and R. D. Spence, The oblate spheroidal wave functions, J. Franklin Inst. 249, 299-321 (1950). [21.6] J, Meixner and F. W. Schafke, Mathieusche Funktionen und Sphiiroidfunktionen (SpringerVerlag, Berlin, Gottingen, Heidelberg, Germany, 1954).

[21.7] P. M. Morse and H. Feshbach, Methods of theoretical physics (McGraw-Hill Book Co., Inc., New York, N.Y., 1953). [21.8] L. Page, The electrical oscillations of a prolate spheroid, Phys. Rev. 65, 98-117 (1944). [21.9] J. A. Stratton, P. M. Morse, L. J. Chu and R. A. Hutner, Elliptic cylinder and spheroidal wave functions (John Wiley & Sons, Inc., New York, N.Y., 1941). [21.10] J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, and F. J. Corbat6, Spheroidal wave functions (John Wiley & Sons, Inc., New York, N.Y., 1956).

760

SPHEROIDAL Table 21.1

WAVE FUNCTIONS

EIGENVALUES-PROLATE

AND OBLATE

PROLATE

k(c)--m(mf1)

0:879933 1.127734

1 2.000000 2.593084 3.172127 3.736869 4.287128

2 6.000000 6.533471 7.084258 7.649317 8.225713

3 12.000000 12.514462 13.035830 13.564354 14.100203

4 20.000000 20.508274 21.020137 21.535636 221054829

1.357336 1.571155 1.771183 1.959206 2.136732

4.822809 5.343903 5.850492 6.342739 6.820888

8.810735 9.401958 9.997251 10.594773 11.192938

14.643458 15.194110 15.752059 16.317122 16.889030

22.577779 23.104553 23.635223 24.169860 24.708534

2.305040 2.465217 2.618185 2.764731 2.905523

7.285254 7.736212 8.174189 8.599648 9.013085

11.790394 12.385986 12.978730 13.567791 14.152458

17.467444 18.051962 18.642128 19.237446 19.837389

25.251312 25.798254 26.349411 26.904827 27.464530

3.041137 3.172067 c-t)3

9.415010 9.805943 (-92

14.732130 15.306299 t-32

20.441413 21.048960

28.028539 28.596854

3 5.26224 5.25133 5.25040 5.26046 5.28251

4 7.14921 7.05054 6.96237 6.88638 6.82460

5.31747 5.36610 E: 5:59516

6.77941 6.75360 6.75030 6.77286 6.82451

4.324653 4.381878 4.436798 4.489168 4.539096

5.69566 5.80359 5.91452 6.02383 6.12806

6.90779 7.02356 7;16962 7.33916 7.52035 7.69932 7.86638 8.01951 8.16148 8.29538 8.42315 8.54594 8.66452 8.77945 8.89116 9.00000 $314

G/n

0 0.000000

f x'K40

32

4

10 :: ::

*

[ 1

[ 1

[

1

[(-;I91

[(-p1

c-%0&1

od:: 0:21

0 0.793016 0.802442 0.811763 0.820971 0.830059

1 2.451485 2.477117 2.503218 2.529593 2.556036

0.20 0.19 0.18 0.17 0.16

0.839025 0.847869 0.856592 0.865200 0.873698

2.582340 2.608310 2.633778 2.658616 2.682743

0.15 0.14 0.13 0.12 0.11

0.882095 0.890399 0.898617 0.906758 0.914827

2.706127 2.728784 2.750762

0.10 0.09 0.08 0.07 0.06

0.922830 0.930772 0.938657 0.946487 0.954267

2.813346 2.833316 2.852927 2.872213 2.891203

4.586895 4.632927 4.677506 i%slet; .

6.22577 6.31730 6.40385 6.48655 6.56618

0.05 0.04 0.03 0.02 0.01 0.00

0.961998 0.969683 0.977324 0.984923 0.992481 1.000000 (342

2.909920 2.928382 2.946608 2.964611 2.982404 3.000000

4.804519 4.845033 4.884779 4.923820 4.962212 5.000000 (-$I6

6.64326 6.71812 6.79104 6.86221 6.93182 7.00000 c--y

c-l\n

0.25 0.24

[: 1

%19?1 .

[ (-yJ1

2 3.826574 3.858771 3.895890 EZ .

[

1

[

1

[ 1

761

SPHEROIDAL WAVE FUNCTIONS EIGENVALUES-PROLATE

AND OBLATE

Table

21.1

OBLATE

k,,(-ic)-m(mf1) X04- ic) G\n 10 f 4

15 16

0 0.000000 -0.348602 -0.729391 -1.144328 -1.594493

1 2.000000 1.393206 0.773097 +0.140119 -0.505243

-2.079934 -2.599668 -3.151841 -3.733981 -4.343292

-1.162477 -1.831050 -2.510421 -3.200049 -3.899400

-4.976895 -5.632021 -6.306116 -6.996903 -7.702385 -8.420841 -9.150793 (y)4

II 1

*

3 12.000000 11.492120 10.990438 10.494512 10.003863

4 20.000000 19.495276 18.994079 18.496395 18.002228

2:923796 2.578730 2.251269

9.517982 9.036338 8.558395 8.083615 7.611465

17.511597 17.024540 16.541110 16.061382 15.585448

-4.607952 -5.325200 -6.050659 -6.783867 -7.524384

1.938419 1;637277 1.345136 1.059541 0.778305

7.141427 6.673001 6.205705 5.739084 5.272706

15.113424 141645441 14.181652 13;722230 13.267364

-8.271795 -9.025710 c-y

0.499495 0;221407 (-;I3

4.806165 4.339082

12.817261 121372144 6gW

c 1

‘-I

6.OO;OOO 5.486800 4.996484 4.531027 4.091509

z%~~

[ 1

[:t-y1

[ 1

c-2[Xon( GC)]

0.25 0.24 0.23 0.22 0.21

0 -0.571924 -0.585248 -0.599067 -0.613349 -0.628058

1 -0.564106 -0.579552 -0.595037 -0.610591 -0.626242

2 +0.013837 -0.009136 -0.031481 -0.053477 -0.075480

3 0.271192 0.213225 0.157464 0.103825 0.052196

4 0.77325 0.67822 0.58772 0.50191 0.42099

0.20 0.19 0.18 0.17 0.16

-0.643161 -0.658625 -0.674418 -0.690515 -0.706891

-0.642016 -0.657938 -0.674031 -0.690310 -0.706792

-0.097943 -0.121428 -0.146603 -0.174201 -0.204894

+0.002437 -0.045635 -0.092251 -0.137692 -0.182301

0.34521 0.27490 0.21043 0.15215 0.10020

0.15

-0.723530 -0.740416 -0.757541 -0.774896 -0.792476

-0.723486 -0.740399 -0.757535 -0.774894 -0.792476

-0.239109 -0.276886 -0.317881 -0.361548 -0.407352

-0.226469 -0.270627 -0.315206 -0.360594 -0.407081

0.05428 +0.01332 -0.02476 -0.06337 -0.10723

-0.810279 -0.828301 -0.846539 -0.864992 -0.883657

-0.810279 -0.828301 -0.846539 -0.664992 -0.883657

-0.454896 -0.503937 -0.554337 -0.606021 -0.658931

-0.454839 -0.503928 -0.554337 -0.606021 -0.658931

-0.16065

-0.902532 -0.921616 -0.940906 -0.960402 -0.980100 -1.000000 t-f)6

-0.902532 -0.921616 -0.940906 -0.960402 -0.980100 -1.000000

-0.713025 -0.768262 -0.824608 -0.882031 -0.940503 -1.000000 (-$14

-0.713025 -0.768262 -0.824608 -0.882031 -0.940503 -1.000000 (-S4)3

1:;;;;;

c-L\n

if:; 0:12 0.11 0.10 KG 0:07 0.06 0.05 0.04 0.03 0.02 0.01 0.00

[

1

[

1

c 1

:g;;;; -0:37117 -0.45125

-0:71218 -0.80533 -0I90131 -1.00000

c(-y 1

762

SPHEROIDAL

Table

WAVE

FUNCTIONS

EIGENVALUES-PROL.4TE

21.1

AND OBLATE

PROLATE

x,.(c)-m(?n-t-l)

cz\n 10 s 4

1110 12 13 14 15 16

Xl”(C)

*

1 0.000000 0.195548 0.382655 0.561975 0.734111

2 4.000000 4.424699 4.841718 5.251162 5.653149

3 10.000000 10.467915 10.937881 11.409266 11.881493

4 18.000000 18.481696 18.965685 19.451871 19.940143

28.000000 28.488065 28.977891 29.469456 29.962738

0.899615 1.058995 1.212711 1.361183 1.504795

6.047807 6.435272 6.815691 7.189213 7.555998

12.354034 12.826413 13.298196 13.768997 14.238466

20.430382 20.922458 21.416235 21.911569 22.408312

30.457716 30.954363 31.452653 31.952557 32.454044

1.643895 1.778798 1.909792

7.916206 8.270004 8.617558

9%t: .

%XE! .

14.706292 15.172199 15.635940 16.097297 16.556078

22.906311 23.405410 23.905451 24.406277 24.907729

32.957080 33.461629 33.967652 34.475109 34.983956

2.281832 2.399593

9.624450 9.948719

17.012115 17.465260 (-;I4

25.409649 25.911881 t-y3

35.494147 36.005634 (-y

4 6.47797 6.38296 6.29522 6.21556 6.14494

5 9.00140 8.80891 8.62445 8.44916 8.28436

4.279522 4.279366 4.285495 4.297965 4.316672

6.08438 6.03498 5.99788 5.97420 5.96496

8.13163 7.99282 7.87010 7.76598 7.68328

4.341320 4.371397 4.406191 4.444844 4.486445

5.97090 5.99230 6.02874 6.07889 6.14051

7.62508 7.59446 7.59407 7.62539 7.68773

4.530151 4.575277 4.621329 4.667984 4.715031

6.21063 6.28624 6.36482 6.44473 6.52505

7.77728 7.88714 8.00897 8.13579 8.26355

4.762333 4.809790 4.857332 4.904906 4.952472 5.000000

6.60532 6.68528 6.76480 6.84378 6.92219 7.00000 (-;I2

8.39048 8.51592 8.63963 8.76153 8.88164 9.00000 (-?)4

IIc-p1

[1(-$)l1

c-l\n 0.25 0.24 0.23 0.22 0.21

1 0.599898 0.613295 0.627023 0.641073 0.655431

2 2.487179 2.491544 2.497852 2.506130 2.516383

0.20 0.19 0.18 0.17 0.16

0.670084 0.685014 0.700204 0.715632 0.731281

2.528591 2.542705 2.558644

0.15 0.14 0.13 0.12 0.11

0.747129 0.763159 0.779353 0.795696 0.812174

2.616135 2.637968

0.10 0.09 0.08 0.07 0.06

0.828776 0.845493

2.733891 2.759305 2.785099

0.05 0.04 0.03 0.02 0.01 0.00

0.913352 0.930535 0.947796 0.965129 0.982531 1.000000 c-p4

%% 0:896251

[

*See page x1.

--2

*

1

;*z:192 .

I'%69 2:708934

%%%E . 2.864224 2.891056 2.918069 2.945243 2.972558 3.000000 c-y

[ 1

[L 1

c-'IX&)-21 3 4.366315 4.338520 4.315609 4.297923 4.285792

1(-;I8 1

*

[ 1

[

1

5

c 1

[

1

SPHEROIDAL WAVE FUNCTIONS EIGENVALUES-PROLATE

AND OBLATE

Table

21.1

OBLATE x,,(-ic)--77)2(772+1)

c2\n 10 f

4

xI,~-ic)-2 3

* *

9.534818 9.073104 8.615640 8.163245

4 18.000000 17.520683 17.043817 16.569461 16.097655

5 28.000000 27.513713 27.029223 26.546548 26.065706

1.758534 1.286300 0.806045 +0.317782 -0.178458

7.716768 7.277072 6.845015 6.421425 6.007074

15.628426 15.161786 14.697727 14.236229 13.777252

25.586715 25.109592 24.634357 24.161031 23.689634

-2.593577 -2.934882 -3.293803 -3.670646 -4.065548

-0.682630 -1.194673 -1.714511 -2.242055 -2.777205

5.602649 5.208724 4.825732 4.453947 4.093464

13.320743 12.866634 12.414640 11.965266 11.517803

23.220190 22.752726 22.287271 21.823856 21.362516

-4.478470 -4.909200 C-$)2

-3.319848 -3.869861

3.744202 3.405903

11.072331 10.628718 c-t)3

20.903290 20.446222 C-2)3

1 0.000000 -0.204695 -0.419293 -0.644596 -0.881446

2 4.000000 mi1 2:678958 2.222747

-1.130712 -1.393280 -1.670028 -1.961809 -2.269420

10.000000

[Ic-y1

[ 1

[(--p’1

c-l\n 0.25 0.24 0.23 0.22 0.21

1 -0.306825 -0.318148 -0.330984 -0.345469 -0.361702

2 -0.241866 -0.266693 -0.291340 -0.315894 -0.340450

3 0.21286 0.17062 0.13125 0.09476 0.06107

4 0.66429 0.57759 0.49460 0.41533 0.33974

5 1.2778 1.1420 1.0120 0.8879 0.7697

0.20 0.19 0.18 0.17 0.16

-0.379735 -0.399564 -0.421125 -0.444308 -0.468974

-0.365113 -0.389998 -0.415222 -0.440907 -0.467166

0.03001 +0.00127 -0.02563 -0.05142 -0.07710

0.26779 0.19942 0.13449 0.07282 +0.01411

0.6575 0.5515 0.4520 0.3591 0.2735

0.15 0.14 0.13 0.12 0.11

-0.494976 -0.522180 -0.550474 -0.579775 -0.610027

-0.494104 -0.521805 -0.550335 -0.579732 -0.610016

-0.10406 -0.13412 -0.16924 -0.21076 -0.25868

-0.04205 -0.09625 -0.14929 -0.20210 -0.25572

0.1958 0.1271 0.0680 +0.0183 -0.0250

0.10 0.09 0.08 0.07 0.06

-0.641193 -0.673251 -0.706186 -0.739985 -0.774638

-0.641191 -0.673251 -0.706186 -0.739985 -0.774638

-0.31185 -0.36901 -0.42934 -0.49242 -0.55807

-0.31111 -0.36888 -0.42932 -0.49242 -0.55807

-0.0685 -0.1219 -0.1907 -0.2714 -0.3598

0.05 0.04 0.03 0.02 0.01 0.00

-0.810135 -0.846468 -0.883628 -0.921608 -0.960401 -1.000000 c-y

-0.810135 -0.846468 -0.883628 -0.921608 -0.960401 -1.000000

-0.62616 -0.69657 -0.76923 -0.84406 -0.92100

-0.62616 -0.69657 -0.76923 -0.84406 -0.92100 -1.00000 C-l)5

-0.4542 -0.5540 -0.6588 -0.7682 -0.8820 -1.0000 (-72

c 1

*see page II.

c 1

*

cc-y1

cc-y1

-1.00000

c 1

1 1

II 1

764

SPHEROIDAL

Table

21.1

WAVE

FUNCTIONS

EIGENVALC’ES-PROLATE

.4ND OBLATE

PROLATE

hi.(c)-m(m+1) C-+8 10 3

4

2 0.000000 0.140948 0.278219 0.412006 0.542495

3 6.000000 6.331101 6.657791 6.980147 7.298250

b,(c) -6 4 14.000000 14.402353 14.804100 15.205077 15.605133

0.669857 0.794252 0.915832 1.034738 1.151100

7.612179 7.922016 8.227840 8.529734 8.827778

1.265042 1.376683 1.486122 1.593469 1.698816 1.802252 1.903860

[c-y1

* * 5 24.000000 24.436145 24.872744 25.309731 25.747043

6 36.000000 36.454889 36.910449 37.366657 37.823486

16.004126 16.401931 16.798429 17.193516 17.587093

26.184612 26.622373 27.060261 27.498208 27.936151

38.280913 38.738910 39.197451 39.656510 40.116059

9.122052 9.412636 9.699610 9.983052 10.263039

17.979073 18.369377 18.757932 19.144675 19.529549

28.374023 28.811761 29.249302 29.686584 30.123544

40.576070 41.036514 41.497364 41.958589 42.420160

10.539650 10.812958 C-t)‘5

19.912501 20.293486 C-j’2

30.560125 30.996267 C-2)6

42.882048 43.344222 C-j)8

[

1

c 1

c-‘Ix,“(c) -61

ll 1

[ 1

*

0.25 0.24 0.23 0.22 0.21

2 0.475965 0.489447 0.503526 0.518220 0.533551

3 2.703239 2.683149 2.665356 2.650003 2.637236

4 5.073371 4.994116 4.919290 4.849313 4.784640

5 7.74906 7.58138 7.41971 7.26479 7.11743

6 10.8360 10.5536 10.2781 10.0103 9.7512

0.20 0.19 0.18 0.17 0.16

0.549534 0.566185 0.583513 0.601526 0.620224

2.627196 2.620017 2.615819 2.614701 2.616735

4.725757 4.673177 4.627427 4.589031 4.558480

6.97858 6.84931 6.73081 6.62442 6.53155

9.5023 9.2649 9.0409 8.8323 8.6417

0.15 0.14 0.13 0.12 0.11

0.639604 0.659659 0.680376 0.701737 0.723722

2.621954 2.630349 2.641862 2.656384 2.673764

4.536196 4.522485 4.517479 4.521086 4.532956

6.45371 6.39236 6.34878 6.32389 6.31794

8.4718 8.3260 8.2078 8.1208 8.0678

0.10 0.09

0.746308 0.769471 0.793186 0.817429 0.842175

2.693817 2.716339 2.741120 2.767960 2.796673

4.552484 4.578871 4.611219 4.648642 4.690346

6.33030 6.35935 6.40263 6.45738 6.52096

8.0507 8.0688 8.1184 8.1932 8.2864

0.867402 0.893087 0.919209 0.945747 0.972684 1.000000 C-j'9

2.827089 2.859059 2.892449 2.927138 2.963019 3.000000 c-t)4

4.735658 4.784022 4.834980 4.888160 4.943252 5.000000

6.59127 6.66670 6.74607 6.82849 6.91330 7.00000 (33’2

8.3919 8.5057 8.6249 8.7477 8.8730 9.0000

c-l\n

F% 0:06 0.05 0.04 0.03 0.02 0.01 0.00

1 1

*See

page

II.

[

1

[c-y1

[ 1

765

SPHEROIDAL WAVE FUNCTIONS EIGENVALUES-PROLATE

AND

ORLATE

Table

21.1

OBLATE

2 0.000000 -0.144837 -0.293786 -0.447086 -0.604989

X,,(-ic)--m(m+l) &,a(--id-6 3 4 6.000000 14.000000 5.664409 13;597220 5.324253 13.194206 4.979458 12.791168 4.629951 12.388328

-0.767764 -0.935698 -1.109090 -1.288259 -1.473539

4.275662 3.916525 3.552475 3.183450 2.809393

:i

-1.665278 -1.863838 -2.069595 -2.282933 -2.504245

15 16

-2.733927 -2.972375

G\n 10

2 3 4

10 ::

[C-f)1 1

* * 5 24.000000 23.564371 23.129322 22.694912 22.261201

6 36.000000 35.545806 35.092330 34.639597 34.187627

11.985928 11.584224 11.183489 10.784014 10.386106

21.828245 21.396098 20.964812 20.534436 20.105013

33.736444 33.286069 32.836522 32.387826 31.940000

2.430250 2.045970 1.656508 1.261822 0.861875

9.990084 9.596286 9.205059 8.816762 8.431761

19.676587 19.249195 18.822869 18.397640 17.973532

31.493066 31.047043 30.601952 30.157814 29.714648

0.456635 0.046076 (-p7

8.050424 7.673121 (-:)5

17.550565 17.128753

29.272476 28.831317

[ 1

[ 1 * II(-pl1 ~-~[X~,(-ic)-6]

[Ic-y1

c-l\n 0.25 0.24 0.23 0.22 0.21

2 -C.185773 -0.190754 -0.196680 -0.203790 -0.212386

3 +0.002879 -0.030028 -0.062228 -0.093813 -0.124893

4 0.47957 0.41280 0.34933 0.28933 0.23297

5 1.07054 0.95365 0.84167 0.73461 0.63251

6 1.8019 1.6261 1.4577 1.2965 1.1428

0.20 0.19 0.18 0.17 0.16

-0.222841 -0.235596 -0.251126 -0.269873 -0.292149

-0.155607 -0.186120 -0.216631 -0.247375 -0.278624

0.18049 0.13215 0.08816 0.04864 +0.01342

0.53537 0.44322 0.35607 0.27389 0.19662

0.9964 0.8574 0.7260 0.6022 0.4863

0.15 0.14 0.13 0.12 0.11

-0.318047 -0.347414 -0.379928 -0.415213 -0.452947

-0.310677 -0.343847 -0.378432 -0.414688 -0.452800

-0.01813 -0.04727 -0.07609 -0.10778 -0.14643

0.12409 +0.05600 -0.00822 -0.06954 -0.12937

0.3785 0.2795 0.1901 0.1120 +0.0470

0.10 0.09 0.08 0.07 0.06

-0.492902 -0.534942 -0.578991 -0.625006 -0.672956

-0.492871 -0.534937 -0.578991 -0.625006 -0.672956

-0.19508 -0.25333 -0.31876 -0.38955 -0.46494

-0.18959 -0.25217 -0.31861 -0.38955 -0.46494

-0.0051 -0.0517 -0.1076 -0.1844 -0.2768

0.05 0.04 0.03 0.02 0.01 0.00

-0.722813 -0.774556 -0.828164 -0.883618 -0.940902 -1.000000 C-i)5

-0.722813 -0.774556 -0.828164 -0.883618 -0.940902 ,1.000000 C-l’2

-0.54456 -0.62821 -0I71571 -0.80691 -0.90171 -1.00000

-0.54456 -0.62821 -0.71571 -0.80691 -0.90171 -1.00000 (536

-0.3791 -0.4895 -0.6073 -0.7319 -0.8629 -1l0000 (-i)3

II 1

*see page II.

[

1

IIc-p1

[ 1

[ 1

766

SPHEROIDAL

Table 21.2

ANGULAR

WAVE

FUNCTIONS

FUNCTIONS-PROLATE PROLATE

AND OBLATE

Smn(c, cos 0)

10”

20”

30”

40”

50”

60”

70

80”

0.8525 015431 0.2815 0.1312 0.0585

0.8651 015772 0.3242 0.1689 0.0861

0.8847 016320 0.3967 0.2379 0.1419

0.9091 0.7032 0.4980 0.3442 0.2380

0.9354 0.7842 0.6226 0.4885 0.3839

0.9606 0.8654 0.7571 0.6589 0.5742

0.9815 0.9355 0.8805 0.8271 0.7776

0.9952 0.9831 0.9682 0.9530 0.9383

90” 1.000 1.000 1.000 1.000 1.000

: 4 5

0.9046 0.6681 0.4034 0.2042 0.0916

0.8936 016665 0.4099 0.2138 0.1001

0.8602 0.6598 0.4273 0.2415 0.1262

0.8035 0.6429 0.4489 0.2833 0.1703

0.7225 0.6081 0.4630 0.3294 0.2279

0.6169 0.5472 0.4543 0.3618 0.2840

0.4878 0.4540 0.4068 0.3566 0.3104

0.3381 0.3270 0.3110 0.2929 0.2752

0.1731 0.1717 0.1695

i 0

OdE .

i

: 4 5

1.022 1.064 1.041 0.8730 0.6018

0.9795 1.030 1.023 0.8768 0.6233

0.8553 0.6621 0.9271 0.7579 0.9640 0.8497 LT.8787 0.8513 0.6792 0.7407

0.4198 0.5296 0.6660 0.7549 0.7537

0.1556 -0.0988 -0.3105 0.2602 -0.0192 -0.2668 0.4104 +0.1061 -0.1938 0.5553 0.2512 -0.0998 0.6494 0.3844 +0.0008

0.9892 0.9590 0.9090 0.8197 0.6650

0.9042 0.8864 0.8546 0.7877 0.6560

0.6692 0.6816 0.6957 0.6868 0.6183

0.3400 -0.0045 -0.2816 0.3840 +0.0560 -0.2261 0.4485 0.1501 -0.1364 0.5087 0.2591 -0.0215 0.5245 0.3482 +0.0971

0.1578 0.1194 0.0776 0.0449 0.0239

0.3134 0.2437 0.1654 0.1018 0.0588

0.4643 0.3757 0.2724 0.1832 0.1179

0.6067 0.5149 0.4030 0.2994 0.2162

0.7355 0.6562 0.5546 0.4537 0.3650

0.8450 0.7892 0.7144 0.6353 0.5602

0.9290 0.9000 0.8597 0.8150 0.7698

0.9819 0.9740 0.9627 0.9497 0.9361

0.4788 0.3896 0.2780 0.1762 0.1011

0.9054 0.7509 0.5538 0.3683 0.2254

1.232 1.052 0.8148 0.5813 0.3896

1.417 1.253 1.030 0.7968 0.5906

1.435 1.316 1.149 0.9643 0.7879

1.276 1.212 1.118 1.008 0.8957

0.9562 0.9335 0.8992 0.8575 0.8127

0.5119 0.5088 0.5039 0.4979 0.4911

i 5

0.9928 0.9559 0.8745 0.7393 0.5662

1.745 1.710 1.611 1.418 1.146

2.075 2.092 2.063 1.934 1.691

1.903 1.998 2.097 2.128 2.047

1.280 1.432 1.640 1.841 1.975

0.3775 0.5298 0.7606 1.032 1.299

-0.5521 -0.4541 -0.2972 -0.0951 +0.1319

; 4 5

0.0844 0.0690 0.0500 0.0328 0.0198

0.3295 0.2744 0.2051 0.1405 0.0898

0.7111 0.6092 0.4773 0.3487 0.2414

1.189 1.054 0.8738 0.6876 0.5212

1.710 1.572 1.380 1.171 0.9701

2.211 2.101 1.944 1.764 1.580

0.4222 0.3597 0.2765 0.1934 0.1244

1.570 1.358 1.070 0.7758 0.5226

3.116 2.755 2.255 1.723 1.243

4.596 4.175 3.576 2.909 2.269

5.530 5.170 4.641 4.025 3.395

5.548 5.327 4.994 4.588 4.150

0"

m

n

c\e

0

01

0.8481 0.5315 32 0.2675 0.1194 54 0.0502

011

0

0

21

31

2 : 5

111

2 3 54

12

1 2 i 5

13

2

2

1 2

21

31 32 :

From C. Flammer, permission).

Spheroidal

wave functions. Stanford

Univ.

-0.4259 -0.3907 -0.3319 -0.2514 -0.1575

-0.4509 -0.4385 -0.4171 -0.3879 -0.3542

-0.4085 -0.2467 -0.3949 -0.2447 -0.3714 -0.2412 -0.3376 -0.2361 -0.2952 -0.2293

-0.5000 -0.5000 -0.5000 -0.5000 -0.5000 ii 0 00

1.000 1.000 1.000 llOO0 1.000 0 : 0 0

-1.244 -1.214 -1.174 -1.097 -1.017

-1.500 -1.500 -1.500 -1.500 -1.500

2.627 2.566 2.475 2.367 2.251

%3e 2:859 2.827 2.791

3.000 3.000 3.000 3.000 3.000

4.501 4.417 4.286 4.122 3.936

2.522 2.510 2.491 2.466 2.437

Press, Stanford,

Calif., 1957 (with

e

SPHEROIDAL

ANGULAR

WAVE

FUN4 ZTIONS-PROLATE OBLATE &m(-k,

m

0

n

01

: 5

21

21.2

7)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.002 1.008 1.022 1.047 1.083

1.007 1.032 1.089 1.191 1.341

1.016

1.044 1.210 1.617 2.452 3.952

1.064

:*i;: 1:449 1.835

1.028 1.132 1.377 1.854 2.648

1.088 1.434 2.366 4.557 9.211

1.115 1.585 2.923 6.323 14.23

1.147 1.767 3.648 8.837 22.11

1.0 1.183 1.986 4.589 12.42 34.48

:

0.1001

0.2034 0.2009

i%:

0.4274 0.4065

0.5128

0.6952 0.6222

0.8530 1.035

0.9760 1.243

1.484 1.105

i

0:1016 K~

0.2079

0.3526 0:3273

0.4664

ii-:::; 0:7681

1.096 0.8398

2.195 1.425

3.105 1.842

4.396 2.378

5”

0

0.1032

K::; .

0.3884

FE: .

0.9804

1.525

3.684

5.741

8.970

-0.4863 -0.4897 -0.4943

2 4' 5

1:;;;; .

4 5

-0.1477 2f:g -0:1495 -0.1504

: 4 5

0.9961 0.9994 1.006 1.020 1.041

O 3 3: 111

l 2 2’ :

5 131 : :

-1.500 -1.500 -1.500 -1.500 -1.500

-1.421 -1.431 -1.447 -1.467 -1.486

-0.4450 -0.4585 -0.4766 -0.4966 -0.5234

-0.3757 -0.4052 -0.4448 -0.4891 -0.5495

-0.2779 -0.3277 -0.3952

-0.1507 -0.2231 -0.3223

r:;,'g .

-0.5977 -0.4356

-0.2810 -0.2839 -0.2885

-0.3855 -0.3947 -0.4097 -0.4306 -0.4589

-0.4466 -0.4668 -0.4998

-0.4491 -0.4839 -0.5421

1;;::: .

1;:;;; .

+0.0070 -0.0872 -0.2183 -0.3681 -0.5869

0.1965 +0.0849 -0.0721 -0.2485 -0.5067

0.4197 0.2999 +0.1311 -0.0458 -0.2880

0.6784 0.5660 0.3845 0.2868 0.1892

0.9749 0.8930 0.7958 0.8201 1.132

-0.7489 -0.6270

-0.3768 -0.4275 -0.5140 -0.6432 -0.8356

-0.2130 -0.2757 -0.3841 -0.5540 -0.8080

+0.0600 -0.0015 -0.1091 -0.2765 -0.5447

0.4613 0.4274 0.3711 0.2912 0.1715

1.011 1.051 1.138 1.327 1.723

i%: i135 1.498 2.242

0.8299 0.9340 1.172 1.708 2.878

0.7506 0.8802 1.188 1.920 3.642

0.6402 0.7864 1.149

0.4731 0.6118 0.9724

0 0 0

2.067 4.400

1.950 4.651

i

:'E

1.247 1.487

0

2:200

2.000

:

3.092 4.786

3.033 5.138

:

0.9838 0.9973 1;025

0.9628 0.9923 1;05sm

:%i .

:*:i: .

0.9316 0.9827 1.093 1.319 1.776

0.5897 0.5950 0.6043 0.6213 0.6400

i%:: . 0.9140 0.9640 1.040

1.113 1.153 1.228 1.349 1.537

1.322 1.398 1.541 I.780 2.165

:-2: 1:837 2.250 2.947

:%I 2:082 2.723 3.868

-0.3165 -0.4427 -0.6502 -0.9148 -1.198

0.2710 +0.1060 -0.1738 -0.5415 -0.9435

0.9015 0.7174 +0.3916 -0.0538 -0.5506

1.501 El 0:5403 0.0161

1.946 1.826 1.572 1.177 0.7471

1.951 1.988 1.834. 1.619 1.439

: 0 0 0

2.291 2.425 2.693

1.970

1.585

0.6041

0

%F 3:157 4.564

z1" 2:966 4.746

1.131 1.305 :z:

1:615 %E

:

4:460

3.188

i

5.877

5.503

t-9':: 8:132 10.07

5.982 6.904 8.515 11.28

4.477 4.990

2.683 3.077

i

6.008 7.857 11.21

5.408 3.879 8.354

: 0

-1.189

-0.8136 -0.8941

+%; rg;.

-1.184 -1.024 -1.353

2 2 2’ :5

2.972 2.979 2.992 3.013 3.052

2.889

2.748

2.965 2.915

ZE

%4 2:830

x.

3.469 3:111

x.

3.200 4.202

2

1.486 1.488 :-:2 iso9

x

4.115 4.180 4.295 4.475 4.738

5.086 5.226 5.482 5.891 6.515

5.704 5.954 6.413 7.166 8.347

3.

Table

:

011

0

AND OBLATE

1.000 1.000 1.000 1.000 1.000

C\l

2

767

FUNCTIONS

1 : :

2.996 21943 3.073

768

SPHEROIDAL PROLATE

‘Table 21.3

RADIAL

WAVE

FUNC!l’IONS

FUNCTIONS-FIRST

AND SECOND

R(l) mn (c’ t) wl

n

0

0

0

1

0

0

C\E 1.005

1.020

1.044

JP)nm (c f E)

1.077

3

-1 3.249 -1 5.308

-1 3.328 -1 5.311

-1 5.162 5.786 -1 4.125

1: 54'::; -1 3:137

-1 4.413 -2 4.444 1.833 4.954 3.421

-2 5.373 -1 3.976 4.293 3.509 1.947

-1 1.287 -1 12.323

11 3 4

s ; : 5. 13

2

1.005

1.020

1.044

1.077

I II II II

2

12

KINDS

II II II II II1;:-:;: II -2 -3 2.378 6.503 -2 4.658

-2 1.322 4.802 -2 9.296

-2 2.012 7.227 -1 1.372 1.960

-2 2.754 9.738 -1 1.798

15 $$::

1: :-:g .

-1 2.376

1; :-g .

II II II -10 -2.077 -4.885 -1.075 -7.294 -6.911

-10 -1.417 -2.874 -7.453 -4.734 -4.585

-10 -1.071 -1.248 -5.480 -3.432 -2.924

-1 3:553 -2 7.089 1.108

2

-2 6.612 1: -4 -3 1:372 :3';2" 2,566

2

3

From C. F’lammer, Spheroidal wave functions. Stanford Univ. Press, Stanford, Calif., 1957 (with permission).

SPHEROIDAL

OBLATE

RADIAL

WAVE

FUNCTIONS-FIRST

769

FUNCTIONS

AND SECOND

KINDS

Table 21.4

R@) mn(- GC, a) nk 0

0

n 0

0.75

7864 9707 :z: 2189 0356 3758

0

1

0

2

1

1

1

2

1

3

2

2

PROLATE

JOINING

FACTORS-FIRST

KIND

$&)

Table

21.5

From C. F’lammer, Spheroidal wave functions. Stanford Univ. Press, Stanford, Calif., 1957 (with permission).

22. Orthogonal

Polynomials

URS W. HOCHSTRASSER

*

Contents

Page

Mathematical Properties .................... 22.1. Definition of Orthogonal Polynomials

22.2. 22.3. 22.4. 22.5. 22.6. 22.7. 22.8. 22.9. 22.10. 22.11. 22.12. 22.13.

22.14. 22.15. 22.16.

22.17.

773 773 774 775 777 777 781 782 783 783 784 785 785 785 786 787 787 788 788 788 790 791

..........

Orthogonality Relations ................ Explicit Expressions .................. Special Values. .................... Interrelations ..................... Differential Equations ................. Recurrence Relations ................. Differential Relations ................. Generating Functions ................. Integral Representations ............... Rodrigues’ Formula. ................. Sum Formulas .................... Integrals Involving Orthogonal Polynomials ....... Inequalities ..................... Liiit Relations ................... Zeros ........................ Orthogonal Polynomials of a Discrete Variable ......

Numerical

Methods

......................

Use and Extension of the Tables ............ 22.19. Least Square Approximations ............. 22.20. Economization of Series ................

22.18.

References

792 793

..........................

Table 22.1. Coefficients

for the Jacobi Polynomials

.....

Ppfl)(z)

n=0(1)6 22.2. Coefficients for the Ultraspherical forz”inTermsof C:)(z). ................... n=0(1)6

Table

Table 22.3. Coefficients

s”inTermsof n=0(1)12

T&r)

for the Chebyshev .....................

Polynomials

C’j? (5) and 794

Polynomials

Z’,(z) and for 795

Table 22.4. Values of the Chebyshev

Polynomials T,(z) ....... ~=..(..)l. 10D Table 22.5. Coefficients for the Chebyshev Polynomials U,(z) and for z”in Terms of U,,,(z) .................... n=0(1)12 ...... Table 22.6. Values of-the Chebyshev Polynomials U,(z) n=0(1)12, x=.2(.2)1, 10D

795

n=0(1)12,

* Guest Worker, Nationd Bureau of Standards, ently, Atomic Energy Commission, Switzerland.)

from The American

University. 771

796 796 (Pres-

772

ORTHOGONAL

POLYNOhUALS

Page Table 22.7. Coefficients z”inTermsof Cm(z) n=0(1)12

for the Chebyshev Polynomials C,(z) and for . . . . . . . . . . . . . . . . . . . . .

797

Table 22.8. Coefficients for the Chebyshev Polynomials S,(z) and for Z* in Terms of S,,,(z) . . . . . . . . . . . . . . . . . . . . . n=0(1)12

797

Table 22.9. Coefficients for the Legendre Polynomials P,(z) and for 9’ in Terms of P,(z) . . . . . . . . . . . . . . . . . . . . . . n=0(1)12

798

Table 22.10. Coefficients for the Laguerre Polynomials L,(z) and for 2” in Terms of L,(z) . . . . . . . . . . . . . . . . . . . . . . T&=0(1)12

799

Table 22.11. Values of the Laguerre Polynomials n=0(1)12, x=.5, 1, 3, 5, 10, Exact or 10D

L,(s)

. . . . . . .

800

Table 22.12. Coefficients for the Hex-mite Polynomials H,,(z) and for 2” in Terms of H,(z) . . . . . . . . . . . . . . . . . . . . . . T&=0(1)12

801

Table 22.13. Values of the Hermite Polynomials H,,(z) n=0(1)12, x=.5, 1, 3, 5, 10, Exact or 11s

. . . . . . .

802

22. Orthogonal Mathematical 22.1.

Definition

of Orthogonal

Polynomials Properties where g(x) is a polynomial

Polynomials

A system of polynomialsj,(x), degree [jn(x)]=n, is called orthogonal on the interval a<x_
of SZ.

consists again of orthogonal

The system polynomials.

in x independent

-

(1.5 P”(’

22.1.1 b

s

w(~>$(~)j&)dx=0

3

a

(n#m;n,

m=o, 1,2,. . .)

The weight function w(x)[w(x) >O] determines the system j*(x) up to a constant factor in each polynomial. The specification of these factors is referred to as standardization. For suitably standardized orthogonal polynomials we set 22.1.2 b w(x)j~(x)dx=h,, s@

jn(x)=k,x”+k:x~-‘+

. . .

(T&=0,1,2,.

...)

These polynomials satisfy a number of relationThe most ships of the same general form. important ones are: /

Differential

22.1.3

L

Equation

sz(x>f~+m(x>j~+~nfn=o

where gZ(x), m(x) are independent constant depending only on n. Recurrence

/=

of 72 and a,, a

Relation

\

22.1.4 .fn+‘=

3

b,+~b,)j,-d-~

/

where 22.1.5

\

I-

Rodrigues’

22.1.6

$=Lenw(x)

Formula

d”I~(~N9(~~1”~

dx,

FIGURE

22.1. a=1.5,

Jacobi Polynomials /3=-.5,

Pfi 8)(x) ,

r&=1(1)5. 773

22.2.

Orthogonality

Relations =

f.(z)

(ame of Polynomia

a

b

Standardization

w(z)

Remarks

h. --

22.2.1

P$@

22.2.2

G.(P,

22.2.3

(2)

q, 4

C”‘(I) = )

Jacobi

-1

Jacobi Ultraspherical (Gegenbauer)

0 -1

1

(l--2)*(1

1

(1 --2)D-a~a-1

1

(1--2*)a-t

+q

P$#‘(l)=

(

n+an

2n+a+o+1

T&l

Chebyshev of the first kind

-1

1

w21-*r(n+2a) n!(n+a)[r(41*

C”‘(1) n

(l---z’)-1

n+2a-1 n (

> bf0)

U”(4

Chebyshev of the second kind

-1

1

(1 --zqt

22.2.6

cvd4

Chebyshev of the first kind

-2

2

(

22.2.7

S.(z)

Chebyshev of the second kind

-2

2

22.2.8

T:(4

Shifted Chebyshev of the first kind

0

22.2.9

Shifted Chebyshev of the second kind

0

22.2.10

Legendre (Spherical)

22.2.11

Shifted Legendre

*See page 11.

T.(l)

n#O

= 1

n=O

22.2.5

1

(

U,(l)

=n+

1-q 29 -4

C,(2)=2

l-2

S,(z)=n+

>

4>

(z--z’)-+

+

1 n#O n=O 1 n#O

Tf(1) = 1

n=O

-1 0

1 1 1

(x-x’)b

1)

1)

II>-1,8>-1 P1-q>

(2n+~)r*Ch+~)

a=0

22.2.4

r(n+a+ l)r(n+B+ n!r(n+a+8+

n!r(fl+q)r(n+p)r(n+p-q+l)

k,=l

=

*+8+1 >

q(1)

=n+

*

1

P.(l)=1 1 2n+l

a#0

a>--+

- 1, q>o

* *

22.2.

-

-

Orthogonality

Relations-Continued

Generalized Laguerre

0

m

e-Q?

k =(-1)” n n!

r(a+n+ n!

0

co

e-*

k n -C--l)” n!

1

e-zz

e,=(-1)”

&i2nn!

e,=(-1)”

&n!

22.2.13

L"(Z)

Laguerre

22.2.14

fin(Z)

Hermite

--m

co

22.2.15

He,(z)

Hermite

--a0

m

-

,-‘?t

1)

a>-1

22.3.

Explicit

Expressions

f”w=d”mgc.h(z) --

= N

f”(Z)

d.

22.3.1

P’“.I @)(2)

n

22.3.2

P-6) ” (J)

n

lYa+n+ n!lYa+@+n+

22.3.3

&(P, 4, d

n

r(q+n) r(p+W

22.3.4

(72’ (2)

22.3.5

cy (2)

[I; [Iz

22.3.6

T.(z)

22.3.7

Un(4

22.3.8

E-‘.(4

22.3.9

Lb) (2)

n

22.3.10

H.(z)

[IT

n!

22.3.11

He.64

n L-15

n!

Bn (4 ("lfr")

n

5 L-1

n 3

[Iz [I;z n n

1) 1)

n 0m

>

r(a+8+n+m+l) . 2=r(a+m+l)

(x-l)“@

1 2n+a+8 .jz ( n

>

r(p+2n-m) r(q+n-m)

zm-m

1

P-q>-1,

2" r(a+n) -arG) 2" n#O n

a>-f>

(-‘)“,

(3

(-l)n

rb+n-74 m!(n-2m)!

(22) “-*-

(-,)m

(n-m-l)! tn!(n-2m)!

(22) --*m

l)*

1

--

2n+a+B n

(z- I)“-++

&)

Remarks

k. 5

(

a>--I,@>-1 a>-1,

(22) n-h

2nd

(n-m)! t---11* m!(n-2m)!

(2x) n--l*

2"

C---l)-

(;,)

~“-*“I

(2n) ! 2n(n!)*

(--ljm

(n"-';)

2”

(-l)n n!

a>-1

(2.z)"-+m

2"

ser! 22.11

Z"-*m

1

(--1)m

,$-

' m!(n-2m)! : (-- llm m!2m(n-2m)!

q>o a#0

n#O, CA”(l)=

(- l)m (n-mI)! m!(n-2m)!

(,,,,,)

s>--1

-

1

776

ORTHOOONAL

POLYNOMIALS Explicit

Espressione

Involving

f.(cose)=~

m-0

fs(ms@I a4.0n

-X

Trigonometric

am COB

Function

(n--an@

0,

Remarka

a#0

22.3.12

C”’ I (CO8 e)

rbSm)r(a+n-m) mun-m)![Iya)]’

22.3.13

P,(cose)

$ (“m”) (2:r:)

22.3.14

CiO)(c4x3e>=i cosne

22.3.15

T,(cos8)=cos nB (n+l)e U,(cos f3)= sin sine

22.3.16

FIQURE 22.2. Jacobi Polyrwmiata P!..@)(x), a=1(.2)2, 8=-J, n=5.

-I

FIQURE

22.3.

Jacobi Polynomiala

a=1.5, fl=-.8(.2)0,

n=5.

P>@(x),

FIQURE

22.4.

mid

Gegenbaw (Ultrmpherical) C$)(x), a=.5, n=2(1)5.

PO&W.+

ORTHOGONAL

POLYNOMIALS

22.4. Z :cial Values f”(Z)

f.(-4

22.4.1

P$m (2)

(-

l)“Pp(z)

22.4.2

C’“’ I (2)

(-

l)q’(z)

22.4.3

P’” (2)

a#0

(-

l)“C~O’(Z)

fvm n+a * ( n > n+2a-

(

fled ;[a--B+(a+Bf2)2]

10, n=2m+

I n

1

1

2az

(- I)d* r(a+n’2), n=2m r(a) (n/2) !

(-1)1

2 it n#O

f.(O)

9 n=2m#O

m

I

2x

IO, n=2m+

1

22.4.4

T,(z)

(-

l)“T.(z)

1

(-l)m,n=2m I0, n=2m+ 1

22.4.5

U.(z)

t-

l)nU”(z)

n+l

(-1) m,n=%m 10, n=2m+ 1

22.4.6

P” (4

(-

l)np”(z)

1

:(-lin

2

2x

n=2m*

2m , ( m >

4”1

x

.O, n=2m+l 22.4.7

I

Lp (2)

(

22.4.8

~I&)

(-

n+a n

-x+a+ 1

>

cm! iII (-1)” ~8

l)“H.(Z)

-

n=2m

2x

(0, n=2m+l

22.5.

Interrelations

Between

Interrelations Orthogonal Same Family

Jacobi

Polynomials

Polynomiab

22.5.1 ppyz>=

r b+a+b+l)

G.

nW+a+B+l)

(

a+B+l,B+l,T)

22.5.2 Q*(p, *,+n!r(n+p)

~y*--1y22-1)

r(2n+P)

(see [22.21]). 22.5.3 F,(p, q,z)=(-l)“n!

= r(n+n>

P;~-~*Q-“(2r-l)

(see [22.13]). Ultraspherical

22.5.4

c:‘(z)=lim; Chebyshev

cp’(z) Polynomiab

22.5.5

L-l.5 FIGURE 22.5. Gegenbauer (Uhasphmhzl) mials cy(z), a=.2(.2)1, n=5.

Polynomials

PO&LO22.5.6

T”(z)=u”(z)--zu”-l(4

of the

ORTHOGONAL

778 22.5.7

T”(z)=sU.-l(x)-Un-2(~)

22.5.8

T&)=3 [u,(z)-u”-2(4l

22.5.9

u.(r)=s.(2it)=u:(~)

POLYNOMIALS

PA--‘* -*)(x)=& (F) T,(x)

12.5.23 12.5.24

P!“*O)(x)=P,(x) Ultraspherical

Polynomials

i2.5.25 22.5.10 U.-I(X)=~-~J22.5.11

C,(z)=

22.5.12

7:“,‘(x)=

[z~“(z)--*+1(41

2T” (;)=2

r(a+Qb!2** r(a)(2n)!

Pp-,)

(22*-l)

(am

22.5.26

T*, (y)

c”(x>=&(z~--s,-*~~~

WO) 12.5.27

22.5.13

s.(x)=u.(;)=u:(q

22.5.14

2’*,(s)=2’,(22-1)=3

C,,(4z-2)

(a#O)

22.5.28

(see [22.22]). 22.5.15

U~(cc)=5.(4~-2)=U,(2z--) Cbebyshev

Polynomials

(see [22.22]). Generalized

Lph)=(-l)m Hermite

22.5.18

T,.+,(z)=&

zPk-“t’(2~2--1)

22.5.30

n!fi U*n(“)=r(n++)

Polynomials

Ltp’(x)=L,(x)

22.5.16 22.5.17

Laguerre

22.5.29

gm [L+mb$l Polynomials

He,(x)=2-“‘*H,

22.5.31

2!!@T”(@=r(n+4j

22.5.32

(n+l)!fi ua(2)=2r(n+*)

p$ -1, (2&- 1) pc-t. z 4) @) Pi** 1)(x)

z 04

(see [22.201). 22.5.19

H,(~)=2~l*Ele.(xJZ)

(see [22.13], [22.20]). Interrelations

Between Difllerent Jacobi

Orthogonal Families

Polynomials

of

Polynomials

22.5.20

22.5.21

22.5.22

FUXJRE

22.6.

Chebyshev Polynomials %=1(l)&

*See page xx.

T*(X),

ORTHOGONAL POLYNOMIALS

779 Legendre

Polynomials

22.5.35

P.(x)=P$yz)

22.5.36

P, (5) = cy)

(z)

22.5.37 g

[P&)1=1.3

. . . (2m-l)C$!!_+,+)(z)

Generalized

Laguerre

Cm 234

Polynomials

22.5.38

I2 FIWRE

22.7.

-

Ciieby~hu Polynomials

22.5.39

U,(x),

W”‘(x)

=?&&j

JJ2.+1(da

T&=1(1)5. Hermite

T.(x)=; CkO) (2) U,(x>=C~‘)(x)

22.5.33 22.5.34

Orthogonal

22.540

as Hypergeometric

fn(x>=@(a,

For each of the listed polynomials geometric functions.

(-l)‘“2k+1m!zLg’2)(z2)

H2,+,(5)= Functions

(z2)

(see

chapter 15)

6; c; dx))

there are numerous d

f. (2)

Ha,(z) = (- 1)m2”m!L$1/2)

22.5.41

Polynomials

Polynomials

other representations

a

b

in terms of hyper-

VW

C

l.

22.5.42

Py(x)

-n

n+cr+Is+l

r+1

1-Z -3-

22.5.43

P$8'(2)

-n

-n--o!

-2n--a--8

2 1--2

22.5.44

P$"(x)

--A

--n-B

a+1

x-l z+i

22.5.45

P'.".@(z)

-n

-n-a

8+1

x+1 z-i

22.5.46

C"'$8(x)

-n

n+2a

a++

l-2 -T

22.5.47

T.(x)

1

-n

n

f

l-2 -3-

22.5.43

U.(x)

n+l

-n

n+2

%

l-2 2

22.5.49

P,(x)

-n

n+l

1

l-x 2

22.5.50

P"(X)

-n

--n

-2n

2 1--z

22.5.51

P,(z)

-- n

l-n

i-n

1 2

Un+2a) n!l-(2a)

2

22.5.52

Pdx)

n CW! (- 1) 22qny

22.5.53

%+I(4

(- 1)“(2n+ 111 -2qGpZ

2

*

-n

n+&

4

29

-n

n+t

9

39

780

ORTHOGONAL

Orthogonal

Pojynomials Functions

as Confluent (see chapter

Hypergeometric 13)

‘OLYNOMIALS

22.5.58 H,(x)

22.5.54

L:a’(x)=~;a)M(-n,

a+l,

=2*~2ez2/2D,,(JZx)=2n~2ez2/2U

r) 22.5.59

Orthogonal

Polynomials Functions

as Parabolic (see chapter 19)

He,(x)-ee’2/4D,(x)=e”2/4U

Cylinder Orthogonal

22.5.55 22.5.56

H, (5) =2”U

(

;-;

n, ;, x2 >

cp (2) = r(a+i>r(2cY+n) n!Iy242)

22.5.57 * Hzm+l(~)=(--l)~

Functions

22.5.60

H,,(z)=(-l)-@$M(-m,;rzz)

(2m+l)! m!

Polynomials as Legendre (see chapter 8)

2xM(-m,

1 [z (22-q-~

P$q$,

iJ x2)

P”(X)

I

FIGURE 22.9.

FIGURE

22.8.

Legendre Polynomials n=2(1)5.

P,(x) ,

FIGURE 22.10.

Lagumre Polynomials n=2(1)5.

Hermite n=2(1)5.

Polynomials

L,,(x),

H&9 7)

(5)

ORTHOGONAL

22.6.

781

POLYNOMIALS

Differential

Equations

g2(2)y”+g1(x)y’+go(x)y=o -__-.--sdz)

Y

91(z) _-

22.6.1

P$@ (2)

22.6.2

(1 --z)“(l

22.6.3

-+I (1 --z);;-(1

22.6.4

n(nta+P+

1)

l-22

8-a-((2+B+2).r

1-x*

a-P+(a+B-2)x

1

ID

(,i”g+ycos,y++Py (cos 2)

1

ID

22.6.5

C”‘( I =)

1-Z

- (2,-l- I)2

22.6.6

(1 -zpc(q

l--22

(2a - 3)x

22.6.7

(,-~*)~+~

1

0

22.6.8

(sin r)%$“‘(cos

1

0

22.6.9

Tdr)

1-9

-X

n*

22.6.10

T,(cost)

1

0

n*

22.6.11

-

l-22

-3x

n*-

22.6.12

lJ.(z)

l-22

-3x

n(n+2)

22.6.13

Pm(x)

1-Z

-2%

n(n+

1)

22.6.14

4i=3P.(z)

1

0

n(n+

1)

22.6.15

L;'(z)

x

a+1--2

n

22.6.16

e-rxa’2L~Q)(5)

1;

&+I

n+;+1-g

22.6.17

e-z12z(atI)12q4 (r)

1

0

___ 2n+a+i+1--(r2 2x

22.6.18

,-**/2,“+4L’,“’

1

0

4n+2a+3-x*+7

22.6.19

H.(z)

1

-2.z

2n

22.6.20

2 e-yH,(s)

1

0

2n+

22.6.21

He.(x)

1

-X

n

+z)~P(*~~)(.z) n

Sz)

8&l * fy’(2)

n x) Cc,)(x)

x)

T"(Z) ; v,-lb)

(x’)

*

*

-

n(n+2a)

1

l-xl

+(Lxy*

1 -x1

__--

4x2

41 1--a*

782

ORTHOGONAL

POLYNOMIALS

22.7. Recurrence Recurrence

Relations

al,fn+l(x)= fn

With

Relations Respect

Degree

n

-a4,fn-l(Z)

(aan+aanr)fn(x)

aIn

to the

ah

ah

--

ah

@n+a+Bh

22.7.1

2(n+a)b+i3) Ghfa+B+2)

22.7.2

Gnh

-[2n(n+p)+q(p-11))

q, 2)

(2n+p-2)4

(2n+p72)3

n(n+r-l)b+p-1) (n+P-dm+P+l)

@n-!-p-l)

22.7.3

Cl”‘(x)

n+1

0

W+ff)

n+2n-1

22.7.4

T,(x)

1

0

2

1

22.7.5

U,(x)

1

0

2

1

22.7.6

S”(X)

1

0

1

1

22.7.7

cm

1

0

1

1

22.7.8

T.’ (4

1

-2

4

1

22.7.9

Kc4

1

-2

4

1

22.7.10

p. (4

n+l

0

2n+l

n

22.7.11

p:(x)

n+l

-2n-1

4n+2

n

22.7.12

L!“‘(x)

n+l

2n+(u+1

-1

n+a

22.7.13

He64

1

0

2

2n

22.7.14

He,(z) -

1

0

1

n

-

Miscellaneous Jacobi

Recurrence

Relations

Ultraspherical

Polynomials

Polynomials

22.7.21

22.7.15

2~(1--;c2)C~_:“(2)=(2a+n--)C~,(2)--n2C~’(z)

(

n+g+i+1)

(l-z)P~+‘+~)(z) = (n+a+

l)P,‘a*@

(2) - (n+ l>P’&B,

(z)

22.7.22 =(n+2cY)zC~'(z)

22.7.16

-(n+lm%(~)

n+E+-+1 B (

2 2

22.7.23

(l+z)PpJ+l)(,)

(n+(~)C~~~‘(2)=(ru-l)[c~:1(5)-C~l(~)]

>

=(n+B+I)P~s)(2>+(n+l)P~~Pi)(~)

22.7.17

Chehyshev

2 T,(z) T&l

(l--)P~+‘,s)(2)+(1+2)P%‘8+1’(2)=2P~8)(s)

Polynomials

22.7.24

= T?l+m(z) + ~,-I&$

*

(n>m>

22.7.25

22.7.18

2(sc2-1)um-~(2)un-1(5)=T,+,(s)--T,_,(s)

(2n+cu+B)P~-',8'(2)=(n+(y+B)P~,8'(~)

(n2

22.7.26

-(n+Bv%P(~)

2T,(2)Un-1(2)=Un+,-1(2>+Un-m-*(2)

22.7.19

22.7.27

(2n+a+P)P~,8-"(2)=(n+a!+B)P~,8'(s) + (n+cY>PgyJ(az)

22.7.20

Plp,S-l)(,)-P~-l.S)(,)=P~~)(z)

(n>m)

2T,(z)U,-,(s)=U,+,-l(s)--u,-,-*(2) 22.7.28 *See page II.

2T,(z)U,-,(5)=Uz,-l(z)

(n>m)

m>

ORTHOGONAL

General&d

Laguerre

Polynomials

783

POLYNOMIAL&

22.7.31

22.7.29

Lp+yz)=~

Lp+yI)==; 22.7.30

[(z--n)L~“‘(z)+(ar+n)Ljp_‘,(z)]

22.7.32 Q-')(z)=

L~-')(~)=L,='(z)--L~*(z) 22.8.

[(72+Ly+l)L~)(2)-((12+l)L~,(2)]

Ihfferential

&a

[(nt-l)~~1(2)-(nSl--z)L~'(z)l

Relations

Ql(z)~f.(z)=g*(z)f"(z)+no(z)f"-~(~) f.

61

B1

go v

(zn+a+B)(l-z*)

7&-B-

c!=) (2)

1-Z’

--Rz

22.8.3

T&)

1-X’

-nx

n

22.8.4

U.(z)

l-9

-7l.l

n+l

22.8.5

PA4

1-Z’

-TIX

n

22.8.6

L!“) (2)

2

n

- (n+a)

22.8.7

HI&)

1

0

2n

22.8.8

He,@)

1

0

22.8.1

Pp’

22.8.2

(2)

(2n+a+BM *

2(n+a)b+f9) n+2a-1

n

22.9.

Generating

Functions R= Jl-2zz+z2

f&)

Remarks

g(w)

-22.9.1

P$fl(x)

R-‘(l--e+R)-“(l+e+R)-8

22.9.2

cp (2)

22.9.3

cp (2)

22.9.4

c#y (2)

1

22.9.5

C”I (2)

rb+t)rcb+4

22.9.6

T,(z)

Id<1 bl
NW 2 42

2n

R-h

bl
--In R’

bl
eam*~(;sin*)L-al.-t(asinB)

x=cos 9

(9+1)

--1<2<1

R-‘(l--ZL+

R)*fi

-l<:
T&J

Gn (

‘J”.W

1 n

1-4

22.9.9

1

l-22 R’

-l<x
22.9.10

1

R-1

- l
22.9.11

42 2n+2 pi ( n+1 >

$ (1--zz+R)-“2

22.9.7 22.9.8

>

l4<1 Id<1

In R*

ao= I -l
l4<1 14<1

*see page n.

*

-l
Id<1

784

ORTHOGONAL

22.9. Generating

POLYNOMIALS

Functions-Continued

A(4 /I

P*(z)

1

22.9.13

P”(4

1 J

22.9.14

u4

1

22.b.15

L”’I) (2)

1

22.9.16

LA=’ (4

22.9.17

H,(z)

1 IYnSa+l) 1 ii

22.9.18

Hd4

(-.I)” (2n) !

ez co9 (226)

22.9.19

Han+* (4

C-1)” (2n+ l)!

rl’*e* sin (2~4)

22.9.12

Remarks

g@, 4

4 A

-l
R-1 /J e* 0”’ ‘J&e sin 0)

z=cos 6

(l-zz+z’)-’

-2<2<2

IKl

Id<1 (l--r)-*+ *

exp (5)

M
(x~)-t~e~~.(2(xz)*~~l elZ,-”

* t

22.10. Integral Representations Contour Integral Representationa 2(t, z)ds where C is a closed contour f&J)

b

BOW

taken around gdwz)

Bl (64

22.10.1

(1-x)$1 +2p

21-l 2(2-x)

(1 - E)‘(l+ 2-X

22.10.2

1

l/z

(1--22~+z’)-9-’

22.10.3

Tdz)

112

22.10.4

U.(x)

1

22.10.5

Pm

1

22.10.6

Pm

1 5

22.10.7

Lyx) I

e97

22.10.8

L,$qz)

22.10.9

H,(z)

z=a

l/t

a

z)@

sense Remarks

f 1 outside C

1 - 2’ e(l-2Zz+E’)

Both zeros of 1 - 2zr+ a’outside C, a>0 Both zeros of I-2zz+z*outaideC

1 2(1-222+r*)

Both zeros of 1 - 22~ + z* outside C

i (l-Ls+e’)-“~

Both zeros of 1 - 2zc+ ESoutside C

21-l I-2 e E-2

-,“,

1

1+;

e-*

?Z!

l/S

- 1 Z-2 Zero outside C

e-’ (

1+:

>

Y/z

z= --z outside C

esz*-rl

Miacdaneoue

in the positive

2

Integral

Representations

(a>O)

ORTHOGONAL

22.10.12

P,(cos

* (cos O+i sin e co9 fp)U#~

e) =-;

s0

785

POLYNOMIALS

22.10~14 ~g)(x)=$

m e-It”+; .s

J,@&)dt

0

22.10.15

*

22.10. 13 P,(cos e,d

* sin (n+wJd4 n- s 0 (co9 e-cos +)*

H,(x)=e’*

22.11.

The polynomials this formula.

I

given in the following

f”(Z)

22.11.1

PFp 8) (2)

22.11.2

cp

(-

T&J

22.11.4

Us&)

x*i . . 22.11.7 22.11.8

Pd4 L?‘(x) H,(z) He,(z)

;{;,+I

(-1)“2”

1/;;

y;;;,

(fi:“:,%

(1 --z)“(l

*

n! [I:{”

22.12.

Sum

II

Formulas

Christoffel-Darboux

22.13.

d4

+iry

1-Z’

(l-@a-t

1-Z

(1 -cl?)-’

1-Z’

(l-21)’

1-Z

1 e-97 p’ &2

l-24 2 ;

Integrals

Formula

22.12.1

which satisfy

PC4

r(n+2a) f

polynomials

I

rcwr(a+n+3) 1)“2”n!

dt

0

table are the only orthogonal

1)“2%!

(-

srn e-l*t” cos (2&-i,)

Formula

a,

I

(2)

22.11.3

Rodrigues’

T

Involving nomials

Orthogonal

Poly-

22.13.1

2n ‘(l--y)“(l+y)Bp:“,8)(y)dy s Miscellaneous

Sum

Formulas Is Given

22.12.2

Q2&~=t[l+u*n(41

22.12.3

g;

22.12.4

Tl isou2"(4'

I

(Only Here.)

Selection

~~lp_~‘,B+l’(~)-(~~x~~+l(l+x~~+lp~~~l~~+l~~~~

22.13.2

~*m+1(4=3U*n-1(5)

=cyp(O)-

l-T*n+2(4

2(1-x*)

22.12.5

22.12.6

a Limited

22.13.3

Tn Mdy r’ JTl (y-x)~=Tu~-l(x)

22.13.4

_cl vT=i%&)dy -

J-I

(l-2yqy(2!)

=-nTn,xl

*

(Y--X)

~oLl’(x)~~~~(y)=L~+B’.‘(z+y) 22.13.5

’ (1-x s

* Pz.(cos @de=&

22.13.6

s0

22.12.8 - . , *See page Ii.

22.13.7

)-l/*P.(x)dx=&

*

-1

S

r P2,,+,(~0s e) cos

0

p>’

786

ORTHOGONAL

POLYNOMIALS

22.14. Inequalities

22.13.8

22.14.1

( > “;t’

Ic?~)(x)I

i 2’ maximum

(n+i)fi

lP,(t)dt S2 q=q-(n+a;Fx s

-$t

if *<--l

8-a

point nearest

to a+P+l

22.14.2

[T&l

(a>01

la%M

1 +Tn+l(41

22.13.11

22.13.12

IP$e)(z’)I

(“‘y>

22.13.10 “P,odt= s -1 &3

(a, /3) 2 -l/2

b+--l,8>-1)

5

22.13.9

CA>-2)

wag, if q=max

x’=O if n=2m; if n=‘2m+l

[~n(x)--n+1(91

{ Ia@W>l

(-f
x’=maximum

point

nearest

zero

22.14.3 ~W(cos

zm e-*Lr)(t)dt=e-“[L~‘(X)-L$)P-‘1(2)]

22.13.13

@1<2’-=

(O<(ll
22.14.4

IT&)ll1

22.14.5

dTn - (4
(-11x11)

22.14.6

IUn(x)l In+1

(-l
r(a+p+n+l)Joz (x-t)@-+LP)(t)dt =r(cu+n+i)r(~))+BLP+B)(2)

(sinne;‘;(a)

o<e<4

(-15x11)

(A%% rel="nofollow">--1, m>w 22.13.14 s0

= Lm(t)Ln(x-t)dt =

S

(-l<x
L,+,(t)dt=L,+“(x)---L,+“+l(x)

0

22.13.15

IPn(4 II 1

22.14.1

z

S S0)

22.14.8

’ e-f2~~(t)dt=~n-l(O>-e-Z2~,-~(x)

dp,(xl ----&-I

I

1 Ip(n+l)

(-l~s~l)

0

22.13.16

22.14.9

0

22.13.17

IPn(x)l 2

z H,(t)dt=

S-m e-f2H2,(tx)dt=fi

(---l
22.14.10 y

~‘.o-P~-l(,)P~+l(x)<3~n~l)(-15x11)

(9-l)”

22.14.11

me-f2tH2,+l(tx)dt=~ Pm+l)! mrx(x2~l)m S-m /rm

22.13.18

Pn2(x)-PAdPn+1(x~2

22.13.19

2 --iJ 7m4&?

1

22.14.12

e-“tnH,(Xt)dt--~!P,(X)

1 -P’,(x)

(2n-l)(n+1)

(-l<x
ILd4 I I fP

(x2 0)

J-m

22.14.13 22.14.14

22.13.20 cos (xt)dt=Ji;2n-‘n!e+L, Sme-r2[H,(t)]2 0

0

g

1Lp) (x) I 5 LfrTr:.:)

ezj2

620,

x20)

ORTHOGONAL

22.14.15

I&,(z)

15 ez2’222nr~!

(x20> H,,(x) 1<ez212k2”f2&!

22.15. Limit

22.15.1

787

22.15.2 fi

22.14.16 22.14.17

POLYNOMIALS

[;

L:) (;)]=r-a/v,(2&

22.15.3 lim W,“dEH 4”n! n+m

k= 1.086435

Relations

22.15.4 lim n-t-

r In zfi

HI

1 =-&cm

5

-(-ljnH 4%!

22.15.5

=LP’ (2)

22.16. Zeros For tables of the zeros and associated weight factors necessary for the Gaussian-type quadrature formulas see chapter 25. All the zeros of the orthogonal polynomials are real, simple and located in the interior of the interval of orthogonality. Explicit

and

Notations: xg)mth zero of fn(x) (xj”)<xP< e:)=arccos

Asymptotic

. . . <x?))

~$~+~(o<ejn)<ep<

Formulas

and

Inequalities

j, %, mth positive zero of the Bessel function Jol(x)

. . . -0X4

O<j, l<ja,2<

= f”(Z) 22.16.1

Py)(cos

e)

22.16.2

C’“‘( n x)

22.16.3

C’“’n (co9 I?)

22.16.4

T&)

22.16.5

U.(x)

22.16.6

P,(cos

Relation

-lim nf$?=j,,, Ta+-

&>-I,

s>-1)

&pcos2m--1 * 2n

e)

xg)=cos 2% * n+l 2m-1 2m -2n+l r
i

22.16.7

P.(x)

22.16.8

L’“’II (2)

k,=T+q +()(+)

For error estimates see [22.6].

. - .

788

ORTHOGONAL

22.17.

Orthogonal

Polynomials Variable

of

POLYNOMIALS

Tw*(xi)

a Discrete

is finite.

The constant

factor which is

In this section some polynomialsf,(x) are listed which are orthogonal with respect to the scalar product

still free in each polynomial when only the orthogonality condition is given is defined here by the explicit representation (which corresponds to the Rodrigues’ formula)

22.17.1

22.17.2

KJm’=~

w*(xtYn(x*)f,(xi>.

The xi are the integers in the interval a<x& b and 20*(x<) is a positive function such that =

=

-

Name

=

a

--

w*(x)

b

--

--

=

=

--

_-

N-l

1

Krawtchouk

N

p-J”-=

Charlier

OD

e-au= x!

Meixner

03

c=r(b+x) r(b)x!

C”

Hahn

0)

r(b)r(c+z)r(d+z) s!r(b+x)r(c)r(d)

n!

-

list of the properties

Use and

Remarks

_-

P, q>o;

(-l)ndann!

x! (z-n)!

a>0

X! (x-n)!

b>O, O
p+q=l

x!r(b+x) (x-n)!r(b+z-n)

-

-

of these polynomials

see [22.5] and [22.17].

Numerical 22.18.

dx, 4

qv! (x-n)!

-

For a more complete

=

l)nn!

(-

x

and

(3 (“n”>

l/n!

0

A”[w*(xMx, n)l

n

where g(x, n)=g(x)g(x-1) . . . g(x-n+l) g(x) is a polynomial in x independent of n.

Chebyshev

-

fn(x)=--&

Methods Extension

of the Tables

Evaluation of an orthogonal polynomial for which the coe&ients are given numerically. Example

Horner

Evaluate

1.

La(1.5) and its first and second derivative

using Table

22.10

and the

scheme. -

= I

-36

450

2=1.5

1. 5 1

-51.75

-34.5

1.5

- 2400 597.375

398. 25 1. 5

5400

- 1802.

-49.5

- 2703.

625

- 4320 9375

2696.0625

523. 125

- 1919.

25

720

4044.09375

- 413. 859375

- 275.90625

306. 140625

1165.

21875

L -306. 6=.

1

733.0

1.5

348. 75

1. 5

1

-47.

-31.5

- 1279.

25

776. 8125

452. 250

301. 50

-

500

-827.

-

- 1240.

250

875

- 464.0625

-

140625 720 42519 53

889. 3125 L,=889. 3125 6 720 = 1.23515 625 L,,=2 [ - 464.06251 (I 720 =-I. 28906 25

ORTHOGONAL

Evaluation of an orthogonal polynomial given numerically. If an isolated value of the orthogonal expression rewritten in the form

using

b,, c,,.f(z)

f*(x)

when the coe&ients

is to be computed,

use the proper

(m=n,

n-l,

are listed in the following

= I

bm

table:

G&

cl”,’

n+a ( > (-n. $?

C’“’ an+1

(-1)” kp

T an

C-1)”

Z(n--m+l)(n+m-1)

m(2m-

T 2?I+1

(-1)n(2n+1)2

2(n-m+l)(n-tm)

m(2m+ 1)

u In

(-1)”

2(n-m+l)(n+m)

m(2m-

1)

u 2”fl

(-1)“2(n+l)z!

2(n-m+l)(n+m+l)

m(2m+

1)

Pl,

(-1)” -( 4”

(n-m+1)(2n+2m-l)

m(2m-

1)

(n-m+1)(2n+-2m+l)

m(2mS

1)

Lb’n

n-m+1

m(a+m)

Hzn

2(n-m+

1)

m(2m-

1)

2(n-m+

1)

m@m+

1)

Example m

i: %I

-I

(n+l)x

%+I 4”

H ail+1

.,&

2n n >

0”

P a*+1

n

(

>

@n+l)‘&

(-1)”

n!

2.

(n-m+l)(a+B+n+m)

2m(a+m)

Z(n-m+l)(a+n+m-1)

m(2m-

1)

P(n-m+l)(a+n+m)

m(2m+

1) 1)

=3.33847, f(2)=-1.

Compute Z’~1’2*3i2)(2). Here de= 8

1 1;:

7

6

5

4

3

2

1

1. 132353 1::

1.366667 48 78

1.841026 60 55

3.008392 70 36

6.849651 78 21

26.44156 84 10

223. 1091 88

n

2

3

4

5

50.87648

207.0649

6

-___ *5)

Check: Compute @‘(2.5)

3. 65625

13.08594

by the method of Example 2.

0

3

~~*‘2~3’2’(2)=d~,(2)=(3.33847)(6545.533)=21852.07 Evaluation of orthogonal polynomials by means of their recurrence relations Example3. Compute@)(2.5)for n=2,3,4,5,6. From Table 22.2 C$)= 1, C?= 1.25 and from 22.7 the recurrence relation is

cq2 n

explicit

. . ., 2, 1, a,(z)=l).

of this chapter

dn(4

p’u, ” 8)

are not

=&(4dx~

-~jCW.Cx)

for the polynomials

fn(4

representation

where

a,-l(x)==l The d,(x),

the explicit

polynomial fkd

and generate a,,(r) recursively,

789

POLYNOMIALS

867.7516

6545.

533 90

0

790

ORTHOGONAL Change

POLYNOMIALS

of Interval

of Orthogonality

In some applications it is more convenient to use polynomials orthogonal on the interval [0, 11. One can obtain the new polynomials from the ones given in this chapter by the substitution x=2:1. The coefficients of the new polynomial can be computed from the old by the following recursive scheme, provided the standardization is not changed. If f:(z)=~~(2z-l)=~~u~zm

“f”(X) =gowm, then the n: are given recursively u~)-~u~-~‘-u~~,; (-‘)=a,/2, am u$)=2&,j=O,

7n=n-1,

n-2,

. . ., j; j=O,

m=O, 1, 2, . . ., n 1, 2, . . ., n and u~~,“‘=uZ; m=O,

Example

5

4

&”

0

the relations 1, 2, . . ., n

1, 2, . . ., n.

Given Ts(z)=5z-20z3+16z5,

4.

‘m j

by the a,,, through

find T:(z). 3

2

1

0

0

2.5=a;-*)

0

\ f+

-1 0 1

-16 -64 - 192 -512 - 1280=a;

it

P

1;: 256 512=a;

t

-lo=&-” -4 3;: 1120=a;

-1=a; -4: -4oo=a;

5L;

Hence, Z’,*(z)=512xs-1280x4+1120z3-400x2+50x---1. 22.19. Least

Square

Approximations

D a Continuous

Problem: Given a function j(z) (analytically or in form of a table) in a domain D (which may be a continuous interval or a set of discrete p~ints).~ Approximate J(Z) by a polynomial F,,(z) of given degree n such that a weighted sum of the squares of the errors in lJ is least. Solution: Let w(z) 20 be the weight function chosen according to the relative importance of the errors in different parts of D. Let j,,,(x) be orthogonal polynomials in D relative to w(z), i.e. Cj,,,, j,,) =0 for m #n, where

Example

5.

Find a least square polynomial J in the interval

degree 5 for j(z)=kx using the weight

Interval

21x15,

function 1 W(Z)=,&x-2)(5-x)

which stresses the importance ends of the interval. Reduction

to interval

[-1,

of the errors

at the

11, t=2q

WMf(4!JW~ if D is a continuous

interval

w(x(t)>2 1

3 Jl-t2

if D is a set of N discrete points x ,,,.

From 22.2, jm(t) = Z’,,,(t) and

Then

where

am= (j, jmMj~9 &de *f(z) *See

4’1 v um=3?r -1 Jl.+Z

1 T,(t)& t-/-3

2 l -- 1 u”=3?, s -, Jiq

t+3

S

* haa to be page

II.

square integrable, see e.g.

(22.17).

of

di!

(m #O>

OR!l’HOGONAL

Evaluating

the integrals

numerically

1 1+x

POLYNOMIAL&

we get

+ .013876Tz (F)-

--.235703-.080880Tl

791

.00238OT, rq)

D a Set of Discrete

Points

If x,,,=m(m=O, 1, 2, . . ., N) and w(x)=l, use the Chebyshev polynomials in the discrete such that range 22.17. It is convenient to introduce here a slightly different standardisation

Recurrence

relation:

jO(x) = 1 ,ji(2) = 1 -g

(n+l>(N-~)j”+l(~)=(2~+1)(~--x)j”(~)--n(~+~+l)j~-l(x) in the least square sense the function Example 6. Approximate by a third degree polynomial.

2 I f(z) I

z=-

z-

10

fo@)

2

I

10 ::

.3162 : 2673 2887

0 1

:

::

: 2357 2500

i

:1

hcf) 5 1.3579 (f t fn>

.271580

an=(fn,

fit23

j(x) given in the following

fi@)

1,; -l/1

fz(z)

I

I

-1,; -7);

-1

table

4 $

1

-1

fl (3

j*m

f2m

2. 5

3.5

10

.09985

.01525

.0031

.039940

.0043571

.000310

j(x)-.27158+.03994(3.5-.25x)+.0043571(23.5-3.5x+.125;Ga)+.00031(266-59.8333x +4.375x2-.10417x33) j(x) +.59447-

.043658x+

.00190092-

22.20. lkonomization Problem:

Given j(x)=

2

9000322921

of Series

a,x’

in the interval

m=O

-11x11

and R>O.

b,,,xmwith N m=O as small as possible, such that /y(x)-j(x) I
1 Then, since IT,(s)l11(-l<x
Findy(x)=g

f(x) =m$obmT,(x)

within the desired accuracy if

I% lbml
w-N+1

j(x) is evaluated most conveniently recurrence relation (see 22.7).

by using the

792

ORTHOGONAL

Example

7.

+jc’[4+x4/5+$/6 From

Table

Economize

with

f(z)

= 1 +x/2

+xz/3

POLYNOMIALS

so

R=.05.

22.3

&I=&,

W’o(4+32Tz(41+&

F’6~1(~)+11~d~~l

.I(~)=~~~149~e(z)+32~~(~)+3~,(~)1 since

+~[76~~(~)+llT,(z)+~~(~)l

References Texts [22.1] Bibliography on orthogonal polynomials, Bull. of the National Research Council No. 103, Washington, D.C. (1940). [22,2] P. L. Chebyshev, Sur l’interpolation. Oeuvres, vol. 2, pp. 59-68. [22.3] R. Courant and D. Hilbert, Methods of mathematical physics, vol. 1, ch. 7 (Interscience Publishers, New York, N.Y., 1953). [22.4] G. Doetsch, Die in der Statistik seltener Ereignisse auftretenden Charlierschen Polynome und eine damit zusammenhiingende Differentialdifferenzengleichung, Math. Ann. 109, 257-266 (1934). [22.5] A. ErdBlyi et al., Higher transcendental functions, vol. 2, ch. 10 (McGraw-Hill Book Co., Inc., New York, N.Y., 1953). [22.6] L. Gatteschi, Limitazione degli errori nelle formule asintotiche per le funzioni speciali, Rend. Sem. Mat. Univ. Torina 16,83-94 (1956-57). [22.7] T. L. Geronimus, Teorla ortogonalnikh mnogochlenov (Moscow, U.S.S.R., 1950). [22.8] W. Hahn, tuber Grthogonalpolynome, die qDifferenzengleichungen gentigen, Math. Nachr. 2, 4-34 (1949). [22.9] St. Kaozmarz and H. Steinhaus, Theorie der Orthogonalreihen, ch. 4 (Chelsea Publishing Co., New York, N.Y., 1951). [22.10] M. Krawtchouk, Sur une g6nt%lisation des polynomes d’Hermite, C.R. Aced. Sci. Paris 187, 620-622 (1929). [22.11] C. Lanczos, Trigonometric interpolation of empirical and analytical functions, J. Math. Phys. 17, 123-199 (1938). [22.12] C. Lanczos, Applied analysis (Prentice-Hall, Inc., Englewood Cliffs, N.J., 1956). [22.13] W. Magnus and F. Oberhettinger, Formeln und Siitze fiir die speziellen Funktionen der mathemat&hen Physik, ch. 5, 2d ed. (SpringerVerlag, Berlin, Germany, 1948).

[22.14] J. [22.15] [22.16] [22.17] 122.181

Meixner, Orthogonale Polynomeysteme mit einer besonderen Gestalt der erzeugenden Funktion, J. London Math. Sot. 9, 6-13 (1934). G. Sansone, Orthogonal functions, Pure and Applied Mathematics, vol. IX (Interscience Publishers, New York, N.Y., 1959). J. Shohat, ThQrie g&&ale des polynomes orthogonaux de Tchebichef, MBm. Sot. Math. 66 (Gauthier-Villars, Paris, France, 1934). G. Szego, Orthogonal polynomials, Amer. Math. Sot. Colloquium Publications 23, rev. ed. (1959). F. G. Tricomi, Vorlesungen tiber Orthogonalreihen, chs. 4, 5, 6 (Springer-Verlag, Berlin, Germany, 1955). Tables

[22.19] British Association for the Advancement of Science, Legendre Polynomials, Mathematical Tables, Part vol. A (Cambridge Univ. Press, Cambridge, England, 1946). P,(z), 2=0(.01)6, n=1(1)12, 7-8D. [22.20] N. R. Jorgensen, Undersbgelser over frekvensflader og korrelation (Busck, Copenhagen, Denmark, 1916). He,(z), 2=0(.01)4, n=1(1)6, exact. [22.21] L. N. Karmazina, Tablitsy polinomov Jacobi (Izdat. Akad. Nauk SSSR., Moscow, U.S.S.R., 1954). C.(p, q, z), z=O(.Ol)l, q=.l(.l)l, p=1.1(.1)3, n=1(1)5, 7D. [22.22] National Bureau of Standards, Tables of Chebyshev polynomials S.(Z) and C,,(r), Applied Math. Series 9 (U.S. Government Printing Office, Washington, D.C., 1952). 2=0(.001)2, n=2(1)12, 12D; Coeflicients for T.(Z), U,,(z), C.(z), S,(z) for n=0(1)12. [22.23] J. B. Russel, A table of Hermite functions, J. Math. Phys. 12, 291-297 (1933). eMsa”H.(z), x=0(.04)1(.1)4(.2)7(.5)8, n=O(l)ll, 5D. [22.24] N. Wiener, Extrapolation, interpolation and smoothing of stationary time series (John Wiley & Sons, Inc., New York, N.Y., 1949). L,(Z), n=0(1)5, z=0(.01).1(.1)18(.2)20(.5)21(1)26(2)30, 3-5D.

Coeffieients a.

(z-1)0

p$z* P,

1

1

p+ b, p;a. pl p+ P, pia*8) pia. P, p& 8)

2 8

4((r+l),

(z-

1)1

Polynom,ials

(x-l)’

P$@(z) (z-

=a;’

2 c,,,(zIn=0

1)s

1)”

(z-l)'

Table (x-

1)s

22.1

(z-l)‘1 E

384

16(a+

1)'

a+!9+2 4(a+@+W(a+% 12(a+8+4(a+% Wa+8+5)(a+%

3840

32(a+

l)s

wa+k3+6)(a+2)4

46080

64(a+

l)a

48

for the Jacobi

2ta+ 1)

8(a+l),

lQ%a+8+7)b+2)r

4 z: (a+8+3)r

6(a+8+4Ma+3) Wa+8+5h(a+% Wa+8+6Ma+3h 240(a+8+7h(a+3)4 (m).=m(m+l)(m+2)

$ (a+8+4),

lWa+8+7)s(a+4)a

(a+l9+5)4

Wa+8+6Ma+5) Wa+8+7)da+5h

[(8)s(z-l)~+10(8),(6)(z-l)4+40(8),(5)~(z-l)s+8O(8),(4),(z-l)’+80(8)(3)4(z-l1)+32(2)s]

Pf*“(z)=&

[9504O(z-1)6+4752OO(z-

1)‘+6912OO(z-

zi 2

(a+B+Ws Wa+8+7)s(a+6)

. . . (m+n-1)

F:*“(z)=&

1)‘+864OOO(z-

$

8(a+8+5Ma+4 4O(a+8+6Ma+4)r

(a+@+716

2 $ z:

1)*+2304OO(z-

1) +23040]

-

L

Table

Coefficients

22.2

for the

Ultraspherical C(z) (z) = a;’

b.

-Cc!=)

a, 1

Cl”’

1

Ci=)

1

cp

3

C$’

6

20

2’

1

2a

1

Polynomials

5 c,,,zm and ?I’=0

fl=

%a)3

4(&

3(a+

2(a), -

15

I

1 15(&

Cp

90

) -15b)a

1

(a?-)

39

15(a+

(a)“=a(a+l)(a+2)

2,

CP wa+

4(& -

55s

. * * (a+n-l)

[4(2)#-6(2)~]

-A*=4(2),

Ckyz)

=;

[969-36%)

9’9s

1

[3(3)G3’(2) [9G”(Z)

+3cp(z)]

I Cp

+ 3cpyz)l

I

Cp 90

864s

d

=;

4)

30

W&

c?(z)

I$1

3)

6

-2’%h

I

1) (a+4)

3fxa+ 46x)4

I cc?

1 45(a+2)(a+5)

3

w34

a9

I

15&+4)(a+5)

G+ ‘3 4(a)s

I z” I %’

3Y

I

I

1)

- W4a

Cp

d,&‘!?(z)

of C~)(Z)

4(d4

1

6(43

3b)3

for 2” in terms

3a(a+3)

1

--Q

$,

and

z’

a 2a

b;’

39

23

1

C:‘(Z)

I

a9

I Cp

I

I

I

ORTHOGONAL

795

POLYNOMIALS

Table Coefficients

for the

Chebyshev

Polynomials

T,(x) = 2 c&P m=O

b. 1 __-------------To 1 1

1

2

---------~~~~~~Tl 1 1

4

8

1

4

Ta -1 2 1 __-_______-___-~______~_______-3 4 1 Ta

------~~---___~~--1 T4 -8

--------~~~~~~5 -20 TS --------____--__________-

T8

-1

T7

pp------p----pp 1 TS

18

-7

-------___-~ TO

9

TIO -1

50

-72

-112

:7 12

35

432 - 1232

.

22.3

36

495 T,

45

T5 220 Ta

55

---~______

10

256 1

T7 66

11 512 1

-2816 6912

T3

165

9

2816

792 Ta

120

- 1280

TI

330

8

- 3584

462

84

128 1

2048 462 T,,

210

28

1120

To

12 1024 1

-6144

Ts

TIO Tll

2048 1

TIS

Table

22.4

1 z+=~ [10To+15T~1+6T,+ Tel

Polynomials

0.4 + 1.0000000000 + 0.4000000000 -0.68000 00000 -0.94400 00000 -0.07520 00000 +0.88384 00000 i-0.78227 20000 -0.25802 24000 -0.98868 99200 -0.53292 95360 +0.56234 62912 +0.98280 65690 +0.22389

126 126

-576

-400

512

56

-256

Chebyshev

I 9

1024

64 1

T&z) = 329 - 48++18z*- 1

4 5 6

256

35

32 1

840

128

7

16 1

0.2 + 1.0000000000 + 0.2000000000 -0.92000 00000 -0.56800 00000 +0.69280 00000 +0.84512 00000 -0.35475 20000 -0.98702 08000 -0.04005 63200 +0.97099 82720 +0.42845 56288 -0.79961 60205 -0.74830 20370

64

6

160

-------~-~____~-___-11 220 5-11 ----___--~~~-~________1 Tl3 -----------~~___-

8 1

- 120

of T,(x)

21

-48

-32

-------______~-~~______

~~~~-~~15

5

56

for x” in terms

m=O

10 10

and

~=b&d,T,(z)

32

3 3

------

------_______--~~___--

16

T,(Z)

89640

T,(X)

0.6 + 1.0000000000 + 0.6000000000 - 0.2800000000 -0.93600 00000 -0.84320 00000 -0.67584 00000 +0.75219 20000 $0.97847 04000 +0.42197 24800 -0.47210 34240 -0.98849 65888 -0.71409 24826 +0.13158 56097

0.8 + 1.0000000000 +0.80000 00000 +0.28000 00000 -0.35200 00000 -0.84320 00000 - 0;9971200000 -0.75219 20000 -0.20638 72000 +0.42197 24800 +0.88154 31680 $0.98849 65888 +0.70005 13741 $0.13158 56097

1.0

1 1 : 1

796

ORTHOGONAL

POLYNOMIALS

Table 22.5 Coefficients

-------

20

xl

for.the

9

Chebyshev

39

b,

1

2

CJO 1

1

1 __~__-____----_______1 .2

Ul ---__-----------_ ua -1 -P----P-ua

2

---~--

4

4

2’

8

Polynomials

a?

27

zs

32

64

128

256

2

5

xl0

512

1024

2”

2’9

2048

4096 132 ---

42 28

4

---

of U,(z)

42

14 9

1

-----

x9

14

5 3

8

and for X* in terms

26 -----

16

1

-4

U,,(s)

132

14

297

48

165

--

U6 1 ------~___US ---us -1 --_____---___--___U7 ----__---~___--~ ua 1 -----P-----P---

---ppppp-p -12 6

-32

-8

5

-40

32

60

-84

1792

1120

~~-~~____4608

- 5376

275 110 154 44

1

54

1

- 2304

10 1024

-5120

11 2048

4096

1

-~20

23

2”

21

u,(z)

Table 22.6

39

z’

26

27

1

9=&

=64a+8Od+24&-

Chebyshev

n/x 0 1 : 4 5

0.2

2’0

[5~o+9ua+5u4+

Polynomials

0.4

X9

28

2’1

2’1

U,]

U,, (2) 0.6

0.8

UlO

U11

1

- 11264

11520

P-----P--

u,

--us

1

----

u, U7

____~

9 512

U, U5

-----

8

- 1024

- 1792

280

1 256

672

35

ua Ua

---_

-------

7

-448

-560

-12

27

1 128

240

75

6 64

-160

-1

20

1

-192

80

10

Ull --Y----P-1 Ul2

1

---_________-80

24

us -------------

UlO ---__--~

16

Ul

90

-----

CL,

1.0

+ 1.00000 +0.40000 - 0.84600 -0.73600 +0.54560 +0.95424

00000 00000 00000 00000 00000 00000

+ 1 .ooooo +0.80000 -0.36000 - 1.08800 -0.51040 $0.67968

00000 00000 00000 00000 00000 00000

+ 1.00000 +1.20000 +0.44000 -0.67200 - 1.24640 -0.82368

00000 00000 00000 00000 00000 00000

$;.ogo$

go0

1

+1:56000 +0.89600 -0.12640 - 1.09824

00000 00000 00000 00000

4 5 6

-0.16390 - 1.01980 -0.24401 +0.92219 $0.61289 -0.67703

40000 16000 66400 49440 46176 70970

+ 1.05414 +0.16363 -0.92323 -0.90222 +0.20145 + 1.06338

40000 52000 58400 38720 67424 92659

+0.25798 + 1.13326 + 1.10192 $0.18905 -0.87506 - 1.23913

40000 08000 89600 39520 42176 10131

- 1.63078 - 1.51101 -0.78683 +0.25207 + 1.19015 + 1.65217

40000 44GOO 90400 19360 41376 46842

:: 12

-0.88370

94564

$0.64925

46703

-0.61189

29981

+ 1.45332

53571

13

3” i 9

UN

ORTHOGONAL

797

POLYNOMIALS

Table Coefficients

for the

Chebyshev

Polynomials

C,(z) = -&c&P m=o

C,(z)

and

2”= b;;l5

11 11

cz

-2

-------~C3 -----

I

1

31

I

1

10

22.7

of C,(Z)

d&,(x)

m-0

x8 _---------

Cl

for 2” in terms

20

X’O

1 -~~.~~ 35 -~~-~

i

4

2’2

1

1

1

1

126

462

Co

462

126

-~-~-~ 56

15

2”

Cl 792

210

c2

-~~~~~

-3

1

2

c4

c5* -___----C6

1

-4

1

51

I

-2 ------~-

C3 495

120

cc

-~~~~

11 -___- 11 1

14

330

84

-~~___~~ 28

6

-6

-7

21

1

-51

9

c7

--------~CS --------CO

5

-___-

1

-7

36

-~~~~-8 -~~~~~ 1

C5

165 45

1

220

9

Co

55

c7

--~~~-

2

-16

20

9

-8

-----

-30

1 1 --------

27

-9

-______-~~ - 10

10 1

66

1

cs

11 1

CO

1

12

Cl0

-____--~___

-11

-~~~~54

1

1

Cl1

-12

1

1

Cl2

---~~----28

‘See

page

ii.70

2”

Z’O

x’

f

II.

Table Coefficients

Chebyshev

1 1 --------~~~~~

Sl

-----

for the

sz

-1

2 1

ss

Polynomials

1

-2

5

for 5” in terms

132

28

4

of L!&,(Z)

42

9

1

and

14

3 ----_____---~~1

S,,(z)

Sr

90

14

22.8

297

48

Sl

165

53

------___--~~-~-

1

s4

------------------

-3

S5

1

3

S6

----

-1

---------P----P St

1

-4

6

5 1

1

-5

-4

10

20 6

~-~~~~~1 1

-6

75 27

S5

35

1

S4

110

7 1

275

154

8

Sa

44

St

----P-i--------

1

88 --P------P-----

-10

15

5

SO

so

-20

-1

-P-P---------

28

35

1

-8

-35

-6

Sll

1

21

15

~---

-7

9 1

-9

-56

36

54

1

698

SO

10 1

1

~-

-10

1

11

SIO

1

Sll

----___--~~~~-

Sl2

1

--P----P-

x0

-21 2’

29

70 23

S&r) *see

Page

n.

x4

=ti-524+6z*--

-84 25

39

1

45

---z’

28

39=5so+9s*+5s4+ss

____-

39

-11

xl0

1 al

----

2"

Xl2

-

SIZ

.

Table

22.9

Coefficients

for the

Legendre

P,(z)

20

2’

x2

a9

39

=a;’

Polynomials

P,(z) x”=b,’

2 CmX~ m=o

a9

26

and

for

2“

in terms

of P,(X)

5 d,P,(z) m=o

29

29

20

2'0

2"

2'2

-----~---

---_I-_--PO

an

bm

2

----~~ Pa P4

8

----~PS

8

---PO

1

16

P,,

3

3

~-

- 1260 315

-63

6930 - 4620

3465

231

15015 - 18018

960 6435

90090 - 90090

225225

218790

46189

2078505

-----

P&) For

values

of

P,(z),

see chapter

8.

=$

[231ti-3lW+

105z2-51

2432

a+=&

[33Po+llOPa+72P4+16P~]

10752 88179

- 1939938

Ps PO

256

- 230945

- 1021020

50048

128

- 109395

Ps p7

2176 12155

133952 10080

128

- 25740

P4

PS

7904

16

- 12012

- 30030

-693

429

PI

220248 23408

832

PO

Pa

15504 2992

16

18018

31654

2160

b.

Pl 208012

4760

88

- 693

52003

16150

182

231

315

35

256 1024

-315

-35

128

256

105

676039

20349

2600

8

88179

3315

72 63

46189 4199

143

8

-70

12155

715

28 35

6435

110

2

-30

429

33

20

15

16

231

27

2 5

-5

63

3

-3

_______~~____ PO 128 ---PlO ---PII ----

35 7

1

-1

2

---

5

1

1

---PZ

--P7

3

1

111

------PI

---~~~ PO

1

256

PI0 PI1

676039

1024

Pll

Coefficients

for the

Laguerre

Polynomials

L,(z) =a.-l

&I

-. --

ti

1

-. --

-

Lt L4 4 Lc

2

6

6

24

24

120

120

7m

tm

5040

5040

403m

403m

362360

362880

3628800

36m6oo

39916690

39916mo

479991599

479991590

-. -~ -. -~ __--_-~ -_-~

4

_---

Ll -

__~-

-

LIO Lll 41 -

__-._-.- --

0”

-

1

2

L7

-

1

1

-. -____ -

._

1

-. --

1

-1

-1

_-

-4

-_

-18

_-

-96

__

-600

__

-43m

__

-35230

.-

-32ZMl

.-

-32659m

...-

z(

-m233ooo --

=&

-4

_1

9

.-

72

.-

600

._

5400

._

52920

._ ._ ...-

564460 6531340 31343990 199771#)0

15397om3Llo

-13 18

__ -1

-6

__

-16,

__

-mo

__

-2400

__ __ __...-

2)

-

6

_. __

2

._

.-

-5743918200

2

and cc”= 2

5 Gl&m-0

for 5” in terms

of L,(Z)

Table

d,L,(z)

-

-29400 -378320 -5030320 -7257m

-199771w99 -17663382ooo

-.

__

-_

__

-_

l44I

.-

-96I

1 12001 10600 -1299

I

-14409

I

105849 -176400(

1

1123939 -2257920

1

13063880~

1 -39431920

183293Q99 I

-435453999

1 I

2m424o99 -5535212ooo

[1p-336ab+45021-24OW+54OW-43202+720]

479901509 -5743918200

__

-105360352ooo

._

__

._

-_

.-

450 I

-36

( 1

7mI

3-l

1123959

1

39431m~

76!xwm

13441561699

31614105699

__

23710579!Mm

‘-379359257mcl

__

.-

__

._

_.

._

_.

.-

_.

.-

-.

.-

_.

-

-

442597473409 -379359267200 2371057m -105330352UOLl 31614105699 -5743919200

1

zl

-

-21~

-_

__

22.10

m-0

21

_.

__

II

-

-

_-

_-

.-

L&c)

21

L,(X)

47cam3cKJ

__LO

__4

__L,

__4

__LC

__-

LS

__-

Ls

__4

__4

-_-

L*

__-

Ll0

__-

41

_.-

Ll

_.-

2’1

ti=720L,,-4320L,+10800L~-

14400~+108OOL,-432OLs+72OL1

--

800

ORTHOGONAL

Table 22.11

Laguerre

POLYNOMIALS

Polynomials

L,(z)

n/x

0.5

1.0

3.0

0 ::

+1.00000 00000 +0.50000 00000 to.12500

+1.00000.00000 -0.50000 0.00000 00000

+1.00000 00000 -2.00000 00000 -0.50000

3 4 5

-0.14583 33333 -0.33072 91667 -0.44557 29167

-0.66666 66667 -0.62500 00000 -0.46666 66667

;

-0.51833 49653 -0.50414 92237

i 10 11

-0.49836 29984 -0.45291 95204 -0.38937 44141 -0.31390 72988

-0.25694 -0.04047 to.15399 +0.30974 to.41894 $0.48013

12

-0.23164 96389

+0.49621 22235

44444 51905 30556 42681 59325 41791

5.0

10.0

+1.00000 00000 +1.37500 00000 +0.85000 00000

+1.00000 -4.00000 +3.50000 +2.66666 -1.29166 -3.16666

00000 00000 00000 66667 66667 66667

+1.00000 -9.00000 +31.00000 -45.66666 +11.00000 +34.33333

00000 00000 00000 66667 00000 33333

-0.01250 -0.74642 -1.10870 -1.06116 -0.70002 -0.18079

-2.09027 +0.32539 +2.23573 +2.69174 +1.75627 to.10754

77778 68254 90873 38272 61795 36909

-3.44444 -30.90476 -16.30158 +14.79188 f27.98412 +14.53695

44444 19048 73016 71252 69841 68703

00000 85714 53571 07143 23214 95130

+0.34035 46063

-1.44860 42948

-9.90374 64593

.

Coefficients

for the

Hermite

Polynomials

H,(z) = 5

20

2’

22

2’

39

b, HQ

1

-.

x

-.

-48 120

-_ H6 H7

-_

30240

-_

HlI Hl2

-13440

1680

-30240

302400

-_

-665280

-665280

-. 2217600

- 7983360

2’1

2048

4096

332640

22.12

277200 25200

1

55440

72 I

-9216 -23040

-1774080

3960

110 1024

-56320 1520640

5940

1

132

-135168

Ha

Ha HO

1 2048

H,

Hr

90 512

Hs

HS 110880

--

HQ

H3 831600

2520

b,

HI 1995840

1512

256

161280

665280

75600

56

13305600

--

1

H,o HlI

4096

1

HI2

-_ XQ

-

-403200

-.

2”

30240

10080

1

-3584

1024 *

840

128

48384

512

3360

___

42 64

13440

-.

2’0

15120

420

-1344 .

-. --80640

840

1

-480

-.

XQ

256

30 I 32

-. 3360

180

1

-.

1

1680

2. I 16

39

128

-1

-I

-.

--

720 -1680

12

1

-160

-120

-_

H0 -_ HO HI0

8

1% ? ----I

60

-.

’

120

-

-12

-.

64

12

-

41

2’

32

6

-2

Table

of H,(x)

-___

16

-

2

for zn in terms

m~o&Jf&~

26

--

8

21

-.

H4

4

11

-.

HI -. HZ Ha

2

x”=by’

zs

-

--

and

m=O

-

-

c,,,z’

H,(z)

2'

2'

z

a?

zb

26

27

39

29

2'0

2"

2"

H,(z)=642s-480z'+720zs-120

a?=&[120Haf

lSOHri-3OH~+&I

l wr

page

II.

802

ORTHOGONAL POLYNOMIALS

Table 22.13 4% 1" i I!

Hermite 0.5 + 1.ooooo t :-g$gj - 5:ooooo + 1.ooooo (1) +4.10000

1.0 $EE

Polynomials

H,(Z)

3.0

5.0

10.0

++S.OOOOO 1.ooooo 00

(1) 1.00000 00000

'~:~~~~ (1) -2:OOOOo (0) - 8.00000

(2) (1) ++3.40000 1.80000 00 (2) +8.76000 00 (3) +3.81600 00

(1)9.80000 00000 (2)9.40000 (3)8.81200 00000 (4)8.06000 00000

(4) + 1.41360 (4) + 3.90240 (4) +3.62400 (5) - 4.06944 (6) -3.09398 (7) - 1.04250

(6) +5.51750 40

t :

(1)+3.10000 (2) -4.61000 (2) -8.95000 (3) $6.48100

::

(5) +2.25910 (4) - 1.07029

(2) + 1.84000 (2) +4.64000 (3) - 1.64800 (4) - 1.07200 (3) +8.22400 (5) + 2.30848

12

(5) -6.04031

(5) +2.80768

00 00 00 00 40 24

(5)7.17880 (6)6.21160 (7)5.20656 (8)4.21271 (9)3.27552 (10)2.43298

00000 00000 80000 20000 97600 73600

(11)1.71237 08128

1.00000 (1)2.00000 (2)3.98000 (3)7.88000 (5) 1.55212 (6)3.04120

00000 00000 00000 00000 00000 00000

(7)5.92718 (9)1.14894 (10)2.21490 (11)4.24598 (12)8.09327 (14) 1.53373

80000 32000 57680 06240 82098 60295

(15)2.88941 99383

23.

Bernoulli and Euler PolynomialsRiemann Zeta Function EMILIE

V. HAYNSWORTH

l AND KARL

GOLDBERG

z

Contents Mathematical 23.1. 23.2.

Properties

. . .

. . . . . . . . . . . . . . . . .

Bernoulli and Euler Polynomials and the Euler-Maclaurin Formula . . . . . . . . . . . . . . . . . . . . . . Riemann Zeta Function and Other Sums of Reciprocal Powers. , . . . . . . . . . . . . . . . . . . . . .

References

. . . . . . . . . . . . . . . . . . . . . . . . . .

Page 804 804 807 808

Table

Coe5cients of the Bernoulli and Euler Polynomials B,(z) and E,,(z), n=0(1)15

. . .

809

Table

Bernoulli and Euler Numbers . . . . . . . . . . . . . B, and E,, n=O, 1, 2(2)60, Exact and B, to 10s

810

Sums of Reciprocal Powers . . . . . . . . . . . . . .

811

Table

23.1.

23.2.

23.3.

n=1(1)42 Table

23.4.

Sums of Positive Powers . . . . . . . . . . . . . . .

k% k”, n=l(l)lO, Table

23.5.

9/n!,

m=1(1)100 s=2(1)9,

The authors acknowledge checking of the tables.

‘National ZNational

813

n=1(1)50,

10s . . . . . . . . . . . .

the assistance of Ruth E. Capuano

Bureau of Standards. Bureau of Standards.

(Presently,

Auburn

in the preparation

818 and

University.) 803

23.

Bernoulli

and

Euler Polynomials-Riemann Function Mathematical

23.1.

Bernoulli

and

Euler

Properties

Polynomials

and

Generating

Bernoulli 23.1.2

B,=&(O)

23.1.3

BO=l,

Bz=;,

the

Euler-Maclaurin

Formula

Functions

and

Euler

Numbers

E,,=2”E,

n=o, 1,. . .

B,=-;,

Zeta

E,=l,

B,=-&

0

n=o, 1,. . .

a = integer

ES=-1,

E,=5

(For occurrence of B, and E, in series expansions of circular functions, see chapter 4.) Sums

of Powers

2

k=l

k=l

m,n=l,

Derivatives

23.1.5

B;(z)=nB,-l(s)

23.1.6

B&+1)-B,(z)

m,n=1,2,.

2,. . . and

Differences

n=l,

n.=l, 2, . . . 1 E:(r)=nEn-l(z)

=m?-’

1 E&+1)

n=O,l,...

..

+E&)

2, . . .

n=O, 1,. . .

=2f’

Expansions

23.1.7 B.(x+~)=$~

@

BdG”-n

n=o, 1,. . .

E.(z+h)=&

@

n=O,l,...

Eh)h”-’

n=O, 1, . . . Symmetry 23.1.8

B,(l-z)=(-l)“B,(z)

23.1.9

(-l)“B,(-z)=B,(x)+m”-l

E,,(l-x2)=(-l)“E,(s)

n=O,l,...

n=o, 1,. . .

(-1)“+1E,(-x)=E~(z)-2P

n=o, 1,. . .

Multiplication

Theorem

n=O,l,...

23.1.10

B,(m)=

m-1 mn-’ go B,

n=O, 1,. . . m=l,

E,(ms)=m”

z

(-1)“E,

(x+i)

n=O, 1,. . . m=l,

2,. . . E,(ms)=--

nil

mn m& (-l)kB,+~

3,. . .

(x+f) n=O,l,.

..

m=2,4,.

..

BERNOULLI

AND

EULER

POLYNOMIALS,

RIEMANN

Integrals

23.1.11

= B,(t)&= sa

23.1.12

S

&

805

FUNCTION

En+1b)--ErL+*(a) SazEn(t)&= n+l S’E,(wMt)dt

R+1(d--B,+lb) n+l

1 B,(t)B,(t)dt=(-l)“-’

ZETA

Bm+,

0

0

m,n=l,

2,. . .

=(-1)“4(2m+n+z-1)

(my;;2)!

I (The polynomials

Bm+n+z m,n=O,

are orthogonal

for m+n

1,. . .

odd.)

Inequalities

23.1.13

IBz,J>lBzn(z)I

n=l,

2,. . .,

l>z>O

4-“IE*,I>(-l)“~*,(2)>0

n=1,2,.

. .,

a>z>o

. .)

+>2>0

23.1.14 4(2y)! n=l,

2, . . .)

(1+2~)>(-l)~E2:2.--1(x)>o n=1,2,.

*>x>o

23.1.15 4”+‘(2n)! rzn+ 1 >wn&&>4’::!2,T-)!

(1+31-,-2.) n=O, 1 , . . .

Fourier

Expansions

23.1.16 B*(x)=-2

m cos (2nkz-irn) k” zl

&

- sin ((2k+l)nz-$rn) (2k+l)“+’ n>l,l>z>O n=l,

23.1.17 B2n-l(z),(-1)“2(2n-1)! (2*)2”-’

23.1.18 B2n(x)-(-1)n-‘2(2n)! (2n)2”

l>z>O

n>o,

l>z>O

n=O,

l>s>O

I 5 sin %rx k=, ~ k2”-’

E2n-l(z)=(-1)“4(2n-1)! lrzn

n>l,

l>i>O

n=l,

l>z>O

- cos 2krx g k2” n=1,2,.

E2n(z)=(-1)“4(2n)! =2n+1

. .,

5 cos (2k+l)rz ,c=,, (2k+1)2” n=1,2,. . .,

5 sin (2k+l)ra: k=,, (2k+1)2n+1 n>o,

l>s>O _ _

1>210

122>0

n=O, 1>2>0 Special

23.1.19

Bzn+,=O

23.1.20

B,(0)=(-l)nB,,(l) =B,,

23.1.21

’

B,(i)=-(l-F”)B,, I.

Values

n=1,2,...

E 2n+1- -0

n=O, 1 ) . . .

n=o,

En(O) = ---E,(l) =-2(n+l)-1(2”+‘-l)B,+,

n=1,2

n=O,l,

1, . . .

. . . 1 E,(i)

=2-“E,

n=O,l,.

,... . .

806 23.1.22

BERNOULLI

AND

EULER

POLYNOMIALS,

B,(+)=(-l)“&,($)

RIEMANN

=-(2n)-1(l-31-2n)(22”-1)Bzn

n=1,2,.

23.1.24

FUNCTION

~2n-1W=--E2,-1G)

=-2-n(l-21-“jB,-n4-~~~.-l

23.1.23

ZETA

Bz,(f)=&,($) ~-2-‘(1-3’-~“)B~,,

n=1,2,.

..

. .

n=O,l,...

&,,(~)=&&j) =21’(1--21-2”)(1-31-2”)B2,

n=O,l,... Symbolic

23.1.25

Pvw)

+ 1) -P(%4)

23.1.26

B&?-t

h) = (B(x) + h) *

Herep

Between

P(Jw

=P’(4

denotes a polynomial Relations

Operations

the

n=o,

1, . . .

+ 1) +P(G))

=2p(4

n=o,

a&(x+h)=(E(x~+h)”

in z and after expanding we set {B(z)}~=B,(z) Equivalent

Polynomials

1, . . .

and {E(cc)}“=E,,(z).

to this is

23.1.27

IL,(z)=;{ B, (q-Bn =-z{ B,(z)--2”B,

6)) (g)}

23.1.31

n=1,2,.

..

1 2+a F(t)dt=; fl z

S

{ F(x+h) + F(z) }

23.1.28 En-*(z)=2

G)-’

g

(9

(2n-n-1)B,-kB&(s) n=2,3,...

-&!

23.1.29

Euler-Maclaurin

Let F(z) have on an interval m equal parts some 8, 1 >e>O, we have

Formulae

its first 2n derivatives continuous (a, b). Divide the interval into and let h=(b-u)/m. Then for depending on Fc2”)(z) on (a, b),

BznF’2”‘(x+Bh)

Let &r) =B,(z-[a$. Formula is

b-hlxza

The Euler Summation

23.1.32

z1 F(a+kh+wh)=k se

sb F(t)dt B

23.1.30

-5

l1 fip(u-tt)

{ z1 F”‘(a+kh+th)}dt p_<2n, 120>0

BERNOULLI

23.2. Riemann

23.2.1 23.2.2

AND

EULER

POLYNOMIALS,

Zeta Function and Other of Reciprocal Powers

r(s)=&

=:

Sums

ZETA

807

FUNCTION

S’(O) = - $ In 257

23.2.13

f(-2n)

23.2.14 9s>l

k-”

RIEMANN

23.2.15

{(l-2n)=-2

23.2.16

f(2n)=2s

=0

n=l,

2, . . .

n=1,2

,...

B%>l

o-p-“)-’

n=1,2,...

1Bznl 1

(product over all primes p). 23.2.17 23.2.3

1(2n+l)=-(-1)“+1(2?r)2”+1 2(2n+l)!

1 B2n+1(z) cot (m)ds s0 n=1,2,.

* 23.2.4

z-w s$&?dZ

Sums

Powers

The sums referred to are

23.2.5

23.2.18

where

23.2.19

m (In k)”

of Reciprocal

..

(In m)“+’

q(n)=2

b(n) =gl

n=2,3

k-”

).. .

n=1,2,.

(-1)L-1k-n=(1-21-“)r(n)

..

n+l

g’s>0 23.2.6

=2%Y1

x(n)=&

sin (+rs)r(l-s){(l-s)

23.2.7 23.2.8

&?s>l ODz’-1 s o e’+l

=(1-2&(s)

n=l, =exp

(In 2r-1-$7)~ w--l)r(fS+1)

2,. . ., Sits>0 n p (

1-~

e;

23.2.11 23.2.12

Values

23.2.21 n=l,

(-1)“(2k+l)-”

These sums can be calculated from the Bernoulli and Euler polynomials by means of the last two formulas for special values of the zeta function (note that q(l)=111 2), and 23.2.22

f!?(2n+l)=($Jj~~~’

n=O, 1, . . .

IE2.1

23.2.23 fl(271)=(-~)‘+” 4(2n-l)!

’ E2n-l(2) sO

sec(7rx)dx

B(2) is known as Catalan’s other special values are 23.2.24

1 s(z)=1+~+35+

1

n=l,

constant. 7r2 * * * =x

r(O)=-3 rw

==

2,. * .

P>

product over all zeros p of t(s) with 3?p>O. The contour C in the fourth formula starts at infinity on the positive real axis, circles the origin once in the positive direction excluding the points f2nir for n=l, 2, . . ., and returns to the starting point. Therefore l(s) is regular for all values of s except for a simple pole at s=l with residue 1. Special

n=2,3,...

@k+l)‘“=(l--2-“){(n)

o(n)=&

dx

23.2.9

23.2.10

23.2.20

23.2.25

7r4 *=cz

2, . . .

Some

BERNOULLI

808 23.2.26 23.2.27 23.2.28

?@I=1 -$+$d4)‘l

-1+L 24

AND

..

34-

A(2)=1+$+$+

EULER

POLYNOMIALS,

7r2 *=12

71r4 * - * =m

RIEMANN

FUNCTION

23.2.29

x(4)=1+$+$+.

23.2.30

j3(1)=1-;+;-

. . =g . . . =;

p(3)+&++-

23.2.31

..

ZETA

. . . =g

References Tables

Texta

[23.1] G. Boole, The calculus of finite differences, 3d ed. (Hafner Publishing Co., New York, N.Y., 1932). [23.2] W. E. Briggs and S. Chowla, The power series coefficients of r(s), Amer. Math. hlonthly 62, 323-325 (1955). [23.3] T. Fort, Finite differences (Clarendon Press, Oxford, England, 1948). [23.4] C. Jordan, Calculus of finite differences, 2d ed. (Chelsea Publishing Co., New York, N.Y., 1960). [23.5] K. Knopp, Theory and application of infinite series (Blackie and Son, Ltd., London, England, 1951). [23.6] L. M. Milne-Thomson, Calculus of finite differences (Macmillan and Co., Ltd., London, England, 1951). [23.7] N. E. Norlund, Vorlesungen iiber Differenzenrechnung (Edwards Bros., Ann Arbor, Mich., 1945). [23.8] C. H. Richardson, An introduction to the calculus of finite differences (D. Van Nostrand Co., Inc., New York, N.Y., 1954). [23.9] J. F. Steffensen, Interpolation (Chelsea Publishing Co., New York, N.Y., 1950). [23.10] E. C. Titchmarsh, The zeta-function of Riemann (Cambridge Univ. Press, Cambridge, England, 1930). [23.11] A. D. Wheelon, A short table of summable series, Report No. SM-14642, Douglas Aircraft Co., Inc., Santa Monica, Calif. (1953).

[23.12] G. Blanch and R. Siegel, Table of modified Bernoulli polynomials, J. Research NBS 44, 103-107 (1950) RP2060. [23.13] H. T. Davis, Tables of the higher mathematical functions, vol. II (Principia Press, Bloomington, Ind., 1935). [23.14] R. Hensman, Tables of the generalized Riemann Zeta function, Report No. T2111, Telecommunications Research Establishment, Ministry of Supply, Great Malvern, Worcestershire, England (1948). f(s, a), s= - lO(.l)O, a=0(.1)2, 5D; (s-l){(s, a), s=O(.l)l, a=0(.1)2, 5D. [23.15] D. H. Lehmer, On the maxima and minima of Bernoulli polynomials, Amer. Math. Monthly 47,

533-538

(1940).

[23.16] E. 0. Powell, A table of the generalized Riemann Zeta function in a particular case, Quart. J. Mech. Appl. Math. 5, 116-123 (1952). f(& a), a=1(.01)2(.02)5(.05)10, 10D.

BERNOULLI

COEFFICIENTS

AND

EULER

bk OF THE

POLYNOMIALS,

BERNOULLI 4

RIEMANN

ZETA

POLYNOMIALS

5

b

809

FUNCTION

& (+&b& a

7

Table

9

10

&5:

2

11

12

13

23.1

14

15

&A(Q)

3 1

4

1

5

-3 +

6

0

7

0

10 11

-.

1

0

0

11

0

0

22

n\k

0

0

1

ek OF THE

2

EULER 4

3

0

1

5-J+

-4

1

0

266 7

0

-y

11

-+2

-b

0

y

1

13-3

0-q 0

POLYNOMIALS 5

-5 0

0 0

Y

COEFFICIENTS

- 11

1001

0

3

0

0 0

I

-4

b

-7

--%

9.

15

1

-4

0

5

O

14

A+

-25

-3

13

1

0

-Y

v-'

0 -3

l "Jp$

1

En (x)=kcoe& a

7

b

9

10

11

12

13

14

15

1 -L

1

1 2

2

0

3

-1

$

4

0

10

0

12

0

0

2073

0 0

0 9

255

0

0

0

21

0

0

30

-126

2y

-38227

0

573:05

0 0

0 O-

-3410

2 26949

0

62881 0

4943215

7 0

-31031

1. -9

I 0

0

-5

&5 4

- 396

7293 0

0 -v

-4

- 231

1683

T

1

- +

14

-b3

-9

.5y

14

0 0

0 0

0

1

-3

5

-28

153 -T

-155

691 T

0

0 0

0

1

- a

5

-4

17

-5

1 0

0 0

0

a

-2

+

-3

+

11

0

0

b

9

1

-f

1

-+

7

1 0

0

5

15

0

+

9

13

-+ 3

a

12

1

-3

0 0

0

_

0

8109395

0

-"

1

55

1287 2

7293

1

0

0

9

0

30032

-1001

-6

1 o-+

0

1

91 0

o-7 y

1 "-A+

1

BERNOULLI

810 Table

AND EULER

POLYNOMIALS,

BERNOULLI

23.2

AND

EULER

RIEMANN

ZETA

FUNCTION

NUMBERS

B,,= NID

N

D

1

1

Bf&

(

0) 1.0000 00000

-1

-:

2 6 2

- 1 -5.0000 00000 d - 1 1.6666 bbb67 1 I - 2 I -3.3333 2.3809 33333 52381: 2.

-:

30

- 2 -3.3333 33333 '(

330 138 2730

I

b7 234I

-2.7298 -8.6580 -5.2912 1.4255 42424 6.1921 25311 7\ 23107 17167 23188

87: 14322 510

I II

19 1919: 6 t20

-2 15 20097 61082 64391 71849 80708 64491 22051 02691

146 48

-278 5964 51111 33269 57930 59391 21632 10242 35023 77961 -560 94033 68997 81768 62491 27547 -8011649 50572 57181 05241 35489 07964 95734 82124 79249 77525 91853

:x

;46 58

-247929 39292 14996 93132 36348 84862 26753 68541 42141 81238 57396 12691 63229 84483 61334 88800 41862 04677 59940 36021

60

-121 52331 40483 75557 20403 04994 07982 02460 41491

I 10 148I -1.5116 13 11 -1.3711 4.8833 08739 4.2961 6.0158 46431 31577 23190 :p'I 65521

13530 1806 690 282 46410

23 I -1?9296 19 17 16 21 -4.0338 -1.2086 2.1150 30476,~ 8.4169 57934 c% 62652 74864; 07185 ii

1::; 870 354

32 -2.8498 26 30 28 24 -5.0387 2.3865 66746 3.6528 7.5008 77648 42750 76930 78101

567 86730

( 34)-2.1399

94926

-,:I:

1385'2

- 50521.! 27>02765 -1 -1993 60981 5 1 93915112145 ‘ -240 48796175441 ?

iii

;20 24 26 28

.M - --..----

-44fl5438 93249 79539 28943 02310 bb647 45536 82821:; 89665’.’ 41222 -80 72329,92358 17751193915 87898 02122 34707 06216 82474 96712 53281 590453~C

320 2

38

-234 89580 06033195177 52017 82857 52704,31082 i-----------1 8511 50718 11498 00178 77156 78140 -1036 6227 33519 61211 93979 57304 74518 7 94757 4225 97592 70360 80405 10088 07061

40 42 44

44;

60 -6667 96278

_,-. 50 52 54 56 58

60

11882 37525 'h

-69

-60532

650 61624 -126

-7 9420 22019

54665 32189 25180

99390

64202

64556 53751 ---,-- ._.___l_l!

6055 5421

58691 44977

68574 43502

28768 84747

73748 43153

19752 97653

61989 47741.1 50266 84425't 59763 10201 't 95192 738056-1 41076 90444

35.185”. 046611

85248 18862 18963 14383 78511 16490 88103 49822 51468 15121~!

86684

b0884

77158

08739 41204

09806 20228

14325 62376

70634 65889

08082 73674

29834

83644

23676

53855

76565’:

90583

22720

93888

52599

64600

93949

45581 05945

42122 40024 71169 90586

62187 19903 40923 72874 89255 48234 10611 91825 59406 99649 20041

181089 11496 57923 04965 45807 74165 21586 88733 48734 92363 14106 00809 54542 31325

From H. T. Davis,Tablesof the highermathematicalfunctions,vol. II. PrincipiaPress,Bloomington, Ind., 1935(with permission).

BERNOULLI

AND

EULER

SUMS

POLYNOMIALS,

OF RECIPROCAL

n

13

;:

33 332 36 37 3398 4410 42

ZETA

811

FUNCTION

POWERS

Table

23.3

s(w)=&-l)Wc-n 1.64493 406:8 48226 43647

12

RIEMANN

0.82246 0.69314 71805 70334 24113 59945 21824 30942 5* ';;,%

1.20205 1.08232 1.03692 1.01734

69031 32337 77551 30619

59594 11138 43369 84449

28540--yL3) 0.90154 19152 0.94703 92633 0.97211 13971 0.98555

26773 28294 97704 10912

69695 97245 46909 97435

71405 91758 c 30594 10410

1.00834 1.00407 1.00200 1.00099 1.00049 1.00024

92773 73561 83928 45751 41886 60865

81922 97944 26082 27818 04119 53308

82684 33938 21442 08534 46456 04830

0.99259 0.99623 0.99809 0.99903 0.99951 0.99975

38199 30018 42975 95075 71434 76851

22830 52647 41605 98271 98060 43858

28267 89923 33077 56564 75414 19085

1.00012 1.00006 1.00003 1.00001 1.00000 1.00000

27133 12481 05882 52822 76371 38172

47578 35058 36307 59408 97637 93264

48915 70483 02049 65187 89976 99984

0.99987 85427 0.99993 91703 0.99996 95512 0.99998 47642 0.99999 23782 0.919999 61878

63265 45979 13099 14906 92041 69610

11549 71817 23808 10644 01198 11348

1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

19082 09539 04769 02384 01192 00596

12716 62033 32986 50502 19925 08189

55394 87280 78781 72773 96531 05126

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

80935 90466 95232 97616 98808 99403

08171 11581 58215 13230 01318 98892

67511 52212 54282 82255 43950 39463

1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

00298 00149 00074 00037 00018 00009

03503 01554 50711 25334 62659 31327

51465 82837 78984 02479 72351 43242

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

99701 99850 99925 99962 99981 99990

98856 99231 49550 74753 37369 68682

96283 99657 48496 40011 41811 28145

1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

00004 00002 00001 00000 00000 00000

65662 32831 16415 58207 29103 14551

90650 18337 50173 72088 85044 92189

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

99995 99997 99998 99999 99999 99999

34340 67169 83584 41792 70896 85448

33145 89595 85805 39905 18953 09143

1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

00000 00000 00000 00000 00000 00000

07275 03637 01818 00909 00454 00227

95984 97955 98965 49478 74738 37368

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

99999 99999 99999 99999 99999 99999

92724 96362 98181 99090 99545 99772

04461 02193 01084 50538 25268 62633

t(n+l)=~[ltr(n)] ~(n+l)=~[l+d~~] From H. T. Davis, Tables of the higher mathematical functions, vol. II. Principia Press, Bloomington, Ind., 1935 (with permission). For C-42,

'

812

BERNOULLI

Table

AND

EULER

SUMS

23.3

POLYNOMIALS,

RIEMANN

OF RECIPROCAL

ZETA

FUNCTION

POWERS

81633 55941 61462 45517 78280 52222

97448 77219 59369 41105 77088 18438

310015-O 380 336 064 135

1.01467 1.00452 1.00144

80316 04192 05455 37627 95139 61613"~+&q 70766 40942 12191

.91596 0.96894 0.98894 0.99615 0.99868

1.00047 1.00015 1.00005 1.00001 1.00000 1.00000

15486 51790 13451 70413 56660 18858

52376 25296 83843 63044 51090 48583

55476.-&b? 11930 'Lo 77259 82549 10935 11958

0.99955 0.99984 0.99994 0.99998 0.99999 0.99999

45078 99902 96841 31640 43749 81223

90539 46829 87220 26196 73823 50587

909 657 090 877 699 882

1.00000 1.00000 1.00000 1.00000 1~00000 1.00000

06280 02092 00697 00232 00077 00025

55421 40519 24703 37157 44839 81437

80232 21150 12929 37916 45587 55666

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

93735 97910 99303 99767 99922 99974

83771 87248 40842 75950 57782 19086

841 735 624 903 104 745

1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

00008 00002 00000 00000 00000 00000

60444 86807 95601 31866 10622 03540

11452 69746 16531 77514 20241 72294

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

99991 99997 99999 99999 99999 99999

39660 13213 04403 68134 89377 96459

745 274 029 064 965 311

1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

00000 00000 00000 00000 00000 00000

01180 00393 00131 00043 00014 00004

23874 41247 13740 71245 57081 85694

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

99999 99999 99999 99999 99999 99999

98819 99606 99868 99956 99985 99995

768 589 863 288 429 143

1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

00000 00000 00000 00000 00000 00000

00001 00000 00000 00000 00000 00000

61898 53966 17989 05996 01999 00666

0.99999 0.99999 0.99999 0.99999 0.99999 0.99999

99999 99999 99999 99999 99999 99999

99998 99999 99999 99999 99999 99999

381 460 820 940 980 993

1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

00000 00000 00000 00000 00000 00000

00000 00000 00000 00000 00000 00000

00222 00074 00025 00008 00003 00001

0.99999 0.99999

99999 99999 998 99999 99999 999

F-~

BERNOULLI

Ah’D

EULER

SUMS

m\n : 3 4 5

! 8 9 10 11

1 1 2 10 15 21 28 36 5455

POLYNOMIALS,

OF

RIEMANN

POSITIVE

POWERS

2

3

I 14 30 55

; 1:: 225

91 140 204 285 385

441 784 1296 2025 3025

2275 4676 8772 15333 25333

4356 6084 8281 11025 14400

39974 60710 89271 1 27687 1 78312

ZETA

813

FUNCTION

Table

skkn

23.4

k=l

4 1

5 1 33 276 1300 4425

i87 354 979

6

.

6: 794 4890 20515

12201 29008 61776 1 20825 2 20825

1 4 9 19

67171 84820 46964 78405 78405

:32 14 15

1:: 120

506 650 819 1015 1240

:‘: 18 19 20

136 153 171 190 210

1496 1785 2109 2470 2870

18496 23409 29241 36100 44100

2 3 4 5 7

43848 27369 32345 62666 22666

33 47 66 91 123

47776 67633 57201 33300 33300

472 713 1054 1524 2164

60136 97705 09929 55810 55810

231 253 276 300 325

3311 3795 4324 4900 5525

53361 64009 76176 90000 1 05625

9 11 14 17 21

17147 51403 31244 63020 53645

164 215 280 359 457

17401 71033 07376 70000 35625

3022 4156 5636 7547 9988

21931 01835 37724 40700 81325

351 378 406 435 465

6201 6930 7714 a555 9455

1 1 1 1 2

23201 42884 64836 89225 16225

26 31 37 44 52

10621 42062 56738 63999 73999

576 719 891 1096 1339

17001 65908 76276 87425 87425

13077 16952 21771 27719 35009

97101 17590 07894 31215 31215

496 528 561 595 630

10416 11440 12529 13685 14910

2 2 3 3 3

46016 78784 14721 54025 96900

61 72 84 97 112

97520 46096 32017 68353 68978

1626 1961 2353 2807 3332

16576 71008 06401 41825 63700

43884 54621 67536 82984 1 01367

34896 76720 44689 49105 14730

666 703 741 2 780 820

16206 17575 19019 20540 22140

4 4 5 6 6

43556 94209 49081 08400 72400

129 148 169 192 217

48594 22755 07891 21332 81332

3937 4630 5423 6325 7349

29876 73833 09001 33200 33200

1 1 1 2 2

23134 48792 78901 14089 55049

97066 23475 59859 03620 03620

861 903 946 990 1035

23821 25585 27434 29370 31395

7 8 8 9 10

41321 15409 94916 80100 71225

246 277 311 348 389

07093 18789 37590 85686 86311

8507 9814 11284 12934 14779

89401 80633 89076 05300 33425

3 5 4 4 5

02550 57440 20654 93217 76254

07861 39605 02654 16510 82135

63767 43448 51864 16665 66665 functions,

16838 19132 21680 24505 27630 vol. II.

96401 41408 45376 20625 20625 Principia

3: E43 25 2276 28 3; 31 33; ;z z; 38 2 :: 43 44 45

!ii

1081 33511 11 68561 434 1128 35720 t; 12 72384 483 48 1176 38024 536 1225 40425 49 :: EE 594 50 1275 42925 16 25625 656 From H. T. Davis, Tables of the higher maihematical Ind., 1935 (with permission).

3 6 10 15 22

81876 30708 02001 39825 99200

37 67 115 190 304

49966 35950 62759 92295 82920

6 70997 79031 7 78789 94360 9 01095 84824 10 39508 72025 11 95758 72025 Press, Bloomington,

BERNOULLI

814 Table

23.4

AND EULER POLYNOMIALS, SUMS

OF

POSITIVE

RIEMANN

POWERS

ZETA FUNCTION

&kw

1 1326 1378 1431 1485 1540

2 45526 48230 51039 53955 56980

17 18 20 22 23

3 58276 98884 47761 05225 71600

724 797 876 961 1052

4 31866 43482 33963 37019 87644

31080 34882 39064 43656 48688

45876 49908 45401 10425 94800

13 15 17 20 23

6 71721 69427 91071 39020 15826

59826 69490 30619 41915 82540

1596 1653 1711 1770 1830

60116 63365 66729 70210 73810

25 27 29 31 33

47216 32409 27521 32900 48900

1151 1256 1369 1491 1620

22140 78141 94637 11998 71998

54196 60213 66776 73925 81701

26576 18633 75401 99700 99700

26 29 33 37 42

24236 67201 47888 69693 36253

61996 09245 01789 35430 35430

1891 1953 2016 2080 2145

77531 81375 85344 89440 93665

35 38 40 43 46

75881 14209 64256 26400 01025

1759 1906 2064 2232 2410

17839 94175 47136 24352 74977

90147 99309 1 09233 1 19971 1 31573

96001 28833 65376 07200 97825

47 53 59 66 73

51457 19459 44694 31889 86078

09791 45375 47584 24320 14945

5

2211 2278 2346 2415 2485

1 1 1 1

98021 02510 07134 11895 16795

48 51 55 58 61

88521 89284 03716 32225 75225

2600 2802 3015 3242 3482

49713 00834 82210 49331 59331

1 1 1 1 2

44097 57598 72137 87778 04585

30401 55508 89076 20425 20425

82 91 101 111 123

12617 17201 05876 85057 61547

64961 47130 29754 92835 92835

2556 2628 2701 2775 2850

1 1 1 1 1

21836 27020 32349 37825 43450

65 69 72 77 81

33136 06384 95401 00625 22500

3736 4005 4289 4589 4905

71012 44868 43109 29685 70310

2 2 2 2 3

22627 41976 62707 84897 08627

49776 67408 39001 45625 92500

136 150 165 181 199

42550 35691 49033 91098 70883

76756 46260 72549 62725 78350

2926 3003 3081 3160 3240

1 1 1 1 1

49226 55155 61239 67480 73880

85 90 94 99 104

61476 18009 92561 85600 97600

5239 5590 5961 6350 6760

32486 85527 00583 50664 10664

3 3 3 4 4

33983 61051 89922 20693 53461

17876 02033 76401 32800 32800

218 239 262 286 312

97883 82106 34102 64977 86417

06926 87015 87719 43240 43240

3321 3403 3486 3570 3655

1 1 1 2 2

80441 87165 94054 01110 08335

110 115 121 127 133

29041 80409 52196 44900 59025

7190 7642 8117 8615 9137

57385 69561 27882 15018 15643

4 5 5 6 6

88329 25403 64793 06614 50985

17201 15633 56276 75700 28825

341 371 404 439 477

10712 50779 20183 33163 04658

79721 51145 24514 56130 71755

3741 3828 3916 4005 4095

2 2 2 2 2

15731 23300 31044 38965 47065

139 146 153 160 167

95081 53584 35056 40025 69025

9684 10257 10856 11484 12140

16459 06220 75756 17997 27997

6 7 8 8 9

98027 47870 00643 56483 15532

99001 08208 27376 86825 86825

517 560 607 657 710

50331 86593 30633 00446 14856

06891 07900 94684 85645 85645

4186 4278 4371 4465 4560

2 2 2 2 2

55346 63810 72459 81295 90320

175 183 191 199 207

22596 01284 05641 36225 93600

12826 13542 14290 15071 15885

02958 42254 47455 22351 72976

9 10 11 11 12

77936 43844 13413 86803 64181

08276 23508 07201 47425 56800

766 827 892 961 1034

93549 57099 27001 25699 76617

37686 39030 22479 03535 94160

4656 4753 4851 4950 5050

2 3 3 3 3

99536 08945 18549 28350 38350

216 225 235 245 255

78336 91009 32201 02500 02500

16735 17620 18542 19503 20503

07632 36913 73729 33330 33330

13 14 15 16 17

45718 31592 21984 17083 17083

83776 24033 32001 32500 32500

1113 1196 1284 1379 1479

04195 33915 92339 07141 07141

83856 88785 69649 19050 19050

.

BERNOULLI

AND EULER POLYNOMIALS, SUMS

OF POSITIVE

RIEMANN POWERS

$,kn

Table

8

7

815

ZETA FUNCTION 23.4 9

1 129 2316 18700 96825

25: 6818 72354 4 62979

76761 00304 97456 80425 80425

21 42595 79 07396 246 84612 677 31333 1677 31333

123 13161 526 66768 1868 84496 5743 04985 15743 04985

375 67596 733 99404 1361 47921 2415 61425 4124 20800

3820 90214 8120 71910 16278 02631 31035 91687 56664 82312

39322 52676 90920 33028 1 96965 32401 4 03575 79185 7 88009 38560

3 12 32 80 180

5113 20196 2 82340 22 35465

6808 56256 10911 94929 17034 14961 25972 86700 38772 86700

1 2 4 7

99614 49608 69372 07049 79571 67625 49407 30666 05407 30666

14 75204 15296 26 61082 91793 46 44675 82161 78 71552 79940 129 91552 79940

56783 75241 81727 33129 1 15775 58576 1 61640 30000 2 22675 45625

10 16 24 35 50

83635 90027 32394 63563 15504 48844 16257 63020 42136 53645

209 34353 26521 330 07045 44313 510 18572 05776 774 36647 46000 1155 83620 11625

3 4 5 7 9

02993 55801 07597 09004 42526 37516 15025 13825 33725 13825

71 30407 18221 99 54702 54702 137 32722 53038 187 35186 65999 252 96186 65999

1698 78656 90601 2461 34631 75588 3519 19191 28996 4969 90651 04865 6938 20651 04865

12 15 19 25 31

08851 27936 52448 66304 78633 09281 03866 59425 47259 56300

338 25097 03440 448 20213 31216 588 84299 49457 767 42238 54353 992 60992 44978

9582 16872 65536 13100 60593 54368 17741 75437 56321 23813 45365 22785 31695 01751 94660

39 48 60 73 90

30901 20396 80219 97529 24375 80121 96685 86800 35085 86800

1274 72091 52434 1625 96886 06355 2060 74807 44851 2595 94900 05332 3251 30900 05332

41851 01318 63076 54847 18716 58153 71368 79729 21001 92241 63340 79760 1 18456 03340 79760

109 82628 60681 132 88021 93929 160 06208 05036 191 98986 14700 229 35680 67825

4049 80152 34453 5018 06672 30869 6186 88675 08470 7591 70911 33686 9273 22165 24311

1 1 2 3 3

51194 22684 73721 91861 36523 23193 42120 62642 60036 03932 81037 69540 79600 87463 47665

272 93857 25041 323 60088 45504 382 30771 87776 450 13002 60625 528 25502 60625

11277 98287 56247 13659 11154 18008 16477 03958 47064 19800 33264 16665 23706 58264 16665

4 5 7 8 10

71819 89090 16721 83732 93821 19488 18993 48427 14176 81834 84406 24625 77147 34406 24625

816

BERNOULLI

Table

POLYNOMIALS,

OF

POSITIVE

:23 2:

66 67 is 70 71 72 zi 75

81 E 88:

z: 93 94 95 1 1 1 1

RIEMANN

POWERS

ZETA

FUNCTION

&kn

617 720 838 972 1124

99609 80326 27437 16689 41042

38476 41004 80841 90825 25200

28283 33629 34855 47085 55458

8 37709 34995 31899 51512 90891

87066 18522 29883 69019 59644

13 15 19 23 27

10563 88554 18530 08961 69498

86137 44973 80891 40014 05854

15076 50788 52921 66265 50640

1297 1492 1713 1962 2242

11990 60965 40807 27322 20922

74736 67929 35481 20300 20300

65130 76273 89079 1 03762 1 20559

64007 55578 86395 90771 06771

33660 45661 63677 67998 67998

33 39 46 55 65

11115 46261 89027 55326 63096

00335 19889 07286 65472 25472

95536 79593 24521 79460 79460

2556 2908 3302 3742 4232

48350 64496 54303 34768 57047

56321 62529 01696 12800 03425

1 1 1 2 2

39729 61563 86379 14526 46391

79901 80957 38760 88527 36656

65279 50175 17696 28352 18977

77 63 106 124 145

32510 86219 49600 51040 22232

86401 51863 93432 78527 06906

13601 77151 30976 12960 03585

4778 5384 6056 6801 7624

08654 15770 45658 09190 63490

04481 09804 28236 80825 80825

2 3 3 4 4

82395 23002 68718 20098 77746

42718 19494 51890 35635 36635

88673 45314 98690 27331 27331

168 196 227 262 303

98500 19153 27863 73072 08433

07044 51006 53971 32327 02327

03521 98468 28036 04265 04265

8534 9537 10641 11857 13191

14692 20822 94807 07610 91497

39216 43504 62601 35625 07500

5 6 6 7 8

42321 14542 95188 85107 85220

71947 13310 14229 61631 53135

73092 81828 75909 79685 70310

348 400 459 526 601

93283 93152 80311 34352 42821

09511 87653 54736 62487 25280

53296 82288 50201 29625 26500

14656 16261 18017 19938 22035

43442 28675 84364 23453 38653

79276 46129 01041 87200 87200

9 11 12 14 15

96524 20097 57109 08819 76592

01010 63925 07632 95731 11731

25286 72967 56103 62664 62664

686 781 888 1007 1142

01885 17055 03947 89106 10879

63746 08237 17370 77196 57196

04676 76113 60721 79040 79040

24323 26815 29529 32480 35686

06578 92048 52558 42905 19993

42161 98929 88556 44300 72425

17 19 21 24 27

61894 66308 91537 39413 11903

13620 22206 44528 33638 86142

14505 69481 08522 91018 81643

1292 1459 1646 1854 2086

20343 82298 76323 97898 59593

10166 14263 66939 52248 15080

78161 86193 26596 56260 59385

39165 42938 47024 51447 56230

47815 02610 78207 91556 88456

94121 81904 18896 14425 14425

30 33 36 40 45

11121 39333 98967 92626 23094

78853 46007 98488 86545 07545

47499 84620 39916 41997 41997

2343 2629 2945 3296 3683

92334 46750 94588 30228 72277

88197 30627 48916 85991 75991

23001 52528 18576 03785 03785

61398 66976 72993 79478 86462

49475 96076 96947 74541 11837

50156 73804 34561 53825 63200

49 55 60 66 73

93346 06565 66147 75716 39136

60306 47620 28587 22441 65570

93518 69134 19535 30351 20976

4111 4583 5104 5677 6307

65257 81394 22502 21982 46923

77288 10154 40039 62325 59571

92196 48868 36161 52865 62240

93976 02056 10737 20058 30058

59315 42160 67693 33041 33041

74016 52129 76801 67500 67500

80 88 96 106 116

60526 44269 95032 17777 17777

23468 59412 61670 31113 31113

59312 36273 54129 33330 33330

7000 7760 8593 9507 10507

00323 23429 98205 49930 49930

17816 04362 25663 00499 00499

42496 07713 57601 98500 98500

7

51

z78 99 100

EULER

SUMS

23.4

m\n

96

AND

9

BERNOULLI

Ah-D

EULER

POLYNOMIALS,

SUMS

OF

POSITIVE

RIEMANN

POWERS

ZETA

817

FUNCTION

skkn

Table

23.4

k=l

613 757 932 1143 1396

10 38941 75112 94452 34603 83199 38258 66451 30907 95967 52098

62626 19650 32699 53275 93900

40451 15700 57524 41925 41925

:76 58

1700 2062 2493 3004 3608

26516 29849 10270 21945 88121

43060 57628 26623 59629 59629

08076 99325 05149 46550 46550

4322 5161 6146 7299 8645

22412 52349 45378 37528 64962

76258 34941 53759 99828 44456

29151 69375 60224 07200 97825

10213 12036 14150 16596 19421

98650 82430 74713 94119 69368

53564 99082 00654 07202 07202

93601 55050 65674 25475 25475

93723 84347 46930 45970 81117

17301 43545 40581 14140 23612

06676 94100 51749 29125 94750

1 1025 60074 11 08650 108 74275 713 3538 14275 49143 1 49143

m\n 51 52 53 54 55

10

59 60

4 10 24 52 110

08517 27691 06276 98822 65326

66526 30750 22599 77575 68200

61

220 422 779 1392 2416

60442 20381 25054 35716 35716

95976 96425 23049 80850 80850

66

4084 6740 10882 17223 26760

34526 33754 98866 32676 06992

59051 50475 64124 29500 70125

71 77: 7754

22676 26420 30718 35642 41273

40876 61465 91085 1 33156 1 92205

77949 89270 56937 29270 29270

23501 18150 13574 13775 13775

:76 78 79 80

47702 55029 63365 72833 83570

70010 38057 15640 43249 85073

47012 72874 85236 11504 11504

36126 36775 36199 83400 83400

1 1 1 1

95728 09473 24989 42479 62166

51619 31932 36051 48338 92382

02074 38034 10093 76074 16796

12201 70825 24274 16050 81675

1 2 2 2 3

84297 09139 36989 68171 03039

08171 42312 52073 24066 08467

04827 96263 05665 05327 05327

52651 21500 33724 17325 17325

3 3 4 4 5

41980 85419 33817 87679 47552

69648 54190 77262 28403 97795

23434 47066 26359 21259 59638

62726 76550 94799 64975 55600

6 6 7 8 9

14036 87778 69485 59924 59924

24155 65424 93493 14243 14243

51139 46067 33614 42419 42419

60176 86225 75249 24250 24250

2 3 5 7 10

74168 86758 39916 46353 22208

12139 11208 01061 78601 52136

94576 37200 01649 61425 77050

13 18 24 33 43

87824 68682 96503 10544 59120

36537 80261 98741 59593 59593

40026 57875 46099 37700 37700

57 74 95 122 156

01386 09406 70554 90290 95353

52694 33911 57044 66428 55588

90101 67925 52174 70350 85975

199 251 316 396 494

37428 97341 89847 69074 34699

30416 52774 73860 36836 36836

62551 92600 37624 49625 49625

64 65

2; 69 70

81 82 2 85

91 92

818 *

BERNOULLI

Table

AND

EULER

POLYNOMIALS,

23.5

RIEMANN

FUNCTION

x%/n!

4

2

n\x

ZETA

0)4.0000 o)a.oooo 1.0666 1.0666

: : 5

5 00000 00000 66667 66667

0 5.0000 00000 11 1.2500 00000 2.0833 33333

; a 9 10 11 12 13

- 5)5.1306

71797

0 1.2232 1 5.0968 1 1.9603 2 7.0011 2 I 2.3337

47480 64499 32500 87499 29166

-17)3.9118 -la 5.2863 -19 6.9556 -20 8.9175 -20 I 1.1146

75343 18031 81620 40538 92567

''2.0527 69883 4.8300 46785 (- 9j1.4331

79137

2.0473 98767

(-24)4.0479

-38)6.5735 -39 3.1302 -40 1.4559 -42 6.6179 I -43 I 2.9412

93393

20725

-15)6.3028 -16 7.0031 -17 7.5033 -18 7.7620 I -19 I 7.7620

01010 12233 34535 70209 70209

-11)1.1167

10881

-20 7.5116 -21 I 7.0422 -22 6.4020 -23 5.6488 (-24j4.8418

80847 00794 00722 24167 49286

-16)5.6083 -17 7.0104 -18 a.4975 -19 9.9971 I -19 I 1.1425

86851 83563 55835 24511 28516

-25 4.0348 -26 3.2715 -27 2.5827 I -28 / 1.9867 (-29 1.4900

74405 19788 78780 52907 64681

-20)1.2694

76128

(-24)1.4816

80567

64027 68584 38876 03984 90659 -31 8.9989 09998 -32 I 7.6586 46807 -33 6.3822 05673

44; 48 49 50

For

-11)7.2586

4.1679 71052 x = 1, see Table

6.3.

BERNOULLI

AND

EULER

POLYNOMIALS,

RIEMANN

ZETA

X”/n!

6 0)6.0000 1)1.8000 1 3.6000 1 5.4000 1 6.4800

!I I

Table

23.5

8

7

00000 00000 00000 00000 00000

819

FUNCTION

0)7.0000 00000 1)2.4500 00000 115.7166 66667

0)8.0000 1 3.2000 11 8.5333 2 1.7066 2 2.7306

00000 00000 33333 66667 66667

2)1.2150

00000

1 5.5542 6.4800 00000 85714 1 2.7771 4.1657 14286 42857

t

10 (

ljl.6662

85714 1)4.9536 1 I 2.8896 1 1.5559 0 7.7797 0 3.6305

20388 11893 44865 24327 38019

( ( (

26 27 %i 30 31 (-10 I 1.6131 21179 :9 I -11 -12 5.4992 3.0246 02210 76746 34 -13 9.7046 06022 35 -13 1.6636 46746

I - 567I 9.4206 3.5589 65421 1.3049 4.6397 46701 10976 34025 (- 7)2.4224 52009

36 (-14 2.7727 445-78 37 -15 4.4963 42559 ;: -16 7.0994 1.0922 88251 28962 40

I -17 I 1.6383 43442

41 t: 154

I

46

(-17 I 6.3324 19956

t; 49 50

I -17 -20 4.6927 -19 -18 2.9329 58716 1.7964 1.0778 18219 48887 31193

24. Combinatorial K. GOLDBERG,~

Analysis

M. NEWMAN,~

E. HAYNSWORTH

3

Contents Page

Mathematical

Properties.

...................

Basic Numbers .................... 24.1.1 Binomial Coefficients .............. 24.1.2 Multinomial Coefficients ............. 24.1.3 Stirling Numbers of the First Kind ........ 24.1.4 Stirling Numbers of the Second Kind ....... 24.2. Partitions ...................... 24.2.1 Unrestricted Partitions .............. 24.2.2 Partitions Into Distinct Parts ........... 24.3. Number Theoretic Functions .............. 24.3.1 The Mobius Function .............. 24.3.2 The Euler Function ............... 24.3.3 Divisor Functions ................ 24.3.4 Primitive Roots ................. 24.1.

References Table

24.1.

n550, Table

24.2. nll0

..........................

Binomial Coefficients m125 Multinomials

m

. . . . . . . . . . . . . .

(Including a List of Partitions)

. . . . . .

822 822 822 823 824 824 825 825 825 826 826 826 827 827 827 828

831

Stirling Numbers of the First Kind S$“). . . . . . . . .

833

Stirling Numbers of the Second Kind SaQ) . . . . . . .

835

Table

24.5. Number of Partitions and Partitions Into Distinct Parts . . p(n), n(n), n1500

836

Table

24.6. Arithmetic Functions o(n), us(n), n(n), n_
. . ‘. . . . . . . . . . . . . .

840

Table

24.7. Factorizations n<10000

. . . . . . . . . . . . . . . . . . . .

844

Table

24.3.

,

n125 Table

24.4.

nS25

Table n< Table

24.8. Primitive 10000 24.9.

Roots, Factorization

of p- 1 . . . . . . . . .

864

Primes . . . . . . . . . . . . . . . . . . . . . . .

870

p_<105 ‘2 * National a National

Bureau of Standards. Bureau of Standards.

(Presently,

Auburn

University.) 821

24. Combinatorial Mathematical In each sub-section a fixed format wliicli rrlctl1otls

OF

T11c format

of this chapter en~pliasizes ttlr

rrtcntling

ttlc

follows

this forIll:

we use

Properties use

nlltl

tnl)lPS.

:l~~~~lll~~i1ll~~ill~

1. Drfinitions

A. (‘onit~inatorial B. Generating functions C. Closed form II.

Analysis n special and easily recognizable symbol, and We have yet that s~riit~ol must bc c:ls.v to write. ict tlctt on a script capital 3 without any certainty that WC tiavc settled this question pernianently. FVe feel that the subscript-superscript notation cmphasizrs ttlc genernting functions (which are p6wers of mutually inverse functions) from which most of the important relations flow.

Relations

24.1.

A. Rrcurrenccs B. Checks in computing C. Basic use in numerical III.

Asymptotic

and

Special

24.1.1

for

the

Stirling

This chapter (24.21 Fort 124.71 Jordan [24.10] hfoser and Wyman [24.9] &filne-Thomson [24.15] Riordan [24.1] Carlitz [24.3] Gould 1 Miksa (Unpublished tables) (24.171 Gupta

Kind

S’I’ n s& s:: (,:,I;)

Second

(1-x)

---I=$m

(-l)~~~Sl(n-l,n-m)

(;)

*

. . .

x”-m

C. Closed form n n.1 m =G!(n-m)!=

n n-m ( >

0

II.

*

n2 m

.. .

-

Relations

A. Recurrences

B:::’

S(n, m) &(m,

n)

(;)

n=O,l,

m!(n’-m+g

n-m)

=(~)+(7~~:)+. . +(,lim)

n2m

ns”

B. Checks u(n, m)

We feel that a capital S is natural for Stirling numbers of the first kind; it is infrequenLly used for other notation in this contest. But once it is used we have difficulty finding a suitable symbol for Stirling numbers of the second kind. The numbers are sufficiently important to warrant 822

* (l+d”=~o (;) Xrn

Kind

@:: g.“I

s(n, m) S(n-m+l,

is the number of \yays of choosing m n m objects from a collection of n distinct objects witllotlt regard to order. 13. Generating functious 0

=n(n-1)

cgr

Bb”l,

Coeflicients

A.

Numbers

Firs1

Numbers

1. Definitions

Values

Reference

Binomial

analysis

In general the notations used are standard. This includes the difference operator A defined on functions of s by Aj(z)=f(cr+ 1)-f(x), A”+!{(x) =A(A’j(s)), the Rroneckrr delta hiI, the Riemann zet,a function l(s) and the greatest common divisor symbol (m, n). The range of the summands for a summation sign without limits is explained to the right of the formula. The notations which are not standard are those for the multinomials which are arbitrary shorthand for use in this chapter, and those for the Stirling numbers which have never been standardized. A short table of various notations for these numbers follows : Notations

Basic

‘SW pngr II.

so (A)

(n,” m>=(‘t”)

go c--l)“+

(?rJ=(‘,,‘>

r+s>n r> /I,+ 1

COMBINATORIAL

ANALYSIS

823

where n=e

nnpk,

m=g

k=O

m,pk k=O

C. Numerical

=&

(--l)s-k

analysis

III.

rif:lc’> Special

A”f(z-s) Values

(>

2n =2”(2n-1)(2n-3) I n 12.

Multinomial I.

s
. . . 3.1

Coefficients Definitions

. . . +&,, different objects A. (n; n,, .n2, . . ., n,) is the number of ways of putting n=nl+n,+ into m different boxes with nk in the k-th box, k=l, 2, . . ., m. is the number of permutations of n=al+2a2+ . . . +nu, symbols composed (75 al, a2, . . ., 4* of a,, cycles of length k for k=l, 2, . . ., n. a set of n=u,+2a,+. . . +na, dif(n; al, a,, . . ., a,)’ is the number of ways of partitioning fereut objects into a, subsets containing k objects for k= 1, 2, . . ., n. B. Generating functions . . . +x,)“=Z;(n;

(r1faS

nl, n2, . . ., n,)ci$%$ .

. z:m

summed over n,+n,+

. . . +n,=n

summed over u1+2u2+ . . . +na,=n (~~tk)l=m!~~~Z(n;u~,u2,...,u.)‘~:~2;~...2:~

andu1+u2+“‘+u’=m

C. Closed forms (n; nl, n2, . . ., n,) =n!/n,!n,!

. . . n,!

a2, . . ., u,)*=n!/1%,!2”~~!

haI,

. .+n,=n

nl+n2+.

. . . n%z,!

u,+2u2+

(n;al,a2, . . ., a,)’ =n!/( l!)%,! (2!)%z2! . . . (n!)“nun! II.

. . .+na,=n

ul+2a2+

. . . +nu,=n

Relations

A. Recurrence

(n+m;nl+l,

n2+1, .

.

.,n,+l)=c

(ntm-1;

nl+l,

.

.

.,nk-1+1,

nk,nk+l+l,

.

.

.

k=l

,%i+1>

B. Checks ml1 * Z(n;n,,%,

all TLi2 1

. . .,n,)=

summed over ‘n, -:-n,+ . . . +n,=n

1m! sLrn’ .

.,

J

&J*=(-l)n-mS;~)

~:(7L;a,,a?,

.

Z(n;u,,az,

. . ., a,)‘=

summed over aI+2%+

. . . +nu,=n

g;Lmm,

C. Numerical analysis (Faa di Bruno’s formula) ~l(s(z))=~~of’“‘(g(z))~(n;a,, summed over al+2a2+ *see

page

II.

6, . . .) Un)‘{g’(z)}“‘{g”(2))=z. . . . +na,=n

and al+%+.

. . {g’“‘(z)j”n

. . +an=m.

and ~,+a~+.

. . +un=m

824

COMBINATORIAL

PIlO

ANALYSIS

. ..o

Pz

P,

2

...

.

Pa

P,

PI

...

.

...

.

...

0

...

n-l

...

P,

P,-1

P,

Pn--l

=Z(-l)“-m:(n;

al, &, . . ., u”)*P:‘P;z

. . . P>

summed over a1+2a2+ . . . $non=n; e.g. if Pn=Zj,rx~ for k=l, 2, . . ., n then the determinant and sum equal n!Zx1x2 . . . x,, the latter sum denoting the n-th elementary symmetric function of x,, x2, . . ., x,. 24.1.3

Stirling

Numbers I.

of the

First

Kind

III.

Asymptotics

and

Special

Values

~S~~~~-(n-l)!(-y+lnn)~-l/(m-l)!

De&&ions

for m=o(ln

A. (-1)“~“Sag’ is the number of permutations of n symbols which have exactly m cycles. B. Generating functions x(x-1)

. . . (x-n+l)=m&

{ln (l+x)}“=m!

$,

S$@ $

Sim)xm SO’76 = 60,

bl
S~l)=(-l)“-l(n-l)!

C. Closed form (see closed form for 8;Am1”‘)

fp-l),, 78

n 0

II.

2

fp’ n = ]

Relations

A. Recurrences

24.1.4

#?C?),=S$n-l)-nS(m)

n)

Stirling

Numbers I.

n

of the

Second

Kind

De.finitioxle

A. ~~“‘5s the number of ways of partitioning set of n elements into m non-empty subsets. B. Generating functions B. Checks x-g,

n>l

%ijl”‘x(x-1)

. . . (x-m+l>

2 (-l)“-mS~m)=n! m=O

Y(l-x)-1(1-2x)-1

. . . (l’-mx)-l=,Cn

gP)x”-n J

v C. Numerical

(x)<m-l

analysis C. Closed form

if convergent.

g~)=-$&

. -

(--I)“-”

(;)

k”

a

COMBINATORIAL II.

825

ANALYSIS

B. Generating

Relations

function

A. Recurrences g$y= (y)

m s ncm)+ 5 h-1) II

$3p’=,~$-,

@

n>m>_l n2 m>r

&&c:-‘, sp-”

c.

Closed form

B. Checks

s(h, k)=;# ((x))=x-[x] =o

((g))

-+ if 2 is not an integer if x is an integer II.

Relations

A. Recurrence c= sgpn)(2)

A’“j(x)=m!

ikI=&k! k-0

III.

gig,“, c;;)

=

g kmZ=j$ a

if convergent

B$‘x’ $

Asymptotics

and

lim n+-

m-s

B. Check

{‘s} Special

sp)=

Values

(-l)k3yp(n-v)=ul(n)

C 1<3klf<$& 2

-

m!-l III.

gp)

*+m

rv-

mzn 2”f7&!

24.2.2

=

BP’=

1

Into

Distinct

Parts

A. q(n) is the number of decompositions of n into distinct integer summands without regard to so that q(5)=3. order. E.g., 5=1+4=2+3

goq(n)xn=nil

24.2. Partitions

.I.

4n@

Partitions

B. Generating

;p-lq)

Unrestricted

1rn&

I. Definitions

gf’=f&l, p,

Asymptotics

p(n)--le

for n= o(mt)

b 2&m n-t-

24.2.1

p(n)+

Partitions

Definitions

A. p(n) is the number of decompositions of n into integer summands without regard to order. E.g.,5=1+4=2+3=1+1+3=1+2+2=1+1+ l-t-2=1+1+1+1+1 so that p(5)=7.

function (l+x’)=nil

(1-x2”-‘)-’

Ixl
C. Closed form k p(n)=;&

A,,-,(n)

&Jo

(&-&-&iii$

where Jo(x) is the Bessel function of order 0 and AznM1(n) was defined in part I.C. of the previous subsection.

826

COMBINATORIAL II.

Relations

ANALYSIS

g(x) =gI

f(nz) for all x>O if and only if

A. Recurrences

j(x)=2

p(n)g(nx)

for all 2>0

n-1

o<&k
(7w3F)=(-1)’

if n=3Pfr

-

a@>=1

=0 otherwise

andif gl gl If(mnx) )=gl The cyclotomic .pFl)““‘d’

polynomial

B. Check

III.

a<3~~<,(-l)14(~-(3~~fk))=l -

24.3.

24.3.1

Theoretic

The

Mtibius

Functions

24.3.2

=o B. Generating

/4nJn-a= l/Z(S)

T&=1

Euler

Totient

Function

A. q(n) is the number of integers not exceeding and relatively prime to n. B. Generating functions

1) 2 (p(n)n-*= i+-n-1 T(s)

functions 5

Asymptotics

I. Definitions

if n=l if n is the product of k distinct primes if n is divisible by a square >l.

=(-l)*

The

Function

I. Definitions

A. &)=l

n is

- r*(n) Inn=-1 c+I n

Asymptotics

Number

of order

if n=‘+ =0 otherwise.

III.

a&) If(nx)) converges.

m p(n)x= 2 cn=l 1--z”=(l_s)2-

as>1

s’s>2

- * b 2.\ ny Jxl
C. Closed form Ixl
q(n) =npyn II.

Relations

A. Recurrence p(mn)=p(m)p(n)

over distinct

II.

if (m,n)>l

n-1

(m, n)=l

dndd)=n 5: for all n

g(z) =g f(x/n) for all x>O if and only if n==1 M

Relations

B. Checks

g(n) =lI f(d) for all n if and only if din f(n) =& g(n/d)fi@’ for all n

f(x)=C

n.

dmn> =dmMn)

cp(n) =

.

>

A. Recurrence

B. Check dn P(d)=&1 7 C. Numerical analysis g(n) = d A f(d) for all n if and only if f= f(n) =g crkOsW>

1 -i

primes p dividing

if (m, n)=l

=o

(

p(n)g(x/n) for all x>O

F n

i+

0

a~(“) = 1 (mod n) III.

Asymptotics

f kgr(k)=;+0(v)

(a, n)=l

COMBINATORIAL

24.3.3

Divisor I.

$ gl(m)=S+O (F)

Functions

Definitions

A. U&J) is the sum of the k-th powers of the divisors of n. Often uo(n) is denoted by d(n); and a(n) by u(n). B. Generating functions n$l dW’=r(4i-(s-k)

B>k+l

C. Closed form uJn)=~

cF= ii p

pl(+“-l

i=l

II.

n=prlptt

Pf--l

=

‘Jkhbkh>

Relations

uk(np)

=

uk(nbk(p>

(m,

1

p prime

-pk.ok(n/p)

III.

$ $I

n)=

Asymptotics

uo(m)=ln n+27-l+O(n+) (y=Euler’s

24.3.4

constant)

Primitive I.

-. Roots

Definitions

The integers not exceeding and relatively prime to a fixed integer n form a group; the group is cyclic if and only if n=2,4 or n is of the form pk or 2pk where p is an odd prime. Then g is a primitive root of n if it generates that group; i.e., if g, g*, . . ., g@tn)are distinct modulo n. There are q((o(n)) primitive roots of n.

. . . pza

A. Recurrences uk(mn>

827

AIVALYSIS

II.

Relations

A. Recurrences. If g is a primitive root of a prime p and gp-’ f l(mod p”) then g is a primitive root of pk for all k. Jf gP-‘=l(mod p”) then g+p is a primitive root of pk for all k. If g is a primitive root of pk then either g or g+pk, whichever is odd, is a primitive root of 2~‘. B. Checks. If g is a primitive root of n t.hen gk is a primitive root of n if and only if (k, v(n)) = 1, and each primitive root of n is of this form.

References Texts

[24.1] L. Carlits, Note on Ntirlunds polynomial BF’, Proc. Amer. Math. Sot. 11, 452-455 (1960). [24.2] T. Fort, Finite differences (Clarendon Press, Oxford, England, 1948). [24.3] H. W. Gould, Stirling number representation problems, Proc. Amer. Math. Sot. 11, 447-451 (1960). [24.4] G. H. Hardy, Ramanujan (Chelsea Publishing Co., New York, N.Y., 1959). [24.5] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 4th ed. (Clarendon Press, Oxford, England, 1960). [24.6] L. K. Hua, On the number of partitions of a number into unequal parts, Trans. Amer. Math. Sot. 51, 194-201 (1942). 124.71 C. Jordan, Calculus of finite differences, 2d ed. (Chelsea Publishing Co., New York, N.Y., 1960). [24.8] K. Knopp, Theory and application of infinite series (Blackie and Son, Ltd., London, England, 1951). [24.9] L. M. Mime-Thomson, The calculus of finite differences (Macmillan and Co., Ltd., London, England, 1951). [24.10] L. Moser and M. Wyman, Stirling numbers of the second kind, Duke Math. J. 25, 29-43 (1958). 124.111 L. Moser and M. Wyman, Asymptotic development of the Stirling numbers of the first kind, J. London Math. Sot. 33, 133-146 (1958). I24.121H. H. Ostmann, Additive Zahlentheorie, vol. I (Springer-Verlag, Berlin, Germany, 1956).

[24.13] H. Rademacher, On the partition function, Proc. London Math. Sot. 43, 241-254 (1937). [24.14] H. Rademacher and A. Whiteman, Theorems on Dedekind sums, Amer. J. Math. 63, 377-407 (1941). [24.15] J. Riordan, An introduction to combinatorial analysis (John Wiley & Sons, Inc., New York, N.Y., 1958). [24.16] J. V. Uspensky and M. A. Heaslet, Elementary number theory (McGraw-Hill Book Co., Inc., New York, N.Y., 1939). Tables

[24.17] British Association for the Advancement of Science, Mathematical Tables, vol. VIII, Number-divisor tables (Cambridge Univ. Press, Cambridge, England, 1940). n 5 10’. [24.18] H. Gupta, Tables of distributions, Res. Bull. East Panjab Univ. 13-44 (1950); 750 (1951). [24.19] H. Gupta, A table of partitions, Proc. London Math. Sot. 39, 142-149 (1935) and II. 42, 546-549 (1937). p(n), n=1(1)300; p(n), n=301 (1)600. 124.201 G. Kavdn, Factor tables (Macmillan and Co., Ltd., London, England, 1937). n 1256,000. [24.21] D. N. Lehmer, List of prime numbers from 1 to 10,006,721, Carnegie Institution of Washington, Publication No. 165, Washington, D.C. (1914). [24.22] Royal Society Mathematical Tables, vol. 3, Table of binomial coefficients (Cambridge Univ. Press, Cambridge, England, 1954). (y).for r
/

828

COMBINATORIAL

Table

BINOMIAL

24.1

n/m

0

1

:

: 8

$11 51

4 5

98 10

11 1

i 10

11 12 13 14 15

1 1 1 1 1

16 17 18 19 20

1 1 1 1 1

2 i8 9 &B 36

ANALYSIS

COEFFICIENTS

(;)

1

11 12 13 14, 15

55 6 8 Q 105

165 220 286 364 455

16 17 18 19 20

120 136 153 171 190

560 680

210 231 253 276 300

1330 1540 1771 2024 2300

330 %!-? 1001 1365 1820 2380 3060 3876 4845

4368 6188 8568 11628 15504

;$-$i.i&

20349 26334 33649 42504 53130 65780 80730 98280 1 18755 1 42506

1 1 1 1 1

26 27 28 29 30

325 351 378 406 435

2600 2925 3276 3654 4060

31 32 33 34 35

1 1 1 1 1

31 32 33 34 35

465 496 528 561 595

4495 4960 5456 5984 6545

31465 35960 40920 46376 52360

1 2 2 2 3

36 37 38 39 40

1 1 1 1 1

36 37 38 39 40

630 666 703 741 780

7140 7770 8436 9139 9880

58905 66045 73815 82251 91390

41 42 43 44 45

1 1 1 1 1

41 42 43 44 45

820 861 903 946 990

10660 11480 12341 13244 14190

101270 111930 123410 135751 148995

i 45

13260 330 ?92-

9 128; 2002 e

26 27 28 29 30

1 8

:: 21Q-

if3

5985 7315 885’5 10626 ,p6%> .A 14950 17550 20475 23751 27405

1035 15180 %67 : fi; 1081 16215 48 1 48 1128 17296 49 1 49 1176 18424 50 1 50 1225 19600 From Royal Society Mathematical Press, Cambridge, England, lY54

1 7

2: 56;-

5 1~ Q

1-t

10

;.

45

8

7

3 L

8008

165 495 1287

38760

11440 19448 31824 50388 77520

75582 ~ 1 25970

54264 74613 1 00947 1 34596 1 77100

2 45157 3 46104 4 80700

4 90314 7 35471 10 81575

2 2 3 4 5

30230 96010 76740 75020 93775

6 8 11 15 20

57800 88030 84040 60780 35800

15 22 31 42 58

62275 20075 08105 92145 52925

69911 01376 37336 78256 24632

7 9 11 13 16

36281 06192 07568 44904 23160

26 33 42 53 67

29575 65856 72048 79616 24520

78 105 138 181 235

88725 18300 84156 56204 35820

3 4 5 5 6

76992 35897 01942 75757 58008

19 23 27 32 38

47792 24784 60681 62623 38380

83 102 126 153 186

47680 95472 20256 80937 43560

302 386 489 615 769

60340 08020 03492 23748 04685

; 9 10 12

z: 62598 86008 21759

44 52 60 70 81

96388 45786 96454 59052 45060

224 269 322 383 453

8i940 78328 24114 20568 79620

955 1180 1450 1772 2155

48245 30185 08513 32627 53195

163185 13 70754 93 66819 535 24680 2609 178365 15 33939 107 37573 628 91499 3144 194580 17 12304 122 71512 736 29072 3773 211876 19 06884 139 83816 859 00584 4509 230300 21 18760 158 90700 998 84400 5368 Tables, vol. 3, Table of binomial coefficients. Cambridge (with permission).

32815 57495 48994 78066 78650 Univ.

COMBINATORIAL

BINOMIAL

n\m

9

COEFFICIENTS

(;)

10

11

1;

2 43 44 45 46 ts7 49 50

12

24.1

13

1

55 220 715 2002 5005

1198 20

Table

.

9 10

16 17

829

ANALYSIS

1144oc , 24310 48620 92378 1 62960

1

1

;i 364 1365

*.-

4368 12376 31824 75582 1 61960 -

iEE

43758* 92378 1 84756

1

;2 455

1fl;

1820 6188 18564 50388 1 25970

560 2380 8568 27132 77520

2 4 8 13 \ 20

93930 97420 17190 07504 42975

3 6 11 19 32

52716 46646 44066 61256 68760

3 7 13 24 44

52716 05432 52078 96144 57400

2 6 13 27 52

93930 4664652078 041.56 00300

31 46 69 100 143

24550 86825 06900 15005 07150

53 84 131 200 300

11735 36285 23110 30010 45015

77 130 214 345 546

26160 37895 74180 97290 27300

96 173 304 518 864

57700 83860 21755 95935 93225

201 280 385 524 706

60075 48800 67100 51256 07460

443 645 925 1311 1835

52165 12240 61040 28140 79396

846 1290 1935 2860 4172

72315 24480 36720 97760 25900

1411 2257 3548 5483 8344

20525 92840 17320 54040 51800

2062 3473 5731 9279 14763

53075 73600 66440 83760 37800

941 1244 1630 2119 2734

43280 03620 11640 15132 38880

2541 3483 4727 6357 8476

86856 30136 33756 45396 60528

6008 8549 12033 16760 23118

05296 92152 22288 56044 01440

12516 18524 27074 39107 55868

77700 82996 75148 97436 53480

23107 35624 54149 81224 1 20332

89600 67300 50296 25444 22880

3503 4458 5639 7089 8861

43565 91810 21995 30508 63135

11210 14714 19173 24812 31901

99408 42973 34783 56778 87286

31594 42805 57520 76693 1 01505

61968 61376 04349 39132 95910

1 1 2 2

78986 10581 53386 10906 87600

54920 16888 78264 82613 21745

1 2 3 5 7

76200 55187 65768 19155 30062

76360 31280 48168 26432 09045

11017 13626 16771 20544 25054

16330 49145 06640 55634 33700

40763 51780 65407 82178 1 02722

50421 66751 15896 22536 78170

1 1 2 2 3

83196 33617 00368 16264 38800

3 5 6 9 12

89106 22514 96685 22637 13996

17655 00851 34468 34836 51100

10 14 19 26 35

17662 06768 29282 25967 48605

30790 48445 49296 83764 18600

33407 74171 25952 91359 73537

2 4 11 24 52

03490 97420 44066 9614400300

104 00600 200'58300 374 42160 678 63915 1197 59850

830 Table

COMBINATORIAL BINOMIAL

24.1

14

ANALYSIS

COEFFICIENTS

(;) 17

18

1:; 969 4845

1; 171 1140

1

20349 74613 2 45157 7 35471 20 42975

5985 26334 1 00947 3 46104 10 81575

15

16

19

1: 120 680 3060 11628 38760 1 3 8 19 44

'1

1;: 816

3876 15504

16280 19770 17190 61256 57400

1 4 13 32

54264 70544 90314 07504 68760

96 200 401 775 1454

57700~ 58300 16600 58760 22675

77 173 374 775 1551

26160 83860 42160.. 58760 17520

53 130 304 678 1454

11735 37895 21755 63915 22675 -

31 84 214 518 1197

24550 36285 74180 95935 54850

2651 4714 8188 13919 23199

82525 35600 09200 75640 59400

3005 5657 10371 18559 32479

40195 22720 58320 67520 43160

3005 6010 11668 22039 40599

40195 80390 03110 61430 28950

2651 5657 11668 23336 45375

82525 22720~ 03110 06220 67650

37962 61070 96695 1 50845 2 32069

97200 86800 54100 04396 29840

55679 93641 1 54712 2 51408 4 02253

02560 99760 86560 40660 45056

1 2 3 6

1 2 5 8

85974 59053 87811 10211 87323

96600 68710 43380 17810 78800

73078.72110 28757 74670 22399 74430 77112 60990 28521 01650

119:

2:

1330 7315 33649

1540 8855 42504 1 77100

f 3stl% 15 46 131 345 864

62275 86825 23110 97290 93225

6 22 69 200 546

57800 20075 06900 30010 27300

2062 4714 10371 22039 45375

53075 35600 58320 6143067650

1411 3473 8188 18559 40599

20525 73600 09200 67520 28950

1 3 6 11

90751 76726 35780 23591 33802

35300 31900 00610 43990 61800

1 3 6 13

85974 76726 53452 89232 12824

966OW 31900 63800 64410 08400

.

3 5 7 11 16

52401 28602 83789 49558 68713

52720 29080 60360 08528 34960

6 9 15 22 34

34322 86724 15326 99116 48674

74896 27616 56696 17056 25584

10 16 26 41 64

30774 65097 51821 67148 66264

46706 21602 49218 05914 22970

15 25 42 68 110

15844 46619 11716 63537 30686

80450 27156 48758 97976 03890

20 35 60 102 171

21126 36971 83590 95306 58844

40600 21050 48206 96964 94940

24 44 80 140 243

46626 67753 04724 88314 83621

70200 10800 31850 80056 77020

23 34 48 67 93

98775 16437 23206 52488 78456

44005 74795 23240 72536 56300

51 75 109 157 225

17387 16163 32600 55807 08295

60544 04549 79344 02584 75120

99 150 225 334 492

14938 32326 48489 81089 36896

48554 09098 13647 92991 95575

174 274 424 649 984

96950 11888 44214 92703 73793

26860 75414 84512 98159 91150

281 456 730 1155 1805

89530 86481 98370 42584 35288

98830 25690 01104 85616 83775

415 697 1154 1885 3040

42466 31997 18478 16848 59433

71960 70790 96480 97584 83200

20

23

.1

24

25

/I ? /’

1

1

232: 1771 10626 53130

2f3 2024 12650

2; 276 2300

2 11 42 143

65780 96010 84040 92145 07150

14950 80730 3 76740 15 60780 58 52925

443 1290 3548 9279 23199

52165 24480 17320 83760 59400

201 645 1935 5483 14763

60075 12240 36720 54040 37800

78 280 925 2860 8344

88725 48800 61040 97760 51800

26 105 385 1311 4172

29575 18300 67100 28140 25900

7 33 138 524 1835

36281 65856 84156 51256 79396

1 2 6 13

55679 28757 87811 23591 12824

02560 74670 43380 43990 08400-

37962 93641 2 22399 5 10211 11 33802

97200 99760 74430 17810 61800

23107 61070 1 54712 3 77112 8 87323

89600 86800 86560 60990 78800

12516 35624 96695 2 51408 6 28521

77700 67300 54100 40660 01650

6008 18524 54149 1 50845 4 02253

05296 82996 50296 04396 45056

26 53 105 201 377

91289 82578 20494 26164 36557

37220 74440 81860 00080 50150

fl 38% 31 08105

100 15005 300 45015

44 45

846 2257 5731 13919 32479

72315 92840 66440 75640 43160

1 3 6 13

73078 59053 35780 89232 78465

72110 68710 00610% 64410 28820

26 51 96 176 316

91289 37916 05669 10393 98708

37220 07420 18220 50070 30126

46 1;

560 82330 07146 1673 976 24796 56794 79106 49896

694 35265 80276 1255 42392 2231 17595 87422 66528

49 '50

2827 75273 46376 4712 92122 43960

3904 99187 16424 6732 74460 62800

24 51 105 210 411

46626 37916 20494 40989 67153

70200 0742081860 63720 63800

789 03711 13950 2738 1483 38976 94226 81648 56572 4969 98965 48176 8874 98152 64600

1 3::

2;

2600 17550 98280 4 75020 20 35800

20 44 96 201 411

21126 67753 05669 26164 67153

40600 10800 18220 0008fk 63800

823 34307 27600 3095 1612 38018 76995 35776 41550 5834 33568 17424 10804 32533 66600

325 2925 20475 1 18755 5 93775

15 35 80 176 377

15844 36971 04724 10393 36557

80450 21050 31850 50070 50150

789 03711 139503224 38018 1612 76036 83100 41550

3:: 3276 23751 1 42506

10 25 60 140 316

30774 46619 83590 88314 98708

46706 27156 48206 80056 30126

694 35265 80276 1483 38976 3095 76995 94226 35776r

6320 53032 18876 6320 53032 18876 12154 86600 36300 12641 06064 37752

COMBINATORIAL

Multinomials

Table

and Partitions

7r=l”l, 2”2, . . ., nun, n=a1+2a2+ Ml=

831

ANALYSIS

. . . +nq,,

(n; nl, n2, . . ., n,) =n!/(l!)al(2!)a2

m=al+az+

24.2

. . . ta,

. . . (n!)“n

M2= (n; al, az, . . ., a,,) *=n!/lala1!2u2a2! . . . &a,! Ma=@;

H

m

al, az, . . ., a,)‘=n!/(l!)0iaI!(2!)%~!

Ml

n

M3

442

1

1

1

1

1

::

2 12

1 2

1 1

1 1

; 6

3” 1

4 193 22 12, 2 14

: 6 12 24

ii 3 6 1

1 5 10 ;i 60 120

6 :: 3

ii 2: 3; 12, 4

1, 293 23 la, 3 12, 22 14, 2 16 7 1, 6

i’45 3

4

+, 5 1,2,4 1, 32 22, 3 la, 4 12, 2, 3 3

5

;1;

6 7

13’ 22 11: 2 1’

8

m 1 2

3

3 1, 2 13

5 1,4 2, 3 12, 3 1, 22 13, 2 1”

. . . (n!)“na,!

88 30 ix 120 180 360 720

: I; 42 105 140 210 210 420 630 840 1260 2520 5040

f;: 20 20 15 10 1 120 144 1: 90 120 :: 45 15 1 720 840 504 420 504 630 280 210 210 420 105 70 105 21 1

; 1 4

1 4 : 1

i 10 10 15 10 1

i 15 10

6 l 9

1 2

3

1, 7 32,: 41 1: 6 1, 2, 5 1, 3,4 22, 4 2, 32 la, 5 12,2, 4 12,32 1, 22, 3 14, 4 la, 2, 3 12, 2a 16, 3 14, 22 16, 2 18 9 1, 8 2, 7 3, 6 4, 5 12, 7 1, 2, 6 1, 3, 5 1, 42 2

4

: 5; 21 105 70 105 35 210 105 35 105 21 1

8

24

5

ix 15 4”; 15 1

7r

5

6

3,354 3; ’ 13,6 12,2, 5 12,3, 4 1, 22, 4 1, 2,32 23, 3 14, 5 13, 2, 4 13, 32 12, 22, 3 1, 24 16,4 14, 2, 3 13, 23 16, 3 16, 22 17,2 19

Ml

M3

442

i!i

28

;i 56 168 280 420 560 336 840 1120 1680 2520 1680 3360 5040 6720 10080 20160 40320 1 3: 84 126 2;; 504 630 756 1260 1680 504 1512 2520 3780 5040 7560 3024 7560 10080 15120 22680 15120 30240 45360 60480 90720 181440 362880

5040 5760 3360 2688 1260 3360 4032 3360 1260 1120 1344 2520 1120 1680 105 420 1120 420 112 210 28 1 40320 45360 25920 20160 18144 25920 30240 24192 11340 9072 15120 2240 10080 18144 15120 11340 10080 2520 3024 7560 3360 7560 945 756 2520 1260 168 378 36 1

1 2: E 28 168 280 210 280 56 420 280 840 105 5;: 420 56 210 28 1

; is

126 36 252 504 315 378 1260 280 7:: 1260 1890 2520 1260 126 1260 840 3780 945 126 1260 1260 84 378 36 1

832

COMBINATORIAL

Table

24.2

n 10

m

lr 10 1,9

::

32'; 4; 6 52 12, 8 172, 7 1,3, 6 1,4,5 22, 6 2,3, a 5 $44 13: 7 12, 2, 6 12, 3, 5 12, 42 1, 22, 5 1, 2, 3,4 1, 33

3

4

*See

page

Multinomials

11.

MI 1 10 45 120 210 252 90 360 840 1260 1260 2520 3150 4200 720 2520 5040 6300 7560 12600 16800

A& 362880 403200 226800 172800 151200 72576 226800 259200 201600 181440 75600 *120960 56700 50400 86400 151200 120960 56700 90720 151200 22400

M3

1 10 45 120 210 126 45 360 840 1260 630 2520 1575 2100 120 1260 2520 1575 3780 12600 2800

ANALYSIS

and

Partitions

n 10

m

?r 3

5

6

7

;z’ ;2 14: 6 13, 2, 5 13, 3, 4 12, 22, 4 12, 2, 32 1, 23, 3 25 15, 5 14, 2, 4 14, 32 13, 22, 3 12. 24 1"; 4 16, 2, 3 ;+4 3

8 9 10

16: g22 18, 2 1'0

MI

Mz

18900 25200 5040 15120 25200 *37800 50400 75600 113400 30240 75600 100800 151200 226800 151200 302400 453600 604800 907230 1814400 3628800

18900 25200 25200 60480 50400 *56700 50400 25200 945 6048 18900 8400 25200 4725 1260 5040 3150 240 630 45 1

A43

3150 6300 210 2520 4200 9450 12600 12600 945 252 3150 2100 12600 4725 210 2520 3150 120 630 45 1

COMBINATORIAL STIRLING

NUMBERS 1

833

ANALYSIS OF THE ( , ,?,

FIRST

KIND

Sirn)

Table 24.3

;

-;

I /

% 24 -120 720 -5040 40320 -3 62880 36 -399 4%0 62270 8 71782

11 12

i3

14 15 16

-225 1624 - 13132 1 18124 -11 72700

28800 16800 01600 20800 91200

-106 1205 - 14864 1 98027 -28 34656

28640 43840 42880 59040 47260

127 -1509 19315 -2 65967 39 21567

53576 17976 59552 17056 97824

-130 76743 68000

433 91630 01600

-616 58176 14720

:;

- 35568 2092 27898 74280 88000 96000

1 -7073 22340 42823 55905 93600 79200

-1 10299 82160 22448 24446 37120 24640

:;

-1216 40237 64510 37057 04088 28000 32000

431 37698 -22 56514 80585 68176 21600 38400

-668 34 01224 60973 95938 03411 22720 53280

21

2432 90200 81766 40000

2": 24

11- 51090 24000 72777 94217 76076 17094 40000 80000 -258 52016 73888 49766 40000

-8752

94803 67616 00000

13803 75975 36407 04000

-411 86244 48476 77933 81078 01702 54547 40000 20000 965 38966 65249 30662 40000 -23427 87216_,39871 85664 00000

-2 67 56146 98631 90286 67377 32163 09306 84000 88000 -1595 39850 27606 68605 44000 39254 95373 27809 77192 96000

>$0,7’10726=& *-Ye

n\m

4

5

4 5

-1;.

1

6 7

85' -735,

9" 10

- 67284 6769 7 23680

:z 15

20 ::

'

-15 175

,'

505 -8707 1 58331 -30 32125 610 11607

69957 77488 39757 40077 57404

03824 75904 27488 19424 91776

-270 4836 - 90929 17 95071 -371 38478

68133 60092 99058 22809 73452

45600 33424 44112 21504 28000

-1 81664 8037 81182 97952 26450 06970 76096 51776

-65 48684 85270 30686 97600

42 80722 86535 71471 42912

:; -39365 1573 61409 75898 13866 28594 15107 31181 32800 31200

-1050 26775 05310 03356 75591 42796 74529 03823 84576 62624

From unpublishedtables of Francis L. Miksa, with permission.

-9 133 -2060 33361 -5 66633

16930 95730 .-.__ 06836 03756 05680

-2 84093 12870 31590 93124 51509 18114 88800 68800

23

-2: 322 -4536 63273

22449 -1960 -2 69325

i.

34 -459 .-. 6572 - 99577 15 97216

11 12

16

i i -j i

6

100 -1886 36901 -7 55152 161 42973 -3599 83637 -___. -20 21687 507 79532 -13237 14091

96721 15670 26492 75920 65301

02055 39535 70150 18786 66760 07080 58880 34384 63024 18960

97951 79476 07200

38169 95448 __-_. ._. .- 02976 .-...

37691 06827 41568 53430 28501 98976 57918 58577 60000

834

COMBINATORIAL Table

STIRLING

24.3

n\m

FIRST

KIND

Sim)

8

-2; 546 -9450

9 10 11

16 17

-27 537 - 11022 2 35312 -52 26090

:i 20

1206 - 28939 7 20308 -185 88776 4969 10165

64780 58339 21644 35505 05554

28032 45234 84661 50405 33625

10680 77960 84200 49984 12720

37803 73354 09246 19497 96448

73360 47760 53696 76576 36800

3 -69 1350 - 26814 5 -114 2487 - 55192 12 95363 -311 7744 -1 99321 53 04713 -1459 01905

33364 65431 97822 71552 52766

-55 1925 - 55770 14 14413 -373 12275

:: :; 15 16 :z: :90 -10 276 -7707 2 20984 -65 08376

-2 57 -1471 38192

9280 30571 79248 07534 20555

95740 59840 94833 08923 02195

14229 01910 40110 45497 17966

98655 92750 12973 94331 81468

11450 35346 61068 17396 50000

1 -37 1103 - 33081 10 14945

13

n\m

-4:

18150 57423 26634 36473 53775

1320 - 32670 7 49463 -166 69653 3684 11615

46311 29553 69012 83528 18452 91936 16815 47048 69899 43896 31613 01695 10661 54458 26492

90640 76800 37360 12976 88000

I0 1

n\m 10

9

-316 870

1 57773 -26 37558 449 90231 -7909 43153 i 44093 22928

:: 14 15

2243 25

OF THE

i

;

21 22

NUMBERS

ANALYSIS

la -430 10241 -2 50385 63 -1634 43714 -12 04749 342 18695

03081 98069 22964 26016 95940

82076 59531 81053 77407 87554

28000 77553 01929 32658 67550

20992 72465 95944 17376 71489

94896 83456 12832 32496 92880

11

12

-6; 2717 - 91091 27 49147

3731 -1 43325

' -6 166 -4628

-785 21850 02026 15733 06477

58480 31420 93980 86473 51910

30753 60053 23088 71136 52782

50105 50868 11859 85742 52146

40395 59745 49736 04996 37300

-7;

48 -1569 48532 -14 75607 446 52267 - 13558 4 15482 -129 00665 4070 38405 -1 30770 92873

51828 38514 98183 70075 67558

99622 524% 22764 03732 57381 99530 30525 31295 69521 73500

16

15

14

1

:z 15

-9; 5005 -2 83 -2996 1 02469 -34 22525

16 17 :i 20 21 ;; 2254

12 -413 13990

18400 94022 50806 37272 11900

-3 138 -5497 2 06929

1131 02769 95381 37310 09998 02531 36304 58470 86207 35671 43013 14056 94520 02391 06865

-75 2718 - 97125 34 70180 -1246 20006

61111 a6118 04609 64487 90702

6580 23680 96582 89282 33630 84500 69881 39913 04206 15000

17

18

:87 19 20

-15: 13566 -9 20550

-17: 16815

-1910

21 22

533 27946 - 27921 67686

-12 56850 797 21796

20615 89765

n\m

05903 ;; -64013 67173 25 29088 66798

36096 57942 67135

1

-10:

24- 12764 45460 -1219 12249

43496 47198 80000

- 72346 1168 69596 96626 41 49085 13800

4 -159 6238 -2 40604 92 44691

20

19

-16

-120 8500 -4 68180 223 23822 -9739 41900 01717 97183 24164 60386 13761

-13: 10812 -6 62796 349 16946

71630 88730 21941 44556 73550

-5

21

- 16722 7 52896 -325 60911 13727 25118 70058 63218

80820 68850 03430 00831 64500

22

23

24

35926 -25: 95000

-27: 42550

-300

25

1

-1

1

-210 25025

-231

1684 -22 40315 23871 12768 42500

-29 32776 30107 2388 10495

-37

1

1

COMBINATORIAL

2 : 1: z:

II\ ,,, 1 : 3 4

5

1 :

1 1 1 1 1

127

10

1

255 511

:: 13

: 1

::

1

lb

1

23

:

b

7 9”

6

1 3;; 966 3025 9330

32767 b5535

71 214 b44

1 31071 2 b2143 5 24287

77215

41686 57825 39010

1 12596

06446

72 291 llba 4677

2; 462 5880 b3987 b 27396 57 :5424 493 29280 4087 41333

70050 23825 74750 34501 38810

60276 a231 4 38264

23799 09572 19991

14948 67440 17305

29987 38518

04540 lb310

149

:“5

227 31 67746 a3248

,I

95028 b5010

91620 47084 00400

22372 31828

21583 11183

68583 20505

190 12

02430

55 1705 39325 7 52752 126 62650

84250 22303

3bb28 2682 634 1OOlb a6425

25008 b8516 56957 13089

89001 70286

44384

b2bbO

1937 27583

95755

54990

34150

3 71121

26805

63803 33785 b4655

33035 24764 a8117

65204 33920 96900

03514 7118 9 59340

37993 71322 12973

91275 77954

llb7 120 b2257 9214P 10929 43260

73005 72500

1203 108 25408 lb339

17849 21753

87500 31500

3200b

72970 75849

491 65620 10178

33391 51300

b

33560 30420 45907 27734

30178 51137 a3930 58260

2249 120 b 40128 88883

07848

74680

1 .i

83405 84530

70000

47 591

1 24196 12327

22324 57118

n,

45225

44725

12

28866

2b2b2

62804

98579 bO3bO

28490

10 b1753 114 4614b 1201 12a2b

b20 27

362

56794

53393

22275 59502

a207 95288

12563 412 1Olbb 34669

35590

01779 95364

1155

04908 29486 17791

802

30alb 60788

51 35130

90799 10153 50199

26558

49898 51039

1

b71

74195

10882b83 24 1 93020 b7216

2658

lb330 99896 b 09023 37 02641

10751

3

60978 36908

75560 51100

95250

45

5120 289

18 :;

25

b90a2

07834 49741

316 750

03480 b2b79

1 14239 13251 9

a4100

9

97510 0932b

1517

12055

baaal

8

83910 94132 08424

:",

:: ::

170

45652

ba401

12625 78219

b4053

2 04159 la 90360

b9 430

379 1913 48500 1 43668

27349

1 75057 11 06872

95545 a4710 90500

9641

12320 27840

21417

90550 51651

56527 2 89580 14 75892 74 920bO

11880 1 59027 99612

209 2lbb

1114

2 19

:: 23

la

08501 75035 bb920

10961

98901 37290 06985 bb950 15901

i

a2604 04786 a3400 34839

55950

77786 b3425 105bb 12897

1 79487 13 23652 93 21312 b34 36373 4206 93273

2 46730 75 400 2107

91745

18 15090

43625 79450 35501 00806 91025

2% 2646 22827

13 79400

32530

1717 6943 27988 4 52321

32818 57081 74624 29246 35540

1050 b951 42525

1 45750 11501

b 25 103 423

1934 48101

1

a8607

lb7

3% 1701 7770 34105

580b 17423 52280 56863 4 70632 14 11979

97151

41 94303 83

1'0

28501 ab526 2 blb25 7 88970 23 75101

10 48575

.’

1 2;

1023 2047 4095 8191 16383

20

ANALYSIS

114

48507

16

1256 1Obab 28924

54638 39160 b0380

babla 49092 b0579 25603

68481 la331 22000 41390

33437

44260

13460

2 49900 6020 a4 63025 2435

1 25

329 14 93040 51652 102lb 91758

36209 b8b2

95811

03608

04500 al331 41000 07115

*

96000

18

Ii

08778 24580 77530

10

1

15

lb

120

:i :;

3 67200 7820 4523 139 29200 16778

1 9996 136

15:

223.50954 5 27136

7

12597 41285

15675 17:

1901 i

:: :: 25

341 56159 30874 19582847

4 29939

02422 94044 46553

,, rw,

b25.90 43200 47080 29331

47200

23611 71824 51616

14041 349

42047 52799

lb1 4 99169 09499

a8803 36915 93110

44464 74004 27264 b5555 95960

4806

33313

21

22

53310

327

9 24849 23648 56785

23435 74629 85369 25445 94925

2:j

13 38807797 62189

lb

24

a9850 19285 al779 39170 b9675

2-9

1

21

2

20b776 52665

20

20

::

2 60465 8099

210

1

Ia 59550 23485 b2201 1169

72779 94750

28336231 lb85 24

19505 54606

1

25: 32

00450 33902

40250276

From unpublished tables of Francis L. Miksa, with permission.

30;

1

COMBINATORIAL

,I

I’(‘0

V(fO

04226 39943 81589 29931 86155

3658 4097 4582 5120 5718

100 101 102 103 104

1905 2144 2412 2712 3048

69292 81126 65379 48950 01365

4 4 5 5 6

44793 a3330 25016 70078 18784

150 151 152 153 154

4 4 4 5 6

08532 35313 50606 24582 96862 88421 47703 36324 03566 73280

194 207 222 238 255

06016 92120 72512 53318 40982

4 5 6 7 8

51276 26823 14154 15220 31820

6378 7108 7917 8808 9192

105 106 107 108 109

3423 3842 4311 4835 5419

25709 16336 49389 02844 46240

6 7 7 8 9

71418 28260 89640 55906 27406

155 156 157 158 159

6 7 8 8 9

64931 32322 06309 87517 76621

28555

273 292 313 335 358

42421 64960 16314 04746 39008

:; 101 135

9 11 13 15 17

66467 21505 00156 05499 41630

10880 12076 13394 14848 16444

110 111 112 113 114

6071 6799 7610 8513 9520

63746 03203 02156 76628 50665

10 10 ii 12 13

04544 a7744 77438 74118 78304

160 161 162 163 164

10 11 12 14 15

74381 81590 99139 27989 69194

59466 68427 04637 95930 75295

383 409 438 468 500

28320 82540 12110 28032 42056

176 231 297 385 4YO

20 23 26 30 35

12558 23520 79689 87735 54345

18200 20132 22250 24576 27130

115 116 117 118 119

10641 44451 11889 08248 13277 10076 14820 74143 16536 68665

14 16 17 18 20

90528 11388 41521 al578 32290

165 166 167 168 169

17 18 20 22 25

23898 93348 78904 82047 04389

00255 22579 20102 32751 25115

534 571 610 651 695

66624 14844 00704 39008 45358

87968 97205 92783 85689 89500

29927 32992 36352 40026 44046

120 121 122 123 124

18443 49560 20561 48051 22913 20912 25523 38241 28419 405001

21 23 25 27 29

94432 68800 56284 57826 74400

170 171 172 173 174

27 30 33 36 39

47686 13848 04954 23268 71250

17130 02048 99613 59895 74750

742 792 045 901 962

36384 29676 43782 98446 14550

07086

:: 22 30

15 :; 18 19

30 :: 33 34 35 36 37 ;t

/<,J)

I’(“) 2 2 2 3 3

7

10 11 12 13 14

ANALYSIS

42

,I

V(JJ)

I’(‘0

I,

82097 43159

64769 78802

V(Jl)

627 792 1002 1255 1575

1:: 122

40 46 53 61 70

1958 2436 3010 3718 4565

142 165 192 222 256

81 92 106 121 138

18264 89091 19863 32164 48650

48446 53250 58499 64234 70488

125 126 127 128 129

31631 35192 39138 43510 48352

27352 22692 64295 78600 71870

32 34 37 40 43

25410 13544 22816

175 176 177 178 179

43 47 52 57 62

51576 67158 21158 17016 58467

97830 57290 31195 05655 53120

1026 14114 1094 20549 1166 58616 1243 54422 1325 35702

5604 6842 8349 10143 12310

296 340 390 448 512

157 180 205 233 265

96476 04327 06255 38469 43660

77312 84756 92864 101698 111322

130 131 132 133 134

53713 59645 66208 13466 81490

15400 39504 30889 29512 40695

46 50 53 58 62

54670 10688 92550 02008 40974

180 181 182 183 184

68 74 81 89 98

49573 94744 98769 66848 04628

90936 11781 08323 17527 80430

1412 31780 1504 73568 1602 93888 1707 27424 1818 10744

14883 21637 26015 31185

585 668 760 864 982

301 342 388 441 499

67357 62962 87673 08109 95925

121792 133184 145578 159046 173682

135 136 137 138 139

1 1 1 1

90358 00155 10976 22923 36109

36076 81680 45016 41831 49895

67 72 77 83 89

11480 15644 55776 34326 53856

185 186 187 188 189

107 117 128 139 152

18237 14326 00110 83417 72735

74337 92373 42268 45571 99625

1935 2060 2193 2334 2484

82642 84096 58315 51098 10816

37338 44583 53174 63261 75175

1113 1260 1426 1610 1816

566 641 725 820 926

34113 12359 33807 10177 69720

189586 206848 225585 245920 267968

140 141 142 143 144

1 1 1 2 2

50658 66706 84402 03909 25406

78135 89208 93320 82757 54445

96 103 110 118 127

17150 27156 86968 99934 69602

190 191 192 193 194

166 182 198 216 236

77274 07011

04093 00652

12768

56363

05469 41845

2642 2811 2990 3179 3381

88462 38048 16608 84256 04630

89134 105558 124754 147273 173525

2048 2304 2590 2910 3264

1046 1181 1332 1501 1692

51419 14304 30930 98136 29875

291874 317788 345856 316256 409174

145 146 147 148 149

2 2 3 3 3

49088 75170 03886 35494 70273

58009 52599 71978 19497 55200

136 146 157 168 181

99699 94244 57502 93952 08418

195 196 197 198 199

258 281 306 334 364

08402 45709 88298 53659 60724

12973 87591 78530 83698 32125

3594 3820 4060 4315 4584

44904 75868 72422 13602 82688

204226

3658

1905

69292

444793

150

4 08532

35313

194 06016

200

397 29990

29388

4870 67746

17971

$2

51021

86271 60227

Values of /JOO from H. Gupta, A table of partitions, Proc. London Math. Sot. 39, 142-149, 1935 and II. 42, 546-549, 193’7 (with permission).

COMBINATORIAL

NUMBER

OF PARTITIONS

p(n)

n

AND PARTITIONS

s(n)

837

ANALYSIS

INTO

DISTINCT

Table 24.5

PARTS

P(rl)

n

200 201 202 203 204

397 432 471 513 559

29990 83636 45668 42052 00883

29388 58647 86083 87973 17495

4870 5173 5494 5834 6195

67746 61670 62336 73184 03296

250 251 252 253 254

23079 24929 26923 29072 31389

35543 14511 27012 69579 19913

64681 68559 52579 16112 06665

85192 89949 94961 1 00243 1 05807

80128 26602 58208 00890 47264

205 206 2.07 208 209

608 662 720 784 852

52538 29877 68417 06562 85813

59260 08040 06490 26137 02375

6576 6980 7408 7862 8341

67584 87424 90786 12446 94700

255 256 257 258 259

33885 36574 39472 42593 45954

42642 95668 36766 30844 57504

48680 70782 55357 09356 48675

1 1 1 1 1

11669 17844 24348 31199 38413

59338 71548 95064 20928 23582

210 211 212 213 214

927 1008 1096 1191 1295

51025 50658 37072 66812 00959

75355 85767 05259 36278 25895

8849 9387 9956 10558 11195

87529 48852 45336 52590 55488

260 261 262 263 264

49574 53471 57667 62183 67044

19347 50629 26749 74165 81230

60846 08609 47168 09615 60170

1 1 1 1 1

46009 54008 62428 71293 80624

65705 01856 82560 59744 90974

215 216 217 218 219

1407 1528 1660 1802 1957

05456 51512 15981 81825 38561

99287 48481 07914 16671 61145

11869 12582 13336 14133 14977

49056 38720 40710 83026 05768

265 266 267 268 269

72276 77905 33961 90476 97483

09536 06295 17303 01083 43699

90372 62167 66814 16360 44625

1 2 2 2 2

90446 00783 11660 23106 35150

44146 03620 75136 91192 17984

220 221 222 223 224

2124 2306 2502 2715 2945

82790 18711 58737 24089 45499

09367 73849 60111 25615 41750

15868 16811 17807 18860 19973

61606 16852 51883 61684 57056

270 271 272 273 274

1 1 1 1 1

05019 13123 21837 31205 41274

74899 85039 43498 18008 95651

31117 38606 44333 16215 73450

2 2 2 2 3

47820 61149 75170 89917 05427

61070 71540 53882 72486 58738

225 226 227 228 229

3194 3464 3756 4071 4413

63906 31263 11335 80636 29348

96157 22519 82570 27362 84255

21149 22392 23705 25091 26556

65120 29960 13986 98528 84608

275 276 277 278 279

1 1 1 1 2

52098 63729 76227 89656 04082

04928 39693 84330 410j5 58525

51175 37171 57269 41584 75075

3 3 3 3 3

21738 38889 56923 75883 95815

19904 46600 20960 26642 57440

230 231 Ef 234

4782 5182 5613 6080 6585

62397 00518 81486 61354 15859

45920 38712 70947 38329 70275

28103 29737 31462 33284 35207

94454 72212 84870 23936 06304

280 281 282 283 284

2 2 2 2 2

19578 36221 54095 73287 93892

63116 91453 25900 31835 97939

82516 37711 45698 47535 29555

4 4 4 4 5

16768 38791 61938 86265 11828

26624 78240 97032 19094 44672

235 236 237 238 239

7130 7719 8356 9043 9786

41855 58926 11039 68396 29337

14919 63512 25871 68817 03585

37236 39379 41639 44025 46543

75326 02688 89458 67324 00706

285 286 287 288 289

3 3 3 3 4

16013 39758 65243 92592 21938

78671 40119 08360 21614 85285

48997 86773 71053 89422 87095

5 5 5 6 6

38689 66911 96562 27710 60430

49522 97084 52987 98024 42088

240 241 242 243 244

10588 11454 12388 13397 14486

22467 08845 84430 82553 76924

22733 53038 77259 44888 96445

49198 52000 54955 58073 61360

87992 62976 97248 01632 27874

290 291 292 293 294

4 4 5 5 6

53425 87203 23437 62299 03976

31269 80564 10697 26919 38820

00886 72084 53672 50605 95515

6 7 7 8 8

94797 30892 68798 08604 50401

40554 09120 39744 19136 45750

245 246 247 248 249

15661 16929 18297 19772 21363

84125 67223 38898 65166 69198

27946 91554 54026 81672 20625

64826 68481 72335 76397 80679

71322 72604 19619 50522 55712

295 296 297 298 299

6 6 7 8 8

48667 96585 47956 03024 62049

41270 01441 50785 83849 62754

79088 95831 10584 43040 65025

8 9 9 10 10

94286 40360 88727 39499 92791

47940 04868 65938 71456 76298

250

23079 35543 64681

85192 80128

300

9 25308 29367 23602

11 48724 72064

COMBINATORIAL

838 Table 24.5

NUMBER

OF PARTITIONS

P(ll)

ANALYSIS

AND PARTITIONS

INTO DISTINCT

PARTS

(1(n)

n

9@> 11 4872472064 12 0742510607 12 6902530816 13 3366383848 14 0148559930

350 351 352 353 354

279 363328483702152 298 330063062758076 318 555973788329084 340 122810048577428 363 117512048110005

126 9182924648 132 9347719190 139 22769 71520 145 8093818816 152 6926715868

13 162217895057704 14 118662665280005 15 142952738857194 16 239786535829663 17 414180133147295

14 7264218618 15 4729217536 16 2560142890 17 0774343642 17 9389964242

355 356 357 358 359

387 632532919029223 413 76618 0933342362 441 622981929358437 471 314064268398780 502 957566506000020

159 8909656578 167 41824 09148 175 2890755072 183 5186738752 192 1228932216

::1” 312 313 314

18 671488299600364 20 017426762576945 21 458096037352891 23 000006655487337 24 650106150830490

18 8425979304 19 7902232212 20 7839472390 21 8259394656 22 9184682870

360 361 362 363 364

536 679070310691121 572 612058898037559 610 898403751884101 651 688879997206959 695 143713458946040

201 1182704478 210 52205 02772 220 3522150410 230 6275150210 241 3675001278

315 316 317 318 319

26 41580 7633566326 28 305020340996003 30 326181989842964 32 488293351466654 34 800954869440830

24 0639052286 25 2647294208 26 5235325352 27 8430235904 29 2260340224

365 366 367 368 369

741 433159884081684 790 738119649411319 843 250788562528427 899 175348396088349 958 728697912338045

252 5925533946 264 3239251488 276 5837686784 289 39517 78822 302 7822257408

27440 5776748077 :“21”37 39 919565526999991 42 748078035954696 :‘2: 45 772358543578028 324 49 005643635237875

30 6755232574 32 1945841664 33 7864488192 35 4544947722 37 2022512608

370 371 372 373 374

1022 141228367345362 1089 657644424399782 1161 537834849962850 1238 057794119125085 1319 510599727473500

316 7700044480 331 3846677248 346 6534741118 362 6048321048 379 26834 76992

325 326 327 328 329

52 462044228828641 56 156602112874289 60 105349839666544 64 325374609114550 68 834885946073850

39 0334057172 40 9518108690 42 9614917632 45 0666531450 47 2716874732

375 1406 207446561484054 376 14984787435905 81081 1596 675274490756791 ::i 1701 169427975813525 379 1812 356499739472950

396 6748730794 414 8565573659 433 8469000206 453 6808055808 474 3946406976

E ::: 334

73 653287861850339 78 801255302666615 84 300815636225119 90 175434980549623 96 45011 0192202760

49 5811828759 51 9999315040 54 5329385792 57 1854313990 59 9628687918

380 381 382 383 384

1930 656072350465812 2056 513475336633805 2190 401332423765131 2332 821198543892336 2484305294265418180

496 0262940968 518 6152380864 542 2025926436 566 8311927092 592 5456572864

335 103 151466321735325 110 307860425292772 ::t 117 949491546113972 126 108517833796355 ::i 134 819180623301520

62 8709513216 65 9156314788 69 1031243770 72 4399192576 75 9327910200

385 386 387 388 389

2645 418340688763701 2816 759503217942792 2998 964447736452194 3192 707518433532826 3398 704041358160275

619 3924614094 647 4200116480 676 6787237064 707 2211032064 739 1018303854

340 144 117936527873832 154 043597379576030 ::: 164 637479165761044 343 175 943559810422753 344 188 008647052292980

79 5888123110 83 4153664940 87 4201606890 91 6112394270 95 9969992704

390 391 392 393 394

3617 712763867604423 3850 538434667429186 4098 034535626594791 4361 106170762284114 4640 713124699623515

772 3778471936 807 1084479444 843 3553742947 881 1829129614 920 6579974150

345 200 882556287683159 346 214 618299743286299 347 229 272286871217150 244 904537455382406 3: 261 578907351144125

100 5862035461 105 38799 77632 110 4119260918 115 6679479970 121 1664556454

395 4937 873096788191655 396 5253 665124416975163 5589 233202595404488 :;8' 5945 790114707874597 399 6324 621482504294325

961 8503143424 1004 8324132444 1049 6798204736 1096 4711585280 1145 2882689344

1,

300 301 :i: 304

9 253082936723602 9 930972392403501 10 657331232548839 11 435542077822104 12 269218019229465

:z $8’ 309

350 279 363328483702152 126 9182924648

400 6727 090051741041926 1196 2163400706

COMBINATORIAL NUMBER

OF PARTITIONS

I’(,,)

,,

AND PARTITIONS

INTO DISTINCT

n

d’l)

839

ANALYSIS PARTS

Table

p(n)

24.5

c/01)

400 401 402 403 404

6727 7154 7608 8091 8603

09005 64022 80284 20027 55175

17410 26539 33398 64844 93486

41926 42321 79269 65581 55060

1196 1249 1304 1362 1422

21634 34404 76365 57124 86674

00706 08000 81998 07808 81438

450 451 452 453 454

1 1 1 1 1

34508 42573 51112 60152 69723

18800 13615 26207 90524 95104

15729 53474 19173 45537 64580

23840 04229 13678 15585 40965

9893 10307 10739 11188 11656

14440 93957 65687 96810 57102

61528 13070 10144 43072 54336

405 406 407 408 409

9147 9725 10339 10990 11682

67906 51251 09726 60006 31627

88591 37420 71239 37759 71923

17602 21729 47241 26994 17780

1485 1551 1619 1691 1765

75420 34186 74236 07292 45549

52794 29884 54282 29128 15430

455 456 457 458 459

1 1 2 2 2

79855 90581 01933 13948 26665

91645 04044 37928 90703 62143

39582 26519 51146 27330 58313

67598 31034 88629 69132 45565

12143 12649 13176 13724 14295

19032 57862 51755 81881 32530

12544 22432 08648 00782 93376

410 411 412 413 414

12416 13196 14023 14902 15834

67740 25896 78888 15629 42088

31511 69254 35188 03099 44881

90382 35702 47344 48968 87770

1843 1923 2008 2096 2187

01696 88934 20999 12178 77334

07104 65516 30208 16576 80960

460 461 462 463 464

2 2 2 2 3

40123 54365 69435 85381 02253

65561 39575 60521 55524 16287

39251 85741 29549 19619 25766

92081 99975 94471 86287 36605

14888 15506 16148 16817 17512

91233 48874 99826 42073 77348

20640 75476 46592 15550 45952

415 416 417 418 419

16823 17873 18987 20170 21424

82278 79296 96426 18301 52136

71392 96898 73316 88059 02556

35544 2283 31930 7Ok 76004 2382 92048,69148 64557 2486 74415 20078 33659 2594q6435 42056 36320 ,2707 76199 52640

465 466 467 468 469

3 3 3 3 4

20103 38987 58963 80095 02447

13615 12724 89376 46876 33986

29932 95254 81628 31205 17114

90544 32549 76613 98477 75160

18236 18988 19771 20585 21431

11274 53505 17881 22576 90268

38194 94524 29024 95744 83034

420 421 422 423 424

22755 24167 25664 27253 28938

29021 05302 64021 16454 03725

65800 14413 38377 62304 70847

25259 63961 14846 21739 98150

2825 2947 3075 3208 3347

32529 84998 53960 60580 26867

77152 62528 09352 00384 45954

470 471 472 473 474

4 4 4 5 5

26088 51092 77535 05499 35069

63801 33635 45970 30531 67535

56524 50960 81641 42046 16072

13417 99864 15593 29558 62125

22312 23228 24180 25171 26201

48299 28849 69117 11509 03821

10884 04960 98586 01902 12696

425 426 427 428 429

30724 32620 34629 36760 39020

98514 06861 70071 66724 14800

70950 74102 39035 18315 02372

51099 32189 75934 27309 59665

3491 3642 3799 3962 4132

75707 30895 17171 60256 86891

60097 45254 07136 14146 79000

475 476 477 478 479

5 5 6 6 7

66337 99397 34350 71304 10369

12186 20478 76365 20389 79823

58055 23018 37870 67318 66282

99675 52926 28583 07232 38005

27271 28385 29543 30747 31998

99448 57585 43443 28468 90573

23232 65430 69603 94368 73738

430 431

41415 43955 46647 49501 52527

73920 47717 86328 89040 07072

71023 05181 42292 94051 91082

58378 16534 67991 50715 40605

4310 4495 4687 4888 5096

24877 03113 51640 01685 85706

85006 72460 62334 40672 20480

480 481 482 483 484

7 7 8 8 9

51666 95317 41457 90222 41761

00419 79841 02874 78495 78911

49931 47582 28236 19280 49976

25591 32180 49455 88294 98055

33300 34652 36059 37521 39040

14373 91433 20520 07873 67468

57056 03468 80640 43946 62530

“4;: 437 438 439

55733 59131 62733 66549 70593

46514 71430 07137 43656 39364

46362 91696 60430 69662 65621

86656 18645 79215 97367 35510

5314 5540 5776 6022 6278

37439 91949 85678 56498 43769

57460 44512 02880 45546 39520

485 486 487 488 489

9 10 11 11 12

96228 53787 14608 78875 46778

80660 07886 77893 49115 71600

85734 24553 64264 57358 12729

11012 46513 84248 02646 19665

40620 42261 43968 45741 47585

21308 99712 41621 94910 16717

45496 45764 12802 51264 64998

440 441 442 443 444

74878 79418 84227 89322 94720

24841 06934 73040 95632 37025

94708 64434 77294 13536 78934

86233 02240 99781 45667 71820

6544 6822 7111 7411 7725

88391 32867 21361 99762 15750

85792 92200 67457 56080 89318

490 491 492 493 494

13 13 14 15 16

18520 94313 74382 58964 48308

40161 50322 57204 37499 54706

22702 44478 03639 49778 61724

33223 16939 53132 06173 38760

49500 51491 53560 55709 57943

73777 42772 10694 75216 45082

62304 84172 36938 10170 47040

00437 06493 12906 19698 26891

54417 05190 52519 71278 54269

17528 52391 91961 27202 09814

47604 18581 03354 05954 18000

8051 8390 8743 9111 9494

18865 60575 94352 75744 62459

81728 94564 40798 62854 05984

495 496 497 498 499

17 18 19 20 21

42678 42351 47619 58791 76192

27774 03350 31798 47204 51543

77609 31598 76580 28849 92874

81187 91466 64007 01563 61625

60264 62675 65181 67784 70489

40509 93600 48774 63214 07325

50309 10788 31176 30326 21792

9893 14440 61528

500

23 00165 03257 43239 95027

::: 434

445 446 447 448 449

1 1 1 1 1

450

1 34508 18800 15729 23840

73298 65212 45024

840

COMBINATORIAL Table

ARITHMETIC

21.6 n

o(n)

9,

51 52 53 54 55

32 24 52 18 40 24 36 28 58 16

8 4 4 2 12

n

v(n)

?I -1

n

(a(n)

100 32 102 48 48

2 8 2 a 8

102 216 104 210 192

151 152 153 154 155

156 72 96 60 120

2 8 6 8 4

152 300 234 288 192

201 202 203 204 205

132 100 168 64 160

4 4 4 12 4

272 306 240 504 252

120 80 90 60 168

106 107 108 109 110

52 106 36 108 40

4 2 12 2 8

162 108 280 110 216

156 157 158 159 160

48 156 78 104 64

12 2 4 4 12

392 158 240 216 378

%

E

208 209 210

96 180 48

4 6 10 4 16

312 312 434 240 576

104 2

152 248 114

161 162 163

132 54 162

104 192 363 2 164

211 212 213

210 104 140

62 212 378 4 288

48 240 144

165 164

80

68 294 288

215 214

168 106

4 324 264

2

4

12 8

1:

6 4

3 4

:: 18

56 57 58 59 60

:'8 14

62 61 63

60 30 36

42 6

96 62 104

112 111 113

72 48 112

Zf

65 64

48 32

74

127 84

115 114

88 36

66 20 67 66 68 32 69 44 70 24

:i

8 16

5 2 6

:t 20

1: 8

2

42

21

12

4

32 36 24

FUNCTIONS

101 102 103 104 105

-1

4 72 6 98 2 54 8 120 4 72

6

8 144 2 68 6 126 4 96 8 144

?I

ml

119 120

96 32

6 6 4 4 16

72 195 74 114 124

121 122 123 124 125

110 60 80 60 100

3 4 4 6 4

133 186 168 224 156

171 172 173 174 175

108 84 172 56 120

126 127 128 129 130

36 12 126 2 64 8 84 4 48 8

312 128 255 176 252

176 177 178 179 180

80 116 88 178 48

10 372 4 240 4 270 2 180 18 546

%

228 88

181 182 183 184 185

180 72 lii 88 144

2 182 8 336 4 248 8 360 4 228

;'3: 233 234 235

::; 232 72 184

8 2 12 4

384 450 234 546 288

8 384 4 216 6 336 8 320 8 360

236 237 238 239 240

116 156 96 238 64

6 4 8 2 20

420 320 432 240 744

27” 258 118

210 182 180 144 360

166 167 168 169 170

82 166 48 156 64

4 2 16 3 8

252 168 480 183 324

216 217 218 219 220

72 180 108 144 80

16 4 4 4 12

600 256 330 296 504

6 6 2 8 6

260 308 174 360 248

221 222 223 224 225

192 72 222 96 120

4 252 8 456 2 224 12 504 9 403

226

112

221

226

72

4 342 2 228 12 560 28 230 432

71 72 73 74 75

70 2 24 12 72 2 36 4 40 6

76 77 78 79 80

36 60 24 78 32

6 140 4 96 a 168 2 80 10 186

81 82 83 84 85

54 40 82 24 64

5 4 2 12 4

121 126 84 224 108

133 134 135

66 72

2 12 4 4 8

:3312 130 40 108

132 336 160 204 240

86 87 88 89 90

42 56 40 88 24

4 4 8 2 12

132 120 180 90 234

136 137 138 139 140

64 136 44 138 48

8 2 8 2 12

270 138 288 140 336

186 187 188 189 190

60 160 92 108 72

Ei 84 78

91 92 93 94 95

72 44 60 46 72

4 112 6 168 4 128 4 144 4 120

141 142 143 144 145

92 70 120 48 112

4 4 4 15 4

192 216 168 403 180

191 192 193 194 195

190 64 192 96 96

2 14 2 4 8

192 508 194 294 336

241 242 243 244 245

240 110 162 120 168

2 6 6 6 6

242 399 364 434 342

;; 42 do 40

1: 252 98 6 171 i 156 9 217

146 147 148 149 150

72 84 72 148 40

4 6 6 2 12

222 228 266 150 372

196 197 198 199 200

84 196 60 198 80

9 2 12 2 12

399 198 468 200 465

246

80

103 124 6 z

'9; 98 $9 100

::; 249 250

zi 164 100

8 4 8 4 8

504 280 480 336 468

i! 26 27 28 29 30

12 18 12 28 8

4 4 6 2 8

42 40

31 32 33 34 35

30 16 20 16 24

2 6 4 4 4

32 63

36 12 37 36 38 18 39 24 40 16

9 2 4 4 8

41 42 43 44 45

40 12 42 20 24

2 8 2 6 6

46 47

22 46

4 2

"4"$ ;$ 50 20

ANALYSIS

:i

72

42

228

From British Association for the Advancement of $+3ence,Mathematical Tables, vol. VIII, Number-divisor tables. Cambridge Univ. Press, Cambridge, England, 1940 (with permission).

59 1

(td

=

186

COMBINATQRIAL

ANALYSIS

ARITHMETIC

841

FUNCTIONS

24.6

Table

01

n

v(n)

-0

UI

252 728 288 384 432

301 302 303 304 305

252 150 200 144 240

4 4 4 10 4

352 456 408 620 372

351 352 353 354 355

q(n)

n

2 18 4 4 8

n

CI

250 72 220 126 128

216 160 352 116 280

8 12 2 8 4

560 756 354 720 432

401 402 403 404 405

400 132 360 200 216

2 8 4 6 10

402 816 448 714 726

451 452 453 454 455

400 224 300 226 288

256 257 258 259 260

128 256 84 216 96

9 511 2 258 8 528 4 304 12 588

306 307 308 309 310

96 306 120 204 120

12 2 12 4 8

702 308 672 416 576

356 357 358 359 360

176 192 178 358 96

6 630 8 576 4 540 2 360 24 1170

406 407 408 409 410

168 360 128 408 160

8 4 16 2 8

720 456 1080 410 756

456 457 458 459 460

144 456 228 288 176

16 1200 2 458 4 690 8 720 12 1008

261 262 263 264 265

168 130 262 80 208

6 4 2 16 4

390 396 264 720 324

311 312 313 314 315

310 96 312 156 144

2 16 2 4 12

312 840 314 474 624

361 362 363 364 365

342 180 220 144 288

3 4 6 12 4

381 546 532 784 444

411 412 413 414 415

272 204 348 132 328

4 6 4 12 4

552 728 480 936 504

461 462 463 464 465

460 120 462 224 240

2 16 2 10 8

462 1152 464 930 768

266 267 268 269 270

108 176 132 268 72

8 4 6 2 16

480 360 476 270 720

316 317 318 319 320

156 316 104 280 128

6 2 8 4 14

560 318 648 360 762

366 367 368 369 370

120 366 176 240 144

8 2 10 6 8

744 368 744 546 684

416 417 418 419 420

192 276 180 418 96

12 4 8 2 24

882 560 720 420 1344

466 467 468 469 470

232 466 144 396 184

4 2 18 4 8

702 468 1214 544 864

271 272 273 274 275

270 128 144 136 200

2 272 10 558 8 448 4 414 6 372

321 322 323 324 325

212 132 288 108 240

4 8 4 15 6

432 576 360 847 434

371 372 373 374 375

312 120 372 160 200

4 12 2 8 8

432 896 374 648 624

421 422 423 424 425

420 210 276 208 320

2 4 6 8 6

422 636 624 810 558

471 472 473 474 475

312 232 420 156 360

4 8 4 il 6

632 900 528 960 620

276 277 278 279 280

88 12 672 2 278 276 4 420 138 180 6 416 96 16 720

326 327 328 329 330

162 216 160 276 80

4 4 8 4 16

492 440 630 384 864

376 377 378 379 380

184 336 108 378 144

8 4 16 2 12

720 420 960 380 840

426 427 428 429 430

140 360 212 240 168

8 4 6 8 8

864 496 756 672 192

476 477 478 479 480

192 312 238 478 128

281 282 283 284 285

280 92 282 140 144

2 282 8 576 2 284 6 504 8 480

‘3:: :2: 333 216 z

2 6 6 4 :6646 4

332 588 494 504 408

381 382 383 384 385

252 190 382 128 240

4 512 4 576 2 384 16 1020 8 576

431 432 433 434 435

430 144 432 180 224

2 20 2 8 8

432 1240 434 768 720

481 482 483 484 485

432 240 264 220 384

286 287 288 289 290

120 240 96 272 112

8 504 4 336 18 819 3 307 a 540

336 337 338 339 340

96 336 156 224 128

20 992 2 338 6 549 4 456 12 756

386 387 388 389 390

192 252 192 388 96

4 582 6 572 6 686 2 390 16 *lo08

436 437 438 439 440

216 396 144 438 160

6 4 8 2 16

770 480 888 440 1080

486 487 488 489 490

162 486 240 324 168

12 1092 2 488 8 930 4 656 12 1026

291 292 293 294 295

192 4 392 144 6 578 292 2 294 84 12 684 232 4 360

341 342 343 344 345

300 108 294 168 176

4 12 4 a 8

384 780 400 660 576

391 392 393 394 395

352 168 260 196 312

4 12 4 4 4

432 855 528 594 480

441 442 443 444 445

252 192 442 144 352

9 8 2 12 4

741 756 444 1064 540

491 492 493 494 495

490 160 448 216 240

2 12 4 8 12

492 1176 540 840 936

296 297 298 299 300

144 180 148 264 80

8 570 8 480 4 450 4 336 18 868

346 347 348 349 350

172 346 112 348 120

4 522 2 348 12 840 2 350 12 744

396 397 398 399 400

120 396 198 216 160

18 2 4 8 15

1092 398 600 640 961

446 447 448 449 450

222 296 192 448 120

4 4 l‘i 2 18

672 600 1016 450 1209

496 497 498 499 500

240 420 164 498 200

10 4 8 2 12

992 576 1008 500 1092

n

251 252 253 254 255

a(n)

an

=n UI

n

9(n)

91

4 6 4 4 a

504 798 608 684 672

12 1008 6 702 4 720 2 480 24 1512 4 4 8 9 4

532 726 768 931 588

COMBINATORIAL,

ANALYSIS

ARITHMETIC 501

332

4

672

502 503 504 505

250 502 144 400

4 2 24 4

504 1560 612

506

220

a

864

507 508 509

312 252

6 6

732 896

510

508 128

2 16

510 1296

511

432

512

256

4 592 10 1023 a 800

513

324

514 515

408

516 517 518 519

168 460 216 344

520

521 522 523 524 525 526 527

256

4

4

756

774 624

551

504

552 553 554 555

468 276 288

556 557

276 556

176

4 600 lb 1440 4 640 4 a34 a 912

4

864 846

611 612

552 192

4 672 la 1638

563 564 565

562

2

564

613

612

2

la4

12

448

4

1344 684

614 615

306 320

4 a

4

a52

616 617 618 619

240 616 204 618

lb 1440 2 618 a' 1248 2 620

620

240

12

396 310 528 192 500

4 4 20 5 4 8

4 4

792 576

568

6 2

280

1080

568 144

2 16

570 1440

570 240

2 12

572 1176

380 240 440

6

744

21 2 6

lb51 578

626 627

312 360

628

312

629 630

576 144

4 768 a looa

160

20

l&i8

529 530

506

3

208

a

553 972

6

780

581

492

4

582 583

192 520

8 4

548

549 550

192 576 272

384 224

968

621 622 623 624 625

528

546

616

10 a

576 577 578 579 580

547

606 288

8

262 480

4 12

921 776

1260 672 1176

12

1260 960

614 924

1008

f@(n)

360

Qn

656

320

10

1302

657 658

432 276

6 8

962 1152

659 658 6bO 160

2 660 24 2016

661

660

2

662

662 663 664 665

330 384 328 432

4 8 8 I3

996 1008 1260 960

666

4 8 2 la 4

686

294

687

456

baa

336

689 690

624 176

a 1200 4 920 10 1364 4 756 16 1728

690 344 360 346 552

12 4 4

4 882 2 588 la 1596 4 640 a loao

636 637 638 639 640

208 504 280 420 256

12 1512 6 798 a 1080 6 936 16 1530

591 592 593 594 595

392 288 592 la0 384

4 792 10 1178 2 594 16 1440 a 864

641 642 643 644 645

640 212

2 642 8 1296

264 336

12 1344 8 1056

691 692 693 694 695

596 597 598 599 600

296

6

288 646 216 580 240

a 2 20 4 12

224

16

4 a

646 647 648 649 650

696

396 264

697 698 699 700

640 348 464 240

4 4 4 la

1134 660

144 546 272 360

lb

1344

200

12

2 6 6

548 966

806 1116

598 160

2 24

1050 a00 1008

600 lab0

642

2418

452 300 682 216 544

292 586 168 540 232

4

30

681 682 683 684 685

586 587 588 589 590

816 728

358 718

6 1106 4 684 24 la72

a 1020 4 720 4 al0 6 684 24 1680 542

718

1281 678 a 1368 4 784 lb 1620

504

2

644

1080 648 la15 720 1302

2 6

1040

1260 960

1240

9 2

a 12

6

6 4 4

312 676 224 576 256

288 288

468

476

676 677 678 679 680

2 632 a 1200 4 a48 4 954 4 768

708 280

816 1680 2 710 8 1296

356

744 2016 674 1014

630

1062

716

4 24 2 4

312 420

4 4 12

717

600 192 672 336 360

632 633 634 635

352 600 232

1524

8 1152

1482

671 672 673 674 675

942 960

14

368

1008

6 4 8

12

320

a

332 444 264

781

711

Cl

2 702 lb lb80 4 760

480

720 1176 896 1224

960 936 720 1736

706 707 708 709 710

un

700 216 648

715

12 4

a

702 703 704 705

o(n)

713 714

216 616

1344

n

701

a 6bO 4 192 16

667 668 669 670

631

316

‘II

8 1024 6 1148 2 654 8 1320 4 792

584 585

2 4

648 1110 1092

8 1224 2 608

n

651

4 583 a 1080 4 648

12 4

432

200

607 608 609

280

524 924 992

360 256

606

320

2 6 12

540 270

300

440

562

522 260 240

541 542 543 544 545

604

561

571 572 573 574 575

356 268 420 144

652 216 520

a a 1116

522

537 538 539 540

324

653 654 655

336 240

1170

264

652

a84 1064 798

610

2 12

536

1056

6 b 6

1488

520 168

480

8

396

4 20

192

176 424

252

603

504 192

569 570

1120

602

605

Ul

602

559 560

696 1260

12

2

12

282 324

216

UQ

600

180

566 567

348

cp(n)

558

1232 576 912

532 533 534 535

n

601

980 558 1248

12 4 a 4 lb

531

FUNCTIONS

912 1152 684 la20 a28

712

719 720

352

192

2

1350 768 1728

1080 720

721

612

4

722 723 724 725

342 480 360 560

6 4 6 6

1143

726 727

220 726

12 2

1596

16

728 lbao

729 730

486

7

1093

4

792 1736 734 1104

728

288 288

8

731

672

732 733 734 735

240 732 366 336

12 2 4 12

352

12

a32 968 1274 930

1332

1368

736 737 738 739 740

240 738 288

1512 816 12 1638 2 740 12 1596

692 1218 1248 1044 a40

741 742 743 744 745

432 312 742 240 592

a a 2 16 4

1800 756

746 747 748 749 750

372 492 320 636 200

4 1122 6 1092 12 1512 4 864 lb la72

1050 936 1736

660

4

1120 1296

744 1920 900

COMBINATORIAL 'ARITHMETIC

ANALYSIS FUNCTIONS

TaIblc: 24.6

Qll Cl 12 1272 2160 954 1:4 2106 1152

751 752 753 754 755

756 752 368 1; 1488 500 4 1008 336 8 1260 600 4 912

801 802 803 804 805

528 6 1170 400 4 1206 720 4 888 264 12 1904 528 8 1152

851 852 853 854 855

792 4 912 280 12 2016 852 2 854 360 8 1488 432 12 1560

901 902 903 904 905

832 400 504 448 720

4 8 8 8 4

972 1512 1408 1710 1092

n 951 952 953 954 955

756 757 758 759 760

216 24 2;;; 756 2 378 4 1140 440 8 1152 288 16 1800

806 807 808 809 810

360 8 536 4 400 8 808 2 216 20

1344 1080 1530 810 2178

856 857 858 859 860

424 8 1620 856 2 858 240 16 2016 858 2 860 336 12 1848

906 907 908 909 910

300 8 906 2 452 6 600 6 288 16

1824 908 1596 1326 2016

956 957 958 959 960

476 6 560 478 4" 816 256 2:

1680 1440 1440 1104 3048

761 762 763 764 765

760 2 762 252 8 1536 648 4 880 380 6 1344 384 12 1404

811 812 813 814 815

810 2 812 336 12 1680 540 4 1088 360 8 1368 648 4 984

861 862 863 864 865

480 8 1344 430 4 1296 862 2 864 288 24 2520 688 4 1044

911 912 913 914 915

910 2 912 288 20 2480 820 4 1008 456 4 1374 480 8 1488

961 962 963 964 965

930 432 636 480 768

993 1596 1404 1694 1164

766 767 768 769 770

382 4 1152 696 4 840 256 18 2;;; 768 2 240 16 1728

816 817 818 819 820

256 20 2232 756 4 880 408 4 1230 432 12 1456 320 12 1764

866 867 868 869 870

432 4 1302 544 6 1228 360 12 1792 780 4 960 224 16 2160

916 917 918 919 920

456 6 1610 780 4 1056 288 16 2160 918 2 920 352 16 2160

966 967 968 969 970

264 16 2304 966 440 1: 1;:; 576 8 1440 384 8 1764

771 772 773 774 775

512 4 1032 384 6 1358 772 2 774 252 12 1;'72" 600 6

821 822 823 824 825

820 2 822 272 8 lb56 822 2 824 408 8 1560 400 12 1488

871 872 873 874 875

792 432 576 396 600

952 1650 1274 1440 1248

921 922 923 924 925

612 4 1232 460 4 1386 840 4 1008 240 24 2688 720 6 1178

971 970 972 972 1'8 2548 973 i% 1120 974 486 i 1464 915 480 12 1736

776 777 778 779 780

384 8 432 8 388 4 720 4 192 24

826 827 828 829 830

348 8 1440 826 2 828 264 18 2184 828 2 830 328 8 1512

876 877 878 879 880

288 12 2072 876 2 878 438 4 1320 584 4 1176 320 20 2232

926 927 928 929 930

462 4 1392 612 6 1352 448 12 1890 928 2 930 240 16 2304

976 977 978 979 980

480 10 1922 976 2 978 1968 22 4" 1080 336 18 2394

781 782 783 784 785

700 4 864 352 8 1296 504 8 1200 336 15 1767 624 4 948

0; 36;; 146 1778 1026 834 276 8 1680 835 664 4 1008

881 882. 883 884 885

880 2 a82 252 18 2223 882 2 884 384 12 1764 464 8 1440

931 932 933 934 935

756 464 620 466 640

6 6 4 4 8

1140 1638 1248 1404 1296

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2.953 22.479 2.32.107 24. i I* 2.7.139

1907

1951 37,53 33.73 7.283 11.181

26.61 2.32.109 3.17-29 2.991 3.3.83

31.7.31 13.151 1973 3.661 1993

2.977 22.491

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165 166 167 168 169

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170 171 172 173 174

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23.83 19.101 3.643 7.277 1949

190 191 192 193 194

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200 201 202 203 204

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210 211 212 213 214

22.3.5a.7 2.5.211 23.5.53 2.3.5.71 2*-5.107

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215 216 217 218 219

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225 226 227 228

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200 201 202 203 204

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205 206 207 208 210 211 212 213 214 215

230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249

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265 266 267 268 269

270 271 272 273 274

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3.7.167 3517 3527 33.131 3547

22.877 2.1759

2.29.61 22.887

112.29 3a-17.23 3529 3539 3-7.132

28.32.72

350 g 351 8 352 353 354

EX 359

2.b-359

53.67 3.1187 3571 3581 38.7.19

360 361 362 363 364

24.32.51 2.5.19 21.5.181 2.3.5.11* 28.5.7.13

13.277 23-157 3.17-71 3631 11.331

2.1801 22.3-7.43 2.1811 24.227 2.3.607

3.1201 3613 3623 3.7.173 3643

22.17.53 2.13.139

5.7.103

E&Y 2%.9;1

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365 366 367 368 369

2.5a.73 21.3.5.61 2.5.367 26.5.23 2.32.5.41

3.1217 7.523 3671 32.409 3691

22.11.83 2.1831 2”.3”.17 2.7-263 22-13.71

13.281 32.11.37 3673 29.127 3.1231

2.32.7-29 24.229 2.11.167 22.3.307 2.1847

5.17.43 5.733 3.52.72 5.11.67 5.739

23.457 2.3.13.47 22.919 2.19.97 2’.3.7.11

3.23.53 19.193 3677 3.1229 3697

2.31.59 22.7,131 2.3.613 23.461 2.432

3659 3.1223 13.283 7.17.31 33.137

370 371 372 373 374

2QY.37 2-5.7.53 28.3.5.31 2.5.373 2a.5.11.17

3701 3.1237 612 7.13.41 3.29.43

2.3.617 27.29 2.1861 22.3.311 2.1871

7.232 47.79 3.17-73 3733 19.197

2’.463 2.3.619 22.72.19 2.1867 25.3a.13

3.5.13.19 5.743 52.149 32.5.83 5.7.107

2.17.109 22.929 2.3’.23 23.467 2.1873

11.337 32.7.59 3727 37.101 3.1249

22.32-103 2.11.132 2’.233 2.3.7.89 22,937

3709 3719 3.11.113 3739 23.163

375 376 377 378 379

2.3.5’ 2’.5.47 2.5.13.29 22.33.5.7 2.5.379

111.31 3761 32.419 19.199 17.223

28.7.67 2.32.11.19 22.23.41 2.31.61 24.3.79

38.139 53.71 73.11 3-13.97 3793

2.1877 22.941 2.3.17.37 2*.11.43 2.7.271

5.751 3.5.251

22.3.313 2.7.269 26.59 2.3.631 22.13-73

13.17” 3767 3.1259 7.541 3797

2.1879 28.3.157 2.1889 22.947 2.32.211

3.7.179 3769 3779 32.421 29.131

28.52.19 2.3.5.127 21.5.191 2g.$.;3

3.7-181 37.103 3821 3.1277 23.167

2.1901 21.953 2.3.72-13 2a.479 2.17.113

3803 3.31.41 3823 3833 32.7.61

22.3.317 2.1907 24.239 2.38.71 22.312

2.11.173 2=.32,53 2.1913 22.7.137 2.3.641

34.47 11.347 43.89 3.1279 3847

25.7.17 2.23.83 22.3.11.29 2.19.101 28.13.37

13.293 3.19.67 7.547 11.349 3.1283

380 381 382 383 384

385 386 387 388 389

2.52.7.11 22-5-193 2.32.5,43 23.5.97 2.5.389

3851 3a.11.13 72.79 3881 3.1297

22.32.107 2.1931 2s.113 2.3.647 22.7.139

3853 3863 3.1291 11.353 17.229

2.41-47 28.3.7.23 2.13.149 22.971 2.3.11.59

3i5i?:7 58.31 3.5.7.37 5.19.41

24.241 2.1933 22.3.17.19 2.29.67 2a.487

7.19-29 3.1289 3877 132.23 32.433

2.3.643, 22.967 2.7.277 2’.3= 2.1949

17.227 53.73 32.431 3889 7.557

385 386 387 388 389

390 391 392 393 394

2a.3.5a.13 2.5.17.23 24.5.72 2.3.5.131 21.5.197

47-83 3911 3.1307 3931 7.563

2.1951 2a.3.163 2.37.53 2a.983 2.33-73

3.1301 7.13.43 3923 32.19.23 3943

26.61 2.19.103 iF.3a.109 2.7.281 28.17-29

5.11.71 38.5.29 52.157 5.787 3.5.263

2.32.7.31 22.11.89 2.13.151 25.3.41 2.1973

3907 3917 3.7.11.17 31.127 3947

22.977 2.3.653 28.491 2.11.179 22.3.7.47

3.1303 3919 3929 3-13.101 11.359

390 391 392 393 394

395 396 397 398 399

2-52.79 2a.32.5.11 2.5.397 22.5.199 2.3.5.7.19

32.439 17.233 11.19 3.1327 13.307

24.13.19 2.7-283 21.3.331 2.11.181 26.499

59.67 3.1321 29.137 7.569 3.11”

2.3.659 22.991 2.1987 2’.3.83 2.1997

22.23.43 2.3.661 2’.7.71 2.1993 22.3a.37

3.1319 3967 41.97 32.443 7.571

2.1979 27.31 2.32.13.17 22.997 2.1999

37.107 34.72 23.173 3989 3.31.43

E

381 382 383 384

2.52.71 23.5.89 2i$.;.;;197

32.389 3511 7.503 3.11.107 3541

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11.17.19 7.509 32.397 3583 3593

2.1777 22.34.11 2.1787 20.7 2.3.599

32.5.79 5.23.31 52.11.13 3.5.239 5.719

22.7.127 2.1783 28.3.149 2.11.163 22.29.31

3557 3.29.41 72.73 17.211 3.11.109

2.3.593 2’.223 2.1789 22.3.13.23 2.7.257

3559 43.83 3.1193 37.97 59.61

355 356 357 358 359

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3607 3617 32.13.31 3637 7.521

28.11.41 2.33.67 22.907 2.17.107 28.3.19

32.401 7.11.47 19.191 3.1213 41.89

360 361 362 363 364 365 366 367 368 369

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2

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2.3.23.29 22.17.59 2.2011 26.32.7 2.43.47

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4051 31.131 3.23.59 7.11.53 4091

22.1013 2.3.677 23.509 2.13.157 22.3.11.31

3.7.193 17.239 4073 3.1361 4093

3.1367 4111 13.317 35.17 41.101

2.7.293 2’.257 2.32.229 22.1033 2.19.109

11.373 32.457 7.19.31 4133 3.1381

2.2027 25.127 2.3.7.97 22.1021 2.23.89 23.3a.19 2.112.17 22.1031 2.3.13.53 2’.7.37

7.593

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4153 23.181 3.13.107 47.89 7.599

2.31.67 22.3.347 2.2087 23.523 2.32.233

2.11.191 22.3’.13 2.2111 23.232 2.3.7.101

32.467 11.383 41.103 3.17.83 4243

5

6

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32.5.89 5.11.73 52.7.23 3.5.269 5.809 5.811 3.5.271 52.163 5.19.43 32.5.7.13

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4007 3.13.103 4027 11.367 3.19.71

23.3.167 2.72.41 22.19.53 2.3.673 2’.11.23

19.211 4019 3.17.79 7.577 4049

400 401 402 403 404

23.3.132 2.19.107 22.1019 2.32.227 2’2

4057 72.83 33.151 61.67 17.241

2.2029 22.32.113 2-2039 23.7.73 2.3.683

32.11.41 13.313 4079 3.29.47 4099

405 406 407 408 409

5.821 5.823 3.53.11 5.827 5.829 3.5.277

2.2053 22.3.73 2.2063 23.11.47 2.3.691

3.372 23.179 4127 3.7.197 11.13.29

22.13.79 2.29.71 25.3.43 2.2069 22.17.61

7.587 3.1373 4129 4139 32.461

410 411 412 413 414

YZ 33.5.31 5.839

22.1039 2.2083 24.32.29 2.7.13.23 22.1049

4157 32.463 4177 53.79 3.1399

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4159 11.379 3.7.199 59.71 13.17.19

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7.601 4217 3.1409 19.223 31.137 32.11.43 17.251 7.13.47 3.1429 4297

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3.23.61 4219 4229 33.157 7.607

415 416 417 418 419 420 421

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4259 3.1423 11.389 4289 3.1433

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3.13.109 4261 4271 3.1427 7.613

22. 1063 2.2131 2’.3.89 2.2141 22.29.37

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25.7.19 2.33.79 22.1069 2.2143 23.3.179

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13.331 19.227 3.11.131 7.619 43.101

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59.73 3.1439 4327 4337 3$.7.23

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31.139 7.617 32.13.37 4339 4349

430 431 432 433 434

3.1451 4363 4373 32.487 23.191

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5.13.67 32.5.97 5G7 3.5.293

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4357 11.397 3.1459 41.107 4397

2.2179 2’.3.7.13 2.11.199 22.1097 2.3.733

3.1453 17.257 29.151 3.7.11.19 53.83 4409 32.491 43.103

435 436 437 438 439

22,52.43 2.5.431 25.33.5 2.5.433 22.5-7-31

435 436 437 438 439

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11.17.23 32.479 29.149 61.71 3‘1447 19.229 72.89 3.31.47 13.337 4391

440 441 442 443 444

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33.163 11.401 4421 3.7.211 4441

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7.17.37 3.1471 4423 11.13.31 3.1481

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5.881 5.883 3.52.59 5.887 5.7.127

2.2203 26.3.23 2.2213 22.1109 2.32.13.19

3.13.113 7.631 19.233 32.17.29 4447

23.19.29 2.472 22.33.41 2.7.317 25.139

445 446 447 448 449

242.89 22.5.223 2.3.5.149 27.5.7 2.5.449

4451 3.1487 17.263 4481 32.499

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61.73 4463 32.7.71 4483 4493

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23.557 2.7.11.29 22.3.373 2.2243 2’.281

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23.563 2.37.61 2a.3.13.29 2.2267 26.71

5.17.53 3.5-7.43 5a.181 5.907 3a.5.101

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4507 4517 32.503 13.349 4547

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33.167 4519 7.647 3.17.89 4549

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29.157 38.13% 17.269 4583 3.1531

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3.72.31 4567 23.199 3.11.139 4597

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47.97 3.1523 19.241 13.353 32.7.73

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3.5.307 5-13.71 58.37 32.5.103 5.929

2.72.47 23.577 2.32.257 22.19.61 2.23.101

17.271 35.19 7.661 4637 3.1549

2’J.32 2.2309 22-13.89 2.3.773 23.7.83

11.419 31.149 3.1543 4639 4649

460 461 462 463 464

465 466 467 468 469

2.3.52.31 2a.5.233 2.5.467 ;.53a+5i;3 . . .

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32.11.47

5.72.19 3.5.311 5a.11.17 5.937 3.5.313

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2.17.137 22.3.389 2.2339 24.293 2.34.29

3.1553

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2.13-179 23.11.53 2.3.19.41 P.1171 2.2347

7h263;g 32.521 37.127

465 466 467 468 469

470 471 472 473 474

22.5a.47 2-3-5-157 24.5.59 2.5.11.43 22.3.5.79

3.1567 7-673 4721 3-19.83 11.431

2.2351 23.19-31 2.3-787 22.7.132 2.2371

4703 3.1571 4723 4733 3a.17.31

25.3.71 2.2357 22.1181 2,32-263 28.593

5.941 5.23,41 33.52.7 5.947 5.13.73

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32.523 53.89 29.163 3.1579 47.101

22.11.107 2.7.337 23.3.197 2.23.103 22.1187

17.277 3-l 12.13 4729 7.677 3.1583

475 476 477

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72.97 11.433 3.37.43 4783 4793

2.2377 22.3.397 2.7.11-31 2’.13.23 2.3.17.47

3.5.317 5.953 52.191 3.5.11.29 5.7.137

22.29.41 2.2383 23-3.199 2.2393 22.11.109

67.71 3.7.227 17.281 4787 32.13.41

2.3.13.61 25.149 2.2389 22.32.7.19 2.2399

4759 19.251 34.59 4789 4799

476 477 478 479

480 481 482 483 484

26.3.5a 2-5.13.37 22.5.241 2-3-5.7.23 25.5.112

4801 17.283 3.1607 4831 47.103

2.7’ 22.3.401 2.2411 2s.151 2.32.269

3.1601 4813 7.13.53 33,179 29-167

22.1201 2.29.83 28.32.67 2.2417 2a.7,173

5.312 32.5.107 52.193 5.967 3.5.17.19

2.33.89 2’.7.43 2.19.127 22.3.13.31 2.2423

11.19.23 4817 3.1609 7-691 37.131

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3-7.229 61.79 11.439 3.1613 13.373

480 481 482 483 484

485 486 487 488 489

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43.113 32.541

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485 486 487 488 489

490 491 492 493 494

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4903 178 32.547 4933 4943

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32.5.109 5.983 57.197 3.5-7.47 5.23.43

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7.701 3.11.149 13.379 4937 3.17.97

22.3.409 2.2459 26.7.1 1 2.3.823 22.1237

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490 491 492 493 494

495 496 497 498 499

2.32.52.11 25.5.31 2.5.7.71 22-3.5.83 2.5.499

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28.619 2-3-827 2a.11.113 2.47-53 2’.3-13

3.13.127 7.709 4973 3.11-151 4993

2.2477 22.17.73 2.3.829 2’.7.89 2.11.227

5.991 3.5.331 52-199 5.997 35.5.37

22.3.7.59 2.13.191 2’.311 2.32.277 2a.1249

4957 4967 32.7.79 4987 19.263

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32.19.29 4969 13.383 3.1663 4999

495 496 497 498%( 499Zh

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5059 37.137 3.1693 7.727 5099

505 506 507 508 509

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22.1277 2.3.853 F.641 2.7.367 22.32.11.13

3.13.131 5119 23.223 32.571 19.271

510 511 512 513 514

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515

41.127 3.37.47 5227 5237 32.11.53

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5209 17.307 32.7.83 132.31 29.181

23.32.73 2.2633 22.1319 2.3-881 24.331

7.751 23.229 3.1759 17.311 5297

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5.1061 5.1063 3.52.71 5.11.97 5.1069

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3.29.61 13.409 7.761 32.593 5347

22.1327 2.2659 2’.32.37 2.17.157 22.7.191

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530 531 532 533 534

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11.487 3.1789 19.283 5387 3.7.257

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23.233 7-13.59 “;;a;;3 5399

535 536 537 538 539

3.1801 5413 11.17.29 3.1811 5443

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32.601 5419 61.89 3.72.37 5449

540 541 542 543 544

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2.33.101 23-683 2.7.17.23

5.1091 5.1093 3.52.73 5.1097 5.7.157

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545 546 547 548 2 549 g

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3.2801 47.179 8423 32.937 8443

22.11.191 2.7.601 23.34.13 2.4217 22.2111

3-5.557 5.7.239 53.67 3.5.13.43 5.23.73 5.41* 32.5.11.17 52.337 5.7.241 3.5.563

2.3*.467 25.263 2.11.383 22.3.19.37 2.41.103

7.1201 19.443 3.53” 11.13.59 8447

23.1051 2.3.23.61 2*.7*.43 2.4219 28.3.11

3.2803 8419 8429 3.29.97 7.17.71

2 847 848 849

2.52.13’ 2*.3*.5.47 2.5.7.11’ 2*.5.53 2.3-5-283

33.313 8461 43.197 3.11.257 7.1213

2*.2113 2.4231 23.3.353 2.4241 2*.11.193

79.107 3.7.13.31 37.229 17.499 3.19.149

2.3.1409 2’.23* 2.19.223 22.3.7.101 2.31.137

5.19.89 5.1693 3.52.113 5.1697 5.1699

23.7.151 2.3.17-83 22.13-163 2.4243 24.31.59

3.2819 8467 72.173

2.4229 22.29.73 2.33.157 23.1061 2.7.607

11.769 3*.941 61.139 13.653 3.2833

800 801 802 803 804

2e.53 2.3*.5.89 2*.5.401 2.5.11.73 23.3.5.67

3*.7.127 8011 13.617 3.2677 11.17.43

805 806 807 808 809

2.5=.7.23 2P.5.13.31 2.3.5.269 24.5.101 2.5.809 2.5.811 23.5.7.29 2.3.5.271 2*.5.11.37

83.97 3.2687 7.1153 8081 32.29.31 8101 8111 3.2707 47.173 7.1163

2.5*.163 25.3.5.17 2.5.19.43 22.5.409 2.32.5.7.13

810 811 812 813 814 815 816 817 818 819 820 821 822 823 824

2X34.52

213

&31 2.3.1399

3ziit

WC3 @ ta .* 4

8 I3

830 831 832 833 834 835 Et 838 839 840 841 842 843 844 845 846 847 ;“4;

00 8

$ zs

g g; 8

2’.53.17 2.5.23.37 23.3.5.71 2-5.853 29.5.7.61

8501 3.2837 8521 19.449 32.13.73

2.3.13-109 26.7.19 2.4261 2x.33.79 2.4271

11.773 8513 3”.947 7.23.53 8543

2a. 1063 2.3’011.43 2’.2131 2.17.251 25-3.89

35.5.7 5.13.131 52.11.31 3.5-569 5.1709

2.4253 22.2129 2.3.72.29 23.11.97 2.4273

47.181 3.17.167 8527 8537 3.7.11.37

22.3.709 2.4259 2’.13.41 2.3.1423 22.2137

67.127 7.1217 3.2843 8539 83.103

850

851 852 853 854 855 856 857 858 859

2.34.51.19 24-5.107 2.5.857 2’.3.5.11.13 2.5.859

17.503 7.1223 3.2857 8581 i la.71

2a.1069 2.3.1427 P.2143 2.7.613 24.3.179

3.2851 8563 8573 3.2861 13.661

2.7.13.47 2z.2141 2.3.1429 28.29.37 2.4297

5.29.59 3.5.571 52.73 5.17.101 32.5.191

22.3.23.31 2.4283 27.67 2.3’.53 2a.7.307

43.199 13.659 32.953 31.277 8597

2.11.389 23.32.7.17 2.4289 22.19.113 2.3.1433

3a.317 11.19.41 23.373 3.7.409 8599

855 856 857 858 859

E 862 863 864

23.52.43 2.3.5.7.41 21.5.431 2.5.863 26.365

3.47.61 79.109 37.233 32.7.137 8641

2.11.17.23 2¶.2153 2.3z.479 23.13.83 2.29.149

7.1229 33.11.29 8623 89-97 3.43.67

2%.32.239 2059.73 24.7z.1 1 2.3.1439 22.2161

5.1721 5.1723 3.53.23 5.11.157 5.7.13.19

2.13.331 23.3.359 2.19.227 22.17.127 2.3.11.131

3.19.151 7.1231 8627 3.2879 8647

25.269 2.31.139 22.3.719 2.7.617 23.23.47

3.131.17 8629 53.163 32.312

860 861 862 863 864

865 -.866 867 868 869

2.52-173 25.5.433 2.3.5.172 23.5.7.31 2.5.11.79

41.211 3.2887 13.23.29 8681 3.2897

22.3.7.103 2.61.71 25.271 2.3.1447 2’.41,53

17.509 8663 3.72.59 19.457 8693

2.4327 2a.3.19* 2.4337 22.13.167 2.3a.7.23

3.5.577 5.1733 52.347 32.5.193 5.37.47

2’.541 2.7.619 22.32.241 2.43.101 2a.1087

11.787 34.107 8677 7.17.73 3.13.223

2.32.13.37 22.11.197 2.4339 2’.3.181 2.4349

7.1237 8669 3.11.263 8689 8699

865 866 867 868 869

870 871 872 873 874

22.3.5*.29 2.5.13.67 24.5.109 2.32.5.97 22.5.19.23

7.11.113 31.281 33.17.19 8731 8741

2.19.229 23.32.1 i* 2.+.8Ci 22.37.59 2.3.31.47

32.967 8713 11.13.61 3.41.71 7.1249

2O.17 2.4357 2=.3.727 2.11.397 23.1093

5.1741 3.5.7.83 52.349 5.1747 3.5.11.53

2.3.1451 22.2179 2.4363 25.3.7.13 2.4373

23.379 3.2909 8737 8747

875 876 877

2.54.7 23.3.5.73 2.5.877 22.5.439 2.3,5.293

3.2917 8761 72.179 3.2927 59.149

2’.547 2.13.337 2*.3.17.43 2.4391 23.7.157

8753 3.23.127 31.283 8783 32.977

2.3.1459 2l.7.313 2.41.107 24.3a.61 2.4397

5.17.103 5.1753 33.52.13 5.7.251 5.1759

22.11.199 2.3l.487 23.1097 2.23.191 22.3.733

32.7.139 11.797 67.131 3.29.101 19.463

2.29.151 26.137 2.3.7.11.19 22.133 2.53.83

19.461 3.37.79 8779 11.17.47 3.7.419

26.52.11 2.5.881

13.677 32.11.89 8821 8831 3.7.421

2.33.163 22.2203 2.11.401 2l.3.23 2.4421

8803 7.1259 3.17.173 112.73 37.239

22.31.71 2.3.13.113 23.1103 2.7.631 2*.3.11.67

3.5.587 5.41.43 52.353 3.5.19.31 5.29.61

2.7.17.37 24.19.29 2.3.1471 22.47e 2.4423

8807 3.2939 7.13.97 8837 32.983

23.3.367 2.4409 2l.2207 2.32.4i+i 24.7.79

23.383 8819 34.109 8839 8849

880 881

53.167 8861 3.2957

2¶.2213 2.3.7.211 23.1109 2.4441 22.32.13.19

3.13.227 8863 19.467 33.7.47 8893

2.19.233 26.277 2.32.17.29 22.2221 2.4447

5.7.11.23 32.5-197 53.71 5.1777 3.5.593

23.33.41 2.11-13.31 22.7.317 2.3.1481 26,139

17.521 8867 3.11.269 8887 7.31.41

2.43.103 22.3.739 2.23.193 23.11.101 2.3.1483

3.2953 72.181 13.683 3.2963 11.809

885 886 887 888 889

:;‘K 3.13.229 8941

2.4451 2’.557 2.3.1487 22.7.11.29 2.17.263

2aO7 3.2971 8923 8933 3.11.271

2a.3.7.53 2.4457 22.23.97 2.3.1489 2’.13.43

5.13.137 5.1783 3.52.7.17 5.1787 5.1789

2.61.73 22.3-743 2.4463 23.1117 2.32.7.71

3.2969 37.241 79.113 33.331 23.389

2*.17.131 2.73.13 25.32.31 2.41.109 2l.2237

59,151 32.991 8929 7.1277 3.19.157

890 891 892 893 894

8951 3.29.103 8971 7.1283 35.37

23.3.373 2.4481 2e.2243 2.32.499 25.281

7.1279 8963 31.997 13,691 17.232

2.112-37 22.33.83 2.7.641 23.1123 2.3.1499 .

30.5.199 5.11.163 52.359 3.5.599 5.7.257

22.2239 2.4483 2’.3.11.17 2.4493 2*.13.173

132.53 3.72.61 47.191 11.19.43 3.2999

2.3.1493 23.19.59 2.672 2*.3.7.107 2.11.409

:T: 880 881

% 884

22.3R5.72 2.5.883 23.5.13.17

885 886 887 888 $89

;;yi5;&9 2.51887 2’.3.5.37 2.5.7.127

890 891 892 893 894

22.51.89 2.3’.5.11 23.5.223 2.5.19.47 22.3.5-149

895 896 897 i%

2.55.179 28.5.7 2.3.5.13.23 2.5.29.31 22.5.449

EXI 30.23.43

22.7.31 1 2.3.1453 23.1091 2.17.257 22.3’

853 854

8

3iE3 7.29.43 32.971 13.673 875 876

E: e.

877 878 879

rj: $

;

:ii 884

A

is d ;;

862 Table 9000

COMBINATORIAL

24.7

Factorizations

ANALYSIS 9499

950 951 952 953 954

.

22.53.19 2.3.5.317 24.5.7.17 2.5.953 22-32.5-53

3.3167 9511 9521 33.353 7.29.47

2.4751 23.29.41 2.32.232 22.2383 2.13.367

13.17.43 32.7.151 89.107 9533 3.3181

25.38.11 2.67.71 22.2381 2.3.7.227 23.1193

5.1901 5.11.173 3.52.127 5.1907 5-23.83

2.72.97 28.3.13.61 2.11.433 26.149 2.3.37.43

xg;

7.1361 3.11.172 9547

21.2377 2.4759 23.3.397 2-19.251 22.7.11-31

37.257 3.19.167 13.733 9539 32.1061

ES 952 O

2.34.59 25.13.23 2.4789 22.3.17.47 2.4799

112.79 7.1367 3.31.103 43.223 29.331

955 956 957 958 959

23.1201

2.61.79 24.32.67

3.3203 9619 9629 34-7.17 9649

960 961 962 963 964 965 966 967 968 969

953 954

955 956 957 958 959

2.5a.191 23.5.239 2.3.5.11.29 22.5.479 2.5.7.137

9551 3.3187 17.563 11.13.67 3.23.139

24.3.199 2.7.683 22.2393 2-3.1597 23.11.109

41.233 73.131 3.3191 7.372 53.181

2.17.281 22.3.797 2.4787 24.599 2.32.13.41

3.5.72.13 5.1913 P.383 33.5.71 5.19-101

22.2389 2.4783 23.32.7.19 2.4793 22.2399

19.503 32.1063 61.157 9587 3.7.457

960 961 962 962 964

27-3.52 2.5.312 22.5.13.37 2.3a.5.107 23.5.241

9601 7.1373 32.1069 9631 31.311

2.4801 22.33.89 2.17.283 25.7.43 2.3.1607

32-11.97 9613 9623 3.13a.19 9643

22.7’ 2.11.19.23 23.3.401 2.4817 22.2411

5.17.113

2.3.1601 24.601 2.4813 22.3.11.73 2.7.13-53

13.739 59.163 3.3209 23.419 11.877

965 966 967 968 969

2-52.193 22.3.5.7.23 2.5.967 24.5.112 2.3.5.17.19

3.3217 9661 19.509 3.7.461 11.881

22.19.127 2.4831 23.3.13.31 2.47.103 22.2423

72.197 3.3221 17.569 23.421 33.359

2.3.1609 26.151 2.7.691 22.32.269 2.37-131

5.1931 5.1933 32.5a.43 5.13.149 5.7.277

23.17.71 2.33-179 22.41.59 2.29.167 25.3. 101

32.29.37 7.1381 9677 3.3229 9697

2.11.439 22.2417 2.3.1613 23.7.173 2.13.373

13.743 3.11.293 9679 9689 3.53.61

970 971 972 973 974

22.5a.97 yg7; 2.5.7.;39 2a.5.487

89.109 32.13.83 9721 37.263 3.17.191

2.3a.7a.ll 24.607 2.4861 22.3.811 2.4871

31.313 11.883 3.7.463 9733 9743

23.1213 2.3.1619 2a.11.13.17 2.31.157 24.3.7.29

3.5.647 5.29.67 3.389 3.5.11-59 5.1949

2.23.211 22.7.347 2.3.1621 23.1217 2.11.443

17.571 3.41.79 71.137 7.13.107 33.192

22.3.809 2.43.113 29.19 2.32.541 22.2437

7.19.73 9719 32.23.47 9739 9749

975 976 977 978 979

2.3.53.13 25.5.61 2.5.977 22.3.5.163 2-5.11.89

72.199 43.227 3.3257 9781 9791

23.23.53 2.3.1627 22.7.349 2.67.73 20.32.17

3.3251 13.751 29.337 32.1087 7.1399

2.4877 22.2441 2.35:181 P-1223 2.59.83

5.1951 32.5.7.31 52.17.23 5.19.103 3.5.653

22.32.271 2.19.257 2’.13.47 2.3.7.233 22.31.79

11.887 9767 3.3259 9787 97.101

2.7.17.41 23-3.11.37 2.4889 22.2447 2-3.23.71

3.3253 9769 7.11.127 3.13.251 41.239

980 981 982 983 984

23.52.72 2.32.5.109 22.5.491 2.5.983 24.3.5-41

2.132.29 22.11.223 2.3.1637 23.1229 2.7.19.37

9803 3.3271 11.19.47 9833 3.17.193

22.3.19.43

!Jz 7.23.61 3.29.113 13.757

~~;‘,o: 2.3.il.149 22-23-107

5.37.53 5.13.151 3.9.131 5.7.281 5-11-179

2.4903 23.3.409 2.173 22.2459 2.32-547

3.7.467 9817 31.317 32.1093 43.229

2’.613 2.4909 22.33.7.13 2.4919 23.1231

17.577 32.1091 9829 9839 3.72.67

980 981 982 983 984

985 986 987 988 989

2.52.197 22.5.17.29 2.3.5.7.47 23.5.13.19 2.5.23.43

9851 3.19.173 9871 41.241 32.7.157

22-3.821 2.4931 2’.617 2.34.61 22.2473

59.167 7.1409 3a.1097 9883 13.761

2.131379 23.32.137 2.4937 22.7.353 2.3.17.97

33.5.73 “.iJEi13 3.5.659 5,1979

27.7.11 2.4933 22.3.823 2.4943 23.1237

9857 3.11.13.23 7.17.83 9887 3.3299

2.3.31.53 22,2467 2.11.449 25.3.103 2.72.101

9859 71.139 3.37.89 11.29.31 19.521

985 986 987 988 989

990 991 992 993 994

22.32.52.11 2.5.991 26.5.31 2.3.5.331 22.5.7.71

9901 11.17.53 3.3307 9931 9941

2.4951 23.3.7.59 2.11a.41 22.13.191 2.3.1657

3.3301 23.431 9923 3.7.11.43 61.163

24.619 2.4957 22.3.827 2.4967 23.11.113

;:;:;:; 52.397 5.1987 32.5.13.17

2.3.13.127 22.37.67 2.7.709 24.33.23 2.4973

9907 47.211 32.1103 19.523 73.29

22.2477 24;3;;9+;9

33.367 7.13.109 9929 3.3313 9949

990 991

995 996 997

23.199 23.3.5.83 2.5.997 22-5.499 2.33.5.37

3+Tm;;7

25.311 2.17.293 2=.3=.277 2.7.23.31 23.1249

37.269 35.41 9973 67.149 3.3331

2.32.7.79 22.47.53 2.4987 28.3.13 2.19.263

5&~“3’ 3.5a.7.19 5.1997 5.1999

22,19.131 2.3.11.151 23.29.43 2.4993 22.3.72.17

3.3319 9967 11.907 3.3329 13.769

2.13.383 2’.7.89 2.3.1663 22.11.227 2.4999

132.59 3a.1109 97.103

.

$5eY 5.41’.47 3.5.643

2a&r22??

2.4969 22.3.829

23.433 y5”8; 7.1427 3a.11.101

t:: 994

8

2 G

864

COMBINATORIAL

Table

Primitive

24.8

Roots,

ANALYSIS

Factorization

of p-l

g, G denote the least positive

and least negative (respectively) primitive roots of p. E denotes whether 10, -10 both or neither are primitive roots. -

P--l

P

P

P--l

--

3

2

::

2%

:; ii _ 2: 31 :: :; x1 6i 67 iA 79 ii1 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 :2 191 2; 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353

2.5 22.3 2’ 2.32 2.11 22.7 2.3.5 g.“5” 2.317 2.23 22.13 2.29 22.3.5 2.3.11 2.5.7 23.32 2.3.13 2.41 23.11

25.3 22.52 2.3.17 2.53 22.33 24.7 2.32.7 2.5.13 23.17 2.3.23 22.37 2.3.52 22.3.13 2.3’ 2.83 22.43 2.89 22.32.5 2.5.19 2O.3 22.72 2.3$.11 2.3.5.7 2.3.37 2.113 22.3.19 23.29 2.7.17 2”.3.5 2.53 2.G 22.67 2.33.5 22.3.23 23.5.7 %ir 2.3&7 2.5.31 23.3.13 22.79 2.3.5.11 2”.3.7 2.173 22.3.29 25.11

359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 i% 613 617 619 631 641 643 647 653 659 661 673 677 683 E 709 719 727 733 739 743 751 757 761 769 773 787 797 809 811

2.179 2.3.61 22.3.31 2.33.7 2.191 22.97 22.32.11 24.52 23.3.17 2.11.19 22.3.5.7 2.5.43 24.3a 2.3.73 2.13.17 26.7 g;; c 2.3.7.11 2.233 2.239 2.3& 2.5.72 % 2;. 127 23.5.13 2.32.29 22.33.5 2.3.7.13 22.139 2.281 23.71 2.3.5.19 26.32 2.293 2’.37 2.13.23 23.3.52 2.3.101 22.32.17 23.7.11 2.3.103 2.32.5.7 27.5 2.3.107 2.17.19 22.163 2.7.47 22.3.5.11 26.3.7 22.132 2.11.31 2.3.5.23

22.52.7

22.3.59 2.359 2.3.112 22.3.61 2.32.41 2.7.53 2.3.53 22.33.7 23.5.19 28.3 22.193 2.3.131 22.199 23.101 2.34.5

--

-

P

P-1

821 823 827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947 953 967 971 977 983 991 997 1009

22.5.41 2.3.137 2.7.59 22.3a.23 2.419 22.3.71 23.107 2.3.11.13 2.431 2=.3.73 2’.5.11 2.32.72 2.443 2.3.151 2.5.7.13 2.33.17 25.29 23.32.13 22.5.47 2.11.43 23.7.17

ix:: 1021 1031 1033 $21 ::x: 1063 1069 1087 1091 ::i; 1.103 1109 1117 1123 1129 1151 1153 1163 1171 1181 1187 1’193 1201 1213 1217 1223 1229 1231 1237 1249 1259 1277 1279 1283 1289 1291 1297

2i;7i273 24.61

2.491 2ip.‘31

&2.7 22.11.23 2.509 22.3.5.17 2.5.103 23.3.43 2.3.173 23.131 2.3.52.7 22.5.53 2.32.59 22.3.89 2.3.181 2.5.109 2a.3.7.13 23.137 2.19.29 22.277 22.32.31 2.3.11.17 23.3.47 2.52.23 27.32 2.7.83 2.32.5.13 22.5.59 2.593 23.149 24.3.5a 22.3.101 28.19 2.13.47 22.307 2.3.5.41 22.3.103 25.3.13 2.17.37 22.11.29 232.71 2.641 23.7.23 2.3.5.43 24.34

-. _

COMBINATORIAL

Primitive

Roots,

865

ANALYSIS

Factorization

of p-l

Table 24.8

g, G denote the least positive and least negative (respectively) primitive roots of p. E denotes whether 10, - 10 both or neither are primitive roots. -

/P

P-1

1301 1303 1307

22.52.13 2.3.7.31 2.653 2.659 2a,3.5.**

E 1327 1361 1367 :z 1399 1409 1423 1427 1429 1433 1439 1447 1451 1453 1459 1471 1481 1483 1487 1489 1493 1499 1511 1523 1531 1543 ::ti 1559 ::t: :iz 1597 1601 1607 1609 1613 1619 1621 1627 1637 1657 % 1669 1693 1697 E 1721 :E 1741 :;:3 E ::i; :~~~ 1811 1823

“i!%:’ ;:683 22.78 22.3.5.23 2.3.233 27.11 2.32.79 2.23.31 22.3.7.17 22”.::99 2.3.241 2.52.29 2a.3.112 2.:::“7s 2x.5.37

2.32.5.17 2.3.257 2a.3a.43 2’,97 2.19.41 2.3a.29 2~5.157 X’K3” 22:3:7.*9 28.52 2.11.73 2a.3.67 22.13.31 2.809 22.34.5 2~3.271 2a.40Q 23.32.23 2.3.277 2.7a.17 2a.3.139 22.32.47 25.53 2.3.283 22.7.61 23.5.43 2.3.7.41 2a.433 22.3.5.29 2.32.97 23.3.73 2.3,293 2’.3.37 2.3’.11 2.19.47 22.3.149 23.32.52 2i%181'

-

-

__ --

.-

c

P

P

P-1

2377 2381 2383 2389 2393 2399 2411 2417 2423 2437 2441 2447 2459 2467 2473 2477 2503 2521 2531 2539 2543 2549 2551

2a.33.11 22.5.7.17 2.3.397 22.3.199 23.13.23 2.11.109 2.5.241 24.151 2.7.173 22.3.7.29 23.5.61 2.1223 2.1229 2.32.137 23.3.103 2%.619 2.32.139 23.32.5.7 2.5.11.23 2.33.47 2.31.41 22.72.13 2%&y

25.&~ 7 2.1013 22.3.132 2.1019 22.33.19 2.1031 22.11.47 25.5.13 2.3.347 2.7.149 23.32.29 2.1049 2.5.211 26.3.11 24.7.19 m7&

z% 2591 2593 2609

2:1289 2.5.7.37 26.3’ 2’.163 23.3.109

24.107 2.32.7,17 23.269 24.33.5 2.32.112

%i 2707 2711 2713 2719 2729 2731 2741 2749

P-1

1831 1847 1861

_. _. _. _. _.

::;: 1873 1877 1879 1889 1901 1907 1913 1931 1933 E 1973 :z 1993 1997 1999 2003 2011 2017 2027 2029 2039 2053 2063 2069 2081 2083 2087 2089 2099 2111 2113 2129 2131 2137 2141 2143 2153 2161 2179 2203 2207 2213 2221 2237 2239 2243 2251 2267 2269 2273 2281 2287 2293 2297 2309 2311 2333 2339 2341 2347 2351 2357 2371

“~%” 2g.k 19 2.953 23.239 2.5.193 22.3.7.23 22.487 2.3.52.13 22.17.29 ;2”;:: 2S.i.83 22.499 2.33.37 2;&15l ;l

22.3.5.37 22.13.43 z%. . 2.32.53 2.11.103 ‘;‘“;I 2G5.19 2.32.127 22.3.191 23.7.41 22.577 2i3,Gi5il 2:7-167 22.32.5.13 2.3.17.23 2.52.47 22.19.31 2.3.5.79

2 2633 2647 2657 2659 2663 2671 Ez; 2687 2689

% 2777 2789 2791 2797 2801 2803 2819 2833 z 2851 2857 2861 2879 2887 2897 2903 2909

El7 . . 2i% 2.31443 2.113 2.3.5.89 22.3.223 2.32.149 2g.‘;;9 22.673 2.19.71 2.3.11.41 2.5.271 23.3.113 2.32.151 2a.ii.31 2.3.5.7.13 22.5.137 22.3.229 26.43 YifE 22.i7.41 2.32.5.31 22.3.233 24.52.7 2.3.467 2.1409 2’.3.59 22.709 2.7a.29 2.3.5a.19 23.3.7.17 22.5.11.13 2.1439 2.3.13.37 24.181 2.1451 2a.727

-

-

.-

_-

COMBINATORIAL

866 Table

Primitive

24.8

Roots,

ANALYSIS

Factorization

of p-1

g, G denote the least positive and least negative (respectively) primitive roots of p. c denotes whether 10, -10 both or neither are primitive roots. -

P 2917 2927 2939 2953 2957 2963 %Y 2999

3001 3011 3019 3023 3037 3041 3049 3061 3067 3079 3083 3089 3109

P-1 22.38 2.7.11.19 2.13.113 23.32.41 22.739 2.1481 23.7.53 2.33.5.11 2.1499 23.3.53 2.5.7.43 ;.gy 22.3.11.23 25.5.19 23.3.127 22.32.5.17 2.3.7.73 2.34.19 2.23.67 24.193 2; :;;$7

fE

2h&l3

E3’ 3167 3169 3181 3187 3191

2;jig

fEi 3217 3221

3301 330;

E’ 332: E 3342 334i

E 3371 337: 338! ;g! 341; 343: 344: 345; 3461 346: 346; E:I E1 351:

25.32.11 22.3.5.53 2.33.59 2.5.11.29 2.1601 23.401 24.3.67 22.5.7.23 22.3.269 2.53.13 22.3.271 23.11.37 2.32.181 2.3.5.109 2.17.97 22.3.52.11 2$l&T;9 2.3.7.79 2.11.151 28.13 s$.&37 2.71239 2.23.73 25.3.5.7 2.5.337 22.3.281 22.7.112 2.3.5.113 2.13.131 22.853 23.3.11.13 23.431 27.33 22.5.173 2.3.577 2.1733 22.3. 172 2.5.349 2.3.11.53 2.33.5.13 22.3.293

._. _.

P

3527 3529 3533 3539 3541 3547 3557 3559 3571 3581 3583 3593 3607 3613 3617 3623 3631 3637 3643 3659 3671 3673 3677 3691 %: 3709 3719 E 3739 3761 3767 3769 3779 3793 3797 3803 3821 3823 3833 3847 3851 3853 3863 3877 3881 3889 3907 3911 3917 3919 3923 3929 3931 3943 3947 3989 4001 4003 4007 4013 4019 4021 4027 4049 4051 4057 4073

P-1 2.41.43 23.33.73

22.883 2.29.61 22.3.5.59 2.32.197 22.7.127 2.3.593 2.3.5.7.17 22.5.179 2.32.199 23.449 2.3.601 22.3.7.43 25.113 2.1811 2.3.5.112 22.32.101 ;:;;“g 2.5.367 23.33.17 22.919 2.32.5.41 2.432 22.52.37 22.32.103 Ei1 22:3:311 2.3.7.89 2’.5.47 2.7.269 23.3.157 2.1889 24.3.79 22.13.73 2.1901 22.5.191 “;$.I’;;” 2.3.641 2.52.7.11 22.32.107 2.1931 21.3.17.19 23.5.97 24.36 2.32.7.31 2.5.17.23 2;“;‘% 2:3i53 23,491 2.3.5.131 2.33.73 2.1973 2.3.661 22.997 25.53 2&-$L;9 22:17.59 2.72.41 22.3.5.67 2$p6; 4 2.3’.5* 23.3.132 23.509

-

-

._

--

-

P-1

0

10 ---_ .---10 .--__ -10 .--__ -10

4079

4091 4093 4099 4111 4127 4129 4133 4139

l :: 9% .--__ 4159 f10

10 .--__ f10

4x ----_ -10 10 ---__ f10 ---__ ---__ ----f10 *lo -10 10 ---__ -------__ 10 ----_ 10 -------__ -10 f10 ----_ f10 10 --------_ 10 -------__ ----_ -10 -10 --_---_--10 -____ --_--:i *:i --___ --_-10 --_-_ 10 ----_ -10 --___ *:: f 10

4177 4201 4211 4217 4219 4229 4231 4241 4243 4253 4259 4261 4271 4273 tX8 4297 4327 t3;; tid; 4363 z 4397 4409 4421 2::

2.2039 2.5.409 22.3.11.31 2.3.683 2.3.5.137 2.2063 26.3.43 22.1033 2.2069 23.3.173 22.1039 2.33.7.11 24.32.29 ;;“.5;; 2k.7.31 2.3.19.37 22.7.151

22.3.5.71

2f;;;3 2i.241 22.1087 22.3*.112 2.3.727 22.1093 2.5.439 22.7.157 23.19.29 22.5.13.17 ;3;1;.;; 2.;2.;3.19

2 t352?

22 4481 a::: % ii:; 4523 4547 4549 4561 4567 4583 4591 4597 4603 4621 4637 4639 4648 4649 4651 4657 4663

2.23.97 27.5.7 2.3a.83 21.1123 2.3.751 2’.3,47 21.1129 2.32.251 2.7.17.19 2.2273 22.3.379 24.3.5.19 2.3.761 2.29.79 2.38.5.17 22.3.383 2.3.13.59 25&5;7.;1 2.3:77’3

c

.-10 10 _..___ 28

10

_..___ -._--*::

-._---._-__ &lo

-.____

l :“o 10

ZtlO -10 _.----10 _.-_--

*:: -10 -,----10 -_-____ *:: *:: ----10 -_--10 -_--_-_ *ix -___ :x f 10 10 -___ -___ -___ -------_-----10 -10 -_--_-:oo -10 ----10 ----_-1:; -_-10 -_e----

COMBINATORIAL

Primitive

Roots,

867

ANALYSIS

Factorization

of p-1

Table

g, G denote the least positive and least negative (respectively) primitive roots of p. e denotes whether 10, -10 both or neither are primitive roots. -

P

P-1

4673 4679 4691 4703 4721 4723 4729 4733 4751 4759 4783 4787 4789 4793 4799 4801 4813 4817 4831 4861 4871 4877 4889 4903 4909 4919 4931 4933 4937 4943 4951 4957 4967 4969 4973 4987 4993

26.73 2.2339 2.5.7.67 2.2351 24.5.59 2.3.787 23.3.197 22.7.132 2.8.19 2.3.13.61

:i:z 5009 5011 5021 5023 5039 5051 5059 5077 5081 E; 5101 5107 5113 5119 “5% 5167 5171 5179 5189 5197 5209 5227 5231 523: 5237 5261 ;;g 528i

ZE 2G.7.19 23.599 2.2399 26.3.5a 22.3.401 2’.7.43 2.3.5.7.23 22.35.5 2.5.487 2a.23.53 23.13.47 2i3f34; i.2459 2.5.17.29 22.32.137 23.617 2.7.353 2.32.52.11 22.3.7.59 2.13.191 23.33.23 22.11.113 2.3=.277 2’.3.13 2.3.72.17 2.41.61 24.313 2.3.5.167 22.5.251 2.3’.31 2.11.229 2.5a.101 2.32.281 2a.33.47 23.5.127 2.2543 2.2549 22.3.52.17 2.3.23.37 23.32.71 2.3.853 2.31.83 25.7.23 2.3a.7.41 2.5.11.47 2.3.863 22.1297 22.3.433 23.3.7.31 2.3.13.67 2.5.523 24.3.109 2a.7.11.17 28.5.263 23.659 yi1;.;; . .

.-

_-

_.

P

P-1

P

P-1

5297 5303 5309 5323 5333 5347 5351 5381 5387 5393 5399 5407 5413 5417 5419 5431 5437 5441 5443 5449 5471 5477 5479 5483 5501 5503 5507 5519 5521 5527 5531 5557 5563 5569 5573 5581 5591 5623 5639 5641 5647 5651 5653 5657 5659 5669 5683 5689 5693 5701 5711 5717 5737 5741 5743 5749 5779 5783 5791 5801 5807 5813 5821 5827 5839 5843 5849 5851 5857 5861

2’.331 2.11.241 22. 1327 2.3.887 2a.31.43 2.35.11 2.52.107 22.5.269 2.2693 2’.337 2.2699 2.3.17.53 22.3.11.41 23.677 2.3=.‘,.43 2.3.5.181 2”.32.151 26.5.17 2.3.907 23.3.227 2.5.547 22.372 2.3.11.83 2.2741 22.53.11 2.3.7.131 2.2753 2.31.89 2’.3.5.23 2.32.307 2.5.7.79 22.3.463 2.38.103 26.3.29 22.7.199 22.32.5.31 2.5.13.43 2.3.937 2.2819 23.3.5.47 2.3.941 2.5a.113 2a.32.157 23.7.101 2.3.23.41 22.13.109 2.3.947 28.31.79 22.1423 2a.3.52.19 2.5.571 21.1429 23.3.239 22.5.7.41 2.32.11.29 22.3.479 2.33.107 2.72.59 2.3.5.193 23.52.29 2.2903 22.1453 22.3.5.97 2.3.971 2.3.7.139 2.23.127 23.17.43 2.3W.13 26.3.61 P.5.293

5867

2.7.419 22.32.163 2.2939 23.3.5.72 23.11.67 2.3.227 2.3a.7.47 2.2963 2.2969 20.3.31 22.5.13.23 2.41.73 2.3.7.11.13 2.5.601 22.11.137 22.3.503 2.3.19.53 2.3023 21.17.89 2.32.337 y&3

5879 5881 5897 5903 5923 5927 5939 5953 5981 5987 6007 6011 6029 6037 6043 6047 6053 6067 6073 6079 6089 6091 6101 6113 6121 6131 6133 6143 6151 6163 6173 6197 6199 6203 6211 6217 6221 6229 6247 6257 6263 6269 6271 6277 6287 6299 6301 6311 6317 6323 6329 6337 6343 6353 E! 6367 6373 6379 6389 6397 6421 6427 6449 6451 EZ 6481

&761 2.3.5.7.29 22.52.61 25.191 23.32.5.17 2.5.613 2;.;%7; 2.3.5a.41 2;p;9 2a:1549 Y14Y~

2.;:&

2;3;;‘60: 2.315.11.19 22.3.523 z%

22:&.7 2.5.631 22.1579 2.29.109 23.7.113 26.32.11 2.3.7.151 24.397 2.11.17a 23.3.5.53

22.3.5.107 2.38.7.17 2’.13.31 2.3,52.43 22.3.7a.11 2a.809 24.34.5

24.8

COMBINATORIAL

868 Table

Primitive

24.8

Roots,

ANALYSIS

Factorization

of p-l

g, G denote the least r>ositive and least negative (respectively) primitive roots of p. e denotes ihether 10, - 10 botkor neither are primitive roots. - - - P

P--l

6491 6521 6529 6547 6551 6553 6563

2.5.11.59 23.5.163 27.3.17

ZE 6577 6581 6599 6607 6619 6637 6653 6659 6661 6673 6679 6689 6691 6701 6703 6709 6719 6733 6737 6761 FY 67;i 6791 6793 6803 6823 6827 % 68;; 6855

6911 691: 694: 66E! SQS’ 6%: 697 697: 698: E; 700 701: 7011 g; 7041 705 706 707 710 710

e

-__ __ _.

“;“;ltg 2*:3*.7:13 2.17.193 2a.821 2i;1;73 **.5.7.47 2.3299 2.3*.367 2.3.1103 22.3.?.79 22.1663 2.3329 2*.3*.5.37 2’.3.139 2.3*.7.53 25.11.19 2.3.5.223 22.52.67 2.3.1117 2*.3.13.43 2.3359 2*.3*.11.17 2’.421 23.5.13” 2i3iJ$;3

7

22.3.5.113 2.5.7.97 23.3.283 2.19.179 2.3l.379 2.3413 21.3.569 2’.7.61 23.32.5. 19 2*.857 2.47.73 21.17.101 2.3.5.229 2.3.31.37 2.3449 2.3.1151 2.5.691 2*.7.13.19 2.23.151 2*.3*.193 2.7*.71 24.3.5.21) 2.3’.43 2.5.17.41 20.109 2.3491 2.3.5.233 2”.3.11.53 23.58.7 22.1753 2.112.29 2.3.1171 2.32.17.23 2.7.503

2’.3*.7*

2*.3.19.31 2.3539 2.53.67 21.1777

_. _. _.

P

P--l

-7121 7127 7129 7151 7159 7177 7187 7193 7207 7211 7213 7219 7229 7237 7243 7247 7253 7283 7297 7307 7309 7321

z 7369 E 7417 7433 7451 7457 E8’ r 7481 7485 748% 749% 750i 751i 752: E( 75ki 754; 754! ;;g’ 757: 757: 758: E 760: 760: 762 763! 764: 764! 766! 767: 768 768 769 769’ E GE

,

e

2’.5.89 2.7.509 23.3’. 11 2;5;.;;;;

3

2a:3:13.i3 2.3593 2*.29.31 2.3.1201 y$l~:”

z 2

2.3*:401 22.13.139 2*.3*.67 2.3.17.71 2.3623 2*.7*.37 2.11.331 27.3.19 2.13.281 2*.3*.7.29 2a.3.5.61 2.5733 2P.3.13.47 2*.11.167 2.3.5*.7* 2a.3.307 25.3.7.11 2;3.$lg”

%

r

-- -

23.929’ 2.52.149 25.233 2.3.11.113 2*.3.7.89 28.5.11.17 2.19.197 2R.3*. 13 2.23.163 2.3*.139 2*.1879 2.3761 28.941 2’.3.157 22.5.13.29 2.73.11 22.3.17.37 2.3779 28.38.5.7 2*.3.631 23.947 2.17.223 2*.7.271 2.3.5.11.23 2.3.7.181 2.3803 22.3.5.127 2.3.19.67 2.3821 2*.239 2*.3*.71 2’.7.137 29.3.5 2.31.7.61 2.5.769 2.3.1283 2.3851 2P.3.643 2.33.11.13 2.3863

: 7

3” ; 2 2 2 2 5 2 2 ; f 2 6 : 7 5 2 5 i: 3 2 it :: ; 2 ; i 2 1; 13

-- __--__---_____ -10 3cl0 -10 i t 10 10 __--_____ t:“o

__--_ -10 10

-_ -10 __---

-10 k 10 -----

-__ __ &lo __.--_ --,.-_

&lo 10 .___

-_

*:o” *lo 10 ____ ____ 10 .--_ 10 -10 _---10 _--_--f 10 -1a _--_ -1a ---_-__

-. 1 _. _.

i il 6 2 5 : 2 3 2 3 15 f 2 2 E

*:: ----1c ---1c -----I( -l( __-_ -_-fl( -_-1E :I l( -_-_ ---_ l(

P

P--l

7741 7753 7757 7759 7789 7793 7817 7823 7829 7841 7853 7867 7873 7877 7879

2*.3*.5.43 23.3.17.19 2*.7,277 2.32.431 22.3.11.59 24.487 28.977 2.3911 22.19.103

2.5.5.72

22.13.151 2.34.19.23 28.3.41 21.11.179 2.3.13.101 2.7.563 2*.5*.79 2v59.67 2.37.107 asp;;

5::: 7;507 7919 7927 7933 7937 7949 7951 7963 7993 8009 8011 8017 8039 8053 8059 8069 8081 8087 8089 8093 8101 8111 8117 8123 81.47 8161 8167 8171

‘$.& 2’. 1987 2.3.5*..53

2..;],3$7

2*.;.1’i.13 2.3*.5.89 24.3.167 2.4019 22.3.11.61 2.3.17.79 21.2017 2’.5.101 gyl; .I .. * 22.7.172

2K34.54

2.5.811 22.2029 “;31i;;l 25-3.5.17 2.3.1361 2.5.19.43 2.3.29.47 2.3”.5.7.13 2’.3a. IQ 2.7.587 2l.3.5.137 2~5.823 23.3.73 21.29.71 2.13.317 2.36.17 2*.3.13.53 2’.11.47

E “,3;; 8223 8231 823: 8235 824: 826: 826E 82;: 8285 8291 829:

8295 8311 831; Ed 836i E 838: 838!

,

2;Y&Y 2Go49 22.2099 2.3.5,277 21.33.7. I 1 28.3.347 25.32.29 2.37.113 2’..523 2*.3.349 2.7.599 2*.3*.233

G - --

6

---f 10 __-_

; 2 2 2 3 3

-10 f10 f 10

: f E 2 2 3 2 3 2 7 2 3 ; LOc 3” 7 5 2 2 4 I 17 2 6 x 3 3 7 1

*:o” ---__--10 f10 -. --10 -10 f 10 -10 -10 10 ---f10 f 10 -10 -10 -_----____ f 10 -10 ---10 *to .---10 .---.-_-.-_-.-__,__--10 -10 .-__: .-__-

: :i 1’:

-.-__-

.-_--

; I

;

IaI I i1 2 2,

1

: 31 2! 3, 21 fi 7 ci 5 2t I 3[ fi

10 .----10 f 10 .-__-10 *:o” f10 10 10 ----rt: 10 -10 ----_-___ f 10 -10 _---f 10 ----f10

COMBINATORIAL

Primitive

Roots,

869

ANALYSIS

Factorization

of p-1

Table

g, G denote the least positive

and least negative (respectively) primitive roots of p. E denotes whether 10, -10 both or neither are primitive roots.

T--

P 8419 8423 8429 8431 8443 8447 8461 8467 8501 8513 8521 8527 8537 8539 8543 8563 8573 8581 8597 8599 8609 8623 8627 8629 8641 8647 8663 8669 8677 8681 8689 2:: 8707 8713 8719 8731 8737 8741 8747 8753 8761 8779 8783 8803 8807 8819 8821 8831 8837 8839 8849 8861 ii:: 8887 8893 8923 8929 8933

P-1 2.3.23.61 2.4211 2”.7=.43 2.3.5.281 2.32.7.67 2.41.103 2=.3’.5.47 2.3.17.83 2*.5’.17 26.7.19 2*.3.5.71 2.3.7*.29 28.11.97 2.3.1423 2.4271 2.3.1427 22.2143 2*.3.5.11.13 2’.7.307 2.3.1433 25.269 2.32.479 m ; lEi 88.j3.5 2.3.11.131 2.61.71 2*.ii.i97 2*.3*.241 2’.5.7.31 2’.3.181 22.41.53 2.4349 2.3.1451 2’.3*.11* 2.3.1453 2.3s.5.97 25.3.7.13 22.5.19.23 2.4373 24.547 2a.3.5.73 2.3.7.11.19 2.4391 2.3’. 163 2.7.17.37 2.4409 -2.5.883 2*.47* . . %“;“,’ 2*.5.443 2.3.7.211 2.11.13.31 2.3.1481 2*.3*.13.19 2.3.1487 25.3S.31 21.7.11.29

-

-

.-

P

__-.i”o -10 -10

10 _-_-10 f 10 f10 __-_--*10 __-_ -:“o ____ _______--____ 2: __-_ -_-10 10 f10 __----------10 -10 ItlO -10 10 __-_ f10 - 10 f10 _-----10 __-. :8 f10 -10 __-_ -10 __-f10 -:: 10 ____ ------____

--

P-1

8941 8951 8963 8969 8971 8999 9001 9007 9011 9013 9029 9041 9043 9049 9059 9067 9091 9103 9109 9127 9133 9137 9151 9157 9161 9173 9181 9187 9199 9203 9209 9221 9227 9239 9241 9257 9277 9281 9283 9293 9311 9319 9323 9337 9341 9343 9349

22.3.5.149 2.5s.179 2.4481 25.19.59 2.3.5.13.23 2.11.409 28.32.5” ;.p:;3

37’: 9391 9397 9403 9413 9419 9421 9431 9433 9437 9439 9461

25.293 2;&&3;3

25.i.751 22.37.61 24.5.113 2.3.11.137 28.3.13.29 2.7.647 2.3.1511 2.31.5.101 2.3.37.41 2*.3*.11.23 2.33.132 2*.3.761 2’.571 2.3.5%.61 2*.3.7.109 23.5.229 22.2293 22.33.5.17 2.3.1531 2.3S.7.73 2.43.107 2a.1151 2%.5.461 2.7.659 2.31.149 2a.3.5.7.11 2a.13.89 2a.3.775 26.5.29 2.3.7.13.17 2*.23.101 2.5.72.19 2.3.1553 2.59.79 25.3.389 22.5.467 2.3”.173 2;3&3$1

2i1.567 22.13.181 2.17.277 2*.3.5.157 2.5.23.41 2’.3*.131 2a.7.337 2.3.11’.13 22.5.11.43

-

Q

-

e

.-,

P

P-1 _-

9463 9467 9473 9479 9491 9497 9511 9521 9533 9539 9547 9551 9587 9601 9613 9619 9623 9629 9631 9643 9649 9661 9677 9679 9689 9697 9719 9721 9733 9739 9743 Ki:: 9769 9781 9787, 9791 9803 9811 9817

2.3.19.83 2.4733 28.37 2.7.677 2.5.13.73 2*. 1187 2.3.5.317 2’.5.7.17 22.2383 2.19.251 2.3.37.43 2.52.191 2.4793 21.3.52 22.33.89 2.3.7.229 2.17.283 2*.29.83 2.3*.5.107 2.3.1607 24.32.67 22.3.5.7.23 22.41.59 2.3.1613 28.7.173 2”.3.101 2.43.113 23.3”.5 2*.3.811 2.3l.541 2.4871 28.2437 2.19.257 23.3.11.37 ;;3;5g L

2:5:1;.89 2.13*.29 2.32.5.169 28.3.409 2*.3a.7,13 2’. 1229 2.4919 2.52.197 27.7.11 2r3.31.53 2.3.5.7.47 2.3’.61 2.4943 21.31.5s. 11 2.3.13.127 2.11*.41 2a.17.73 2.3.5.331 2*.5.7.71 22.3.829 2.3.11.151 2%,3=.277

t% 9839 9851 9837 9859 9871 9883 9887 9901 9907 9923 9929 9931 9941 9949 9967 9973

-

24.8

870

COMBINATORIAL

Talk

24.9 0 1

1

2

547

2

3

4

5

1229

1991

2749

,581

0

i

8

9

1”

11

4421

528,

6143

,001

7927

883,

ANALYSIS PRI#ES 12 13 9739

10663

14

15

10 16411 16417 16421 1642, 16433

2" 1,393 1,401 1741, 1,419 1,431

21 18329 18341 18353 1836, 18371

22 19427 19429 19433 19441 19447

23 20359 20369 20389 20393 20399

24 21391 2139, 21401 2140, 21419

15461 16447 15473 16453 1546, 16451

1,443 1,449 1,467

18379 18397 18401

19457 19463 19469

20407 20411 20431

21433 21481 2146,

1177, 12637 11743 12641 13613 13619 14591 14593 15493 15497 ,647, 16481

1,477 1,471

1842, 18413

1947, 19471

20443 2044,

2148, 21491

11677 12569

:

:

563 557

1231 123,

1999 1997

2753 2767

359, ,583

4441 4423

530, 529,

616, 6151

,019 ,013

,9,, 79,)

8849 8839

9749 9,~)

10687 11681 10667 11689 12577 12583

2

1:

569 571

1249 1259

2011 2003

2789 277,

3613 360,

4451 444,

5323 5309

619, 61,)

,039 ,027

,951 ,949

006, 886,

9769 976,

10709 11699 10691 11701

6 ;

::

57, 593 587

1277 1283 1279

201, 2027 2029

2791 2801 2797

361, 3631 3623

445, 4481 4463

5333 5351 534,

6199 6211 620,

,043 ,069 ,057

,963 8009 ,993

8867 8887 8893

9781 9791 978,

10711 1171, 12611 10729 11731 10723 11719 12613 12619

1;:

:;29

601 599

1291 1284

2039 2053

2819 2803

,637 ,643

4493 4483

538, 5381

621, 6221

710, ,079

8011 801,

8929 8923

9811 9803

1"7,9 10733

12601 12589

16 13513 13523 13537 13553 13567

17 14533 14537 14543 14549 14551

13577 1455, 1359, 14561 13591 14563

1.3 15413 15427 15439 15443 15451

:: 13

:: ::

607 613 617

1297 1,Ol 1303

206, 2069 2"81

2037 2833 2843

3671 3659 3673

4513 4507 451,

5399 5393 5407

624, 6229 625,

,109 ,121 712,

8053 8039 8059

8941 8939 8951

981, 9829 9833

10771 1178, 10753 11779 12647 12653 10781 11789 12659

13633 14621 1,627 1462, 15511 1552, 1648, 16493 13649 14629 15541 16519

1,483 17489 17491

18439 ,843, 18443

19489 19483 19501

20479 20477 20483

21499 21493 21503

:z

4,

631 619

130, 1319

2087 2083

2851 2857

,691 ,677

4519 4523

5413 541,

6263 bZ69

,129 7151

808, 8069

8969 8963

9851 9839

10799 10789

11807 12689 11801 12671 13669 13679 14639 14633 15551 15559 1654, 16529

17509 1,497

18457 18451

19531 19507

20509 2050,

21521 2151,

11813 12697

16

53

641

1321

2089

2861

369,

454,

5419

6271

,159

8087

8971

9857

10831

:;

59

647 643

,361 1327

2111 2099

2807 2879

,709 3701

m4549

543, 5431

628, 627,

,187 ,177

8093 8089

9001 8999

9859 9871

10837 1084,

:;:

t:71

659 653

1373 13b7

2129 2113

2903 289,

3727 3719

4583 456,

5443 5441

6301 6299

,207 ,193

8101 8111

900, 9011

988, 9883

11821 12703 1182, 12713 11833 12721 12739 10853 11831 10859

15569 16553

1,519

18461

19541

20521

21523

13691 1465, 1,687 14669 15583 15581 16561 1656, 13693 14683 13697 14699 15601 1560, lb603 16573

13681 14653

1,539 17551 1,573 17569

18493 18481 18517 18503

,954, 19553 19559 1957,

20533 20543 20551 20549

21529 2155, 21563 21559

:: ::

:; ,"z

673 661 683 67,

1399 1381 1409 1423

213, 2131 2141 2143

2909 291, 292, 2939

,739 373, 3761 3767

459, 4591 4603 4621

5471 5449 5479 547,

631, 6311 6323 6329

,213 ,211 7219 ,229

8123 8117 814, 8161

9013 9029 9041 9043

9901 9907 992, 9929

10867 11839 10861 11863 12743 1275, 10889 11887 10883 11867 12763 12781

13709 14713 13711 14717 13721 14731 13723 14723

15619 16619 15629 16607 15641 16631 15643 16633

1,581 1,579 1,599 1,597

1852> 18521 18541 18539

1958, 1957, 1959, 19603

20593 20563 20599 20611

2157, 21569 21589 2158,

25

9,

691

142,

2153

295,

,769

463,

548,

633,

,237

816,

9049

99.31

10891

13729

1564,

16649

17609

18553

19609

2062,

21599

:; 28

103 101 10,

709 701 719'

1433 1429 1439

2179 2,bl 2203

295, 2963 2969

3793 ,779 ,797

4643 4639 4649

550, 5501 550,

635, 634, 6359

,243 ,247 ,253

8179 8171 8191

9059 906, 9091

9949 9941 996,

10909 11903 10903 11909 10937 11923

:,'

113 109

733 72,

1451 144,

2213 220,

2999 2971

3821 ,803

4651 465,

5521 5519

636, 6361

729, ,283

8219 8209

9109 10007 9103 9973

10949 10939

::,

131 12'

743 739

1459 1453

2221 223,

3011 3001

3823 3833

4673 4663

55,l 552,

6379 637,

,309 ,307

8221 8231

9133 10009 912, 1003,

::

35

139 13, 149

751 757 761

1471 ,481 1483

2243 2239 225,

,019 3023 303,

3851 384, 3853

4691 4679 4703

555, 556, 5569

639, 6389 6421

,331 ,321 ,333

8213 8237 8243

913, lo",9 9151 1006, 915, ,006,

:;

151 15,

773 769

1489 148,

2269 226,

,049 ,041

387, 3863

4723 4721

5581 5573

6449 642,

,,5, ,349

8269 8263

9173 10079 9161 10069

11027 11003

15

173 lb, 163

809 79, 78,

1511 1499 1493

228, 2273

,079 ,067 3061

3889 3881 3907

4751 473, 4729

5639 5623 559,

6473 6451 6469

7411 ,393 7369

828, 827, 8291

9199 10091 918, 9181 10099 10093

11059 1105, 11047

11897 12791

1473,

12809 12799 12821

1375, 14741 13751 1474, 13759 14753

15661 16651 15649 16657 15667 16661

1,627 1,623 1,657

18587 18583 18591

19681 10661 19687

20641 20639 20663

21611 21601 2161,

11933 12823 1192, 12829

13781 14759 1,763 1476,

15671 16691 15679 16673

1,669 1,659

18637 18617

19699 1969,

20693 2068,

2164, 21617

10973 11939 10957 11941 12841 12853

13799 14771 13789 14779 15683 ,572, 16693 lb699

10979 11959 10987 1,953 1099, 11969

1,807 1478, 13829 1479, 13831 14813

1,681 1,683

18671 18661

,971, 19709

2071, 20707

21661 21649

15731 15733 1573,

16703 16729 16741

1,713 1,707 1,729

18679 18691 18701

19727 19,,9 19751

20731 20719 2074,

21613 21683 21701

11981 ,290, 11971 12911

13859 1482, 13841 14821 15749 15739

16747 16759

1,747 1,737

18719 18713

19759 19753

20749 20747

2172, 21713

12011 1291, 1200, 1198, 12923 12919

1,879 1,877 1,873

,484, 14831 14851 15761 15773 16787 1576, 16811 16763

1,783 1,761 1,749

18749 18731 18743

1979, 19763 19777

20771 20759 20753

21751 21739 2173,

12893 12889 12849

::

181 179

821 811'

1523 1531

229, 2293

3083 3089

391, 3911

4783 4759

5641 564,

6491 6481

741, ,433

829, 8293

9209 10111 9203 10103

11071 12041 11069 1203,

15787 15791

16829 16823

1,791 1,789

18773 18757

19813 19801

20789 20773

21767 21757

:: 45

191 193 19,

823 82, 829

1543 1549 1553

2309 2311 2333

3109 3119 3121

3919 ,923 ,929

4787 4789 4793

5651 5653 565,

6521 6529 654,

7451 ,457 ,459

8311 831, 8329

9221 10133 922, 10139 9239 10141

11083 12043 11187 12049 12959 1296, 11093 12071 12973

12941 12953

13903 14879 13907 14887 15797 15803 13913 14891 15809

13883 14869 13901 1486,

16831 16843 16871

1,807 1782, 1,837

1878, 18793 18797

19819 19841 19843

20807 20809 20849

21773 21787 21799

46 2

199 21, 223

839 853 85,

1559 156, 1571

2,39 2341 2347

,l',, 3163 3167

3931 3943 394,

4199 4801 4813

5659 5669 5683

6551 6563 6553

747, ,481 ,487

8353 8363 8369

924, 10151 9277 10159 925, 10163

11113 1111, 11119

12073 12979 12097 12983 12101 13001

13921 13931 13933

1489, 1581, 14923 15823 14929 15859

16879 16883 16889

1,839 1,863 17851

18803 18859 18839

19853 1986, 19861

2085, 20873 20879

21803 21821 21817

:z

229 22,

863 859

1583 1579

2357 2351

3181 3169

396, 3989

4831 481,

5693 5689

6571 6569

7489 ,499

8387 O,,,

9283 1017, 9281 10169

11149 1210, 11131 12109 13003 13007

1396, 13963

1494, 1587, 14939 15881

16901 16903

1,891 1,881

18899 18869

19891 19889

2089, 2088,

21839 21841

51

233

87,

1597

2371

3187

4001

4861

5701

6577

7507

8389

9293 lo181

11159

12113 13009

13997

14951 1588,

16921

17903

18911

19913

20899

21851

2:

239 241

883 881

1607 16Oi

2381 2377

3203 3191

400, 4003

4871 4877

571, 5711

6599 6581

,523 ,517

8423 8419

9319 10193 9311 10211

11171 11161

12119 13037 12143 13033

14009 13999

14969 15901 14957 15889

1692, 16931

1,909 17911

1891, 18913

1992, 19919

20921 20903

21863 21859

2,"

257 251

887 90,

1613 1609

2389 2383

321, 3209

4019 4013

4903 4884

5741 573,

6619 6607

,537 ,529

8431 8429

933, 10223 9323 10243

11177 12149 11173 1215,

2

269 263

911 919

,621 1619

2399 2393

3221 3229

402, 4021

4919 4909

5749 5743

665, 663,

,547 ,541

844, 8443

934, 10247 9341 10253

11213 lU9,

13049 13043

14011 14029

15013 15907 14983 15913

16943 16937

17923 17921

18947 18919

1993, 19949

20939 20929

21881 21871

12163 13063 12161 13093

14051 14033

150,l 1501,

16963 16979

1,939 1,929

18973 18959

19961 19963

20947 20959

21893 21911

15923 15919

ii

27, 271 281

94, 929 93,

1657 162, lb,,

242, 2411 241,

325, 3253 3251

4057 4051 4049

493, 4931 4933

5791 5779 5783

667, 6661 6659

,561 ,559 ,549

846, 8461 8501

937, 10259 9349 10271 1026,

11251 11243 11239

12211 13099 12203 1219, 1,109 13103

14081 1505, 14071 14057 15061 15073 1593, 15971 15959

16993 16987 16981

1,971 1,957 1,959

19001 18979 19009

19991 19979 19973

20983 20981 20963

21943 21929 21937

2:

293 283

953 947

1667 ,663

2441 243,

927, 9259

4079 4073

4951 4943

580, 5801

6689 6679

,577 ,573

8521 8513

939, 10289 9391 10273

,126l 1125,

12239 1222,

14083 14087 ,410, 14143 14149

1507, 15083 15091 15101 15107

15973 15991 lb"01 16007 lb033

1,011 1,021 1,027 17029 1,033

17977 17981 1798, 17989 18013

19013 19031 19037 19051 19069

19993 1999, 20011 20021 20023

21001 21011 21013 21017 21019

21961 2197, 21991 21997 22003

13121 13127

,":

311 30,

971 967

1693 1669

244, 2459

3299 3301

4093 4091

496, 495,

5821 5813

6701 6691

,589 7583

8537 8527

9413 9403 10301 10303

11279 11273

12251 13151 12241 13147

65

313

977

1697

2467

3307

4099

4969

582,

6703

,591

8539

9419 10313

11287

12253 13159

2: tt

331 31, 337 34,

991 983 1009 997

1709 1699 1721 1723

2473 247, 2521 2503

1319 3313 3323 3329

412, 4111 4129 4133

498, 4973 4999 4993

5839 5843 5851 5849

6719 6709 6737 6733

7607 7603 ,639 ,621

8563 8543 8581 857)

9431 10321 9421 10331 9433 943, 10333 10337

7"

349

1013

1733

2531

3331

4139

5003

5857

6761

,643

859,

9439 10343

14153 11311 11299 12269 12263 13163 13171 14159 1131, 12281 11321 12277 1318> 1,177 14173 1417, 11329 12289 13187 1419,

15121 15131 1513, 15139 15149

lb"57 lb"61 16063 1606, 16069

17041 1,047 1,053 ,707, 1,093

18041 18043 18047 18049 18059

19073 19079 19081 1908, 19121

20029 20047 20051 20063 20071

21023 2103, 21059 Zl"61 21067

22013 2202, 22031 2203, 22039

:: 73

359 353 96,

1021 1019 1031

174, 1747 ,753

2543 2539 2549

1343 ,347 3359

4157 4153 4159

5009 5011 5021

586, 5861 5869

6779 6763 6781

,669 ,649 ,673

8609 8599 8623

9463 10357 9461 10369 9467 10391

1,351 ,135, 12,23 12101 1321, 13219 11369 12329 13229

:z

379 973

1039 1033

1777 1759

255,

,371 1361

4177 4201

5039 5023

5879 5881

6791 6793

,687 ,681

8629 8627

9479 9473 10399 10427

11393 1,383

1234, 12343

14207 14221 14243 14249 1,249 14251 13241

15161 15173 1518, 15193 15199

16073 16087 lb091 1609, 16103

1,099 1,107 1,117 1,123 1,137

18061 18077 18089 18097 18119

19139 19141 19157 19163 19181

20089 20101 20107 20113 20117

21089 21101 21107 21121 21139

22051 22063 22067 22073 22079

76

383

1049

1783

2579

,373

4211

5051

589,

6803

,691

8641

9491 10429

11399

12373 13259

1521,

16111

1,159

:i 8':

189 39, 409 4"l

1051 1061 1063 1069

178, 1789 1811 1801

2591 259, 261, 2609

,389 ,391 3407 3413

421, 4219 4229 4231

5059 5077 5087 5081

5903 592, 5939 592,

6823 682, 6833 6829

,699 ,703 7723 ,717

864, 8663 867, 8669

949, 9511 10433 10453 9533 10457 9521 10459

11411 11423 12377 12379 1326, 13291 14293 14303 1143, 11443 12401 12391 13297 13309 14321 14323

14281

18121

19183

20123

21143

22091

1522, 1612, 15233 16139 1716, 1,183 bW7 18131 15259 16183 15241 16141 17191 17189 18143 18133

1920, 19211 19219 19213

20129 20143 20149 20147

21149 21157 21169 21163

22093 22109 22123 22111

Lz

421 419

1091 ,087

1831 1823

262, 2633

,449 3433

4243 4241

5101 5099

5953 5981

6857 6841

,741 ,727

8689 868,

954, 10463 9539 1047,

1146, 1144,

12409 12413

,": 85

431 433 439

1093 109, 1103

1861 184, 186,

265, 264, 2659

3457 3461 3463

4253 4259 4261

510, 5113 5119

6007 5987 6011

6863 6869 6871

,753 775, 7759

8693 8699 8707

9551 10487 958, 10499 9601 10501

11471 11483 11489

12421 13331 12433 13337 12437 13339

1432, 14.341 1434, 14369 14387

15263 lb187 15269 16189 15271 16193 1527, 16217 1528, 16223

17203 1,207 1,209 1,231 1,239

18149 18169 18181 18191 18194

19231 19237 19249 19259 19267

20161 20173 20177 20183 20201

21179 2118, 21191 21193 21211

22129 22133 22147 22153 22157

86 2

443 449 457

1109 111, 112,

1871 ,873 187,

266, 2671 2677

,467 ,469 3491

4271 4273 4283

514, 516, 5153

6029 6043 6037

6883 6899 690,

,789 ,793 ,817

8713 8719 8731

9613 10513 9619 10529 9623 10531

1,491 1149, 11503

12451 13367 1245, 13381 12473 1339,

14389 14401 14407

15289 16229 1,257 15299 16249 1530, 16231 17291 17293

18211 18217 18223

19273 19301 19289

20219 20231 20233

21221 21227 21247

22159 22189 22171

,";

463 461

1129 1151

1889 1879

2683 268,

3511 ,499

4289 429,

5179 5171

6053 604,

6917 6911

,829 ,823

8741 8737

963, 10559 9629 10567

1152, 11519

12479 1248,

15319 16253 15313 1626,

1,317 1,299

182.33 18229

19319 19309

20261 20249

2127, 21269

22229 22193

15,29

1,321

18251

19333

20269

21283

2224,

1332, 1,313

13411 14411 13399 14419 14423

2 ,':

46, 479 491 48,

1151 1163 1181 1171

1901 190, 1931 1913

2693 2689 270, 2699

352, 351, ,533 ,529

432, 433, 4349 4339

5189 5197 5209 5227

6073 606, 6079 6089

694, 6949 6961 6959

7841 ,853 7873 ,867

8753 8747 876, 8779

9649 10589 9643 10597 9677 10601 9661 10607

95

499

1187

1933

2711

,539

435,

523,

609,

696,

7877

8783

I," 9'9" 100

509 503 523 521 541

120, 1193 1217 ,213 1223

1949 1951 1979 1973 198,

2719 2713 2729 2731 2141

,541 3547 3559 355, ,571

4373 4363 439, 4391 4409

5237 5233 5273 5261 5279

6113 6101 6131 6121 6133

697, 6971 6991 6983 6997

7883 ,879 7907 7901 ,919

8807 8803 8819 8821 883,

11551 11549 1158, 1,579

16273

9679 10613

12491 13417 1249, 13421 14431 12511 1,451 12503 13441 1443, 1444, 11593 1251, 13457 14449

15331 lb301 15349 16319 15359 16333 15361 16339

1X327 1,333 1,341 17351

18253 18257 18269 18287

19373 19379 19301 19387

2028, 2029, 20323 2032,

21313 21317 21319 21323

22259 22271 22273 2227,

9689 9697 10631 10627 9721 10639 9719 10651 9733 10657

11617 1159, 11633 11621 ,165,

,537, 16349 15373 16361 1,359 ,,,,, 15391 16363 153.83 lb369 1,383 1,387 15401 16381 17389

18301 18289 18311 18307 18313

19403 1939, 19421 19417 19423

20341 20333 20353 20347 20357

21347 21341 21177 21379 21383

22283 22279 22303 22291 22307

12539 13463 ,252, 13469 14461 14479 12541 1,487 1254, 13477 14489 14503 12553 13499 14519

COMBINATORIAL

871

ANALYSIS PRIME:S

25

26

27

28

19

B”

:a

6 1 8 9 10

22391 22397 22409 22433 22441

23371 23399 23417 23431 23447

24373 24379 24391 24407 24413

25453 2545, 25463 25469 25471

26449 26459 26479 26489 26497

27527 2,529 2,539 2,541 21551

28549 28559 2857, 28573 28579

11 12 13 14 15

22447 22453 22469 22481 22483

23459 23473 23497 23509 23531

24419 24421 24439 24443 24469

25523 2553, 25541 25561 25577

26501 26513 26539 26557 26561

2,581 2,583 2,611 2,617 2,631

I6 17 18 19 20

22501 22511 22531 22541 22543

2~35~37 23539 23549 2355, 23561

24473 24481 24499 24509 2451,

25579 25583 25589 25601 25603

26573 26591 2659, 2662, 26633

21 22 23 24 25

22549 22567 22571 22573 22613

23563 2356, 23581 23593 23599

2452, 24533 2454, 24551 2457,

25609 25621 25633 25639 25643

26 2, 28 29 30

22619 22621 22637 22639 22643

23603 23609 23623 23627 23629

24593 24611 24623 24631 24659

31 32 33 34 35

22651 22669 22679 22691 2269,

23633 23663 23669 23671 2367,

36 3, 38 39 40

22699 22709 22717 22721 22727

41 42 43 44 45

:12

55

36

57

Tddc-

:a

34

s8

39

4”

41

12

43

29531 2953, 2956, 29569 29573

30649 30661 30611 3061, 30689

3,663 31667 31687 31699 31721

3268, 32693 ,270, 32713 32717

33641 3364, 33679 3,703 33713

34693 3470, 34721 34729 34739

35831 35837 35839 35851 35863

36833 36847 36857 36871 3687,

37879 37889 3,897 37907 37951

38911 38993 39019 39023 39041

40031 4003, 40039 40063 40087

41149 41161 41117 41179 41183

42157 42169 42179 42181 4218,

2859, 2859, 28603 28607 28619

29581 2958, 29599 2961, 29629

30697 30703 30707 30713 ,072,

31723 3,727 31729 ,174, 31751

32719 32749 ,2,,1 32779 ,278,

,3,21 3,739 3,749 3,751 3,757

,474, 3475, 34759 34763 34781

35869 35879 35897 35899 35911

36887 36899 36901 36913 36919

3795, 3,963 3796, 3,987 37991

39043 3904, 39079 39089 39097

40093 40099 40111 40123 40121

41189 41201 41203 41213 41221

2,647 2,653 2,673 2,689 27691

28621 2862, 2863, 2864, 28649

29633 29641 2966, 29669 29671

30751 30763 30713 30781 30803

31769 ,I,,1 31793 31799 ,181,

32789 3279, 32801 32803 32831

33767 33769 ‘3,,,, 3379, 3,797

3480, 34819 ,4841 34843 3484,

35923 35933 35951 35963 35969

36923 36929 36931 36943 3694,

37993 37991 38011 38039 38047

39103 39101 39113 39119 39133

40129 40151 40153 40163 40169

26641 2664, 26669 26681 26683

2,697 2,701 2,733 2,737 27739

2865, 28661 28663 28669 2868,

2968, 29717 29723 29741 29753

30809 30811 30829 30839 30841

3,847 31849 31859 31873 31883

,28,3 32839 32843 32869 32881

3,809 33811 3382, 3,829 33851

,4849 34871 ,487, 34883 34897

35977 35983 35993 35999 36001

36973 36979 36997 31003 3,013

38053 38069 38083 38113 38119

39139 39157 39161 39163 39181

2565, 2566, 25673 25679 25693

26681 26693 26699 26701 26711

21743 2,749 2,751 2,763 2,767

2869, 28703 28711 28723 28729

29759 29761 29789 29803 29819

,085, 3085, 30859 30869 30811

31891 ,190, 31957 31963 31973

32909 3291, 32917 329,3 32939

3385, 33863 33871 33889 33893

34913 34919 34939 34949 34961

36011 36013 36017 36131 36061

3,019 3,021 31039 37049 3,057

38149 38153 3816, 3817, 38183

24671 2467, 24683 24691 2469,

25703 25717 25733 25741 25747

26713 2671, 26723 26729 26731

2,773 2,779 2,791 27793 27799

28751 28753 28759 28771 28789

29833 2983, 29851 29863 2986,

30881 30893 30911 30931 3093,

3198, 31991 32003 32009 32027

32941 ,295, 32969 32971 32983

33911 3392, 33931 33937 33941

34963 34981 35023 35027 35051

3606, 36073 36083 36097 36107

3,061 3,087 31097 31117 37123

23687 23689 23719 23741 23743

24709 24733 24749 24763 2476,

25759 25763 25771 25793 25799

26737 26759 26777 26783 2680,

2,803 2,809 2,817 2,823 2,827

28793 2880, 28813 2881, 2883,

29873 29879 29881 29917 29921

30941 30949 30971 3097, 30983

32029 32051 3205, ,2059 32063

32981 ,299, 32999 3,013 3,023

33961 ,396, 33997 34019 340,l

35053 35059 ,5069 35081 ,508)

36109 36131 36137 36151 36161

22739 22741 22751 22769 2277,

2374, 23753 23761 23767 2,773

24781 24793 24799 24809 24821

25801 25819 25841 25847 25849

26813 26821 26833 26839 26849

2,847 2,851 27883 2,893 2,901

28843 28859 2886, 28871 28879

2992, 2994, 29959 29983 29989

31013 31019 31033 31039 31051

32069 3207, 32083 32089 32099

33029 3303, 33049 33053 33071

34033 34039 3405, 34061 34123

35089 35099 35107 35111 3511,

46 4, 48 49 5"

22783 22787 22807 22811 22817

23789 23801 23813 23819 23827

24841 2484, 24851 24859 24877

2586, 25873 25889 25903 25913

26861 26863 26879 26881 26891

2,917 2,919 27941 27943 27947

28901 28909 28921 2892, 28933

30011 30013 30029 3004, 30059

31063 31069 31079 31081 31091

3211, 32119 32141 32143 32159

33073 33083 33091 33107 33113

3412, 34129 34141 3414, 3415,

51 52 53 54 55

22853 22859 22861 22871 22877

23831 23833 23857 23869 23873

24889 24907 24917 24919 24923

25919 25931 25933 25939 25943

26893 26903 26921 2692, 26947

2,953 27961 2,967 2,983 21997

28949 28961 28979 29009 29017

30011 30089 30091 3009, 30103

,I121 31123 31139 3,147 31151

,211, 32183 32189 32191 32203

33119 33149 33151 33161 33179

56 57 58 59 60

22901 22907 22921 22937 22943

23879 23887 23893 23899 23909

24943 24953 24967 24971 24977

25951 25969 25981 2599, 25999

26951 26953 26959 26981 2698,

28001 28019 28027 28031 28051

29021 29023 2902, 29033 29059

30109 30113 30119 30133 30137

31153 3,159 3117, 31181 31183

32213 32233 32237 32251 3225,

61 62 63 64 65

22961 22963 22973 22993 23003

23911 2391, 23929 2395, 23971

24979 24989 25013 25031 25033

26003 26017 26021 26029 26041

26993 2,011 2,011 2,031 2,041

28057 28069 28081 28087 2809,

29063 2907, 29101 29123 29129

30139 3016l ,OlbP 30181 30187

31189 31193 31219 31223 31231

66 67 68 69 70

23011 2301, 23021 23027 23029

2397, 23981 23993 24001 24007

25037 2505, 25073 25087 2509,

26053 26083 26099 2610, 26lll

27059 2,061 2,067 2,073 21077

28099 28109 28111 28123 28151

29131 2913, 29141 29153 2916,

30197 30203 30211 30223 30241

71 12 73 74 75

23039 23041 23053 2305, 23059

24019 24023 24029 24043 24049

25111 2511, 25121 2512, 2514,

26113 26119 26141 26153 26161

2,091 2,103 27107 2,109 2,127

28163 28181 28183 28201 28211

29173 29179 29191 29201 29207

76 77 18 79 80

23063 23071 23081 2308, 23099

24061 24071 2407, 24083 24091

25153 25163 25169 25171 25183

26171 2617, 26183 26189 26203

2,143 2,179 27191 2,197 2,211

28219 28229 28277 28279 28283

81 82 83 84 85

2311, 23131 23143 23159 2316,

2409, 24103 2410, 24109 24113

25189 25219 25229 25237 25243

26209 26227 2623, 26249 26251

2,239 2,241 2,253 27259 2,271

86 8, 88 89 90

23173 23189 23197 23201 23203

24121 24133 24137 24151 24169

2524, 25253 25261 25301 25303

26261 26263 26267 26293 26297

91 92 93 94 95

23209 23227 2325, 23269 23279

24179 24181 2419, 24203 24223

2530, 25309 25321 25339 25343

96 9, 98 99 100

23291 23293 23291 23311 23321

24229 24239 2424, 24251 24281

25349 25357 2536, 25373 25391

44

45

P-I.9

40

li

.t”

4Y

43133 43151 43159 4317, 43189

44263 4426, 44269 44273 44279

45343 4536, 4537, 45389 45403

46499 4650, 46511 46523 46549

41591 47599 4,609 4,623 47629

42193 4219, 42209 42221 42223

43201 43207 43223 4323, 43261

44281 44293 44351 4435, 44371

45413 4542, 45433 45439 45481

46559 4656, 46573 46589 46591

4,6,9 4,653 4,657 4,659 41681

41227 41231 41233 41243 41257

42227 42239 42257 42281 42283

43271 43283 43291 43313 43319

44381 44383 44389 4441, 44449

45491 4549, 45503 45523 45533

46601 46619 46633 46639 46643

4,699 47701 4,111 4,713 4,717

40177 40189 40193 40213 40231

41263 41269 41281 41299 41333

42293 42299 4230, 42323 42331

43321 43331 43391 4339, 43399

44453 44483 44491 4449, 44501

45541 45553 45557 45569 4558,

46649 46663 46679 46681 4668,

4,737 4,741 47743 4,777 4,779

39191 39199 39209 39211 3922,

40237 40241 40253 4027, 40283

41341 41351 4135, 41381 41387

4233, 42349 42359 42373 42379

43403 43411 43427 43441 43451

44507 44519 44531 44533 4453,

45589 45599 45613 45631 45641

46691 46703 46723 46727 4674,

41791 4,797 4780, 4,809 4,819

38189 3819, 38201 38219 38231

39229 39233 39239 39241 39251

40289 40343 40351 4035, 40361

41389 41399 41411 41413 41443

42391 4239, 42403 42407 42409

4345, 43481 43487 43499 43517

44543 44549 44563 44579 44587

45659 4566, 45673 4567, 45691

46751 46757 46769 46771 46807

4,831 4,843 4,857 4,869 4,881

3,139 37159 37171 3,181 3,189

38237 38239 38261 38273 38281

39293 39301 39313 39317 39323

4038, 40423 4042, 40429 40433

41453 4146, 41479 41491 4150,

42433 42437 42443 42451 4245,

43541 4354, 43573 43577 43579

4461, 44621 44623 44633 44641

4569, 4510, 45737 45751 4575,

46811 4681, 46819 46829 46831

4,903 4,911 4,917 479,) 4,939

3618, 36191 36209 36217 36229

3,199 37201 31217 3,223 3,243

3828, 38299 38303 38317 38321

39341 39343 39359 39367 39311

40459 40471 40483 4048, 40493

41513 41519 41521 41539 41543

42461 42463 42467 42473 4248,

43591 4359, 43607 43609 43613

4464, 44651 4465, 44683 44687

45763 45767 45779 4581, 45821

46853 46861 46867 46877 46889

4,947 4,951 4,963 4,969 4,977

35129 ,5141 35149 35153 35159

36241 36251 36263 36269 36277

3,253 3,273 3121, 3,307 31309

3832, 38329 38333 38351 38371

39373 39383 3939, 39409 39419

40499 4050, 40519 40529 40531

41549 4,579 41593 4159, 41603

4249, 42499 42509 42533 4255,

4362, 43633 43649 43651 43661

44699 44701 44711 44729 44741

45823 4582, 45833 45841 45853

46901 46919 46933 4695, 46993

4,981 48017 48023 48029 48049

34159 34171 34183 34211 34213

,517, 35201 35221 ,522, 35251

,6293 36299 36307 36,13 36319

,731, 3,321 3,337 3,339 3,357

38,,, 38393 38431 38441 38449

39439 39443 39451 39461 39499

4054, 40559 40577 40583 40591

41609 41611 4161, 41621 4,627

42569 42571 4257, 42589 42611

4,669 43691 43711 43717 43721

44753 44771 44773 44777 44789

45863 45869 45887 45893 45943

4699, 47017 47041 4,051 4,057

4807, 48079 48091 48109 48119

3,181 33191 33199 33203 33211

3421, 3423, 34253 34259 34261

3525, 3526, 35279 35281 35291

36341 36343 36353 36373 ,6383

3736, ,,363 31369 ,13,9 3,397

38453 38459 38461 38501 38543

39503 39509 39511 39521 39541

40597 40609 40627 40631 40639

4,641 4164, 4,651 41659 41669

42641 42643 42649 4266, 4267,

43753 43759 4371, 43781 43783

44797 44809 44819 44839 44843

45949 45953 45959 45971 45979

4,059 4,087 4,093 4,111 47119

48121 48131 4815, 48163 48179

32261 3229, 32299 32303 32309

33223 3324, 3328, 33289 33301

3426, 34273 34283 34291 34301

35311 35317 35323 35327 35339

36389 36433 3645, 3645, 36461

,,409 3,423 3,441 3144, 3,463

3855, 38561 38567 38569 38593

3955-l 39563 39569 19581 3960,

4069, 4069, 40699 40709 40739

41681 4168, 41119 41129 4173,

42683 42689 4269, 4270, 42703

43787 43789 43793 43801 43853

44851 44867 44879 4488, 44893

45989 46021 4602, 46049 46051

4,123 47129 4,137 4,143 41147

48187 48193 4819, 48221 48239

31237 3124, 31249 31253 31259

32321 32323 32327 32341 32353

33311 3331, 33329 33331 33343

34303 34313 34319 34321 34331

35353 35363 35381 35393 35401

36469 36473 36479 36493 3649,

3,483 3,489 3,493 3,501 3,507

38603 38609 38611 38629 38639

39619 ,9623 39631 39659 3966,

40751 40159 40763 40771 40787

41759 4,761 41771 41777 41801

42709 42719 4272, 4273, 42743

43867 43889 43891 43913 43933

44909 4491, 4492, 44939 44953

46061 46073 46091 46093 46099

41149 4,161 47189 4,207 4,221

48247 48259 48271 48281 48299

30253 30259 30269 30211 30293

3126, 31271 31277 31307 31319

32359 32363 32369 32371 3237,

3334, 33349 33353 33359 3337,

34351 34361 3436, 34369 34381

35407 35419 35423 35431 3544,

3652, 3652, 36529 ,6541 36551

3,511 3,517 37529 3,537 3,547

38651 38653 ,8669 38671 38677

39671 39679 39703 39709 39719

40801 40813 40819 40823 40829

41809 41813 41843 41849 41851

42751 42767 42773 4278, 42793

43943 43951 43961 43963 43969

44959 44963 44971 44983 4498,

46103 46133 46141 46141 46153

4,237 4,251 47269 47279 4,287

48311 48313 48337 48341 48353

29209 29221 29231 29243 29251

3030, 30313 30319 30323 30341

31321 31327 31333 31337 31351

32381 32401 32411 32413 32423

33391 33403 33409 33413 33427

34403 34421 34429 34439 3445,

35449 35461 35491 3550, 35509

36559 36563 ,6511 36583 3658,

3,549 3,561 3156, 3157, 3,573

38693 38699 38707 3871, 38713

3972, 39733 39149 39761 39769

4084, 4084, 40849 40853 4086,

4186, 41879 4,887 41893 41891

4279, 42821 42829 42839 42841

4,973 4398, 43991 4399, 44017

4500, 45013 45053 45061 4501,

4617, 46181 46183 4618, 46199

4729, 4,297 4,303 4,309 4,317

48371 48383 4839, 4840, 48409

28289 2829, 2830, 28309 28319

29269 2928, 2929, 29303 29311

3134, 3036, 30389 30391 30403

31379 31387 31391 31393 31397

32429 32441 32443 32467 32479

33457 33461 33469 33479 3348,

34469 34411 34483 34487 34499

35521 35527 35531 35533 35537

,6599 3660, ,6629 3663, ,664,

31519 3,589 ,759, 3,607 37619

38723 38729 3873, ,874, 38149

39779 39791 39199 3982, 39821

40879 40883 4089, 40903 4092,

41903 41911 4192, 41941 41941

42853 42859 42863 42899 42901

44021 4402, 44029 44041 4405,

45083 45119 45121 4512, 451,1

46219 46229 4623, 46261 46271

4,339 4,351 4,353 4,363 47381

48413 4843, 48449 48463 4847,

2127, 2,281 27283 2,299 2,329

28349 28351 2838, 28393 28403

2932, 29333 29339 29347 29363

3042, 30431 30449 3046, 30469

31469 31471 31481 31489 31511

32491 32497 32503 3250, 32531

33493 33503 33521 33529 33533

34501 34511 34513 34519 34537

35543 35569 35573 35591 35593

3665, 3667, 3667, 36683 36691

3,633 3,643 ,1649 ,765, 3,663

3876, 38783 38791 3880, 38821

39829 39839 39841 ,984, 3985,

40933 40939 40949 40961 40973

41953 4,957 41959 4,969 41981

42923 42929 42937 42943 42953

44059 44071 4408, 44089 44101

4513, 45139 45161 45179 45181

46273 46279 46301 46307 46,09

4,387 41389 4,407 4,417 4,419

48479 48481 4848, 4849, 4849,

26309 26317 26321 26339 2634,

2,337 27361 2,367 2,397 2740,

28409 28411 28429 28433 28439

29383 29387 29389 29399 29401

30491 30493 3049, 30509 30517

31513 3151, 31531 31541 31543

32533 3253, 32561 32563 32569

33547 33563 33569 3357, 33581

34543 34549 34583 34589 34591

35597 35603 35617 35671 35677

36691 16709 36713 36721 36739

3,691 3,693 31699 3,717 ,714,

38833 38839 3885, 38861 38861

39863 39869 3987, 39883 3988,

40993 41011 4,017 4,023 41039

4,983 41999 42013 42017 42019

42961 42961 42979 42989 43003

44111 44119 44123 44129 44131

4519, 4519, 45233 45247 45259

4632, 4633, 46349 46351 46381

4743, 4,441 4,459 4,491 4,497

48523 4852, 48533 485,9 48541

2635, 26371 2638, 26393 26399

2,409 27427 2,431 27437 21449

2844, 28463 28477 28493 28499

29411 29423 29429 2943, 29443

30529 30539 30553 3055, 30559

3154, 3156, 31513 31583 31601

32573 32579 3258, 32603 32609

33587 33589 33599 33601 33613

34603 34607 34613 34631 34649

35729 35731 35747 35753 35759

36749 36161 36767 36779 3678,

3,781 3,783 3,799 3,811 3,813

38873 38891 38903 38911 38921

39901 39929 39937 3995, 39971

4,047 41051 41057 4107, 41081

42023 42043 42061 42071 42073

43013 43019 43037 43049 43051

44159 44171 44179 44189 4420,

45263 45281 45289 45293 4530,

46399 46411 46439 4644, 4644,

4,501 4,507 4,513 L7521 4,527

48563 48571 48589 48593 48611

872

COMBINATORIAL

‘I’ulh

m.9

ANALYSIS PRIhlES 02 83

63

54

56

66

67

58

58

84

61

a5

86

67

68

88

70

71

72

7%

48619 49667 4862, 49669 48647 49681 48649 49697 4866, 49711

50767 51817 50773 51827 50777 51829 50789 51839 50821 5,853

52937 52951 52957 5296, 5296,

54001 54011 54013 54037 54049

55109 55117 55127 55147 55163

56197 56207 56209 562,7 56239

5719, 57203 57221 5722, 57241

58243 58271 58309 58313 58321

59369 59377 59187 59J93 59399

60509 60521 60527 60539 60589

61637 6164, 61651 61657 61667

62791 62801 62819 62827 62851

6,823 6,839 63841 6,853 6,857

65071 65089 65099 65101 65111

66107 66109 66137 66161 66169

67247 67261 67271 67213 67289

68389 68399 60437 6844, 68447

69497 69499 695,9 695,7 6959,

7066, 70667 70687 70709 70717

71719 71741 71761 71777 71789

72859 72869 72871 7288, 72889

7,999 74017 74021 74027 74047

75083 75109 7513, 75149 75161

6 7 8 9 10

4867, 48677 48679 48731 48733

49727 49739 49741 49747 49757

5083, 50839 50849 50857 50867

52973 52981 52999 53003 5,017

54059 54083 54091 54101 54121

55171 55201 55207 55213 55217

56249 57251 5626, 57259 56267 57269 56269 57271 56299 5728,

58337 SD63 58367 58369 58379

59407 60601 59417 60607 59419 60611 59441 606l7 5944, 60623

6167, bl68l 61687 61703 61717

62861 62869 6287, 62897 6290,

b,a63 6,901 6,907 6,913 6,929

65119 6512, 65129 65141 65147

6617, 66179 66191 66221 66239

67JO7 67339 6734, 67349 67369

ba449 6847, 68477 68483 ba489

6962, 69653 69661 69677 69691

70729 7075, 70769 7078, 7079,

71807 71809 71821 71837 7184,

7289, 72901 72907 72911 7292,

74051 74071 74077 7409, 74099

75167 75169 75181 75193 75209

11 12 1, 14 15

48751 48757 48761 48767 48779

4978, 50873 49787 50891 49789 5089, 49801 50909 49807 50923

51907 53047 51913 5,051 51929 5,069 5,941 5,077 51949 5,087

54133 54139 54151 5416, 54167

55219 56311 55229 56,,3 55243 56359 55249 56369 55259 56377

57287 57301 57329 573,1 57,47

60631 60637 60647 60649 60659

61723 61729 61751 61757 6178,

62921 6,949 62927 6,977 62929 6,997 62939 64007 62969 64013

65167 66271 65171 6629, 6517, 66301 65179 66337 65183 6634,

67391 68491 67399 68501 67409 68507 67411 68521 67421 68531

69697 69709 697,7 69739 69761

7082, 70841 7084, 70849 7085,

71849 71861 71867 7,879 71881

72931 72937 72949 7295, 72959

74101 74131 7414, 74149 74159

75211 75217 7522, 75227 75239

lb 17 10 19 20

4878, 48787 48799 48809 48817

49811 4982, 49831 4984, 4985,

50929 50951 50957 50969 5097,

5,971 5197, 51977 5,991 52009

5,089 53093 5,101 5,113 5,117

54181 54193 54217 54251 54269

5529, 55313 55331 5533, 55337

56383 5639, 56401 56417 56431

57349 58427 57367 58439 57373 58441 57383 58451 57389 58453

59497 59509 5951, 59539 59557

6066, 60679 60689 60703 60719

blal, 61819 61837 61843 61861

62971 64019 62981 6403, 62983 64037 62987 6406, 62989 64067

6520, 66347 6521, 66359 65239 66361 65257 66373 65267 66377

67427 68539 67429 6854, 6743, 68567 67447 685a1 6745, 68597

6976, 69767 69779 69809 69821

70867 70877 70879 70891 70901

71887 71899 71909 71917 719,3

7297, 72977 72997 7,009 7,013

74161 74167 74177 74189 74197

7525, 75269 75277 75289 75307

21 22 2, 24 25

48821 4882, 48847 4885, 48859

4987, 4987, 49891 49919 49921

50989 50993 5,001 51031 5104,

52021 52027 52051 52057 52067

5,129 5,147 5,149 5,161 5,171

54277 5428, 5429, 54311 54319

55339 5534, 55351 5537, 55381

5643, 5644, 56453 56467 5647,

57397 5847, 5741, 58481 5742, 58511 57457 58537 57467 5854,

59561 59567 59581 59611 59617

60727 6073, 60737 60757 60761

61871 61879 61909 61927 61933

63029 63031 6,059 6,067 63073

65269 65287 6529, 65,09 6532,

66383 66403 66413 66431 66449

67477 67481 67489 6749, 67499

68611 68633 68639 68659 68669

69827 69829 69833 69847 69857

7091, 70919 70921 70937 70949

71941 71947 ,196, 71971 71983

73019 73037 73039 7,043 7,061

74201 7420, 74209 74219 74231

7923 75329 75337 75347 75353

26 2, 28 29 30

48869 48871 4888, 48889 4890,

49927 49937 49939 4994, 49957

51047 51059 5,061 51071 51109

52069 52081 52103 52121 5212,

53173 53109 5,197 5,201 5,231

5432, 54331 5434, 54361 54967

55399 55411 55439 55441 55457

5647, 56479 56489 5650, 56503

57487 5749, 57503 5752, 57529

58549 58567 5857, 58579 58601

59621 59627 59629 59651 59659

60763 6077, 60779 60793 60811

61949 63079 61961 6,097 6196, 6,103 61979 6,113 61981 6,127

6415, 65327 64157 65,5, 64171 65X7 64187 65371 64189 65381

66457 6646, 66467 66491 66499

67511 6752, 67531 67537 67547

6868, 6868, 68699 68711 6871,

69859 69877 69899 69911 69929

70951 70957 70969 70979 70981

7198, 71993 71999 72019 720,1

7,063 7,079 73091 7,121 73127

74257 74279 74287 74293 ,429,

75367 75377 75389 75391 75401

,, 32 3, 34 35

48947 4895, 4897, 48989 4899,

49991 51~1 4999, 5,113 49999 51137 50021 5115, 5002, 51157

52147 52153 5216, 52177 52181

5,233 54371 532,9 54377 5326, 54401 5,269 54403 53279 54409

55469 56509 5,557 55487 56519 5,559 55501 56527 5,571 55511 56531 57587 55529 56533 57593

58603 5861, 58631 58657 5866l

59663 60821 59669 60859 59671 60869 5969, 608a7 59699 60889

61987 63131 61991 63149 62003 63179 62011 63197 62017 63199

64217 6539, 64223 65407 64231 6541, 64237 65419 64271 6542,

66509 6652, 66529 66533 66541

67559 67567 67577 67579 67589

60729 69931 60737 69941 68743 69959 68749 69991 68767 69997

70991 ,099, 70999 71011 7102,

72043 72047 72053 72073 72077

,313, 73141 73181 73189 7,237

74311 74317 74323 74353 74357

75403 75407 75431 75437 75479

36 37 38 39 40

49003 50033 49009 5004, 49019 50051 49031 5005, 4903, 50069

51169 5119, 51197 51199 51203

52183 52189 52201 5222, 52237

5,281 5,299 5,,09 5,,23 5,827

5441, 54419 54421 54437 5444,

55541 55547 55579 55589 55603

56543 56569 56591 5659, 56599

57601 57637 57641 5,649 5765,

58679 5970, 58687 59723 5869, 59729 58699 5974, 58711 5974,

62039 62047 62053 62057 62071

64219 64283 64301 64303 64319

65437 65447 65449 65479 65497

66553 67601 66569 67607 66571 67619 66587 67631 66593 67651

68771 68771 68791 68813 68819

70001 70003 70009 70019 70039

71039 71059 71069 71081 71089

72089 72091 72101 ,210, 72109

7,243 7,259 7,277 73291 ,330,

7436, 74377 74381 ,4,8, 74411

75503 75511 75521 75527 7553,

41 42 4, 44 45

4903, 4904, 49057 49069 49081

5007, 5008, 5009, 50101 50111

5,217 51229 5,299 51241 5,257

52249 5225, 52259 5226, 52289

5,353 53359 53,,, 5,381 53401

54449 54469 5449, 5449, 54499

55609 55619 55621 55631 5563,

56611 56629 5663, 56659 5666,

5766, 58727 57679 5873, 57689 58741 5,697 58757 57709 5876,

46 4, 48 49 50

4910, 49109 49117 49121 4912,

50119 5012, 50129 50131 50147

5,263 52291 5,283 52301 5,287 5231, 5,307 52,2l 51329 52361

5,407 5,411 53419 53437 5,441

54503 54517 54521 54539 54541

55639 55661 5566, 55667 5567,

56671 57713 56681 57719 56687 57727 56701 57731 56711 5773,

51 52 5, 54 55

49139 49157 49169 49171 49177

5015, 50159 5017, 50207 50221

51341 52,6, 5,343 52369 51347 52379 51349 5238, 51361 52391

5,453 5,479 5,503 53507 5352,

5454, 54559 5456, 5457, 54581

55681 55691 55697 55711 55717

56713 5,751 56731 5777, 56737 57781 56747 57787 56767 57791

56 5, 58 59 60

49193 49199 4920, 4920, 49211

50227 50231 50261 5026, 50273

5138, 51407 51413 51419 51421

52433 5245, 5245, 52489 52501

5,549 54583 53551 54601 5,569 54617 53591 5462, 5,593 54629

61 62 6, 64 65

4922, 4925, 49261 49277 49279

5028, 50291 50311 50321 50329

51427 51431 51437 514,9 5,449

5251, 5,597 5251, 5,609 52529 5,611 52541 5,617 5254, 5,623

66 67 68 69 70

49297 49,07 4933, 49333 49339

5033, 50341 50359 50363 50377

51461 5255, 5147, 5256, 51479 52567 51481 52571 5,487 52579

71 72 7, 74 75 76 77 78 79 80

49363 49367 49369 4939, 49393 49409 49411 49417 49429 4943,

50,8, 50,8, 50411 50417 50423 50441 50459 50461 5049, 5050,

5,503 5151, 51517 5,521 51539 5,551 5,563 5,577 51581 51593

5258, 52609 5262, 52631 52639 52667 5267, 5269, 5269, 52709

8, 82 8, a4 85

4945, 49459 4946, 49477 49481

5051, 50527 50539 50543 50549

51599 5,607 5161, 5,631 5,6,,

5” 1 2 3 4 5

61

86 49499 5055, 88 87 49529 4952, 5058,

62

51859 51869 51871 5189, 51899

5975, 59771 59179 59791 59797

60923 62081 60937 62099 60943 62119 6095, 62129 60961 62131

6,299 64327 63311 64333 63313 64373 63317 64381 63,,l 64399

65519 65521 65537 65539 6554,

66601 66617 66629 66643 66653

67679 67699 67709 67723 67733

68821 68863 68879 68881 68891

70051 70061 70067 70079 70099

71119 71129 7114, ,114, 71153

72139 72161 72167 72169 72173

73,09 7,327 73331 73351 7,361

74413 74419 74441 14449 7445,

75539 75541 75553 75557 75571

58771 58787 58789 58831 58889

59809 59833 5986, 59879 59887

6lOOl 6100, 61027 61031 6104,

62137 62141 62143 62171 62189

63337 64403 63347 64433 6335, 64439 6,361 64451 6,367 6445,

65551 65557 6556, 65579 65581

6668, 6669, 66701 66713 66721

67741 67751 67757 67759 6776,

68897 68899 68903 68909 68917

70111 70117 70121 70123 70139

71161 7116, 71171 71191 71209

72211 72221 7222, 72227 72229

73363 7,369 73379 7,387 7,417

74471 74489 74507 74509 74521

75577 7558, 75611 75617 75619

5889, 58901 58907 58909 58913

59921 61051 59929 bl057 59951 61091 59957 61099 59971 61121

62191 62201 62207 62213 62219

6,377 63389 6,391 6,397 6,409

6448, 64489 64499 64513 64553

65587 6671, 65599 66739 65609 66749 65617 66751 65629 6676,

67777 68927 6778, 68947 67709 68963 67801 68993 67007 69001

70141 70157 70163 70177 70181

71233 71237 71249 71257 71261

72251 7225, 72269 72271 72277

7,421 7,433 7345, 7,459 7,471

74527 74531 74551 74561 74567

75629 75641 7565, 75659 75679

62233 63419 62273 6,421 62297 6,439 62299 6344, 62303 6,463

64567 64577 64579 64591 64601

65633 65647 65651 65657 65677

66791 66797 66809 66821 66841

67819 69011 67829 69019 6784, 69029 67053 69031 67867 69061

7018, 70199 70201 70207 ,022,

7126, 71287 7129, 71317 71327

72287 72107 72313 7233, 72341

7,477 7,483 73517 ,352, 73529

7457, 7458, 74597 74609 74611

7568, 75689 75703 75707 75709

55799 55807 55813 55817 55819

5681, 56821 56827 5684, 56857

57847 58979 60037 57853 58991 60041 57859 58997 60077 57881 59009 60083 57899 59011 60089

61211 61223 61231 6125, 61261

62311 62323 62327 62347 62351

6,467 63473 6348, 6,493 6,499

64609 65687 64613 65699 64621 65701 64627 65707 6461, 6571,

66851 6685, 66863 6687, 66883

6788, 67891 67901 67927 6,931

69067 69073 69109 69119 69127

70229 70237 70241 70249 70271

i1329 71333 71339 ,I,41 71347

7235, 7236, 72,79 72383 72421

7,547 7,553 7,561 7,571 73583

7462, 7465, 74687 74699 7470,

75721 75731 75743 75767 7577,

5,629 5,633 53639 5,653 5,657

54709 55823 54713 55829 5472, 55837 54727 55843 54751 55849

56873 56891 5689, 56897 56909

57901 59021 57917 59023 57923 59029 57943 59051 57947 59053

60091 60101 6010, 60107 60127

61283 62383 61291 62401 61297 62417 61331 62423 61333 62459

63521 6,527 6353, 6,541 6,559

64661 65717 64663 65719 64667 65729 64679 657,1 6469, 6576,

66889 67933 66919 6,9,9 6692, 6,943 66931 67957 6694, 67961

6914, 69149 69151 6916, 69191

70289 70297 70309 70313 70321

,135, 71359 7136, 71387 71,89

72431 72461 7246, 72469 72481

73589 7,597 7,607 7,609 73613

7471, 74717 74719 74729 74731

75781 75787 7579, ,579, 75821

5,681 5369, 5,699 5,717 53719 5,731 5,759 5,773 5,777 5,783

54767 5477, 54779 54787 54799 54829 548,) 54851 54869 5487,

55871 55889 55897 55901 55903 55921 55927 55931 55933 55949

56911 5797, 56921 57977 56923 57991 56929 58013 56941 58027 56951 58031 5695, 58043 5696, 58049 56983 58057 56989 58061

59063 59069 59077 59083 59093 59107 59113 59119 59123 59141

60133 60139 60149 60161 60167 60169 60209 60217 60223 60251

61,,9 61343 61357 61363 61379 61381 61403 61409 61417 61441

62467 6247, 62477 6248, 62497 62501 62507 62533 62539 62549

6357, 6,587 6,589 6,599 6,601 6,607 6,611 6,617 6,629 6,647

64709 64717 6474, 6476, 64781 64783 6479, 64811 6481, 64049

6694, 66949 66959 6697, 66977 6,003 6702, 670,) 6,043 67049

6796, 67979 6,987 6799, 68023 68041 68053 68059 68071 6&,0,

69193 69197 69203 69221 6923, 69239 69247 69257 69259 6926,

70327 70351 ,037, 70,79 70381 ,039, ,042, 70429 704)9 70451

71399 71411 ,141, 71419 71429 7,437 7,144, ,145, 71471 7147,

7249, 72497 ,250, 7253, 72547 72551 72559 7257, 72613 72617

7,697 ,364, 73651 7,673 7,679 7,681 7369) 7,699 7,709 7,721

,474, 74759 74761 74771 74779 ,479, 74821 7482, 74031 74043

,583, 7585, 75869 7588, ,591, 759,1 75937 75941 75967 75979

52711 5,791 52721 5,813 52727 53819 52733 538,, 52747 5,849

5488, 54907 5491, 54919 54941

55967 55987 55997 56003 56009

5699, 56999 5,037 57041 57047

59149 59159 59167 59183 59197

60257 60259 60271 60289 6029,

6146, 61469 61471 61483 61487

65881 6705, 65899 6,061 65921 6707, 6592, 67079 65929 6710, 62617 6,691 6490, 65951 67121 62633 62627 63697 64919 65957 6,129 6370, 64921 65963 6,139 62639 63709 64927 65981 6714, 62653 6,719 6493, 6598, 67153 62603 62659 6,727 64951 6599, 67157 63737 64969 66029 6,169 62701 62687 6,743 64997 66037 67101 62723 6,761 65003 66041 67107 6,773 65011 66047 67189

60099 68111 6811, 68141 68147

69313 69317 69337 6934, 69371

70457 70459 70401 70487 70489

71479 ,148, 71503 71527 ,l5,,

7262, ,264, 72647 72649 72661

7,727 7,751 7,757 7,771 7)70,

74857 74861 74869 7487, 7408,

75903 75909 7599, ,599, 76001

68,6l 69379 @I,1 69383 6820, 69389 68209 69401 68213 6940,

70501 70507 ,,,529 70537 70549

71549 71551 71563 71569 ,159,

72671 7267, 72679 72609 72701

7,819 ,382, ,384, 7,049 7,059

74891 7489, ,490, 7492, 74929

,600, 760,1 76039 76079 76,381

60219 6942, 6022, 69.,,1 68239 69439 68261 69457 68279 6946,

70571 705,) 70583 70509 ,"6,,7

,159~ ,163, 71647 71663 71671

,270~ 72719 72727 7273, ,27,9

,386, 7,877 ,388; 7,897 ,,9,,,

749,) 74941 74959 75011 7501,

,609, 76099 7610; 7612, 76129

69467 70619 6947) 70621 69481 70627 69491 70639 6949, ,065,

7169, 71699 ,I,", 71711 71713

7276, 72767 ,279, 72017 7282,

7,939 7,943 7,951 7,961 7397,

,501, 75029 ,503~ 75041 75079

7614, 76157 76159 7t 16; 7 207

51647 52757 52769 5167, 51659 52783

546X 54647 5466, 5467, 54679

55721 5677, 55733 56779 55763

92 91 49549 49547 5062, 50599 5171, 51691

5283, 52817 5,899 5389,

52889 52883 52901 5290, 52919

58067 58073 58099 58109 58111

53857 54949 56039 57059 58129 59207 60317 61493 5,881 5,861 5497, 54959 5605, 56041 5707, 57073 58151 58147 59219 59209 60337 60331 61511 61507 5,087 54979 56081 57089 53891 54983 56087 57097 55009 5500, 56099 5609,

93 94 49597 49559 50651 50647 51721 51719 52861 52859 53923 53917 55049 55021 56113 56101 95 4960, 50671 5,749 52879 5,927 55051 5612, 50707 50683 51769 51767 50723 51787 50741 51797 5075, 5180,

6,211 63241 63247 63277 63281

61129 61141 61151 61153 61169

5280, 5281,

49627 4961, 49633 49639 4966,

60899 60901 6091, 60917 60919

64081 64091 64109 6412, 64151

74

57793 58921 59981 57803 58937 59999 56783 57809 58943 60013 55787 56807 57829 58963 60017 55793 56809 57839 58967 60029

89 49531 5059, 51679 90 49537 50593 5168,

97 96 98 99 100

58391 59447 5839, 5945, 5840, 59467 58411 59471 58417 5947,

M

53951 5,939 5,959 5,987 5,993

55061 5505, 56149 56131 5507, 56167 55079 56171 5510, 56179

58153 59221 58169 59233

57119 5710, 58189 58171

60343 61519 60353 61543

59243 59239 60383 60373 61553 6154,

57139 57131 58199 58193 5927, 59263 60413 60397 61561 61559 5,143 58207 59281 60427 6158, 5716, 57149 57173 57179 57191

58217 58211 58229 58231 58237

59341 59333 59351 59357 59359

60449 60443 61609 6160, 60457 61613 60493 61627 60497 61631

6256, 62581 62591 62597 62603

62731 6274, 62753 62761 62773

65777 65789 65809 65827 65831 65837 65839 65843 65051 65867

6,649 64853 6,659 64871 6,667 64877 6,671 64879 6,689 64091

6,781 6,793 6,799 6,803 6,809

6502, 66067 65029 66~71 65";; 6608, 65053 66009 65063 6610,

6,211 67213 6,217 67219 67231

68201 68311 68329 68351 68371

COMBINATORIAL

873

ANALYSIS

rdh 94

I ‘Hlhl BO

80779 80783 80789 80803 80809

8188, 81899 81901 81919 81929

81971 81973 8200, 8200, 82009

81

x2

83

x4

82903 8291, 82939 8296, 82981

84131 84137 84143 84163 84179

85243 8524, 85259 8529, 85303

86?m 86389 86399 86413 8642,

8880, 88811 88813 88.91, 88819

82997 83003 83009 83023 8,047

84181 84191 84199 84211 8422,

85313 85331 85333 8536, 85363

8644, 86453 8646, 8646, 8647,

8884, 8885, 88861 88867 88873

83093 83101 8,117 8,137 8,177

84299 84307 84X3 8017 84319

85439 8544, 85451 85453 85469

86539 8656, 86573 86579 8658,

86

Hi 8986, 89891 8989, 89899 89909 95483 95507 9552, 95531 95539

88951 88969 88993 8899, 89003

21.9 95

9,829 9,841 9784, 97847 9,849

9895, 98963 98981 98993 98999

9,859 97861 9787, 9,879 9,883

99013 99017 99023 9904, 99053 99079 9908, E; 99109

95549 9556, 95569 95581 9559,

96737 96739 EG 96763

9,919 9,927 979,1 9,943 97961

95603 95617 95621 95629 956,,

96769 96779 96787 9679, 96799

9,961 9,973 9,987 98009 98011

99119 99,,, 99133 9913, 99139

95651 95701 95707 95713 95717

96821 96823 9682, 9684, 96851

98017 9804, 9804, 98057 98081

99149 9917, 99181 99191 99223

26 27 28 29 30

,650, 76511 76519 76537 76541

77591 77611 ,761, 7,621 7,641

7878, 78787 78791 ,879, 78803

79889 79901 7990, 799P7 799s

81001 8,013 81017 8,019 51023

82051 82067 8207, 82129 82139

83231 8,233 8324, 83257 8,267

84401 84407 84421 84431 84437

85549 85571 85577 8559, 85601

86693 8671, 86719 86729 86743

89057 89069 89071 89083 89087

90089 90107 90121 90127 90149

9572, 95731 9573, 9574, 95773

9685, 9689, 96907 9691, 96931

98101 9812, 98129 98143 98179

99233 99241 99251 9925, 99259

31 32 33 34 35

7654, 76561 76579 76597 76603

,764, 77659 77681 77687 77689

78809 ,882, 78839 78853 78857

,994, 79967 7997, 79979 7998,

810,1 81041 8,043 81047 81049

82141 8215, 8216, 82171 82183

83269 8327, 83299 83311 833,9

8444, 84449 8445, 8446, 8446,

8560, 85619 8562, 85627 85639

8675, 8676, 8677, 86783 8681,

89101 8910, 8911, 89119 89123

90163 90173 9018, 9019, 9019,

95783 95789 95791 95801 95803

96953 96959 96973 96979 96989

98207 E: 98227 98251

99277 99289 9931, 9934, 99349

K: 89189 89203 89209

90199 9020, 90217 90227 90239

95813 95819 95857 95869 95873

96997 9,001 9700, 9,007 9,021

9825, 98269 98297 98299 98317

9936, 99371 99377 99391 9939,

89213 8922, 89231 89237 89261

90247 90263 90271 90281 90289

95881 95891 95911 9591, 9592,

9,0,9 97073 9,081 9,103 9,117

9832, 9023 9832, 98347 98369

9940, 99409 994,1 99439 99469

89269 89273 892% 89303 89317

WI,,, 90,53 90359 90,,1 90,7,

95929 9594, 9595, 95959 95971

9,127 9,151 9715, 97159 9,169

9x377 98x7 98x9 98407 98411

9948, 99497 9952, 99527 99529

89329 8936, 8937, 89381 89387

90,,9 90397 9040, 90403 90407

95987 95989 96001 96013 96017

9717, 9,177 9,187 9,213 9723,

98419 98429 9844, 9845, 98459

99551 99559 9956, 9957, 9957,

89393 90437 90439 EE 90469

96043 96053 96059 96079 96097

97241 97259 9,283 97301 9,301

98467 98473 98479 98491 98507

9958, 99607 9961, 9962, 9964, 99661 9966, 99679 99689 99707

83563 8,579 83591 8,597 8,609

84719 84131 84737 84751 84761

85889 85903 85909 85931 85933

8,071 8,083 8710, 8,107 8,119

8941, 8943,

9047, 90481

61 6Z 63 64 65

7691, 76919 ,694, 76949 ,696,

77983 7,999 ,800, ,801, 78031

79187 7919, 79201 79229 ,923,

96137 96149 9615, 96167 96179

97327 9,367 9,369 ::::z

98519 98533 9854, 98561 9856,

66 6, 68 69 70

7696, 76991 77003 ,701, 7,023

,804, 78049 78059 78079 78101

7924, 79259 ,927, 79279 ,928,

96181 96199 96211 96221 96223

97,81 9,387 9739, 9742, 97429

98573 9859, 9862, 9862, 98639

99709 99713 99719 99721 99,,,

71 72 73 74 75

7,029 7,041 77047 77069 ,708,

,812, 78137 78119 ,815, ,816,

96233 96259 96263 96269 96281

9,441 9745, 97459 9,463 9,499

98641 98663 98669 98689 98711

99761 99767 99787 99793 99809

96289 9629, 96,2, 96329 96331

9,501 9,511 9752, 9,547 9,549

9871, 9871, 98729 987,l 987,7

9981, 9982, 99829 9983, 99839

96337 96353 9637, 96401 96419

97553 9,561 97571 9,577 97579

98773 98779 98SOl 98807 98809

99859 9987, 99877 99881 99901

96431 9644, 96451 96457 96461

9,583 97607 9,609 9761, 97649

9&a, 98849 98867 98869 9887,

99907 9992, 99929 99961 9997,

96469 96479 9648, 9649, 96497

9765, 9767, 9,687 97711 9,729

9888, 9889, 98897 98899 98909

99989 9999,

9651, 9652, 9655, 9655, 96581

9,771 9,777 9,787 9,789 97813

98911 9892, 98929 98939 9894,

89533 E: 8956, 89591 8,813 83833 8,843 83857 83869

81 cl* 8, 84 85

,716, 7,171 77191 77201 ,721,

,820, 78229 ,82,, 78241 78259

,9,9, 79399 ,941, ,942, ,942,

86 8, 88 89 90

7723, ,,2,9 77243 7,249 77261

,827, 7828, ,830, ,830, 78311

794,) ,945, 79481 79493 ,953,

96 9, 98 99 100

7,317 77323 7,339 7,347 7,351

,842, 7843, 78439 ,846, 78479

79589 ,960, 79609 ,961, 79621

84979 84991 85009 85021 8502,

8616, 86171 86179 8618, 86197

87293 87299 8731, 8,317 8732,

EG 89689 89753 89759

25. Numerical PHILIP

Interpolation, and Integration J. DAVIS l’ AND IVAN

Differentiation, POLONSKY

2

Contents Formulas

Page

25.1.

25.2. 25.3. 25.4. 25.5.

Differences ...................... ..................... Interpolation .................... Differentiation ...................... Integration Ordinary Differential Equations

.............

. . . . . . . . . . . . . . . . . . . . . . . . .

898

Coefficients (3 In 58) . .

900

References

. .

Table

n-Point

25.1.

877 878 882 885 896

Lagrangian

Interpolation

n=3, 4, p= -pq]

(.01) E],

Exact

n=5,

6, p=-pq]

(.Ol) E],

10D

n=7,

8, p= -p$]

(.l)

10D

E-j,

*Point Coefficients for k-th Order Differentiation (15k55) . . . . . . . . . . . . . . . . . . . . . . . . . . k=l, n=3(1)6, Exact k=2(1)5, n=k+l(l)6, Exact

Table

Table

25.2.

25.3. n-Point Lagrangian Exact

Integration

Coefficients

(3 in 5 10) . .

914

915

n=3(1)10, Table

25.4. Abscissas and Weight Factors for Gaussian Integration . . . . . . . . . . . . . . . . . . . . . . . . . n=2(1)10, 12, 15D n=16(4)24(8)48(16)96, 2lD

(2In196).

Table

25.5. Abscissas

(2In59)

n=2(1)7,

for

Equal

Weight

Chebyshev

Integration

. . . . . . . . . . . . . . . . . . . . . . . . . .

9,

916

920

10D

25.6. Abscissas and Weight Factors for Lobatto Integration (3lnslO). . . . . . . . , . . . . . . . . . . . . . . . . . n=3(1)10, 8-1OD

920

25.7. Abscissas and Weight for Integrands with a Logarithmic n=2(1)4, 6D

920

Table

Table

1 National 2 National

Bureau of Standards. Bureau of Standards.

Factors for Gaussian Singularity (2in14)

(Presently,

Integration . . . . .

Bell Tel. Labs., Whippany,

N.J.) 875

NUMERICAL

876

ANALYSIS

Table 25.8. Abscissas and Weight Factors for Gaussian Integration of Moments (l
Page 921

25.9. Abscissas and Weight Factors for Laguerre Integration (25~~115). . . . . . . . . . . . . . . . . . . . . . . . . . n=2(l)lO, 12, 15, l2D or S

923

Abscissas and Weight Factors for Hermite Integration . . . . . . . . . ‘. . . . . . . . . . . . . . . . 12, 16, 20, l3-15D or S

924

Table

Table

25.10.

(2
25.11. Coefficients for Filon’s Quadrature Formula (0 _<0_<1) . , 0=0(.01).1(.1)1, 8D

924

25. Numerical

Interpolation,

Numerical analysts have a tendency to accumulate a multiplicity of tools each designed for highly specialized operations and each requiring special knowledge to use properly. From the vast stock of formulas available we have culled the present selection. We hope that it will be useful. As with all such compendia, the reader may miss his favorites and find others whose utility he thinks is marginal. We would have liked to give examples to illuminate the formulas, but this has not been feasible. Numerical analysis is partially a science and partially an art, and short of writing a textbook on the subject it has been impossible to indicate where and under what circumstances the various formulas are useful or accurate, or to elucidate the numerical difficulties to which one might be led by uncritical use. The formulas are therefore issued together with a caveat against their blind application.

Differentiation,

and Integration Central

Differences

25.1.2

&=At+-k) Forward

if n and

Diflerences

Central

fl

Xl

A0

x2

f2

53

f3

6-t

Ai AI

Notation: Abscissas: xO<xI< . . .; functions: valy f(xi> =J;iJ~;;4=~fl. J, f@~cissis Q;,,;, *$t . . . are equally )spac)ed,x t+l--zf=h an’d f,=f(~+pW d”,i”,“d”,.ecessarily integral). R, R, indicate re-

Forward

G

G

Xl

fl

x2

fz

6::

4

6 312

A2

25.1.3

6:

fo 4

A;

Mean

25.1. Differences

Differences

x-1 f-l

x0 f0 Formulas

are of same parity.

k

Differences

dfn)=wn+l+fn-t)

Differences

25.1.1 Divided

Differences

A(fn>=An=At=fn+l-fn 25.1.4

A:=A:+,-A;=fn+r-2fn+l+fn

[xo,xl]=fo-fl=[xl,xol x0--21

A:=A”,+,-Ai=fn+r3fn+zf3fn+l-fn

[x0,x1,. . .,x&,1=[x0, * Divided

A:=&:-A:-‘=$

(-W(;)

fn+k-,

25.1.5

Differences

* *,xk-I]-[%,

in Terms

[X00,%. * .,x,1&

* * .,xk:l

x0-Xk

of Functional

Values

&k-0 Tn(Xk) 877

878

NUMERICAL

where n,(z)=(r-ZO) and n:(z) is its derivative:

(r-21)

25.1.6

ANALYSIS

. . . (z-2,)

Remainder

in

Lagrange

Interpolation

Formula

25.2.3

25.1.7

?r:,(GJ=(a--x0:0>* * . (zk--k-1)(~f-~k+1) . * . cw-&) Let D be a simply connected domain with a piecewise smooth boundary C and contain the points zo, . . ., z, in its interior. Let j(z) be analytic in D and continuous in D+ C. Then,

=?r,(x) .f3 @+I)!

(%<5<Xnn)

25.2.4

IR,(x) 15 “Tn$));+’

max (jcn+l)(x) 1 * iz,IzIz,

25.2.5 25.1.8

[zo,zl, . . .,zJ=&

25.1.9

A:=h”f’“’

(t)

The conditions

(xo<E<xJ

Lagrange

25.1.10

of 25.1.8 are assumed here.

Interpolation,

Equally

n Point

[x0,x1,. . *,xn]-LLf’“‘(E) n!h” n!

(xo<.i<&z)

25.2.6

f(s+ph)=T

Spaced

Abscissas

Formula

&x~)f,+Rn-~

2S.l.11 * * *,&lo, * * .++Bao”

Ix-,,x-,+I,

For n even,

-f

(W-2)

For n odd,

-f

(n-l)
Sk<

f n).

h2"(2n)!

Reciprocal

Differences

i (n-1)).

25.1.12

P(Xo,21)=yy

o-

25.2.7 1

AixP)=(,;2 x0--22 P2(xo,%52)=

);-(~~;)!,, fl (p++t)

+.f1 P(Xoco, X11-P

(a, x0-

P3(~0:0,%~2,x3)=

22)

n even.

x3 -tPh

s)

P2~~0,~1,~2)-~2~~1,~2,~3~

Pnbo,&~ * .,x,)=

X0--% Pn-1(x0,* *

.,&&-d-&*(x1, +Pn-zh

25.2.1 25.2.2

Interpolation

* * .,&a)

p+y-t

>

n odd.

>

25.2.8

* * .,x7&-1>

R,_I=$

25.2. Interpolation Lagrange

n-l fIo A

y (p-k)Pj(“)(t)

Formulas

---$ y (p-k)Af

(xo

k has the same range as in 25.2.6. Lagrange

rn (4 z’(x)=(x-x*)R;(xl)

Two Point Interpolation (Linear Interpolation)

(xZJ *(x~-X*-,)(xi-xt+l)*.‘.-. (x--t-*)(x--i+*) =~x~z?~:. (x,-x,) 25.2.9

25.2.10

j(xo+ph)

= (1 -p)jo+pfi+R,

R,(p) =.125h2f’2’(5)

=.125A2

Formula

NUMERICAL Lagrange

Three

Point

Interpolation

879

ANALYSIS

25.2.18

Formula

MP) =

25.2.11

.0049h6j’E’(t) = .0049A6

KKP
f(xo+phl =A-,f-~+Aofo+Afi+Rz

.0071h6j(6)(~) = .0071As

(--l
.024hBf6'(.f) =.024As

P(P-1) “~j-I+(l--pZ)jo+p~jl

l
(-2<~<--1,2<~<3) (2-2
25.2.12 Lagrange

R2(p) = .065h3fc3’(i) = .065A3

(IPILl)

Lagrange

Formula

Four

Point

Interpolation

25.2.19

25.2.13

25.2.14 .024/-&j'"(() =.024A4 .042hy'"(l) =.042A4 Lagrange

Five

R,(P)

(P) =

25.2.21

l
j-2-(~-1)~(~2-4)f-

+(P”-l)(P2-4) 4

jo-(P+l)P(p2-4~ 6

j

1

+(P2-l)P(p+2)

24

25.2.16

R4(p)

.031hyf’s’(f) =.031As

(l
2

Interpolation

Formula

,-P(P~-I)(P-~)cP-3) 120

Formula

AJi+R7 (O
I

.0014h*j’8’ (c$)n. .0014A8

(--l
.0033hsj@‘(E)= .0033A8

(-2
.016h8j@‘([) = .016A8

c--3cp.e2) (3
2

24 j(P2-4) (P-3) 1

(P2-4) (P-3) jl-P(P2-l)

12

Interpolation

.OO1lhsj’s’(~)=.0011A8

fbhJa,x1>=

j-

f(4xo,x2)=

_ (P”- 1) (P2-4) (P-3) f 0 12 +P(p+l)

Point

Iteration

Method

25.2.23

Atf,+Rs

+PlP-1)

(2
Let f(4~o,21,. . .,x,J denote the unique polynomial of kth degree which coincides in value with j(x) at x0, . . ., xk.

k2<5
25.2.17 =ik2

Eight

Aitken’s

Ma

Six Point

f

=

.Olahy(s)([) = .012As

Lagrange

R7 (P) =

1

6

(l
N .019A’

25.2.22 c

24

.019h’j”‘([)

(IPl
.0046A'

fbo+ph) =i$,

Formula

z(~2-l)P(~-2)

Formula

(2-3
25.2.15

fh+ph)

.0046h’j”’ (0 =

Lagrange

Interpolation

Interpolation

.0025h’j”’ ([) A .0025A’

1

=

(O
Point

Rs

Point

f (xoofph) =i&3 Adi+&

25.2.20

j(~+ph)=A-,f-,+Aojo+A,f,+A,f,+R3 ,-P(P-WP--2) f-I+(P2-1)(P--2)f 0 2 6 -P(P+NP--2) j1+P2!j2 2

Seven

f(4xo,%%)=-

(p+2) (P-3) 24 f

+P(PW

(P2-4)

120

“f3

2

fc+o, Xl,x2,

x&l?

::A

AoI-gI

:::I

1 x2-21 1

x3 rel="nofollow"> =-

X3-Q

f(~l~O,~l)

x1-x

f(xlxo,

x2)

22-z

f(xlxo,

Xl,

x2>

x2-x

f(xlxo,

Xl,

23)

x3-x

880

NUMERICAL Taylor

ANALYSIS

Everett’e

Expansion

25.2.24

Formula

25.2.31

f(xo+ph)=(1-p)jo+p~l-p(p-13)l(p-2) + (x-XOP n!

+(P+lMP--l)

P++Rn

s2+

1

3!

R,=

25.2.25

+(

$$)

6;

F’-t-

. . .-(p22;1)

R2n

=(l-~)fo+pj,+E26~+F26~+E,6~

@3<5<4

-l-F&Newton’s

Divided

Difference

Interpolation

Formula x0

fo

Xl

2,

25.2.26

s: 61

0l=zf0+$~

n+--l(4 [a,xl,

s”,

. . .,4+R,

. . . +Rz,

q 8;

s:

e:

25.2.32 %I fo

RZn=h2”+2 ( &y2>f(2”+2)

[x0,XII

Xl f*

[X0,%X21 h%zl

x2

[~0,~1,~2,&1

j2

6)

-(;+“2) [“““;A”“]

[%X2,X31

(x-,
[x2:2,231 x3

f3

Relation

25.2.27

Between

Everett

and

Lagrange

25.2.33

R,(x)=*,(x)

[xo, . . .,x.,x]=r&)‘~

(xo<~<x?J

(For T, see 25.1.6.) Newton’s

Forward

Difference

Ez=A’_,

E4=AB-*

EB=Af8

F2=A:

F,=A:

F,,=&

Everett’s Formula With Throwback (Modified Central Difference)

Formula

25.2.34

25.2.28 g

f(Xo+Ph)=fo+pA~+

&+

f(xo+ph)=(l--p)fo+pf,+E2~a,,o+F2C.,+R

. . . +@A;+&

0

25.2.35

x0 fo Ao Xl fl x2

j2

x3

f3

25.2.36

& Al

G

6f,,=62--.18464 R=.00045~p6;(+.00061~6;(

25.2.37

A: A2

f(s+ph)=(l--p)fo+pf,+E26~+Fzs:

25.2.29

+E&,,o+F&,+R 25.2.38

Relation

Between

Newton

and

Lagrange

(xo
25.2.39

Coefficients

25.2.4Q

25.2.30

0

Coefficients

; =&(P)

fbo+ph)

@=-At,(p)

0

; =A;(l--p)

0

; =A;(2--p)

6:=64-.20769-

. . .

R= .000032~p1s~~+.000052~s’,~

= (1 -p).fo+pfi +E&+ F& +E4~+F,s:+E,sa,,o+FgbB,.l+R

25.2.41

68,=6°-.21868+.049610+

25.2.42

R =.0000037[rs;~+

... . . .

NUMERICAL Simultaneous

881

ANALYSIS

25.2.54

Throwback

m=2n+l

25.2.43 ak=Gl

f(xo:o+ph)=(1-p)fo+pjl+~2~~,0+~*~~.1

g j, cos kx,;

b,=2&

$’ jr sin kx,

+J%G~~-F,$,I+R

(k=O,l,

25.2.44

6;=62-

.0131266+.004368-.001610

25.2.45

6;=a4-

.27827P+ .06856*- .01661°

25.2.55

R = .00000083 I/d;1 +

25.246

2

Formula

With

1 2n-1 jr

COS

kx,;

bk=;

2

jr

sin

kx7

.00000946’ (k=O,l,

Bessel’s

m=2n

1 2n-1 at=;

. . .,n)

. . .,n)

(k=O,l,

. . .,%-I)

Throwback

25.2.47

b, is arbitrary. Subtabulation

f(zo+ph)=(l--p)fo+pfl+B2(6~,o+6K.1)

+B38;+R, B2-p(p;l), 25.2.48

B3,p(p-l;

k-3)

6;=62-.18464

25.2.49

Interpolation

initially in intervals of to subtabulate f(z) in Let A and x designate to the original and the

final intervals respectively.

R-.00045(&[+.00087/6~[ Thiele’s

Let j(x) be tabulated width h. It is desired intervals of width h/m. differences with respect

-.f(z,). Assuming that differences are zero,

Formula

25.2.50

Thus xo=j the original

x0+: ( ) 5th order

25.2.56

f(x) =.mJ + x-x,

x0=$

(l-m)(l-2m) 6m3

Ao+ &:A;+

ti

P(z1,~2)++~2

+(l-m)(l-2m)(l-3m) 24m4

Pz(z,,22,23)-f(21)+2--23

(

P3(%~2,5,&) -P(c92)+.

**>

ti

(For reciprocal differences, p, see 25.1.12.) Trigonometric Gauss’

25.2.51

Interpolation Formula

f(x) -@klk(s)=tn(z)

25.2.52 sin #(z-G) . . . sin $(2--x~-,) ln(x>= sin ~(x~-zo) . . . sin +(z~-x+~) sin $(2-~~+~) . . . sin i(z-x2J sin $(xk--4+1) . . . sin 3(~~-2~~)

From this information we may construct the final tabulation by addition. For m=lO, 25.2.57 L\o=.1Ao-.045A~+.0285~-.02066A~

t,(x) is a trigonometric polynomial of degree n such that tn(zn) =fn (k=O,l, . . .,2n) Harmonic

~=.OlA$.OO9A:+.OO7725& Z=.OOlG-.00135&

Analysis

z4= 0 *OO01A4 0

Equally spaced abscissas 51, * * .,&-l,X~cm=27r

x0=0,

Linear Find

25.2.53 f(X) dj

Inverse

P, givenj,(=j(~+ph)). Linear

a0+2

(ak cos kx+bk sin kz)

25.2.58

Interpolation

882

NUMERICAL Quadratic

Inverse

ANALYSIS

Bivariate

Interpolation

25.2.59

Three

Interpolation

Point

Formula

(Linear)

25.2.65 Inverse

25.2.60

Interpolation

by

Reversion

Given f(xo+ph) =jp=&

of Series

spk

25.2.61

+

. . ., A=(jl,-ao)/al

p=X+c2X2+cJ3+

f(s+ph,yofq~)=(l-p---)jo,o

25.2.62 c2= -a2/al

+pfi,o+efo,l+O(h2) Four

Point

Formula

25.2.66 0

Inversion

of Newton’s

Forward

Difference

+

Formula

f(xo++h,yo+qk)=(l-p)(l--)jo,o+p(l--q)fl,o

25.2.63 ao=fo

+a(l-p)jo,1+pefi,1+O(h2) Six Point

Formula

25.2.67 &

A;

a2=T-;i-+24+

A: A: a”=x-z+ G a’=%+

Inversion

...

0

... +

...

(Used in conjunction 25.2.64

llg

f(xo+ph,yo+qk)=q~

with 25.2.62.) of Everett’s

Formula

joy,+‘p

+u+P!J-P2-!12)fo,o

ao=jo

+P(P--2P+l)

a,=a,-$-$+!t+!i+

+!zk2Pfl)

25.3. Differentiation

25.3.1 ...

with 25.2.62.)

.f(x)=$o

Formula

z;(djk+R:@)

(See 25.2.1.) 25.3.2

(Used in conjunction

fo,l+Pafl,l+ow)

2

Lagrange’s

-S:+S: 120

f LO

2

...

g s; a2=2-zi + . . .

a5=-+

j-l.0

?mw

1; (x) =& j;rk”

(x-xk)

(x-+ddxk)

NUMERICAL

25.3.3

883

ANALYSIS

25.3.10

h$t”=&+&+++;+

...

25.3.11

h3j,++;~;+&+!?~+. Equally

Spaced

..

Abscissas 25.3.12

Three

Points

h~j++2$,+!..&++

25.3.4

...

25.3.13

h~j~“‘=&d~+~&+~+ Four

Points

Everett’s

25.3.5

Formula

25.3.14

j;=jys+*h)+{

-3p2-p+2j-l j

+3p* -4p-1

Five

3p2-2p-2 2

f1

Pointa

A:+?

j6=j’(~+ph),;(2P3-3~~-P+l

5p4-20p3+15p2+10p-6 120

,,+5$-15p2+4 0 120

hj; c: -jo+j1-,

j-*

A,,=hj;+; +2p3+3pe-p-1

For; numerical values of differentiation cients see Table 25.2.

of Derivatives

n)+;

jia)+$

je)+;

coeffi25.3.18

Formulae Formula

in Terms

s:+231 %+a1 8

25.3.17

fi)+R;

12

Difference

1

25.3.16

_4pa+3p*--8p-4 6 -f1

Forward

1 6-g

Differences

f4 r-l+Qfpjo

Markoff’e

Differentiated)

25.3.7

25.3.19

25.3.20

+3~‘-6~+2~+ 6

25.3.21

. . . +$

@

&]+R:

25.3.8

+

(a0
hj;=A,,-a@+;

6:

25.3.15

25.3.6

4p3-3p*----8p 6

~-jo+jl-3p2-~+2

hj’(G+ph)

0-

2

(Newton’s

...

A;-$+

...

ajo 0 z=%

1

(ho-j-l.o rel="nofollow">+O(h*)

j:“’

r.’ 1

NUMERICAL

884

ANALYSIS

25.3.26

25.3.22

wo0 1 (j1,1-j-l.l+fl.-l-f-l.-l)+O(h2) *=a

a2jo 0 --L=~ bxby 4h2 Cr,,l-ji,-I-f-l.l+j-l.-1)+0(h2)

25.3.27

25.3.23

0

0

t

+

woo 1 Lbx2 h* (fl,o-2f,,o+f-,,,)+O(~2)

a2jo )=-

0

bxby

1

-1

2h2

Ci1,o+j-l,o+fo,l+jo,

-1

-?fo,o--f,,,-j-1,

-J+0(h2)

25.3.24 25.3.28

+ wo

d=L 32

0

12h2(-fi,o+l6j~o--3Ojc,,o +Xj-l,o-j-z,o)+O(h4)

a4.fo 0 d ==+, bX4

.f-~~>+

25.3.29

25.3.25 0

0

0

0

0

0

l

0

+

+

b”fo 0 g=gp

-?f-LO+

V2.0-4j~.0+6j0,0

1

avo o i

----=- h4 cfl,l+f-l,l+fl.-l+j-l,-1 bx2by2

~~l.~-2~o.~+j-~.~+j~,o-2jo,o+j~~,o +.fi. -I--?fo,

-&-I,

-JfO(h2)

--2f,,o-

2f-~,o--jo,~--2jo,-,+4jo,o)+O(h2)

O(h2)

NUMERICAL

885

ANALYSIS

25.3.33

Laplacian

25.3.30

v4uo,o=& I---(uo,3+uo.-2+u2,o+u-2,0)

25.3.31

25.4. Integration Trapezoidal

Rule

25.4.1

J)m=; uo+x,-; SC’ (t--5) (21--t)j”(t)dt v2uo.o=1&2 [--GOuo.o+l6(a,,+u,,,+u-l,,+u,,-1) (u2,o+uo.2+u-2.o+uo,

-2)l+w4)

Extended

Trapezoidal

Biharmonic

Rule

f.

25.4.2 Operator

25.3.32 Error l

0

0

+

0

Term

in Trapezoidal Functions

Formula

for

Periodic

If f(z) is periodic and has a continuous ktb derivative, and if the integral is taken over a period, then constant 25.4.3 IError I mk Mod&d

25.4.4

Trapezoidal

Rule

NUMERICAL

886 Simpson’s

ANALYSIS

Rule

s

zyj(x)dx=;

[fo+4h+.f*l +y-“’

z~‘~b9dx=h

Newton-Cotes

(x~-t)*(xl-t)p(t)dt

Formulas

(For Trapezoidal 25.4.6.) 25.4.13

(x,-t)*(xl-t)p(t)dt =1 =;

Extended

f -& =-

*

A,(m>f<+Rn

(See Table 25.3 for A,(m) .)

20 +;J-“’

[!I

S

15.4.12

25.4.5

(Closed

Type)

and Simpson’s Rules see 25.4.1(Simpeon’e

i rule)

[fo+4h+j2l-~j”‘(E)

Simpson’s

Rule

25.4.6

+32j3+7fi)-8’;$h’ 25.4.15 Euler-Maclaurin

Summation

Formula

S

‘sj(z)dx=$8

25.4.7

wh+75j1+50j2+50

20

-

+w4+19fa)

275j (O)(5) h’ 120g6

25.4.16 (41f,+216f1+27f2+272fa

Rlr=~R2w2h2k+3 zo!$~*“~f(2k+2’

(2/t+2)!

(For B,,, If jw+a q,
Cx) 1)

(-1gEl)

Bernoulli numbers, see chapter 23.) (2) andf’2R+4)(x) do not change sign for then IRzkl is less than the first neglected f(2k+2)(5) does not change sign for lRzkl is less than twice the first neglected

+27j~+216j~+41j,kg';;~~hs 25.4.17

S

d’J(x)dx=&

+2989f,+2989j4+1323js+3577jLJ Lagrange

Formula

+751j7) -

25.4.8 s

abj(4dx=&

(L:“‘(b)--LI”‘(a))fi+R,

=s z. ef(ddx=&

518400

w

(989j,,+5888jr928js

+10496f,-4540j4+10496j,-928js+5888j7

25.4.9

2368fW +9@! 8)_ 467775

?, dt= ’ Z,(t)dt s 20 t--2, s 20 25.4.19

S

25.4.10 R,=- &!

:r.(rlf’“+“(~(x))dx

Equally

Spaced

S

z)4dx=&o

Abscissae

25.4.11

z;f(x)dx=$,

8183j’q~)hQ

25.4.18

S

(See 25.2.1.)

S

(751j,,+3577fI+1323fi

{2857(fo+fB)

+15741(fl+fa)+1080U2+~7)+19344(fa+fe)

go I j, &.

s” T,(2> dx+R, * IO x-x<

*See page Ix.

+5778(f4+fs)}

-&gl”(Uh”

([)h"

NUMERICAL

25.4.20

Io-h

210 =.

zo+h s

ANALYSIS

I16067 Cfo+fm)

f(z)d~=&~

mz=;

887

I24f(zo)

+4wo+w

+f(zo--h)]

-[f(2o+ih)+f(zo--h)ll

s

+R

/RI 2 ls90 lh” Max ~f~“~(z) I, S designates the square rrs with vertices z,+Ph(k=O, 1,2,3); hcan be complex. Chebyshev’s Newton-Castes

Formulae

(Open

Equal

Weight

Integration

Formula

Type)

25.4.21

Abscissas: xr is the jth zero of the polynomial of

part

25.4.22 (See Table 25.5 for z+)

For n=8 complex. Remainder:

25.4.23

R,= 25.4.24

S:“f(x)dx=$

and n> 10 some of the zeros are

S

_:’ &!

f(“+l) (Odx

-2

n(n+l)!

(1lj~-l4j2+26j3-l4f,+l~.f3)

+4lj'"(~)h' 140

where E=E(x) satisfies OI.$<x (i=l,

25.4.25 Integration

of Gaussian

Type

see chapter 22)

Formula

1

-Ij(z)dz=$

25.4.29 s

and O<&<X~

Polynomials Gauss’

25.4.26

j-1

. . . ) n)

Formulas

(For Orthogonal

A x:+If’“+“(&)

z;j(rc)dz=s5

(46Oj~-954f,+2196jr2459j,

+2196f,-954f,+46Of,)+Five

Point

Rule

for

Analytic

l~l;;r(3)

we

Related ortbogonal polynomials: nomials P,(z), P,(l)=1

Weights:

w,=2/(1-ti)

20

h

zo+

Gauss’

h 25.4.30

+ to-

Legendre

poly-

*

[P:(cc~)]~

(See Table 25.4 for xt and w,.)

ih

R,= (2n2;;J1;‘;;;l 2,-

+Rn

Abscissas: x1 is the it” zero of P,,(Z)

Functions

25.4.27 z,+

w.f(4

s

ih *See page II.

Formula,

!13fzn)

(4)

Arbitrary

(-l
Interval

NTJME RICAL

888

*

AN-ALY SIS

25.4.33

Related orthoLcona1 polynomkk: P,(Z), P,(l)=1 Abscissas: xi is the it” zero of P,(x) Weights: wl=2/(l-x:) [Pb(x31”

s0 Related

orthogonal

(2”)

R = (b-u,2n+yn!)4

n (27&+1)[(2n)!]3f

’ Y(z)dx=&

mf (4 +R.

polynomials:

qn(x)=4k+27L+1P;“‘0)(1-2x)

(o

(For the Jacobi polynomials Radau’e

Integration

Formula

Abscissas:

25.4.31

xi is the i* zero of q,,(x) l f!xMr=~f-l+n~

WJ(XJSR”

s -1 Related

Pir,o’ see chapter 22.)

Weights:

polynomials:

*

wr= { z [*,(zr)l’} -I

Pn-l(x~)+Pn(d x+1

(See Table 25.8 for x1 and w,.) Remainder:

Abscissas: CC*is tho z? zero of p,-,(x)+P&> x+1

25.4.34

Weights: 1 wi=2

1-x<

1

1

[P,-l(xJ2=iq

[P:-,(,,)I2

Remainder:

Related

R,_[(;;-I;;L!13

[ (7&-1)!]4f(2*-1)(,$)

Lobatto’s

Integration

(---1<,E
Formula

25.4.32 s

‘,f (ddx=

%q&) vu)+f(--l)l n-l +g

wf(xJ

+Rn

v’l--zdx=& wJ(xt)+Rn s.‘f(d orthogonal --A&i

polynomials:

PZn+l

(a=),

P2,+1(l)=l

Abscissas: xr= 1 -f$ where .$* is the jth positive zero of Psn+,(x). Weights: ~~=25!~1~~+l’ where w12”+l) are the Gaussian weights of order 2n+ 1. Remainder: 24”+3[ (2n+ l)!]’ (O
Related

polynomials:

Ph-,(x)

Abscissas: xt is the (i-l)llt

s

abf(y) db-ydy=(b-a)3’2

zero of P:-,(x)

w&/r)

yi=a+ (b--ah

Weights: 2 W’=n(n-l)[P,-,(z~)]Z

Related

(XiZfl)

(See Table 25.6 for xI and wl:)

(2~+-1)[(2~-2)!13

41-x

f""-"(G (---1<E
*see p*ge II.

orthogonal 1P2,+1

Remainder: R =-n(n-1)322n-l[(n-2)!14 n

fi

polynomials:

w--2),

P2,+,(1)=l

Abscissas: x1= 1-5: where lr is the ith positive zero of P2n+1(x). w*&?gWy+l where wi2n+1) are the Weights: Gaussian weights of order 2n + 1.

NUMERICAL

25.4.36 Related

s

o1ex

as=8

orthogonal

ANALYSIS

889

Abscissas:

wJ(xJ +R,

(2&l)* xr=CoS 2n

polynomials:

Weights: w,2

PZn(-+Qz), Pzn(l)=l Abscissas: xl=1 -Et where tr: is the ?” positive zero of Ptn(x). Weights: w1=2wj2”), w/*“) are the Gaussian weights of order 2n. Remainder: z4”+’

[cw

Rn=4rL+1

(6)

!I” j(2m)

(O<W)

[(47&l!]*

25.4.40

n

+1 --1 f(x) +x*~~=&d(x3+Rn

s

Related orthogonal polynomials: Chebyshev nomials of Second Kind sin [(n+l) arccos x] U,(x)=

sin (arccos

Poly-

2)

Abscissas: 25.4.31

’&

dy=&a

s a db-y

5 w&y,) i-1

+ R,

x*=cos .-I.-

yt=a+(b--ah Related

orthogonal

7r

71+1

Weights:

polynomials:

wi

P2nG--LPzn(l)=l

=- lr

sin* - i

n+l

‘]r

n+l

Remainder:

Abscissas: where tr is the it” positive

q=l--[:

Weights: ~,=2wj*~), of order 2n.

zero of P,,(x).

WI*“’ are the Gaussian weights s.‘&/-4

S

25.4.38

+’ f(z)

-1

dx=g

(b--7J)f WY=(~)*

3

Chebyshev

y.=b+a 8 2+2

PolyRelated

orthogonal

mx~,w1)=&

xf=cos n+l

Weights:

Remainder:

25.4.42 (--l
(b- y) y_b+a f T+T”’

Related orthogonal

=& wJ(yi)+R, i-1 b-a

polynomials:

2)

=

i - lr sin* WL=n+ 1 n+l

n

S’ f(y)dy

sin (arccos

i

Weights:

R.=~2n~~2n-,f(2nY~)

sin [ (75+ 1) arccos x]

Abscissas:

xI=cos w-1)s 2n w,=L

b-a x*

polynomials:

U&> =

Abscissas:

= &,+a)

wtf (yr) +Rn

w, j(x,) +R,

414

Related orthogonal polynomials: nomials of First Kind

25.4.39

8

s Related

1 of(x)

J

orthogonal

ex

dx=&

* w.f@i> +Rn

polynomials: $

T*n+I(m

Abscissas: x*=cos*

2i-1 y-&j

A -2

Weights: 2r w=2n+l

T,(z),T.(1)=& *See

pageII.

xi

NUMERICAL

890

ANALYSIS

Remainder:

Remainder:

25.4.43

Filon’s

Integration

Formula

a

25.4.41 x--a dx= (b-u)

2

i=l

wJ(yJ SR,

S

Zhf(x) co9 tx dx=h

y,=a+(b--ah Related

orthogonal

20

[

a(th) ( f2n sin tq,

-fosin t3 +Nth) -C,,+r(th) *C2+1

polynomials:

+;

th4S:,_,]-

R,

25.4.48

Abscissas: x

t-

--cos2

Weights: w=2n+l

2i-l -.-

g

2n+l

2

2s

25.4.49

xi

R c2n-1=~f26-1

cos

txn-1

i=l

25.4.44

S 0

l In xf(x)dz=&

wJ(x,>

25.4.50

+R,

S4,-l=C

Related orthogonal polynomials: polynomials orthogonal with respect to the weight function --In x Abscissas: See Table 25.7 Weights: See Table 25.7 25.4.45

n

f#,

sin tx2t-1

i=l

25.451 R.=~~hy”‘(~)+O(th’) 25.4.52 1 sin 20 2 sin2 8 a(e)=s+~-fP

w,f(xi> +& So-a-Y(x)dx=Z$

Related orthogonal polynomials: Laguerre nomials L,(x). Abscissas: x1 is the it” zero of L,(x) Weights:

polyFor small e we have 25.4.53

283 2e5 2e7 cu=jg-~5+4725-

** -

(See Table 25.9 for xr and wt.) Remainder :

y=~-g+~o-&~+ 25.4.54

25.4.46

S

=rnf(x) sin

Related orthogonal polynomials: Hermite nomials H,(x). Abscissas: xt is the ith zero of H,(x) Weights: 2”-‘n! & n2EL-1(~012 (See Table 25.10 for xr and wt.)

...

tx dx=h[a(th)

(f.

cos

IS-fin

+B&+Y&,,--~+~

poly-

CO9

1

th4CL,

--Rn

25.4.55 S2n=I&

f 2i sin (tx2d -f

[fzn

sin

(tx2.)+f0

t&J

sin

(tx0)l

a For certain difficulties associated with this formula, see the article by J. W. Tukey, p. 400, “On Numerical Approximation,” Ed. R. E. Langer, Madison, 1959.

NUMERICAL

25.4.56

fJ2.-l=~j2c-1

ANALYSIS

891

(Xr,Yr)

sin (taf-J

(0,

25.4.57

Wi

112

0)

(*ho),

(0,khh)

R=0(h4)

l/8

(See Table 25.11 for a, /3, y.) Iterated

Integrals

25.4.58 s 0 z at,

s 0

In d&e, . . . s 0 13dt, s 0 “f(t,)dtl (x- t)“-tf(t)dt

25.4.59

s s =dtn 0

” dtn-, . . . l3 dt, 12f(tl)dtl a sa sa l =- (x-a)n t”-‘f(z(n-l)! o

S

Multidimensional

Circumference

(x-a) t)dt

Integration

of Circle

h,Yf)

wr

(+;)

l/4

(Xf,YJ

Wf

(0, 0)

Ai2

(&O)

l/12

R=0(h4)

r: d+y2=h2.

25.4.60

’ sr.f(z,y)ds=& 2*h

$f(h

cos Et h sin z) +O(h*m-a)

Circle C: z2+y2
1 *h2 ss cjCw)duly=$

l/12

w.f(x,,yJ

+R

R= O(h’)

892

NUMERICAL

hY3 (0,0)

WC

CM, 0)

l/24

(0,fh)

l/24

Square4

*i

(0, 0)

l/4

(zt$h,O)

l/s

S:

lzl
25.4.62

116

(Xr,Yt)

ANALYSIS

R = O(he)

(%Yc)

*f

(O,O)

419

( fh, fh)

l/36

(fh,o)

l/9

(0, f h)

l/9 0

w--B

R=O(he)

r-l

1

0

i 0 1 0 I ..a--

(Xi,Yr)

*i

(&A&,

;th,@

l/4

(XOYJ

16/M

4 For regions, such as the square, cube, cylinder, etc., which are the Cartesian products of lower dimensional regions, one may always develop integration rules by “multiplying together” the lower dimensional rules. Thus if

(0, 0)

6-d6h (J

10

cos g,

I- o’f(z)dz

(k=l, 6+*h 10

2?rk 6+& cos loy J 1.

R=O(h4)

*i

NO) kG,YJ

R=0(h4)

h

sin

2d 10

. . .,lO>

is a one dimensional

=&

Wifh)

rule, then

-IS--&

360

R=O(hlO)

becomes a two dimensional rule. necessarily the most “economical”.

Such rules are not

NUMERICAL

(l

$

h,f $

h)

‘ANALYSIS

893

251324 R=O(h6)

(,,h$

lO/Sl

h)

lo/81

(+,o) Equilateral

Radius

Triangle

of Circumscribed

T

Circle=h

25.4.63 1 a&h2

Tf(x,~)d2dy=I&

ss

270/1200

wif(x,,~r)+R ((9)

h,,O >

I

155-yml 1200

I

h,

((-T+‘) --w-

+-

--

h(v)

W

D I

R=O(he)

314

((-i+l)

,Bh)

40 >

155+fl

h,*(F)

((T)

@h)

1200

R=0(h3) Regular

l/12 Radius

Hexagon

of Circumscribed

H

Circle=h

25.4.64 1 ssH

;fihz

Rx, y)drdy=&

wJ(xr,

YJ +R

I 1 --u--me t---lI --I I

I I

(Xi,YJ

Wi

o-40)

27160

(h,o)

3160 3160

R=0(h4)

(X*,Y*)

Wi

(0,O)

21/36

8160 (f;,&; 8160

(fho)

$3)

5172 5172

R=O(h4)

NUMERICAL

894

0

ANALYSIS

I I

I

---

:4 --

0

(&&h,+,O)

l --*

--

(*&h,o,+)

’

l/15

0

R=O(he) 258/1008 125/1008

>

R=Oth9

125/1008 Surface of Sphere Z:

l/30

(0, *ho)

1+y2+z2=h2

(o,o, fh) WI

(Xi,Yl,%)

(*&h,$h,&&h)

27/840

25.4.65

(&$h,O,&&h)

32/840

R=O(h”)

(fh,O,O) (0, f&O)

40/840

to,% f h) Sphere S: za+y2+z2Sh2 25.4.66

(k&O,@

WJ

(o,khh,O)

l/6

R=O(h4)

1 4 -rh3 3

-SSS

.f(w,

,g

dd=Wz=&

wJ(xi,yi,

4 +R

NUMERICAL

(Xf,Yf,Zf)

Wf

((40,

215

0)

(fh,o,o)

l/10

(0, *hh,o)

l/l0

(o,o, fh)

l/l0

ANALYSIS

895

~ji=sum

of values of j at the 6 points midway from the center of C to the 6 faces. ~ji=sum of values of j at the 6 centers of the faces of C. cjO=sum of values of j at the 8 vertices of C. ~j~=sum of values of j at the 12 midpoints of edges of C. ~jd=sum of values of j at the 4 points on the diagonals of each face at a distance of

R=O(h’)

Cube 6 C: Izj_
i&h from the center of the face.

Ivl Sh

Y

bllh 25.4.67

1 8ha sss f(z,y,z)dzdydz=&

wJ(xi,yr,4+R

C

Tetrahedron: bf,Yf, zr>

Wf

(hhh,o,o)

l/6

(0, f h, 0)

116

(0, 0, *h)

l/6

5

25.4.70

R=O(h”) +terms

of 4th order

+terms

of 4th order

25.4.68

where

=ko [-496j~+128~j,+8r,j,+5CfD1+O(he)

C jD : Sum of values of the function of ..F. I

25.4.69 =ko

V: Volume of F

[91~f,--40~j~+16Cj~l+O(h6)

where f,,,=f(O, 0,O). 6 See footnote

to 25.4.62.

Cf.:

at the vertices

Sum of values of the function at midpoints of the edges of 37 cj,: Sum of values of the function at the center of gravity of the faces of 9Y jm: Value of function at center of gravity of .F.

NUMERICAL

896 25.5. Ordinary

Differential

Equations”

First Order: y’=f(z, Point

25.5.1

Slope

y)

ANALYSIS

25.5.9 Y.+I=Y.+;

Formula

k,+;

ka+O(h’)

k,=hf(~n,yJ,kz=hf

(xv,+;

h,y%+; kl)

Yn+l=Yn+~Y:,+W) ka=hf

25.5.2

h, y,+i

(x%+%

k2)

Ytl+l=y,-*+2hY:,+wa) Trapezoidal

Fourth

Formula

Order

25.5.10 25.5.3

Y.+*=Y.+; Adams’

(Y:+l+y:)+o(ha)

Y.+I=Y~+;

Extrapolation

Formula

k,+;

kz+; k,+;

kl=hf@n,yn).,kz=hf

k,+O(h*)

(~,a+; h,y.+;

kl)

25.5.4 ka=hf (~-l-a

h,y.+!j

kt), k4=hf(z,+h,y,+k,)

(55yh-59y:,4+37y:,4-9yL>+O(h*)

Yn+l=Y.+&

25.5.11 Adams’

Interpolation

Formula

Y.+I=Y.+;

k,+;

k,+;

k,+;

k,+O(h*)

25.5.5 kl=h~(r.,y.),4=hf(a+5),k~=h~(~~+~ h,y,+; Y.+l=Yn+$

kl)

(~:+1+19Yl-5y:-1+Yh_z)+O(h6>

ka=hj (xn+; Runge-Kutta

h, y,r;

k,+k,),

L=hf(z,+h,y,+kl-k~+k,)

Methods

Gill’s Second

Order

25.5.12

25.5.6

Yn+l=yn+;

h=hf(s,

25.5.7

h,y,+;

(1-d)

kz

kl)

ka+kd)+O(hb)

YJ

kz=hf (z.+;

Yn+l=yn+kz+O(ha)

Third

(k,+2

i-2 (I+&)

kl=hf(~n,Yn),ka=hj(x,+h,y,+kl>

kl=hf(r.,y,),ka=hj(x,+~

Method

ht y,+;

kl)

ka=hf(~n+~h,y.+(-~+&)kl

Order

+(1-&h)

25.5.8 h=hj Y~+I=YII+~

k,+$ k,+k ka+O(h*)

(

x,+h, yn -&,f(l+&) Predictor-Corrector

kl=hf(Zn,yn),kz=hf

(z,s+;

h,y.+;

kl)

ka=hfb+h,y,-kl+2k2) ‘The reader is cautioned against possible instabilities especially in formulas 25.5.2 and 25.5.13. See, e.g. [25.11], [25.12].

Milne’s

ka) Methods

Methods

25.5.13 P:

Yn+,=Yn-a+4i

c:

Y.+l=Y.-I+;

(2y~-yy:-1+2y:-,)+O(h*) (y:-l+4y:+y:+1)+O(h*)

NUMERICAL

897

ANALYSIS

,25.5.14

p: Y.+l=Y.-5+;~ (1ly:-14Y:-I +26y:-,--14y:-,+lly:-,)+O(h’) c:

ka=hf(x,th,y,+k,,z,tz3)

2h

Yn+l=Yyn-3+rs (7YJ+,t32y:

Z4=hg(z,+h,y,t~~,z,+zr) Second

+12y:-~+32y:-*+7y:-3)+O(h7) Formulas

Using

Higher

p:

c:

R+l-Yn+;

Method

Yk+~=y:-3+$

(2y~k--y~.q+2y’,)+O(h5)

(2: Yk+l=Y:-,t!j (Yh+l+yh) -&

y, y’)

25.5.19

25.5.15

yn+,=yn-2+3(yn-yn-J+h2(y::-yL)+O(hs)

y”=f(z,

Milne’s

Derivatives

P:

Order:

(y::-l-t-4y;+y;+&t-O(h6)

(d’+:+1---y3+W5) Runge-Kutta

25.5.16

Method

25.5.20

p:

y.+l=y~-3+3(y~-y,-~)+~(y~~+y~~~)

c:

Yyn+l=Ys+g (Yb+l+Yb)-pj

h

tO(h') h2

yln+l=yn+h 1ybt; (k,tkz+kJ]+O(h9 Yk+l=Y:t;

(!/ii+1 -Y3

(h+%+%+k,)

kl=Mxn,y,,y’,>

+go

@k’L+yk”)

+O(h’)

y:+,,y,+$)

kz=hf(z,+;h,y,+; Systems of Differential First

Order:

Equations

y’=f(x,y,z),

Second

Order

ka=h.f (xn+;

z’=g(x,y,z).

Runge-Kutta k4=hf

25.5.17 ~n+l=~n+;

(h+kJ

2 .+1=2.+; kl=VCwwn),

vl+z2) +O(h3)

Order: Milne’s

Fourth

Order

c:

~y::+10yi,‘-:-l-ty;-:-,)+O(h6)

Runge-Kutta Runge-Kutta

Method

(k,+21ez+21c,+k,)+O(h6), 2.+I=%+;

h=hf(~n,Yn,Gb)

k=hf

(5y::t2y~-:_,t5y~-:-,)+O(h6)

Y.=2yn-ryn-St;

25.5.18 Y.+l=Yn+;

y”=f(x,y) Method

t; z,+Z,)

k3,y;tk3)

Yn+l=Yn+Yn-2-yn-3

kz=hf(x,+h,y,+kl,z,+zl), 4=Mxn+hy,+k,,

k,,y;,t+)

25.5.21 p:

h=hg(x,,yy,,zn)

y;+;

(zn+h,y.+hy:+; Second

+O(h3),

h,y.t;

(-.+a

(Z1+2Z,+2Z3+ZJ+O(h5)

Y:+l=yht;

k,-t;

k,+;

k3

4=Mxn,yn,Zn) by,+;

k,,z,+f

k2=h.f(x.t;~y.t;

zl) z.+;,

y,+;,

y:t;

k,=hf (z.+h,y.+hy:t~kz).

z,,+$) *see

pnge

II.

kl)

NUMERICAL

898

ANALYSIS

References Texts (For

textbooks

on numerical analysis, see texts in chapter 3)

[25.1] J. Balbrecht and L. Collats, Zur numerischen Auswertung mehrdimensionaler Integrale, Z. Angew. Math. Mech. 38, 1-15 (1958). 125.21 Berthod-Zaborowski, Le calcul des integrales de 1

la forme:

[25.3] [25.4] [25.5] [25.6] [25.7]

[25.8]

[25.9] [25.10] [25.11] 125.121 [25.13] [25.14] [25.15] [25.16] [25.17]

[25.18]

s

f(x) log x dx.

H. Mineur, ‘Techniques de calcul numerique, pp. 555-556 (Librairie Polytechnique Ch. B&anger, Paris, France, 1952). W. G. Bickley, Formulae for numerical integration, Math. Gas. 23, 352 (1939). W. G. Bickley, Formulae for numerical differentiation, Math. Gaz. 25, 19-27 (1941). W. G. Bickley, Finite difference formulae for the square lattice, Quart. J. Mech. Appl. Math., 1, 35-42 (1948). G. Birkhoff and D. Young, Numerical quadrature of analytic and harmonic functions, J. Math. Phys. 29, 217-221 (1950). L. Fox, The use and construction of mathematical tables, Mathematical Tables vol. I, National Physical Laboratory (Her Majesty’s Stationery Office, London, England, 1956). S. Gill, Process for the step-by-step integration of differential equations in an automatic digital computing machine, Proc. Cambridge Philos. Sot. 47, 96-108 (1951). P. C. Hammer and A. H. Stroud, Numerical evaluation of multiple integrals II, Math. Tables Aids Comp. 12, 272-280 (1958). P. C. Hammer and A. W. Wymore, Numerical evaluation of multiple integrals I, Math. Tables Aids Comp. 11,59-67 (1957). P. Henrici, Discrete variable methods in ordinary differential equations (John Wiley & Sons, Inc., New York, N. Y., 1961). F. B. Hildebrand, Introduction to numerical analysis (McGraw-Hill Book Co., Inc., New York, N.Y., 1956). Z. Kopal, Numerical analysis (John Wiley & Sons, Inc., New York, N.Y., 1955). A. A. Markoff, Differenzenrechnung (B. G. Teubner, Leipzig, Germany, 1896). S. E. Mikeladze, Quadrature formulas for a regular function, SoobBE. Akad. Nauk Gruzin. SSR. 17, 289-296 (1956). W. E. Milne, A note on the numerical integration of differential equations, J. Research NBS 43, 537-542 (1949) RP2046. D. J. Panov, Formelsammlung zur numerischen Behandlung partieller Differentialgleichungen nach dem Differenzenverfahren (Akad. Verlag, Berlin, Germany, 1955). R. Radau, Etudes sur les formules d’approximation qui servent il calculer la valeur d’une integrale definie, J. Math. Pures Appl. (3) 6, 283-336 (1880).

[25.19] R. D. Richtmeyer, Difference methods for initialvalue problems (Interscience Publishers, New York, N.Y., 1957). [25.20] M. Sadowsky, A formula for approximate computation ‘of a triple integral, Amer. Math. Monthly 47, 539-543 (1940). [25.21] H. E. Salzer, A new formula for inverse interpolation, Bull. Amer. Math. Sot. 50, 513-516 (1944). [25.22] H. E. Salzer, Formulas for complex Cartesian interpolation of higher degree, J. Math. Phys. 28,~fZOO-203 (1949). [25.23] H. E. Salzer, Formulas for numerical integration of first and second order differential equations in the complex plane, J. Math. Phys. 29, 207-216 (1950). [25.24] H. E. Salzer, Formulas for numerical differentiation in the complex plane, J. Math. Phys. 31, 155-169 (1952). [25.25] A. Sard, Integral representations of remainders, Duke Math. J. 15,333-345 (1948). [25.26] A. Sard, Remainders: functions of several variables, Acta Math. 84, 319-346 (1951). [25.27] G. &hula, Formelsammlung eur praktischen Mathematik (DeGruyter and Co., Berlin, Germany, 1945). [25.28] A. H. Stroud, A bibliography on approximate integration, Math. Comp. 15, 52-80 (1961). [25.29] G. J. Tranter, Integral transforms in mathematical physics (John Wiley & Sons, Inc., New York, N.Y., 1951). [25.30] G. W. Tyler, Numerical integration with several variables, Canad. J. Math. 5, 393-412 (1953).

Tables

[25.31] L. J. Comrie, Chambers’ six-figure mathematical tables, vol. 2 (W. R. Chambers, Ltd., London, England, 1949). [25.32] P. Davis and P. Rabinowitz, Abscissas and weights for Gaussian quadratures of high order, J. Research NBS 56, 35-37 (1956) RP2645. [25.33] P. Davis and P. Rabinowitz, Additional abscissas and weights for Gaussian quadratures of high order: Values for n=64, 80, and 96, J. Research NBS 60,613-614 (1958) RP2875. [25.34] E. W. Dijkstra and A. van Wijngaarden, Table of Everett’s interpolation coefficients (Elcelsior’s Photo-offset, The Hague, Holland, 1955). [25.35] H. Fishman, Numerical integration constants, Math. Tables Aids Comp. 11, l-9 (1957). [25.36] H. J. Gawlik, Zeros of Legendre polynomials of orders 2-64 and weight coefficients of Gauss quadrature formulae, A.R.D.E. Memo (B) 77/58, Fort Halstead, Kent, England (1958). [25.37] Gt. Britain H.M. Nautical Almanac Office, Interpolation and allied tables (Her Majesty’s Stationery Office, London, England, 1956).

NUMERICAL

[25.38] I. M. Longman, Tables for the rapid and accurate numerical evaluation of certain infinite integrals involving Bessel functions, Math. Tables Aids Comp. 11, 166-180 (1957). [25.39] A. N. Lowan, N. Davids, and A. Levenson, Table of the zeros of the Legendre polynomials of order l-16 and the weight coefficients for Gauss’ mechanical quadrature formula, Bull. Amer. Math. Soc.48, 739-743 (1942). [25.40] National Bureau of Standards, Tables of Lagrangian interpolation coefficients (Columbia Univ. Press, New York, N.Y., 1944). [25.41] National Bureau of Standards, Collected Short Tables of the Computation Laboratory, Tables of functions and of zeros of functions, Applied Math. Series 37 (U.S. Government Printing Office, Washington, D.C., 1954). [25.42] P. Rabinowitz, Abscissas and weights for Lobatto quadrature of high order, Math. Tables Aids Comp. 69, 47-52 (1966). [25.43] P. Rabinowitz and G. Weiss, Tables of abscissas and weights for numerical evaluation of integrals of the form “0, Math. Tables Aids Comp. 68, 285-294 (1959).

ANALYSIS

899

[25.44] H. E. Salzer, Tables for facilitating the use of Chebyshev’s quadrature formula, J. Math. Phys. 26, 191-194 (1947). [25.45] H. E. Salzer and R. Zucker, Table of the zeros and weight factors of the first fifteen Laguerre polynomials, Bull. Amer. Math. Sot. 55, 10041012 (1949). [25.46] H. E. Salzer, R. Zucker, and R. Capuano, Table of the zeros and weight factors of the first twenty Hermite polynomials, J. Research NBS 48, 111-116 (1952) RP2294. [25.47] H. E. Salzer, Table of coefficients for obtaining the first derivative without differences, NBS Applied Math. Series 2 (U.S. Government Printing Office, Washington, D.C., 1948). [25.48] H. E. Salzer, Coefficients for facilitating trigonometric interpolation, J. Math. Phys. 27,274-278 (1949). [25.49] H. E. Salzer and P. T. Roberson, Table of coefficients for obtaining the second derivative without differences, Convair-Astronautics, San Diego, Calif. (1957). [25.50] H. E. Salzer, Tables of osculatory interpolation coefficients, NBS Applied Math. Series 56 (U.S. Government Printing Office, Washington, D.C., 1958).

NUMERICAL

900 Table

25.1

THREE-POINT

LAGRANGJAN

A&)=(-ly+’ Ao

ANALYSIS

INTERPOLATION P(P2-1)

(l+k) ! (l-k) ! (p-k) A-1

o.spo

P 0.00

A-1 -0.00000

OIOl 0.02 0.03 0.04

-0I00495 -0.00980 -0101455 -0.01920

1.00000 0.99990 0.99960 0.99910 0.99840

0.05 0.06 0.07 0.08 0.09

-0.02375 -0.02820 -0.03255 -0.03680 -0.04095

0.99750 0.99640 0.99510 0.99360 0.99190

0.02625 0.03180 0.03745 0.04320 0.04905

0.55 0.56 0.57 0.58 0.59

0.10 0.11 0.12 0.13 0.14

-0.04500 -0.04895 -0.05280 -0.05655 -0.06020

0.99000 0.98790 0.98560 0.98310 0.98040

0.05500 0.06105 0.06720 0.07345 0.07980

0.15 0.16 0.17 0.18 0.19

-0.06375 -0.06720 -0.07055 -0.07380 -0.07695

0.97750 0.97440 0.97110 0.96760 0.96390

0.20 0.21 0.22 0.23 0.24

-0.08000 -0.08295 -0.08580 -0.08855 -0.09120

0.25 0.26 0.27 0.28 0.29

AI 0.00000

COEFFICIENTS

Ao

AI

0.75000 0.73990 0.72960 0.71910 0.70840

0.37500 0.38505 0.39520 0.40545 0.41580

-0.12375 -0.12320 -0.12255 -0.12180 -0.12095

0.69750 0.68640 0.67510 0.66360 0.65190

0.42625 0.43680 0.44745 0.45820 0.46905

0.60 0.61 0.62 0.63 0.64

-0.12000 -0.11895 -0.11780 -0.11655 -Oil1520

0.64000 0.62790 0.61560 0.60310 0.'59040

0.48000 0.49105 0.50220 0.51345 0.52480

0.08625 0.09280 0.09945 0.10620 0.11305

0.65 0.66 0.67 0.68 0.69

-0.11375 -0.11220 -0.11055 -0.10880 -Oil0695

0.57750 0.56440 0.55110 0.53760 0.52390

0.53625 0.54780 0.55945 0.57120 0.58305

0.96000 0.95590 0.95160 0.94710 0.94240

0.12000 0.12705 0.13420 0.14145 0.14880

0.70 0.71 0.72 0.73 0.74

-0.10500 -0.10295 -0.10080 -0.09855 -0.09620

0.51000 0.49590 0.48160 0.46710 0.45240

0.59500 0.60705 0.61920 0.63145 0.64380

-0.09375 -0.09620 -0.09855 -0.10080 -0.10295

0.93750 0.93240 0.92710 0.92160 0.91590

0.15625 0.16380 0.17145 0.17920 0.18705

0.75 0.76 0.77 0.78 0.79

-0.09375 -0.09120 -0.08855 -0.08580 -0.08295

0.43750 0.42240 0.40710 0.39160 0.37590

0.65625 0.66880 0.68145 0.69420 0170705

0.30 0.31 0.32 0.33 0.34

-0.10500 -0.10695 -0.10880 -0.11055 -0.11220

0.91000 0.90390 0.89760 0.89110 0.88440

0.19500 0.20305 0.21120 0.21945 0.22780

0.80 0.81 0.82 0.83 0.84

-0.08000 -0.07695 -0.07380 -0.07055 -0.06720

0.36000 0.34390 0.32760

i%:i .

0.72000 0.73305 0;74620 0.75945 0.77280

0.35 0.36 0.37 0.38 0.39

-0.11375 -0.11520 -0.11655 -0.11780 -0.11895

0.87750 0.87040 0.86310 0.85560 0.84790

0.23625 0.24480 0.25345 0.26220 0.27105

0.85 0.86 0.87 0.88 0.89

-0.06375 -0.06020 -0.05655 -0.05280 -0.04895

0.27750 0.26040 0.24310 0.22560 0.20790

0.78625 0.79980 0.81345 0.82720 0.84105

0.40 0.41 0.42 0.43 0.44

-0.12000 -0.12095 -0.12180 -0.12255 -0.12320

0.84000 0.83190 0.82360 0.81510 0.80640

0.28000 0.28905 0.29820 0.30745 0.31680

0.90 0.91 0.92 0.93 0.94

-0.04500 -0.04095 -0.03680 -0.03255 -0.02820

0.19000 0.17190 0.15360 0.13510 0.11640

0.85500 0.86905 0.88320 0.89745 0.91180

0.45 0.46 0.47 0.48 0.49

-0.12375 -0.12420 -0.12455 -0.12480 -0.12495

0.79750 0.78840 0.77910 0.76960 0.75990

0.32625 0.33580 0.34545 0.35520 0.36505

0.95 0.96 0.97 0.98 0.99

-0.02375 -0.01920 -0.01455 -0.00980 -0.00495

0.09750 0.07840 0.05910 OiO3960 0.01990

0.92625 0.94080 0.95545 0.97020 0.98505

0.50

-0.12500

0.75000

0.37500

1.00

-0.00000 A1

0.00000 Ao

1.00000 A-1

-P

AI

Ao

0.00505 0.01020 0.01545 0.02080

A-1

0.51 0.52 0.53 0.54

-P

-0.12500 -0.12495 -0.12480 -0.12455 -0.12420

*

See 25.2.6. Compiled from National Bureau of Standards, Tables of Lagrangian interpolation coefficients. Columbia Univ. Press, New York, N.Y., 1944 (with permission).

NUMERICAL FOUR-POINT

o.:o

LAGRANGIAN

A-1 0.00000

00

INTERPOLATION

Ao 1.00000

901

ANALYSIS COEFFICIENTS

00

Al 0.00000

0.01004 0.02019 0.03043 0.04076

Table

25.1

A2

0.00000

00

95 60 65 80

-0.00166 -0.00333 -0.00499 -0.00665

65 20 55 60

1.00 0. 99 0.98 0.97 0.96

00

0. 01 0. 02 0. 03 0. 04

-0.00328 -0.00646 -0.00955 -0.01254

35 80 45 40

0.99490 0.98960 0.98411 0.97843

05 40 35 20

0.05 0. 06 0.07 0. 08 0. 09

-0.01543 -0.01823 -0.02094 -0.02355 -0.02607

75 60 05 20 15

0.97256 0.96650 0.96027 0.95385 0.94726

25 80 15 60 45

0.05118 0.06169 0.07227 0.08294 0.09368

75 20 85 40 55

-0.00831 -0.00996 -0.01160 -0.01324 -0.01487

25 40 95 80 85

0.95 0.94 0.93 0.92 0.91

0.10 0.11 0.12 0.13 0.14

-0.02850 -0.03083 -0.03308 -0.03524 -0.03732

00 85 80 95 40

0.94050 0.93356 0.92646 0.91919 0.91177

00 55 40 85 20

0.10450 0.11538 0.12633 0.13735 0.14842

00 45 60 15 80

-0.01650 -0.01811 -0.01971 -0.02130 -0.02287

00 15 20 05 60

0.90 0. 89 0.88 0.87 0.86

0.15 0.16 0.17 0.18 0.19

-0.03931 -0.04121 -0.04303 -0.04477 -0.04642

25 60 55 20 65

0.90418 0.89644 0.88855 0.88051 0.87232

75 80 65 60 95

0.15956 0.17075 0.18199 0.19328 0.20462

25 20 35 40 05

-0.02443 -0.02598 -0.02751 -0.02902 -0.03052

75 40 45 80 35

0.85 0.84 0.83 0.82 0.81

0.20 0. 21 0.22 0.23 0.24

-0.04800 -0.04949 -0.05090 -0.05224 -0.05350

00 35 80 45 40

0.86400 0.85553 0.84692 0.83818 0.82931

00 05 40 35 20

0.21600 0.22741 0.23887 0.25036 0.26188

00 95 60 65 80

-0.03200 -0.03345 -0.03489 -0.03630 -0.03769

00 65 20 55 60

0.80 0.79 0.78 0.77 0.76

0.25 0.26 0.27 0.28 0.29

-0.05468 -0.05579 -0.05683 -0.05779 -0.05868

75 60 05 20 15

0.82031 0.81118 0.80194 0.79257 0.78309

25 80 15 60 45

0.27343 0.28501 0.29660 0.30822 0.31985

75 20 85 40 55

-0.03906 -0.04040 -0.04171 -0.04300 -0.04426

25 40 95 80 85

0.75 0.74 0.73 0.72 0.71

0.30 0.31 0. 32 0. 33 0.34

-0.05950 -0.06024 -0.06092 -0.06153 -0.06208

00 85 80 95 40

0.77350 0.76379 0.75398 0.74406 0.73405

00 55 40 85 20

0.33150 0.34315 0.35481 0.36648 0.37814

00 45 60 15 80

-0.04550 -0.04670 -0.04787 -0.04901 -0.05011

00 15 20 05 60

0.70 0.69 0.68 0.67 0.66

0.35 0.36 0.37 0.38 0.39

-0.06256 -0.06297 -0.06332 -0.06361 -0.06383

25 60 55 20 65

0.72393 0.71372 0.70342 0.69303 0.68255

75 80 65 60 95

0.38981 0.40147 0.41312 0.42476 0.43639

25 20 35 40 05

-0.05118 -0.05222 -0.05322 -0.05418 -0.05511

75 40 45 80 35

0. 65 0.64 0.63 0.62 0.61

0.40 0.41 0.42 0.43 0. 44

-0.06400 -0.06410 -0.06414 -0.06413 -0.06406

00 35 80 45 40

0.67200 0.66136 0.65064 0.63985 0.62899

00 05 40 35 20

0.44800 0.45958 0.47115 0.48269 0.49420

00 95 60 65 80

-0.05600 -0.05684 -0.05765 -0.05841 -0.05913

00 65 20 55 60

0.60 0.59 0.58 0.57 0.56

0. 45 0.46 0.47 0.48 0.49

-0.06393 -0.06375 -0.06352 -0.06323 -0.06289

75 60 05 20 15

0.61806 0.60706 0.59601 0.58489 0.57372

25 80 15 60 45

0.50568 0.51713 0.52853 0.53990 0.55122

.75 20 85 40 55

-0.05981 -0.06044 -0.06102 -0.06156 -0.06205

25 40 95 80 85

0.55 0. 54 0.53 0.52 0.51

0.50

-0.06250

00

0.56250 00

0.56250 00

-0.06250

00

AI

Ao

0.50 P

A2

A-1

902

NUMERICAL Table 25.1

FOUR-POINT

LAGRANGIAN

ANALYSIS INTERPOLATION

65 20 55 60

Ao 0.00000 00 -0.00994 95 -0.01979 60 -0. 02953 65 -0.03916 80

Al 1.00000 1.00489 1.00959 1.01408 1.01836

0.00831 0.00996 0.01160 0.01324 0.01487

25 40 95 80 85

-0.04868 75 -0.05809 20 -0.06737 85 -0.07654 40 -0.08558 55

1.10 1.11 1.12 1.13 1.14

0.01650 0.01811 0.01971 0.02130 0.02287

00 15 20 05 60

-0.09450 -0.10328 -0.11193 -0.12045 -0.12882

1.15 1.16 1.17 1.18 1.19

0.02443 0.02598 0.02751 0.02902 0.03052

75 40 45 80 35

-0.13706 -0.14515 -0.15309 -0.16088 -0.16852

1.20 1.21 1.22 1.23 1.24

0.03200 0.03345 0.03489 0.03630 0.03769

1.25 1.26 1.27 1.28 1.29

P 1. 00

A-1 0.00000

00

1.01 1.02 1. 03 1. 04

0.00166 0.00333 0.00499 0.00665

1. 05 1.06 1. 07 1.08 1.09

COEFFICIENTS

A2

00 95 60 65 80

0.00000 0.00338 0.00686 0.01045 0.01414

00 35 80 45 40

1.02243 1.02629 1.02992 1.03334 1.03653

75 85 40 55

0.01793 0.02183 0.02584 0.02995 0.03417

75 60 05 20 15

00 45 60 15 80

1.03950 1.04223 1.04473 1.04700 1.04902

00 45 60 15 80

0.03850 0.04293 0.04748 0.05214 0.05692

00 85 80 95 40

0.10 0.11 0.12 0.13 0.14

25 2,o 35 40 05

1.05081 25 1.05235 20 1.05364 35

25

1.05547 05

0.06181 0.06681 0.07193 0.07717 0.08252

55 20 65

0.15 0.16 0.17 0.18 0.19

00 65 20 55 60

-0.17600 00 -0; 18331 95 -0.19047 60 -0.19746 65 -0.20428 80

1.05600 00 1.05626 95 1.05627 60 lt 05601 65 1.05548 80

0.08800 .oo 0.09359 35 0.09930 80 0.10514 45 0.11110 40

0.20 0.21 0.22 0.23 0.24

0.03906 0.04040 0.04171 0.04300 0.04426

25 40 95 80 85

-0.21093 -0.21741 -0.22370 -0.22982 -0.23575

75 20 85 40 55

1.05468 1.05361 1.05225 1.05062 1.04870

75 20 85 40 55

0.11718 0.12339 0.12973 0.13619 0.14278

75 60 05 20 15

0.25 0.26 0. 27 0.28 0.29

1.30 1.31 1.32 1.33 1.34

0.04550 0.04670 0.04787 0.04901 0.05011

00 15 20 05 60

-0.24150 -0.24705 -0.25241 -0.25758

1.04650 1.04400 1.04121 1.03813 1.03474

00 45

-0.26254

00 45 60 15 80

15 80

0.14950 0.15634 0.16332 0.17043 0.17768

00 85 80 95 40

0.30 0.31 0.32 0.33 0.34

1.35 1.36 1.37 1.38 1.39

0.05118 0.05222 0.05322 0.05418 0.05511

75 40 45 80 35

-0.26731 -0.27187 -0.27622 -0.28036 -0.28429

25 20 35 40 05

1.03106 1.02707 1.02277 1.01816 1.01324

25 20 35 40 05

0.18506 0.19257 0.20022 0.20801 0.21593

25 60 55 20

0.35 0.36 0.37 0.38 0.39

1.40 1.41 1. 42 1.43 1. 44

0.05600 0.05684 0.05765 0.05841 0.05913

00 65 20 55 60

-0.28800 -0.29148 -0.29475 -0.29779

1.00800 1.00243 0.99655 0.99034 0.98380

00 95 60 65 80

0.22400

00

0.23220 35 0.24054 80 0.24903 45

-0.30060

00 95 60 65 80

1.45 1.46 1.47 1.48 1.49

0.05981 0.06044 0.06102 0.06156 0.06205

25 40 95 80 85

-0.30318 -0.30553 -0.30763 -0.30950 -0.31112

75 20 85 40 55

0.97693 75

1.50

0.06250 00

-0.31250 00 Al

A2

1.05468

20

40

60

60

65

0.25766

40

0.26643

75

0.00

0.01 0.02 0.03 0.04 0. 05 0.06

0. 07 0.08 0.09

0.40 0.41 0.42 0.43 0.44

0.95430 40 0.94607 55

0.27535 0.28442 0.29363 0.30299

60 05 20 15

0.45 Oi46 0. 47 0.48 0.49

0.93750 00

0.31250 00

0.50

Ao

A-1

0.96973 0.96218

20 85

-P

NUMERICAL FOUR-POINT

1.;0 1. 51 1.52 1.53 1.54

LAGRANGIAN

INTERPOLATION

A-I

Ao

0.06250 00 0.06289 15

-0.31250 00 -0.31362 45 -0.31449 60 -0.31511 15 -0.31546 a0

0.06323 0.06352 0.06375

20 05 60

1.55 1.56 1.57 1.58 1.59

0.06393 0.06406

75 40

1.60 1.61 1.62 1.63 1. 64

0.06400 00

1. 65 1.66 1.67 1.68 1.69

0.06256

1.70 1.71 1.72 1.73 1.74

0.05950 00 0.05868 15

-0.28350 00 -0.27899 45

0.05683 05

-0.26904 15 -0.26358 80

1.75 1.76 1. 77 1.78 1.79 1.80 1.81 1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89

0.06413 45 0.06414 a0 0.06410 35

-0.31556 -0.31539 -0.31495 -0.31424 -0.31326

25 20 35 40 05

0.06383

65

-0.31200 00 -0.31045 95

0.06332 0.06297

55 60

-0.30412 a0

0.06361 20

0.06208 0.06153 0.06092 0.06024

0.05779 0.05579

25

40 95 80 85

20

60

0.05468 75

0.05350 0.05224 0.05090 0.04949

40 45

a0 35

0.04aoo 00 0.04642 0.04477 0.04303

65 20 55

0.04121 60 0.03931 25 0.03732 0:03524

40 95

0.03308 80 0.03083 a5

-0.30863 -0: 30652

60 65

-0.30143 -0.29845 -0.29516 -0.29158 -0.28769

75

-0.27417

20 a5 40 55

60

-0.25781

25

-0i25171 -0.24528 -0l23852 -0.23143

20 35 40 05

-0.22400 -0.21622 -0.20811 -0.19965 -0.19084

00 95 60 65 80

0. 51 0.52 0.53 0.54 0.55

0.37273

60

0.38331 55 0.39405 0.40494

20 65

0. 56 0. 57

0.83200 00 0.81940 95

0.41600 0.42721 0.43858 0.45012 0.46182

00 35 a0 45 40

0.60 0. 61 0.62 0.63 0.64

0.76518 75

75

a5 0.72038 40

0.47368 0.48571 0.49791 0.51027 0.52280

0. 65 0. 66 0. 67

0.68850 00 0.67194 45

0.53550 00 0.54836 a5 0.56140 a0

0.75065 0.73571

60 65 80

20

0. JO464 55

0.65497 0.63759

60

15 0.61978 a0 0.60156 25 0.58291 20 0.56383 35 0.54432

40

0.52438 05 0.50400 00 0.48317 95 0.46191 60

0.57461

60 05 20 15

95

0.58800 40 0.60156 25 0.61529 60

0.58 0.59

0.68 0.69 0.70

0. 71 0.72 0.73 0.74 0.75 0.76 0.77

0.62920 0.64329 0.65755

55 20 65

0.67200 0.68662

00 35

0.80 0. al 0.82 0.83 0.84

0.78 0. 79

-0.16229 a5 -0.15206 40 -0.14146 55

75 20

0.34884 a5 0.32486 40 0.30041 55

0.74693 0.76247

75 60

0.77820 05 0.79411 20 0.81021 15

0.85 0.86 0.87 0.88 0.89

-0.13050 00 -0.11916 45 -0.10745 60 -0.09537 15 -0.08290 a0

0.27550

00

0.82650 0.84297 0.85964 0.87650 0.89356

00 a5 80 95 40

0.90 0.91 0.92 0.93 0.94

-0.07006

25

0.14381 25 0.11603 20

0.91081 25 0.92825 60 0.94589 55

0.95 0.96

-0.02920

40

20

-0.01480 05

2.00

0.00000 00

0.00000 00

A2

40

0.50

0.39543 0.37237

-0.18168 75 -0.17217

0.00328 35

45 80

0.35204

00 a5 a0 95

0.36231 25

0.80643 0.79307 0.77932

25 20 35 40 05

0.31250 0.32215 0.33196 0.34192

0.41804 80

0.01543 75 0.01254 40 0.00955 0.00646

A2 00 45

0.91929 60 0.90966 15 0.89966 80 0.88931 0.87859 0.86750 0.85604 0.84421

25.1

0.70142 a0 0.71641 45 0.73158 40

1.95 1.96 1.97 1.98 1.99

0.01823 60

AI 0.93750 0.92857

Table

65

0.02850 00 0.02607 15 20 05

COEFFICIENTS

0.44020

1.90 1.91 1.92 1.93 1.94

0.02355 0.02094

903

ANALYSIS

-0.05683 20 -0.04321 35

AI

0.25011 45 0.22425 0.19792 0.17110

0.08776 0.05900 0.02975

60

15 a0

35 40 05

o*oooAooo O0

0.96373

20

0.98176 65 I.. """P

1O0

0.97

0.98 0.99 1.00 -P

904 Table

NUMERICAL 25.1

FIVE-POINT

LAGRANt

A;(P)=(

o.io

;IAN

ANALYSIS

INTERPOLATION

COEFFICIENTS

P(P2-l)(P2--4)

-1)k+2 (2+k)!(2-k)!(p-k) AI Ao 1.00000 00000 0.00000 00000 0.99987 50025 0.00673 31650 0.99950 00400 0.01359 86400 0.99887 52025 OiO2059 53650 0.99800 06400 0.02772 22400

A2

0.01 0.02 0.03 0.04

0.00000 0.00082 0.00164 0.00246 0.00326

A-I 0.00000 00000 00000 90838 -0.00659 98350 93400 -0.01306 53600 02838 -0.01939 56350 14400 -0.02558 97600

0. 05 0.06 0.07 0. 08 0.09

0.00405 0.00483 0.00560 0.00635 0.00710

23438 25400 15838 90400 44838

-0.03164 -0.03756 -0.04334 -0.04898 -0.05448

68750 61600 68350 81600 94350

0.99687 0.99550 0.99388 0.99201 0.98989

65625 0.03497 81250 -0.00426 01563 32400 OiO4236 18400 -0.00513 14600 10025 0.04987 21650 -0.00600 79163 02400 0.05750 78400 -0.00688 89600 14025 0.06526 75650 -0.00777 40163

0.05 0.06 0.07 0.08 0.09

0.10 0. 11 0.12 0.13 0.14

0.00783 0.00855 0.00926 0.00995 0.01063

75000 76838 46400 79838 73400

-0.05985 -0.06506 -0.07014 -0.07508 -0.07987

00000 92350 65600 14350 33600

0.98752 0.98491 0.98205 0.97894 0.97559

50000 16025 18400 64025 60400

0.07315 0.08115 0.08927 0.09751 0.10587

00000 37650 74400 95650 86400

-0.00866 -0.00955 -0.01044 -0.01134 -0.01223

25000 38163 73600 25163 86600

0.10 0.11 0.12 0.13 0.14

0.15 0.16 0.17 0.18 0. 19

0.01130 0.01195 0.01258 0.01320 0.01381

23438 2‘6400 78838 77400 18838

-0.08452 -0.08902 -0.09338 -0.09760 -0.10167

18750 65600 70350 29600 40350

0.97200 0.96816 0.96408 0.95976 0.95520

15625 38400 38025 24400 08025

0.11435 0.12294 0.13164 0.14045 0.14937

31250 14400 19650 30400 29650

-0.01313 -0.01403 -0.01492 -0.01582 -0.01671

51563 13600 66163 02600 16163

0.15 0.16 0.17 0.18 0.19

0.20 0.21 0.22 0.23 0.24

0.01440 0.01497 0.01552 0.01606 0.01658

00000 17838 69400 51838 62400

-0.10560 -0.10938 -0.11301 -0.11650 -0.11984

00000 06350 57600 52350 89600

0.95040 0.94536 0.94008 0.93457 0.92882

00000 12025 56400 46025 94400

0.15840 0.16753 0.17676 0.18610 0.19554

00000 23650 82400 57650 30400

-0.01760 -0.01848 -0.01936 -0.02024 -0.02110

00000 47163 50600 03163 97600

0.20 0.21 0.22 0.23 0.24

0.25 0.26 0.27 0.28 0. 29

0.01708 0.01757 0.01804 0.01849 0.01892

98438 57400 36838 34400 47838

-0.12304 -0.12609 -0.12900 -0.13176 -0.13438

68750 89600 52350 57600 06350

0.92285 0.91664 0.91020 0.90353 0.89664

15625 24400 36025 66400 32025

0.20507 0.21470 0.22443 0.23425 0.24415

81250 90400 37650 02400 63650

-0.02197 -0.02282 -0.02367 -0.02451 -0.02534

26563 82600 58163 45600 37163

0.25 0.26 0.27 0.28 0.29

0.30 0.31 0.32 0.33 0. 34

0.01933 0.01973 0.02010 0.02046 0.02079

75000 13838 62400 18838 81400

-0.13685 -0.13917 -0.14135 -0.14338 -0.14527

00000 40350 29600 70350 65600

0.88952 0.88218 0.87462 0.86683 0.85884

50000 38025 14400 98025 08400

0.25415 0.26422 0.27439 0.28463 0.29495

00000 89650 10400 39650 54400

-0.02616 -0.02697 -0.02776 -0.02854 -0.02931

25000 01163 57600 86163 78600

0.30 0.31 0.32 0.33 0.34

0.35 0.36 0.37 0.38 0.39

0.02111 0.02141 0.02168 0.02194 0.02218

48438 18400 89838 61400 31838

-0.14702 -0.14862 -0.15008 -0.15139 -0.15256

18750 33600 14350 65600 92350

0.85062 0.84219 0.83356 0.82471 0.81565

65625 90400 04025 28400 86025

0.30535 0.31582 0.32636 0.33697 0.34765

31250 46400 75650 94400 77650

-0.03007 -0.03081 -0.03153 -0.03224 -0.03293

26563 21600 55163 18600 03163

0.35 0.36 0.37 0.38 0.39

0.40 0. 41 0.42 0. 43 0.44

0.02240 0.02259 0.02277 0.02292 0.02306

00000 64838 25400 80838 30400

-0.15360 -0.15448 -0.15523 -0.15584 -0.15631

00000 94350 81600 68350 61600

0.80640 0.79693 0.78727 0.77742 0.76737

00000 94025 92400 20025 02400

0.35840 0.36920 0.38006 0.39098 0.40195

00000 35650 58400 41650 58400

-0.03360 -0.03425 -0.03487 -0.03548 -0.03607

00000 00163 94600 74163 29600

0.40 0.41 0.42 0.43 0.44

0.45 0.46 0.47 0.48 0.49

0.02317 0.02327 0.02334 0.02339 0.02342

73438 09400 37838 58400 70838

-0.15664 -0.15683 -0.15689 -0.15681 -0.15659

68750 97600 56350 53600 98350

0.75712 0.74669 0.73607 0.72527 0.71428

65625 36400 42025 10400 70025

0.41297 0.42404 0.43516 0.44632 0.45751

81250 82400 33650 06400 71650

-0.03663 -0.03717 -0.03768 -0.03817 -0.03863

51563 30600 57163 21600 14163

0.45 0. 46 0.47 0.48 0.49

0.50

0.02343 75000

-0.03906 25000

0.50 -P

A-2

A2

-0.15625 00000 Al

0.70312 50000 Ao

0.46875 00000 A-I

0.00000 -0.00083 -0.00168 -0.00253 -0.00339

A-2

00000 74163 26600 52163 45600

0. 00

0.01 0.02 0.03 0. 04

NUMERICAL

FIVE-POINT

LAGRANGIAN

INTERPOLATION

Ah)=(-1)k+2 A-Z

P

A-I

905

ANALYSIS

P(P2--1)

COEFFICIEM

TS

Table

25.1

(P2-4)

(2+k)!(2-k)!(p-k) Ao

Al

A2

0.50 0. 51 0. 52 0.53 0.54

0.02343 0.02342 0.02339 0.02334 0.02327

75000 -0.15625 00000 70838 -0.15576 68350 58400 -0.15515 13600 37838 -0.15440 46350 09400 -0.15352 77600

0.70312 0.69178 0.68027 0.66860 0.65675

50000 80025 90400 12025 76400

0.46875 0.48001 0.49131 0.50263 0.51398

00000 61650 26400 63650 42400

-0.03906 -0.03946 -0.03983 -0.04017 -0.04048

25000 44163 61600 67163 50600

0.50 0. 51 0.52 0.53 0. 54

0.55 0. 56 0.57 0. 58 0.59

0.02317 0.02306 0.02292 0.02277 0.02259

73438 30400 80838 25400 64838

la750 al600 78350 21600 24350

0.64475 0.63258 0.62026 0.60779 0.59516

15625 62400 50025 12400 84025

0.52535 0.53673 0.54814 0.55955 0.57097

31250 98400 11650 38400 45650

-0.04076 -0.04100 -0.04120 -0.04137 -0.04150

01563 09600 64163 54600 70163

0.55 0.56 0.57 0.58 0.59

0.60 0.61 0. 62 0.63 0.64

0.02240 0.02218 0.02194 0.02168 0.02141

00000 -0.14560 00000 31838 -0.14384 62350 61400 -0.14197 25600 89838 -0.13998 04350 la400 -0.13787 13600

0.58240 0.56948 0.55644 0.54325 0.52994

00000 96025 08400 74025 30400

0.58240 0.59382 0.60525 0.61667 0.62808

00000 -0.04160 67650 -0.04165 14400 -0.04166 05650 -0.04163 06400 -0.04156

00000 33163 58600 65163 41600

0.60 0. 61 0.62 0.63 0.64

0.65 0.66 0.67 0.68 0.69

0.02111 0.02079 0.02046 0.02010 0.01973

48438 al400 la838 62400 13838

-0.13564 -0.13330 -0.13085 -0.12829 -0.12562

68750 85600 a0350 69600 70350

0.51650 0.50293 0.48925 0.47545 0.46154

15625 68400 28025 34400 28025

0.63947 0.65085 0.66222 0.67355 0.68486

81250 94400 09650 90400 99650

76563 58600 76163 17600 71163

0. 65 0.66 0.67 0. 68 0. 69

0.70 0.71 0.72 0.73 0.74

0.01933 0.01892 0.01849 0.01804 0.01757

75000 47838 34400 36838 57400

-0.12285 -0.11996 -0.11698 -0.11389 -0.11070

00000 76350 17600 42350 69600

0.44752 0.43340 0.41918 0.40487 0.39046

50000 42025 46400 06025 64400

0.69615 0.70739 0.71860 0.72976 0.74088

00000 -0.04016 25000 53650 -0.03975 67163 22400 -0.03929 85600 67650 -0.03878 68163 50400 -0.03822 02600

0.70 0.71 0. 72 0.73 0.74

0.75 0.76 0.77 0.78 0.79

0.01708 0.01658 0.01606 0.01552 0.01497

98438 62400 51838 69400 17838

-0.10742 -0.10404 -0.10056 -0.09699 -0.09334

la750 09600 62350 97600 36350

0.37597 0.36140 0.34675 0.33203 0.31725

65625 54400 76025 76400 02025

0.75195 0.76296 0.77392 0.78481 0.79564

31250 70400 27650 62400 33650

-0.03759 -0.03691 -0.03617 -0.03538 -0.03452

76563 77600 93163 10600 17163

0.75 0.76 0.77 0.78 0.79

0.80 0.81 0. a2 0.83 0.84

0.01440 0.01381 0.01320 0.01258 0.01195

00000 18838 77400 78838 26400

-0.08960 -0.08577 -0.08185 -0.07786 -0.07379

00000 10350 a9600 60350 45600

0.30240 0.28749 0.27253 0.25752 0.24246

00000 la025 04400 08025 78400

0.80640 0.81708 0.82768 0.83820 0.84863

00000 19650 50400 49650 74400

-0.03360 -0.03261 -0.03156 -0.03044 -0.02926

00000 46163 42600 76163 33600

0.80 0. al 0.82 0.83 0.84

0.85 0.86 0.87 0.88 0. a9

0.01130 0.01063 0.00995 0.00926 0.00855

23438 73400 79838 46400 76838

-0.06964 -0.06542 -0.06113 -0.05677 -0.05234

68750 53600 24350 05600 22350

0.22737 0.21225 0.19709 0.18192 0.16673

65625 20400 94025 38400 06025

0.85897 0.86922 0.87936 0.88940 0.89933

a1250 26400 65650 54400 47650

-0.02801 -0.02668 -0.02529 -0.02382 -0.02228

01563 66600 15163 33600 08163

0. a5 0. 86 0.87 0.88 0.89

0.90 0.91 0. 92 0.93 0.94

0.00783 0.00710 0.00635 0.00560 0.00483

75000 44838 90400 15838 25400

-0.04785 -0.04329 -0.03868 -0.03401 -0.02929

00000 64350 41600 58350 41600

0.15152 0.13631 0.12109 0.10588 0.09068

50000 24025 82400 a0025 72400

0.90915 0.91884 0.92841 0.93786 0.94717

00000 65650 98400 51650 78400

-0.02066 -0.01896 -0.01719 -0.01533 -0.01340

25000 70163 29600 89163 34600

0.90 0. 91 0.92 0.93 0.94

0.95 0. 96 0.97 0.98 0.99

0.00405 23438 OiOO32614460 0.00246 02838 0.00164 93400 0.00082 90838

-0.02452 -0.01970 -0.01483 -0.00992 -0.00498

la750 0.07550 15625 17600 0.06033 66400 66350 0.04519 a2025 93600 0.03009 20400 28350 0.01502 40025

0.95635 0.96538 0.97427 0.98300 0.99158

31250 62400 23650 66400 41650

-0.01138.51563 -0.00928 25600 -0.00709 42163 -0.00481 a6600 -0.00245 44163

0.95 0.96 0.97 0.98 0.99

1.00

0.00000

1.00000 00000 A-I

0.00000 00000

1.00

00000 A2

-0.15252 -0.15138 -0.15012 -0.14874 -0.14723

0.00000 00000 A1

0.00000 00000 Ao

-0.04144 -0.04128 -0.04107 -0.04082 -0.04051

A-2

-P

306

NUMERICAL

Table

25.1

FIVE-POINT

LAGRANGIAN

ANALYSIS INTERPOLATION

0.00000

A-Z 00000

0.00000

A-1 00000

:IENTS

P(P2-1) (P2-4) (2+k)!(2-k)!(p-k)

4(P)=-k+2 P 1.00

COEFFIC

Ao 00000

Al

A2 00000

00000 91650 66400 73650 62400

60838 53400 92838 94400

1. 00 1. 01 1.02 1.03 1.04

0.00000

1.01 1.02 1.03 1.04

-0.00083 -0.00168 -0.00253 -0.00339

74163 26600 52163 45600

61650 -0.01497 39975 26400 -0.02989 19600 63650 -0.04474 77975 42400 -0.05953 53600

1.oocoo 1.00824 1.01632 1.02422 1.03194

1.05 1.06 1.07 1.08 1.09

-0.00426 -0.00513 -0.00600 -0.00688 -0.00777

01563 0.02535 31250 -0.07424 84375 14600 0.03048 98400 -0.08888 07600 79163 0.03564 11650 -0.10342 59975 89600 0.04080 38400 -0.11787 77600 40163 0.04597 45650 -0.13222 95975

1.03947 1.04681 1.05396 1.06089 1.06763

81250 0.01367 73438 78400 0.01670 45400 01650 0.01983 25838 98400 0.02306 30400 15650 0.02639 74838

1.05 1.06 1.07 1.08 1.09

1.10 1.11 1.12 1.13 1.14

-0.00866 -0.00955 -0.01044 -0.01134 -0.01223

25000 38163 73600 25163 86600

0.05115 00000 -0.14647 50000 0.05632 67650 -0.16060 73975 0.06150 14400 -0.17462 01600 0.06667 05650 -0.18850 65975 0.07183 06400 -0.20225 99600

1.07415 1.08044 1.08652 1.09237 1.09798

00000 97650 54400 15650 26400

1.15 1.16 1.17 1.18 1.19

-0.01313 -0.01403 -0.01492 -0.01582 -0.01671

51563 13600 66163 02600 16163

0.07697 0.08210 0.08722 0.09230 0.09736

81250 94400 09650 90400 99650

1.20 1.21 1.22 1.23 1.24

-0.01760 -0.01848 -0.01936 -0.02024 -0.02110

00000 47163 50600 03163 97600

0.10240 0.10739 0.11235 0.11726 0.12213

00000 -0.28160 00000 53650 -0.29422 77975 22400 -0.30666 63600 67650 -0.31890 83975 50400 -0.33094 65600

1.25 1.26 1.27 1.28 1.29

-0.02197 -0.02282 -0.02367 -0.02451 -0.02534

26563 82600 58163 45600 37163

0.12695 0.13171 0.13642 0.14106 0.14564

31250 70400 27650 62400 33650

1.30 1.31 1.32 1.33 1.34

-0.02616 -0.02697 -0.02776 -0.02854 -0.02931

25000 0.15015 00000 -0.39847 50000 01163 0.15458 19650 -0.40887 51975 57600 0.15893 50400 -0.41901 05600 86163 0.16320 49650 -0.42887 31975 78600 0.16738 74400 -0.43845 51600

1.35 1.36 1.37 1.38 1.39

-0.03007 -0.03081 -0.03153 -0.03224 -0.03293

26563 21600 55163 18600 03163

0.17147 0.17547 0.17936 0.18315 0.18683

1.40 1.41 1.42 1.43 1.44

-0.03360 -0.03425 -0.03487 -0.03548 -0.03607

00000 00163 94600 74163 29600

1.45 1.46 1.47 1.48 1.49

-0.03663 -0.03717 -0.03768 -0.03817 -0.03863

51563 0.20635 31250 30600 0.20913 62400 57163 0.21177 23650 21600 0.21425 66400 14163 0.21658 41650

1.50

-0.03906 25000 A2

0.00501 0.01006 0.01513 0.02023

0.00000

-0.21587 -0.22934 -0,24265 -0.25580 -0.26879

0.02983 0.03338 0.03704 0.04080 0.04468

75000 46838 06400 69838 53400

1.10 1.11 1.12 1.13 1.14

34375 1.10335 31250 0.04867 01600 1.10847 74400 0.05278 31975 1.11334 99650 0.05700 55600 1.11796 50400 0.06135 01975 1.12231 69650 0.06581

73438 46400 88838 17400 48838

1.15 1.16 1.17 1.18 1.19

00000 0.07040 00000 83650 0.07510 87838 62400 0.07994 29400 77650 0.08490 41838 70400 0.08999 42400

1.20 1.21 1.22 1.23 1.24

1.12640 1.13020 1.13373 1.13697 1.13992

48438 77400 46838 74400 77838

1.25 1.26 1.27 1.28 1.29

1.15115 1.15188 1.15227 1.15232 1.15201

00000 0.12333 75000 49650 0.12937 83838 90400 0.13556 22400 59650 0.14189 08838 94400 0.14836 61400

1.30 1.31 1.32 1.33 1.34

84375 49600 65975 51600 23975

1.15135 1.15032 1.14891 1.14713 1.14496

31250 06400 55650 14400 17650

0.15498 98438 0.16176 38400 0.1686.8 99838 0.17577 01400 0.18300 61838

1.35 1.36 1.37 1.38 k.39

0.19040 00000 -0.48960 00000 0.19384 65650 -0.49698 95975 0.19716 98400 -0.50403 27600 0.20036 51650 -0.51072 09975 0.20342 78400 -0.51704 57600

1.14240 1.13943 1.13607 1.13229 1.12809

00000 0.19040 00000 95650 0.19795 34838 38400 0.20566 85400 61650 0.21354 70838 98400 0.22159 10400

1.40 1.41 1.42 1.43 1.44

1.12347 1.11842 1.11293 1.10699 1.10060

81250 42400 13650 26400 11650

81250 26400 65650 54400 47650

-0.34277 -0.35438 -0.36576 -0.37691 -0.38781

0.00254 0.00518 0.00791 0.01074

34375 1.14257 81250 15600 1.14492 50400 33975 1.14696 17650 13600 1.14868 22400 77975 1.15008 03650

-0.44774 -0.45674 -0.46543 -0.47381 -0.48187

-0.52299 -0.52857 -0.53375 -0.53853 -0.54291

84375 03600 27975 69600 39975

0.21875 00000 -0.54687 50000 1.09375 00000 Al

Ao

A-I

0.09521 0.10056 0.10605 0.11167 0.11743

0.22980 0.23818 0.24673 0.25545 0.26436

23438 1.45 29400 1.46 47838 1.47 98400 1.48 00838 1.49

0.27343 75000 A-2

1.50 -P

NUMERICAL FIVE-POINT

LAGRANGIAN

907

ANALYSIS INTERPOLATION

A;(P)

=(--l)k+Q &g

COEFFICIENTS

Table

25.1

) (p2-4)

-k)!(p-k)

1.50 1.51 1.52 1.53 1.54

-0.03906 -0.03946 -0.03983 -0.04017 -0.04048

25000 44163 61600 67163 50600

0.21875 0.22074 0.22257 0.22422 0.22569

00000 91650 66400 73650 62400

-0.54687 -0.55041 -0.55351 -0.55617 -0.55837

50000 09975 29600 17975 83600

Al 1.09375 00000 1.08643 21650 1.07864 06400 1.07036 83650 1.06160 82400

0.27343 0.28269 0.29213 0.30175 0.31155

75000 40838 18400 27838 89400

1.50 1.51 1.52 1.53 1.54

1.55 1.56 1.57 1.58 1.59

-0.04076 -0.04100 -0.04120 -0.04137 -0.04150

01563 09600 64163 54600 70163

0.22697 0.22806 0.22896 0.22964 0.23013

81250 78400 01650 98400 15650

-0.56012 -0.56139 -0.56219 -0.56249 -0.56230

34375 77600 19975 67600 25975

1.05235 1.04259 1.03232 1.02154 1.01023

31250 58400 91650 58400 85650

0.32155 0.33173 0.34210 0.35267 0.36343

23438 50400 90838 65400 94838

1.55 1.56 1.57 1.58 1.59

1.60 1.61 1.62 1.63 1.64

-0.04160 -0.04165 -0.04166 -0.04163 -0.04156

00000 33163 58600 65163 41600

0.23040 0.23044 0.23027 0.22987 0.22923

00000 97650 54400 15650 26400

-0.56160 -0.56037 -0.55863 -0.55634 -0.55351

00000 0.99840 00000 93975 0.98602 27650 11600 0.97309 94400 55975 0.95962 25650 29600 0.94558 46400

0.37440 0.38556 0.39692 0.40848 0.42025

00000 01838 21400 79838 98400

1.60 1.61 1.62 1.63 1.64

1.65 1.66 1.67 1. 68 1. 69

-0.04144 -0.04128 -0.04107 -0.04082 -0.04051

76563 58600 76163 17600 71163

0.22835 0.22722 0.22584 0.22421 0.22231

31250 74400 99650 50400 69650

-0.55012 -0.54616 -0.54163 -0.53651 -0.53079

34375 71600 41975 45600 81975

0.43223 0.44443 0.45683 0.46945 0.48228

98438 01400 28838 02400 43838

1.65 1.66 1.67 1.68 1.69

1.70 1.71 1.72 1.73 1.74

-0.04016 -0.03975 -0.03929 -0.03878 -0.03822

25000 0.22015 00000 67163 0.21770 83650 85600 0.21498 62400 68163 0.21197 77650 02600 0.20867 70400

-0.52447 -0.51753 -0.50996 -0.50176 -0.49290

50000 0.84915 00000 . 0.49533 75000 47975 0.83097 13650 0.50861 17838 73600 0.81217 02400 0.52210 94400 23975 0.79273 87650 0.53583 26838 95600 0.77266 90400 0.54978 37400

1.70 1. 71 1.72 1.73 1.74

1.75 1.76 1.77 1.78 1.79

-0.03759 -0.03691 -0.03617 -0.03538 -0.03452

76563 77600 93163 10600 17163

0.20507 0.20117 0.19696 0.19243 0.18758

81250 50400 17650 22400 03650

-0.48339 -0.47321 -0.46235 -0.45081 -0.43856

84375 0.75195 31250 0.56396 48438 85600 Oi73058 30400 0.57837 82400 93975 0.70855 07650 0.59302 61838 03600 0.68584 82400 0.60791 09400 07975 0.66246 73650 0.62303 47838

1.75 1.76 1.77 1.78 1.79

1.80 1.81 1.82 1.83 1.84

-0.03360 -0.03261 -0.03156 -0.03044 -0iOi926

00000 46163 42600 76163 33606

0.18240 0.17688 0.17102 0.16482 0.15826

00000 49650 90400 59650 94400

-0.42560 -0.41191 -0.39750 -0.38234 -0.36642

00000 0.63840 00000 0.63840 00000 71975 0.61363 79650 0.65400 88838 15600 0.58817 30400 0.66986 37400 21975 0.56199 69650 0.68596 68838 81600 0.53510 14400 0.70232 06400

1.80 1.81 1.82 1.83 1.84

1.85 1.86 1.87 1.88 1.89

-0.02801 -0.02668 -0.02529 -0.02382 -0.02228

01563 0.15135 31250 66600 0.14407 06400 15163 0.13641 55650 33600 0.12838 14400 08163 0.11996 17650

-0.34974 -0.33229 -0.31404 -0.29500 -0.27515

84375 19600 75975 41600 03975

0.50747 0.47911 0.45001 0.42015 0.38953

81250 0.71892 73438 86400 0.73578 93400 45650 0.75290 89838 74400 0.77028 86400 87650 0.78793 06838

1.85 1.86 1.87 1.88 1.89

1.90 1.91 1.92 1.93 1.94

-0.02066 -0.01896 -0.01719 -0.01533 -0.01340

25000 0.11115 00000 70163 0.10193 95650 29600 0.09232 38400 89163 0.08229 61650 34600 0..07184 98400

-0.25447 -0.23296 -0.21061 -0.18740 -0.16332

50000 65975 37600 49975 87600

0.35815 0.32598 0.29302 0.25927 0.22472

00000 0.80583 75000 25650 0.82401 14838 78400 0.84245 50400 71650 0.86117 05838 18400 0.88016 05400

1.90 1.91 1.92 1.93 1.94

1.95 1.96 1.97 1.98 1.99

-0.01138 -0.00928 -0.00709 -0.00481 -0.00245

51563 25600 42163 86600 44163

-0.13837 -0.11252 -0.08577 -0.05811 -0.02952

34375 0.18935 31250 73600 0.15316 22400 87975 0.11614 03650 59600 Or07827 86400 69975 0.03956 81650

P

2.00

A-1

A-2

0.00000 00000 AZ

0.06097 0.04967 0.03793 0.02574 0.01310

-40

81250 42400 13650 26400 11650

0.00000 00000 A1

0.00000

00000

Ao

0.93097 0.91579 0.90002 0.88367 0.86671

81250 54400 89650 10400 39650

A2

0.89942 0.91897 0.93880 0.95891 0.97931

73438 34400 12838 33400 20838

1.95 1.96 1.97 1.98 1.99

0.00000 00000 1.00000 00000 A-1 A-2

2.00 -P

908

NUMERICAL SIX-POINT

Table 25.1

LAGRANGIAN

INTERPOLATION

&p)+l)k+” A-1

Ao

COEFFICIENTS

P(P2-l)(P2-4) (P-3) (2+k) ! (3-k) ! (p-k) AI 0.00000 0.01006 0.02026 0.03058 0.04102

00000 60817 19736 41170 89152

0.00000 -0.00250 -0.00501 -0.00752 -0.01004

00000 38746 43268 95922 78976

0.00000 0.00033 0.00066 0.00099 0.00133

A3 00000 32917 63334 88752 06675

1.00 0.99 0.98 0.97 0.96

19531 31752 04458 66336 46604

0.05159 0.06227 0.07306 0.08396 0.09496

27344 19048 27217 14464 43071

-0.01256 -0.01508 -0.01760 -0.02011 -0.02262

74609 64924 31946 57632 23873

0.00166 0.00199 0.00231 0.00264 0.00296

14609 10065 90557 53606 96742

0.95 0.94 0.93 0.92 0.91

0.95460 0.94879 0.94276 0.93652 0.93006

75000 81771 97664 53917 82248

0.10606 0.11726 0.12855 0.13994 0.15140

75000 71904 95136 05758 64552

-0.02512 -0.02761 -0.03008 -0.03255 -0.03500

12500 05290 83968 30217 25676

0.00329 0:OOjbl 0.00392 0.00424 0.00455

17500 13426 82074 21011 27815

0.90 0.89 0.88 0.87 0.86

18359 88576 13273 00868 60096

0.92340 0691652 0.90945 0.90217 0.89470

14844 84352 23870 66936 47517

0.16295 0.17457 0.18627 0.19803 0.20986

32031 68448 33805 87864 90158

-0.03743 -0.03984 -0.04224 -0.04461 -0.04695

51953 90624 23240 31332 96417

0.00486 0.00516 0.00546 0.00575 0.00604

00078 35405 31416 85746 96051

0.85 0.84 0.83 0.82 0.81

-0.07392 -0.07629 -0.07854 -0.08067 -0.08269

00000 29929 59532 98752 57824

0.88704 0.87918 0.87114 0.86292 0.85452

00000 59183 60264 38830 30848

0.22176 0.23370 0.24570 0.25775 0.26984

00000 16492 78536 64845 93952

-0.04928 -0.05157 -0.05383 -0.05606 -0.05826

00000 23583 48668 56760 29376

0.00633 0.00661 0.00689 0.00716 0.00743

60000 75284 39614 50719 06355

0.80 0.79 0.78 0.77 0.76

94141 15055 18513 04314 72328

-0.08459 -0.08637 -0.08804 -0.08960 -0.09104

47266 77876 60729 07168 28802

0.84594 0.83720 0.82828 0.81920 0.80996

72656 00952 52783 65536 76929

0.28198 0.29415 0.30635 0.31858 0.33083

24219 13848 20892 03264 18746

-0.06042 -0.06254 -0.06463 -0.06667 -0.06868

48047 94324 49783 96032 14711

0.00769 0.00794 0.00819 0.00843 0.00866

04297 42345 18324 30086 75510

0.75 0.74 0.73 0.72 0.71

0.01044 0.01061 0.01077 0.01092 0.01106

22500 54844 69446 66459 46105

-0.09237 -0.09359 -0.09470 -0.09571 -0.09660

37500 45385 64832 08458 89124

0.80057 0.79102 0.78132 0.77148 0.76150

25000 48096 84864 74242 55448

0.34310 0.35538 0.36768 0.37998 0.39229

25000 79579 39936 63433 07352

-0.07063 -0.07254 -0.07441 -0.07622 -0.07798

87500 96127 22368 48054 55076

0.00889 0.00911 0.00932 0.00953 0.00973

52500 58993 92954 52378 35295

0.70 0.69 0.68 0.67 0.66

t::.

0.01119 0.01130 0.01140 0.01149 0.01157

08672 54515 84054 97774 96219

-0.09740 -0.09809 -0.09867 -0.09916 -0.09955

19922 14176 85435 47468 14258

0.75138 0.74113 0.73075 0.72024 0.70962

67969 51552 46195 92136 29842

0.40459 0.41688 0.42917 0.44144 0.45369

28906 85248 33480 30664 33833

-0.07969 -0.08134 -0.08293 -0.08447 -0.08594

25391 41024 84077 36732 81254

0.00992 0.01010 0.01028 0.01044 0.01060

39766 63885 05783 63626 35618

0.65 0.64 0.63 0.62 0.61

0.40 0.41 0.42 0.43 0.44

0.01164 0.01170 0.01175 0.01178 0.01180

80000 49786 06306 50351 82765

-0.09984 -0.10003 -0.10012 -0.10013 -0.10004

00000 19092 86132 15915 23424

0.69888 0.68802 0.67706 0.66599 0.65482

00000 43508 01464 15155 26048

0.46592 0.47811 0.49028 0.50241 0.51450

00000 86167 49336 46520 34752

-0.08736 -0.08870 -0.08998 -0.09120 -0.09234

00000 75421 90068 26598 67776

0.01075 0.01089 0.01102 0.01114 0.01125

20000 15052 19094 30487 47635

0.60 0.59 0.58 0.57 0.56

0.45 0.46 0.47 kE .

0.01182 0.01182 0.01181 0.01179 0.01176

04453 16375 19546 15034 03961

-0.09986 -0.09959 -0.09923 -0.09879 -0.09826

23828 32476 64892 36768 63965

0.64355 0.63220 0.62075 0.60922 0.59762

75781 06152 59108 76736 01254

0.52654 0.53854 0.55048 0.56236 0.57418

71094 12648 16567 40064 40421

-0.09341 -0.09441 -0.09534 -0.09619 -0.09696

96484 95724 48621 38432 48548

0.01135 0.01144 0.01153 0.01160 0.01166

68984 93025 18292 43366 66877

0.55 0.54 0.53 0.52 0.51

0.50

0.01171 87500

-0.09765

62500

0.58593 75000 Al

0.58593 75000 Ao

-0.09765

62500

0.01171 87500

0.50

A-2

o.‘do 0.01 0.00049

00000

0.02 0.03 0.04

57921 0.00098 30066 0.00146 14085 0.00193 07725

0.00000 -0.00493 -0.00973 -0.01440 -0.01893

00000 33767 36932 12590 64224

1.00000 0.99654 0.99283 0.98888 0.98469

0.05 0.06 0.07 0.08 0.09

0.00239 0.00284 0.00328 0.00371 0.00413

08828 15335 25281 36794 48096

-0.02333 -0.02761 -0.03175 -0.03576 -0.03964

95703 11276 15567 13568 10640

0.98026 0.97559 0.97069 0.96555 0.96019

0.10 0.11 0.12 0.13 0.14

0.00454 0.00494 0.00533 0.00571 0.00608

57500 63412 64326 58827 45585

-0.04339 -0.04701 -0.05050 -0.05387 -0.05710

12500 25223 55232 09296 94524

0.15 0.16 0.17 0.18 0.19

0.00644 0.00678 0.00712 0.00744 0.00776

23359 90995 47422 91654 22787

-0.06022 -0.06320 -0.06607 -0.06881 -0.07142

0.20 0.21 0.22 0.23 0.24

0.00806 0.00835 0.00863 0.00890 0.00915

40000 42553 29786 01118 56045

0.25 0.26 0;27 0.28 0.29

0.00939 0.00963 0.00985 0.01006 0.01025

0.30

0.00000

E 0:33 0.34 0.35 0.36 037

'

ANALYSIS

A3

A2

00000 20858 67064 64505 39648

A2

A-1

A-2

P

NUMERICAL SIX-POINT

LA( ;RANGIAN

INTERPOLATION

P 1.00 1.01 1.02 1.03 1.04

-0.00033 32917 -0.00066 63334 -0.00099 88752 -0.00133 06675

1.05 1.06 1.07 1.08 1.09

A-2

Ao

A-1

COEFFICIENTS

P@-l)(P*-4) (2+k) ! (3-k)

&)+l)k+3

909

ANALYSIS Table

25.1

(Y-3) ! (p-k)

AI

-42

A3

0.0024955421 0.0049810068 0.0074546597 0.0099147776

0.0000000000 -0.00993 27517 -0;01972 86936 -0.02938 43870 -0.03889 64352

1.0000000000 1.0032079192 1.0061633736 1.0088639545 1.0113073152

0.0000000000 0.0050667067 0.0102669732 0.0156009890 0.0210689024

0.0000000000 -0.00050 41246 -0.00101 63266 -0.00153 63410 -0.00206 38925

0.00

-0.00166 14609 -0.00199 10065 -0.00231 90556 -0.00264 53606 -0.00296 96742

0.0123596484 0.0147875724 0.0171968621 0.0195858432 0.0219528547

-0.04826 14844 -0.05747 62248 -0.06653 73917 -0.07544 17664 -0.08418 61771

1.0134911719 1.0154133048 1.0170715592 1.0184638464 1.0195881446

0.0266708203 0.0324068076 0.0382768866 0.0442810368 0.0504191940

-0.00259 86953 -0.00314 04535 -0.00368 88606 -0.00424 35994 -0.00480 43420

0.05 0.06 0.07 0.08 0.09

1.10 1.11 1.12 1.13 1.14

-0.00329 17500 -0.00361 13426 -0.00392 82074 -0.00424 21011 -0.00455 27815

0.0242962500 0.0266143965 0.0289056768 0.0311684892 0.0334012476

-0.09276 75000 -0.10118 26604 -0.10942 86336 -0.11750 24458 -0.12540 11752

1.0204425000 1.0210250279 1.0213339136 1.0213674133 1.0211238552

0.0566912500 0.0630970523 0.0696364032 0.0763090596 0.0831147324

-0.00537 07500 -0.00594 24737 -0.00651 91526 -0.00710 04152 -0.00768 58785

0.10 0.11 0.12

1.15 1.16 1.17 1.18 1.19

-0.0048600078 -0.00516 35405 -0.00546 31415 -0.00575 85746 -0.00604 96051

0.0356023828 -0.13312 19531 0.0377703424, -0.14066 19648 0.0399035915 -0.14801 84505 0.0420006132 -0.15518 87064 0.0440599092 -0.16217 00858

1.0206016406 1.0197992448 1.0187152180 1.0173481864 1.0156968533

0.0900530859 0.0971237376 0.1043262572 0.1116601668 0.1191249396

-0.00827 51484 -0.00886 78195 -0.00946 34747 -0.01006 16854 -0.01066 20112

0.15 0.16 0.17 0.18 0.19

1.20 1.21 1.22 1.23 1.24

-0.00633 60000 -0.00661 75284 -0.00689 39614 -0.00716 50719 -0.00743 06355

0.0460800000 0.0480594258 0.0499967468 0.0518905435 0.0537394176

-0.16896 00000 -0.17555 59192 -0.18195 53736 -0.18815 59545 -0.19415 53152

1.0137600000 1.0115364867 1.0090252536 1.0062253220 1.0031357952

0.1267200000 0.1344447229 0.1422984332 0.1502804052 0.1583898624

-0.01126 40000 -0.01186 71878 -0.01247 10986 -0.01307 52443 -0.01367 91245

0.20 0.21 0.22 0.23 0.24

1.25 1.26 1.27 1.28 1.29

-0.00769 04297 -0.00794 42345 -0.00819 18324 -0.00843 30086 -0.00866 75509

0.0555419922 0.0572969124 0.0590028458 0.0606584832 0.0622625385

-0.19995 11719 -0.20554 13048 -0.21092 35592 -0.21609 58464 -0.22105 61446

0.9997558594 0.9960847848 0.9921219267 0.9878667264 0.9833187121

0.1666259766 0.1749878676 0.1834746029 0.1920851968 0.2008186102

-0.01428 22266 -0.01488 40255 -0.01548 39838 -0.01608 15514 -0.01667 61653

0.25 0.26 _~0.27 0.28 0.29

1.30 1.31 1.32 1.33 1.34

-0.00889 52500 -0.00911 58993 -0.00932 92954 -0.00953 52378 -0.00973 35295

0.0638137500 0.0653108802 0.0667527168 0.0681380729 0.0694657876

-0.22580 25000 -0.23033 30279 -0.23464 59136 -0.23873 94133 -0.24261 18552

0.9784775000 0.9733427954 0.9679143936 0.9621921808 0.9561761352

0.2096737500 0.2186494685 0.2277445632 0.2369577758 0.2462877924

-0.01726 72500 -0.01785 42169 -0.01843 64646 -0.01901 33784 -0.01958 43305

0.30 0.31 0.32 0.33 0.34

1.35 1.36 1.37 1.38 1.39

-0.00992 39766 -0.01010 63885 -0.01028 05783 -0.01044 63626 -0.01060 35618

0.0707347266 0.0719437824 0.0730918752 0.0741779532 0.0752009929

-0.24626 16406 -0.24968 72448 -0.25288 72180 -0.25586 01864 -0.25860 48533

0.9498663281 0.9432629248 0.9363661855 0.9291764664 0.9216942208

0.2557332422 0.2652926976 0.2749646735 0.2847476268 0.2946399558

-0.02014 86797 -0.02070 57715 -0.02125 49379 -0.02179 54974 -0.02232 67544

0.35 t:; 0138 0.39

1.40 1.41 1.42 1.43 1.44

-0.01075 20000 -0.01089 15052 -0.01102 19094 -0.01114 30487 -0.01125 47635

0.0761600000 0.0770540096 0.0778820868 0.0786433273 0.0793368576

-0.26112 00000 -0.26340 44867 -0.26545 72536 -0.26727 73220 -0.26886 37952

0.9139200000 0.9058544542 0.8974983336 0.8888524895 0.8799178752

0.3046400000 0.3147460392 0.3249562932 0.3352689215 0.3456820224

-0.02284 80000 -0.02335 85111 -0.02385 75506 -0.02434 43676 -0.02481 81965

0.40 0.41 0.42 0.43 0.44

1.45 1.46 1.47 1.48 1.49

-0.01135 68984 -0.01144 93025 -0.01153 18292 -0.01160 43366 -0.01166 66877

0.0799618359 0.0805174524 0.0810029296 0.0814175232 0.0817605223

-0.27021 58594 -0.27133 27848 -0.27221 39267 -0.27285 87264 -0.27326 67121

0.8706955469 0.8611866648 0.8513924942 0.8413144064 0.8309538796

0.3561936328 0.3668017276 0.3775042192 0.3882989568 0.3991837265

-0.02527 82578 -0.02572 37575 -0.02615 38871 -0.02656 78234 -0.02696 47286

0.45 0.46 0.47 0.48 0.49

0.00000

00000

0.00000

00000

0.01 0.02 0.03 0.04

FiE .

1.50 -0.01171 87500 0.0820312500 -0.27343 75000 0.8203125000 0.4101562500 -0.02734 37500 0.50 A2 A-1 -43 Al Ao A-2 -P

910

NUMERICAL

Table 25.1

SIX-POINT

LAGRANGIAN

ANALYSIS

INTERPOLATION

COEFFICIENTS

P 1.50 1.51 1;52 1.53 1.54

A-2 -0.01171 87500 -0.01176 03961 -0.01179 15034 -0.01181 19546 -0.01182 16375

A-1 0.08203 12500 0.08222 90640 0.08235 33568 0.08240 35567 0.08237 91276

P(P2-VW4) (P-3) (2+k) ! (3-k) ! (p-k) Ao AI -0.27343 75000 0.82031 25000 -0.27337 07954 0.80939 19629 -0.27306 63936 0.79819 40736 -0.27252 41808 0.78672 07483 -0.27174 41352 0.77497 40152

1.55 1.56 1.57 1.58 1.59

-0.01182 -0.01180 -0.01178 -0.01175 -0.01170

04453 82765 50350 06306 49786

0.08227 0.08210 0.08185 0.08152 0.08112

95703 44224 32590 56932 13767

-0.27072 -0.26947 -0.26797 -0.26624 -0.26428

63281 09248 81855 84664 22208

0.76295 0.75066 0.73811 0.72529 0.71221

60156 90048 53530 75464 81883

0.46625 0.47769 0.48921 0.50080 0.51244

08984 84576 59897 06868 96721

-0.02893 -0.02919 -0.02942 -0.02962 -0.02980

97109 26835 13812 48294 20377

0.55 0.56 0.57 0.58 0.59

1.60 1.61 1.62 1.63 1.64

-0.01164 -0.01157 -0.01149 -0.01140 -0.01130

80000 96219 97774 84054 54515

0.08064 0.08008 0.07944 0.07873 0.07793

00000 12933 50268 10110 90976

-0.26208 -0.25964 -0.25697 -0.25406 -0.25092

00000 24542 03336 44895 58752

0.69888 0.68528 0.67143 0.65734 0.64299

00000 58217 86136 14570 75552

0.52416 0.53592 0.54775 0.55962 0.57155

00000 86554 25532 85377 33824

-0.02995 -0.03007 -0.03016 -0.03022 -0.03025

20000 36943 60826 81108 87085

0.60 0.61 0.62 0.63 0.64

1.65 1.66 1.67 1.68 1.69

-0.01119 -0.01106 -0.01092 -0.01077 -0.01061

08672 46105 66459 69446 54845

0.07706 0.07612 0.07509 0.07399 0.07280

91797 11924 51133 09632 88061

-0.24755 -0.24395 -0.24012 -0.23606 -0.23178

55469 46648 44942 64064 18796

0.62841 0.61358 0.59851 0.58322 0.56769

02344 29448 92617 28864 76471

0.58352 0.59553 0.60758 0.61967 0.63179

37891 63876 77354 43168 25427

-0.03025 -0.03022 -0.03015 -0.03004 -0.02990

67891 12495 09703 48154 16318

0.65 0.66 0.67 0.68 0.69

1.70 1.71 1.72 1.73 1.74

-0.01044 -0.01025 -0.01006 -0.00985 -0.00963

22500 72328 04314 18513 15055

0.07154 0.07021 0.06879 0.06730 0.06573

87500 09477 55968 29404 32676

-0.22727 -0.22253 -0.21758 -0.21241 -0.20702

25000 99629 60736 27483 20152

0.55194 0.53597 0.51978 0.50338 0.48678

75000 65304 89536 91158 14952

0.64393 0.65610 0.66830 0.68050 0.69272

87500 92010 00832 75083 75124

-0.02972 -0.02949 -0.02923 -0.02893 -0.02858

02500 94834 81286 49649 87545

0.70

1.75 1.76 1.77 1.78 1.79

-0.00939 -0.00915 -0.00890 -0.00863 -0.00835

94141 56045 01118 29786 42553

0.06408 0.06236 0.06056 0.05869 0.05674

69141 42624 57427 18332 30604

-0.20141 -0.19559 -0.18956 -0.18332 -0.17688

60156 70048 73530 95464 61883

0.46997 0.45296 0.43575 0.41836 0.40079

07031 14848 87205 74264 27558

0.70495 0.71718 0.72942 0.74165 0.75387

60547 90176 22061 13468 20883

-0.02819 -0.02776 -0.02727 -0.02674 -0.02616

82422 21555 92045 80814 74609

0.75 0.76 0.77 0.78 0.79

1.80 1.81 1.82 1.83 1.84

-0.00806 -0.00776 -0.00744 -0.00712 -0.00678

40000 22787 91654 47422 90995

0.05472 0.05262 0.05045 0.04821 0.04589

00000 32771 35668 15948 81376

-0.17024 -0.16339 -0.15635 -0.14911 -0.14168

00000 38217 06136 34570 55552

0.38304 0.36511 0.34702 0.32876 0.31035

00000 45892 20936 82245 88352

0.76608 0.77827 0.79043 0.80258 0.81469

00000 05717 92132 12540 19424

-0.02553 -0.02485 -0.02411 -0.02332 -0.02247

60000 23376 50946 28741 42605

0.80 0.81 0.82 0.83 0.84

1.85 1.86 1.87 1.88 1.89

-0.00644 -0.00608 -0.00571 -0.00533 -0.00494

23359 45585 58826 64326 63412

0.04351 0.04106 0.03853 0.03594 0.03328

40234 01324 73971 68032 93898

-0.13407 -0.12627 -0.11829 -0.11013 -0.10180

02344 09448 12617 48864 56471

0.29179 0.27309 0.25425 0.23528 0.21619

99219 76248 82292 81664 40145

0.82676 0.83879 0.85078 0.86272 0.87460

64453 98476 71516 32768 30590

-0.02156 -0.02060 -0.01957 -0.01848 -0.01733

78203 21015 56336 69274 44750

0.85 0.86 0.87 0.88 0.89

1.90 1.91 1.92 1.93 1.94

-0.00454 -0.00413 -0.00371 -0.00328 -0.00284

57500 48096 36794 25281 15335

0.03056 0.02777 0.02492 0.02201 0.01904

62500 85315 74368 42242 02076

-0.09330 -0.08464 -0.07582 -0.06684 -0.05770

75000 45304 09536 11158 94952

0.19698 0.17766 0.15823 0.13871 0.11910

25000 04979 50336 32833 25752

0.88642 0.89817 0.90985 0.92145 0.93297

12500 25173 14432 25246 01724

-0.01611 -0.01483 -0.01347 -0.01205 -0.01056

67500 22067 92806 63882 19265

0.90 0.91 0.92 0.93 0.94

1.95 1.96 1.97 1.98 1.99

-0.00239 -0.00193 -0.00146 -0.00098 -0.00049

08828 07725 14086 30066 57921

0.01600 0.01291 0.00976 0.00656 0.00330

67578 53024 73265 43732 80442

-0.04843 -0.03900 -0.02945 -0.01975 -0.00994

07031 94848 07205 94264 07558

0.09941 0.07964 0.05981 0.03992 0.01998

03906 43648 22880 21064 19233

0.94439 0.95573 0.96696 0.97809 0.98910

87109 23776 53223 16068 52046

-0.00899 -0.00735 -0.00563 -0.00383 -0.00195

42734 17875 28077 56534 86242

0.95

0.00000 00000 A-2

1.00 -P

&)+l)k+3

2.00

0.00000 00000 -43

0.00000 00000 A2

0.00000 00000 Al

0.00000 00000 Ao

A2 0.41015 62500 0.42121 41848 0.43235 51232 0.44357 65921 0.45487 60524

-0.02734 -0.02770 -0.02804 -0.02836 -0.02866

37500 40202 46566 47617 34225

0.50 0.51 0.52 0.53 0.54

1.00000 00000 A-1

-43

2:: 0:73 0.74

% 0:98 0.99

NUMERICAL

SIX-POINT

LAGRANGIAN

INTERPOLATION

&+(4)k+3

P

A-1

A-2

911

ANALYSIS

Ao

COEFFICIENTS

Pk-l)(P2-4) (P-3) (2+k) ! (3-k) ! (p-/c) Al

Table 25.1

A2

A3

2.00 2.01 2.02 2.03 2.04

0.00000 0.00050 0.00101 0.00153 0.00206

00000 41246 63266 63410 38925

0.00000 -0.00335 -0.00676 -0.01021 -0.01371

00000 80392 42932 69214 40224

0.00000 0.01005 0.02022 0.03049 0.04087

00000 74108 59064 97755 31648

0.00000 -0.02001 -0.04005 -0.06011 -0.08017

00000 52433 52264 12080 42848

1.00000 1.01076 1.02140 1.03190 1.04226

00000 97879 82732 90702 57024

0.00000 0.00204 0.00416 0.00638 0.00868

00000 19592 90134 29427 55475

1.00 1.01 1.02 1.03 1.04

2.05 2.06 2.07 2.08 2.09

0.00259 0.00314 0.00368 0.00424 0.00480

86953 04535 88605 35994 43420

-0.01725 -0.02083 -0.02445 -0.02810 -0.03179

36328 37276 22191 69568 57264

0.05134 0.06189 0.07252 0.08323 0.09401

00781 43752 97708 98336 79854

-0.10023 -0.12028 -0.14031 -0.16031 -0.18027

53906 52952 46033 37536 30179

1.05247 1.06252 1.07240 1.08211 1.09165

16016 01076 44679 78368 32752

0.01107 0.01356 0.01614 0.01881 0.02159

86484 40865 37232 94406 31417

1.05 1.06 1.07 1.08 1.09

2.10 2.11 2.12 2.13 2.14

0.00537 0.00594 0.00651 0.00710 0.00768

07500 24737 91526 04151 58785

-0.03551 -0.03926 -0.04304 -0.04684 -0.05066

62500 61847 31232 45921 80524

0.10485 0.11575 0.12669 0.13767 0.14868

75000 15021 29664 47167 94248

-0.20018 -0.22003 -q.23981 -0.25951 -0.27911

25000 21346 16864 07492 87448

1.10100 1.11016 1.11912 1.12787 1.13641

37500 21335 12032 36409 20324

0.02446 0.02744 0.03052 0.03370 0.03699

67500 22100 14874 65686 94615

1.10 1.11 1.12 1.13 1.14

2.15 2.16 2.17 2.18 2.19

0.00827 0.00886 0.00946 0.01006 0.01066

51484 78195 34747 16854 20112

-0.05451 -0.05837 -0.06224 -0.06612 -0.07002

08984 04576 39898 86868 16721

0.15972 0.17078 0.18185 0.19292 0.20399

96094 76352 57120 58936 00767

-0.29862 -0.31801 -0.33728 -0.35642 -0.37541

49219 83552 79445 24136 03092

1.14472 1.15281 1.16066 1.16827 1.17562

88672 65376 73385 34668 70208

0.04040 0.04391 0.04754 0.05129 0.05515

21953 68205 54091 00546 28726

1.15 1.16 1.17 1.18 1.19

2.20 2.21 2.22 2.23 2.24

0.01126 0.01186 0.01247 0.01307 0.01367

40000 71878 10986 52443 91245

-0.07392 -0.07782 -0.08172 -0.08561 -0.08950

00000 06554 05532 65377 53824

0.21504 0.22606 0.23706 0.24801 0.25892

00000 72433 32264 92080 62848

-0.39424 -0.41289 -0.43137 -0.44966 -0.46773

00000 96758 73464 08405 78048

1.18272 1.18954 1.19609 1.20235 1.20832

00000 43042 17332 39865 26624

0.05913 0.06324 0.06747 0.07182 0.07631

60000 15959 18414 89394 51155

1.20 1.21 1.22 1.23 1.24

2.25 2.26 2.27 2.28 2.29

0.01428 0.01488 0.01548 0.01608 0.01667

22266 40255 39838 15514 61653

-0.09338 -0.09724 -0.10109 -0.10492 -0.10872

37891 83876 57353 23168 45427

0.26977 0.28055 0.29126 0.30188 0.31240

53906 72952 26033 17536 50179

-0.48559 -0.50322 -0.52060 -0.53772 -Ok55457

57031 18152 32358 68736 94504

1.21398 1.21934 1.22438 1.22908 1.23346

92578 51676 16841 99968 11915

0.08093 0.08568 0.09057 0.09559 0.10076

26172 37145 06999 58886 16184

1.25 1.26 1.27 1.28 1.29

2.30 2.31 2.32 2.33 2.34

0.01726 0.01785 0.01843 0.01901 0.01958

72500 42169 64646 33784 43305

-0.11249 -0.11624 -0.11994 -0.12361 -0.12723

87500 12010 80832 55083 95124

0.32282 0.33312 0.34329 0.35333 0.36323

25000 41346 96864 87492 07448

-0.57114 -0.58741 -0.60337 -0.61900 -0.63429

75000 73671 52064 69817 84648

1.23748 1.24115 1.24446 1.24739 1.24994

62500 60498 13632 28571 10924

0.10607 0.11152 0.11712 0.12287 0.12878

02500 41668 57754 75053 18095

1.30 1.31 1.32 1.33 1.34

2.35 z76 2138 2.39

0.02014 0.02070 0.02125 0.02179 0.02232

86797 57715 49379 54974 67544

-0.13081 -0.13434 -0.13781 -0.14121 -0.14456

60547 10176 02060 93468 40883

0.37296 0.38253 0.39191 0.40111 0.41010

49219 03552 59445 04136 23092

-0.64923 -0.66380 -0.67798 -0.69177 -0.70513

52344 26752 59770 01336 99417

1.25209 1.25384 1.25519 1.25610 1.25659

65234 94976 02548 89268 55371

0.13484 0.14105 0.14743 0.15397 0.16067

11641 80685 50458 46426 94293

1.35 1.36 1.37 1.38 1.39

2.40 2.41 2.42 2.43 2.44

0.02284 0.02335 0.02385 0.02434 0.02481

80000 85111 75506 43676 81965

-0.14784 -0.15104 -0.15416 -0.15720 -0.16016

00000 25717 72132 92540 39424

0.41888 0.42743 0.43574 0.44380 0.45160

00000 16758 53464 88405 98048

-0.71808 -0.73057 -0.74260 -0.75416 -0.76522

00000 47083 82664 46730 77248

1.25664 1.25623 1.25536 1.25401 1.25219

00000 21204 15932 80027 08224

0.16755 0.17459 0.18181 0.18920 0.19677

20000 49727 09894 27162 28435

1.40 1.41 1.42 1.43 1.44

2.45 2.46 2.47 2.48 2.49

0.02527 0.02572 0.02615 0.02656 0.02696

82578 37575 38870 78234 47286

-0.16302 -0.16579 -0.16845 -0.17101 -0.17345

64453 18476 51516 12768 50590

0.45913 0.46637 0.47331 0.47993 0.48623

57031 38152 12358 48736 14504

-0.77578 -0.78580 -0.79529 -0.80421 -0.81256

10156 79352 16683 51936 12829

1.24986 1.24704 1.24370 1.23983 1.23542

94141 30276 08004 17568 48077

0.20452 0.21245 0.22058 0.22889 0.23739

40859 91825 08967 20166 53552

1.45 1.46 1.47 1.48 1.49

2.50

0.02734 37500

-0.17578

12500

0.49218 75000

-0.82031

25000

1.23046t~7500

0.24609 37500

1.50

-43

A2

AI

Ao

1

-1-2

-P

912

NUMERICAL fjIX.POINT

Table 25.1

ANALYSIS

LAGRANGIAN

INTERPOLATION

&+(-l)k+3

P(P2-l)(P2-4)(P-3)

COEFFICIENTS

(2+k) ! (3-k) !(p-k) -40 Al 0.4921875000 -0.82031 25000 0.4977893671 -0.82745 11996 0.5030232064 -0.83395 95264 0.5078749817 -0.83981 94142 0.5123304648 -0.84501 25848

A2 1.2304687500 1.2249522660 1.2188639232 1.2121921734 1.2049253524

A3 0.2460937500 0.2549900635 0.2640871834 0.2733880221 0.2828955175

1.50 1.51 1.52 1.53 1.54

2.50 2.51 2.52 2.53 2.54

0.0273437500 0.0277040203 0.0280446566 0.0283647616 0.0286634225

A-1 -0.17578 12500 -0.17798 45173 -0.18005 94432 -0.18200 05246 -0.18380 21724

2.55 2.56 2.57 2.58 2.59

0.0289397109 0.0291926835 0.0294213812 0.0296248294 0.0298020377

-0.18545 87109 -0.18696 43776 -0.18831 33223 -0.18949 96068 -0.19051 72046

0.5163752344 0.5199946752 0.5231739770 0.5258981336 0.5281519417

-0.84952 05469 -0.85332 45952 -0.85640 58095 -0.85874 50536 -0.86032 29742

1.1970516797 1.1885592576 1.1794360710 1.1696699868 1.1592487533

0.2926126328 0.3025423565 0.3126877026 0.3230517106 0.3336374461

1.55 1.56 1.57 1.58 1.59

2.60 2.61 2.62 2.63 2.64

0.0299520000 0.0300736943 0.0301660826 0.0302281107 0.0302587085

-0.19136 00000 -0.19202 17879 -0.19249 62732 -0.19277 70702 -0.19285 77024

0.5299200000 0.5311867083 0.5319362664 0.5321526730 0.5318197248

-0.86112 00000 -0.86111 63408 -0.86029 19864 -0.85862 67055 -0.85610 00448

1.1481600000 1.1363912367 1.1239298532 1.1107631190 1.0968781824

0.3444480000 0.3554864894 0.3667560574 0.3782598730 0.3900011315

1.60 1.61 1.62 1.63 1.64

2.65 2.66 2.67 2.68 2.69

0.0302567891 0.0302212495 0.0301509704 0.0300448154 0.0299016317

-0.19273 16016 -0.19239 21076 -0.19183 24679 -0.19104 58368 -0.19002 52752

0.5309210156 0.5294399352 0.5273596683 0.5246631936 0.5213332829

-0.85269 13281 -0.84837 96552 -0.84314 39008 -0.83696 27136 -0.82981 45154

1.0822620703 1.0669016876 1.0507838166 1.0338951168 1.0162221240

0.4019830547 0.4142088905 0.4266819134 0.4394054246 0.4523827520

1.65 1.66 1.67 1.68 1.69

2.70 2.71 2.72 2.73 2.74

0.0297202500 0.0294994834 0.0292381286 0.0289349650 0.0285887545

-0.18876 37500 -0.18725 41335 -0.18548 92032 -0.18346 16409 -0.18116 40324

0.5173525000 0.5127031996 0.5073675264 0.5013274142 0.4945645848

-0.82167 75000 -0.81252 96321 -0.80234 86464 -0.79111 20467 -0.77879 71048

0.9977512500 0.9784687823 0.9583608832 0.9374135896 0.9156128124

0.4656172500 0.4791123003 0.4928713114 0.5068977188 0.5211949855

1.70 1.71 1.72 1.73 1.74

2.75 2.76 2.77 2.78 2.79

0.0281982422 0.0277621555 0.0272792044 0.0267480814 0.0261674609

-0.17858 88672 -0.17572 85376 -0.17257 53385 -0.16912 14668 -0.16535 90208

0.4870605469 0.4787965952 0.4697538095 0.4599130536 0.4492549742

-0.76538 08594 -0.75084 01152 -0.73515 14420 -0.71829 11736 -0.70023 54067

0.8929443359 0.8693938176 0.8449467873 0.8195886468 0.7933046696

0.5357666016 0.5506160845 0.5657469793 0.5811628586 0.5968673228

1.75 1.76 1.77 1.78 1.79

2.80 2.81 2.82 2.83 2.84

0.0255360000 0.0248523376 0.0241150946 0.0233228741 0.0224742605

-0.16128 00000 -0.15687 63042 -0.15213 97332 -0.14706 19865 -0.14163 46624

0.4377600000 0.4254083408 0.4121799864 0.3980547055 0.3830120448

-0.68096 00000 -0.66044 05733 -0.63865 25064 -0.61557 09380 -0.59117 07648

0.7660800000 0.7378996529 0.7087485132 0.6786113352 0.6474727424

0.6128640000 0.6291565462 0.6457486454 0.6626440097 0.6798463795

1.80 1.81 1.82 1.83 1.84

2.85 2.86 2.87 2.88 2.89

0.0215678203 0.0206021015 0.0195756335 0.0184869274 0.0173344751

-0.13584 92578 -0.12969 71676 -0.12316 96841 -0.11625 79968 -0.10895 31915

0.3670313281 0.3500916552 0.3321719008 0.3132507136 0.2933065154

-0.56542 66406 -0.53831 29752 -0.50980 39333 -0.47987 34336 -0.44849 51479

0.6153172266 0.5821291476 0.5478927329 0.5125920768 0.4762111402

0.6973595234 0.7151872385 0.7333333502 0.7518017126 0.7705962087

1.85 1.86 1.87 1.88 1.89

2.90 2.91 2.92 2.93 2.94

0.0161167500 0.0148322068 0.0134792806 0.0120563881 0.0105619265

-0.10124 62500 -0.09312 80498 -0.08458 93632 -0.07562 08571 -0.06621 30924

0.2723175000 0.2502616321 0.2271166464 0.2028600467 0.1774691048

-0.41564 25000 -0.38128 86646 -0.34540 65664 -0.30796 88792 -0.26894 80248

0.4387337500 0.4001435985 0.3604242432 0.3195591059 0.2775314724

0.7897207500 0.8091792770 0.8289757594 0.8491141956 0.8695986135

1.90 1.91 1.92 1.93 1.94

2.95 2.96 2.97 2.98 2.99

0.0089942734 0.0073517875 0.0056328077 0.0038356534 0.0019586242

-0.05635 65234 -0.04604 14976 -0.03525 82547 -0.02399 69268 -0.01224 75371

0.1509208594 0.1231921152 0.0942594420 0.0640991i'36 0.0326874067

-0.22831 61719 -0.18604 52352 -0.14210 68745 -0.09647 24936 -0.04911 32392

0.2343244922 0.1899211776 0.1443044035 0.0974569068 0.0493612858

0.8904330703 0.9116216525 0.9331684760 0.9550776866 0.9773534596

1.95 1.96 1.97 1.98 1.99

0.00000 00000

1.0000000000 2.00

P

A-2

3.00 0.0000000000 0.00000 00000 -43 A2

0.00000 00000

Al

0.00000 00000

Ao

A-1

A-2

-P

NUMERICAL

SEVEN-POINT

LAGRANGIAN

INTERPOLATION

A-3

A-2

COEFFICIENTS

Table

25.1

(p2-4)(p2-9) (3+k) !(3-k) ! (p-k)

&+~l)k'3Ppl)

P

913

ANALYSIS

A-1

0.0 0.1 0.2 0.3 0.4

0.00000 -0.00159 -0.00295 -0.00400 -0.00465

00000 10125 68000 28625 92000

0.00000 0.01409 0.02580 0.03445 0.03960

00000 18250 48000 94250 32000

0.00000 -0.06725 -0.11827 -0.15241 -0.16972

00000 64375 20000 66875 80000

1.00000 0.98642 0.94617 0.88062 0.79206

Ao 00000 77500 60000 97500 40000

0.00000 0.08220 0.17740 0.28305 0.39603

Al 00000 23125 80000 95625 20000

0.00000 -0.01557 -0.03153 -0.04662 -0.05940

A2 00000 51750 92000 15750 48000

0.00000 0.00170 0.00337 0.00489 0.00609

00000 07375 92000 23875 28000

0.0 0.1 0.2 0.3 0.4

0.5 0.6 0.7 0.8 0.9

-0.00488 -0.00465 -0.00400 -0.00295 -0.00159

28125 92000 28625 68000 10125

0.04101 0.03870 0.03291 0.02407 0.01283

56250 72000 24250 68000 78250

-0.17089 -0.15724 -0.13068 -0.09363 -0.04898

84375 80000 16875 20000 64375

0.68359 0.55910 0.42315 0.28089 0.13788

37500 40000 97500 60000 77500

0.51269 0.62899 0.74052 0.84268 0.93074

53125 20000 95625 80000 23125

-0.06835 -0.07188 -0.06835 -0.05617 -0.03384

93750 48000 65750 92000 51750

0.00683 0.00698 0.00643 0.00510 0.00295

59375 88000 93875 72000 47375

0.5 0.6 0.7 0.8 0.9

A3

1.0 1.1 :*: 1:4

0.00000 0.00170 0.00337 0.00489 0.00609

00000 07375 92000 23875 28000

0.00000 -0.01349 -0.02661 -0.03824 -0.04730

00000 61750 12000 95750 88000

0.00000 0.04980 0.13719 0.09676 0.16755

00000 73125 95625 80000 20000

0.00000 -0.12678 -0.32365 -0.23654 -0.38297

00000 22500 02500 40000 60000

1.00000 1.04595 1.05186 1.06444 1.00531

00000 35625 80000 33125 20000

0.00000 0.04648 0.10644 0.18031 0.26808

00000 68250 94250 48000 32000

0.00000 -0.00367 -0.00788 -0.01237 -0.01675

00000 00125 48000 48625 52000

1.0 1.1 1.2 1.3 1.4

1.5 1.6 1.7

0.00683 0.00698 0.00643 0.00510 0.00295

59375 88000 93875 72000 47375

-0.05273 -0.05358 -0.04907 -0.03870 -0.02227

43750 08000 85750 72000 41750

0.18457 0.18547 0.16813 0.13132 0.07488

03125 20000 95625 80000 73125

-0.41015 -0.40185 -0.35606 -0.2723'3 -0.15240

62500 60000 02500 40000 22500

0.92285 0.80371 0.64853 0.45964 0.24130

15625 20000 83125 80000 35625

0.36914 0.48222 0.60530 0.73543 0.86869

06250 72000 24250 68000 28250

-0.02050 -0.02296 -0.02328 -0.02042 -0.01316

78125 32000 08625 88000 20125

1.5 1.6 1.7 1.8 1.9

2.0 2.1 2.2 2.3 2.4

0.00000 -0.00367 -0.00788 -0.01237 -0.01675

00000 00125 48000 48625 52000

0.00000 0.02739 0.05857 0.09151 0.12337

00000 0'3250 28000 64250 92000

0.00000 -0.09056 -0.19219 -0.29812 -0.39916

00000 64375 20000 16875 80000

0.00000 0.17825 0.37273 0.57031 0.75398

00000 77500 60000 97500 40000

0.00000 -0.25523 -0.51251 -0.75677 -0.96940

00000 26875 20000 04375 80000

1.00000 1.12302 1.23002 1.31173 1.35717

00000 38250 66000 54250 12000

0.00000 0.02079 0.05125 0.09369 0.15079

00000 67375 12000 53875 68000

2.0 2.1 2.2 2.3 2.4

2.5 2.6 2.7 2.8 2.9

-0.02050 -0.02296 -0.02328 -0.02042 -0.01316

78125 32000 08625 88000 20125

0.15039 0.16773 0.16940 0.14810 0.09508

06250 12000 54250 '38000 88250

-0.48339 -0.53580 -0.53797 -0.46771 -0.29867

84375 80000 66875 20000 b4375

0.90234 0.98918 0.98296 0.84633 0.53555

37500 40000 97500 60000 77500

-1.12792 -1.20556 -1.17089 -0.98739 -0.61307

96875 80000 04375 20000 26875

1.35351 1.28593 1.13743 0.88865 0.51770

56250 92000 64250 28000 58250

0.22558 0.32148 0.44233 0.59243 0.77655

59375 48000 63675 52000 87375

2.5 2.6 2.7 2.8 2.9

3.0

0.00000

00000 A3

0.00000

00000 A2

00000

0.00000

00000

0.00000

00000

1.00000

00000

3.0 -p

0.00000 Al

EIGHT-POIKT

LAGRANGIAN

A;(p)=(pl)kt’ A-3

P

0 00000

A-2 00000

if -0'00086 0:2. -0:00160 0.3 -0.00211 0.4 -0.00239 0.5 -0.00244

64213 51200 57988 61600 14063

0.00000

A-1 00000

0.00915 0.01634 0.02124 0.02376 0.02392

0.00000

96863 30400 99787 19200 57812

-0.05246 -0.08988 -0.11278 -0.12220 -0.11962

00000

0.00000

Ao

A-1

A-3

A-2

INTERPOLATION

COEFFICIENTS

P(P2-1)(P2-4)(P2-g)(p4) (3+k)!(4-k)!(p-k) Ao

4

A2

A4

A3

00213 67200 83487 41600 89062

1.00000 0.96176 0.89886 0.81458 0.71285 0.59814

00000 70563 72000 25188 76000 45312

0.00000 0.10686 0.22471 0.34910 0.47523 0.59814

00000 30063 68000 67938 84000 45313

0.00000 -0.03037 -0.05992 -0.08624 -0.10692 -0.11962

00000 15913 44800 99137 86400 89062

0.00000 0.00663 0.01284 0.01810 0.02193 0.02392

00000 28763 09600 18337 40800 57812

0.00000 -0.00070 -0.00135 -0.00188 -0.00226 -0.00244

00000 45913 16800 70638 30400 14062

00000

E 0:6 0.5

1.0 1.1

0.00000 0.00070

00000 45912

0.00000 -0.00652

00000 31512

0.00000 0.02888

00000 82412

0.00000 -0.09191

00000 71312

1.00000 1.01108

00000 84438

0.00000 0.06740

00000 98962

0.00000 -0.01064

00000 30362

0.00000 0.00099

00000 61462

:*: 1:4

0.00135 0.00188 0.00226

16800 70638 30400

-0.01241 -0.01721 -0.02050

23088 85600 04800

0.05419 0.07408 0.08712

77638 00600. 70400

-0.21846 -0.16558 -0.24893

08000 39188 44000

0.99348 0.94667 0.87127

48000 69812 04000

0.14902 0.24343 0.34850

27200 12238 81600

-0.03341 -0.02207 -0.04356

74400 21288 35200

0.00300 0.00202 0.00382

75200 53238 97600

J-1" -0:2 -0.3 -0.4

;:; 1:8 1.9

0.00239 0.00244 0.00211 0.00160 0.00068

61600 14062 57988 51200 64213

-0.02197 -0.01881 -0.02143 -0.01419 -0.00779

26562 34538 23200 26400 59613

0.09228 0.07734 0.08902 0.05778 0.03145

51562 41988 65600 43200 26712

-0.24111 -0.20473 -0.25634 -0.14981 -0.08001

76562 46438 36000 12000 11812

0.76904 0.49721 0.64296 0.33707 O.lb891

29688 27062 96000 52000 24938

0.57867 0.46142 0.69609 0.80898 0.91212

26400 57812 77888 04800 74662

-0.05126 -0.05354 -0.05511 -0.04494 -0.02764

95312 59838 16800 33600 02263

0.00439 0.00432 0.00459 0.00350 0.00206

45312 35888 26400 20800 83163

-0.5 -0.6 -0.7 -0.6 -0.9

2.0 2.1 2.2 2.3 2.4

0.00000 -0.00099 -0.00202 -0.00300 -0.00382

00000 61462 75200 53238 97600

0.00000 0.00867 0.01757 0.02592 0.03290

00000 37612 18400 96538 11200

0.00000 -0.03441 -0.06918 -0.10136 -0.12773

00000 52462 91200 13738 37600

0.00000 0.08467 0.16773 0.24238 0.30159

00000 24312 12000 58938 36000

0.00000 -0.16164 -0.30750 -0.42883 -0.51701

00000 73600 72000 65812 76000

1.00000 1.06687 1.10702 1.11497 1.08573

00000 26338 59200 51112 69600

0.00000 0.03951 0.09225 0.15928 0.24127

00000 38012 21600 21588 48800

0.00000 -0.00267 -0.00585 -0.00936 -0.01292

00000 38662 72800 95388 54400

-1.0 -1.1 -1.2 -1.3 -1.4

2.5 2.6 2.7 2.8 2.9

-0.00439 -0.00459 -0.00432 -0.00350 -0.00206

45312 26400 35888 20800 83162

0.03759 0.03913 0.03670 0.02962 0.01743

76562 72800 45088 17600 29512

-0.14501 -0.15002 -0.13987 -0.11225 -0.06570

95312 62400 39388 08600 88162

0.33837 0.34621 0.31946 0.25390 0.14727

89062 44000 51688 08000 83812

-0.56396 -0.56259 -0.50738 -0.39495 -0.22479

48438 84000 58562 68000 33188

1.01513 0.90015 0.73933 0.53319 0.28473

67188 74400 36762 16800 82038

0.33837 0.45007 0.57503 0.71092 0.85421

89062 87200 73038 22400 46112

-0.01611 -0.01837 -0.01895 -0.01692 -0.01109

32812 05600 72738 67200 36962

-1.5 -1.6 -1.7

-2.0 -2.1 -2.2 -2.3 -2.4

1:::

3.0 3.1 3.2 3.3 3.4

0.00000 0.00267 0.00585 0.00936 0.01292

00000 38662 72800 95388 54400

0.00000 -0.02238 -0.04888 -0.07796 -0.10723

00000 70762 57600 16338 32800

0.00000 0.08354 0.18157 0.28827 0.39481

00000 20162 56800 67388 34400

0.00000 -0.18415 -0.39719 -0.62605 -0.85155

00000 17562 68000 55438 84000

0.00000 0.27184 0.57774 0.89825 1.20637

00000 30688 06000 36062 44000

0.00000 -0.31138 -0.63551 -0.95353 -1.24084

00000 38788 48800 07512 22400

1.00000 1.14174 1.27102 1.37732 1.44764

00000 08888 97600 21962 92800

0.00000 0.01812 0.04539 0.08432 0.13787

00000 28712 39200 56400 13600

;,;.

0.01837 0.01611

05600 32812

-0.15155 -0.13330

07812 71200

0.55351 0.48876

29600 95312

-1.17877 -1.04736

32812 76000

1.63215 1.46630

36000 85938

-1.59134 -1.46630

97600 85938

1.41453 1.46630

31200 65930

0.30311 0.20947

42400 26562

:-'8 3:9

0'001692 01895 0:OllOS

72738 67200 36962

-015598 -0' 13891 -0:09081

17788 58400 78862

0.56750 0.50356 0.32805

81738 99200 64462

-1.20148 -1.06014 -0.68695

12688 72000 58062

1.64647 1.43877 0.92383

43312 12000 71188

-1.34285 -1.56899 -0.84604

31200 31862 03088

1.00713 1.27013 0.59536

73412 98400 16988

0.57550 0.42337 0.76546

91138 84800 50412

1:; -2:e -2.9

4.0

0.00000

00000

0.00000

00000

0.00000

00000

0.00000 00000 A-2

1.00000

00000

-3.0

00000 A4

0.00000 A3

00000 A2

0.00000 4

00000 Ao

0.00000 A-1

A-3

-2.5

P

914 Table

NUMERICAL

25.2

COEFFICIENTS

Differentiation FlRST

DERIVATIVE

4 0

DIFFERENTlATlON

dkf (,l) I xx $ Aif -,,,&Y ,z=zj / It~!hki,o

Formula:

THIRD A- r)

Three Point (m=2) -3 -1 l-4

FOR

(k=l)

4

* j

ANALYSIS

-1 1 3

j

!!!Error k! l/3 -l/b l/3

h3f

10 2

(3)

3

A0

A; -6 9

-:

-9 -18

-:

;

-10

4 (4)

-:::2h

z;

*

2 6

Five Point (~11=4) -50

-;t -1612

-72 36 -36 0

-32

72

-6

-: b

32 -12 16

-6

l/5 -l/20 1/30 -l/20 l/5

-:

560

600

-46 -2:

-130 -60 30 -40 150

-600

240 -40 -120 120 -400

SECOIVD *

j

.-I()

400 -120 120 -2:: 600

DERIVATIVE 112

AI

-150 40 -30 b0 130 -bOO

24 -b

-85

Three Point (m=2) 0

1

-2

1

1

-2

1

2

1

-2

1

t

-i

-15-b 1:

123

-3

-2

11/24 -1124 -1/24h 11/24

i b

-40

1 h3f(3) h4f

: 1 1

-36

-590 -170 50 -1:: -490

490 110 -70 -50 170 590

DERIVATIVE A-

-1 I

(4)

-4 -4 -4

-4:s

b 6

-4 -4

6

r;

i

-4 -4

h3ft3)

2

160

-205 -35 35

35

2

-5/lb -l/48 l/48 -l/46 l/48 5/16

A

EError

2 5

-12: -355

h6f~6)

(L4) -44

*

-l/12 -l/24 -l/144 l/24 l/12

:

-4 :

1

h5f(5) h’f(‘) ,5*(S)

Six Point (t/1=5)

Four Point (m=3) 0 1 2 3

48

7/24 l/24 -l/24 h5fc5) l/24 7/24

-2 -6

Five Point (7)~=4)

k!

1/2

-28

FOURTH

h” Error

-l/24

:; -2

2;:

l/6

-l/2

1

h4f(4)

6 (6) *

j

A;

-4* -24 240

-3: -2 -35

(k=2)

A:{

2 -1;

355 125

-1:

-l/b lj30 -l/60 l/60h -1130

227:

-l/4 -l/12 l/12 l/4

:

*

h” Error iz

Six Point (1~5) h5fc5)

Six Point (!)1=5) -274 -24

A;

-44

:

-3 -3

3 3

1:

(k=3)

Five Point (m=4)

l/12

1:

1:

;

-l/4 -11 -2

Al & -4i Four Point (m=3)

r:

Four Point @=3)

DERIVATIVE

15 10 5

4 c4) *

-; -10

-70 -45 -20

130 80 -2

5 30 55

-70 -120

-120 -70 -20 30 80 130

55 30

-10 -5

-2: -45 -70

; 10 15

17/144 5/144 -l/144 -l/144 5/144 17/144

-I,

$ Error

h6 *(b)

Five Point (ttf-4) 0 1 2 3 4

11

-104 -20

114

-1 11 -1

lb -5:

-3: 11:

-it -104

225 50 -5

-770 -75 80 -5 -30 305

1070 -20 -150 80 70 -780

-780 70 80 -150 -20 1070

::

-56 4

-5/12 ,5,(s) l/24 1/180h6f’b) 5/12 h5f (5) -l/24

1: 11 35

j

.I0

FIFTH

DERIVATIVE

-41

A-

:I:{

(k -44

5)

*

Six Point (N -=5) 10 2 3 4 5

-50

305 -30 -5

-50 2

l/180 -5

-87; -770

137/360 -13/360

2%

-1 h6f

(6)

l/180 -13/360 137/360

Compiled from W. G. Bickley, Formulae for numerical differentiation, *See page II.

1: -1 -1 -1

: 5

-10 -10

10 too

1: -5

5 :

-10 -10

10 10

1: -5

: 1 1 1 1

-i/48 -l/80 -l/240 l/24(! l/60 l/48

Math. Gaz. 25, 19-27, 1941 (with permission).

h6f(6)

NUMERICAL

LAGRANGIAN

915

ANALYSIS

INTEGRATION

COEFFICIENTS

Jzy+‘f(,r)dr = /b 2 Ak(rn)f@k)

Table 25.3

*

k

DAD4 n

m\k

4

n = odd 0

1

5

8

-1

251 -19

646 346

-264 456

106 -74

-19 11

19087 -863 271

65112 25128 -2760

-46461 46989 30819

37504 -16256 37504

-20211 7299 -6771

6312 -2088 1608

-863 271 -191

4467094 1:;;;;; 36394

-4604594 3244786 1638286 -216014

5595358 -1752542 2631838 1909858

-5033120 -833120 1317280 2224480

3146338 -755042 397858 425762

-1291214 -142094 294286 126286

312874 -68906 31594 -25706

-33953 -3233 7297 2497

2

1

0

-3

-4

-2

-3

-1

3 -1

5 1:

7 -3 1,'

9 -4 1; -1

1070017 -3;;;; -3233

4

3

2

3

4

D

0

-1

i

-2

12 720

60480

2' 1 0

3628800

k\m

n = even n m\k

-4

-3

-2

-1

0

1

2

-':

19 13

-5 13

-:

475 -27 11

1427 637 -93

-798 1022 802

482 -258 802

-173

4 -1 0 6 -2 -1 0

41

8 -3

36799

139849

-121797

123133

-88547

41499

1; 0

-1375 351 -191

-4183 47799 1879

101349 57627 -9531

-44797 81693 68323

-20227 26883 68323

-11547 7227 -9531

1.O -4 -3 1; 0

3

4

5

-2 11 -11351 -1719 2999 1879

-342136 3133688 162680

3609968 5597072 -641776

-2166334 4763582 -1166146 1295810 4134338 4134338

-617584 462320 -141304 206072 -641776 162680

24

2

1440

i 1375

;

-351 191 -191

ii

57281 2082753 9449717 -11271304 16002320 -17283646 13510082 -7394032 2687864 -583435 -57281 2655563 6872072 -4397584 3973310 -2848834 1481072 -520312 110219 -1;;;; -3969 -163531 10625 50315 2497 -28939

D ii

-42187 27467 -28939

-2497 2497

4 ; 1 0

5 4 3 2 1 0 -1 -2 -3 -4 k\m Compiledfrom National Bureau of Standards,Tablesof Lagrangianinterpolation coefficients. Columbia Univ. Press,New York, N.Y., 1944(with permission).

*SeepageII.

120960

7257600

916

NUMERICAL Table 25.4

Abscissas=iri *xi

ABSCISSAS

AND

WEIGHT

ANALYSIS

FACTORS

02691 a9626

0.00000 00000 00000 0.77459

Weight Factors=w;

10435 84856 63115 94053

0.00000 00000 00000 0.53846 0.90617

93101 98459

05683 38664

0.23861 0.66120 0.93246

91860 93864 95142

83197 66265 03152

1.00000

00000

00000

0.88'388 0.55555

88888 55555

88889 55556

0.65214 0.34785

51548 48451

0.56888 0.47862 0.23692

88888 88889 86704 99366 68850 56189

0.46791 0.36076 0.17132

39345 15730 44923

n= 5

n=6

0.00000

0.40584 0.74153 0.94910

00000

00000

m-7 ‘” ~'

51513 77397 11855 99394 79123 42759

0.41795 0.38183 0.27970 0.12948

0.18343 0.52553 0.79666 0.96028

46424 95650 24099 16329 64774 13627 98564 97536

0.00000

00000

00000

0.32425 0.61337 0.83603 0.96816

34234 14327 11073 b2395

03809 00590 26636 07626

014887 0:43339 0.67940 0.86506 0.97390

43389 81631 53941 29247 95682 99024 33666 88985 65285 17172

0.12523 0.36783 0.58731 0.76990 0.90411 0.98156

34085 14989 79542 26741 72563 06342

72691 48139 79170

--i%: o:a3911 0.91223 0.96397 0.99312

65211 58511 60887 70019 36807 19064 69718 44282 19272 a5991

33497 41645 15419 50827 26515 60150 22218 51325 77913 85094

333755 078080 560673 098004 025453 792614 823395 905868 791268 924786

0.06405 0.19111 031504 0.43379 0.54542 0.64809 0.74012 0.82000 0.88641 0.93827 0.97472 0.99518

68928 80674 26796 35076 14713 36519 41915 19859 55270 45520 a5559 72199

62605 73616 96163 26045 88839 36975 70554 73902 04401 02732 71309 97021

626085 309159 374387 138487 535658 569252 364244 921954 034213 758524 498198 360180

n=10

11469 98180 86617 94305 70475 46719

A= 16 440185 913230 386342 748447 033895 743880 576078 932596

0.36268 0.31370 0.22238 0.10122

37833 78362 66458 77887 10344 53374 85362 90376

0.33023 0.31234 0.26061 0.18064 0.08127

93550 70770 06964 81606 43883

0 29552 0:26926 0.21908 0.14945 0.06667

42247 14753 67193 09996 63625 15982 13491 50581 13443 08688

0.24914 0.23349 0.20316 0.16007 0.10693 0.04717

70458 25365 74267 a3285 93259 53363

01260 40003 02935 94857 61574

a=12

91836 73469 00505 05119 53914 89277 49661 68870

37637 79258 57227 02643 55003 87831 73232 91649

8

n=g

62546 37454

*xi 25098 35507 67776 62444 44083 12023 50230 09349

,Q+O9501 A28160 Q45801 CA1787 o&75540 La6563 0.94457 0.98940

u)i IL =

n=3

66692 41483

INTEGRATION

*xi

IL=4

0.33998 0.86113

GAUSSIAN

(Zeros of Legendre Polynomials) Wi n==Z

0.57735

FOR

13403 38355 23066 43346 95318 86512

wi 0.18945 0.18260 0.16915 0.14959 0.12462 0.09515 0.06225 0.02715

06104 ;5068 34150 44923 65193 95002 59888 16576 89712 55533 85116 a2492 35239 38647 24594 11754

496285 588867 538189 732081 872052 784810 892863 094852

IL=20 0.07652 0.22778 0.37370 0.51086

0.15275 33871 0.14917 29864 0.14209 61093 0.13168 86384 0.11819 45319 0.10193 01198 0.08327 67415 0.06267 20483-_ _~._~ ~. 0.04060 14298 0.01761 40071

30725 72603 18382 49176 61518 17240 76704 34109 _ .~.. 00386 39152

850698 746788 051329 626898 417312 435037 748725 063570 ______ 941331 118312

0.12793 0.12583 0.12167 0.11550 xl0744 0.09761 0.08619 0.07334 0.05929 0.04427 0.02853 0.01234

46752 46828 27803 53725 15965 04113 31953 11080 15436 17419 28933 99987

156974 296121 91204 6 1353 63 "s, 783 888270 275917 305734 780746 806169 663181 199547

n=24 81953 74563 04729 56680 42701 86521 01615 64814 85849 74388 13886 12297

Compiled from P. Davis and P. Rabinowitz, Abscissas and weights for Gaussian quadratures of high order, J. Research NBS 56, 35-37, 1956, RP2645; P. Davis and P. Rabinowitz, Additional abscissas and weights for Gaussian quadratures of high order. Values for ~~=64, 80, and 96, J. Research NBS 60, 613-614,1958, RP2875; and A. N. Iowan, N. Davids, and A. Levenson, Table of the zeros of the Legendre polynomials of order 1-16 and the weight coefficients for Gauss’ mechanical quadrature formula, Bull. Amer. Math. Sot. 48,739-743, 1942 (with permission).

NUMERICAL ABSCISSAS

AND

WEIGHT

FACTORS

FOR

GAUSSIAN

Abscissas= fzi (Zerosof LegendrePolynomials) n=32 0.048307665687738316235 0.144471961582796493485 0.239287362252137074545 0.331868602282127649780 0.421351276130635 345364 0.506899908932229390024 0.587715757240762329041 0.663044266930215200975 0.732182118740289680387 0.794483795967942406963 0.849367613732569970134 0.896321155766052123965 0.934906075937739689171 0.964762255587506430774 0.985611511545268335400 0.997263861849481563545

917

ANALYSIS Table 25.4 INTEGRATION

Weight Factors=wi Wi

0.096540088514727800567 0.095638720079274859419 0.093844399080804565639 0.091173878695763884713 0.087652093004403811143 0.083311924226946755222 0.078193895787070306472 0.072345794108848506225 0.065822222776361846838 0.058684093478535547145 0.050998059262376176196 0.042835898022226680657 0.034273862913021433103 0.025392065309262059456 0.016274394730905670605 0.007018610009470096600

n=40 0.038772417506050821933 0.116084070675255208483 0.192697580701371099716 0.268152185007253681141 0.341994090825758473007 0.413779204371605001525 0.483075801686178712909 0.549467125095128202076 0.612553889667980237953 0.671956684614179548379 0.727318255189927103281 0.778305651426519387695 0.824612230833311663196 0.865959503212259503821 0.902098806968874296728 0.932812808278676533361 0.957916819213791655805 0.977259949983774262663 0.990726238699457 006453 0.998237709710559 200350 0.032380170962869362033 0.0970046992 09462698930 0.161222356068891718056 0.224763790394689 061225 0.287362487355455576736 0.348755886292160 738160 0.408686481990716729916 0.4669029047 50958404545 01523160974722233033678 0.577224726083972 703818 0.628867396776513623995 0.677872379632663905212 0.724034130923814654674 0.767159032515740 339254 0.807066204029442 627083 0.843588261624393530711 0.876572020274247885906 0.905879136715569672822 0.931386690706554333114 0.952987703160430860723 0.970591592546247250461 0.984124583722826857745 0.993530172266350757548 0.998771007252426118601

0.077505947978424811264 0.077039818164247965588 0.076110361900626242372 0.074723169057968264200 0.072886582395804059061 0.070611647391286779695 0.067912045815233903826 0.064804013456601038075 0.061306242492928939167 0.057439769099391551367 0.053227846983936824355 0.048695807635072232061 0.043870908185673271992 0.038782167974472017640 0.033460195282547847393 0.027937006980023401098 0.022245849194166957262 0;016421058381907888713 0.010498284531152813615 0.004521277098533191258 nF48

0.064737696812683922503 0.064466164435950082207 0.063924238584648186624 0.0631141922 86254025657 0.062039423159892663904 0.060704439165893880053 0.059114839698395635746 0.057277292100403215705 0.055199503699984162868 0.052890189485193667096 0.0503590355 53854474958 0.047616658492490474826 0.044674560856694280419 0.041545082943464749214 0.038241351065830706317 0.0347772225 64770438893 0.0311672278 32798088902 0.027426509708356948200 0.023570760839324379141 0.019616160457355527814 0.015579315722943848728 0.011477234579234539490 0.0073275539 01276262102 0.003153346052305838633

918

NUMERICAL Tttble 25.4 ABSCISSAS

AND

Abscissas=*q

WEIGHT

FACTORS

ANALYSIS FOR

GAUSSIAN

(Zeros of Legendre Polynomials) *xi

Weight Factors=wi 21 )i

n=64

0.02435 0.07299 0.12146 0.16964 0.21742 0.26468 0.31132 0.35722 0.40227 0.44636 0.48940 0.53127 0.57189 0.61115 0.64896 0.68523 0;719ai 0.75281 0.78397 0.81326 0.84062 0.86599 0.88931 0.91052 0.92956 0.94641 0.96100 0.97332 0.98333 0.99101 0.99634 0.99930

02926 31217 28192 44204 36437 71622 28719 01583 01579 60172 31457 94640 56462 53551 54712 63130 18501 99072 23589 53151 92962 93981 54459 21370 91721 13748 87996 68277 62538 33714 01167 50417

63424 a7799 96120 23992 40007 08767 90210 37668 63991 53464 07052 19894 02634 72393 54657 54233 71610 60531 43341 22797 52580 54092 95114 78502 31939 58402 52053 89910 a4625 76744 71955

432509 039450 554470 818037 084150 416374 956158 115950 603696 087985 957479 545658 034284 250249 339858 242564 826849 896612 407610 559742 362752 819761 105853 805756 575821 816062 718919 963742 95693I 320739 279347

0.01951 0.05850 0.09740 0.13616 0.17471 0.21299 0.25095 0.28852 0.32566 0.36230 0.39839 0.43387 0.46869 0.50280 0.53614 0.56867 0.60033 0.63107 0.66085 0.68963 0.71736 0.74400 0.76950 0.79383 0.81695 0.83883 0.85943 0.87872 0.89667 0.91326 0.92845 0.94224 0.95459 0.96548 0.97490 0.98284 0.98929 0.99422 0.99764 0.99955

13832 44371 83984 40228 22918

56793 52420 41584 09143 32646 57666 92272 84511 47701 99487 81969 31756 70544 88784 97131 22709 29751 46871 86119 42027 62099 a3597 35041 04605 81463 80255 63111 78213 38770 71757 72445 09872 43634 43799 85727 38629 99755 65688 98237 51630

997654 668629 599063 886559 812559 132572 120493 853109 914614 315619 227024 093062 477036 987594 932020 784725 743155 966248 801736 600771 880254 272317 373866 449949 470371 275617 096977 828704 683194 654165 795953 674752 905493 251452 793386 070418 531027 277892 688900 629880

INTEGRATION

0.04869 0.04857 0.04834 0.04799 0.04754 0.04696 0.04628 0.04549 0.04459 0.04358 0.04247 0.04126 0.03995 0.03855 0.03705 0.03547 0.03380 0.03205 0.03023 0.02833 0.02637 0.02435 0.02227 0.02013 0.01795 0.01572 0.01346 0.01116 0.00884 0.00650 0.00414 0.00178

09570 54674 47622 93885 01657 81828 47965 16279 05581 37245 35151 25632 37411 01531 51285 22132 51618 79283 46570 96726 74697 27025 01738 48231 17157 60304 30478 a1394 67598 44579 70332 32807

09139 41503 34802 96458 14830 16210 81314 27418 63756 29323 23653 42623 32720 78615 40240 56882 37141 54851 72402 14259 15054 68710 08383 53530 75697 76024 96718 60131 26363 68978 60562 21696

720383 426935 957170 307728 308662 017325 417296 144480 563060 453377 589007 528610 341387 629129 046040 383811 609392 553585 478868 483228 658672 873338 254159 209372 343085 719322 642598 128819 947723 362856 467635 432947

0.03901 0.03895 0.03883 0.03866 0.03842 0.03812 0.03777 0.03736 0.03689 0.03637 0.03579 0.03516 0.03447 0.03373 0.03294 0.03210 0.03121 0.03027 0.02928 0.02825 0.02718 0;02607 0.02492 0.02373 0.02250 0.02124 0.01995 0.01862 0.01727 0.01589 0;01449 0.01306 0.01162 0.01016 0.00868 0.00719 0.00569

78136 83959 96510 17597 49930 97113 63643 54902 77146 37499 43939 05290 31204 32149 19393 04986 01741 23217 83695 98160 82275 52357 25357 18828 50902 40261 06108 68142 46520 61835 35080 87615 41141 17660 39452 29047 09224

EE 0:00114

% 49500

56306 62769 59051 74076 06959 14477 62001 38730 38276 05835 53416 44747 51753 84611 97645 73487 88114 59557 83267 57276 00486 67565 64115 65930 46332 15782 78141 08299 56269 83725 40509 92401 20797 41103 69260 68117 51403 24694 89512 03186

654811 531199 968932 463327 423185 638344 397490 490027 008839 978044 054603 593496 928794 522817 401383 773148 701642 980661 847693 862397 380674 117903. 491105 101293 461926 006389 998929 031429 306359 688045 076117 339294 826916 064521 858426 312753 198649 895237 681669 941534

n=80

2%; 80548 43707 47534 34058 53708 66151 41118 59208 12681 06228 57730 98989 76443 51853 02975 24201 27175 41386 14735 14066 25676 55794 31025 98771 27613 07663 50890 91405 85727 13024 75409 98643 38226

NUMERICAL ABSCISSAS

AND WEIGHT

919

ANALYSIS

FACTORS

FOR GAUSSIAN

Table 25.4 INTEGRATION

S_:lf(x)dx= 2 wif(xi) i=l

Abscissas= &xi (Zeros of Legendre Polynomials) *Xi

Weight Factors=wi Wi

n=96

0.01627 0.04881 0.08129 0.11369 0.14597 0.17809

67448 29851 74954 58501 37146 68823

49602 36049 64425 10665 54896 67618

969579 731112 558994 920911 941989 602759

0.03255 0.03251 0.03244 0.03234 0.03220 0.03203

06144 61187 71637 38225 62047 44562

92363 13868 14064 68575 94030 31992

166242 835987 269364 928429 250669 663218

0.21003 0.24174 0.27319 0.30436 0.33520 0.36569

13104 31561 88125 49443 85228 68614

60567 63840 91049 54496 92625 72313

203603 012328 141487 353024 422616 635031

0.03182 0.03158 0.03131 0.03101 0.03067 0.03029

87588 93307 64255 03325 13761 99154

94411 70727 96861 86313 23669 20827

006535 168558 355813 837423 149014 593794

0.39579 Oii2547 0.45470 0.48345 0.51169 0.53938

76498 89884 94221 79739 41771 81083

28908 07300 67743 20596 54667 24357

603285 545365 008636 359768 673586 436227

0.02989 0.02946 0.02899 0.02849 0.02797 0.02741

63441 10899 46141 74110 00076 29627

36328 58167 50555 65085 16848 26029

385984 905970 236543 385646 334440 242823

0.56651 -0.59303 I).61892 0.64416 0.66871 0.69256

04185 23647 58401 34037 83100 45366

61397 77572 25468 84967 43916 42171

168404 080684 570386 106798 153953 561344

0.02682 0.02621 0.02557 0.02490 0.02420 0.02348

68667 23407 00360 06332 48417 33990

25591 35672 05349 22483 92364 85926

762198 413913 361499 610288 691282 219842

0.71567 0773803 0.75960 0178036 0.80030 0.81940

48967 44400 76647 67433 39140 37931

626225 132851 498703 217604 817229 675539

0.02273 0.02196 0.02117 0.02035 0.01951 0.01866

70696 66444 29398 67971 90811 06796

58329 38744 92191 54333 40145 27411

374001 349195 298988 324595 022410 467385

0.83762 0.85495 0.87138 0.88689 0.90146 0.91507

68123 06437 23411 90438 87441 03107 A-. 35112 90334 85059 45174 06353 14231

28187 121494 34601 455463 09296 502874 02420 416057 15852 341319 2089e74206

0.01778 0.01688 0.01597 0.01503 0.01409 0.01312

16045 64245 02562 26994 72314 66961

260838 172450 291381 938006 860916 572637

0.92771 0.93937 0.95003 0.95968 0196832 0.97593

24567 03397 27177 82914 68284 91745

22308 52755 84437 48742 63264 85136

690965 216932 635756 539300 212174 466453

0.01215 0.01116 0.01016 0.00914 0.00812 0.00709

25023 54798 05629 87210 09417 82295 -.. 16046 21020 07705 86712 68769 64707

71088 99838 35008 30783 25698 91153

319635 498591 415758 386633 759217 865269

0.98251 0.98805 0.99254 0.99598 0.99836 0.99968

72635 41263 39003 18429 43758 95038

63014 29623 23762 87209 63181 83230

677447 799481 624572 290650 677724 766828

0.00605 0.00501 0.00396 0.00291 0.00185 0.00079

85455 42027 45543 07318 39607 67920

04235 42927 38444 17934 88946 65552

961683 517693 686674 946408 921732 012429

920

NUMERICAL

Table 25.5

ABSCISSAS

FOR

EQUAL

ANALYSIS

WEIGHT

CHEBYSHEV

INTEGRATION

Abscissas= *.~i f.Ci

,I

2

0.57735 02692

3

0.70710 67812 0.00000 00000

4

0.79465 0.18759 Compiled from H. Phys. 26,191-194, Table

fR’i

5

0.83249 74870 0.37454 14096 0.00000 00000

J_:l.fw

3

*xi

1

0.88386 17008 0.52965 61153 0.32391 18105 0.00000 00000

0.91158 93077 0.60101 86554 0.86624 68181 0.52876 17831 6 44723 0.42251 86538 0.16790 61842 24141 0.26663 54015 0.00000 00000 E. Salzer, Tables for facilitating the use of Chebyshev’s quadrature formula, J. Math. 1947 (with permission). WEIGHT

FACTORS

FOR n--l

I < I E w,f( -l)+;ga

Abscissas= * .~‘i * .r’i Uli

1,

n

9

ABSCISSAS AND

25.6

,I

LOBATTO

w;f(.ri) + G!(l) I, 7

1.00000 000 0.00000 000

0.33333 333 1.33333 333

4

1.00000 000 0.44721 360

0.16666 667 0.83333 333

5

1.00000 000 0.65465 361 0.00000 000

0.10000 000 0.54444 444 0.71111 111

INTEGRATION

Weight Factors=wi * .ri I1.i 1.00000 000 0.04761 904 0.83022 390 0.27682 604 0.46884 a79 0.43174 538 0.00000 000 0.48761 904

a

1.00000 0.87174 0.59170 0.20929

000 015 018 922

0.03571 0.21070 0.34112 0.41245

428 422 270 880

9

1.00000 0.89975 0.67718 0.36311 0.00000

00000 79954 62195

0.02177 0.16549 0.27453 0.34642 0.37151

77778 53616 a7126 a5110 92744

74638 00000

10

6

1.00000 0.76505 0.28523 Compiled from permission).

Table

25.7

1.00000 00000 0.02222 22222 0.91953 39082 0.13330 59908 000 0.06666 667 0.73877 38651 0.22488 93420 0.47792 49498 0.29204 26836 532 0.37847 496 152 0.55485 a38 0.16521 89577 0.32753 97612 Z. Kopal, Numerical analysis, John Wiley & Sons, Inc., New York, N.Y., 1955 (with

ABSCISSAS FOR

AND WEIGHT FACTORS FOR GAUSSIAN INTEGRATION INTEGRANDS WITH A LOGARITHMIC SINGULARITY

Jo’ f(,v) 1n.r (/.r =ig, wif(.ri) +‘gy Abs&sas=.ri *

K~I

Weight Factors- ,,I;

?, .)'I -W K,, 4 0.041448 0.383464 0.00001 0.245275 0.386875 0.556165 0.190435 0.848982 0.039225 Compiled from Berthod-Zaborowski, Le calcul des integrales de la forme $ j(r) log ,” rl.l,. H. Mineur, Techniques de calcul numerique, pp. 555-556. Librairie Polytechnique Ch. B&anger, Paris, France, 1952 (with permission).

71 .ri O;rlws;39 Kn 2 0.112009 0.00285 0.602277 0.281461

*See page 11.

11 .r; -W ICI, 3 0.063891 0.513405 0.00017 0.368997 0.391980 0.766880 0.094615

*

NUMERICAL

ABSCISSAS

AND WEIGHT

FACTORS

FOR GAUSSIAN

INTEGRATION

OF MOMENTS

Table

25.8

Weight Factors=wi

Abscissas=z; k=O

921

ANALYSIS

k=l

k=2

n

1

o.5000criooooo

l.ooooowiooooo

o.66666zi66667

2

0.21132 48654 0.78867 51346

0.50000 00000 0.50000 00000

3

0.11270 16654 0.50000 00000 0.88729 83346

4

0.06943 0.33000 0.66999 0.93056

5

o.750002iooooo

0.33333%33

0.35505 10257 0.18195 86183 0.84494 89743 0.31804 13817

0.45584 81560 0.87748 51773

0.10078 58821 0.23254 74513

0.27777 77778 0.44444 44444 0.27777 77778

0.21234 05382 0.59053 31356 0.91141 20405

0.29499 77901 0.65299 62340 0.92700 59759

0.02995 07030 0.14624 62693 0.15713 63611

18442 94782 05218 81558

0.17392 0.32607 0.32607 0.17392

74226 25774 25774 74226

0.13975 0.41640 0.72315 0.94289

98643 0.03118 09710 95676 0.12984 75476 69864 0.20346 45680 58039 0.13550 69134

0.20414 0.48295 0.76139 0.95149

85821 27049 92624 94506

0.01035 0.06863 0.14345 0.11088

22408 38872 87898 84156

0.04691 0.23076 0.50000 0.76923 0.95308

00770 53449 00000 46551 99230

0.11846 0.23931 0.28444 0.23931 0.11846

34425 43352 44444 43352 34425

0.09853 0.30453 0.56202 0.80198 0.96019

50858 57266 51898 65821 01429

79145 88701 69871 46381 15902

0.14894 0.36566 0.61011 0.82651 0.96542

57871 65274 36129 96792 10601

0.00411 0.03205 0.08920 0.12619 0.08176

38252 56007 01612 89619 47843

6

0.03376 0.16939 0.38069 0.61930 0.83060 0.96623

52429 53068 04070 95930 46932 47571

0.08566 0.18038 0.23395 0.23395 0.18038 0.08566

22462 07865 69673 69673 07865 22462

0.07305 0.23076 0.44132 0.66301 0.85192 0.97068

43287 0.00873 83018 61380 0.04395 51656 84812 0.09866 11509 53097 0.14079 25538 14003 0.13554 24972 35128 0.07231 03307

0.11319 0.28431 0.49096 0.69756 0.86843 0.97409

43838 88727 35868 30820 60583 54449

0.00183 0.01572 0.05128 0.09457 0.10737 0.06253

10758 02972 95711 71867 64997 87027

7

0.02544 0.12923 OI29707 0.50000 0.70292 0.87076 0.97455

60438 0.06474 24831 44072 0.13985 26957 74243 0.19091 50253 00000 0.20897 95918 25757 0.19091 50253 55928 0.13985 26957 39562 0.06474 24831

0.05626 0.18024 0135262 0.54715 0.73421 0.88532 0.97752

25605 0.00521 43622 06917 0.02740 83567 47171 OIO6638 46965 36263 0.10712 50657 01772 0.12739 08973 09468 0.11050 92582 06136 0.05596 73634

0.08881 0.22648 0.39997 0.58599 0.75944 0.89691 0.97986

68334 27534 84867 78554 58740 09709 72262

0.00089 0.00816 0.02942 0.06314 0.09173 0.09069 0.04927

26880 29256 22113 63787 38033 88246 65018

8

0.5000~00000

0.06982 69799 0.22924 11064 0.20093 19137

0.01574 0.07390 0.14638 0.16717 0.09678

0.01985 50718 0.05061 42681 0.04463 39553 0.00329 51914 0.07149 10350 0.00046 85178 0.10166 67613 0.11119 05172 0.14436 62570 0.01784 29027 0.18422 82964 0.00447 45217 0.28682 47571 0.04543 93195 0.33044 77282 0.01724 68638 0.23723 37950 0.15685 33229 0.40828 26788 0.18134 18917 0.45481 33152 0.07919 95995 0.49440 29218 0.04081 44264 0.59171 73212 0.18134 18917 0.62806 78354 0.10604 73594 0.65834 80085 0.06844 71834 0.76276 62050 0.15685 33229 0.78569 15206 0.11250 57995 0.80452 48315 0.08528 47692 0.89833 32387 0.11119 05172 0.90867 63921 0.09111 90236 0.91709 93825 0.07681 80933 0.98014 49282 0.05061 42681 0.98222 00849 0.04455 08044 0.98390 22404 0.03977 89578 Compiled from H. Fishman, Numerical integration constants, Math. Tables Aids Comp. 11, l-9,1957 (with permission).

922 Table

NUMERICAL 25.8

ABSCISSAS

AND

WEIGHT

ANALYSIS

FACTORS

FOR

Abscissas=s

GAUSSIAN

INTEGRATION

OF

MOMENTS

Weight Factors=wi k=4

o.*3333zi33333

k=5 o.20000wiooooo

0.857142i8571~

0.1666766667

0.52985 79359 0.06690 52498 0.89871 34927 0.18309 47502

0.58633 65823 0.04908 24923 0.91366 34177 0.15091 75077

0.63079 15938 0.03833 75627 0.92476 39617 0.12832 91039

0.36326 46302 0.01647 90593 0.69881 12692 0.10459 98976 0.93792 41006 0.12892 10432

0.42011 30593 0.01046 90422 0.73388 93552 0.08027 66735 0.94599 75855 0.10925 42844

0.46798 32355 0.00729 70036 0.76162 39697 0.06459 66123 0.95221 09767 0.09477 30507

0.26147 77888 0.53584 64461 0.79028 32300 0.95784 70806

0.00465 83671 0.04254 17241 0.10900 43689 0.09379 55399

0.31213 54928 0.57891 56596 0.81289 15166 0.96272 39976

0.00251 63516 0.02916 93822 0.08706 77121 0.08124 65541

0.35689 37290 0.61466 93899 0.83107 90039 0.96658 86465

0.00153 44797 0.02142 84046 0.07205 63642 0.07164 74181

5

0.19621 20074 0.41710 02118 0.64857 00042 0.84560 51500 0.96943 57035

0.00152 06894 0.01695 73249 0.06044 49532 0.10031 65045 0107076 05281

0.23979 20448 0.46093 36745 0.68005 92327 0.86088 63437 0.97261 44185

0.00069 69771 0.01021 05417 0.04402 44695 0.08271 27131 0.06235 52986

0.27969 31248 0.49870 98270 0.70633 38189 0.87340 27279 0.97519 38347

0.00036 97155 0.00672 96904 0.03376 77450 0.07007 13397 0.05572 81761

6

0.15227 31618 0:33130 04570 0.53241 15667 0.72560 27783 0.88161 66844 0.97679 53517

0.00056 17109 0:00708 53159 0.03052 61922 0.06844 32818 0.08830 09912 0.05508 25080

0.18946 95839 0.37275 11560 0.56757 23729 0.74883 64975 0.89238 51584 0.97898 52313

0.00021 94140 0.00372 67844 0.01995 62647 0.05223 99543 0.07464 91503 0.04920 84323

0.22446 89954 0.40953 33505 0.59778 90484 0.76841 36046 0.90135 07338 0.98079 72084

0.00010 13258 0.00218 79257 0.01396 96531 0.04148 63470 0.06445 88592 0.04446 25560

7

0.12142 71288 0.26836 34403 0.44086 64606 0.61860 40284 0.78025 35520 0.90636 25341 0.98176 99145

0.00022 99041 0.00314 75964 0.01531 21671 0.04099 51686 0.06975 00981 0.07655 65614 0.04400 85043

0.15324 14389 0.30632 65225 0.47654 00930 0.64638 93025 0.79771 66898 0.91421 99006 0.98334 38305

0.00007 70737 0.00144 70088 0.00892 69676 0.02854 78428 0.05522 48742 0.06602 18459 0.03975 43870

0.18382 87683 0.34080 75951 0.50794 05240 0.67036 34101 0.81258 84660 0.92085 64173 0.98466 74508

0.00003 11046 0:00075 53838 0.00566 04137 0.02095 92982 0.04510 49816 0.05790 76135 0.03624 78712

8

0.09900 17577 0.22124 35074 0.36912 39000 0.52854 54312 0.68399 32484 0.82028 39497 0.92409 37129 0.98529 34401

0.00010 24601 0.00148 56841 0.00785 50738 0.02363 15807 0.04745 43798 0.06736 18394 0.06618 20353 0.03592 69468

0.12637 29744 0.25552 90521 0.40364 12989 0.55831 66758 0.70600 95429 0.83367 15420 0.92999 57161 0.98646 31979

0.00002 97092 0.00059 89500 0.00407 79241 0.01490 99334 0.03471 99507 0.05491 00973 0.05800 05653 0.03275 28699

0.15315 06616 0.28726 44039 0.43462 74067 0.58451 85666 0.72512 64097 0.84518 94879 0.93504 35075 0.98746 05085

0.00001 05316 0.00027 83586 0.00233 53415 0.01004 46144 0.02648 53011 0.04588 56532 0.05153 42238 0.03009 26424

NUMERICAL

ABSCISSAS

AND WEIGHT

FACTORS

923

ANALYSIS

FOR LAGUERRE

s,$-zf(~)~-i~, wf(xJ

INTEGRATION

Jomg(x)d-i;,w+g(xi)

Abscissas=xi (Zeros of Laguerre Polynomials) xi

Weight Factors =wi

wie’i

wi

Xi

1.53332 603312 4.45095 733505

0.15232 0.80722 2.00513 3;78347 6.20495 9.37298 13.46623 18.83359 26.37407

22277 00227 51556 39733 67778 52516 69110 77889 18909

32 42 19 31 77 88 92 92 27

0.13779 0.72945 1.80834 3.40143 5.55249 8.33015 11.84378 16.27925 40422 08 21.99658 29.92069 787360 324237 690092 635435

34705 45495 29017 36978 61400 27467 58379 78313 58119 70122

40 03 40 55 64 64 00 78 81 74

0.11572 0.61175 0.57353 55074 23 1.51261 1.36925 259071 2.83375 2.26068 459338 4.59922 3.35052 458236 6.84452 4.88682 680021 9.62131 7.84901 594560 13.00605 17.11685 22.15109 28.48796 37.09912

21173 74845 02697 13377 76394 54531 68424 49933 51874 03793 72509 10444

58 15 76 44 18 15 57 06 62 97 84 67

78120 17403 54120 95262 27217 66274

17 02 71 04 51 14

62266

13

n=3 0.41577 45567 83 2.29428 03602 79 6.28994 50829 37

-2)1.03892

565016

n=4 0.32254 1.74576 4.53662 9.39507

76896 11011 02969 09123

0.83273 2.04810 3.63114 6.48714

19 58 21 01

n=5 0.26356 1.41340 3.59642 7.08581 12.64080

03197 30591 57710 00058 08442

18 07 41 59 76

3.98666 811083 3.61175 867992 2.33699 723858

n=6 0.22284 1.18893 2.99273 5.77514 9.83746 15.98287

66041 21016 63260 35691 74183 39806

79 73 59 05 83 02

0.19304 1.02666 2.56787 4.90035 8.18215 12.73418 19.39572

36765 48953 67449 30845 34445 02917 78622

60 39 51 26 63 98 63

0.17027 0.90370 2.25108 4.26670 7.04590 10.75851 15.74067 22.86313

96323 17767 66298 01702 54023 60101 86412 17368

05 99 66 88 93 81 78 89

1.07769 285927 2.76214 296190 5.60109 462543

n=7

0.67909 1.63848 2.76944 4.31565 7.21918

0.49647 1.17764 1.91824 2.77184 3.84124 5.38067 8.40543

n=8 -1 3.69188 -1 4.18786 -1 1.75794 -2 3.33434 -3 2;79453 -5 9.07650 -7 8.48574 -9 I 1.04800

589342 780814 986637 922612 623523 877336 671627 117487

91238 38 243845 630582 508441

75975 40 306086 978166 863623 912249 820792 248683

0.09330 0.49269 1.21559 2.26994 3.66762 5.42533 7.56591 0.43772 34104 93 10.12022 1.03386 934767 13.13028 1.66970 976566 16.65440 2.37692 470176 20.77647 3.20854 091335 25.62389 4.26857 551083 31.40751 5.81808 336867 38.53068 8.90622 621529 48.02608

wiezi

Wi

n=2 0.58578 64376 27 3.41421 35623 73

Table 2.59

n=9

- 1 3.36126 421798 - 1 4.11213 980424 - 1 1.99287 525371 - 4I 3.05249 - 6 6.59212 - 8 4.11076 -11 3.29087

767093 302608 933035 403035

n=lO

85680 24821 77083 88994 42267 91697 33064 55726

19 76 30 49 29 54 86 86

-

1 3.08441115765 1 14.01119 929155 1 2.18068 287612 2 6.20874 560987

7.53008 - 5 2.82592 - 7 4.24931 - 9 I 1.83956 -13 9.91182

388588 334960 398496 482398 721961

n=12

-

n=15

11 2.18234 1 3.42210 1 12.63027 1 1.26425 2 4.02068 3 8.56387 3 1.21243

885940 177923 577942 818106 649210 780361 614721

0.39143 0.92180 1.48012 2.08677 2.77292 3.59162 4.64876 6.21227 9.36321

11243 16 50285 29 790994 080755 138971 606809 600214 541975 823771

0.35400 0.83190 1.33028 1.86306 2.45025 3.12276 3.93415 4.99241 6.57220 9.78469

97386 07 23010 44 856175 390311 555808 415514 269556 487219 248513 584037

0.29720 0.69646 1.10778 1.53846 1.99832 2.50074 3.06532 3.72328 4.52981 5.59725 7.21299

96360 44 29804 31 139462 423904 760627 576910 151828 911078 402998 846184 546093

0.23957 0.56010 0.88700 1.22366 1.57444 1.94475 2.34150 2.77404 3.25564 3.80631 4.45847 5.27001 6.35956 8.03178 11.52777

81703 11 08427 93 82629 19 440215 872163 197653 205664 192683 334640 171423 775384 778443 346973

Compiled from H. E. Salzer and R. Zucker, Table of the zeros and tieight factors of the first fifteen Laguerre polynomials, Bull. Amer. Math. Sot. 55, 1004-1012, 1949 (with permission).

E8i2

NUMERICAL

924

Table

25.10

ABSCISSAS

AND

WEIGHT

Abscissas =l xi (Zeros of Hermite *Xi Wi

FACTORS

ANALYSIS

FOR

HERMITE

Polynomials) 2 WieZi

INTEGRATION

Weight

Factors=w;

*-xi

0.70710

E%

67811

Ko3

86548

(-1)8.86226

Kii

92545

[- 01 12.95408 1.18163

28

1.46114

11826

611

n=3 59006 97515

04 09

1.32393 1.18163

11752 59006 136 037

0.34290

13272

1.03661 1.75668

n=lO

6.10862 2.40138

23705

08297 89514 36492 99882

2.53273

16742

3.38743 1.34364 b)7.64043

32790

3.43615

91188 37738

0.31424 0.94778

03762 83912

n=4 1”:Et:

22’:: E%

I- 12 18.04914 8.13128

0.00000 00000 00000 0.95857 2.02018

24646 28704

35447 09000 25 55

1.24022 1.05996

58176 44828 958 950

n=5 -1)9.45308 1.99532

0.94530

87204

32315 22

29

0.98658

09967 514

829

42059

05

1.18148

86255

360

44 29 09

0.87640 0.93558

13344 05576

362 312

1.13690

83326 745

63373 53 61108 23

0.68708 0.70329

18539 513 63231 049

94455 57467 28552

48 81 33

0.74144 0.82066 1.02545

19319 16913

657

25

0.62930

78743

695

29 88

0.66266 0.70522 0.78664 0.98969

27732 03661 39394 90470

669 122 633 923

378 675

436

61264 048

n=12

1.59768

72048

3.93619

13819 56086

W&

Wi

n=2

54359 40164

- 1 I 5.70135 - 1 2.60492 -

26351 52605

2 5.16079 3 1 3.90539 5 8.57368 7)2.65855

23626

31026 42 85615 88 05846 70435

0.63962

12320 203

2.27950 3.02063 3.88972

70805 70251 48978

OlObO 20890 69782

0.27348 0.82295 1.38025 1.95178

10461 14491 85391 79909

3815 4466 9888 1625

0.54737 0.55244

52050 19573

0.56321 0.58124

78290 882 72754 009

2.54620 3.17699 3.86944 4.68873

21578 91619 79048 89393

4748 7996

0.60973 0.65575 0.73824 0.93687

69582 56728

560

44928

841

0.24534 0.73747

07083 37285

009 454

1.23407 1.73853

62153 953 77121 lbb

0.49092 0.49384 0.49992 0.50967

15006 33852 08713 90271

667 721 363 175

2.25497 2.78880 3.34785 3.94476

40020

893

45673

832

03509 17423 24428

486 644 525

4.60368

40401 156 24495 507

0.52408 0.54485 0.57526

5.38748

08900

0.89859

19614

532

16843 56

n-6 0.43607 1.33584 2.35060

74119 90740 49736

-1 I 7.24629

27617 13697 74492

-1 -3

59522 32032 99055

1.57067 4.53000

n=7 0.00000 0.81628

00000 78828

00000 58965

1.67355 2.65196

16287 67471 13568 35233

-1 I 8.10264 -1 4.25607 -2 5.45155 -4)9.71781

b1755 68 25261 01 82819 13

0.81026 0.82868 0.89718

24509

95

1.10133

82 49 41 14

0.76454 0.79289

46175 568 73032 836 46002 07296

252 103

fZ=lfi

6012 0582

n=8 0.38118

1.15719 1.98165 2.93063

69902

07322

37124 46780 67566 95843

74202

57244

0.00000 00000 00000

-1) 6.61147 I -1 1 2.07802 -2 1.70779 (-4)1.99604

01255 32581 83007 07221

10187

52838

1.46855

32892 lb668

7.20235 -1 4.32651

2.26658 3.19099

05845 32017

31843 81528

-2 8.84745 -3)4.94362 -5)3.96069

0.72355

00483 26065 01442

1.07193

864 634 480

n=9 1 -1 I

21560 61 55900 26 27394 42755 77263

38 37 26

n=20

41286 517

0.86675

0.72023

52156 061 24527 451

0.73030 0.76460 0.84175 1.04700

81250 27014 35809

946 787 767

- 1 4.62243 - 1 2.86675 - 1 I 1.09017

66960 06 50536 28 20602 00

- 2 I 2.48105 - 3 3.24377 - 4 2.28338 - 6 7.80255 - 7 1.08606

20887 33422

60584 281

-10 4.39934

112

(-13

I 2.22939

64785

46 38 b3 32

36455

34

63601

93707 69 09922 73

0.62227 0.70433

Compiled from H. E. Salzer, R. Zucker, and R. Capuano, Table of the zeros and weight factors of the first twenty Hermite polynomials, J. Research NBS 48,111-116,1952, RP2294 (with permission). Table

25.11

COEFFICIENTS e 0. 00 0. 01

FOR

FILON’S

0.0000~

000

0.00000 0.00000 0.00000 0.00000

004 036 120 284

0.00000

555

0.00000 0.00001

961 524

0.00002 0.00003

274 237

0.00004 0.00035

438 354

0.00118

467

0.00278 0.00536

012 042

Xl

0.00911 0.02076 0.01421

797 156 151

1":;

0.03850 0.02884

683 188

0.02 0.03 0.04 0. 05 0.06 0.07

2 i; :: 0: 3

0.6

See 25.4.47.

761 56222 777

QUADRATURE 0.66666 0.66668 0.66671

B

667 000 999

0.66678

664

0.66687

990

0.66699 0.66714

976 617

0.66731

909

0.66751 0.66774

844 417

0.66799

619

0.67193

927

0.67836

065

FORMULA 7 1.33333 1.33332 1.33328 1.33321 1.33312

333 000 000 334 001

1.33300 1.33285 1.33268 1.33248 1.33225

003 340 012 020 365

1.33200 1.32800 1.32137

048 761 184

0.68703

909

1.31212

154

0.69767

347

1.30029

624

0.70989

111

0.72325 0.73729 0.76525 0.75147

813 136 168

1.28594 1.26913 1.24992

638 302 752

1.22841

118

831

1.20467

472

86961 914 29611 769

26. Probability MARVIN

Functions

ZELEN l AND NORMAN

C. SEVERO 2

Contents Page 927 927 931 936 940 944 946 948

Mathematical Properties .................... 26.1. Probability Functions: Definitions and Properties ..... 26.2. Normal or Gaussian Probability Function ........ 26.3. Bivariate Normal Probability Function ......... 26.4. Chi-Square Probability Function ............ 26.5. Incomplete Beta Function ............... 26.6. F-(Variance-Ratio) Distribution Function ........ 26.7. Student’s t-Distribution ................ Numerical Methods ....................... 26.8. Methods of Generating Random Numbers and Their Applications ........................ 26.9. Use and Extension of the Tables .............

949

References

961

..........................

Table 26.1. Normal Probability Function and Derivatives (0 1255). P(x), Z(x), Z”‘(x), 15D zqx)) 10D; Z(*)(x), n=3(1)6, 8D x=0(.02)3 P(x), 1OD; Z(x), 10s; Z@)(x), n=1(1)6, 8s 2=3(.05)5

.

949 953

966

Table 26.2. Normal Probability Function for Large Arguments (552<500) . . . . . . . . . . . . . . . . . . * . . . . . . -log Q(x), x=5(1)50(10)100(50)500, 5D

972

Table 26.3. Higher Derivatives of the Normal Probability Function (O<s<5) . . . . . . . . . . . . . . . . . . . . . . . . . . Z(“)(x), n=7(1)12, x=0(.1)5, 8S

974

Table 26.4. Normal Probability Function-Values of Z(x) in Terms of P(x) and Q(x) . . . . . . . . . . . . . . . . . . . . . . . . Q(z>=0(.001).5, 5D

975

Table 26.5. Normal Probability Function-Values of x in Terms of P(x) and Q(x) . . . . . . . . . . . . . . . . . . . . . . . . ‘Xx)=0(.001).5, 5D

976

l National Bureau of Standards. z National

Bureau

of Standards.

(Presently, (Presently,

National Institutes of Health.) University of Buffalo.)

925

PROBABILITY

FUNCTIONB

Table 26.6. Normal Probability Function-Values of 2 for Extreme Values of P(z) and Q(z) . . . . . . . . . . . . . . . . . . . . ~(~)=0(.0001).025, 5D m=4(1)23, 5D Q(z) = lo-“, Table 26.7. Probability Function, Cumulative

Integral of x2-Distribution, Incomplete Gamma Sums of the Poisson Distribution . . . . . .

Page

977

978

x”=.001(.001).01(.01).1(.1)2(.2)10(.5)20(1)40(2)76 Y= 1(1)30, 5D Table

26.8. Percentage

TermsofQandu

Points of the x2-Distribution-Values of x2 in . . . . . . . . . . . . . . . . . . . . . . .

984

Q(x2(u)=.995, .99, .975, .95, 2, .75, .5, .25, .l, .05, .025, .Ol, .005, .OOl, .0005, .OOOl v==1(1)30(10)100, 5-6s Table

26.9. Percentage

TermsofQ,

yl, La.

Points of the F-Distribution-Values of F in . . . . . . . . . . . . . . . . . . . . . .

Q(Fjv1, Y&=.5, .25, .l, .05, .025, .Ol, .005, .OOl

986

v1=1(1)6, 8, 12, 15, 20, 30, 60, a V2=1(1)30, 40, 60, 120, Q), 3-5s 26.10. Percentage Points of the t-Distribution-Values of t in TermsofAandv . . . . . . . . . . . . . . . . . . . . . . .

Table

990

A(tlv)=.2,.5,.8,.9,.95,.98,.99,.995,.998,.999,.9999,.99999,.999999 v=1(1)30,

40, 60, 120, OJ, ZD

Table 26.11. 2500 Five Digit

Random

Numbers

. . . . . . . . . .

991

The authors gratefully acknowledge the assistance of David S. Liepman in the prep aration and checking of the tables and graphs and the many helpful comments received from members of the Committee on Mathematical Tables of the Institute of Mathematical Statistics.

26. Probability Mathematical 26.1.

Functions Properties

Probability

Univariate

Cumulative

: Definitions

Distribution

and

Functions

A real-valued function F(Z) is termed variate) cumulative distribution function or simply distribution function if

a (uni(c.d.f.)

i) F(z) is non-decreasing, i.e., F(z1)5F(a) XII% ii) F(s) is everywhere continuous from right, i.e., F(s) =2+ F(z+t) iii) F(-

=)=O,

for the

F(a)=l.

The function F(s) signifies the probability of the event “X
which

need only

X=s,}

be subject

20

26.1.1

FCd=WXSx?

Properties

distribution

function

can

=zqz~. ”

* Commenton notationand conventions. a. We follow the customary convention of denoting a random variable by a capital letter, i.e., X, and using the corresponding lower case letter, i.e., z, for a particular value that the random variable assumes. b. For statistical applications it is often convenient to have tabulated the “upper tail area,” 1 -F(z), or the c.d.f. for 1x1, F(z)-F(-z), instead of simply the c.d.f. F(z). We use the notation P to indicate the c.d.f. of X, Q = 1 -P to indicate the “upper tail area” and A = P-Q to denote the c.d.f. of 1x1. In particular we use P(z), Q(z), and A(z) to denote the corresponding functions for the normal or Gaussian probability function, see 26.2.2-26.2.4. When these distributions depend on other pnrametcrs, say ~91 and 82, we indiwtr this by writing P(rJh, 021, Q(zl01, et), or .4(rl&, 0,). For csamplc the chisquare distribution 26.4 depends on the parameter Y and the tabulated function is n ritten Q(x~~v).

’

where the summation is over all values of z for which ~5s. The set { 2,) of values for which p,>O is termed the domain of the random variable X. A discrete distribution of a random variable is called a lattice distribution if therL exist numbers a and b #O such that every possible value of X can be represented in the form a+ bn where n takes on only integral values. A summary of some properties of certain discrete distributions is presented in 26.1.19-26.1.24. Continuous Distributions. Continuous distributions are characterized by F(x) being absolutely continuous. Hence F(z) possesses a derivative F’(s)=j(z) and th e c.d.f. can be written 26.1.2

The

S

’ f(t)dt. -0D

F(z)=Pr{X
derivative

j(z) is termed the probability (p.d.f.) or frequency junction, and the values of 5 for which J(r)>0 make up the domain of the random variable X. A summary of some properties of certain selected continuous distributions is presented in 26.1.25-26.1.34.

density junction

Multivariate

to the restriction

l$Pa=l.

The corresponding then be written

Functions

The real-valued defines an n-variate tion if

Probability

Functions

function cumulative

F(z,,

q, . . . z,)

distribution

func-

i) F(s, x2, . . . 2,) is a non-decreasing function for each 5; ii) F(s, q, . . . 2,) is continuous from the right in each sl; i.e., F(s, ~2, . . . z,) =lim F(s, . . ., zf+t, . . ., s,) 4+ when x2, . . . x.)=0 F(~D, co, . . ., a~)=l.

iii) F(xl, iv)

any q=--o3;

F(s,, z2, . . ., x,) assigns nonnegative

ability

to

the

x2<X21x2+h2,

event

t,<X,<

*

probqfh,,

. . .,x,<-KA~,+h, for x:,+M---F(s+h,,

all x1, q, . . ., zn and all nonnegative h,, hz, . . ., h,, e.g., for n=2, F(x,+hl, x2>+

sz+M--F(G

F(z,, z2> 20 and in general for s,<X& sr+hr (i=l, 2, . . ., n), the kth order

difference k=l,

AIF(q,

G, . . ., s,)>O

2, . . ., n.

927

for

PROBABILITY

928 The

probability of the event XI
x25x2,

joint *

* ‘,

Characteristics

of diutribution

functione:

ntb moment about 0rWn

26.1.4

meau

vectors (21, 22, . . ., s,) and continuous distributions are characterized by absolute continuity of F(G, x2, . . ., CC,).

Moments,

Contlauous

26.1.2

FUNCTIONS

funclions,

characteristic

distdbutions

=

cumulants

Discrete distributlona

--

26.13 moment

73th central

26.1.6 26.1.7 26.1.8 26.1.9

chartitoristic U(x)

function

26.1.11

InversIon fowluls

01 PI--

Relation

of

26.1.11

the

Characterirtic Ahout the

4’“‘(O)=

[$9(t)]

Cumulant

26.1.12

Function Origin

to

rP8-“+4(f)df

(lsttice dlstrlbutions

CoeiBcientr

Momenta

only)

of Skewneee

26.1.15

and

Exceu

(skewness)

,_o=i”p: 26.1.16

Function

In +(t) -2

b

2r J -r/b

n-0

K$$

Occasionally (or kurtosis)

(excess) coefficients of skewness and excess are given by

K,, is called the 72”’ cumulant. 26.1.17

26.1.13 Relation

26.1.14

K1=?n,

of

Central

Im--8

K~=c?,

K8=p2,

Momenta Origin

K4=p4-3pi

to Moments

(Y) (-l)n-‘p~mn-5

About

the

26.1.18

A=d=($)

(skewness)

82=%+3=$

(excess or kurtosis)

PROBABILITY

F’UNCTIONS

: 3

929

Some one-dimensional Probabilfty Function

Nl3UM

Density f(z)

wntinuous

distribution

co 8

functions

Skewmaw 71

MWJI

CUIIIUlSnt.3

0

1 5i3

D

NOrUN

(II

z

D

0

i&e I c

~*=m. x:=6,

26.1.27

Cauchy

not de5cd

not defined

not defined

not defined

,i.H-#I II

not de5ed

26.1.28

Exponential

a

2

rt=a+8,

26.1.29

Lapbce, ordouble

w

0

L[*=a, r2=2p

26.1.2s

Error

26.1.26

function

exponential

26.1.30

26.131

s”+*-ilJ

; expt-r-e-q

Extreme-Value,’ (Fisher-Tim&t Type I or doubly exponential)

Pearson

Type

with

-(4)’ 6

1.3

2.4

Iy1-i&p*

VP

(1-fl)-p

seefcotnote 6.

M(a,

-1.2

$h (T) sin

..=o

for

K.=pr(n)

‘”

?a>2

for n>l

‘) ~olgr. n for n=1.2.

i!

. . .

y-=T

III %-

26.1.32

Oamma

distribution 5

26.133

Beta

distribution

-

B(a, 1 L

26.136

1 b)

z*‘(l-z)*I

2(a-b)

(a+b+2) 0

xlop,K.=pr(n)

o+b.

fore4

it) x,=nl, n.+1=0 hz=Bt. ““=-iii-

*

B,. (Bernoulli B,=%.

* -t (Euler’s

catstad)~67721

56649

. . . .

nUmbers). B,=-

3fr. . . .

g m P 3 cc

PROBABILJTY

Inequalities (F@) denotes

the C.&f. of the random

variable

Xand

1denotes

R positive

931

FUNCTIONS

for distribution cmstmt;

further

functions m is always

assumed

to be fmite and all expectations

are assumed

to exist.)

Conditions

Inequality

_-

26.1.35

Prig(X) It1 S%?(X)I/t

26.1.36

Pr(

(9 8(x) 20 (i) PrlX
X>t) <m/t

F(1)21-y Pr( IX-rnl>tu)

26.1.37

(i) E(X)= m (ii) E(X- rn)?=G


F(m+to)-F(m-tc)

*

21-i

26.1.38

(i) E(XJ =mi E(Xi-Tlnli)‘=U? (iii) E([Xi-mmil[Xi-mil) (ii)

=O(i#$

26.1.39

(i) E(X-m)‘=2 (ii) F(z) is a continuous c.d.f. (iii) F(z) is unimodal at ~0’

26.1.40

Pr( IX--ml

(i) E(X-m)a=a* F(z) is a continuous c.d.f. (iii) F(z) is unimodal at xoe (iv) m=zo

>tu) <4/9tS

(ii)

PY( IX--ml

26.1.41

>to)

Fb+td-F(m--tul

(i) JC(X-m)a=2 (ii) E(X-rn)‘=p4

< pr+;;f2pd

11-p,+;;;f2p,

620is mlcll that EV(ZO)>F’(I)forZZIO.

26.2. Normal or Gaussian Probability

Function

Pr{X<x}=-L

26.2.1 26.2.2 P(z) =&

26.2.4

&(x)=+~

n.m

J

S

.

S nm

=- e-t1’2dt=

g/2*

--(t-m)’ ’ e 24 dt --

_=(DZ(t)dt

J

The corresponding

2- Z(t)dt

probability

density

function

is

A(x) -1 -JZ;SI.L-t’12dt=SI;Z(t)dt

26.2.5 26.2.6 26.2.7 Probability

S

;. e -“12dt =

.

26.2.3

26.2.8

26.2.9

P(x) + Q(x) = 1 PC-x)=&(4 A(x)=2P(z)-1 Integral

with

Mean

and is symmetric m and

Variance

around m, i.e.

d

A random varirtble X is said to be normally distributed with mean m and variance 2 if the probability that X is less than or equal to z is given by

The infiexion points of the probability function are at m f u. *see page n.

density

PROBABILITY

932 Power

Series

FUNCTIONS Polynomial

(z 20)

and

Rational and

Approximations Z(x)

05x<

7 for

P(x)

-

26.2.16 26.2.11

- 1

P(z)=l--Z(x)(aIt+a2t"+a~t~)+B(x),

t=l+px

le(s)1<1X10-6 Asymptotic

Expansions

p= .33267

(x>O)

26.2.12

al=.43618 36 us=-. 12016 76 a3= .93729 80

26.2.17

Q(x)=?{

I-$+y+.

.. P~x)=l-z(x)(b,t+b2t2+b2t2+blt4+b~t~)+t(Z),

+(-1)“l.

3.. . (2?2--1) +R n x2n 1

1 t=l+pz

where

[e(z) [<7.5x

R,=(-l)“+‘1.3

. . . (2n+l$-

g

0%

10-s

p=.23164 19 .31938 1530 b4=-1.82125 5978 b,=-.35656 3782 bb= 1.33027 4429 ba= 1.78147 7937 6,=

which is less in absolute value than the first neglected term.

26.2.18

26.2.13

p(x)=&;

-(x2+2)

je(x)(<2.5X 1O-4 cl=.196854 c3= .000344 c2=.115194 cd=.019527

(s”a;4) (x2+6) + . * . )

where al=l, a2=l, a3=5, a4=9, aS=129 and the general term is

(1+c~x+c2~+c3~++c4x4)-4+,(~)

26.2.19

u,=col -3 . . . (2~1) +2cIl -3 . . . (2~33) +22c21-3 . . . (2n-5) + . . . +2”-‘c,-, +d4x4+d3x3+d3x3)-13+t(x)

and cli is t,he coefficient of t”-” in the expansion of trt-1) . . . (t-n+l). Continued

Fraction

Expansions

26.2.14

Q(d)=%)

Je(x))<1.5XlO-7

dl=.04986 &=.02114

10061

ds=.00327

76263

26.2.20

{

&&$$&.

. .}

(XX)

26.2.15

Q(x~=;--z(x) 1

2 x2 23? 3x2 4;c2 -_-_-l-3$5-77+9-.”

)

(x20)

73470

d,=.00003 80036 d,=.OOOO4 88906 da=.00000 53830

Z(X)=(U~+U~X~+~Z~X~+CZ~X')-'+~(~)

Je(z)1<2.7X 1O-3 a,=2.490895 a4= - .024393 a, = 1.466003 a6= .178257 1 Basedon approximationsin C. Hastings,Jr., Approximations for digital computers. Princeton Univ. Press, Princeton, N.J., 1955 (with permission).

PROBABILITY

FUNCTIONS

933

26.2.25

26.2.21 z(x)=(bo+b2x2+b4x4+bax6+bsx8+b,020)-1+e(x) (e(~)[<2.3XlO-~ 3,=2.50523

67

bs=

bz=1.28312

04

b,=-.02024

b4= .22647 18 Rational

b,O=

Approximations

7 for

90

.00391 32 Q(x,)=p

xp where

Cx>O)

P(x) 2 I P4(x) = 1-i

.13064 69

(2n)+e-xz2/2

b>2.2)

See Figure 26.1 for error curves.

o
+6(p),

k(P) 1<3X lO-3 ao=2.30753

bl = .99229

al=

b2=.04481

.27061

P>l-PM

FIGURE

26.2.23 co+C1t+c2tZ

2p=t-

l +d,t

+e(p),

/

- ,006

t=

+d3t”+d3t3

Error

26.1.

curves for distribution.

bounds

on

normal

In-l-

P2

J-

Derivatives

of the

Normal

Probability

Density

Function

Jc(p))<4.5x10-4

Bounds

q=2.515517

d, = 1.432788

cl = .802853

d,=

c2= .010328

da= .001308

Useful

26.2.26

Z(m)(2) =d$m Z(x)

.189269

as Approximations Distribution Function

to

Differential

the

Normal

26.2.27

Equation

Zcm+2~(x)+xZ(m+‘~(x)+(m+1)Z’“‘(x)=.0 Value

26.2.24

at x=0

26.2.28 P,(x)=++;

(l--e-tiz/r)+

(x>O)

(- lym!

P(x) 2 p2(x) = I-

(4+22Y--s 2

(27r)-+e-22/2 (x>l.4)

for m=2r, r=O, 1, . . .

&i2m’2 (g)!

Z(m)(0) = i

1

0

for odd m>O

PROBABILITY

934 Relation

of P(r)

and

FUNCTIONS Z(m)(z)

to Other

Functions

Relation

Function

26.2.29

26.2.30

Error function

erf 2=2P(zJz)-1

Incomplete

y~ ;,x =[2P

gamma

function

(special case)

(

rf

(x20)

>

0 n

26.2.31

Hermite

polynomial

26.2.32

“

26.2.33

Hh function

26.2.34

“

26.2.35

Tetrachoric

26.2.36

Confluent case)

(x20)

(&GC) - l]

Z’“‘(x)

He&$=(-1)

z(x)

2(*)(x&?) &-Jz)

H,(z)=(-1)“2”‘2

VC4 H&(x)

=y

72.

Hh-l(x)

-& (@)

(00)

function hypergeometric

function

(XX)

(special

26.2.37

I‘

(x>O)

26.2.38

‘I

(x2 0)

26.2.39

II

(x20)

function

(G-0)

26.240

Parabolic

cylinder

Repeated

Integrals

26.2.41

I,(x)=

of the

Normal

mIn-l(t)dt sz

Probability

Integral

(n2 0)

where I-l(z>=Z(x) 26.2.42

(n2 -1) 26.2.43

(-g+x

~-+(x)=0

26.2.44 ~n+l>~,+l(x)+x~~(x)--I,-l(x)=O

(n>-1)

PROBABILITY

26.2.45 I,(x)

=

935

FUNCTIONS Asymptotic

S

Let

z

26.2.46

I,(O)=I-n(O)= 0

Asymptotic Density

Expansion Arbitrary

Expansions Function

l s+2 ; !2 2

of an Arbitrary and Distribution

Let Y, (i=1,2,

for the Distribution

the cumulative

Inverse Function Function

distribution

of an

function

of

(n>--1)

Y=$,

(n even)

l&he;) asymptotic expansion with respect to n for the value of yp such that Q&=1--p is

Probability Function

. . ., n) be n

Y, be denoted by F(y).

26.2.49

Then the (Cornish-

ypmfm

where wx+

LYlh (x> I

independent random variables with mean m,, variance I& and higher cumulants K~, c Then asymptotic expansions with respect to n for the probability density and cumulative distribution function of Zi!i (Yi---mJ + are

+r:hI(dl+

and &(x>=p,

26.2.47

Y,-2=$2,

t-=3,4,.

*** ..

26.2.50

j(x)-Z(x)-k P’(x)]+~g 2’4’(x)+~Z’e) (x)] h,(x)=: - [z. P’(x)+Tz; 2’7’ (x)+g6Z’@(x)] +[goz’“‘(x)+& .z@‘(x)+~~; P)(x) +& 2’10 (x)+aM P’(x)]+... Md=& h2Cd

He2(xj

=A

Hea

[2Hfh(d+Hh(dl

h,(X)=-&

We4(41

26.2.48

F(x) -P(x) -pi P’(x)]+~; P(x)+g2’”(XjJ - LgoP(x)+yg;2’6’ (2,+& 2’~’ (x)] +[go2’6’ (x)+A 2’7’(x) +?S2(7’(x) +g z(yx)+&4 zyx)]+... where

h2(a)=-& $ll(x~=~4

We4W

1

%-2=nf,-1

0

K7 -p

He&4

[3He6(4 -I-6He3(d -I-‘JHel(dl

h22@l=-&4

IL,~~(z)=~&

1

[12He4(x)+l9~e2(x~l

h,(~)=$~

h13(z)=--

Terms in brackets are terms of the same order with respect to n. When the Y, have the same distribution, then mt=m, d=u2, K,,t=K, and

+He2(41

1 180

[2Heds) +3He&)l

D4He6(d +37He3(4 +@Jel(dl

hill@) = -17776[252He6(x)+832He3(x)

+227Hel(x) 1 Terms in brackets in 26.2.49 are terms of the same order with respect to n. The He,(x) are the Hermite polynomials. (See chapter 22.)

936

PROBABILJTY

Z(n)(x) El t- 1)” z(z)=n! mgo 2mm!(n-2m)! P-2m

*&)=(-l)”

26.2.51 In the following

auxiliary

table, the polynomial

functions h,(z), h,(z) . . . /Lag,,

are tabulated

for

.l, .05, ,025, .Ol, .005, .0025, .OOl, .0005.

p=.25, Auxiliary

FUNCTIONS

coe$cientaB for use with Corn&h-Fisher

26.2.49

asymptotic expansion.

.2&i

--

* 81449

‘:“I%!!

-:i%i

.00372 -. 14607 -. 04410

:EE .00282

GE .14704 -.06333

3%

26.3. Bivariate

Normal

to mathematical 4, l-14 (lQ37) (with

Probability

statlstlcs. Paper permission).

22;

30 (with

3% :34331

: 67070

-:E -%z -3% -. 174Q3 l.wH45 -1.3QlQQ

-‘:%2

-‘:Z

-%% :04sQ1 -.5@00

-t

x2 -3: 32’108

E. A. Comlsh)

Extralt

-

23: : 1OQsO

r:g; 7123307 -5.40702

de la Revue

de

Pr{XSh,

The probability 26.3.4

Q--h,

-k, p)=

’ dx ’ g(x, y, p)dy s -aD s -(D

26.3.5

L(-h,

k, -p)=

’ dx ?J(x, y, p)dy f -0 sk

L(h, -k, -p)=

26.3.6

26.3.7

W,

26.3.8

U--h,

26.3.9

L(-h,

s

=dr h

s

k, d-Uk,

k, P)+U&

-k, p)-L(h,

&$j=%g

’ g(x, y, p)dy -0D

Probability

Function uZ, &

uy ‘p >

where

k, -p)=Q(lc)

k, p)=P(k)-Q(h)

-p>+W)-Q(k)l---l A = dx k s -h s -k With Means and Correlation

m,,

my,

Normal

Probability

Density

Function

26.3.13 dx, Y, PMY

1 x-mm, ;2”9 ( -’ u

Variances

p

The random variables X, Y are said to be distributed as a bivariate Normal distribution with page

( a,

h, P)

Circular

h

is

x-mm, y-m,

26.3.10 k, p)+Uh,

density function

26.3.12 2rc**v1&7exp

*Bee

.alOs

.-

26.3.11

g(x,y, p)=(l-p*)-iz(x)z

26.3.3

+ WW,

.oOl

means and variances (m,, my) and (4, ~“y) and correlation p if the joint probability that X is less than or equal to A and Y less than or equal to k is given by

Function

26.3.1

26.3.2

“:&%

-:IIE .3wd6 -.4tx34

-: .ZE

8 From R. A. Fisher. Contributions 1’Institute Intematlonal de Statlstique

.0025

.38012 -. 58171 . OeJxo -.53531 . bQ767 -. 02821

$!+I$

-:i%

.Qo.5

y-mm, -10 u

>

=

&* exp- (x-m2+(y-mv~2 2V2

XI. hJA

PROBABILITY

Special

26.3.14

Values

of L(h,

k, P)

L(h,

L(h, k, -l)=O

(h+k
Uh, k, l)=&(h)

(klh)

26.3.18

Uhk, l)=&(k)

(k>N

p)=;+arc2y

+L

-

.h

.

0

13

p

where sgn h=l

P 0

k o W--h)kn

0, p)

>

if hk>O or hk=O and h+k>O otherwise

if h>O and sgn h=-1

.I"

.LC

k)

’ ’ dh2-2phk+k2

26.3.17

L(O,O,

of L(h,

L(h, k, p)=L

(h+kZ 0)

26.3.16 L(h, k, - 1) =P(h) -Q(k)

26.3.19

k, p) as a Function

26.3.20

Uh,kQ)=Q(h)Q(k)

26.3.15

937

FUNCTIONS

.I"

if h
-.I

-*2

-.3

-.4

l/l

IX

l/IA

l/l

I/I

/VI .03

-.6

r

-v I/

h

FIWJB~ 26.2. Vabs

L(h,

0,

p) for 0
for h
using

and -1

L(h, 0, --p)+-L(-h,


PROBABILITY

l/l

0

.OS

.I0

.I5

.20

.25

I/I

.30 FIQIJRE

.35

Y

.40

26.3.

Valuesforh
.45

L(h,

FUNCTIONS

I n

30 0,

l/l

.ss

.60

p) for O
beobtainedusing

L(h.0,

I/I

.65

30

.75

and O
p).

.60

.65

.90

.95

1.00

PROBABILITY

939

FUNCTIONS

P .I4

.I2

.I0

.09

.06

.07

.06

.05

.04

.03

.02

.Ol

-.I /’ /’

/

-.2

/

A’ 1

/’

/’

/. /

/ /

/’

/

/ -.3

/

I

/’

I

/

/’

/’

/’ 0’

/ /’

-.4 / /’

4.0

-rh 1.00 LO5 I.10

I.15 120

L25

1.30 1.35

I.40 1.45 1.50 I.55

FIQURE 20.4.

I.60

1.65 1.70 I.75

1.60 I.65 1.90 t95 200

205

2JO 2.15 2.20 225

L(h, 0, p) fo7 h21 and -1 1~51.

Values for h
0, P.)

2.30

2.35 2.40

245

2.50

940

PROBABILITY

Integral

Over

an Ellipse

With

Center

at (m,,

FUNCTIONS Approximation

m,)

to P(pI

2,s)

26.3.25

26.3.21

Approximation

where A is the area enclosed by the ellipse (Y- mu>

%4x-d

WG +(y), Integral

Over

Condition

26.3.26

P(xl)

R>l

26.3.27

P(x2)

RX

=d(l-$)

an Arbitrary

Region

26.3.22

=ss A’(& I)

where A*(s, t) is the transformed from the transformation se-

1

--

t-JzA-p

g(s, t, o)dsdt

region obtained

(

x-m+ ---

Q(h)-Fk

&---k>O,

k, d<&(h)

Series

O
26.3.29

Q > Probability v=l Over

Z(k) [Q (j+2)-@h

where

y-mm,

a,

Integral of the Circular Normal With Parameters m.=n,=O, Rounded by y=O, y=ox, x=h

Inequality

Qv >

gz

*

26.3.28

-x- mZ+y-- m,

4 42+2p

R, r both large

x2=R-,fi

L(h, k, p) =Q(h) Q(k) +$

Function the Triangle

26.4. Chi-Square

m

Z’n~~l;;(k)

Probability

p”+’

Function

26.4.1 26.3.23 P(x~J,)+~I V(h, ah)=&

=;

(;)]-‘12

(t)+e-+dt

sh s”: e-f(z2+v2)dxdy 0 0 +Uh,

0, P)--L(O,

0, p)-

(Olx2< a) 26.4.2

; Q(h)

Q(x"lv>=1-P(x"lv)

where

+I

r (!L)]-lsx; Relation

Integral of Circular Normal Distribution Over Circle With Radius Ru and Center a Distance h9 m,)

an O&et ru From

26.3.24 0) dxdy=P(R2(2, r’)

where P(R212, r”) is the c.d.f. of the non-central x2 distribution (see 26.4.25) with v=2 degrees of freedom and noncentrality parameter 9.

(t)+-‘e-t

to Normal

dt

Distribution

Let XI, X2, . . ., X, be independent and identically distributed random variables each following a normal distribution with mean zero and unit variance.

t ‘T,

@9"< ->

Then X2=$X:

is said to follow the

chi-square distribution with v degrees of freedom and the probability that x25x2 is given by P(x’~v). Cumulants

26.4.3

K,,+l=2%!Y

(n=O, 1, . . .)

PROBABILITY Series

941

FUNCTIONS Approximations

Expansions

,to

the

Cl&Square Large Y

26.4.4 26.4.13

QW>

Distribution

Approzimation x&E+i%z = &(a,,

for

Condition (v>lOO)

26.4.14 (x*/v,l~3-( l-2)

(V odd) and x=fi Q(x'l4 =&(x4,

26.4.5

(v>30)

x2= 4s

26.4.15

Q(x*lv) =Qb2+hA,

hv=T

(v even)

hao

(u>30)

Values of hm

26.4.6

Approximations

26.4.7

for

the

Inverse

Function

If Q(x’,~v) =p and Q(z,> = 1--P(g)

p(X*lv)

=p,

for

then Condition

Approximation Recurrence

26.4.9

and

Differential

b"Q(x*ld =F 1 $ (7) b(X2)" Continued

26.4.16

xl, =;

{ x.+Ji;=r)s

(v>lOO)

26.4.17

x;~:Y{ ~-$+x~G}~

(v>30)

26.4.18

x”, = v.

(v>30)

Relations

lG+

(5 -h”)g}”

(-1~“+‘Q(x’l~--2~~ where h, is given by 26.4.15.

Fraction

Relation

26.4.19

Incomplete

to Other

Functions

gamma

function

y=?#a,x2=2x

~$g=P(x’Iv,, 1

l-v/2

1

2-v/2

2

- *) { x2/2+1+x2/2+1+@% Asymptotic

26.4.11

Distribution

P(x*lv) -P(x) Asymptotic

for

Large

where z=Expanxionx

for

~=Q(x*lv, 26.4.20

Y

Pearson’s incomplete

x2-u Jz;

Large

26.4.21

x2

2,+1

x2=2u 4g

&@*I~> =&lj

cm

5)

c=i,

n=$

(-1)’ r

1-i (

(x2)’ >

function

Poisson distribution C-l

r (++j) *seepageIL

gamma

v=2(p+l),

26.4.12

Q(x2,.)~~~~,~;~~;;s&

Large

Q(~~~v>-Q(~~(v-2)=e-‘”

&!

(v even)

Y

942

PROBABILITY

26.4.22

FUNCTIONS Non-Central

Pearson Typo III

X*

Distribution

Function

26.4.25 v=2abf2, 26.4.23

Incomplete

moments

P (X’21v, A)=g

x2=2b(s+a)

of Normal

distribu-

where Xl0 eter.

tion (n- 1) !! ‘9

(n even)

@$?P(xzlv) T

(n odd)

Generalized

ng (-lP+’ n!L(“,) (2)s PO

Laguerre (“;‘>

SS

Polynomials

A

g(x, y, O~dx&/=P(x2=R2~v=2,

Q(x21v+2--237

Approximations

=1-g

to the

Non-Central

a=v+h

b=-

x2 distribution

P(X’2lv, M-P

26.4.28

Normal

distribution

P(x’2lv, A) =P(x),

26.4.29

Normal

distribution

P(x’2(v, A) =P(x),

Approximations

l+b

to the X*

If Q(x~lv, X)=p, Q(xilv*)=p, x2

26.4.31

Normal

26.4.32

Normal

x v+x

( x2I>*

26.4.27

26.4.30

Distribution

X*

Approximation

Function

Variable

param-

26.4.26

x=x2/2, cz=v/2

Approximating

the non-centrality

vsn+l

2”[Q(x21v+2> - &(x2141

Approximating

P(Xt21v+2j)

Relation of Non-Central xz Distribution With v=2 to the Integral of Circular Normal Distribution (u*=l) Over an O&et Circle Having Radius R and Center a Distance r=h From the Origin. (See 26.3.2446.3.2’7.)

x2=$,

26.4.24

is termed

emhiy

and Q(x~)=~ Approximation

v )

v*=&

x+cJ-[~-l]

Inverse Function Distribution

of Non-Central

then to the

x;=(l+b)x;

Inverse Function

A) $7

Q(R2(2+2$

PROBABILITY

FTJNCTIONS

943

944

PROBABILITY

26.5. Incomplete

FUNCTIONS Continued

Beta Function

26.5.1

Iz(u,b)=B(;,b)

s

O=tyl--t)L-ldl

(Olxil)

b+m)(a+b+m)

dh+l=-

Zz(u,b)=l-Z,-,(b,u) Chi-Square

Distribution

If X: and Xi are independent random variables following chi-square distributions 26.4.1 with Y, and v2 degrees of freedom respectively, then X is said to follow a beta distribution with x:+x: vI and v2 degrees of freedom and has the distribution function

Best results

are obtained

zQ(l-x)b-’ uB(u b) f

b)=

e2m=-

Expansions

&f!, b-5

.2

l+.g b)

mJSl,n+l) n-o @+b, n+l)

x”+l

1

e,=l

(u+m-l)(b-m) (u+2m--2)(u+2m-1)

(u+2m-l)(u+2m) Recurrence

Z,(u, +“(‘-x)b aNa,

* * *

.122

m(u+b-l+m) e2d=

(O<x
26.5.4 *

el ez e3 C 1+ 1+ 1+ -

xc1

*

Series

when x<s2.

26.5.9

26.5.3

2

’

Also the 4m and 4 m+ 1 convergents are less than Zz(u, b) and the 4m+2, 4m+3 convergents are greater than Z,(u, b).

z&,

=L(a,b)

X

(a+2m)(a+2m+l) m(b-m) (u+2in- l)(u+2m)

d2m= to the

*

---. 1 4 4 .. Zz(u, +f(l-x)b aB(a, b) 1+ 1+ 1+

26.5.2

Relation

Fractions

26.5.8

II,

Relations

26.5.10

Z,(a,b)=xZ,(u-l,b)+(l-x)Z,(u,b-1) 26.5.11

26.5.5

Zz(a,b)=

xyl -Zy-’

Zz(u,b)=;{Zz(u+l,b)-(l-x)Zz(u+l,b-1))

uB(u, b)

26.5.12 *

[ 1

Lb, b)= ucl-;,+bi

=2(1--r)b-’ Wa, b)

bzz(d’+l)

+u(l-x)Z,(u+l,b-1)

1

26.5.13

+Zz(a+s, b-s) 26.5.6

l-Z&,

I,(a,b)=;;-

{uZ,(u+l,b)+bZ~(~,b+l)}

26.5.14 b)=Z,-,(b,u)

Jl-x)b Bm

I

z&,u)=; I,-,t (u, f)!

O--l (-l)i C i-0

(“;i’)

5:)

(integer a>

x)=4 (x-;y[x

26.5.15

26.5.7 1-Z,(u, =

b) =I,-r(b,

(1s-@+b-’

a.)

g

(“+t-‘)

26.5.16

(+r)i

(integer a)

Z&b)=

-- r(“+b) r(a+l)r(b)

P(l-r)b+Z,(u+l,b)

s ;I

PROBABILITY Asymptotic

Expansions

and y is taken negative when x< ~ u-l

26.5.17 r@

l-lz(a,b)=I~-*(b,a)---

945

FUNCTIONS

u+b-2

Y)

r(b) Approximations

26.5.20

If (u+b-1)(1-x)5.8

I&

Q(x'lv)+ e,

b) =

]c]<~X~O-~

if u+b>6

x’=(u+b-l’)(l-x)(3-x)-(l-x)(b-1), v=2b

26.5.18

If (a+b-1)(1-x)2.8

26.5.21

Lb,

b)=W+e,

(ej<5X10W3

if u+b>6

26.5.19 Lb, b) -Q/)--z(y)

[u,f”‘f$--$ +%(l+?m+

. . .

1 +a2

ul=;

wz=[u(l-x)]“”

~~=(bx)“~,

(b--a)[(u+b-2)(u-l)(b-

1

Approximation

to the

Inverse

Function

,1)1--M

26.5.22 If &(a,

1

and Q(y,)=p

b)=p

then

U 5”,+be2”

,=Yv@+v’

+(b-1)

In

b-l

(u+b-1)(1-x)

Relations

to Other

1

h=2

Functions

and

h,Yz-3 6 Distributions

Function

26.5.23

Hypergeometric

26.5.24

Binomial

Relation

function

&;

F(u, 1-b;

9

u+l;

n n p”(l--p)“-‘=l,(u, ,-II s c(>

distribution ‘I

26.5.25

-

h

X

b)

x)=1&

n-‘u+l)

r pa(l--p)“-“=I,(u,n-u+l)--l&2+1,

n-a)

0

26.5.26

Negative

binomial

26.5.27

Student’s

distribution

265.28

F-(variance-ratio)

distribution

2

(“‘:-‘)

p”q’=I,(u,

;[l-&j+;

distribution

1, (;$,

Q(F/vl, vJ=Iz

(‘II, ;;?)P

n) x=--f-

x=2 vz+vr

*See

page

II.

*

v+t2

F

*

946 26.6.

F-(Vdance-Ratio)

PROBABILITY

FUNCTIONS

Function

26.6.5

Distribution

26.6.1

Q(F~v,,v,)=1-(l-2)"'2[l+~

Ftf(“1-2)(Y2+Ylt)-f(“I+“l)dt

PWIVI,d=

* . * (V2+Vl--4)

+vlh+2) V'2

2.4.

. . (~~-2)

Q(Flv,, vJ=1-(l-2)

2

0)

26.6.2

x+w

x2+...

1

xT

(v2 even)

26.6.6 Q(Fjv,, vJ=l-P(FIvI,

vJ=Iz

(

z,;

>

where x=-----

v2

v2+v1

Relation

to the

F

Chi-Square

Distribution

If X,” and Xi are independent random variables following chi-square distributions 26.4.1 with v1 and v2 degrees of freedom respectively, then the distribution of F=m2WV1 is said to follow the variance ratio or F-distribution with degrees of freedom. The corresponding tion function is P(FJv,, v2). Statistical

26.6.3

26.6.7

and v2 distribu-

v1

Properties

QU%, VZ

mean :

m=-

variance:

g2=vl(v2-q2(v24)

v2)=1-4tlv2)+/3(~1,

bJP>2)

vz-2

2&vl+v2-2)

23.54 -**(‘=I;;

4t\v2y=

h>4)

.

cos --2e]

. . .

;

( > Vz--l

,

$

\sinecosv~e

l+

>

v2+1 -sin*f3++..+

P(v1, Vz)=

3

(V2+1)(V2+3).

. . (~~+V2-4)

Oforv,=l

*

where

Expansions

f3= arctan

v2 vz+vP

26.6.4

+v2b2+2)

‘See

Reflexive ~2)=s”~~[lf;(l--r)++9(1

. . . (vz+v~--4) 2.4.. * h-2)

page II.

q-3

e

for v2>l

l$(t)=E(P)=M ( 2, 2, -z it>

* Q(Flv,,

Sin

3*5...(v1-2)

functioni

xc-----

v2>1

{ (

Series

for 1

F for v2=l

about the origin:

characteristic

(VI, v2 odd)

~2)

O+sin B[cosO+~ cos3 O+ . . . +

third central moment:

moments

(v2 even)

26.6.8

-x)2+.

Y,-2

(l-x)2

1

. .

(IQ even)

Relation

If Fp(vl, v2) and F,-,(v2, v,) satisfy

QV'vh, &VI-ph,

V2) lV1, V2) =P VAIV2, Vd=l---p

PROBABILITY

26.6.9

947

FUNCTIONS

26.6.15

then

x/113 (l-ia-(l-&)

Q(Flv,, ~2) =Q&(d, Relation

to Student’s

26.6.10

t-Distribution

Q(FIvI=l,

Function

(See 26.7)

J

t=@

v,)=l-A(ljvz)

Approximation Limiting

Inverse

Function

Forma

26.6.16

26.6.11

~2)=Q(x21~~),

lim QU%, *a+-

to the

$+F2j3 &

If Q(FvIv,, v2)=p, then

F, = e2wwhere w is given by 26.5.22, with

x2=vIF

vI=2b, v2=2a

26.6.12 x2=-

lim Q(FIv,, v~)=P(~~ld,

n-f-

Non-Central

;

P’ P(F’IvI,

vz,N=

s0

26.6.13

(vl

- v2 v,--2

J

~(tlv,,vz,~)dt=l-Q&(F’Iv~,v2,X)

where

F-Y& v*-z

x=

and v2 large)

Function

26.6.17

Approximations

QV'Iv,, v2> =&Cd,

F-Distribution

-

2(v,+w--2) 5 h-4)

26.6.14

J

v*+u--2

(2v2-1) ; F-d-

&U%,

~2)

x=

=Q(d,

xt

I-

d

l+:F

Relation

of

Non-Central

- b,+2j+v,)/2

[v2+~v1+2j)tl

and A>0 is termed the non-centrality F-Distribution

Function

to Other

Functions

P(F’Iv,,

F-distribution

v2,A)= F. evN2 y

.

P(F’I ~1,~2,x=O)=P(F’(VI,

26.6.19

Non-central

tdistribution

26.6.20

Incomplete

Beta function

P(F’JvI=l, P(F’Iv,,

v2,X)=P(t’lv,

S), t’=dF’,

eMAl yIz

v,)=%

P(F’lvl+2j,v2)

v2)

v=vg, S=J);

(;+j,

2)

j-0

2-7

vlF’ * v,F -4-a

pa

--1

26.6.21

parameter.

Relation

Function

26.6,18

2

Confluent hypergeometric

function

P(F’IvI,

~2,

A)=

2ebN2

2x

i-o (vl+v2)B

(

.

v2

(?,i,l,

even and x=-

;-i)”

v2

v$“+vz >

23

PROBABILITY Series

FUNCTIONS

is called Student’s

Expansion

26.6.22

freedom.

-x 0-a

P(F’Ivl,vz,N=e

z~b,+y2-2)

2

‘2

Ti

(v2 even)

t-distribution

The probability

with

that

& in absolute value than a fixed constant

degrees of

v

will be less is

t

26.7.1

where To=1 T,=;

(v,+vq-2+Xz)

T t =q x=-

e

[(vl+vz--2i+Xs)T,-,+X(1-z)T,-,]

where

VP v,F’+v, Limiting

v-l-i*

Forms Statistical

26.6.23 x’*=v,F’,

lim P(F’Iv~,v~,X)=P(X’*IV,X), vr+m 26.6.24

x2=v2(1

lim P(F’~v,,v~,X)=&(~*~V),

“I-+”

~-

v=v,

+c*1 F’

Approximations

to the

P(F’lv,,v,,M

Proper&s

26.7.2

mean :

m=O

variance:

g&Y

(v>2)

v-2

skewness: r,=O

where X/v,-+* as vl+ Q).

26.6.25

V

xc-

Non-Central

F-Distribution

cxccss:

(v, and v2 largr)

=P(q),

where

r*=&

(v>4)

moments: ~*..-(vl;)~v~~~)

(2n-l)v” . . . (v-2n)

(v>W

VI (v2 -a

x1=

CL*n+1=O

g+v,+2h

characteristic

I

function:

26.6.26 PiF’lv~,vn,V

-W7b:,v2),

F-

‘1 v,+x

F’,

v~-(v1+X’2

Series

v1+2x

(

@=arctan -!4 >

26.6.27 PW’h,v2,M

4x2:,),

26.7.3

,=[&l"" [I-&I-[l-29(;1,:31 (& q3] [

~{0+sin0[cos8+~ros30+. + 2.4 . . . (v-3) 1.3.. . (v-2)

; 55i+&

A(+)=.

26.7.

Student’s

then the distribution

of the ratio x Jx*Iv

..

2 -8 n-

*

cosy-2 0.

(v>l

t-Distribution

If X is a random variable following a normal distribution with mean zero and variance unity, and X* is a random variable following an independent chi-square distribution with v degrees of freedom,

Expansions

II

and odd)

(V’l)

26.7.4

A(tlv)=sin +

0

1 1.3 1+z cos* e+m c-OS4 e+ ,

1.3.5 . . . (v-3) 2.4~6.. . (v-2)

cosy-* g 1

(V

even) *

PROBABILITY Asymptotic

Expansion

If A(t,(v)=l-2p

for

the

Inverse

and Q(z,)=p,

949

FUNCTIONS Approximation

Function

26.7.8 +!Jl(X,)

sP(xP)+gdxP)

7+7

P

--$-+

p

&Iv)

=2P(x)-1,

x=

I-

+2

- * *

!a)=; (xa+x) gz(z)=&

Large

then

26.7.5

t -x P

for

Non-Central

t-Distribution

e-s2f2 9

1z ($i+j),

26.7.9

(5scb+l62’+3s)

P (t/Iv, S) = g&)=&

(32’+191+1721-152) (79x9+776x7+ 1482x5- 192023-945x) Limiting

Distribution

=1-s

26.7.6

x-

where 6 is termed the non-centrality Approximation

a, b,

Large

Values

of

t

and

v<S

Approximation

A(tlv) E31-2 { ;+$}

26.7.7 Y

for

1

9

.3183 .oooo

.4991 .0518

3

1.1094 - .0460

to the

v , + v+t”

parameter.

Non-Central

t-Distribution

26.7.10 4 3.0941 -2.756

i!’

.5

P(t’Jv, 6) =zqx)

9.948

where x=

- 14.05

Numerical 26.8. Methods of Generating

Methods

Random Numbers and Their Applications

Random digits are digits generated by repeated independent drawings from the population 0, 1, 2 9 where the probability of selecting any * *, digit is one-tenth. This is equivalent to putting 10 balls, numbered from 0 to 9, into an urn and drawing one ball at a time, replacing the ball after each drawing. The recorded set of numbers forms a collection of random digits. Any group of n successive random digits is known as a random number. Several lengthy tables of random digits are available (see references). However, the use of random numbers in electronic computers has resulted in a need for random numbers to be generated in a completely deterministic way. The numbers so generated are termed pseudo-random numbers. The quality of pseudo-random numbers is determined by subjecting the numbers to several statistical tests, see [26.55], [26.56]. The purpose of these statistical tests is to detect any properties of the pseudo-random numbers which are different from the (conceptual) properties of random numbers. 0 The authors wish to express their appreciation Professor J. W. Tukey who mnde many penetrating helpful suggestions in this section.

9

Experience has shown that the congruence method is the most preferable device for generating Let the sequence random numbers on a computer. of pseudo-random numbers be denoted by {X,} , n=o, 1, 2, . . . . Then the congruence method of generating pseudo-random numbers is X,,+,=aX,,+b(mod T) where b and T are relatively prime. The choice of T is determined by the capacity and base of the computer; a and b are chosen so that: (1) the resulting sequence {X,,} possesses the desired statistical properties of random numbers, (2) the period of the sequence is as long as possible, and (3) the speed of generation is fast. A guide for choosing a and b is to make the correlation ’ between

the numbers

be near zero, e.g., the correla-

tion between X,, and X,,+* is +e

where al=aa (mod T)

to and

b,=(l+a+a2+

lel
pa&-e II.

. . . +a’-‘)b

(mod 7’)

950

PROSABILITY

which

FUNCTIONS

a= 1 (mod 4). When T= 10q, b need only be not divisible by 2 or 5, and a=1 (mod 20). The most convenient choices for a are of the form a=2”+1 (for binary computers) and a=lO”+l (for decimal computers). This results in the fastest generation of random numbers as the operations only require a shift operation plus two additions. Also any number can serve as the starting point to generate a. sequence of random digits. A good summary of generating pseudo-random numbers is [26.51]. Below are listed various congruence schemes and their properties.

occur in Xn+s=asXn+h

(mod T rel="nofollow">

When a is chosen so that a= PIP, the correlation p1z T-l’“. The sequence defined by the multiplicative congruence method will have a full period of T numbers if (i) (ii) (iii)

b is relatively prime to T a= 1 (mod p) if p is a prime factor of a= 1 (mod 4) if 4 is a factor of T.

T

Consequently if ?‘=2q, b need only be odd, and Congruence

= = a

methods

for

X.+,=aX.+b(mod T

b

Period

generating

X0

Special

_1+t*

odd

T=tp

O<Xo
26.8.2

r2’fl (r odd, 812)

0

T=to

relatively

26.8.3

r2*fl (r odd, 822) -pa

T=tcfl

Fme relatively

T= 100

pTrime tc relatively

T=lO*

Fme relatively

26.8.5

10x0 given

*

point

for random

numbers

when

,

I

Generation

of Random

Deviates

Let {X} be a generated sequence of independent random numbers having the domain (0, T). Then { U} = { T-lx} is a sequence of random deviates (numbers) from a uniform distribution on the interval (0, 1). This is usually a necessary preliminary step in the generation of random deviates having a given cumulative distribution function F(y) or probability density function f(y). Below are summarized some general techniques

‘See

page U.

cases for which

random numbers tests for randomness

have 10

passed

statistical

T=218, Xa unknown; a=27+1. b=l; T=247, a=Z@+l. b=2974109625 8473. X0=76293 94531 25. T=W, 211, X0=1; a=5”(8=2) T=2as, X0=1; T=219, X0=1-2-19, 5478126193; a=5’#(8=2) T=286, X0=1; a=515(8=2) T=aas+l, X0=10,987,654,321; 0=23; period = 106 T=los+l, X0=47,594,118; a=23; period=5.8XlOs T=lOlo, T=lOl$

to

Prime to

is the starting

to

X0=1; X0=1;

a=7 a=71:

-

statistical

When the numbers are generated using a congruence scheme, the least significant digits have short periods. Hence the entire word length cannot be used. If one desired random numbers with as many digits as possible, one would have to modify the congruence schemes. One way is to generate the numbers mod T* 1. This unfor1 iunatelv reduces the neriod. .

numbers prime

_-

26.8.1

26.8.4

random

T), T and b relatively

tests were made.

for producing arbitrary random deviates. (In what follows { U} will always denote a sequence of random deviates from a uniform distribution on the interval (0, ., l).) 1. Inverse

Method

The solutions {y) of the equations {u=F(y)) form a sequence of independent random deviates with cumulative distribution function F(y). (If F(y) has a discontinuity at y= yo, then whenever u is such that F(y,-O)
a Discrete

Random

Variable

Let Y be :I discrete random variable with point probabilities p,=Pr{ Y=y,] for i=l, 2, . . . .

PROBABILITY

The direct way to generate Y is to generate and put Y=y, if

FUNCTIONS

{ lJ]

. * - +Pi. * * * +p*-l
P,=lO-’

6,* for r=l,

i=l

II.=&

r=O

2, . . . , k, and

lO’P,, s=l,

2, . . . , k.

Number

the computer memory locations by 0, 1, 2, . , . ) Q-1. The memory locations are divided into k mutually exclusive sets such that the 8th set consists of memory locations II,.-,, q-,+1, . . . ) Q-1. The information stored in the memory locations of the sth set consists of y1 in 6,, locations, y2 in a#2 locations, . . . , yn in 6,, locations. Denote the decimal expansion of the uniform deviates generated by the computer by u = .d,d,d, . . . and finally let a {m} be the contents of memory location m. Then if .-I nut .¶-I

y=a

d,d2 . . . ds+us-l--os

g

Pi

.

This method is perhaps the best all-around method for generating random deviat#es from a discrete distribution. In order to illustrate this method consider the problem of generating deviates from the oinomial distribution with point probabilities

PI=0y for n=5

and p=.20.

py1-py

The point probabilities

thus

P,,=O,

K=43, divided

Set 1 2 3 4

;

: 5

p,= p,= p,=

p,=

.2048 .0512 .0064 .0003

Ps=.027,

Memory Locations 0,1,...,8 9, 10, . . . , 15 16, . . . , 42 43, . . . , 72

Frequency Frequency Frequency Frequency

(set (set (set (set

1)

2) 3) 4)

0

1

2

3

4

5

3 2

4 0

2 0

7 7

9 6

4 8

0 5 1 2

0 0 6 4

0 0 0 3

Then to generate the random 0
y=a{dld2-81} y=a{ddld2d3-954) y=a{d,d,d,d,-9927}

.97
variables if

y=aIdlj

Put

a Continuous

Random

Variable

The method for generating deviates from a discrete distribution can be adapted to random variables having a continuous distribution. Let F(y) be the cumulative distribution function and assume that the domain of the random variable (If the is (a,b) where the interval is finite. domain is infinite, it must be truncated at (say) the points a and b.) Divide the interval (b-a) into n sub-intervals of length A (nA=b-a) such that the boundary of the ith interval is (yl+ yr) where yr=a+iA for i=O, 1, . . . , n. Now define a discrete distribution having domain

to

zt-Yf+Yf-l 2

{ Point Probabilities p0=0.3277 p,= .4096

P2=.07,

Among the nine memory locations of set 1, zero is stored &=3 times, 1 is stored 6,,=4 times, 2 is stored a12=2 times; the seven locations of set 2 store 0 a2,=2 times and 3 823=5 times; etc. A summary of the memory locations is set out below: Value of Random Variable

4 D are Value of Random Variable 0

P,=.9,

from which II,=O, I&=9, 112=16, 114=73. The 73 memory locations are into 4 mutually exclusive sets such that

P,=.OO30

PI+P2-k

PO-O,

and

951

)

with point probabilities pi=F(yr) Finally, let W be a random variable uniform

distribution

done by setting

on

.

-F(y+J.

having

a

This can be Then

random

952

PROBABILITY

deviates from the distribution function F(y), can be generated (approximately) by setting y=z+w 1 =z+A u-. This is simply an approximate 2 ( > decomposition of the continuous random variable into the sum of a discrete and continuous random variable. The discrete variable can be generated quickly by the method described previously. The smaller the value of A the better will be the approximation. Each number can be generated by using the leading digits of U to generate the discrete random variable Z and the remaining digits forming a uniformly distributed deviate having (0,l) domain. 4. Acceptance-Rejection

Methods

In what follows the random variable Y will be assumed to have finite domain (a, b). If the domain is infinite, it must be truncated for computational purposes at (say) t’he points a and b. Then the resulting random deviates will only have this truncated domain. a) Let j be the maximum of j(y). Then the procedure for generating random deviates is: (1) generate a pair of uniform deviates VI, U, ; (2) compute a point y=a+(b--a)uz in (a, b); accept y as the random deviate, (3) if u~
b) Let F(y) be such that f(y)=f,(~)j&) where the domain of y is (a, b). Let ji and ji be the maximum of ji(y) and ji(y) respectively. Then the procedure for generat’ing random de-

FUNCTIONS

viates having the probability density function j(y) is: (1) generate U,, U,, US; (2) define z=u + (b-ah;

(3)

if both ul<

take z as the random another sample of three acceptance ratio of this and can be increased by intervals as in the previous c) Let the probability f(y) =saBs(~, W,

‘y

and uz< j?,

deviate: otherwise uniform deviates. method is [(b-u)jlji]-’ dividing (a, b) into case. density function of

(6 t 93,

(sly<

take The subY be

b).

Let g be the maximum of g(y, t). Then the procedure for generating random deviates having the probability density function j(y) is: (1) generate Ul, U2, U2; (2) define a=~+@--(Y)u~; ~=a+ (b-u)u,;

(3) if u,< $-%

take z as the random

deviate; otherwise take another sample of three. The acceptance ratio for this method is [ (b-u)g]-’ and can be increased by dividing t)he domain of t and y into sub-domains. 5. Composition

Method

Let g,(y) be a probability density function which depends on the parameter z; further let H(z) be the cumulative distribution function for z. In order to generate random deviates Y having the frequency function f(Y)

=Jrn

g,(yMH(z) -m

one draws a deviate having the cumulative distribution function H(z); then draws a second sample having the probability density function gp(y). 6. Generation

a. Normal

of Random Deviates Distributions

From

Well

Known

distribution

(1) Inverse method: The inverse method depends on having a convenient approximation to the inverse function x=P-l(u) where UC

&)

-l/2

S =

-m

e-1212dt.

Two ways of performing this operation are to (i) 112 use 26.2.23 with t= In r or (ii) approximate U2 > ( x=P-l(u) piecewise using Chebyshev polynomials, see [26.54]. (2) Sum of unijorwz deviates: Let U,, U2, . . ., U, be a sequence of ‘n uniform deviates. Then

PROBABILITY

xn=(& CL-;) (;)-I” will be distributed asymptotically as a normal random deviate. When n=12, the maximum errors made in the normal deviate are 9X 10e3 for jXl<2, 9X10-l for 2
as the normal deviate where az, are suitable coefficients. These coefficients may be calculated using (say) Chebyshev polynomials or simply by making the asymptotic random deviate agree with the correc normal deviate at certain specified points. When n=12, the maximum error in the normal deviate is 8X 10e4 using the coefficients * a6= (-7) -5.102 * a0 = 9.8746 * *

a:! = (- 3)3.9439 a,= (-5)7.474

*

ax=

(-7)1.141

(3) Direct method: Generate a pair of uniform deviates (U,, U,). Then Xi= (-2 In IU,)“~ cos 2*UZ, X*=(--2 In U1)1’2 sin 27rUz will be a pair of independent normal random deviates with mean zero and unit variance. This procedure can be modified by calculating cos 21rl.J and sin 2rU using an acceptance rejection method; e.g., (l)generate(UI, U,);(2)if (2U,-1)*+(2U~-l)*
Yl=(--ln

u3J1’*$$$yZ=*2(-ln

random). deviates. *

213)*12a (h I Both y, and yz are the desired random

(4) Acceptunce-rejection method: 1) Generate a pair of uniform deviates (U,, Uz); 2) compute z = -In u1 ; 3) if e-H(z-na >u2 (or equivalently (z-1)*5 -2 (In N) accept 2, otherwise reject the

953

FUNCTIONS

pair and start over. The quantity will be the required normal deviate with mean zero and unit variance. b. Bivariate

normal

distribution

Let {XI, X2] be a pair of independent normal deviates with mean zero and unit variance. Then {X,, pX,+ (1 -p2)“*XZ} represent a pair of deviates from a bivariate normal distribution with zero means, unit variances, and correlation coefficient p. c. Exponential

distribution

(1) Inverse method: Since F(z) =e-I’*, X= -0 In U will be a deviate from the exponential distribution with parameter 0. (2) Acceptance-rejection method: 1) Generate a pair of independent uniform deviates (U,, U,); 2) if Ul
an exponential

26.9. Use and Extension of the Tables Use of Probability

Example 1. Let X be a random finite mean and variance equal respectively. Use the inequalities functions 26.1.37, 40, 41 to place on A(t)=F(f)-F(-t)=P

for t=1(1)4. *See page 11.

variable with to m and 2, for probability lower bounds

IX-ml
Function

Inequalities

Lower bounds on A(t) = F(t) - F( - t) Remarks

0 .7500 .8889 .9375 no knowledge F(t);

.5556 .8889 .9506 .9722 >

0 .8182 .9697 .9912

of

26.1.37

is unimodal and continuous; 26.1.40 F(t) is such that F(t)

/~,=3; 26.1.41

954

PROBABILITY

It is of interest to note that the standard normal distribution is unimodal, has mean zero, unit variance pL4=3, is continuous, and such that

A(t)=P(t)-P(-t) =.6827, for t=l,

for

values we can write x=x0+.01 and a two-term Taylor series is P(x) = P(z,) +Z(Z,) 10p2. Thus one need only multiply 2(x,) by 1O-2 and add the result to P&J. Calculation

.9545, .9973, and .9999

2, 3 and 4 respectively. Interpolation

FUNCTIONS

P(z)

in Table

26.1

Example 2. Compute P(Z) for x=2.576 to fifteen decimal places using a Taylor expansion. Writing x=x0+8 we have P(x)=P(x,)+Z(x,)e+Z(l~(x~)

;

of P(z)

for

z Approximate

Example 4. Using Table 26.1, find P(x) for Z= 1.96, when there is a possible error in z of & 5 x 10-3. This is an example where the argument is only The question arises as to known approximately. how many decimal places one should retain in P(x). If As and AP(z) denote the error in 2 and the resulting error in P(x), respectively, then AP(x) = Z(x) Ax

+.r2’(x())~+2’3’(xo) e”+ ..* 41 3! Taking x0=2.58 and 13=-4X10-~ the successive terms to l6D

we calculate

Hence AP(l.960) =3 X low4 which indicates tha6 P(1.960) need only be calculated to 4D. Therefore P(1.960) = .9750. Inverse

-j-.99505 5 -

99842 42230 72204 35976 2952 57449 8 63097

4 9

84265

7

-

.99500 The result correct

24676

Example

for

3.

P(s)

to l7D is

P(2.576)=.99500 Calculation

for

5. Find the value of z for which P(x)=.97500 00000 00000 using Table 26.1 and determining as many decimal places as is consistent with the tabulated function. For inverse interpolation the tibulated function P(x) may be regarded as having a possible error of .5X 1C1-l~. Hence L.

6 6 8

1439

-

Interpolation

Example

24676

Arbitrary

84264

Mean

and

98

Variance

Find the value to 5D of

Let P(x,J correspond to the closest tabulated value of P(x). Then a convenient formula for inverse interpolation is x=xo+t+ig+%p

using 26.2.8 and Table This represents the variable being less than distribution with mean Using 26.2.8 we have

26.1. probability of the random or equal to .5 for a normal m=l and variance a2=4.

p

where ,=w

-ml) a&J

If only the first two terms (i.e., r=x,,+t)

are used,

the error in x will be bounded by g X lop4 and the

F{X<.5j=P Since P(--2)=1--P(z),

(

q

>

=P(-.25)

we have

P(-.25)=1-P(.25)=1-.59871=.40129 where a two-term Taylor series was used for interpolation. Note that when interpolating for P(x) for a value of x midway between the tabulated

true value will always be greater than the value thus calculated. With respect to this example, Ax= 10-l’ and thus the interpolated value of x may be in error by one unit in the fourteenth place. The closest value to P(x)=.97500 0000000000 is P(x,)=.97500 21048 51780 with x0=1.96. Hence using the preceding inverse interpolation formulas with

’

PROBABILITY

t=-.00003

and carrying sive terms

I ,

60167 31129

fifteen decimals 00000 60167 12

00000 31129 71261 68 0

+ 1.95996

39845

40064

Asymptotic

Q(X’jv)=l--F(t)=l--F

where (2v-1)f

and 1-d

are the mean and vari-

ance to terms of order v-’ of 4s The values of y1 and yz for 42x2 are

(see 26.4.34).

Thus we obtain

Expansion

Example 6. Find the Edgeworth asymptotic expansion 26.2.49 for the c.d.f. of chi-square. Method 1. Expansion for x2

Let /

(y;--ijl)‘)

we have the succes-

+ 1.96000 - .00003 + -

Edgeworth

955

FUNCTIONS

F(t)--P(t)--f [g (1+;)P(t)] +$[~z’“‘(t)+~(1+~)22(5)(t)]

For numerical

&(X2lv)=l--F(t)

examples using these expansions

see Example 12.

where

Calculation

x2-v te-.----

@VP

Example 7.

Since the values of y1 and y2 26.4.33 are

Using

26.3.20

J.O9=.3

L(.5, .4, ,8)=L(.5,O,O)+L(.4,0,

-.S)

Reference to Figure 26.2 yields

WV,

we obtain, by using the first two bracketed of 26.2.49

k, p)

Find L(.5, .4, .8). -

,/h2-2phk+k2=

y1=2&p Y2=

of L(h,

terms

L(.5,O,O)+L(.4,0,

-.6)=.16+.08=.24

The answer to 3D is L(.5, .4, .8)=.250. Calculation

+; [; P(t)+; m(t)] The Edgeworth expansion is an asymptotic expansion in terms of derivatives of the normal distribution function. It is often possible to transform a random variable so that the distribution of the transformed random variable more closely approximates the normal distribution function than does the distribution of the original random variable. Hence for the same number of terms, greater accuracy may be achieved by using the transformed variable in the expansion. Since the distribution of 42x2 is more closely approximated by a normal distribution than x2 itself (as judged by a comparison of the values of y1 and y2), we would expect that the Edgeworth asymptotic expansion of 42x2 would be superior to that of x2. Method b. Expansion for @. Let

of the

Bivariate Function

Normal

Probability

Example 8. Let X and Y follow a bivariate normal distribution with parameters m,=3, mv=2, a,=4 , uv=2, and p= -.125. Find the value of P,{X>2, Y>4} using 26.3.20 and Figures 26.2, 26.3. SinceP,{X>h,Y>k}=L

have P{X>2,

Y24j

( =L(-.25,

‘*,F,p)

we

1, -.125).

Using

26.3.20 L(-.25,1,

-.125)=L(-.25,0,

.969) +L(l,

0, .125)-i

Figure 26.2 only gives values for h>O, however, using the relationship 26.3.8 with k=O, L( - h, 0, p) =$-L(h, 0, -p) and thus L(-.25,0, .969) =&L(.25,0, --.969). Therefore L(-.25,1, -.125) =-L(.25,0,-.969)+L(l,

0, .125)=-.01+.09=.08. l., -.125)=.080.

The answer to 3D is L(-.25,

956 Integral

PROBABILITY of

a Bivariate

Normal Polygon

Distribution

Over

FUNCTIONS a

For the following

two configurations

we define

Example 9. Let the random variables X and Y have a bivariate normal distribution with parameters m,=5, uZ=2, mv=9, uy=4, and p=.5. Find the probability that the point (X, Y) be inside the triangle whose vertices are A= (7, S), B=(9, l3), and C=(2,9). When obtaining the integral of a bivariate normal distribution over a polygon, it is first necessary to use 26.3.22 in order to transform the variates so that one deals with a circular normal distribution. The polygon in the region of the transformed variables is then divided into configurations such that the integral over any selected configuration can be easily obtained. Below are listed some of the most useful configurations. Y

(0 I ,b2)

(02

,b2

1

(al,

(a2

SKI

1

bt)

FIGURE

ss

m

FIGURE

26.5

FIGURE

26.6

az

SS

0 dx, Y, OM+=

0

$7(x, AAOB

26.8

Y, w~Y=v@,

k2)-W,

k,)

kz)+V(h,

k,)

arctan a 2* FIGURE

ssdx, AAOB

26.9

Y/, o>d~Y=v(h,

Using the circularizing transformation for our example results in

h

ss0 *I See 26.3.23

$7(x, 0

for

Y, w~Y=v(h,

definition

of V(h,

w

k).

26.3.22

PROBABILITY

The vertices of the triangle in the (s, t) coordinates become A=(&/4, -5/4), B=(&, -1) and (q-g).

These points

are plotted

957

FUNCTIONS

ss

AAOB

below.

From the figure it is seen that the desired probability is the sum of the probabilities that the point having the transformed variables as coordinates is inside the triangles AOB, AOC, and

BOC.

= i+L(1.31,

O,-.76)--L(O,

O,-.76)-;Q(l.31)]

-

0,~.14)--L(O,

O,-.14)-i&(1.31)]

afL(1.31,

=L(1.31,

O,-.76)--L(O,

--L(1.31,0,-.14)+L(0,

O,-.76) o,-.14) =.OO-.ll-.04+.23=.08

t,O)*dt=V(g,g)+V ssg(s,

AAOC

= ;+L(.14, L-

o,-.99)--L(O,

+[;+L(.14,0,--1)

o,--.99)-i

&(.14)]

Q(.14)]

--L(O, 0,-l)-;

=.01+.02=.03

t,O)dsdt=v ssg(s,

FIQURE 26.10

ABOC

For these three triangles we have h

AAOB

= [;+L(.48,

o,-.97)--L(O,

o,-.97)-k

&(.48)]

m4 +[;+L(.48,0,-.96)--L(O,O,--.96)-k

Q(.48)] =.05+.04=.09

Thus adding all parts, the probability that X and Yare in triangle ABCis =.08+.03+.09=.20. The answer to 3D is .21l.

ABOC From the graph it is seen that the probability over AOB may be found in the same manner as that over Figure 26.8, and over AOC and BOC the probabilities may be found as that over Figure

26.9.

Hence

!J@, ?A*5)de/=

ss

A

=

ss

AAOB

ss AABC

g(8, t, 0)dsdt

gb, t, oMdt+

g(s;t, 0)ddt SS

AAOC

Calculation

of a Circular Normal Offset Circle

26.2

Over

an

Example 10. Let X and Y have a circular normal distribution with u=lOOO. Find the probability that the point (X, Y) falls within a circle having a radius equal to 540 whose center is displaced 1210 from the mean of the circular normal distribution. In units of u, the radius and displacement from

the center are, respectively, 1210 =1000=1.21.

and consequently using 26.3.23 and Figure

Distribution

R=go=.54

and T

The problem is thus reduced to

finding the probability of X and Y falling in a circle of radius R=.54 displaced r= 1.21 from the center of the distribution where u=l.

958

PROBABILITY

Since R
2(.54)2

r2)=4+(.54)2

26.3.25 is used.

FUNCTIONS

For this example Ax*= &5X10m4 and %=25. results in

This

-2(1.21)2 exP 4+ (.54)2 as the possible error in Q(x21v).

The answer to 5D is .06870. Interpolation

for

Calculation Q(Xa 1Y)

Example 11. Find Q(25.298120) using the interpolation formula given with Table 26.7. Taking x2=25, 8=.298 and applying the interpolation formula results in

Q(25)16)82+Q(25)18)

(48-282)

+Q(25120)(8--4e+e2) 1 (.06982)(.088804) + (.12492) (1.014392) + (.20143) (6.896804) }

of Q($lv)

A&(x”/ Y)x ‘%$)

the

Range

of Table

26.7

Example 12. Find the value of Q(84172). Since this value is outside the range of Table 26.7 we can approximate &(84/72) by (1) using the Edgeworth expansion for Q(x21u) given in Example 6, (2) the cube root approximation 26.4.14, (3) the improved cube root approximation 26.4.15 or (4) the square root approximation 26.4.13. The results of using all four methods are presented below: 1. Edgeworth

expansion

The successive terms of the Edgeworth expansion for the distribution of chi-square result in l-&(84172)=.841345 .oooooo .001120

= .19027 A less accurate interpolate may be obtained by setting e2 equal to zero in the above formula. This results in the value .19003. The correct value to 6D is &(25.298120)=.190259. On the other hand if x2=25.298 is assumed to have an error of +5X 10d4, then how large an error arises in &(x2(y) ? Denoting the error in x2 by Ax* and the resulting error in Q(x21v) by A&(x”~v), we then have the approximate relationship

Outside

.842465 Hence &(84)72)=.15754. The successive terms of the Edgeworth sion for the distribution of v’@ result in l-&(84172)=.842544 - .000034 - .000138

expan-

.842372

Ax2 Hence Q(84172) =.15764.

Using 26.4.8 we can write

aQ(x21v) 1 [Q(x"lu-a>-Q(x'lu)] -=ija9 and

AQ(x'lu)-f [Q(x2(~-2)-Q(x21~)]~2 For practical purposes it is sufficient to evaluate the derivative to one or two significant figures. Consequently we can write

2. Cube

root

26.4.14

Using the cube root approximation &@4172)=

where

QC@ 0046 *

23 gw

This results R(1.0046)=.15754. The volves Linearly below -.0006

we have

,=(g~‘3 I1-6314 L-1

3. Improved

where xi is the closest value to x2 for which Q is tabulated. Hence

approximation

cube

in root

&(84172)=&(1.0046)=Japproximation

26.4.15

improved cube root approximation incalculating a correction factor h, to 2. interpolating for h,, (which appears 26.4.15) with x=1.0046 results in hao= and hence

PROBABILITY

959

FUNCTIONS

60

2. Cube

approximation

26.4.17

Taking

x.,,, =2.32635

we have

Thus &(84/72)=&(1.0046-.0005)=&(1.0041) =l-I’(1.0041)=.15766 4. Square

root

approximation

26.4.13

Using the square root approximation &(84/72)=Q(s) where

we have

cube

he,=.0012

&(84~72)=&(1.0032)=1--P(1.0032)=.15788 The value correct to 6D is &(84)72)=.157653. Generally the improved cube root approximation will be correct with a maximum error of a few units in the fifth decimal and is recommended for calculations which are outside the range of ~2

for

Q(xZlv) Outside Table 26.8

26.4.18

the

Range

x2=144

(.0012)=.00049

[

2 1-- g(144)

J 1

approximation

x2=;

3

2 g(144)

+(2.32635-.00049)

of

Example 13. Find the value of X* for which Q(x”l144)=.01. Since v=l44 is outside the range of Table 26.8, we can compute it by using (1) the Cornish-Fisher asymptotic expansion 26.2.50, for x2, (2) the cube approximation 26.4.17, (3) the improved cube approximation 26.4.18, or (4) the square approximation 26.4.16. We shall compute the value by all four methods.

and thus h,,,=g

Hence

4. Square

Table 26.7. of

approximation

From the table for h,, we obtain using linear interpolation with x=2.33 (approximately)

This results in

Calculation

3. Improved

-186.394

26.4.16

[2.32635+42(144)-1]*=185.616

The correct answer to 3D is x2=186.394. Generally the improved cube approximation will give results correct in the second or third decimal for v>30. Calculation

Example

of the

Incomplete

Gamma

Function

Find the value of

14.

9

t’.5e-‘dt

-y(2.5,.9)= s 0

1. Cornish-Fisher

x2

asymptotic

expansion

26.2.50

The Cornish-Fisher asymptotic expansion with v=144 can be written as

for

making use of 26.4.19 and Table 26.7. Using 26.4.19 we have y(2.5,.9)=I’(2.5)f’(1.8/5j=I’(2.5)[1-Q(1.815)] y(2.5,.9)=$1-.87607]=.16475

+s

8

[6h,(z)+3h,2(z)+2h,,,(2)1+

@123 [30h,(z) Example

+9~,,(z)+12h,3(r)+6h1,*(2)+4h1~1,(2)1

Hence using the auxiliary table following with p=.Ol we have 144.0000 39.4794 2.9413 -. 0242 -. 0019 +. 0002 x2= 186.395 *see page II.

Poisson

15.

26.2.51

Distribution

Find the value of m for which

y$ e-n z&99 using 26.4.21 and Table 26.8. From Table 26.8 with v=2c=8 and Q=.99 we have x2=1.646482. Hence m=x2/2=.823241. Inverse

Incomplete

Beta

Function

16. Find the value of z for which S)=.lO using Table 26.9 and 26.5.28. * 26.5.28 we have

Example

I,(lO, Using

of the

960

PROBABILITY

FUNCTIONS

where z=2&F

Iz(10,6)=&(F~12,20)=.10

From Table 26.9 the upper 10 percent point of F with 12 and 20 degrees of freedom is F=1.89. Hence 20

Y=

3[(1.8469)(.98942)-(1.8566)(.99306)]=-.0668 (1.8469)2+(1.8566)2 4 10.5 16 [

1

and interpolating

“=20+12(1.89)=‘46g

P(-.0668)=1-P(.O668)=.47336

The correct value to 4D is x=.4683. Calculation

of IJo,

b) for

The answer correct to 5D is I.so(16, 10.5)=.47332.

a or b Small

Integers

Calculate 1.,,(3, 20). Values of I,(a, b) for small integral a or b can conveniently be calculated using 26.5.6 or 26.5.7. Using 26.5.6 we have Example

(.11O39OX1O-2)=.62OO4O

Binomial

Example

18.

Distribution

Find the value of p which satisfies

cc>

p=l-p

using 26.5.24 and Table 26.9. Combining 26.5.24 and 26.5.28 we have

n n a=0 co s P??‘-~=Q(F(~,, ~2) where vl=2(n---a+l),

Example

20.

for

v2=2(u), andp=

Fin

Table

26.9

Find the value of F for which

Q(Fj7, 20) =.05 using Table 26.9. Interpolation in Table 26.9 is approximately linear when the reciprocals of the degrees of freedom (vl, v2) are used as the interpolating variable. For this example it is only necessary to interpolate with respect to l/vi. Thus linear interpolation on l/v1 results in F=2.51 which is the correct interpolate. Calculation

2o 50 p’qW--“=.95, s=o s *

Interpolation

17.

.121576 =.21645OX1O-3

in Table 26.1 gives

of

F for

Q(F[v1,vJ>.50

Find the value of F for which using 26.6.9 and Table 26.9. Table 26.9 only tabulates values of F for which Q(FIv,, v2)=p where p=.500, .250, .lOO, ,050, .025, .OlO, .005, .OOl. However making, use of Table 26.9 we can find the values of Fp for which p=.75, .9, .95, .975, .99, .995, .999. For this example we have Example

21.

Q(F14,8) =.90

a a+@--a+l)F

1

F.80(4'8)=F,10(8,4)

Hence

m 50 cc >p’p~-%+~ 0 50

r-0

s

a=21

s

and referring to the table for which Q(F(vl, v2)=.lO

Pa!Zso-”

=l-

gives

Q(F(60,42)=.95

Harmonic interpolation on v2 in the table for which Q(FI vl, v2)= .05 results in F= 1.624 for 42 vl=60, v2=42, and thus p= 42+60(1.624)=*301’

The correct answer to 4D is p= .3003: Approximating

the

Incomplete

Beta

Function

Example 19. Find I,so(16, 10.5) using 26.5.21. Values of Z(a, b) can conveniently be calculated with good accuracy using the approximation given by 26.5.20 or 26.5.21. For this example (atb-l)(l-x)=10.20 which is greater than .8 and hence 26.5.21 will be used. Thus

F.,,(8,4)=3.95

and

thus

P,(4,8)=&

=.253. Calculation

of Q(Fjv~,vz)

for

Small

Integral

VI or vz

Example 22. Compute Q(2.514, 15) using 26.6.4. Values of Q(FIv,, v2) can be readily computed for small v1 or v2 using the expansions 26.6.4 to 26.6.8 inclusive. We have using 26.6.4

15 2=15+4(2.5O)='6o and Q(2.50j4,15)=(.6)7~5[1+$

(.4)]=.086

735

PROBABILITY Approximating

Example 26.6.15.

23.

Q(F(v1,

Calculate

~2)

using

&(1.714/10,40)

961

FUNCTIONS

h=2

The approximation given by 26.6.15 will result in a maximum error of .0005. For this example we have (1.714)“3 (l-g@

>-ha=,

2= &+(1.714)z/3 C

Interpolating

2222

&-)I”

.

-(f-ii) [

On the other hand the approximation given by 26.6.14 which is usually less accurate results in

and interpolating

U.714,-~=l

1+g

2210

(1.714)

in Table 26.1 gives

of F Outside

the

Range

and thus F=e2”=7.23.

The correct

the Non-Central

Approximating

1

answer is

F-Distribution

Example 25. Compute P(3.7113, 10,4) using the approximation 26.6.27 to the non-central Fdistribution. Using 26.6.27 with v1=3, vz=lO, x=4, F’=3.71 we have x=

of Table

[(&)

we have a=;=lO,

(3.7q3[1-&]-[1-;$$$]

26.9

Example 24. Find the value of F for which Q(FjlO,20) G .OOOl using 26.6.16 and 26.5.22.

For this problem

2 1.8052+.8333-3~,2~2143~

F=7.180.

&(1.714110,40) =Q(1.2210)=1--P(1.2210)=.1112 Calculation

8052 ’

w=.9889

The correct value to 5D is &(1.714)10,40)=.11108.

J

’ ’ +=12.2143 B+g

w=3 71go (12.2143+1.8052)* 12.2143

&(1.714~10,40)=&(1.2222)=1-P(1.2222)=.1108

(;)

sub-

[ 1

X=3*71go2-3=1 6

in Table 26.1 results in

x~+@Wl -

Hence

y=3.7190 (i.e., &(3.7190)=.0001). stituting in 26.5.22 gives

b=:=5,

p=.OOOl. Th e value of the normal deviate which cuts off .OOOl in the tail of the distribution is

[

5 g&+&J

[(&)

(3.7lJl”‘]t = .675

and interpolating

in Table 26.1 gives

P(3.71(3,10,4)

@(.675)=.750

The exact answer is P(3.7113,10,4)

=.745.

References Texts

(26.11 H. Cramer, Mathematical methods of statistics (Princeton Univ. Press, Princeton, N.J., 1951). [26.2] A. Erdelyi et al., Higher transcendental functions, ~01s. I, II, III. (McGraw-Hill Book Co., Inc., New York, N.Y., 1955). [26.3] W. Feller, Probability theory and its applications, 2d ed. (John Wiley & Sons, Inc., New York,

N.Y., 1957). (26.41 R.

A. Fisher, Contributions to mathematical statistics, Paper 30 (with E. A. Cornish), Moments and cumulants in the specification of distributions (John Wiley&Sons, Inc., New York, N.Y., 1950). [26.5] C. Hastings, Jr., Approximations for digital computers (Princeton Univ. Press, Princeton, N.J., 1955). [26.6] M. G. Kendall and A. Stuart, The advanced theory of statistics, vol. I, Distribution theory (Charles Griffin and Co. Ltd., London, England, 1958).

Tables

[26.7] R. A. Fisher and F. Yates, Statistical tables for biological, agricultural and medical research (Oliver and Boyd, London, England, 1949). [26.8] J. Arthur Greenwood and H. 0. Hartley, Guide to tables in mathematical statistics (Princeton Univ. Press, Princeton, N.J., 1962). (Catalogues a large selection of tables used in mathematical statistics). [26.9] A. Hald, Statistical tables and formulas (John Wiley & Sons, Inc., New York, N.Y., 1952). [26.10] D. B. Owen, Handbook of statistical tables (Addison-Wesley Publishing Co., Inc., Reading, Mass., 1962). [26.11] E. S. Pearson and H. 0. Hartley (Editors). Biometrika tables for statisticians, vol. I (Cambridge Univ. Press, Cambridge, England, 1954).

962 [26.12]

PROBABILITY K.

Pearson (Editor), Tables biometricians, parts I and Press, Cambridge, England, Normal

[26.13]

[26.14]

[26.15]

[26.16]

[26.17]

[26.18]

Probability

[26.20]

126.211

[26.22]

end

and Univ.

Normal

Probability

(1956). a=.25(.25)1,

h=0(.25)3; 3(.05)3.5(.1)4.7,

u)=$

arctan

h=0(.01)2(.02)3; a=.l,

.2(.05).5(.1).8, 6D.

B. Owen, The bivariate normal probability function, Office of Technical Services, U.S. Department of Commerce (1957). T(h, a) =

[26.24]

Non-Central Gamma

Chi-Square, Incomplete

[26.25]

a-V@,

ah) for

a=O(.ol)l, 1, o),

UJ, h=

a-V(h,

arctan

G.

Function,

m;

a=0(.025)1,

6D. Part II of [26.12]. L(h, p=-1(.05)1, 6D for

Chi-Square, Poisson

Probability Distribution

h=

k, p) p>O

Integral,

A. Campbell, Probability curves showing Poisson’s exponential summation, Bell System Technical Journal, 95-l 13 (1923). Tabulates values 2D;

of

wz=:

.OOOl,

.Ol,

3D;

3D;

.999999,

.99, .9999, [26.26]

ah) for

0(.01)3.5(.05)4.75, Tables VIII and IX, for h, k=0(.1)2.6, and 7D for p
for

which .I,

Q(x2]~)=.000001,

.25,

.5,

.75,

2D for c=i=

Table IV of [26.7]. Tabulates Q(xalu) = .OOl, .Ol, .02, .05, .9, .95, .98, .99 and ~=1(1)30,

.I,

.9,

4D;

l(l)lOl.

values of xz for .2, .3, .5, .7, .8, 3D or 35.

[26.27]

E. Fix, Tables of noncentral xa, Univ. of California Publications in Statistics 1, 15-19 (1949). Tabulates X for P(x’z]~, X) =.1(.1).9, Q(x’~]v)= .Ol, .05; ~=1(1)20(2)40(5)60(10)100, 3D or 35.

[26.28]

H. 0. Hartley and E. S. Pearson, Tables of the x2 integral and of the cumulative Poisson distribution, Biometrika 37, 313-325 (1950). Also reproduced as Table 7’ in [26.11]. P(x2]u) for v=1(1)20(2)70, x2=0(.001).01(.01).1(.1)2(.2)10 (.5)20(1)40(2)134, 5D.

[26.29]

T.

[26.30]

E. C. Molina,

Integral

Tables for computing bivariate normal Ann. Math. Statist. 27, 1075-1090

T(h,

D.

$

Bell Aircraft Corporation, Table of circular normal probabilities, Report No. 02-949-106 (1956). Tabulates the integral of the circular normal distribution over an off-set circle having its center a distance T from the origin with radius R; R=0(.01)4.59, r=0(.01)3, 5D. National Bureau of Standards, Tables of the bivariate normal distribution function and related functions, Applied Math. Series 50 (U.S. Government Printing Office, Washington, D.C., 1959). L(h, k, p) for h, k=0(.1)4, p=O(.O5).95 (.Ol)l, 61); L(h, k, -p) for h, k=O(.l)A, p=0(.05).95(.01)1whereAissuchthatL<.5.10-7, 7D; V(h, ah) for h=O(.Ol)4(.02)4.6(.1)5.6, a, 7D; V(ah, h) for a=.l(.l)l, h=0(.01)4(.02)5.6, a, 7D. C. Nicholson, The probability integral for two variables, Biometrika 33, 59-72 (1943). V(h, ah) for h=.1(.1)3, ah=.1(.1)3, 03, 6D. D. B. Owen, probabilities,

[26.23]

Derivatives

J. R. Airey, Table of Hh functions, British Association for the Advancement of Science, Mathematical Tables I (Cambridge Univ. Press, Cambridge, England, 1931). Harvard University, Tables of the error function and of its first twenty derivatives (Harvard 1952). Univ. Press, Cambridge, Mass., P(z)-+, Z(r), Z(“)(z), n=1(1)4 for 2=0(.004) 6.468, 6D; Z(n)(z), n=5(1)10 for 2=0(.004) 8.236, 6D; Zen)(r), 72=11(1)15 for 2=0(.002) 9.61, 7s; Z(n)(z), n=16(1)20 for 2=0(.002) 10.902, 75 or 6D. T. L. Kelley, The Kelley Statistical Tables (Harvard Univ. Press, Cambridge, Mass., 1948). z for P(x) = .5(.0001) .9999 and corresponding values of Z(z), 8D. National Bureau of Standards, A guide to tables of the normal probability integral, Applied Math. Series 21 (U.S. Government Printing Office, Washington, D.C., 1951). National Bureau of Standards, Tables of normal probability functions, Applied Math. Series 23 (U.S. Government Printing Office, Washington, D.C., 1953). Z(z) and A(z) for s=O(.OOOl) 1(.001)7.8, 15D; Z(z) and 2[1-P(s)] for z=6(.01)10, 7s. W. F. Sheppard, The probability integral, British Association for the Advancement of Science, Mathematical Tables VII (Cambridge Univ. Press, Cambridge, England, 1939). A (x)/Z(z) for 2=0(.01)10, 12D; z=O(.l)lO, 24D. Bivsriate

[26.19]

Integral

for statisticians II (Cambridge 1914, 1931).

FUNCTIONS

Kitagawa, Tables of Poisson distribution (Baifukan, Tokyo, Japan, 1951). e+m*/a! for m=.001(.001)1(.01)5, 80; m=5(.01)10, 7D. (D. 1940).

Van

Poisson’s Nostrand

e-mm*/s!

m=x2/2=0(.1)16(1)100, (.01)3, 7D. [26.31]

[26.32]

exponential Co.,

and

Inc., P(xz]v)=e 6D;

New j=e

binomial

limit

York,

N.Y.,

e-mmi/j!

for

m=O(.OOl).Ol

K.

Pearson (Editor), Tables of the incomplete p-function, Biometrika Office, University College (Cambridge Univ. Press, Cambridge, England, for p=-1(.05)0(.1)5(.2)50, 1934). Z(%P) u=O(.l) Z(u,p)=l to 7D; p=-l(.Ol)-,.75, u=O(.1)6, 5D; ln[Z(u,p)]up+*], p= - 1(.05)0 (.l)lO, u=O(.1)1.5, 8D; [2”?’ r(p+l)l-‘r(p,z), p=-l(.Ol)-.9,z=O(.O1)3, 7D. E. E. Sluckii, Tablitsy dlya. vy%oleniya nepolnoI p-funktsii i funktsii veroyatnosti x2. (Izdat. Akad. Nauk SSSR, Moscow-Leningrad, U.S.S.R., 1950).

l?(x2,v)=

; x2 -‘I2 P @Iv), 9 (&Y) = ( > Q(x+), II@, 2) =Q(x~]v) where 1=(2x2)*(2r)*, ~=(~/2)-+. r(x*,v), x~=o(.05)2(.1)10, y=o(.05) 2(.1)6; Q(xz]v), x2=0(.1)3.2, v=O(.O5)2(.1)6; xz=3.2(..2)7(.5)10(1)35, v=O(.1).4(.2)6; g(t,~), t=-4(.1)4.8, ~=6(.5)11(1)32; Il(t,z): t=-4.5 (.1)4.8, r=O(.02).22(.01).25, 5D.

PROBABILITY Incomplete

Beta

Function,

Binomial

Distribution

[26.33] Harvard University, Tables of the cumulative binomial probability distribution (Harvard Univ. Press, Cambridge, Mass., 1955). gc (:)p*(l-pp)n-*

for p=.O1(.01).5,

l/16, l/12,

l/8, l/6, 3116, 5/16, l/3, 3/S, 5112, 7116, n=1(1)50(2)100(10)200(20)500(50)1000, 5D. [26.34] National Bureau of Standards, Tables of the binomial probability distribution, Applied Math. Series 6 (U.S. Government Printing Office, Washington, D.C., 1950). (:)pa(l-pp)“-e and n a for p-.01(.01).5, n=2(1)49, c C)pa(l-P)n8=C 7D. [26.35] K. Pearson (Editor), Tables of the incomplete beta function, Biometrika Office, University College (Cambridge Univ. Press, Cambridge, Z,(a,b) for z=.Ol(.Ol)l; England, 1948). a,b=.5(.5)11(1)50, &b, 7D. 126.361 W. H. Robertson, Tables of the binomial distribution function for small values of p, Office of Technical Services, U.S. Department of Commerce (1960). go (Jn)~*(l-pp)~-*

for p=.OO1(.001).02,

n=2(1)

p=.O21(.001).05, 100(2)200(10)500(20)1000; n=2(1)50(2)100(5)200(10)300(20)600(50) 1000, 5D. [26.37] H. G. Romig, 50-100 Binomial tables (John Wiley & Sons, Inc., New York, N.Y., 1953).

0 r

P*(~-P)“-*

and

2

z

p*(l-p)n-’

for

0

p=.O1(.01).5 and n=50(5)100, 6D. [26.38] C. M. Thompson, Tables of percentage points of the incomplete beta function, Biometrika 32, 151-181 (1941). Also reproduced as Table 16 in [26.11]. Tabulates values of z for which * I&, b) =.005, .Ol, .025, .05, .l, .25, .5; 2a=1(1)30, 40, 60, 120, 0~; 2b=l(l)lO, 12, 15, 20, 24, 30, 40, 60, 120, 5D. [26.39] U.S. Ordnance Corps, Tables of the cumulative binomial *probabilities, ORDP 20-1, Office of Technical Services, Washington, D.C. (1952). n n p’(l-p)“-’ for p=.O1(.01).5 and n=l 8-c co s (1)150, 7D. F (Variance-Ratio)

126.401 Table Z=i

end

V of [26.7].

Non-Central

Tabulates

In F for Q(Flv,, &=.2,

F Distribution

values

of F and

.l, .05, .Ol, .OOl;

~i=1(1)6, 8, 12, 24, m; vz=1(1)30, 40, 60, 120, a, 2D for F, 4D for 2. [26.41] E. Lehmer, Inverse tables of probabilities of errors of the second kind, Ann. Math. Statist. 15, 388-398 (1944). 4=&(vi+l) for ~i=l(l)lO, 12, 15, 20, 24, 30,.40, 60, 120, 0~; vz=2(2)20, 24, 30, 40, 60, 80, 120, 240, m and P(F’jui, sat 4) = .2, .3 where Q(F’]ri, ~1) = .Ol, .05, 3D or 3s.

963

FUNCTIONS

[26.42] M. Merrington and C. M. Thompson, Tables of percentage points of the inverted beta (F) distribution, Biometrika 33, 73-88 (1943). Tabulates values of F for which Q(Flv,, ~~)=.5, .25, .l, .05, ,025, .Ol, .005; v,=l(l)lO, 12, 15, 20, 24, 30, 40, 60, 120, 0~ ; vz= 1(1)30, 40, 60, 120, a~. [ 26.431 P. C. Tang, The power function of the analysis of variance tests with tables and illustrations of their use, Stat,istical Research Memoirs II, 126149 and tables (1938). P(F’jq, YZ, +) for Y,= 1(1)8, ,,=2(2)6(1)30, 60, ~0 and +=dG= 1(.5)3(1)8 where Q(F’]v,, ~2)= .Ol, .05, 3D. Student’s

t and

Non-Central

t-Distributions

[ 26.441 E. T. Federighi, Extended tables of the percentage points of Student’s t-distribution, J. Amer. Statist. Assoc. 54, 683-688 (1959.) Values of t for which

Q(tlv)=

i [I-A((Ijv)]=.25XlO-“,

.1X lo-“, n=0(1)6, .05X lo-“, n=0(1)5, Y= 1(1)30(5)60(10)100, 200, 500, 1000, 2000, 10000, a; 3D. [26.45] Table III of [26.7]. Values of t for which A(t]v) = .1(.1).9, .95, .98, .99, .999 and r=1(1)30, 40, 60, 120, ~0; 3D. (26.461 N. L. Johnson and B. L. Welch, Applications of the noncentral t-distribution, Biometrika 31, 362-389 (1939). Tabulates an auxiliary funotion which enables calculation of 8 for given t’andp,ort’ for given 6 and p where P(t’jv,b) =p= .005, .Ol, .025, .05, .1(.1).9, .95, .975, .99, .995. [26.4?,] J. Neyman and B. Tokarska, Errors of the second kind in testing Student’s hypothesis, J. Amer. Statist. Assoc. 31, 318-326 (1936). Tabulates 6 for P(t’lv,@=.Ol, .05, .1(.1).9; v=1(1)30, a; Q(t’Iv) = .Ol, .05. [26.48] Table

9 of [26.ll].P(tjv)=a

[l+A(l]v)]

for t=

0(.1)4(.2)8; v=1(1)20, 5D; t=0(.05)2(.1)4, 5; v=20(1)24, 30, 40, 60, 120, 00, 5D. [26.49] G. S. Resnikoff and G. J. Lieberman, Tables of the noncentral t-distribution (Stanford Univ. Press, Stanford, Calif., 1957). dP(t’Iv, s)/at’and P(t’I v,6) for v=2(1)24(5)49, 6= m z, where Q(z,)= p=.25, .15, .l, .065, .04, .025, .Ol, .004, .0025, .OOl and f/J Y covers the range of values such that throughout most of the table the entries lie between 0 and 1, 4D. Random

Numbem

and

Normal

Deviates

[26.50] E. C. Fieller, T. Lewis and E. S. Pearson, Correlated random normal deviates, Tracts for Computers 26 (Cambridge Univ. Press, Cambridge, England, 1955). [26.51] T. E. Hull and A. R. Dobell, Random number generators, Sot. Ind. App. Math. 4. 230-254 (1962). [26.52] M. G. Kendall and B. Babington Smith, Random sampling numbers (Cambridge Univ. Press, Cambridge, England, 1939).

964

PROBABILITY

[26.53] G. Marsaglia, Random variables and computers, Proc. Third Prague Conference in Probability Theory 1962. (Also as Math. Note No. 260, Boeing Scientific Research Laboratories, 1962). [26.54] M. E. Muller, An inverse method for the generation of random normal deviates on large scale computers, Math. Tables Aids Comp. 63. 167174 (1958).

FUNCTIONS

[26.55] Rand Corporation, A million 100,000 normal deviates Glencoe, Ill. 1955).

random digits with (The Free Press,

[26.56] H. Wold, Random normal deviates, Tracts for Computers 25 (Cambridge Univ. Press, Cambridge, England, 1948). I

966

PROBABILITY

NORMAL

Table 26.1

FUNCTIONS

PROBABILITY

FUNCTION

P(r)

.,’

AND DERIVATIVES

Z(x)

Z(l) (.1.) 0.00000 00000 00000

0.50000 0.50797 0.51595 0.52392 0.53188

00000 83137 34368 21826 13720

00000 16902 52831 54107 13988

0.39894 0.39886 0.39862 0.39822 0.39766

22804 24999 32542 48301 77055

01433 23666 04605 95607 11609

-0.00797 -0.01594 -0.02389 -0.03181

72499 49301 34898 34164

98473 68184 11736 40929

0.12 0.14 0.16 0.18

0.53982 0.54775 0.55567 0.56355 0.57142

78372 84260 00048 94628 37159

77029 20584 05907 91433 00901

0.39695 0.39608 0.39505 0.39386 0.39253

25474 02117 17408 83615 14831

77012 93656 34611 68541 20429

-0.03969 -0.04752 -0.05530 -0.06301 -0.07065

52547 96254 72437 89378 56669

47701 15239 16846 50967 61677

0.20 0.22 0.24 0.26 0.28

0.57925 Oi58706 0.59483 0.60256 0.61026

97094 44226 48716 81132 12475

39103 48215 97796 01761 55797

0.39104 0.38940 0.38761 0.38568 0.38360

26939 37588 66151 33691 62921

75456 33790 25014 91816 53479

-0.07820 -0.08566 -0.09302 -0.10027 -0.10740

85387 88269 79876 76759 97618

95091 43434 30003 89872 02974

0.30 0.32 0.34 0.36 0.38

0.61791 0.62551 0.63307 0.64057 0.64802

14221 58347 17360 64332 72924

88953 23320 36028 17991 24163

0.38138 0.37903 0.37653 0.37391 0.37115

78154 05261 71618 06053 38793

60524 52702 33254 73128 59466

-0.11441 -0.12128 -0.12802 -0.13460 -0.14103

63446 97683 26350 78179 84741

38157 68865 23306 34326 56597

0.40 0.42 0.44 0.46 0.48

0.65542 0.66275 0.67003 0.67724 0.68438

17416 72731 14463 18897 63034

10324 51751 39407 49653 83778

0.36827 0.36526 0.36213 0.35889 0.35553

01403 26726 48824 02910 25285

03323 22154 13092 33545 05997

-0.14730 -0.15341 -0.15933 -0.16508 -0.17065

80561 03225 93482 95338 56136

21329 01305 61761 75431 82879

0.50 0.52 0.54 0.56 0.58

0.69146 0.69846 0.70540 0.71226 0.71904

24612 82124 14837 02811 26911

74013 53034 84302 50973 01436

0.35206 0.34849 0.34481 0.34104 0.33717

53267 25127 80014 57886 99438

64299 58974 39333 30353 22381

-0.17603 -0.18121 -0.18620 -0.19098 -0.19556

26633 61066 17207 56416 43674

82150 34667 77240 32997 16981

0.60 0.62 0.64 0166 0.68

0.72574 0.73237 0.73891 0.74537 0.75174

68822 11065 37003 30853 77695

49927 31017 07139 28664 46430

0.33322 0.32918 0.32506 0.32086 0.31659

46028 39607 22640 38037 29077

91800 70765 84082 71172 10893

-0.19993 -0.20409 -0.20803 -0.21177 -0.21528

47617 40556 98490 01104 31772

35080 77874 13813 88974 43407

0.70 0.72 0.74 0.76 0.78

0.75803 0.76423 0.77035 0.77637 0.78230

63477 75022 00028 27075 45624

76927 20749 35210 62401 14267

0.31225 0.30785 0.30338 0.29887 0.29430

39333 12604 92837 24057 50297

66761 69853 56300 75953 88325

-0.21857 -0.22165 -0.22450 -0.22714 -0.22955

77533 29075 80699 30283 79232

56733 38294 79662 89724 34894

0.80 0.82 0.84 0.86 0.88

0.78814 0.79389 0.79954 0.80510 0.81057

46014 19464 58067 54787 03452

16604 14187 39551 48192 23288

0.28969 0.28503 0.28034 0.27561 0.27086

15527 63584 38108 82471 39717

61483 89007 39621 53457 98338

-0.23175 -0.23372 -0.23548 -0.23703 -0.23836

32422 98139 88011 16925 02951

09186 60986 05281 51973 82537

0.90 0.92 0.94 0.96 0.98

0.81593 0.82121 0.82639 0.83147 0.83645

98746 36203 12196 23925 69406

53241 85629 61376 33162 72308

0.26608 0.26128 0.25647 0.25164 0.24680

52498 63012 12944 43410 94905

98755 49553 25620 98117 67043

-0.23947 -0.24038 -0.24108 -0.24157 -0.24187

67249 33971 30167 85674 33007

08879 49589 60083 54192 55702

1.00

0.84134

47460

68543

0.24197

07245

19143

-0.24197

07245

19143

0.00

0.02 0.04 OiO6 0.08

0.10

[ 1 (q3

Z(,r) =$

T

e-v

P(x) =Jy m z(t)dt

Z(“)(x)

=g

2(X!)

Hen(x) = (- 1)wqx)

/Z(Lt$

PROBABILITY

NORMAL

PROBABILITY

ZW(x)

x 0.00 0.02 0.04 0.06 0.08

-0.39894 -0.39870 -0.39798 -0.39679 -0.39512

0.10 0.12 0.14 0.16 0.18

967

FUNCTIONS

FUNCTION

AND

DERIVATIVES

z(*)(x)

22804 29549 54570 12208 26322

Z(3)(x) 0.00000 0.02392 0.04780 0.07159 0.09523

000 856 928 445 664

1.19682 1.19563 1.19204 1.18607 1.17774

-0.39298 -0.39037 -0.38730 -0.38378 -0.37981

30220 66567 87267 53315 34631

0.11868 0.14190 0.16483 0.18744 0.20967

881 445 771 353 776

0.20 0.22 0.24 0.26 0.28

-0.37540 -0.37055 -0.36528 -0.35961 -0.35353

09862 66169 98981 11734 15588

0.23149 0.25286 0.27372 0.29405 0.31380

0.30 0.32 0.34 0.36 0.38

-0.34706 -0.34021 -0.33300 -0.32545 -0.31755

29121 78003 94659 17909 92592

0.40 0.42 E 0:48

-0.30934 -0.30083 -0.29202 -0.28294 -0.27361

0.50 0.52 0.54 0.56 0.58

Tal,lc* 20.1

Z(“)(x)

Z’“‘(X)

684 029 400 800 897

0.00000 -0.11962 -0.23891 -0.35754 -0.47516

000 684 887 249 649

-5.98413 -5.97575 -5.95066 -5.90893 -5.85073

421 893 325 742 151

1.16708 1.15410 1.13884 1.12136 1.10169

019 144 890 503 839

-0.59146 -0.70610 -0.81878 -0.92919 -1.03701

327 997 968 252 674

-5.77625 -5.68577 -5.57961 -5.45815 -5.32182

460 399 395 435 895

727 011 555 426 836

1.07990 1.05604 1.03017 ii06237 0.97272

350 063 556 941 834

-1.14196 -1.24376 -1.34214 -1.43683 -1.52759

980 938 434 568 737

-5.17112 -5.00657 -4.82876 -4.63831 -4.43591

356 387 317 979 441

0.33295 0.35144 0.36926 0.38637 0.40274

156 923 849 828 947

0.94130 0.90818 0.87347 0.83725 0.79963

327 965 711 919 298

-1.61419 -1.69641 -1.77405 -1.84692 -1.91485

723 762 617 643 840

-4.22225 -3.99809 -3.76420 -3.52140 -3.27051

716 459 646 244 871

69179 03372 55692 91055 78339

0.41835 0.43316 0.44716 0.46033 0.47264

488 939 995 566 779

0.76069 0.72055 0.67932 0.63709 0.59398

880 987 193 291 256

-1.97769 -2.03531 -2.08758 -2.13440 -2.17570

904 269 144 537 278

-3.01241 -2.74796 -2147807 -2.20363 -1.92557

439 802 382 810 548

-0.26404 -0.25426 -0.24426 -0.23409 -0.22375

89951 01373 90722 38293 26107

0.48408 0.49464 0.50430 0.51306 0.52090

982 748 874 383 525

0.55010 0.50556 0.46048 0.41496 0.36913

207 372 050 574 279

-2.21141 -2.24148 -2.26589 -2.28463 -2.29771

033 307 443 613 801

-1.64480 -1.36224 -1iO7881 -0.79543 -0.51298

520 740 949 249 749

0.60 0.62 0.64 0.66 0.68

-0.21326 -0.20264 -0.19191 -0.18109 -0.17020

37459 56463 67607 55308 03472

0.52782 0.53382 0.53890 0.54306 0.54630

777 841 643 327 259

0.32309 0.27696 0.23085 0.18486 0.13911

457 332 017 483 528

-2.30516 -2.30703 -2.30336 -2.29426 -2.27980

783 091 981 388 875

-0.23237 +0.04554 Oi31990 0.58988 0.85469

218 255 581 999 355

8.:; 0:74 0.76 0.78

-0.15924 -0.14826 -0.13725 -0.12624 -0.11524

95060 11670 33120 37042 98497

0.54863 0.55005 0.55058 0.55023 0.54901

016 386 359 127 073

0.09370 0.04874 +o.o043TI -0.03944 -0.08247

741 473 808 465 882

-2.26011 -2.23531 -2.20553 -2.17094 -2.13170

583 162 714 715 944

1.11354 1.36570 1.61045 1.84714 2.07512

405 074 709 311 746

0.80 0.82 0.84 0.86 0.88

-0.10428 -0.09337 -0.08253 -0.07177 -0.06110

89590 79110 32179 09916 69120

0.54693 0.54402 0.54030 0.53578 0.53049

765 952 551 644 467

-0.12468 -0.16597 -0.20625 -0.24546 -0.28351

324 047 697 336 458

-2.08800 -2.04002 -1.98796 -1.93204 -1.87248

401 228 617 726 587

2.29381 2.50267 2.70117 2.88887 3.06536

943 061 643 745 044

0.90 0.92 0.94 0.96 0.98

-0.05055 -0.04013 -0.02985 -0.01972 -0.00977

61975 35759 32587 89163 36558

0.52445 0.51768 0.51022 0.50209 0.49332

403 968 310 689 478

-0.32034 -0.35587 -0.39005 -0.42282 -0.45413

003 378 463 627 732

-1.80951 -1.74335 -1.67426 -1.60247 -l.52824

008 486 103 436 456

3.23025 3.38325 3.52407 3.65250 3.76836

923 538 854 673 628

0.48394 145

-0.48394

145

-1.45182

435

3.87153 159

1.00

PROBABILITY

968 Table

26.1

l.“oO 1.02

NORMAL

PROBABILITY

P(x) 0.84134 47460 68543

FUNCTIONS FUNCTIOX

.\ND

DERIVATIVES

Z(I) (2)

Z(x) 0.24197 0.23713 0.23229 0.22746 0.22265

07245 19520 70047 96324 34987

19143 19380 43366 57386 51761

-0.24197 -0.24187 -0.24158 -0.24111 -0.24046

07245 45910 88849 78104 57786

19143 59767 33101 04829 51902

73947 74443 08806 28104 84808

35806 88804 53704 17281 76986

1.04 1.06 1.08

0.84613 0.85083 0.85542 0.85992

57696 00496 77003 89099

27265 69019 36091 11231

1.10 1.12 1.14 1.16 1.18

0.86433 0.86864 0.87285 0.87697 0.88099

39390 31189 68494 55969 98925

53618 57270 37202 48657 44800

0.21785 21770 32551 0.21306 91467 75718 0.20830 Oi20357 0.19886

77900 13882 31193

47108 90759 87276

-0.23963 -0.23863 -0.23747 -0.23614 -0.23465

1.20 1.22 1.24 1.26 1.28

0.88493 0.88876 0.89251 0.89616 0.89972

03297 75625 23029 53188 74320

78292 52166 25413 78700 45558

0.19418 0.18954 0.18493 0.18037 0.17584

60549 31580 72809 11632 74302

83213 91640 63305 27080 97662

-0.23302 -0.23124 -0.22932 -0.22726 -0.22508

32659 26528 22283 76656 47107

79856 71801 94499 66121 81008

1.30 1.32 1.34 1.36 1.38

0.90319 0:906?8 0.90987 0.91308 0.91620

95154 24910 73275 50380 66775

14390 06528 35548 52915 84986

0.17136 0.16693 0.16255 0.15822 0.15394

85920 70417 50552 47903 82867

47807 41714 25534 70383 62634

-0.22277 -0.22035 -0.21782 -0.21518 -0.21244

91696 68950 37740 57149 86357

62150 99062 02216 03721 32434

1.40 1.42 1.44 1.46 1.48

0.91924 0.92219 0.92506 0.92785 0.93056

33407 61594 63004 49630 33766

66229 73454 65673 34106 66669

0.14972 0.14556 0.14145 0.13741 0.13343

74656 41300 99652 65392 53039

35745 37348 24839 82282 51002

-0.20961 -0.20670 -0.20370 -0.20062 -0.19748

84518 10646 23499 81473 42498

90043 53034 23768 52131 47483

1.50 1.52 1.54 1.56 1.58

0.93319 0.93574 0.93821 0.94062 0.94294

27987 45121 98232 00594 65667

31142 81064 88188 05207 62246

0.12951 0.12566 0.12187 0.11815 0.11450

75956 46367 75370 72950 48002

65892 89088 32402 59582 59292

-0.19427 -0.19101 -0.18769 -0.18432 -0.18091

63934 02479 14070 53802 75844

98838 19414 29899 92948 09682

1.60 1.62 1.64 1.66 1.68

0.94520 0.94738 0.94949 0.95154 0.95352

07083 38615 74165 27737 13421

00442 45748 25897 33277 36280

0.11092 0.10740 0.10396 0.10058 0.09728

08346 60751 10953 63684 22693

79456 13484 28764 27691 31467

-0.17747 -0.17399 -0.17049 -0.16697 -0.16343

33354 78416 61963 33715 42124

87129 83844 39173 89966 76865

1.70 1.72 1.74 1.76 1.78

0.95543 0.95728 0.95907 0.96079 0.96246

45372 37792 04910 60967 20196

41457 08671 21193 12518 51483

0.09404 0.09088 0.08779 0.08477 0.08182

90773 69790 60706 63613 77759

76887 16283 10906 08022 92143

-0.15988 -0.15632 -0.15276 -0.14920 -0.14565

34315 56039 51628 63959 34412

40708 08007 62976 02119 66014

1.80 1.82 1.84 1.86 1.88

0.96406 0196562 0.96711 0.96855 0.96994

96808 04975 58813 72370 59610

87074 54110 40836 19248 38800

0.07895 0.07614 0.07340 0.07074 0.06814

01583 32736 68125 03934 35661

00894 96207 81657 56983 01045

-0.14211 -0.13858 -0.13506 -0.13157 -0.12810

02849 07581 85351 71318 99042

41609 27097 50249 29989 69964

1.90 1.92 1.94 1.96 1.98

0.97128 0.97257 0.97381 0.97500 0.97614

34401 10502 01550 21048 82356

83998 96163 59548 51780 58492

0.06561 0.06315 0.06076 0.05844 0.05618

58147 65614 51689 09443 31419

74677 35199 54565 33451 03868

-0.12467 -0.12126 -0.11788 -0.11454 -0.11124

00480 05979 44277 42508 26209

71886 55581 71856 93565 69659

0.05399

09665 rc--6)91

13188

-0.10798

19330

26376

0.97724

Z(x) =&e-f+

[‘y’ 1 98680

51821

P(x) =$I

II 1 ‘3”

L‘ 10' J m Z(f)&

Z(n)(x) =g

Z(x)

He,(x)=(-l)"Z(")(s)/Z(z)

PROBABILITY

NORMAL

PROBABILITY

969

FUNCTIONS

FUNCTION

Table

AND DERIVATIVES

26.1

1.00 1.02 1.04 1.06 1.08

2 (2)(x) 0.00000 00000 0.00958 01309 0.01895 54356 0.02811 52466 0.03704 95422

0.48394 145 0.47397 745 0.46346 412 0.45243 346 0.44091 805

-0.48394 -0.51219 -0.53886 -0.56392 -0.58734

145 739 899 521 012

-1.45182 -1.37346 -1.29343 -1.21197 -1.12934

435 846 272 312 487

3.87153 159 3.96192 478 4.03951 497 4.10431 754 4.15639 308

1.10 1.12 1.14 1.16 1.18

0.04574 89572 0.05420 47909 0.06240 90139 0.07035 42718 0.07803 38880

0.42895 094 0.41656 552 0.40379 549 0.39067 467 0.37723 697

-0.60909 -0.62916 -0.64755 -0.66424 -0.67924

290 776 390 543 129

-1.04580 -0.96159 -0.87697 -0.79217 -0.70744

155 420 050 397 317

4.19584 622 4.22282 430 4.23751 585 4.24014 894 4.23098 941

1.20 1.22 1.24 1.26 1.28

0.08544 18642 0.09257 28784 0.09942 22822 0.10598 60955 0.11226 09995

0.36351 629 0.34954 639 0.33536 083 0.32099 285 0.30647 534

-0.69254 -0.70416 -0.71411 -0.72240 -0.72907

515 524 427 928 143

-0.62301 -0.53910 -0.45594 -0.37373 -0.29268

100 399 161 571 993

4.21033 894 4.17853 305 4.13593 896 4.08295 339 4.02000 029

1.30 1.32 1.34 1.36 1.38

0.11824 43285 0.12393 40598 0.12932 88019 0.13442 77819 0.13923 08305

0.29184 071 0.27712 083 0.26234 695 0.24754 965 0.23275 873

-0.73412 591 -0.73760.168 -0.73953 132 -0.73995 087 -0.73889 953

-0.21299 916 -0.13484 911 -0.05841 584 +0.01613 459 0.08864 645

3.94752 847 3.86600 921 3.77593 384 3.67781 128 3.57216 556

1.40 1.42 1.44 1.46 1.48

0.14373 83670 0.14795 13818 0.15187 14187 0.15550 05559 0.15884 13858

0.21800 319 0.20331 117 0.18870 986 0.17422 548 0.15988 325

-0.73641 -0.73255 -0.72735 -0.72087 -0.71315

957 600 645 087 137

0.15897 463 0.22698 486 0.29255 386 0.35556 954 0.41593 103

3.45953 335 3.34046 152 3.21550 469 3.08522 283 2.95017 891

1.50 1.52 1.54 1.56 1.58

0.16189 69946 0.16467 09400 0.16716 72298 0.16939 02982 0.17134 49831

0.14570 730 0.13172 067 0.11794 528 0.10440 190 0.09111 010

-0.70425 -0.69422 -0.68313 -0.67103 -0.65798

193 823 742 785 890

0.47354 871 0.52834 425 0.58025 051 0.62921 147 0.67518 208

2.81093 657 2.66805 791 2.52210 132 2.37361 937 2.22315 681

11% 1:64 1.66 1.68

0.17303 65021 0.17447 04284 0.17565 26667 0.17658 94284 0.17728 72076

0.07808 827 0.06535 359 0.05292 202 0.04080 829 0.02902 592

-0.64405 -0.62928 -0.61375 -0.59751 -0.58062

073 410 011 005 516

0.71812 810 0.75802 588 0.79486 211 0.82863 352 0.85934 661

2.07124 871 1.91841 857 1.76517 671 1.61201 862 1.45942 351

1.70 1.72 1.74 1.76 1.78

0.17775 27562 0.17799 30597 0.17801 53128 0.17782 68955 0.17743 53495

0.01758 718 +0.00650 315 -0.00421 632 -0.01456 254 -0.02452 804

-0.56315 -0.54516 -0.52670 -0.50785 -0.48864

647 459 954 061 614

0.88701 729 0.91167 051 0.93333 988 0.95206 725 0.96790 228

1.30785 296 1.15774 966 1.00953 633 0.86361 469 0.72036 463

1.80 1.82 1.84 1.86 1.88

0.17684 83546 0.17607 37061 0.17511 92921 0.17399 30717 0.17270 30539

-0.03410 -0.04329 -0.05208 -0.06047 -0.06846

647 263 243 285 193

-0.46915 -0.44942 -0.42952 -0.40949 -0.38940

342 853 621 971 073

0.98090 203 0.99113 045 0.99865 794 1.00356 087 1.00592 110

0.58014 345 0.44328 526 0.31010 045 0.18087 536 +0.05587 197

1.90 1.92 1.94 1.96 1.98

0.17125 72766 0.16966 37866 0.16793 06209 0.16606 57874 0.16407 72476

-0.07604 -0.08323 -0.09001 -0.09640 -0.10238

873 327 655 044 771

-0.36927 -0.34918 -0.32915 -0.30925 -0.28950

924 347 976 250 408

1.00582 548 1.00336 537 0.99863 613 0.99173 666 0.98276 891

-0.06467 -0.18054 -0.29155 -0.39754 -0.49836

2.00

0.16197 28995 C-i)4

-0.10798 193 C-i)7

-0.26995 483 ‘-32

0.97183 740 C-i)4

-0.59390 063

X

[ 1

2(3)(x)

.x(4) (2)

[ 1

-

[ 1

2 (5)

(x)

1 1

Z@)(x)

219 414 530 137 204

[c-y1

970

PROBABILITY

Table 26.1

NORMAL

FUNCTIONS

PROBABILITY

FUNCTION

AND DERIVATIVES

2.00 2.02 2.04 2.06 2.08

P(x) 0.97724 98680 51821 0.97830 83062 32353 0.97932 48371 33930 0.98030 07295 90623 019812372335 65062

Z(x) 0.05399 09665 13188 0.05186 35766 82821 0.04980 00877 35071 0.04779 95748 82077 0.04586 10762 71055

-0.10798 -0.10476 -0.10159 -0.09846 -0.09539

Z(1)(2) 19330 26376 44248 99298 21789 79544 71242 57079 10386 43794

2.10 2.12 2.14 2.16 2.18

0.98213 55794 37184 0.98299 69773 52367 0.98382 26166 27834 0.98461 36652 16075 0.98537 12692 24011

0.04398 35959 80427 0.04216 61069 61770 0.04040 75539 22860 0.03870 68561 47456 0.03706 29102 47806

-0.09236 -0.08939 -0.08647 -0.08360 -0.08079

55515 58897 21467 58953 21653 94921 68092 78504 71443 40218

2.20 2.22 2.24 2.26 2.28

0.98609 65524 86502 0.98679 06161 92744 0.98745 45385 64054 0.98808 93745 81453 0.98869 61557 61447

0.03547 45928 46231 0.03394 07631 82449 0.03246 02656 43697 0.03103 19322 15008 0.02965 45848 47341

-0.07804 -0.07534 -0.07271 -0.07013 -0.06761

41042 61709 84942 65037 09950 41882 21668 05919 24534 51938

2.30 2.32 2.34 2.36 2.38

0.98927 58899 78324 0.98982 95613 31281 0.99035 81300 54642 0.99086 25324 69428 0.99134 36809 74484

0.02832 70377 41601 0.02704 80995 46882 0.02581 65754 71588 0.02463 12693 06382 0.02349 09853 58201

-0.06515 -0.06275 -0.06041 -0.05812 -0.05590

21868 05683 15909 48766 07866 03515 97955 63063 85451 52519

2.40 2.42 2.44 2.46 2.48

0.99180 24640 75404 0.99223 97464 49447 0.99265 63690 44652 0.99305 31492 11376 0.99343 08808 64453

0.02239 45302 94843 0.02134 07148 99923 0.02032 83557 38226 0.01935 62767 31737 0.01842 33106 46862

-0.05374 -0.05164 -0.04960 -0.04761 -0.04568

68727 07623 45300 57813 11880 01271 64407 60073 98104 04218

2.50 2.52 2.54 2.56 2.58

0.99379 03346 74224 0.99413 22582 84668 0.99445 73765 56918 0.99476 63918 36444 Oi99505 99842 42230

0.01752 83004 93569 0.01667 01008 37381 0.01584 75790 25361 0.01505 96163 27377 0.01430 51089 94150

-0.04382 -0.04200 -0.04025 -0.03855 -0.03690

07512 33921 86541 10200 28507 24416 26177 98086 71812 04906

2.60 2.62 2.64 2.66 2.68

0.99533 88119 76281 0.99560 35116 51879 0.99585 46986 38964 0.99609 29674 25147 0.99631 88919 90825

0.01358 29692 33686 0.01289 21261 07895 0.01223 15263 51278 0.01160 01351 13703 0.01099 69366 29406

-0.03531 -0.03377 -0.03229 -0.03085 -0.02947

57200 07583 73704 02686 12295 67374 63594 02449 17901 66807

2.70 2.72 2.74 2.76 2.78

0.99653 30261 96960 0.99673 59041 84109 0.99692 80407 81350 0.99710 99319 23774 0.99728 20550 77299

0.01042 09348 14423 0.00987 11537 94751 0.00934 66383 67612 0.00884 64543 98237 0.00836 96891 54653

-0.02813 -0.02684 -0.02560 -0.02441 -0.02326

65239 98941 95383 21723 97891 27258 62141 39135 77358 49935

2.80 2.82 2.84 2.86 2.88

0.99744 48696 69572 0.99759 88175 25811 0.99774 43233 08458 0.99788 17949 59596 0.99801 16241 45106

0.00791 54515 82980 0.00748 28725 25781 0.00707 11048 86019 0.00667 93237 39203 0.00630 67263 96266

-0.02216 -0.02110 -0.02008 -0.01910 -0.01816

32644 32344 17005 22701 19378 76295 28658 94119 33720 21246

2.90 2.92 2.94 2.96 2.98

0.99813 41866 99616 0.99824 98430 71324 0.99835 89387 65843 0.99846 18047 88262 0.99855 87580 82660

0.00595 25324 19776 0.00561 59835 95991 0.00529 63438 65311 0.00499 28992 13612 0.00470 49575 26934

-0.01726 -0.01639 -0.01557 -0.01477 -0.01402

23440 17350 86721 00294 12509 64014 89816 72293 07734 30263

3.00

0.99865 01019 68370 (;i)5

0.00443 18484 11938

-0.01329 55452 35814 (y'7

x

[ 1

Z(x) =&e-w

P(x) =JI mZ(W

[ 1

Z(d(2) =g Z(x)

Hen(z)=(-l)"Z@'(r)/Z(z)

PROBABILITY

SORMAL X

PROBABILITY

Z(Z) (2)

971

FUNCTIONS

FUNCTION Z(3)(x)

AND

Table

DERIVATIVES

Z(4)(x)

26.1

Z@)(x)

Z(5)(x)

2.00 2.02 2.04 2.06 2.08

0.16197 0.15976 0.15744 0.15504 0.15255

28995 05616 79574 27011 22841

-0.10798 -0.11318 -0.11800 -0.12245 -0.12652

193 748 948 372 667

-0.26995 483 -0.25064 297 -0.23160 454 -0.21287 345 -0.19448 137

0.97183 0.95904 0.94451 0.92833 0.91062

740 873 117 417 795

-0.59390 063 -0.68406 360 -0.76878 007 -0.84800 114 -0.92169 927

2.10 2.12 2.14 2.16 2.18

0.14998 0.14734 0.14464 0.14188 0.13907

40623 52442 28800 38519 48644

-0.13023 -0.13358 -0.13659 -0.13925 -0.14158

543 762 143 550 892

-0.17645 779 -0.15882 997 -0.14162 297 -0.12485 967 -0.10856 076

0.89150 0.87107 0.84943 0.82671 0.80301

307 003 890 890 811

-0.98986 750 -1.05251 862 -1.10968 436 -1.16141 446 -1.20777 570

2.20 2.22 2.24 2.26 2.28

0.13622 0.13333 0.13041 0.12746 0.12450

24365 28941 23633 67648 18090

-0.14360 115 -0.14530 204 -0.14670 170 -0.14781 055 -0.14863 922

-0.09274 -0.07742 -0.06262 -0.04834 -0.03460

478 816 527 844 801

0.77844 0.75309 0.72708 0.70050 0.67346

311 866 743 969 314

-1.24885 097 -1.28473 823 -1.31554 947 -1.34140 971 -1.36245 589

2.30 2.32 2.34 2.36 2.38

0.12152 0.11853 0.11554 0.11255 0.10957

29919 55915 46652 50482 13521

-0.14919 851 -0.14949 939 -0.14955 294 -0.14937 032 -0.14896 273

-0.02141 -0.00876 +0.00331 0.01484 0.02581

241 819 989 882 724

0.64604 0.61833 0.59044 0.56243 0.53440

257 976 323 808 589

-1.37883 -1.39070 -1.39823 -1.40159 -1.40097

587 730 661 796 220

2.40 2.42 2.44 2.46 2.48

0.10659 0.10363 0.10069 0.09778 0.09488

79642 90478 85430 01675 74192

-0.14834 137 -0.14751 744 -0.14650 207 -0.14530 633 -0.14394 118

0.03622 0.04607 0.05536 0.06411 0.07231

539 505 942 307 187

0.50642 0.47856 0.45090 0.42350 0.39643

453 812 689 717 129

-1.39654 -1.38851 -1.37705 -1.36239 -1.34470

584 010 991 299 892

2.50 2.52 2.54 2.56 2.58

0.09202 0.08919 0.08639 0.08363 0.08091

35776 17075 46618 50852 54185

-0.14241 -0.14074 -0.13893 -0.13700 -0.13494

744 579 674 058 742

0.07997 0.08710 0.09371 0.09981 0.10541

287 428 533 624 808

0.36973 0.34348 0.31771 0.29247 0.26781

759 039 001 277 102

-1.32420 -1.30109 -1.27556 -1.24781 -1.21804

833 199 010 146 284

2.60 2.62 2.64 2.66 2.68

0.07823 0.07560 0.07301 0.07047 0.06798

79028 45843 73197 77809 74610

-0.13278 711 -0.13052 927 -0.12818 326 -0.12575 818 -0.12326 282

0.11053 0.11517 0.11935 0.12308 0.12638

277 293 186 341 196

0.24376 0.22036 0.19764 0.17563 0.15434

323 399 415 084 760

-1.18644 -1.15321 -1.11853 -1.08259 -1.04556

824 833 985 509 139

2.70 2.72 2.74 2.76 2.78

0.06554 0.06315 0.06082 0.05854 0.05631

76800 95904 41838 22966 46165

-0.12070 569 -0.11809 501 -0.11543 869 -0.11274 431 -0.11001 916

0.12926 0.13173 0.13382 0.13554 0.13690

232 965 945 741 942

0.13381 0.11404 0.09506 0.07686 0.05946

449 817 206 640 846

-1.00761 -0.96890 -0.92961 -0.88988 -0.84986

072 932 727 829 942

2.80 2.82 2.84 2.86 2.88

0.05414 0.05202 0.04996 0.04795 0.04600

16888 39229 15987 48727 37850

-0.10727 020 -0.10450 406 -0.10172 706 -0.09894 520 -0.09616 416

0.13793 0.13862 0.13902 0.13911 0.13894

149 969 007 867 142

0.04287 0.02708 +0.01209 -0.00209 -0.01549

262 053 127 857 465

-0.80970 -0.76951 -0.72943 -0.68959 -0.65008

080 553 954 143 248

2.90 2.92 2.94 2.96 2.98

0.04410 0.04226 0.04048 0.03875 0.03707

82652 81389 31340 28865 69473

-0.09338 928 -0.09062 562 -0.08787 791 -0.08515 058 -0.08244 776

0.13850 0.13782 0.13691 0.13578 0.13446

412 240 166 706 347

-0.02810 -0.03993 -0.05100 -0.06132 -0.07091

482 892 863 737 012

-0.61101 -0.57249 -0.53459 -0.49740 -0.46100

661 036 292 627 520

3.00

0.03545 47873

-0.07977 327 (-65)5

0.13295 545 (-;)7

[(-[)l1

P(-x)=1-P(z)

[ 1

-0.07977 327 (-4)~ 6 z(d(-2)=(-l)nZ@)(x)

[ 1

2-q-2) =2(x)

[ 1

-0.42545 745 (-;)7

[ 1

972

PROBABILITY Table 26.1

NORMAL

PROBABILITY

FUNCTIONS FUNCTION

AND DERIVATIVES

p (4

.J’

3.00 3.05 3.10 3.15 3.20

0.9986501020 0.9988557932 0.9990323968 0.9991836477 0.9993128621

3.25 3.30 3.35 3.40 3.45

0.9994229750 0.9995165759 0.9995959422 0.9996630707 0.9997197067

3.50 3.55 3.60 3.65 3.70

0.9997673709 0.9998073844 0.9998408914 0.9998688798 0.9998922003

3.75 3.80 3.85 3.90 3.95

0.9999115827 0.9999276520 0.9999409411 0.9999519037 0.9999609244

4.00 4.05 4.10 4.15 4.20

0.9999683288 0.9999743912 0.9999793425 0.9999833762 0.9999866543

4.25 4.30 4.35 4.40 4.45

0.9999893115 0.9999914601 0.9999931931 0.9999945875 0.9999957065

4.50 4.55 4.60 4.65 4.70

0.9999966023 0.9999973177 0.9999978875 0.9999983403 0.9999986992

4.75 4.80 4.85 4.90 4.95

0.9999989829 0.9999992067 0.9999993827 0.9999995208 0.9999996289

5.00

0.9999997133 (73

II

-3 -5.68447 -3.58207 75 -4.88674 -6.59440 -4.18954 82 05 52 62

(-6)1.48671 9515

(-6)-7.43359 76

[ 1

Table 26.2 NORMAL

PROBABlLITY

FUNCTION

FOR LARGE

ARGUMENTS

5

6.54265

15

-log Q (.I,) 50.43522

I56

-log Q(x) 137.51475

7"

11.89285 9.00586

16 17

64.38658 57.19458

27

160.13139 148.60624

9"

18.94746 15.20614

:9"

80.06919 72.01140

:i

184.48283 172.09024

10

23.11805

11 12 ::

:72%E 38:21345 44.10827

20 88.56010 I21 106.84167 97.48422

.,

-log Q (4

.I’

5

.7

126.85686 116.63253

From E. S. Pearsonand H. 0. Hartley (editors), Biometrika tablesfor statisticians, ~01. I. CambridgeUniv. Press,Cambridge,England, 1954(with permission).Known error has beencorrected.

PROBABILITY

NORMAL

5.00

(-5)3.56812 NORMAL 3

40 :: 43 44

68

PROBABILITY

(-4)-1.63539

PROBABILITY

-log Q(.c)

349.43701 367.03664 385.07032 403.53804 422.43983

973

FUNCTIONS

FUNCTION

15

(-4)7.10651

FUNCTION

z

20" 70 a0 90

AND

93

FOR LARGE

Table

DERIVATIVES

-3)-4.19931 -3)-3.49521

11 92

(-3)-2.89910

31

ARGUMENTS

-log Q(T)

544.96634 783.90743 1066.26576 1392.04459 1761.24604

.r

(-2)

26.1

1.09422 Table

-log Q(x)

100 150 200 250 300

2173.87154 4aaa.3aalz 8688.58977 13574.49960 19546.12790

350 400 450 500

26603.48018 34746.55970 43975.36860 54289.90830

r(+wi

L 9 J

Hc!.(z)=(-l)nZ(n)(1)/Z(.e)

P(-2)=1-P(s)

Z(-,)=2(c)

Z(~,(-~)=(-l)“Z(~)(r)

56

26.2

974 Table

PROBABILITY

26.3

.I’

HIGHER Z(T) (,I,)

DERIVATIVES

FUNCTIONS

OF THE

Z(W (x)

NORMAL

Z(S) (I)

PROBABILITY ZU~)(J~)

2w

ml)(a)

( 2)-3.77000 46 ( 2)-3.56488 94 (2 ( 2)-2.97583 41 (2 2)-2.07783 39 (3 l)-9.83608 69 (3

1.30711 60 ( 1) 1.58584 37

FUNCTION (.I’)

0.00000 00 4.05782 44 7.59641 48 1.01729 46 1.14847 09 (2)+6.22581 20

0.5 ( 1i 1.40908 65 ( 0)+4.46820 41 ( 2)-1.14961 02 0.6 ( 1 1.39704 30 ( 0I -6.75565 29 0.7 ( 1 1.27812 14 ( 1 -1.67416 58 E 1.0 I.1 1.2 1.3 1.4

0) 1) 7.94982 1.06929 69 72

1 -2.46111 -2.97666 11 59

2) 3.01027 69 (

( 0) 4.83941 45 l)-3.19401 36 ( 0)+1.65937 85 ( O)-1.31434 07 ( ( (

1.5 ( 01-7.05769 71 ( O)-9.09001 03 1.6 ( 0 -7.62276 66 ( O)-2.30231 44 1.7 ( O)-7.54545 38 ( 0)+3.67230 07 0) 8.41240 26 1) 1.16856 49 2.0 ( O)-4.64322 31 2.1 ( O)-3.27029 67 2.4 (-1)+3.13162 82

1) 1) 1 1 0)

2.94236 40 (2)-2.26484 60 2.57621 24 (2)-4.93791 72 2 1.98269 77 2 1.25293 01 i 1I +4.84200 76 ( ( ( ( (

1 1 1) 1 1

7.00965 92 6.46658 36 5.41207 19 4.02950 39 2.50938 72

I 21-6.65963 73 2 -5.14267 14 (2 I -3.28612 11 2 -1.36113 54 1)+3.94747 58

1.34437 51 1.37966 95 1.29729 67 1.12731 97 9.02423 01 ( l)-2.41634 55

(2) 2.97376 42

I 31 3 (3) (3) (3)

1.25562 83 1.73301 70 1.93425 58 1.87567 40 1.60633 92

( 0) 6.53922 01 ( ( 0) 4.08745 39 ( 0) 1.87558 77 4.58182 18 (l)-1.67928 3.0 3.1 3.2 3.3 3.4

( 0 ( 0 ( 0 (-1 (-1

1.75501 20 ( 0)-2.28683 38 ( 1.49720 05 ( O)-2.80440 64 1.20591 21 O)-2.96904 52 9.12450 33 O)-2.86200 69 6.39748 51 O)-2.56761 03

1I 1 1 1 ( 1)

25 (2)-4.55301 20

4.21202 87 3.54198 84 2.71897 33 1.86794 96 1.08280 77

79 Oj-1.71642 80 O)-2.16386 ( O)-1.27559 98 ( 0) 4.24743 76

1)+1.13637 65

(-1) 1.88517 13 (-1 1.63368 76 (-1 1.36227 87

-l)-2.70626

5.0 (-2)-3.73166 60 (-1) 1.09987 51 (-l)-2.51404

Z(.,) &-$2

Z(n) (.I.)=g

Z(.t*)

44

-l)-1.86696 14 -1)+1.00018 72 (0) 2.21617 27

27 (-1) 2.67133 76 (0) 1.17837 39 (O)-8.83034 08

~~~~,,(.,~)=(-l)“Z(~~)(.,~)/Z(.,~)

Z(d (-.I-) = (-1) ‘LZ(n) (J)

PROBABILITY

NORMAL

(,,(.r)

0.000

pR()RARILITY

0.001

FUNCTION-VALI:ES

0.002

0.003

OF Z(z)

0.004

975

FUNCTIONS

0.005

IN TERMS

0.006

OF

0.007

P(x)

AND

0.008

o(x)

0.009

Table

26.1

0.010

0.03 0.01 0.02 0.03 0.04

0.00000 0.02665 0.04842 0.06804 0.08617

0.00337 0.02896 0.05046 0.06992 0.08792

0.00634 0.03123 0.05249 0.07177 0.08965

0.00915 0.03348 0.05449 0.07362 0.09137

0.01185 0.03569 0.05648 0.07545 0.09309

0.01446 0.03787 0.05845 0.07727 0.09479

0.01700 0.04003 0.06040 0.07908 0.09648

0.01949 0.04216 0.06233 0.08087 0.09816

0.02192 0.04427 0.06425 0.08265 0.09983

0.02431 0.04635 0.06615 0.08442 0.10149

0.02665 0.04842 0.06804 0.08617 0.10314

0.99 0.98 0.97 0.96 0.95

0.05 0.06 0.07 0.08 0.09

0.10314 0.11912 0.13427 0.14867 0.16239

0.10478 0.12067 0.13574 0.15007 0.16373

0.10641 0.12222 0.13720 0.15146 0.16506

0.10803 0.12375 0.13866 0.15285 0.16639

0.10964 0.12528 0.14011 0.15423 0.16770

0.11124 0.12679 0.14156 0.15561 0.16902

0.11284 0.12830 0.14299 0.15699 0.17033

0.11442 0.12981 0.14442 0.15834 0.17163

0.11603 0.13133 0.14584 0.15970 0.17292

0.11756 0.13279 0.14726 0.16105 0.17421

0.11912 0.13427 0.14867 0.16239 0.17550

0.94 0.93 0.92 0.91 0.90

0.10 0.11 0.12 0.13 0.14

0.17550 0.18804 0.20004 0.21155 0.22258

0.17678 0.18926 0.20121 0.21267 0.22365

0.17805 0.19048 0.20238 0.21379 0.22473

0.17932 0.19169 0.20354 0.21490 0.22580

0.18057 0.19293 0.20470 0.21601 0.22686

0.18184 0.19410 0.20585 0.21712 0.22792

0.18309 0.19530 0.20700 0.21822 0.22898

0.18433 0.19649 0.20814 0.21932 0.23003

0.18557 0.19765 0.20928 0.22041 0.23108

0.18681 0.19886 0.21042 0.22149 0.23212

0.18804 0.20004 0.21155 0.22258 0.23316

0.89 0.88 0.87 0.86 0.85

0.15 0.16 0.17 0.18 0.19

0.23316 0.24331 0.25305 0.26240 0.27137

0.23419 0.24430 0.25401 0.26331 0.27224

0.23522 0.24529 0.25495 0.26422 0.27311

0.23625 0.24628 0.25590 0.26513 0.27398

0.23727 0.24726 0.25684 0.26603 0.27485

0.23829 0.24823 0.25778 0.26693 0.27571

0.23930 0.24921 0.25871 0.26782 0.27657

0.24031 0.25017 0.25964 0.26871 0.27742

0.24131 0.25114 0.26056 0.26960 0.27827

0.24232 0.25210 0.26148 0.27049 0.27912

0.24331 0.25305 0.26240 0.27137 0.27996

0.84 0.83 0.82 0.81 0.80

0.20 0.21 0.22 0.23 0.24

0.27996 0.28820 0.29609 0.30365 0.31087

0.28080 0.28901 0.29686 0.30439 0.31158

0.28164 0.28981 0.29763 0.30512 0.31228

0.28247 0.29060 0.29840 0.30585 0.31298

0.28330 0.29140 0.29916 0.30658 0.31367

0.28413 0.29219 0.29991 0.30730 0.31436

0.28495 0.29299 0.30067 0.30802 0.31505

0.28577 0.29376 0.30142 0.30874 0.31574

0.28658 0.29454 0.30216 0.30945 0.31642

0.28739 0.29532 0.30291 0.31016 0.31710

0.28820 0.29609 0.30365 0.31087 0.31778

0.79 0.78 0.77 0.76 0.75

i*f: 0:27 0.28 0.29

0 31778 0'32437 0:33065 0.33662 0.34230

0 31845 0'32501 0:33126 0.33720 0.34286

0 31912 0'32565 0:33187 0.33778 0.34341

0 31979 0'32628 0:33247 0.33836 0.34395

0 32045 0'32691 0:33307 0.33893 0.34449

0 32111 0'32754 0:33367 0.33950 0.34503

0 32177 0'32817 0:33427 0.34007 0.34557

0 32242 0'32879 0:33486 0.34063 0.34611

0 32307 0'32941 0:33545 0.34119 0.34664

0 32372 0'33003 0:33604 0.34175 0.34717

0 32437 0'33065 0:33662 0.34230 0.34769

0 74 0'73 0:72 0.71 0.70

0.30 0.31 0.32 0.33 0.34

0.34769 0.35279 0.35761 0.36215 0.36641

0.34822 0.35329 0.35808 0.36259 0.36682

0.34874 0.35378 0.35854 0.36302 0.36723

0.34925 0.35427 0.35900 0.36346 0.36764

0.34977 0.35475 0.35946 0.36389 0.36804

0.35028 0.35524 0.35991 0.36431 0.36844

0.35079 0.35572 0.36037 0.36474 0.36884

0.35129 0.35620 0.36082 0.36516 0.36923

0.35180 0.35667 0.36126 0.36558 0.36962

0.35230 0.35714 0.36171 0.36600 0.37001

0.35279 0.35761 0.36215 0.36641 0.37040

0.69 0.68 0.67 0.66 0.65

0.35 0.36 0.37 0.38 0.39

0.37040 0.37412 0.37757 0.38076 0.38368

0.37078 0.37447 0.37790 0.38106 0.38396

'0.37116 0.37483 0.37823 0.38136 0.38423

0.37154 0.37518 0.37855 0.38166 0.38451

0.37192 0.37553 0.37883 0.38196 0.38478

0.37229 0.37583 0.37920 0.38225 0.38504

0.37266 0.37622 0.37951 0.38254 0.38531

0.37303 0.37656 0.37983 0.38283 0.38557

0.37340 0.37693 0.38014 0.38312 0.38583

0.37376 0.37724 0.38045 0.38340 0.38609

0.37412 0.37757 0.38076 0.38365 0.38634

0.64 0.63 0.62 0.61 0.60

0.40 0.41 0.42 0.43 0.44

0.38634 0.38875 0.39089 0.39279 0.39442

0.38659 0.38897 0.39109 0.39296 0.39457

0.38684 0.38920 0.39129 0.39313 0.39472

0.38709 0.38942 0.39149 0.39330 0.39486

0.38734 0.38964 0.39168 0.39347 0.39501

0.38758 0.38985 0.39187 0.39364 0.39514

0.38782 0.39007 0.39206 0.39380 0.39528

0.38305 0.39028 0.39224 0.39396 0.39542

0.38829 0.39049 0.39243 0.39411 0.39555

0.38852 0.39069 0.39261 0.39427 0.39568

0.38875 0.39089 0.39279 0.39442 0.39580

0.59 0.58 0.57 0.56 0.55

0.45 0.46 0.47 0.4P 0.49

0.39580 0.39694 0.39781 0.39844 0.39882

0.39593 0.39703 0.39789 0.39849 0.39884

0.39605 0.39713 0.39796 0.39854 0.39886

0.39617 0.39723 0.39803 0.3985'3 0.39883

0.39629 0.39732 0.39809 0.39862 0.39890

0.39640 0.39741 0.39816 0.39866 0.39891

0.39651 0.39749 0.39822 0.39870 0.39892

0.39662 0.39758 0.39828 0.39873 0.39893

0.39673 0.39766 0.39834 0.39876 0.39894

0.39683 0.39774 0.39839 0.39879 0.39894

0.39694 0.39781 0.39844 0.39882 0.39894

0.54 0.53 0.52 0.51 0.50

0.010 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 Linear interpolation yields an error no greater than 5 units in the fifth decimal place.

0.000

Compiled from T. L. Kelley, The Kelley Statistical Tables. Harvard Univ. Press, Cambridge, Mass., 1948 (with permission).

I’(.,~)

976

PROBABILITY

Table 26.5 Q(r)

0.00 0.01

0.000

NORMAL

0.001

3.09023 2.32m635 2.29037

PRORARILITY

0.002

0.003

FUNCTIONS

FUNCTION-VALUES

0.004

0.005

0.006

2.65207 2.19729

2.57583 2.17009 1.95996 1.81191 1.69540

2.51214 2.14441

1.91104

1.89570

1.77438

1.76241

1.68494

1.67466

1.66456

1.65463

1.58927 1.50626

1.58047 1.49851

1.43953

1.43250

1.42554

1.36581 1.30469

1.35946 1.29884

1.57179 1.49085 1.41865 1.35317 1.29303

1.56322 1.48328 1.41183 1.34694 1.28727

1.55477 1.47579 1.40507 1.34076 1.28155

0.94 0.93 0.92 0.91 0.90

1.24808 1.19522 1.14551 1.09847 1.05374

1.24264 1.19012 1.14069 1.09390 1.04939

1.23723 1.18504 1.13590 1.08935 1.04505

1.23186 1.18000 1.13113 1.08482 1.04073

1.22653

0.89

1.06252

1.25357 1.20036 1.15035 1.10306 1.05812

1.01943 0.97815 0.93848 0.90023 0.86325

1.01522 0.97411 0.93458 0.89647 0.85962

1.01103 0.97009 0.93072 0.89273 0.85600

1.00686 0.96609 0.92686

0.99446 0.95416 0.91537

0.84 0.83 0.82

0.87055

1.02365 0.98220 0.94238 0.90399 0.86689

0.83450 0.79950 0.76546 0.73228 0.69988

0.83095 0.79606 0.76210 0.72900 0.69668

0.82742 0.79262 0.75875 0.72574 0.69349

0.82390 0.78919 0.75542 0.72248 0.69031

0.66821

0.66508

0.66196

2.01409

1.99539

1.97737

1.85218

1.83842

1.82501

0.04

1.73920

1.72793

1.71689

1.70604

0.05 0.06 0.07 0.08 0.09

1.64485

1.63523

1.55477 1.47579 1.40507 1.34076

1.54643 1.39838

1.62576 1.53820 1.46106 1.39174 1.32854

1.61644 1.53007 1.45381 1.38517 1.32251

1.37866 1.31652

1.37220 1.31058

0.10 0.11 0.12 0.13

1.28155 1.22653 1.17499 1.12639

1.27587 1.22123 1.17000 1.12168

1.27024 1.21596 1.16505

1.26464 1.21072 1.16012

1.25908 1.20553 1.15522

1.08032

1.07584

1.11699 1.07138

1.11232 1.06694

1.10768

0.14

0.15 0.16 0.17 0.18

1.03643 0.99446 0.95416 0.91537

1.03215 0.99036 0.95022 0.91156

1.02789 0.98627 0.94629 0.90777

0.19

0.87790

0.87422

0.20 0.21 0.22 0.23 0.24

0.84162 0.80642 0.77219 0.73885 0.70630

0.83805 0.80296 0.76882 0.73556 0.70309

0.26 0.27

0.25

0.67449 0.64335 0.61281

0.64027 0.60979

0.63719 0.60678

0.60376

0.6310b 0.60076

0.28

0.58284

0.57987

0.57691

0.57395

0.57100

0.29

0.55338

0.55047

0.54755

0.54464

0.30 0.31

0.52440 0.49585

0.52153 0.49302

0.51866 0.49019

0.32 0.33 0.34

0.46770 0.43991 0.41246

0.46490 0.43715 0.40974

0.35 0.36 0.37 0.38

0.38532 0.35846 0.33185 0.30548

0.38262

0.39

0.27932

0.40 0.41 0.42 0.43 0.44

0.25335 0.22754 0.20189 0.17637 0.15097

0.45

0.12566 0.10043 0.07527 0.05015 0.02507 0.010

0.46

0.47 0.48 0.49

0.25076

0.22497 0.19934 0.17383 0.14843 0.12314 0.09791 0.07276 0.04764

0.02256 0.009

1.60725

1.52204 1.44663

1.59819 1.51410

1.94313

2.45726 2.12007

2.40892 2.09693

2.36562 2.07485

0.010

1.92684

2.03352

1.86630

0.35579 0.32921 0.30286 0.27671

0.009

1.78661

2.05375

1.88079 1.75069

0.67135

0.008

1.79912

2.74778 2.22621

0.02

1.33462

0.007

OF P(x) AND Q(x) 2.32635 2.05375 1.88079 1.75069, 1.64485

2.87816 2.25713

0.03

1.46838

OF z IN TERMS

1.17499 1.12639 1.08032 1.03643

0.99 0.98 0.97 0.96 0.95

0.88 0.87 0.86 0.85

1.00271

0.99858 0.95812

0.88901

0.96210 0.92301 0.88529

0.88159

0.84520

0.87790 0.84162

0.81 0.80

0.82038 0.78577 0.75208 0.71923 0.68713

0.81687 0.78237 0.74876 0.71599 0.68396

0.81338 0.77897 0.74545 0.71275 0.68080

0.80990 0.77557 0.74214 0.70952 0.67764

0.80642 0.77219 0.73885 0.70630 0.67449

0.79 0.78 0.77 0.76 0.75

0.65884

0.65573

0.62801

0.62496

0.59477

0.54174

0.59776 0.56805 0.53884

0.65262 0.62191 0.59178 0.56217 0.53305

0.64952 0.61887 0.58879 0.55924

0.64643 0.61584 0.58581 0.55631

0.64335 0.61281 0.58284 0.55338

0.74 0.73 0.72 0.71

0.53016

0.52728

0.52440

0.70

0.51579 0.48736

0.51293 0.48454

0.51007 0.48173

0.50722 0.47891

0.50437

0.50153

0.46211 0.43440 0.40701

0.45933 0.43164 0.40429

0.45654 0.42889 0.40157

0.45376 0.42615 0.39886

0.45099 0.42340 0.39614

0.47610 0.44821 0.42066 0.39343

0.47330 0.44544 0.41793 0.39073

0.49869 0.47050 0.44268 0.41519 0.38802

0.49585 0.46770 0.43991 0.41246 0.38532

0.69 0.68 0.67 0.66 0.65

0.37993 0.35312 0.32656 0.30023

0.37723 0.35045 0.32392

0.37454 0.34779 0.32128

0.37186 0.34513 0.31864

0.29761 0.27151

0.29499 0.26891

0.29237 0.26631

0.36649 0.33981 0.31337 0.28715

0.36381 0.33716 0.31074 0.28454

0.36113 0.33450 0.30811 0.28193

0.35846 0.33185 0.30548 0.27932

0.27411

0.36917 0.34247 0.31600 0.28976 0.26371

0.26112

0.25853

0.25594

0.25335

0.64 0.63 0.62 0.61 0.60

0.24817 0.22240 0.19678 0.17128 0.14590

0.24559 0.21983 0.19422 0.16874 0.14337

0.24301 0.21727 0.19167 0.16620 0.14084

0.24043 0.21470 0.18912 0.16366 0.13830

0.23785 0.21214 0.18657 O.lb112 0.13577

0.23527 0.20957 0.18402 0.15858 0.13324

0.20701 0.18147

0.23269

0.23012 0.20445 0.17892

0.22754 0.20189 0.17637

0.59 0.58 0.57

0.15604

0.15351

0.15097

0.56

0.12061 0.09540 0.07024 0.04513 0.02005

0.11809 0.09288 0.06773

0.11556

0.11304

0.09036

0.08784

0.06522 0.04012 0.01504

0.06271 0.03761 0.01253

0.11052 0.08533 0.06020

0.10799 0.08281 0.05768

0.03510 0.01003

0.03259 0.00752

0.63412

0.04263

0.01755

0.56511

0.53594

0.85239

0.84879

0.13072

0.91918

0.12819

0.12566

0.10547

0.08030 0.05517 0.03008 0.00501

0.008 0.007 0.006 0.005 0.004 0.003 0.002 For Q(~)>O.007,linear interpolation yields an error of one unit in the thil.d decimal interpolation is necessaryto obtain full accuracy. P(z)=l-Q(7.)=J:_

Z(W

Compiledfrom T. L. Kelley, The Kelley Statistical Tables. Harvard Univ. Press,Cambridge, Mass., 1948(with permission).

0.55

0.54 0.53 0.52 0.51 0.50 w

PROBABILITY NORMAL

PROBABILITY

r,Jh) 0.0000 o.o;o 3.0;23

FUNCTION-VALUES

OF z FOR

EXTREME

VALUES

OF P(x)

AND

Q(z)

Table

26.0

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.0008

0.0009

0.0010

2.87816 2.74778 2.65207

3 71902 3:06181 2.86274 2.73701 2.64372

3.54008 3.03567 2.84796 2.72655 2.63555

3.43161 3.01145 2.83379 2.71638 2.62756

3.35279 2.98888 2.82016 2.70648 2.61973

3.29053 2.96774 2.80703 2.69684 2.61205

3.23888 2.94784 2.79433 2.68745 2.60453

3.19465 2.92905 2.78215 2.67829 2.59715

3.15591 2.91124 2.77033 2.66934 2.58991

3.12139 2.89430 2.75888 2.66061 2.58281

3.09023 2.87816 2.74778 2.65207 2.57583

0.999 0.998 0.997 0.996 0.995

0.005 0.006 0.007 3.008 0.009

2.57533 2.51214 2.45726 2.40891 2.36562

2.56897 2.50631 2.45216 2.40437 2.36152

2.56224 2.50055 2.44713 2.39989 2.35747

2.55562 2.49485 2.44215 2.39545 2.35345

2.54910 2.48929 2.43724 2.39106 2.34947

2.54270 2.48377 2.43238 2.38671 2.34553

2.53640 2.47333 2.42758 2.38240 2.34162

2.53019 2.47296 2.42283 2.37814 2.33775

2.52408 2.46765 2.41814 2.37392 2.33392

2.51807 2.46243 2.41350 2.36975 2.33012

2.51214 2.45726 2.40891 2.36562 2.32635

0.994 0.993 0.992 0.991 0.990

0.010 0.011 0.012 0.013 0.014

2.32635 2.29037 2.25713 2.22621 2.19729

2.32261 2.28693 2.25394 2.22323 2.19449

2.31891 2.28352 2.25077 2.22028 2.19172

2.31524 2.28013 2.24763 2.21734 2.18896

2.31160 2.27677 2.24450 2.21442 2.18621

2.30798 2.27343 2.24140 2.21152 2.18349

2.30440 2.27013 2.23832 2.20864 2.18078

2.30085 2.26684 2.23526 2.20577 2.17808

2.29733 2.26358 2.23223 2.20293 2.17540

2.29383 2.26034 2.22921 2.20010 2.17274

2.29037 2.25713 2.22621 2.19729 2.17009

0.989 0.988 0.987 0.986 0.935

0.015 0.016 0.017 0.018 0.019

2.17009 2.14441 2.12007 2.09693 2.07485

2.16746 2.14192 2.11771 2.09467 2.07270

2.16484 2.13944 2.11535 2.09243 2.07056

2.16224 2.13698 2.11301 2.09020 2.06843

2.15965 2.13452 2.11068 2.08798 2.06630

2.15707 2.13208 2.10836 2.08576 2.06419

2.15451 2.12966 2.10605 2.08356 2.06208

2.15197 2.12724 2.10375 2.08137 2.05998

2.14943 2.12484 2.10147 2.07919 2.05790

2.14692 2.12245 2.09919 2.07702 2.05582

2.14441 2.12007 2.09693 2.07485 2.05375

0.984 0.983 0.982 0.981 0.980

0.020 !I;021 0.022 0.023 0.024

2.05375 2;1j3352 2.01409 1.99539 ii97737

2.05169 2;03154 2.01219 1.99356 1;97560

2.04964 2.02957 2.01029 1.99174 1.97384

2.04759 2;02761 2.00841 1.98992 i97208

2.04556 2.02566 2.00653 1.98811 1.97033

2.04353 2.02371 2.00465 1.98630 1.96859

2.04151 2.02177 2.00279 1.98450 1.96685

2.03950 2.01984 2.00093 1.98271 1.96512

2.03750 2.01792 1.99908 1.98092 1.96340

2.03551 2.01600 1.99723 1.97914 1.96168

2.03352 2.01409 1.99539 1.97737 1.95996

0.979 0.978 0.977 0.976 0.975

0.001 0.002 0.003 0.004

0.0001

977

FUNCTIONS

0.0006 0.0005 0.0004 0.0003 0.0002 0.0001 0.0000 0.0010 0.0009 0.0008 0.0007 For Q(1.)\0.0007, linear interpolation yields an error of one unit in the third decimal place; five-point interpolation is necessary to obtain full accuracy.

f)(r)

q (.I.)

.I’

(-4)l.O

3.71902

(-

(-5)l.O

4.26489

(-1O)l.

(-6)l.O

4.75342

(-11)l.

(-7)l.O

5.19934

(-12)l.

(-8)l.O

5.61200

(-13)l.O

.!’

9)l.O

1) (.I.)

5.99781

(-14)l.O

0

6.36134

(-15)l.

0

6.70602

( -1b)l.

0

7.03448 7.34880

P(,r)=l-f)(+JJ

,

(,) (,I.)

.!’

7.65063

(-19)l.

0

9.01327

0

7.94135

(-2O)l.O

9.26234

0

8.22208

(-21)l.O

9.50502

(-17)l.

0

8.49379

(-22)l.O

9.74179

( -18)l.

0

3.75729

(-23)l.O

9.97305

m2(/)111

Compiled from T. L. Kelley, The Kelley Statistical Tables. 1948 (with permission) for Q(X) >(-9)l.

Harvard Univ. Press, Cambridge, Mass.,

I’(V)

978

PROBABILITY

Table

26.7

PROBABILITY

INTEGRAL CUMULATIVE

FUNCTIONS

OF x3-DISTRIBUTION, INCOMPLETE GAMMA SUMS OF THE POISSON DISTRIBUTION

w*=O.OOl

0.002

0.003

0.004

0.005

0.006

0.007

m=O.O005 0.97477 0.99950 0.99999

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

x2=0.01 m=0.005 0.92034 0.99501 0.99973 0.99999

0.96433 0.95632 0.94957 0.94363 0.93826 0.93332 0.99900 0.99850 0.99800 0.99750 0.99700 0.99651 0.99998 0.99996 0.99993 0.99991 0.99988 0.99984 0.99999 0.02 0.03 0.04 0.05 0.06 0.07 0.010 0.015 0.020 0.025 0.030 0.035 0.88754 0.86249 0.84148 0.82306 0.80650 0.79134 0.99005 0.98511 0.98020 0.97531 0.97045 0.96561 0.99925 0.99863 0.99790 0.99707 0.99616 0.99518 0.99995 0.99989 0.99980 0.99969 0.99956 0.99940 0.99999 0.99998 0.99997 0.99995 0.99993

6

0.75183 0.95123 0.99184 0.99879 0.99984

98 10

; 9 10 11 :: 14 15

0.010 0.0050 0.92034 0.99501 0.99973 0.99999

0.10 0.050 0.75183 0.95123 0.99184 0.99879 0.99984

0.9

KO 0.6;472 0.90484 0.97759 0.99532 0.99911

ii-;5 0.58388 0.86071 0.96003 0.98981 0.99764

EO 0.5;709 0.81873 0.94024 0.98248 0.99533

0.99998 0.99985 0.99950 0.99885 0.99997 0.99990 0.99974 0.99998 0.99994 0.99999

E5 0.4;950 0.77880 0.91889 0.97350 0.99212

Eo 0.43858 0.74082 0.89643 0.96306 0.98800

K5 0.40278 0.70469 0.87320 0.95133 0.98297

0.37109 0.67032 0.84947 0.93845 0.97703

0.45 0.34278 0.63763 0.82543 0.92456 0.97022

0.60653 0.80125 0.90980 0.96257

0.99784 0.99945 0.99987 0.99997 0.99999

0.99640 0.99899 0.99973 0.99993 0.99998

0.99449 0.99834 0.99953 0.99987 0.99997

0.99207 0.99744 0.99922 0.99978 0.99994

0.98912 0.99628 0.99880 0.99964 0.99989

0.98561 0.99483 0.99825 0.99944 0.99983

GO 0.31731

x2= 1.1 m=0.55 0.29427 0.57695 0.77707 0.89427 0.95410

1.2

1.3

1.4

1.5

1.6

0.60 0.27332 0.54881 0.75300 0.87810 0.94488

0.65 0.25421 0.52205 0.72913 0.86138 0.93493

0.70 0.23672 0.49659 0.70553 0.84420 0.92431

0.75 0.22067 0.47237 0.68227 0.82664 0.91307

0.80 0.20590 0.44933 0.65939 O-80879 0.90125

0.99999 0.99998 0.99997 0.99995 0.99999 0.99999 1.7 1.8 1.9 2.0 0.85 0.90 0.95 1.00 0.19229 0.17971 0.16808 0.15730 0.42741 0.40657 0.38674 0.36788 0.63693 0.61493 0.59342 0.57241 0.79072 0.77248 0.75414 0.73576 0.88890 0.87607 0.86280 0.84915

0.98154 0.99305 0.99753 0.99917 0.99973

0.97689 0.99093 0.99664 0.99882 0.99961

0.97166 0.98844 0.99555 0.99838 0.99944

0.96586 0.98557 0.99425 0.99782 0.99921

0.95949 0.98231 0.99271 0.99715 0.99894

0.95258 0.97864 0.99092 0.99633 0.99859

0.94512 0.97457 0.98887 0.99537 0.99817

0.93714 0.97008 0.98654 0.99425 0.99766

0.92866 0.96517 0.98393 0.99295 0.99705

0.91970 0.95984 0.98101 0.99147 0.99634

0.99992 0.99987 0.99981 0.99973 0.99998 0.99996 0.99994 0.99991 0.99999 0.99999 0.99998 0.99997 0.99999 0.99999

0.99962 0.99987 0.99996 0.99999

0.99948 0.99982 0.99994 0.99998 0.99999

0.99930 0.99975 0.99991 0.99997 0.99999

0.99908 0.99966 0.99988 0.99996 0.99999

0.99882 0.99954 0.99983 0.99994 0.99998

0.99850 0.99941 0.99977 0.99992 0.99997

::

6

0.009 0.0045 0.92442 0.99551 0.99977 0.99999 0.09 0.045 0.76418 0.95600 0.99301 0.99902 0.99987

0.99999 0.99999 0.99999 0.99998

x2=0.1 m=0.05

6 7

0.008 0.0040 0.92873 0.99601 0.99981 0.99999 0.08 0.040 0.77730 0.96079 0.99412 0.99922 0.99991

FUNCTION

16

Compiled from E. S. Pearson and H. 0. Hartley (editors), Biometrika Cambridge Univ. Press, Cambridge, England, 1954 (with permission).

0.99999 0.99999

tables for statisticians,

vol. I.

PROBABILITY PROBABILITY

INTEGRAL CUMULATIVE

x2=2.2 m=l.l

2.4 1.2

979

FUNCTIONS

OF x~-L)ISTRIBUTION, INCOMPLETE GAMMA SUMS OF THE POISSON DISTRIBUTION

FI’NCTION

2.6 1.3

3.6 1.8

2.8 1.4

3.0 1.5

3.2 1.6

3.4 1.7

Table 26.7

3.8 1.9

4.0 2.0

0.13801 0.33287 0.53195 0.69903 0.82084

0.12134 0.30119 0.49363 0.66263 0.79147

0.10686 0.27253 0.45749 0.62682 0.76137

0.09426 0.24660 0.42350 0.59183 0.73079

0.08327 0.22313 0.39163 0.55783 0.69999

0.07364 0.20190 0.36181 0.52493 0.66918

0.06520 0.18268 0.33397 0.49325 0.63857

0.05778 0.16530 0.30802 0.46284 0.60831

0.05125 0.14957 0.28389 0.43375 0.57856

0.04550 0.13534 0.26146 0.40601 0.54942

0.90042 0.94795 0.97426 0.98790 0.99457

0.87949 0.93444 0.96623 0.98345 0.99225

0.85711 0.91938 0.95691 0.97807 0.98934

0.83350 0.90287 0.94628 0.97170 0.98575

0.80885 0.88500 0.93436 0.96430 0.98142

0.78336 0.86590 0.92119 0.95583 0.97632

0.75722 0.84570 0.90681 0.94631 0.97039

0.73062 0.82452 0.89129 0.93572 0.96359

0.70372 0.80250 0.87470 0.92408 0.95592

0.67668 0.77978 0.85712 0.91141 0.94735

0.99766 0.99903 0.99961 0.99985 0.99994

0.99652 0.99850 0.99938 0.99975 0.99990

0.99503 0.99777 0.99903 0.99960 O-99984

0.99311 0.99680 0.99856 0.99938 0.99974

0.99073 0.99554 0.99793 0.99907 0.99960

0.98781 0.99396 0.99711 0.99866 0.99940

0.98431 0.99200 0.99606 0.99813 0.99913

0.98019 0.98962 0.99475 0.99743 0.99878

0.97541 0.98678 0.99314 0.99655 0.99832

0.96992 0.98344 0.99119 0.99547 0.99774

0.99998 0.99999

0.99996 0.99999

0.99994 0.99998 0.99999

0.99989 0.99996 0.99998 0.99999

0.99983 0.99993 0.99997 il.99999

0.99974 0.99989 0.99995 0.99998 0.99999

0.99961 0.99983 0.99993 0.99997 0.99999

0.99944 0.99975 0.99989 0.99995 0.99998

0.99921 0.99964 0.99984 0.99993 0.99997

0.99890 0.99948 0.99976 0.99989 0.99995

0.99999

0.99999

0.99998 0.99999

x2= 4.2 ne==2.1

4.4 2.2

4.6 2.3

4.8 2.4

5.0 2.5

5.2 2.6

5.4 2.7

5.6 2.8

5.8 2.9

6.0 3.0

0.04042 0.12246 0.24066 0.37962 0.52099

0.03594 0.11080 0.22139 0.35457 0.49337

0.03197 0.10026 0.20354 0.33085 0.46662

0.02846 0.09072 0.18704 0.30844 0.44077

0.02535 0.08209 0.17180 0.28730 0.41588

0.02259 0.07427 0.15772 0.26739 0.39196

0.02014 0.06721 0.14474 0.24866 0.36904

0.01796 0.06081 0.13278 0.23108 0.34711

0.01603 0.05502 0.12176 0.21459 0.32617

0.01431 0.04979 0.11161 0.19915 0.30622

0.64963 0.75647 0.83864 0.89776 0.93787

0.62271 0.73272 0.81935 0.88317 0.92750

0.59604 0.70864 0.79935 0.86769 0.91625

0.56971 0.68435 0.77872 0.85138 0.90413

0.54381 0.65996 0.75758 0.83431 0.89118

0.51843 0.63557 0.73600 0.81654 0.87742

0.49363 0.61127 0.71409 0.79814 0.86291

0.46945 0.58715 0.69194 0.77919 0.84768

0.44596 0.56329 0.66962 0.75976 0.83178

0.42319 0.53975 0.64723 0.73992 0.81526

0.96370 0.97955 0.98887 0.99414 0.99701

0.95672 0.97509 0.98614 0.99254 0.99610

0.94898 0.97002 0.98298 0.99064 0.99501

0.94046 0.96433 0.97934 0.98841 0.99369

0.93117 0.95798 0.97519 0.98581 0.99213

0.92109 0.95096 0.97052 0.98283 0.99029

0.91026 0.94327 0.96530 0.97943 0.98816

0.89868 0.93489 0.95951 0.97559 0.98571

0.88637 0.92583 0.95313 0.97128 0.98291

0.87337 0.91608 0.94615 0.96649 0.97975

0.99851 0.99928 0.99966 0.99985 0.99993

0.99802 0.99902 0.99953 0.99978 0.99990

0.99741 0.99869 0.99936 0.99969 0.99986

0.99666 0.99828 0.99914 0.99958 0.99980

0.99575 0.99777 0.99886 0.99943 0.99972

0.99467 0.99715 0.99851 0.99924 0.99962

0.99338 0.99639 0.99809 0.99901 0.99950

0.99187 0.99550 0.99757 0.99872 0.99934

0.99012 0.99443 0.99694 0.99836 0.99914

0.98810 0.99319 0.99620 0.99793 0.99890

0.99997 0.99999 0.99999

0.99995 0.99998 0.99999

0.99993 0.99997 0.99999 0.99999

0.99991 0.99996 0.99998 0.99999

0.99987 0.99994 0.99997 0.99999 0.99999

0.99982 0.99991 0.99996 0.99998 0.99999

0.99975 0.99988 0.99994 0.99997 0.99999

0.99967 0.99984 0.99992 0.99996 0.99998

0.99956 0.99978 0.99989 0.99995 0.99998

0.99943 0.99971 0.99986 0.99993 0.99997

0.99999

0.99999 0.99999

0.99998 0.99999

+;

Interpolation on x2

px;)

/u="--vo>o

Double Entry Interpolation Q (x2iv)=Q (x;iv0-4)[;

+Q

(x&-2)~-m2-W]+Q

+Q (x;~~~)[l-w~-~+;~~+w#j+Q

-

(x&+l)[;

(x:,1+)[; w2++-w4]

W’-;,u+W+]

980

PROBABILITY

Table 26.7 PROBABILITY

6 ii 9 10 11 12 :i 15 16 :ii 19 20

26 27 28

.2 = 6.2

6.4

m=3.1

3.2

FUNCTIONS

INTEGRAL OF X2-DISTRIBUTION, INCOMPLETE GAMMA FUNCTION CUMULATIVE SUMS OF THE POISSON DISTRIBITTION 6.6 3.3

6.8 3.4

7.0 3.5

7.2 3.6

7.4 3.7

7.6 3.8

7.8 3.9

8.0 4.0

0.01278 0.04505 0.10228 0.18470 0.28724

0.01141 0.04076 0;09369 0.17120 0.26922

0.01020 0.03688 0.08580 0.15860 0.25213

0.00912 0.03337 0.07855 0.14684 0.23595

0.00815 0.03020 0.07190 0.13589 0.22064

0.00729 0.02732 0.06579 0.12569 0.20619

0.00652 0.02472 0.06018 0.11620 0.19255

0.00584 0.02237 0.05504 0.10738 0.17970

0.00522 0.02024 0.05033 0.09919 0.16761

0.00468 0.01832 0.04601 0.09158 0.15624

0.40116 0.51660 0.62484 0.71975 0.79819

0.37990 0.49390 0.60252 0.69931 0.78061

0.35943 0.47168 0.58034 0.67869 0.76259

0.33974 0.45000 0.55836 0.65793 0.74418

0.32085 0.42888 0.53663 0.63712 0.72544

0.30275 0.40836 0.51522 0.61631 0.70644

0.28543 0.38845 0.49415 0.59555 0.68722

0.26890 0.36918 0.47349 0.57490 ,0.66784

0.25313 0.35056 0.45325 0.55442 0.64837

0.23810 0.33259 0.43347 0.53415 0.62884

0.85969 0.90567 0.93857 0.96120 0.97619

0.84539 0.89459 0.93038 0.95538 0.97222

0.83049 0.88288 0.92157 0.94903 0.96782

0.81504 0.87054 0.91216 0.94215 0.96296

0.79908 0.85761 0.90215 0.93471 0.95765

0.78266 0.84412 0.89155 0.92673 0.95186

0.76583 0.83009 0.88038 0.91819 0.94559

0.74862 0.81556 0.86865 0.90911 0.93882

0.73110 0.80056 0.85638 0.89948 0.93155

0.71330 0.78513 0.84360 0.88933 0.92378

0.98579 0.99174 0.99532 0.99741 0.99860

0.98317 0.99007 0.99429 0.99679 0.99824

0.98022 0.98816 0.99309 0.99606 0.99781

0.97693 0.98599 0.99171 0.99521 0.99729

0.97326 0.98355 0.99013 0.99421 0.99669

0.96921 0.98081 0.98833 0.99307 0.99598

0.96476 0.97775 0.98630 0.99176 0.99515

0.95989 0.97437 0.98402 0.99026 0.99420

0.95460 0.97064 0.98147 0.98857 0.99311

0.94887 0.96655 0.97864 0.98667 0.99187

0.99926 0.99962 0.99981 0.99990 0.99995

0.99905 0.99950 0.99974 0.99987 0.99994

0.99880 0.99936 0.99967 0.99983 0.99991

0.99850 0.99919 0.99957 0.99978 0.99989

0.99814 0.99898 0.99945 0.99971 0.99985

0.99771 0.99873 0.99931 0.99963 0.99981

0.99721 0.99843 0.99913 0.99953 0.99975

0.99662 0.99807 0.99892 0.99941 0.99968

0.99594 0.99765 0.99867 0.99926 0.99960

0.99514 0.99716 0.99837 0.99908 0.99949

0.99998 0.99999

0.99997 0.99999 0.99999

0.99996 0.99998 0.99999

0.99994 0.99997 0.99999 0.99999

0.99992 0.99996 0.99998 0.99999

0.99990 0.99995 0.99998 0.99999 0.99999

0.99987 0.99993 0.99997 0.99998 0.99999

0.99983 0.99991 0.99996 0.99998 0.99999

0.99978 0.99989 0.99994 0.99997 0.99999

0.99973 0.99985 0.99992 0.99996 0.99998

:;

x2= 8.2 m -4.1

8.4 4.2

8.6 4.3

8.8 4.4

9.0 4.5

9.2 4.6

9.4 4.7

9.6 4.8

9.8 4.9

10.0 5.0

0.00419 0.01657 0.04205 0.08452 0.14555

0.00375 0.01500 0.03843 0.07798 0.13553

0.00336 0.01357 0.03511 0.07191 0.12612

0.00301 0.01228 0.03207 0.06630 0.11731

0.00270 0.01111 0.02929 0.06110 0.10906

0.00242 0.01005 0.02675 0.05629 0.10135

0.00217 0.00910 0.02442 0.05184 0.09413

0.00195 0.00823 0.02229 0.04773 0.08740

0.00175 0.00745 0.02034 0.04394 0.08110

0.00157 0.00674 0.01857 0.04043 0.07524

0.22381 0.31529 0.41418 0.51412 0.60931

0.21024 0.29865 0.39540 0.49439 0.58983

0.19736 0.28266 0.37715 0.47499 0.57044

0.18514 0.26734 0.35945 0.45594 0.55118

0.17358 0.25266 0.34230 0.43727 0.53210

0.16264 0.23861 0.32571 0.41902 0.51323

0.15230 0.22520 0.30968 0.40120 0.49461

0.14254 0.21240 0.29423 0.38383 0.47626

0.13333 0.20019 0.27935 0.36692 0.45821

O.lJ465 0.18857 0.26503 0.35049 0.44049

1'2

0.69528 0.76931 0.83033 0.87865 0.91551

0.67709 0.75314 0.81660 0.86746 0.90675

0.65876 0.73666 0.80244 0.85579 0.89749

0.64035 0.71991 oi78788 0.84365 0.88774

0.62189 0.70293 0;7729i 0.83105 0.87752

0.60344 0.68576 0.75768 0.81803 0.86683

0.58502 0.66844 0.74211 0.80461 0.85569

0.56669 0.65101 0.72627 0.79081 0.84412

0.54846 0.63350 0.71020 0.77666 0.83213

0.53039 0.61596 0.69393 0.76218 0.81974

:"7 18 19 20

0.94269 0.96208 0.97551 0.98454 0.99046

0.93606 0.95723 0.97207 0.98217 0.98887

0.92897 0.95198 0.96830 0.97955 0.98709

0.92142 0.94633 0.96420 0.97666 0.98511

0.91341 0.94026 0.95974 0.97348 0.98291

0.90495 0.93378 0.95493 0.97001 0.98047

0.89603

0.94974 0:96623 0.97779

0.88667 0.91954 0.94418 0.96213 0.97486

0.87686 0.91179 0.93824 0.95771 0.97166

0.86663 0.90361 0.93191 0.95295 0.96817

0.99424 0.99659 0.99802 0.99888 0.99937

0.99320 0.99593 0.99761 0.99863 0.99922

0.99203 0.99518 0.99714 0.99833 0.99905

0.99070 0.99431 0.99659 0.99799 0.99884

0.98921 0.99333 0.99596 0.99760 0.99860

0.98755 0.99222 0.99524 0.99714 0.99831

0.98570 0.99098 0.99442 0.99661 0.99798

0.98365 0.98958 0.99349 0.99601 0.99760

0.98139 0.98803 0.99245 0.99532 0.99716

0.97891 0.98630 0.99128 0.99455 0.99665

0.99966 0.99981 0.99990 0.99995 0.99997

0.99957 0.99977 0.99987 0.99993 0.99997

0.99947 0.99971 0.99984 0.99991 0.99996

0.99934 0.99963 0.99980 0.99989 0.99994

0.99919 0.99955 0.99975 0.99986 0.99993

0.99902 0.99944 0.99969 0.99983 0.99991

0.99882 0.99932 0.99962 0.99979 0.99988

0.99858 0.99917 0.99953 0.99973 0.99985

0.99830 0.99900 0.99942 0.99967 0.99982

0.99798 0.99880 0.99930 0.99960 0.99977

:: 13

21 22 $43 25

0; 92687

PROBABILITY

FUNCTIONS

Table PROBABILITY

Vm

INTEGRAL CUMULATIVE

x2=10.5 ,= 5.25

11.0 5.5

OF x2-DISTRIBUTION, INCOMPLETE GAMMA SUMS OF THE POISSON DISTRIBUTION

11.5 5.75

12.0 6.0

12.5 6.25

13.0 6.5

13.5 6.75

0.00119 0.00525 0.01476 0.03280 0.06225

0.00091 0.00409 0.01173 0.02656 0.05138

0.00070 0.00318 0.00931 0.02148 0.04232

0.00053 0.00248 0.00738 0.01735 0.03479

0.00041 0.00193 0.00585 0.01400 0.02854

0.00031 0.00150 0.00464 0. 01128 0.02338

0.00024 0.00117 0.00367 0.00907 0.01912

0.10511 0.16196 0.23167 0.31154 0.39777

0.08838 0.13862 0.20170 0.27571 0.35752

0.07410 0.11825 0.17495 0.24299 0.31991

0.06197 0.10056 0.15120 0.21331 0.28506

0.05170 0.08527 0.13025 0.18657 0.25299

0.04304 0.07211 0.11185 0.16261 0.22367

0.48605 0.57218 0.65263 0.72479 0.78717

0.44326 0.52892 0.61082 0.68604 0.75259

0.40237 0.48662 0.56901 0.64639 0.71641

0.36364 0.44568 0.52764 0.60630 0.67903

0.32726 0.40640 0.48713 0.56622 0.64086

0.83925 0.88135 0.91436 0.93952 0.95817

0.80949 0.85656 0.89436 0.92384 0.94622

0.77762 0.82942 0.87195 0.90587 0.93221

0.74398 0.80014 0.84724 0.88562 0.91608

0.97166 0.98118 0.98773 0.99216 0.99507

0.96279 0.97475 0.98319 0.98901 0.99295

0.95214 0.96686 0.97748 0.98498 0.99015

0.99696 0.99815 0.99890 0.99935 0.99963

0.99555 0.99724 0.99831 0.99899 0.99940

x2=15.5

16.0

: 3 4 5 6 8' 1'0 :: :: 15 16 17 :; 20 :: 2 25 26 :; 29 30

m

=o. GE

FUNCTION

14.0 14.5 7.0 7.25 0.000180.00014

15.0 7.5

0.00091 0.00291 0.00730 0.01561

0.00071 0.00230 0.00586 0.01273

0.00011 0.00055 0.00182 0.00470 0.01036

0.03575 0.06082 0.09577 0.14126 0.19704

0.02964 0.05118 0.08177 0.12233 0.17299

0.02452 0.04297 0.06963 0.10562 0.15138

0.02026 0.03600 0.05915 0.09094 0.13206

0.29333 0.36904 0.44781 0.52652 0.60230

0.26190 0.33377 0.40997 0.48759 0.56374

0.23299 0.30071 0.37384 0.44971 0.52553

0.20655 0.26992 0.33960 0.41316 0.48800

0.18250 0.24144 0.30735 0.37815 0.45142

0.70890 0.76896 0.82038 0.86316 0.89779

0.67276 0.73619 0.79157 0.83857 0.87738

0.63591 0.70212 0.76106 0.81202 0.85492

0.59871 0.66710 0.72909 0.78369 0.83050

0.56152 0.63145 0.69596 0.75380 0.80427

0.52464 0.59548 0.66197 0.72260 0.77641

0.93962 0.95738 0.97047 0.97991 0.98657

0.92513 0.94618 0.96201 0.97367 0.98206

0.90862 0.93316 0.95199 0.96612 0.97650

0.89010 0.91827 0.94030 0.95715 0.96976

0.86960 0.90148 0.92687 0.94665 0.96173

0.84718 0.88279 0.91165 0.93454 0.95230

0.82295 0.86224 0.89463 0.92076 0.94138

0.99366 0.99598 0.997 9 0.998 ! 6 0.99907

0.99117 0.99429 0.99637 0.99773 0.99860

0.98798 0.99208 0.99487 0.99672 0.99794

0.98397 0.98925 0.99290 0.99538 0.99704

0.97902 0.98567 0.99037 0.99363 0.99585

0.97300 0.98125 0.98719 0.99138 0.99428

0.96581 0.97588 0.98324 0.98854 0.99227

0.95733 0.96943 0.97844 0.98502 0.98974

16.5

17.0

17.5

18.0

18.5

19.0

19.5

20.0 10.0

9.25

0 i%%?

0.00043 0.00144 0.00377 0.00843

i0:

0.00002 0.00010 0.00035 0.00099 0.00238

0. 0.

0:00026 0.00090 0.00242 0.00555

0 kE

0:00006 0.00022 0.00063 0.00155

0.00001 0.00005 0.00017 0.00050 0.00125

0.00557 0.01034 0.01791 0.02925

;

0.03010 0.01670

9" 10

0.07809 0.05012 0.11487

0.01375 0.02512 0.04238 0.06688 0.09963

0.01131 0.02092 0.03576 0.05715 0.08619

0.00928 0.01740 0.03011 0.04872 0.07436

0.00761 0.01444 0.02530 0.04144 0.06401

0.00623 0.01197 0.02123 0.03517 0.05496

0.00510 0.00991 0.01777 0.02980 0.04709

0.00416 0.00819 0.01486 0.02519 0.04026

0.00340 0.00676 0.01240 0.02126 0.03435

12 11 13

0.21522 0.16073 0.27719

0.19124 0.14113 0.24913

0.16939 0.12356 0.22318

0.10788 0.14960 0.19930

:z

0.41604 0.34485

0.31337 0.38205

0.28380 0.34962

0.25618 0.31886

0.09393 0.13174 0.17744 0.23051 0.28986

0.08158 0.11569 0.15752 0.20678 0.26267

0.07068 0.10133 0.13944 0.18495 0.23729

0.06109 0.08853 0.12310 0.16495 0.21373

0.05269 0.07716 0.10840 0.14671 0.19196

16 :i

0.48837 0.55951 0.62740 0.69033 0.74712

0.45296 0.52383 0.59255 0.65728 0.71662

0.41864 0.48871 0.55770 0.62370 0.68516

0.38560 0.45437 0.52311 0.58987 0.65297

0.35398 0.42102 0.48902 0.55603 0.62031

0.32390 0.38884 0.45565 0.52244 0.58741

0.29544 0.35797 0.42320 0.48931 0.55451

0.26866 0.32853

0.30060 0.24359

0.27423 0.22022

0.39182 0.45684 0.52183

0.42521 0.36166 0.48957

0.33282 0.39458 0.45793

0.79705 0.83990 0.87582 0.92891 0.90527

0.76965 0.81589 0.85527 0.91483 0.88808

0.74093 0.79032 0.83304 0.89912 0.86919

0.71111 0.76336 0.80925 0.84866 0.88179

0.68039 0.73519 0.78402 0.82657 0.86287

0.64900 0.70599 0.75749 0.80301 0.84239

0.61718 0.67597 0.72983 0.77810 0.82044

0.58514

0.55310

0.52126

:4' 25

0.64533 0.70122 0.75199 0.79712

0.61428 0.67185 0.72483 0.77254

0.64191 0.58304 0.69678 0.74683

26 2287 29 30

0.94749 0.97266 0.96182 0.98071 0.98659

0.93620 0.95295 0.97554 0.96582 0.98274

0.92341 0.94274 0.95782 0.96939 0.97810

0.90908 0.93112 0.94859 0.96218 0.97258 +f (X-g>

0.89320 0.91806 0.93805 0.95383 0.96608

0.87577 0.90352 0.92615 0.94427 0.95853 w=v-vo>o

0.85683 0.88750 0.91285 0.93344 0.94986

0.87000 0.83643 0.89814 0.92129 0.94001

0.85107 0.81464 0.88200 0.90779 0.92891

0.83076 0.79156 0.86446 0.89293 0.91654

:i ::

26.7

0.00277

0.04534 0.06709 0.13014 0.09521 0.17193

Interpolation on x2 Q(x~~Y)=Q(~~~~O-4)[~~2]+Q(~~~~-2)[~-$]+Q~~~~~~)[1-~t~~2] Double Entry Interpolation Q(xz~~)=Q~~~~~-4)[~$l+Q(x~~~,1~2)[~-~z-w~]+Q(~~Yn-1)[~~2-~~+~~ +Q(x#o)[l-w2-~+;d+w~+Q(+,+1)[;w2+;w~~~

PROBABILITY

982 Table 26.7

PROBABILITY

y”E21

22 11.0

FUNCTIONS

INTEGRAL OF x2-DISTRIBUTION, INCOMPLETE GAMMA FUNCTION CUMULATIVE SUMS OF THE POISSON DISTRIBUTION 27 30 24 26 28 29 23 25 13.5 15.0 12.5 13.0 14.0 14.5 11.5 12.0

n, = 10.5 0.00001 0.00003 0.00011 0.00032 0.00081

0.00002 0.00007 0.00020 0.00052

0.00001 0.00004 0.00013 0.00034

0.00001 0.00003 0.00008 0.00022

0.00002 0.00005 0.00014

0.00001 0.00003 0.00009

0.00001 0.00002 0.00006

0.00001 0.00004

0.00001 0.00002

0.00001 0..00002

0.00184 0.00377 0.00715 0.01265 0.02109

0.00121 0.00254 0.00492 0.00888 0.01511

0.00080 0.00171 0.00336 0.00620 0.01075

0.00052 0.00114 0.00229 0.00430 0.00760

0.00034 0.00076 0.00155 0.00297 0.00535

0.00022 0.00050 0.00105 0.00204 0.00374

0.00015 0.00033 0.00071 0.00140 0.00260

0.00009 0.00022 0.00047 0.00095 0.00181

0.00006 0.00015 0.00032 0.00065 0.00125

0.00004 0.00010 0.00021 0.00044 0.00086

0.03337 0.05038 0.07293 0.10163 0.13683

0.02437 0.03752 0.05536 0.07861 0.10780

0.01768 0.02773 0.04168 0.06027 0.08414

0.01273 0.02034 0.03113 0.04582 0.06509

0.00912 0.01482 0.02308 0.03457 0.04994

0.00649 0.01073 0.01700 0.02589 0.03802

0.00159 0.00279 0.00471 0.00763 0.01192

0.17851 0.22629 0.27941 0.33680 0.39713

0.14319 0.18472 0.23199 0.28426 0.34051

0.11374 0.14925 0.19059 0.23734 0.28880

0.08950 0.11944 0.15503 0.19615 0.24239

0.06982 0.09471 0.12492 0.16054 0.20143

0.45894 0.52074 0.58109 0.63873 0.69261

0.39951 0.45989 0.52025 0.57927 0.63574

0.34398 0.40173 0.46077 0.51980 0.57756

0.29306 0.34723 0.40381 0.46160 0.51937

0.74196 0.78629 0.82535 0.85915 0.88789

0.68870 0.73738 0.78129 0.82019 0.85404

0.63295 0.68501 0.73304 0.77654 0.81526

,2=31 m = 15.5 0.00001 0.00003 0.00006 0.00014 0.00030

32 16.0 0.00001 0.00002 0.00004 0.00009 0.00020

33 16.5

0.00059 0.00110 0.00197 0.00337 0.00554

0.00460

0.00324

0.00773

0.00553

0.01244 0.01925 0.02874

0.00905 0.01423 0.02157

0.00227 0.00394 0.00655 0.01045 0.01609

0.05403 0.07446 0.09976 0.13019 0.16581

0.04148 0.05807 0.07900 0.10465 0.13526

0.03162 0.04494 0.06206 0.08343 0.10940

0.02394 0.03453 0.04838 0.06599 0.08776

0.01800 0.02635 0.03745 0.05180 0.06985

0.24716 0.29707 0.35029 0.40576 0.46237

0.20645 0.25168 0.30087 0.35317 0.40760

0.17085 0.21123 0.25597 0.30445 0.35588

0.14015 0.17568 0.21578 0.26004 0.30785

0.11400 0.14486 0.18031 0.22013 0.26392

0.09199 0.11846 0.14940 0.18475 0.22429

0.57597 0.63032 0.68154 0.72893 0.77203

0.51898 0.57446 0.62784 0.67825 0.72503

0.46311 0.51860 0.57305 0.62549 0.67513

0.40933 0.46379 0.51825 0.57171 0.62327

0.35846 0.41097 0;46445 0.51791 0.57044

0.31108 0.36090 0.41253 0.46507 0.51760

0.26761 0.31415 0.36322 0.41400 0.46565

34 17.0

35 17.5

36 18.0

37 18.5

38 19.0

39 19.5

40 20.0

0.00001 0.00003 0.00006 0.00013

0.00001 0.00002 0.00004 0.00009

0.00001 0.00003 0.00006

0.00001 0.00002 0.00004

0.00001 0.00003

0.00001 0.00002

0.00001

0.00001

0.00040 0.00076 0.00138 0.00240 0.00401

0.00027 0.00053 0.00097 0.00170 0.00288

0.00019 0.00036 0.00068 0.00120 0.00206

0.00012 0.00025 0.00047 0.00085 0.00147

0.00008 0.00017 0.00032 0.00059 0.00104

0.00006 0.00012 0.00022 0.00041 0.00074

0.00004 0.00008 0.00015 0.00029 0.00052

0.00003 0.00005 0.00011 0.00020 0.00036

0.00002 0.00004 0.00007 0.00014 0.00026

0.00878 0.01346 0.01997 0.02879 0.04037

0.00644 0.01000 0.01505 0.02199 0.03125

0.00469 0.00739 0.01127 0.01669 0.02404

0.00341 0.00543 0.00840 0.01260 0.01838

0.00246 0.00397 0.00622 0.00945 0.01397

0.00177 0.00289 0.00459 0.00706 0.01056

0.00127 0.00210 0.00337 0.00524 0.00793

0.00090 0.00151 0.00246 0.00387 0.00593

0.00064 0.00109 0.00179 0.00285 0.00442

0.00045 0.00078 0.00129 0.00209 0.00327

0.05519 0.07366 0.09612 0.12279 0.15378

0.04330 0.05855 0;07740 0.10014 0.12699

0.03374 iiO462i 0.06187 0.08107 0.10407

0.02613 0.03624 0.04912 0.06516 0.08467

0.02010 0.02824 0.03875 0.05202 0.06840

0.01538 0.02187 0.03037 0.04125 0.05489

0.01170 0.01683 0.02366 0.03251 0.04376

0.00886 0.01289 0.01832 0.02547 0.03467

0.00667 ii00981 0.01411 0.01984 0.02731

0.00500 0.00744 0.01081 0.01537 0.02139

0.18902 0.22827 0.27114 0.31708 0.36542

0.15801 0.19312 0.23208 0.27451 0.31987

0.13107 0.16210 0.19707 0.23574 0.27774

0.10791 0.13502 0.16605 0.20087 0.23926

0.08820 0.11165 0.13887 0.16987 0.20454

0.07160 0.09167 0.11530 0.14260 0.17356

0.05774 0.07475 0.09507 0.11886 0.14622

0.04626 0.06056 0.07786 0.09840 0.12234

0.03684 0.04875 0.06336 0.08092 0.10166

0.02916 0.03901 0.05124 0.06613 0.08394

0.41541

0.36753

0.32254

0.28083

0.24264

0.20808

0.17714

0.14975

0.12573

0.10486

PROBABILITY

PROBABILITY

V

10

INTEGRAL CUMULATIVE

x2=42 m=21

44 22

983

FUNCTIONS

QF xZ-DISTRIBUTION, INCOMPLETE GAMMA SUMS OF THE POISSON DISTRIBUTION

46 23

48 24

2

E

FUNCTION

54 27

56 28

0.00001 0.00002 0.00003 0.00006 0.00012

0.00001 0.00002 0.00003 0.00006

0.00001 0.00001 0.00003

0.00001 0.00001

0.00001

0.00023 0.00040 0.00067 0.00111 0.00177

0.00011 0.00020 0.00034 0.00058 0.00094

0.00005 0.00010 0.00017 0.00030 0.00050

0.00003 0.00005 0.00009 0.00015 0.00026

0.00001 0.00002 0.00004 0.00008 0.00013

0.00001 0.00001 0.00002 0.00004 0.00007

0.00001 0.00001 0.00002 0.00003

0.00001 0.00001 0.00002

0.00001

0.00277 0.00421 0.00625 0.00908 0.01291

0.00151 0.00234 0.00355 0.00526 0.00763

0.00081 0.00128 0.00198 0.00299 0.00443

0.00043 0.00069 0.00109 0.00167 0.00252

0.00022 0.00036 0.00059 0.00092 0.00142

0.00011 0.00019 0.00031 0.00050 0.00078

0.00006 0.00010 0.00016 0.00027 0.00043

0.00003 0.00005 0.00009 0.00014 0.00023

0.00001 0.00003 0.00004 0.00007 0.00012

0.00001 0.00001 0.00002 0.00004 0.00006

s"7 28 29

0.01797 0.02455 0.03292 0.04336 0.05616

0.01085 0.01512 0.02068 0.02779 0.03670

0.00642 0.00912 0.01272 0.01743 0.02346

0.00373 0.00540 0.00768 0.01072 0.01470

0.00213 0.00314 0.00455 0.00647 0.00903

0.00120 0.00180 0.00265 0.00384 0.00545

0.00066 0.00102 0.00152 0.00224 0.00324

0.00036 0.00056 0.00086 0.00129 0.00189

0.00020 0.00031 0.00048 0.00073 0.00109

0.00011 0.00017 0.00026 0.00041 0.00062

30

0.07157

0.04769

0.03107

0.01983

0.01240

0.00762

0.00460

0.00273

0.00160

0.00092

68 34

70 35

:: 13 14

15 16 17 :: 20 21 22 s: 25

V

21 22

x2=62 m=31

64 32

66 33

22; 25

0.00001 0.00001 0.00002 0.00003 0.00006

0.00001 0.00001 0.00002 0.00003

0.00001 0.00001 0.00002

0.00001

26 27 28 29 30

0.00009 0.00014 0.00023 0.00035 0.00052

0.00005 0.00008 0.00012 0.00019 0.00029

0.00003 0.00004 0.00007 0.00011 0.00016

0.00001 0.00002 0.00004 0.00006 0.00009

Interpolation

0.00001 0.00001 0.00002 0.00003 0.00005

74 37

72 36

0.00001 0.00001 0.00002 0.00003

0.00001 0.00001 0.00001

76 38

0.00001

on x3 Q(x2~~)=Q(x~~~0-4)[~~2]+Q(X~/YO-2)[~-~2]+Q(x~~’(1)[1-~+~~2]

Double Entry

Interpolation

Q (+)=Q

(~;~v~-4)[;+~]+Q

(~;~~,-2)[+~~-w~]+4?

(xi1 ~&w~-;~L’+w~]

tQ (x~~~o)[l-w2-@J+~ &u@J]+Q (X;iYO+1)[+2+;

-

Table 26.7

-

w-w9-j

984

PROBABILITY

Table

.\Q

PERCENTAGE

26.8

0.995

0.99

POINTS

OF THE x2 IN TERMS

0.975

FUNCTIONS x2-DISTRIBUTION-VALUES OF Q AND Y 0.95

0.9

OF

0.5 0.75 0.101531 0.454937 0.575364 1.38629 1.212534 2.36597 1.92255 3.35670 2.67460 4.35146

0.25 1.32330 2.77259 4.10835 5.38527 6.62568

0.411740

0.554300

0.831211

0.351846 0.584375 0.710721 1.06362? 1.145476 1.61031

0.675727 0.989265 1.344419 1.734926 2.15585

0.872085 1.239043 1.646482 2.087912 2.55821

1.237347 1.68987 2.17973 2.70039 3.24697

1.63539 2.16735 2.73264 3.32511 3.94030

2.20413 2.83311 3.48954 4.16816 4.86518

3.45460 4.25485 5.07064 5.89883 6.73720

:;

2.60321 3.07382 3.56503 4.07468 4.60094

3.05347 3.57056 4.10691 4.66043 5.22935

3.81575 4.40379 5.00874 5.62872 6.26214

4.57481 5.22603 5.89186 6.57063 7.26094

5.57779 6.30380 7.04150 7.78953 8.54675

7.58412 8.43842 9.29906 10.1653 11.0365

10.3410 11.3403 12.3398 13.3393 14.3389

13.7007 14.8454 15.9839 17.1170 18.2451

16 17 18 19 20

5.14224 5.69724 6.26481 6.84398 7.43386

5.81221 6.40776 7.01491 7.63273 8.26040

6.90766 7.56418 8.23075 8.90655 9.59083

7.96164 9.31223 8.67176 10.0852 9.39046 10.8649 10.1170 11.6509 10.8508 12.4426

11.9122 12.7919 13.6753 14.5620 15.4518

15.3385 16.3381 17.3379 18.3376 19.3374

19.3688 20.4887 21.6049 22.7178 23.8277

21

8.03366 8.64272 9.26042 9.88623 10.5197

8.89720 9.54249 10.19567 10.8564 11.5240

10.28293 10.9823 11.6885 12.4011 13.1197

11.5913 12.3380 13.0905 13.8484 14.6114

13.2396 14.0415 14.8479 15.6587 16.4734

16.3444 17.2396 18.1373 19.0372 19.9393

20.3372 21.3370 22.3369 23.3367 24.3366

24.9348 26.0393 27.1413 28.2412 29.3389

11.1603 11.8076 12.4613 13.1211 13.7867

12.1981 12.8786 13.5648 14.2565 14.9535

13.8439 14.5733 15.3079 16.0471 16.7908

15.3791 16.1513 16.9279 17.7083 18.4926

17.2919 18.1138 18.9392 19.7677 20.5992

20.8434 21.7494 22.6572 23.5666 24.4776

25.3364 26.3363 27.3363 28.3362 29.3360

30.4345 31.5284 32.6205 33.7109 34.7998

40 50 60 70 80

20.7065 27.9907 35.5346 43.2752 51.1720

22.1643 29.7067 37.4848 45.4418 53.5400

24.4331 32.3574 40.4817 48.7576 57.1532

26.5093 34.7642 43.1879 51.7393 60.3915

29.0505 37.6886 46.4589 55.3290 64.2778

33.6603 42.9421 52.2938 61.6983 71.1445

39.3354 49.3349 59.3347 69.3344 79.3343

45.6160 56.3336 66.9814 77.5766 88.1303

190:

59.1963 67.3276

61.7541 70.0648

65.6466 74.2219

69.1260 77.9295

73.2912 82.3581

80.6247 90.1332

89.3342 99.3341

98.6499 109.141

X

-2.5758

-2.3263

-1.9600

-1.6449

-1.2816

-0.6745

0.0000

0.6745

11 113

z3 24 25

From E. S. Pearson and H. 0. Hartley (editors), Biometrika tables for statisticians, Univ. Press, Cambridge, England, 1954 (with permission) for & > 0.0005.

5.34812 7.84080 6.34581 9.03715 7.34412 10.2188 8.34283 11.3887 9.34182 12.5489

vol. I.

Cambridge

PROBABILITY

PERCENTAGE

"\Q

POINTS OF THE x2 IN TERMS

+DISTRIBUTION-VALUES OF Q AND Y

OF

Table

26.8

0.05 3.84146 5.99147 7.81473 9.48773 11.0705

0.025 5.02389 7.37776 9.34840 11.1433 12.8325

0.01 6.63490 9.21034 11.3449 13.2767 15.0863

0.005 7.87944 10.5966 12.8381 14.8602 16.7496

O.001 10.828 13.816 16.266 18.467 20.515

0.0005 12.116 15.202 17.730 19.997 22.105

0.0001 15.137 18.421 21.108 23.513 25.745

8' 9 10

10.6446 12.0170 13.3616 14.6837 15.9871

12.5916 14.0671 15.5073 16.9190 18.3070

14.4494 16.0128 17.5346 19.0228 20.4831

16.8119 18.4753 20.0902 21.6660 23.2093

18.5476 20.2777 21.9550 23.5893 25.1882

22.458 24.322 26.125 27.877 29.588

24.103 26.018 27.868 29.666 31.420

27.856 29.877 31.828 33.720 35.564

;: 13 14 15

17.2750 18.5494 19.8119 21.0642 22.3072

19.6751 21.0261 22.3621 23.6848 24.9958

21.9200 23.3367 24.7356 26.1190 27.4884

24.7250 26.2170 27.6883 29.1413 30.5779

26.7569 28.2995 29.8194 31.3193 32.8013

31.264 32.909 34.528 36.123 37.697

33.137 34.821 36.478 38.109 39.719

37.367 39.134 40.871 42.579 44.263

23.5418 24.7690 25.9894 27.2036 28.4120

26.2962 27.5871 28.8693 30.1435 31.4104

28.8454 30.1910 31.5264 32.8523 34.1696

31.9999 33.4087 34.8053 36.1908 37.5662

34.2672 35.7185 37.1564 38.5822 39.9968

39.252 40.790 42.312 43.820 45.315

41.308 42.879 44.434 45.973 47.498

45.925 47.566 49.189 50.796 52.386

29.6151 30.8133 32.0069 33.1963 34.3816

32.6705 33.9244 35.1725 36.4151 37.6525

35.4789 36.7807 38.0757 39.3641 40.6465

38.9321 40.2894 41.6384 42.9798 44.3141

41.4010 42.7956 44.1813 45.5585 46.9278

46.797 48.268 49.728 51.179 52.620

49.011 50.511 52.000 53.479 54.947

53.962 55.525 57.075 58.613 60.140

35.5631 36.7412 37.9159 39.0875 40.2560

38.8852 40.1133 41.3372 42.5569 43.7729

41.9232 43.1944 44.4607 45.7222 46.9792

45.6417 46.9630 48.2782 49.5879 50.8922

48.2899 49.6449 50.9933 52.3356 53.6720

54.052 55.476 56.892 58.302 59.703

56.407 57.858 59.300 60.735 62.162

61.657 63.164 64.662 66.152 67.633

51.8050 63.1671 74.3970 85.5271 96.5782

55.7585 67.5048 79.0819 90.5312 101.879

59.3417 71.4202 83.2976 95.0231 106.629

63.6907 76.1539 88.3794 100.425 112.329

66.7659 79.4900 91.9517 104.215 116.321

73.402 86.661 99.607 112.317 124.839

76.095 89.560 102.695 115.578 128.261

82.062 95.969 109.503 122.755 135.783

118.136 129.561

124.116 135.807

128.299 140.169

137.208 149.449

140.782 153.167

148.627 161.319

: 43 5 6

40 50 60 ;o" 90 100 X

0.1 2.70554 4.60517 6.25139 7.77944 9.23635

985

FUNCTIONS

107.565 118.498 1.2816

113.145 124.342 1.6449

i

1.9600

2.3263

2.5758

3.0902

3.2905

3.7190

986

PROBABILITY

Table

26.9

PERCENTAGE

FUNCTIONS

POINTS

OF THE

OF F IN TERMS

F-DISTRIBUTION

-VALUES

OF Q, 5, u,

Q(Fjq,vz)=0.5 myl

6 i 9 10 :: :: 15

:: :: 25

1

2

3

4

5

6

8

20

30

1.00 0.667 0.585 0.549 0.528

1.50 1.00 0.881 0.828 0.799

1.71 1.13 1.00 0.941 0.907

1.82 1.21 1.06 1.00 0.965

1.89 1.25 1.10 1.04 1.00

1.94 1.28 1.13 1.06 1.02

2.00 1.32 1.16 1.09 1.05

2.07 1.36 1.20 1.13 1.09

2.09 1.38 1.21 1.14 1.10

2.12 1.39 1.23 1.15 1.11

2.15 1.41 1.24 1.16 1.12

2.17 1.43 1.25 1.18 1.14

2.20 1.44 1.27 1.19 1.15

0.515 0.506 0.499 0.494 0.490

0.780 0.767 0.757 0.749 0.743

0.886 0.871 0.860 0.852 0.845

0.942 0.926 0.915 0.906 0.899

0.977 0.960 0.948 0.939 0.932

1.00 0.983 0.971 0.962 0.954

1.03 1.01 1.00 0.990 0.983

1.06 1.04 1.03 1.02 1.01

1.07 1.05 1.04 1.03 1.02

1.08 1.07 1.05 1.04 1.03

1.10 1.08 1.07 1.05 1.05

1.11 1.09 1.08 1.07 1.06

1.12 1.10 1.09 1.08 1.07

0.486 0.484 0.481 0.479 0.478

0.739 0.735 0.731 0.729 0.726

0.840 0.835 0.832 0.828 0.826

0.893 0.888 0.885 0.881 0.878

0.926 0.921 0.917 0.914 0.911

0.948 0.943 0.939 0.936 0.933

0.977 0.972 0.967 0.964 0.960

1.01 1.00 0.996 0.992 0.989

1.02 1.01 1.01 1.00 1.00

1.03 1.02 1.02 1.01 1.01

1.04 1.03 1.03 1.03 1.02

1.05 1.05 1.04 1.04 1.03

1.06 1.06 1.05 1.05 1.05

0.476 0.475 0.474 0.473 0.472

0.724 0.722 0.721 0.719 0.718

0.823 0.821 0.819 0.818 0.816

0.876 0.874 0.872 0.870 0.868

0.908 0.906 0.904 0.902 0.900

0.930 0.928 0.926 0.924 0.922

0.958 0.955 0.953 0.951 0.950

0.986 0.983 0.981 0.979 0.977

0.997 0.995 0.992 0.990 0.989

1.01 1.01 1.00 1.00 1.00

1.02 1.02 1.02 1.01 1.01

1.03 1.03 1.03 1.02 1.02

1.04 1.04 1.04 1.04 1.03

0.471 0.470 0.470 0.469 0.468

0.716 0.715 0.714 0.714 0.713

0.815 0.814 0.813 0.812 0.811

0.867 0.866 0.864 0.863 0.862

0.899 0.898 0.896 0.895 0.894

0.921 0.919 0.918 0.917 0.916

0.948 0.947 0.945 0.944 0.943

0.976 0.974 0.973 0.972 0.971

0.987 0.986 0.984 0.983 0.982

0.998 0.997 0.996 0.994 0.993

1.01 1.01 1.01 1.01 1.00

1.02 1.02 1.02 1.02 1.02

1.03 1.03 1.03 1.03 1.03

0.468 0.467 0.467 0.466 0.466

0.712 0.711 0.711 0.710 0.709

0.810 0.809 0.808 0.808 0.807

0.861 0.861 0.860 0.859 0.858

0.893 0.892 0.892 0.891 0.890

0.915 0.914 0.913 0.912 0.912

0.942 0.941 0.940 0.940 0.939

0.970 0.969 0.968 0.967 0.966

0.981 0.980 0.979 0.978 0.978

0.992 0.991 0.990 0.990 0.989

1.00 1.00 1.00 1.00 1.00

1.01 1.01 1.01 1.01 1.01

1.03 1.03 1.02 1.02 1.02

0.463 0.461 0.458 0.455

0.705 0.701 0.697 0.693

0.802 0.798 0.793 0.789

0.854 0.849 0.844 0.839

0.885 0.880 0.875 0.870

0.907 0.901 0.896 0.891

0.934 0.928 0.923 0.918

0.961 0.956 0.950 0.945

0.972 0.967 0.961 0.956

0.983 0.978 0.972 0.967

0.994 0.989 0.983 0.978

1.01 1.00 0.994 0.989

1.02 1.01 1.01 1.00

Q(Fly,&)=0.25 5 6 8 12

15

2

3

4

5.83 2.57 2.02 1.81 1.69

7.50 3.00 2.28 2.00 1.85

8.20 3.15 Et

8.82 3.28 2.41 2.07 1.89

8.98 3.31 2.42 2.08 1.89

9.19 3.35 2.44 2.08 1.89

9.41 3.39 2.45 2.08 1.89

9.49 3.41 2.46 2.08 1.89

9.58 S-E

1:se

8.58 3.23 2.39 2.06 1.89

: 10

1.62 1.57 1.54 1.51 1.49

1.76 1.70 1.66 1.62 1.60

1.78 1.72 1.67 1.63 1.60

1.79 1.72 1.66 1.63 1.59

1.79 1.71 1.66 1.62 1.59

1.78 1.71 1.65 1.61 1.58

1.78 1.70 1.64 1.60 1.56

1.77 1.68 1.62 1.58 1.54

1.76 1.68 1.62 1.57 1.53

1.76 1.67 1.61

:: 13 14 15

1.47 1.46 1.45 1.44 1.43

1.58 1.56 1.55 1.53 1.52

1.58 1.56 :-::

1.56 1.54 1.52 1.51 1.49

1.55 1.53 1.51 1.50 1.48

1.53 1.51 1.49 1.48 1.46

1.51 1.49 1.47 1.45 1.44

1.50 1.48 1.46 1.44 1.43

i-:;

1:52

1.57 1.55 1.53 1.52 1.51

1.42 1.42 1.41

1.51 1.51 1.50

1.51 1.50 1.49

1.50 1.49 1.48

1.48 1.47 1.46

1.47 1.46 1.45

1.45 1.44 1.43

1.43 1.41 1.40

1.41 1.40

1.49

1.49 1.48

1.47

1.46 1.45

1.44

1.42

1.40 1.40 1.39 1.39 1.39

1.48 1.48 1.47 1.47 1.47

1.48 1.47 1.47 1.46 1.46

1.46 1.45 :-El 1:44

1.44 1.44 1.43 1.43 1.42

1.43 1.42 1.42 1.41 1.41

:'o

1.38 1.38 1.38 1.38 1.38

1.46 1.46 1.46 1.45 1.45

1.45 1.45 1.45 1.45 1.44

1.44 1.43 1.43 1.43 1.42

1.42 1.42 1.41 1.41 1.41

40 60 120 m

1.36 1.35 1.34 1.32

1.44 1.42 1.40 1.39

1.42 1.41 1.39 1.37

1.40 1.38 1.37 1.35

1.39 1.37 1.35 1.33

v2‘"l

6 7

16 17 :9" 20 :: :: 25 26 :ll

1

12

15

20

60

30

60

-

m

9.67 3-44 2.47 2.08 1.88

9.76 3.46 2.47 2.08 1.87

9.85 3.48 2.47 2.08 1.87

1.75 1.66 1.60 1.55 1.51

z 1:59 1.54 1.50

1.74 1.65 1.58 1.53 1.48

1:45 1.43 1.41

i.48 1.45 1.43 1.41 1.40

1.47 1.44 1.42 1.40 1.38

1.45 1.42 1.40 1.38 1.36

1.41 1.40 1.39

1.40 1.39 1.38

1.38 1.37 1.36

1.36 1.35 :*:;

1.34 1.33 1.32

1.40 1.39

1.38 1.37

1.37 1.36

1.34 1.35

1:32

:-:9" .

1.41 1.40 1.40 1.39 1.39

1.38 1.37 1.37 1.36 1.36

1.37 :-:; 1:35 1.34

1.35 1.34 1.34 1.33 1.33

1.33 1.32 1.32 1.31 1.31

1.31 1.30 1.30 1.29 1.28

1.28 1.28 1.27 1.26 1.25

1.41 1.40 1.40 1.40 1.39

1.38 1.38 1.38 1.37 1.37

1.35 1.35 1.34 1.34 1.34

1.34 1.33 1.33 1.32 1.32

1.32 1.32 1.31 1.31 1.30

:-:: 1:29 1.29 1.28

1.28 1.27 1.27 1.26 1.26

1.25 1.24 1.24 1.23 1.23

1.37 1.35 1.33 1.31

1.35 1.32 1.30 1.28

1.31 1.29 1.26 1.24

1.30 1.27 1.24 1.22

1.28 1.25 1.22 1.19

1.25 1.22 1.19 1.16

1.22 1.19 1.16 1.12

1.19 1.15 1.10 1.00

iO8 1.88

i-2.

Compiled from E. S. Pearsonand H. 0. Hartley (editors), Biometrika tables for statisticians, vol. I. Cambridge Univ. Press, Cambridge, England, 1954(with permission).

PROBABILITY

PERCENTAGE

POINTS

OF THE

OF F IN TERMS

987

FUNCTIONS

F-DISTRIBUTION

-VALUES

Table

26.9

OF Q, u,, v,

vz\vl 1 1 39.86 8.53 : 5.54 4 4.54 5 4.06

2

3

5

6

8

12

15

20

53.59 9.16 5.39 4.19 3.62

30

60

co

49.50 9.00 5.46 4.32 3.78

55.83 9.24 5.34 4.11 3.52

57.24 9.29 5.31 4.05 3.45

58.20 9.33 5.28 4.01 3.40

59.44 9.37 5.25 3.95 3.34

60.71 9.41 5.22 3.90 3.27

61.22 9.42 5.20 3.87 3.24

61.74 9.44 5.18 3.84 3.21

62.26 9.46 5.17 3.82 3.17

62.79 9.47 5.15 3.79 3.14

63.33 9.49 5.13. 3.76 3.10

9" 10

3.78 3.59 3.46 3.36 3.29

3.46 3.26 3.11 3.01 2.92

3.29 3.07 2.92 2.81 2.73

3.18 2.96 2.81 2.69 2.61

3.11 2.88 2.73 2.61 2.52

3.05 2.83 2.67 2.55 2.46

2.98 2.75 2.59 2.47 2.38

2.90 2.67 2.50 2.38 2.28

2.87 2.63 2.46 2.34 2.24

2.84 2.59 2.42 2.30 2.20

2.80 2.56 2.38 2.25 2.16

2.76 2.51 2.34 2.21 2.11

2.72 2.47 2.29 2.16 2.06

:: 13 14 15

3.23 3.18 3.14 3.10 3.07

2.86 2.81 2.76 2.73 2.70

2.66 2.61 2.56 2.52 2.49

2.54 2.48 2.43 2.39 2.36

2.45 2.39 2.35 2.31 2.27

2.39 2.33 2.28 2.24 2.21

2.30 2.24 2.20 2.15 2.12

2.21 2.15 2.10 2.05 2.02

2.17 2.10 2.05 2.01 1.97

2.12 2.06 2.01 1.96 1.92

2.08 2.01 1.96 1.91 1.87

2.03 1.96 1.90 1.86 1.82

1.97 1.90 1.85 1.80 1.76

3.05 3.03 3.01 2.99 2.97

2.67 2.64 2.62 2.61 2.59

2.46 2.44 2.42 2.40 2.38

2.33 2.31 2.29 2.27 2.25

2.24 2.22 2.20 2.18 2.16

2.18 2.15 2.13 2.11 2.09

2.09 2.06 2.04 2.02 2.00

1.99 1.96 1.93 1.91 1.89

1.94 1.91 1.89 1.86 1.84

1.89 1.86 1.84 1.81 1.79

1.84 1.81 1.78 1.76 1.74

1.78 1.75 1.72 1.70 1.68

1.72 1.69 1.66 1.63 1.61

::

2.96 2.95 2.94 2.93 2.92

2.57 2.56 2.55 2.54 2.53

2.36 2.35 2.34 2.33 2.32

2.23 2.22 2.21 2.19 2.18

2.14 2.13 2.11 2.10 2.09

2.08 2.06 2.05 2.04 2.02

1.98 1.97 1.95 1.94 1.93

1.87 1.86 1.84 1.83 1.82

1.83 1.81 1.80 1.78 1.77

1.78 1.76 1.74 1.73 1.72

1.72 1.70 1.69 1.67 1.66

1.66 1.64 1.62 1.61 1.59

1.59 1.57 1.55 1.53 1.52

E 28 29 30

2.91 2.90 2.89 2.89 2.88

2.52 2.51 2.50 2.50 2.49

2.31 2.30 2.29 2.28 2.28

2.17 2.17 2.16 2.15 2.14

2.08 2.07 2.06 2.06 2.05

2.01 2.00 2.00 1.99 1.98

1.92 1.91 1.90 1.89 1.88

1.81 1.80 1.79 1.78 1.77

1.76 1.75 1.74 1.73 1.72

1.71 1.70 1.69 1.68 1.67

1.65 1.64 1.63 1.62 1.61

1.58 1.57 1.56 1.55 1.54

1.50 1.49 1.48 1.47 1.46

40 60 120 m

2.84 2.79 2.75 2.71

2.44 2.39 2.35 2.30

2.23 2.18 2.13 2.08

2.09 2.04 1.99 1.94

2.00 1.95 1.90 1.85

1.93 1.87 1.82 1.77

1.83 1.77 1.72 1.67

1.71 1.66 1.60 1.55

1.66 1.60 1.55 1.49

1.61 1.54 1.48 1.42

1.54 1.48 1.41 1.34

1.47 1.40 1.32 1.24

1.38 1.29 1.19 1.00

;

:7" :9" 20 :: 23

1

2

3

4

6

12

15

20

30

60

161.4 18.51 10.13 7.71 6.61

199.5 19.00 9.55 6.94 5.79

215.7 19.16 9.28 6.59 5.41

4 224.6 19.25 9.12 6.39 5.19

5

1 2 3 4 5

230.2 19.30 9.01 6.26 5.05

234.0 19.33 8.94 6.16 4.95

238.9 19.37 8.85 6.04 4.82

z43.9 19.41 8.74 5.91 4.68

245.9 19.43 8.70 5.86 4.62

248.0 19.45 8.66 5.80 4.56

250.1 19.46 8.62 5.75 4.50

252.2 19.48 8.57 5;i9 4.43

6

5.99 5.59 5.32 5.12 4.96

5.14 4.74 4.46 4.26 4.10

4.76 4.35 4.07 3.86 3.71

4.53 4.12 3.84 3.63 3.48

4.39 3.97 3.69 3.48 3.33

4.28 3.87 3.58

4.00 3.57 3.28 3.07 2.91

3.94 3.51 3.22 3.01 2.85

3.87 3.44

::::

4.15 3.73 3.44 3.23 3.07

:-;: 2177

3.81 3.38 3.08 2.86 2.73

3.74

i 9 10

:*'o: 2179 2.62

3.67 3.23 2.93 2.71 2.54

:: 13 14 15

4.84 4.75 4.67 4.60 4.54

3.98 3.89 3.81 3.74 3.68

3.59 3.49 3.41 3.34 3.29

3.36 3.26 3.18 3.11 3.06

3.20 3.11 3.03 2.96 2.90

3.09 3.00 2.92 2.85 2.79

2.95 2.85 2.77 2.70 2.64

2.79 2.69 2.60 2.53 2.48

2.72 2.62 2.53 2.46 2.40

2.65 2.54 2.46 2.39 2.33

2.57 2.47 2.38 2.31 2.25

2.49 2.38 2.30 2.22 2.16

2.40 2.30 2.21 2.13 2.07

4.49 4.45 4.41 4.38 4.35

3.63 3.59 3.55 3.52 3.49

3.24 3.20 3.16 3.13 3.10

3.01 2.96 2.93 2.90 2.87

2.85 2.81 2.77 2.74 2.71

2.74 2.70 2.66 2.63 2.60

2.59 2.55 2.51 2.48 2.45

2.42 2.38 2.34 2.31 2.28

2.35 2.31 2.27 2.23 2.20

2.28 2.23 2.19 2.16 2.12

2.19 2.15 2.11 2.07 2.04

2.11 2.06 2.02 1.98 1.95

2.01 1.96 1.92 1.88 1.84

4.32 4.30 4.28 4.26 4.24

3.47 3.44 3.42 3.40 3.39

3.07 3.05 3.03 3.01 2.99

2.84 2.82 2.80 2.78 2.76

2.68 2.66 2.64 2.62 2.60

2.57 2.55 2.53 2.51 2.49

2.42 2.40 2.37 2.36 2.34

2.25 2.23 2.20 2.18 2.16

2.18 2.15 2.13 2.11 2.09

2.10 2.07 2.05 2.03 2.01

2.01 1.98 1.96 1.94 1.92

1.92 1.89 1.86 1.84 1.82

1.81 1.78 1.76 1.73 1.71

4.23 4.21 4.20 4.18 4.17

3.37 3.35 3.34 3.33 3.32

2.98 2.96 2.95 2.93 2.92

2.74 2.73 2.71 2.70 2.69

2.59 2.57 2.56 2.55 2.53

2.47 2.46 2.45 2.43 2.42

2.32 2.31 2.29 2.28 2.27

2.15 2.13 2.12 2.10 2.09

2.07 2.06 2.04 2.03 2.01

1.99 1.97 1.96 1.94 1.93

1.90 1.88 1.87 1.85 1.84

1.80 1.79 1.77 1.75 1.74

1.69 1.67 1.65 1.64 1.62

4.08 4.00 3.92 3.84

3.23 3.15 3.07 3.013

2.84 2.76 2.68 2.60

2.61 2.53 2.45 2.37

2.45 2.37 2.29 2.21

2.34 2.25 2.17 2.10

2.18 2.10 2.02 1.94

2.00 1.92 1.83 1.75

1.92 1.84 1.75 1.67

1.84 1.75 1.66 1.57

1.74 1.65 1.55 1.46

1.64 1.53 1.43 1.32

1.51 1.39 1.25 1.00

%\“l

16 17 :9" 20 :: 2 25 26 2 29 30 40 1;: co

8

co 254.3 19.50 8.53 5.63 4.36

988

PROBABILITY

Table

"‘gl

26.9

PERCENTAGE

2

3

1 647.8 38.51 17.44 12.22 10.01

799.5 39.00 16.04 10.65 a.43

864.2 39.17 15.44 9.98

8.81 8.07 7.57 7.21 6.94

7.26 6.54 6.06 5.71 5.46

6.60 5.89 5.42 5.08 4.83

6.72 6.55 6.41 6.30 6.20

5.26 5.10 4.97 4.86 4.77

6.12 6.04 5.98 5.92 5.87

POINTS

FUNCTIONS

OF THE

F-DISTRIBUTION

OF F IN TERMS OF Q, v,, v, Q(F(Y.q)=0.025 4 5 8 12 6 899.6 39.25 15.10 9.60 7.39

921.8 39.30 14.88 9.36 7.15

937.1 39.33 14.73 9.20 6.98

956.7 39.37 14.54 a.98

6.23

5.52 5.05 4.72 4.47

5.99 5.29 4.82 4.48 4.24

5.82 5.12 4.65 4.32 4.07

4.63 4.47 4.35 4.24 4.15

4.28 4.12 4.00 3.89 3.80

4.04 3.89 3.77 3.66 3.58

4.69 4.62 4.56 4.51 4.46

4.08 4.01 3.95 3.90 3.86

3.73 3.66 3.61

3.50 3.44

3.51

::

5.83 5.19 5.75 5.72 5.69

4.42 4.38 4.35 4.32 4.29

3.82 3.78 3.75 3.72 3.69

26 27 28 29 30

5.66 5.63 5.61 5.59 5.51

4.27 4.24 4.22 4.20 4.18

3.67 3.65

40 60 120 m

5.42 5.29 5.15 5.02

6 1 9” 10

11 :: :;

21 22 23

1 2 3 : 6 7 9" 10 11 :: 14 15

21 22 :: 25

40 126: (D

7.16

15

b.76

976.7 39.41 14.34 a.75 6.52

984.9 39.43 14.25 8.66 6.43

5.60

5.37

5.27 4.57 4.10

4.90 4.43 4.10 3.85 3.66 3.51 z 3:20

-VALUES

20

6.33

5.07 4.36 3.89 3.56 3.31 3.12 2.96

39.45 14.17 8.56

4.20 3.87 3.62

3.17

3.52

5.17 4.47 4.00 3.67 3.42

3.43 3.28 3.15 3.05 2.96

3.33 3.18 3.05 2.95 2.86

3.23 3.07 2.95 2.84 2.76 2.62 2.56 2.51 2.46

4.67

30 1001 39.46 14.08 8.46 6.23

993.1

3.48 3.44 3.41 3.38 3.35

3.25 3.22 3.18 3.15 3.13

3.09 3.05 3.02 2.99 2.91

2.07 2.84 2.81 2.78 2.15

2.64 2.60 2.57 2.54 2.51

2.53 2.50 2.47 2.44 2.41

2.42 2.39

3.10 3.08 3.06 3.04 3.03

2.94 2.92 2.90 2.88 2.87

2.73 2.71

3.63 3.61 3.59

3.33 3.31 3.29 3.27 3.25

2.67 2.65

2.49 2.47 2.45 2.43 2.41

2.39 2.36 2.34 2.32 2.31

2.28 2.25 2.23 2.21 2.20

2.09 2.07

4.05 3.93 3.80 3.69

3.46 3.34 3.23 3.12

3.13 3.01 2.89 2.79

2.90 2.79 2.67 2.57

2.52 2.41

2.53 2.41 2.30 2.19

2.29 2.17 2.05 1.94

2.18 2.06 1.94 1.83

2.07 1.94 1.82 1.71

1.82 1.69 1.57

2

3

4

5

4052 98.50 34.12 21.20 16.26

4999.5 99.00 30.82 18.00 13.27

5403 99.17 29.46 16.69 12.06

13.75 12.25 11.26 10.56 10.04

10.92 9.55 8.65 8.02 7.56

9.78 a.45

6.55

9.15 7.85 7.01 6.42 5.99

9.65 9.33 9.07 8.86 a.68

7.21 6.93 6.70 6.51 6.36

6.22 5.95 5.74 5.56 5.42

5.67 5.41 5.21 5.04 4.89

a.53 8.4Q a.29 8.18 8.10

6.23 6.11 6.01 5.93 5.85

5.29 5.18 5.09 5.01 4.94

a.02 7.95 7.88 7.82 7.77

5.66 5.61

4.16 4.72

5.51

4.68

7.72 7.68 7.64 7.60 7.56

5.53 5.49 5.45 5.42 5.39

4.64

7.31 7.08 6.85 6.63

5.18 4.98 4.79 4.61

5.78 5.12

7.59 6.99

5625 99.25 28.71 15.98 11.39

4.77 4.67

4.58 4.50 4.43

4.07

4.31

4.82

4.31 4.26 4.22 4.18

2.74 2.63

Q(Fivl,vz)=O.Ol 6 8

15

20

4.85 4.14

3.78

3.08

3.00

2.88

2.45 2.38

30

2:: 2:40 2.32 2.25 2.19 2.13 2.09 2.04 2.00 1.97 1.94 1.91

2.03 2.00

1.88 1.85 I.83 1.81 1.79

1.98 1.96 1.94

1.94

2.72

2.18 2.14 2.11 2.08

2.05

2.16 2.13 2.11

3.67 3.33

3.45 3.20 2.85

2.31 2.27 2.24 2.21 2.18

6.02

1.80

1.61 1.53 1.39

60

1.00

m 6366 99.50

5859 99.33 27.91 15.21 10.67

5982 99.37 27.49 14.80 10.29

6106 99.42 27.05 14.37 9.89

6157 99.43 26.87 14.20 9.72

6209 99.45 26;69 14.02 9.55

6261 99.47 26;50 13.84 9.38

13.65 9.20

26;13 13.46 9.02

a.75

a.47 7.19

a.10 6.84 b.03

7.40 6.16 5.36

5.06

7.56 6.31 5.52 4.96 4.56

7.23 5.99

6.31

1.72 6.47 5.67 5.11 4.71

4.81 4.41

E 4:25

7.06 5.82 5.03 4.48 4.08

6.88 5.65 4.86 4.31 3.91

4.40 4.16 3.96 3.80 3.67

4.25 4.01 3.82

4.10 3.86 3.66

3.94 3.70 3.51 3.35 3.21

3.78 3.54 3.34 3.18 3.05

3.60 3.36 3.17 3.00 2.87

3.26 3.16

3.10 3.00 2.92

2.93 2.83 2.75 2.67 2.61

2.75 2.65 2.51 2.49 2.42 2.36 2.31 2.26 2.21 2.17

7.46

6313

1.64 1.48 1.31

5764 99.30 28.24 15.52 10.97

99.48 26;32

6.63 6.06 5.64

5.80 5.39

5.32 5.06 4.86

4.69

5.07 4.82 4.62 4.46

4.56

4.32

4.74 4.50 4.30 4.14 4.00

4.44 4.34 4.25 4.17 4.10

4.20 4.10 4.01 3.94 3.87

3.89 3.79 3.71 3.63 3.56

3.55 3.46 3.37 :::t

3.41 3.31 3.23 3.15 3.09

4.04 3.99 3.94 3.90 3.85

3.81 3.76 3.71 3.67 3.63

3.51 3.45 3.41 3.36 3.32

3.17 3.12 3.07 3.03 2.99

3.03 2.98 2.93 2.89 2.85

2.74 2.70

2.54

2.55 2.50 2.45 2.40 2.36

3.82

3.59

3.29

2.96

2.93 2.90 2.87 2.84

2.81 2.78 2.15 2.73 2.70

2.66 2.63 2.60 2.51 2.55

2.50 2.47 2.44 2.41 2.39

2.33 2.29 2.26 2.23 2.21

2.13 2.10 2.06 2.03 2.01

2.50 2.34 2.18

2.66

2.52 2.35 2.19 2.04

2.37 2.20 2.03

2.20 2.03 1.86 1.70

2.02 I.84 1.66 1.47

1.80 1.60 1.38

5.47

4.57 4.54 4.51

4.14 4.11 4.07 4.04 4.02

3.75 3.73 3.70

2:; 3:50 3.41

3.23 3.20 3.17

4.31 4.13 3.95 3.78

3.83 3.65 3.48 3.32

3.51 3.34 3.17 3.02

3.29 3.12 2.96 2.80

2.99 2.82 2.66 2.51

4.60

12

4.96 4.25

2.32 2.27 2.22

:*z 3:29

2.69

8.26

2.57 2.50 2.44 2.39 2.35

2.79 2.72 2.67 2.62 2.57

~0 1018 39.50 13.90

8.36 6.12

2.72 2.61 2.52

2.89 2.82 2.77 2.72 2.68

:% 2:30

13.99

2.84

3.12 3.06 3.01 2.96 2.91

3.56

39.48

2.13 2.64

3.34 3.28 3.22 3.17 3.13

2.68

60 1010

3.18

3.26

3.66

3.52

3.51 3.31

3.08

3.00 2.94 2.88 2.83 2.78

1.88

2.84 2.78

2.12 2.67 2.62 2.58

1.00

PROBABILITY

PEKCEKTAGE

POISTS

OF THE

OF F tS TERW

yz y1 1 2 3 4 5

11 12 13 :;

26 27 28 :; 40 12 m

1

2

3

4

16211 198.5 55.55 11.33 22.78

20000 199.0 49.80 26.28 18.31

21615 199.2 47.47 24.26 16.53

22500 199.2 46.19 23.15 15.56

23056 199.3 45.35 22.46 14.94

la.63 16.24 14.69 13.61 12.83

14.54 12.40 11.04 10.11 9.43

12 ii lo:88 9.60 8.72 8.08

12.03 10.05 8.81 7.96 7.34

11.46 9.52 a.30 7.47 6.87

12.23 11.75 11.37 11.06 10.80

8.91 8.51 a.19 7.92 7.70

7.60 7.23 6.93 6.68 6.48

6.88 6.52 6.23 6.00 5.80

10.58 10.38 10.22 10.07 9.94

7.51 7.35 7.21 7.09 6.99

6.30 6.16 6.03 5.92 5.82

9.83 9.73 9.63 9.55 9.48

6.89 6.81 6.73 6.66 6.60

9.41 9.34 9.28 9.23 9.18 8.83 8.49 8.18 7.88

FUNCTIONS

F-DISTHIBLTIO\

OF Q, y,, Y> Q(Flv,,v,)=0.005

8

12

15

20

30

60

m

23925 199.4 44.13 21.35 13.96

24426 199.4 43.39 20.70 13.38

24630 199.4 43.08 20.44 13.15

24836 199.4 42.78 20.17 12.90

25044 199.5 42.47 19.89 12.66

25253 199.5 42.15 19.61 12.40

25465 199.5 41.83 19.32 12.14

11.07 9.16 7.95 7.13 6.54

10.57 8.68 7.50 6.69 6.12

10.03 8.18 7.01 6.23 5.66

9.81 7.97 6.81 6.03 5.47

9.59 7.75 6.61 5.83 5.27

9.36 7.53 6.40 5.62 5.07

9.12 7.31 6.18 5.41 4.86

a.aa 7.08 5.95 5.19 4.64

6.42 6.07 5.79 5.56 5.37

6.10 5.76 5.48 5.26 5.07

5.68 5.35 5.08 4.86 4.67

5.24 4.91 4.64 4.43 4.25

5.05 4.72 4.46 4.25 4.07

4.86 4.53 4.27 4.06 3.88

4.65 4.33 4.07 3.86 3.69

4.44 4.12 3.87 3.66 3.48

4.23 3.90 3.65 3.44 3.26

5.64 5.5c 5.37 5.27 5.17

5.21 5.07 4.96 4.85 4.76

4.91 4.78 4.66 4.56 4.47

4.52 4.39 4.28 4.18 4.09

4.10 3.97 3.86 3.76 3.68

3.92 3.79 3.68 3.59 3.50

3.73 3.61 3.50 3.40 3.32

3.54 3.41 3.30 3.21 3.12

3.33 3.21 3.10 3.00 2.92

3.11 2.98 2.87 2.78 2.69

5.73 5.65 5.58 5.52 5.46

5.09 5.02 4.95 4.89 4.84

4.68 4.61 4.54 4.49 4.43

4.39 4.32 4.26 4.2( 4.15

4.01 3.94 3.88 3.83 3.78

3.60 3.54 3.47 3.42 3.37

3.43 3.36 3.30 3.25 3.20

3.24 3.18 3.12 3.06 3.01

3.05 2.98 2.92 2.87 2.82

2.84 2.77 2.71 2.66 2.61

2.61 2.55 2.48 2.43 2.38

6.54 6.49 6.44 6.40 6.35

5.41 5.36 5.32 5.28 5.24

4.79 4.74 4.70 4.66 4.62

4.38 4.34 4.30 4.26 4.23

4.10 4.06 4.02 3.98 3.95

3.73 3.69 3.65 3.61 3.58

3.33 3.28 3.25 3.21 3.18

3.15 3.11 3.07 3.04 5.01

2.97 2.93 2.89 2.86 2.82

2.77 2.73 2.69 2.66 2.63

2.56 2.52 2.48 2.45 2.42

2.33 2.29 2.25 2.21 2.18

6.07 5.79 5.54 5.30

4.98 4.73 4.50 4.28

4.37 4.14 3.92 3.72

3.99 3.76 3.55 3.35

3.71 3.49 3.28 3.09

3.35 3.13 2.93 2.74

2.95 2.74 2.54 2.36

2.78 2.57 2.37 2.19

2.60 2.39 2.19 2.00

2.40 2.19 1.98 1.79

xl8 1.96 1.75 1.55

1.93 1.69 1.43 1.00

5

6

23437 199.3 44.84 21.97 14.51

QVI VI, v,) =o *

vpv, 1 2 3 4 5

--\ALUES

1

4 (;;;.;25

Wf53

5 Pi;.;64

6 ';$;.;59

,001

12 8 (5$.9481';$t;o7

15 ';$$.;58

20 W$O9

30 ‘;9;:6’

60 ';$$;13

m ';$2:66

167:0 74.14 47.18

14815 61.25 37.12

141.1 56.18 33.20

13711 53.44 31.09

134.6 51.71 29.75

132:s 50.53 28.84

130:6 49.00 27.64

128:s 47.41 26.42

127:4 46;76 25.91

126:4 46.10 25.39

125:4 45.43 24.87

12415 44.75 24.33

35.51 29.25 25.42 22.86 21.04

27.00 21.69 18.49 16.39 14.91

23.70 la.77 15.83 13.90 12.55

21.92 17.19 14.39 12.56 11.28

20.81 16.21 13.49 11.71 10.48

20.03 15.52 12.86 11.13 9.92

19.03 14.63 12.04 10.37 9.20

17.99 13.71 11.19 9.57 a.45

17.56 13.32 10.84 9.24 8.13

17.12 12.93 10.48 8.90 7.80

16.67 12.53 10.11 a.55 7.47

16.21 12.12 9.73 a.19 7.12

15.75 11.70 9.33 7.81 6.76

19.69 18.64 17.81 17.14 16.59

13.81 12.97 12.31 11.78 11.34

11.56 10.80 10.21 9.73 9.34

10.35 9.63 9.07 8.62 8.25

9.58 8.89 8.35 7.92 7.57

9.05 a.38 7.86 7.43 7.09

8.35 7.71 7.21 6.80 6.47

7.63 7.00 6.52 6.13 5.81

7.32 6.71 6.23 5.85 5.54

7.01 6.40 5.93 5.56 5.25

6.68 6.09 5.63 5.25 4.95

6.35 5.76 5.30 4.94 4.64

6.00 5.42 4.97 4.60 4.31

16.12 15.72 15.38 15.08 14.82

10.97 10.66 10.39 10.16 9.95

9.00 8.73 a.49 8.28 a.10

7.94 7.68 7.46 7.26 7.10

7.27 7.02 6.81 6.62 6.46

6.81 6.56 6.35 6.18 6.02

6.19 5.96 5.76 5.59 5.44

5.55 5.32 5.13 4.97 4.82

5.27 5.05 4.87 4.70 4.56

4.99 4.78 4.59 4.43 4.29

4.70 4.48 4.30 4.14 4.00

4.39 4.18 4.00 3.84 3.70

4.06 3.85 3.67 3.51 3.38

9.77 9.61 9.47 9.34 9.22

7.94 7.80 7.67 7.55 7.45

6.95 6.81 6.69 6.59 6.49

6.32 6.19 6.08 5.98 5.88

5.88 5.76 5.65 5.55 5.46

5.31 5.19 5.09 4.99 4.91

4.70 4.58 4.48 4.39 4.31

4.44 4.33 4.23 4.14 4.06

4.17 4.06 3.96 3.87 3.79

3.88 3.78

:: 25

14.59 14.38 14.19 14.03 13.88

2: 3:52

3.58 3.48 3.38 3.29 3.22

3.26 3.15 3.05 2.97 2.89

:7" 28 29 30

13.74 13.61 13.50 13.39 13.29

9.12 9.02 8.93 8.85 a.77

7.36 7.27 7.19 7.12 7.05

6.41 6.33 6.25 6.19 6.12

5.80 5.73 :z 5:53

5.38 5.31 5.24 5.18 5.12

4.83 4.76 4.69 4.64 4.58

4.24 4.17 4.11 4.05 4.00

3.99 3.92 3.86 3.80 3.75

3.72 3.66 3.60 3.54 3.49

3.44 3.38 3.32 3.27 3.22

3.15 3.08 3.02 2.97 2.92

2.82 2.75 2.69 2.64 2.59

12.61 11.97 11.38 10.83

8.25 7.76 7.32 6.91

6.60 6.17 5.79 5.42

5.70 5.31 4.95 4.62

5.13 4.76 4.42 4.10

4.73 4.37 4.04 3.74

4.21 3.87 3.55 3.27

3.64 3.31 3.02 2.74

3.40 3.08 2.78 2.51

3.15 2.83 2.53 2.27

2.87 2.55 2.26 1.99

2.57 2.25 1.95 1.66

2.23 1.89 1.54 1.00

11 1; :: 15

21 22

40 1;: m

123:5 44.05 23.79

;i’ fi ‘r

990

/

VROBABILITY

FUNCTIONS

IN ‘l’I’I~\lS

0.8

0.9

0.95 0.98

3.078 li886 1.638 1.533 1.476

6.314

12.706

31.821

2.920 2.353 2.132 2.015

4.303

6.965

1.943, 1.895

1.860 1.833 1.812

2.447 2.365 2.306 2.262 2.228

1.796

2.201

1.782

2.179

1

0.325

1.000

2 3

0.289 0.211

0.816 0.765

4

0.271

0.741

5

0.267

0.727

6 7 8 9

0.265 0.263 0.262

0.711

1.415

0.706

10

0.261 0.260

0.703 0.700

1.397 1.383 1.372

11 12 13 14 15

0.260 0.259 0.259 0.258 0.258

0.697 0.695 0.694 0.692

1.363 1.356 1.350 1;345 1.341

16 17 18 19 20

0.258 0,257 0.257 0.257 0.257

0.690 0.689 0.688 0.688 0.687

21 22 23 24 25

0.257 0.256 0.256 0.25.6 0.256

26 27 28 29 30 40 60 120 m

0.718

0.691

1.440

1.771 1.761 1.753

0.99 63.657 9.925

3.182

4.541

2.716 2.571

3.747 3.365

5.841 4.604 4.032

3.143 2.821

3.701 3.499 3.355 3.250

2.764

3.169

2,718 2.681

3.106

2.998 2.896

Ok’ A AND v

0.995

0.998

0.999

0.9999

0.99999

0.999999

127.321 14.089 7.453 5.598 4.713

318.309 22.327 10.214 7.173 5.893

636.619 31.598 12.924 8.610 6.869

6366.198 99.992 28.000 15.544 11.178

63661.977 316.225 60.397 27.771 17.897

636619.772 999.999 130;155 49.459 28.477

4.317 4.029 3.833 3.690 3.581

5.208 4.785 4.501 4.297 4.144

5.959 5.408 5.041 4.781 4.587

9,082 7.885 7.120 6.594 6.211

13.555 11.215 9.782 8.627 8.150

20.047 15.764 13.257 11.637 10.516

4.025 3.930 3.852 3.787 3.733

4.437 4.318 4.221 4.140 4.073

5.921 5.694 5,513 5.363 5.239

7.648 7.261 6.955 6.706 6.502

9.702 9.085 8.604 8.218 7.903

2.160 2.145 2.131

2.650 2.624 2.602

3.012 2.977 2.947

3.497 3.428 3.372 3.326 3.286

2.120 2.110 2.101

2.921 2.898 2.878 2.861 2.845

3.252 3.223 3.197 3.174 3.153

3.686 3.646 3.610 3.579 3.552

4.015 3.965 3.922 3.883 3.850

5.134 5.044 4.966 4.697 4.837

6.330

7.642

%G 51949 5.854

7.421 7.232 7.069 6.927

2.518

3.055

1.325

1.740 1.734 1.729 1.725

2,093 2.086

2.583 2.567 2.552 2.539 2.528

0.686 0.686 0.685 0.685 0.684

1.323 1.321 1.319 1.318 1.316

1.721 1.717 1.714 1.711 1.708

2.080 2.074 2.069 2.064 2.060

2.508 2.500 2,492 2.485

2.831 2.819 2.807 2.797 2.787

3.135 3.119 3.104 3.090 3.078

3.527 3.505 3.485 3.467 3.450

3.819 3.792 3.768 3.745 3.725

4.784 4.736 4.693 4.654 4.619

5.769 5.694 5.627 5.566 5.511

6.802

0.256 0.256 0.256 0.256 0.256

0.684 0.684 0.683 0.683 0.683

1.315 1.314 1.313 1.311

1.706

1.310

1.697

2.056 2.052 2.048 2.045 2.042

2.479 2.473 2.467 2.462 2.457

2.779 2.771 2.163 2.756 2.750

3.067 3.057 3.047 3.038 3.030

3.435 3.421 3.408 3.396 3.385

3.707 3.690 3.674 3.659 3.646

4.587 4.558 4.530 4.506 4.482

5.461 5.415 5.373 5.335 5.299

6.352 6.286 6.225 6.170 6.119

0.255 0.254 0.254 0.253

0.681

1.303

1.684

2.021

2.423

0.679 0.677 0.674

1.296

1.671 1.658 1.645

2.000

1.980 1.960

2.390 2.358 2.326

2.704 2.660 2.617 2.576

2.971 2.915 2.860 2.807

3.307 3.232 3.160 3.090

3.551 3.460 3.373 3.291

4.321 4.169 4.025 3.891

5.053 4.825 4.613 4.417

5.168 5.449 5.158 4.892

1.337 1.333

1.330 1.328

1.289 1.282

1.746

1.703

1.701 1,699

*

*

From E. S. Pearson and H. 0. Hartley (editors), Biometrika tables for statisticians, vol. I. Cambridge Univ. Press, Cambridge, England, 1954 for n 0.999, from E. T. Federighi, Extended tables of the percentage points of Student’s t-distribution, J. Amer. Statist. Assoc. 54, 683-688 (1959) for ,I 0.999 (with permission). *See pageII.

-

6.692

6.593 6.504 6.424

*

PROBABILITY

FIVE

991

FUNCTIONS

DIGIT

R ANDOM

NUMBERS

Table

26.11

53479 97344 66023 99776 30176

81115 70328 38277 75723 48979

98036 58116 74523 03172 92153

12217 91964 71118 43112 38416

59526 26240 84892 83086 42436

40238 44643 13956 81982 26636

40577 83287 98899 14538 83903

39351 97391 92315 26162 44722

43211 92823 65783 24899 69210

69255 77578 59640 20551 69117

81874 19839 09337 31151 67619

83339 90630 33435 58295 52515

14988 71863 53869 40823 03037

99937 95053 52769 41330 81699

13213 55532 18801 21093 17106

30177 60908 25820 93882 64982

47967 84108 96198 49192 60834

93793 55342 66518 44876 85319

86693 48479 78314 47185 47814

98854 63799 97013 81425 08075

61946 04811 05763 73260 54909

48790 64892 39601 56877 09976

11602 96346 56140 40794 76580

83043 79065 25513 13948 02645

22257 26999 86151 96289 35795

11832 43967 78657 90185 44537

04344 63485 02184 47111 64428

95541 93572 29715 66807 35441

20366 80753 04334 61849 28318

55937 96582 15678 44686 99001

42583 27266 49843 29316 30463

36335 27403 11442 40460 27856

60068 97520 66682 27076 67798

04044 23334 36055 69232 16837

29678 36453 32002 51423 74273

16342 33699 78600 58515 05793

48592 23672 36924 49920 02900

25547 45884 59962 03901 63498

63177 41515 68191 26597 00782

75225 04756 62580 33068 35097

28708 13183 60796 13486 34914

84088 50652 76639 46918 94502

65535 94872 30157 64683 39374

44258 28257 40295 07411 34185

33869 78547 99476 77842 57500

82530 55286 28334 01908 22514

98399 33591 15368 47796 04060

26387 61965 42481 65796 94511

02836 51723 60312 44230 44612

36838 14211 42770 77230 10485

28105 59231 87437 29046 62035

04814 45028 82758 01301 71886

85170 01173 71093 55343 94506

86490 08848 36833 65732 15263

35695 81925 53582 78714 61435

03483 71494 25986 43644 10369

57315 95401 46005 46248 42054

63174 34049 42840 53205 68257

71902 04851 81683 94868 14385

71182 65914 21459 48711 79436

38856 40666 40588 78237 98247

80048 43328 90087 86556 67474

59973 87379 37729 50276 71455

73368 86418 08667 20431 69540

52876 95841 37256 00243 01169

47673 25590 20317 02303 03320

41020 54137 53316 71029 67017

82295 94182 50982 49932 92543

26430 42308 32900 23245 97977

87377 07361 32097 00862 52728

69977 39843 62880 56138 90804

78558 23074 87277 64927 56026

65430 40814 99895 29454 48994

32627 03713 99965 52967 64569

28312 21891 34374 86624 67465

61815 96353 42556 62422 60180

14598 96806 11679 30163 12972

79728 24595 99605 76181 03848

55699 26203 98011 95317 62582

91348 26009 48867 39264 93855

09665 34756 12157 69384 93358

44672 50403 73327 07734 64565

74762 76634 74196 94451 43766

33357 12767 26668 76428 45041

67301 32220 78087 16121 44930

80546 34545 53636 09300 69970

97659 18100 52304 67417 16964

11348 53513 00007 68587 08277

78771 14521 05708 87932 67752

45011 72120 63538 38840 60292

38879 58314 83568 28067 05730

35544 60298 10227 91152 75557

99563 72394 99471 40568 93161

85404 69668 74729 33705 80921

04913 12474 22075 64510 55873

62547 93059 10233 07067 54103

78406 02053 21575 64374 34801

01017 29807 20325 26336 83157

86187 63645 21317 79652 04534

22072 12792 57124 31140 81368

Compiled from Rand Corporation, 1955 (with permission).

A million

random

digit s with

100,000 normal deviates.

The Free Press, Glencoe,

Ill.,

992

PROBABILITY Table

26.11

2500

FIVE

DIGIT

FUNCTIONS RANDOM

NUMBERS

26687 60675 45418 69872 03765

74223 75169 98635 48026 86366

43546 24510 83123 89755 99539

45699 15100 98558 28470 44183

94469 02011 09953 44130 23886

82125 14375 60255 59979 89977

37370 65187 42071 91063 11964

23966 10630 40930 28766 51581

68926 64421 97992 85962 18033

37664 66745 93085 77173 56239

a4686 91512 10737 54870 48967

57636 49670 49307 19676 49579

32326 32556 18307 58367 65369

19867 85189 22246 20905 74305

71345 28023 22461 38324 62085

42002 88151 10003 00026 39297

96997 62896 93157 98440 10309

84379 95498 66984 37427 23173

27991 29423 44919 22896 74212

21459 38138 30467 37637 32272

91430 92564 41734 25251 91657

79112 29567 12199 78110 11563

03685 47476 77441 54178 66036

05411 62804 92415 78241 28523

23027 73428 63542 09226 83705

54735 04535 42115 87529 09956

91550 86395 84972 35376 76610

06250 12162 12454 90690 88116

18705 59647 33133 54178 78351

18909 97726 48467 08561 50877

00149 53250 25587 01176 83531

a4745 73200 17481 12182 15544

63222 84066 56716 06882 40834

50533 59620 49749 27562 20296

50159 61009 70733 75456 88576

60433 38542 32733 54261 47815

04822 05758 60365 38564 96540

49577 06178 14108 89054 79462

89049 80193 52573 96911 78666

16162 26466 39391 88906 25353

19902 96516 99417 77699 32245

98866 78705 56171 57853 83794

32805 25556 19848 93213 99528

61091 35181 24352 27342 05150

91587 29064 51844 28906 27246

30340 49005 03791 31052 48263

84909 29843 72127 65815 62156

64047 68949 57958 21637 62469

67750 50506 08366 49385 97048

87638 45862 43190 75406 16511

12874 63899 16255 75553 41772

72753 41910 43271 30207 18441

66469 45484 26540 41814 34685

13782 55461 41298 74985 13892

64330 66518 35095 40223 38843

00056 82486 32170 91223 69007

73324 74694 70625 64238 10362

03920 07865 66407 73012 84125

13193 09724 01050 83100 08814

19466 76490 44225 92041 66785

09270 85058 80222 83901 36303

01245 17815 8'572 88028 57833

81765 71551 62758 56743 77622

06809 36356 14858 25598 02238

10561 97519 36350 79349 53285

10080 54144 23304 47880 77316

17482 51132 70453 77912 40106

05471 83169 21065 52020 38456

82273 27373 63812 84305 92214

06902 68609 29860 02897 54278

91543 14415 82465 27306 91960

63886 33816 07781 39843 82766

60539 78231 09938 05634 02331

96334 87674 66874 96368 08797

20804 96473 72128 72022 33858

72692 44451 99685 01278 21847

08944 25098 84329 92830 17391

02870 29296 14530 40094 53755

74892 50679 08410 31776 58079

22598 07798 45953 41822 48498

59284 10428 65527 59688 44452

96108 96003 41039 43078 10188

91610 71223 79574 93275 43565

07483 21352 05105 31978 46531

37943 78685 59588 08768 93023

96832 55964 02115 84805 07618

15444 35510 33446 50661 12910

12091 94805 56780 18523 60934

36690 23422 18402 83235 53403

58317 04492 36279 50602 18401

87275 94155 26488 37073 83835

82013 93110 76394 34547 89575

59804 49964 91282 88296 55956

78595 27753 03419 68638 93957

60553 85090 68758 12976 30361

14038 77677 89575 50896 47679

12096 69303 66469 10023 83001

95472 66323 97835 27220 35056

42736 77811 66681 05785 07103

08573 22791 03171 77538 63072

PROBABILITY 2500

FIVE

DIGIT

993

FUNCTIONS

RANDOM

Table

NUMBERS

26.11

55034 25521 85421 61219 20230

81217 99536 72744 48390 03147

90564 43233 97242 47344 58854

81943 48786 66383 30413 11650

11241 49221 00132 39392 28415

84512 06960 05661 91365 12821

12288 31564 96442 56203 58931

89862 21458 37388 79204 30508

00760 88199 57671 05330 65989

76159 06312 27916 31196 26675

95776 07603 00645 62950 79350

83206 17344 17459 83162 10276

56144 01148 78742 61504 81933

55953 83300 39005 31557 26347

89787 96955 36027 80590 08068

64426 65027 98807 47893 67816

08448 31713 72666 72360 06659

45707 89013 54484 72720 87917

80364 79557 68262 08396 74166

60262 49755 38827 33674 85519

48339 05842 25855 25272 73003

69834 08439 02209 16152 29058

59047 79836 07307 82323 17605

82175 50957 59942 70718 49298

92010 32059 71389 98081 47675

58446 32910 76159 38631 90445

69591 15842 11263 91956 68919

56205 13918 38787 49909 05676

95700 41365 61541 76253 23823

86211 80115 22606 33970 84892

81310 10024 84671 29296 51771

94430 44713 52806 58162 94074

22663 59832 89124 21858 70630

06584 80721 37691 33732 41286

38142 63711 20897 94056 90583

00146 67882 82339 88806 87680

17496 25100 22627 54603 13961

51115 45345 06142 00384 55627

61458 55743 05773 66340 23670

65790 67618 03547 69232 35109

42166 78355 09552 15771 13231

56251 67041 51347 63127 99058

60770 22492 33864 34847 93754

51672 51522 89018 05660 36730

36031 31164 73418 06156 44286

77273 30450 81538 48970 44326

85218 27600 77399 55699 15729

14812 44428 30448 61818 37500

90758 96380 97740 91763 47269

23677 26772 18158 20821 13333

50583 99485 54676 99343 35492

03570 57330 39524 71549 40231

38472 i0634 73785 10248 34868

73236 74905 48864 76036 55356

67613 90671 69835 31702 12847

72780 19643 62798 76868 68093

78174 69903 65205 88909 52643

18718 60950 69187 69574 32732

99092 17968 05572 27642 67016

64114 37217 74741 00336 46784

98170 25384 02670 86155 36934 42879 56851 12778 05464 28892

03841 56860 81637 24309 14271

23920 02592 79952 73660 23778

47954 01646 07066 84264 88599

10359 42200 41625 24668 17081

70114 79950 96804 16686 33884

11177 37764 92388 02239 88783

63298 82341 88860 66022 39015

99903 71952 68580 64133 57118

15025

20237 08030 905982 40378 81431 99955 21431 59335

63386 81469 05731 52462 58627

71122 91066 55128 67667 94822

06620 88857 74298 97322 65484

07415 56583 49196 69808 09641

94982 01224 31669 21240 41018

32324 28097 42605 65921 85100

79427 19726 30368 12629 16110

70387 71465 96424 92896 32077

95832 99813 77210 13268 44285

76145 44631 31148 02609 71735

11636 43746 50543 79833 26620

80284 99790 11603 66058 54691

17787 86823 50934 80277 14909

97934 12114 02498 08533 52132

12822 31706 09184 28676 81110

73890 05024 95875 37532 74548

66009 28156 85840 70535 78853

27521 04202 71954 82356 31996

70526 88386 83161 50214 97689

45953 11222 73994 71721 29341

79637 25080 17209 33851 67747

57374 71462 79441 45144 80643

05053 09818 64091 05696 13620

31965 46001 49790 29935 23943

33376 19065 11936 12823 49396

13232 68981 44864 01594 83686

85666 18310 86978 08453 37302

86615 74178 34538 52825 95350

994

PROBABILITY Table

26.11

2500 FIVE

DIGIT

FUNCTIONS

RANDOM

NUMBERS

12367 38890 80788 02395 73720

23891 30239 55410 77585 70184

31506 34237 39770 08854 69112

90721 22578 93317 23562 71887

18710 74420 18270 33544 80140

89140 22734 21141 45796 72876

58595 26930 52085 10976 38984

99425 40604 78093 44721 23409

22840 10782 85638 24781 63957

08267 80128 81140 09690 44751

61383 39161 80907 09052 33425

17222 44282 74484 65670 24226

55234 14975 39884 63660 32043

18963 97498 19885 34035 60082

39006 25973 37311 06578 20418

93504 33605 04209 87837 85047

18273 60141 49675 28125 53570

49815 30030 39596 48883 32554

52802 77677 01052 50482 64099

69675 49294 43999 55735 52326

72651 04142 85226 54888 33258

69474 32092 14193 03579 51516

73648 83586 52213 91674 82032

71530 61825 60746 59502 45233

55454 35482 24414 08619‘ 39351

19576 32736 57858 33790 33229

15552 63403 31884 29011 59464

20577 91499 51266 85193 65545

12124 37196 82293 62262 76809

50038 02762 73553 28684 16982

75973 90638 65061 64420 27175

15957 75314 15498 07427 17389

32405 35381 93348 82233 76963

82081 34451 33566 97812 75117

02214 49246 19427 39572 45580

57143 11465 66826 07766 99904

33526 25102 03044 65844 47160

47194 71489 97361 29980 55364

94526 89883 08159 15533 25666

73253 99708 47485 90114 25405

32215 54209 59286 83872 83310

30094 58043 66964 58167 57080

87276 72350 84843 01221 03366

56896 89828 71549 95558 80017

15625 02706 67553 22196 39601

32594 16815 33867 65905 40698

80663 89985 83011 38785 56434

08082 37380 66213 01355 64055

19422 44032 69372 47489 02495

80717 59366 23903 28170 50880

64545 39269 29763 06310 97541

29500 00076 05675 02998 47607

13351 55489 28193 01463 57655

78647 01524 65514 27738 59102

92628 76568 11954 90288 21851

19354 22571 78599 17697 44446

60479 20328 63902 64511 07976

57338 84623 21346 39552 54295

52133 30188 19219 34694 84671

07114 43904 90286 03211 78755

82968 76878 87394 74040 47896

85717 34727 78884 12731 41413

11619 12524 87237 59616 66431

97721 90642 92086 33697 70046

53513 16921 95633 12592 50793

53781 13669 66841 44891 45920

98941 17420 22906 67982 96564

38401 84483 64989 72972 67958

70939 68309 86952 89795 56369

11319 85241 54700 10587 44725

87778 96977 43820 57203 49065

71697 63143 13285 83960 72171

64148 72219 77811 40096 80939

54363 80040 81697 39234 06017

92114 11990 29937 65953 90323

34037 47698 70750 59911 63687

59061 95621 02029 91411 07932

62051 72990 32377 55573 99587

62049 29047 00556 88427 49014

33526 85893 86687 45573 26452

94250 68148 12208 88317 56728

84270 81382 97809 89705 80359

95798 82383 33619 26119 29613

13477 18674 28868 12416 63052

80139 40453 41646 19438 15251

26335 92828 16734 65665 44684

55169 30042 88860 60989 64681

73417 37412 32636 59766 42354

40766 43423 41985 11418 51029

45170 45138 84615 18250 77680

07138 21188 02154 90953 80103

12320 64554 12250 85238 91308

01073 55618 88738 32771 12858

19304 36088 43917 07305 41293

87042 24331 03655 36181 00325

58920 84390 21099 47420 15013

28454 16022 60805 19681 19579

81069 12200 63246 33184 91132

93978 77559 26842 41386 12720

66659 75661 35816 03249 92603

PROBABILITY 2500

FIVE

DIGIT

995

FUNCTIONS RANDOM

NUMB

LERS

Table

26.11

92630 79445 59654 31524 06348

78240 78735 71966 49587 76938

19267 71549 27386 76612 90379

95457 44843 50004 39789 51392

53497 26104 05358 13537 55887

23894 67318 94031 48086 71015

37708 00701 29281 59483 09209

79862 34986 18544 60680 79157

76471 66751 52429 84675 24440

66418 99723 06080 53014 30244

28703 68108 99938 91543 42103

51709 89266 90704 73196 02781

94456 94730 93621 34449 73920

48396 95761 66330 63513 56297

73780 75023 33393 83834 72678

06436 48464 95261 99411 12249

86641 65544 95349 58826 25270

69239 96583 51769 40456 36678

57662 18911 91616 69268 21313

80181 16391 33238 48562 75767

17138 28297 09331 31295 36146

27584 14280 56712 04204 15560

25296 54524 51333 93712 27592

28387 21618 06289 51287 42089

51350 95320 75345 05754 99281

61664 38174 08811 79396 59640

37893 60579 82711 87399 15221

05363 08089 57392 51773 96079

44143 94999 25252 33075 09961

42677 78460 30333 97061 05371

29553 23501 57888 55336 10087

18432 22642 85846 71264 10072

13630 63081 67967 88472 55980

05529 08191 07835 04334 64688

02791 89420 11314 63919 68239

81017 67800 01545 36394 20461

49027 55137 48535 11196 89381

79031 54707 17142 92470 93809

50912 32945 08552 70543 00796

09399 64522 67457 29776 95945

34101 53362 82975 54827 25464

81277 44940 66158 84673 59098

66090 60430 84731 22898 27436

88872 22834 19436 08094 89421

37818 14130 55790 14326 80754

72142 96593 69229 87038 89924

67140 23298 28661 42892 19097

50785 56203 13675 21127 67737

21380 92671 99318 30712 80368

16703 15925 76873 48489 08795

67609 44921 33170 84687 71886

60214 70924 30972 85445 56450

41475 61295 98130 06208 36567

84950 51137 95628 17654 09395

40133 47596 49786 51333 96951

02546 86735 13301 02878 35507

09570 35561 36081 35010 17555

45682 76649 80761 67578 35212

50165 18217 33985 61574 69106

15609 63446 68621 20749 01679

00475 25993 92882 25138 84631

02224 38881 53178 26810 71882

74722 68361 99195 07093 12991

14721 59560 93803 15677 83028

40215 41274 56985 60688 82484

21351 69742 53089 04410 90339

08596 40703 15305 24505 91950

45625 37993 50522 37890 74579

83981 03435 55900 67186 03539

63748 18873 43026 62829 90122

34003 53775 59316 20479 86180

92326 45749 97885 66557 84931

12793 05734 72807 50705 25455

61453 86169 54966 26999 26044

48121 42762 60859 09854 02227

74271 70175 11932 52591 52015

28363 97310 35265 14063 21820

66561 73894 71601 30214 50599

75220 88606 55577 19890 51671

35908 19994 67715 19292 65411

21451 98062 01788 62465 94324

68001 68375 64429 04841 31089

72710 80089 14430 43272 84159

40261 24135 94575 68702 92933

61281 72355 75153 01274 99989

13172 95428 94576 05437 89500

63819 11808 61393 22953 91586

48970 29740 96192 18946 02802

51732 81644 03227 99053 69471

54113 86610 32258 41690 68274

05797 10395 35177 25633 16464

43984 14289 56986 89619 48280

21575 52185 25549 75882 94254

09908 09721 59730 98256 45777

70221 25789 64718 02126 45150

19791 38562 52630 72099 68865

51578 54794 31100 57183 11382

36432 04897 62384 55887 11782

33494 59012 49483 09320 22695

79888 89251 11409 73463 41988

27. Miscellaneous

Functions

IRENE A. STEGUN l Contents 27.1. Debye

= t”dt o -cl-l.

S

Functions

n=1(1)4,

. , . . . . . . . . . . . . . .

z=0(.1)1.4(.2)5(.5)10,

27.2. Planck’s

Radiation

Function

6D

z-s(el’z-

1) -I . . . . . . . . . .

x=.o5(.oos).l(.ol).2(.o2).4(.o5).9(.1)1.5(.5)3.5, Gum, [email protected]>, 27.3. Einstein x2e2

999

3D

9-10s

Functions

. . . . . . . . . . . . . . . . . . . .

X -’e”-1

(e”-1)2’

Page 998

z=O(.O5)1.5(.1)3(.2)6,

27.4. Sievert Integral

&-In

In (l-e-“),

999

(l-e-“)

5D 8

S

e-zsec%

. . . . . . . . . . . . . . . .

1000

0

z=O(.1)1(.2)3(.5)10,

~=10°(100)600(150)900,

. . . . . . . . .

1001

. . . . . . . . . . . . . . . . . . . . .

1003

tme-c2-f dt and Related

27.5. jm(x> =s,-

fm(x) 9

m=l,

f3 (,+,

z=O(.2)8(.5)15(1)20,

27.6. j(x)=S,f(4 f(4

dt

x=0(.05)1 +ln x, x=1(.1)3(.5)8, 9

27.7. Dilogarithm

Integrals 4D

.l(.l)l, 4-5D

4D

(Spence’s Integral)

2=0(.01).5,

j(z)=-1

2

dt . . . . . .

1004

. . . . . . . . .

1005

. . . . . . . .

1006

9D

27.8. Clausen’s

Integral

f@>+eIn 8, f(e),

2, 3; s=0(.01).05,

2;

6D

and Related

Summations

e=o”(10)150

e=15°(10)300(20)900(50)1800,

6D

27.9. Vector-Addition Coefficients (jljzmlm21 j,j,jm) Algebraic Expressions for j,=1/2, 1, 312, 2 Decimal Values for j2=1/2, 1, 3/2, 5D 1 National Bureau of Standards.

997

--

27. Miscellaneous 27.1.

Debye

Series

s

Functions

1

Representations

Functions Relation

27.1.3 [27.1]

o= g=f&&)+g

(z~~~;~~)!l

s

[27.2]

Function

E.

Die Abhilngigkeit reiner Metalle (5) 16, 530-540

20 r=

42

x=0(.01)24,

27.1

Debye

Functions

--1 = t&i 2 I- eel-1

2 = Pdt 2? s- 0 et-l

3 = Pdt 2 s- 0 et-1

4 = t’dt 2 s- 0 et-1

1.000000 0.975278 0.951111 0. 927498 0.904437

1.000000 0.967083 0.934999 0.903746 0. 873322

1.000000 0. 963000

1.000000 0.960555 0.922221 0.884994 0.848871

0. 881927 0. 859964 0. 838545 0.817665 0.797320

0.843721 0.814940 0.786973 0. 759813 0. 733451

0. 824963 0. 792924

0.777505 0.758213 0.739438 0. 721173 0. 703412

;

0. 669366 0.637235 0. 606947 0.578427 0. 551596

0.570431 0. 530404 0. 493083 0.458343 0.426057

0. 524275 0. 481103 0. 441129

0. 626375 0. 502682 0.480435 0.459555 0.439962

0.396095

0. 338793 0. 309995 0.283580

0. 421580 0. 404332 it %%i 0: 358696

0. 276565 0. 257835 0. 240554 0.224615 0.209916

0. 0. 0. 0. 0.

0. 196361 0. 183860 0. 172329 0.147243 0. 126669

345301 332713 320876 294240 271260

0. 251331 0. 233948 0. 218698 0.205239 0. 193294

y8;;

;: memo” 0: 613281

it Ez 0: 318834 0.296859

x: E%:: 0.084039 0.074269 0.066036

0. 182633 0. 173068 0. 164443 [ y5] 998

[ ‘-;I”]

: EEt 0: 857985

it %E 0: 702615 0.674416 0.647148 0.620798 0. 595351 0. 570793

8: %%4

x: %E

it %E 0: 181737 0. 166396 0.152424 0. 139704 0. 128129 0.117597

i:XE x:Ei:

0. 813846 0. 779911 0.747057 0.715275 0.684551 0.654874 0. 626228 it FEZ; 0: 546317 0. 497882 0. 453131

x:%Zi: I:pf$ 0. 186075 t :%z 0: 137169 0. 123913 0.111957 0. 101180 0.091471

:: 8%

0.043655 0.036560 0.030840

0.043730 0.034541 0.027453 0.021968 0.017702

0.026290 0. 022411 0.019296

0.014368 0.011747 0.009674

[ 5961

[ (-;I61

6s.

des elektrischen von der Temperatur, (1933).

z=0(.1)13(.2)18(1)20(2)52(4)80.

Table

23)

of the Debye energy J. Math. Phys. 6,

L2 [l”e$-e&], zJ

Griineisen, Widerstandes Ann. Physik. t’dt

(see chapter

o- g=n!‘(n+l).

3 z-~ 28s 0 eu-1

numbers B,,, see chapter 23.)

27.1.2

Zeta

J. A. Beattie, Six-place tables and specific heat functions, l-32 (1926).

(I4<2~Pfi2.1)

(For Bernoulli

to Riemann

49.

MISCELLANEOUS

Planck’s

2 0.050 0.055 0.060 0.065 0.070 0.075 0.080 0.085 0.090 0.095 0. 100

) f(z) )I

=

0.007 0.025 0.074 0. 179 0.372

0. 0. 0. 0. 0.

10 11 12 13 14

0. 682 :* E

0. 0. 0. 0. 0. 0.

15 16 17 18 19 20

I II 2: 531 3.466 4.540

II

[27.3]

.20140

Miscellaneous functions Government 1941).

2

4 540 6.998 9.662 12.296 14 710 16.780 18.446 19.692 20. 539 21.025 21. 199

[ (-j)2] 2 m.x=

Radiation f(z) =ci+(eYs-

f(z) 0. 0. 0. 0. 0.

999

FUNC’l’IONS

Function

f(z)

20 22 24 26 28

Table

II

2

21. 199 20.819 19. 777 18.372 16. 809 15. 224 13.696 12.270 10.965

EX 0: 34 0. 36 0. 38 0. 40

1

f(z)

2

f(z)

0. 0. 0. 0. 0.

40 45 50 55 60

8.733 6. 586 5.009 3.850 2.995

0. 9

::;

0.831 0. 582 0.419 0.309 0.233

0. 0. 0. 0. 0. 0.

65 70 75 80 85 90

2.356 1.875 1.508 1. 225 1.005 0.831

::it 2: 28

0. 178 0. 139 0. 048 0.021 0.010 0.006

:::

[ (-52’51 52353

and

[(-y]

f&,.x)=21.20143

58.

Physical Tables, Planck’s electronic functions, MT Printing Office, Washington,

radiation 17 (U.S. D.C.,

Table

NA=2rcX-4(ee*fAT-

R&h= 1)-i,

Rx&

s0

No+=

for

0. 0. 0. 0. 0.

x6e* (es- 1)’

00 05 10 15 20

0. 25 0. 30 0. 35 x: 2

1

x: E%3’

In (l-t?) --In

es-l (1-e-m)

--oD

- 3.02063 -2.35217 -1.97118 - 1.70777

: 99584 3. 30300 2.89806 2. 61110 2. 38888 2. 20771. 2. 05491 1.92293 1.80690

: %Z 0: 98677 0.98329

: EE $ ;E

- 1: 21972 -1. 10963 - 1.01508.

0.77075

-0.93275

1.70350 1.61035 1. 52569 1.44820 1.37684

-0.63935

1.31079 1.24939 1. 19209 1. 13844 1.08809

it %%28 0: 70996 0.69050

8: z

it E 0: 93515 0.92807 0.92067

0.67144

1. 25 :: k%i:

El%E 0: 84178 0.83185 [ (-35)5]

‘2

1.04065 0.99592 0.95363 0.91358 0.87560

: Eg” 0: 89671 0.88817 -0.33758 -0.31818 1; ggy; -0: 26732 -0.25248

- NO-X N o--co

NA m.x

it %io2 0: 77253 0.74139 0.71168 0.68331

for K”.

No-1

(T=

1000° microns.

Table

1; pg;

0.95441

:*2 1: 56

2

e*-

-3 NA

ZZZ: NX for X=[.25(.05)1.6(.2)3(1)10] T=[1000°(5000)35000 K and 6000’

0.88020

0. 75

:: ii

&,

0.99481

EG 0: 70

:~~ 1: 10 1. 15 1. 20

RA A m.x

Functions

1.00000 0.97521 0.95083

0. 50 0. 55

X2

R

Table ZZ: RA, R&k, NA, X=[.5(.01)1(.05)4(.1)6(.2)10(.5)20]

Tabb microns,

A Nxdx s6

Einstein x

I:

R o-m XT=[.05(.001).1(.005).4(.01).6(.02)1(.05)2]cm

A RI=c,X-6(eC’IAT-1)-1,

27.2

1)-i

K)

K].

27.3

1000

MISCELLANEOUS

Table

FUNCTIONS

Einstein

27.3

1. 6

2

2

zaea (es- l)s

5

Functions

In (l-e-5)

e=-1

0.81143

lj.po;; 0: 74657 0. 72406 0.70127 t EE 0: 63200 0. 60889

es-l --In (1-e-l)

: %i 0: 35646 0.33416 0. 31304

-0.22552 -0.20173 -0. 18068 -0. 16201 -0. 14541

0.63027 0. 58171 0.53714 0.49617 0.45845

0. 29304 0.27414 0.25629 0.23945 0.22356

-0. 13063 -0.11744 -0. 10565 -0.09510 -0.08565

0.42367 0.39158 0.36194 0. 33465 0.30921

0. 20861 0. 19453 0. 18129

0. 58589

ii: :E 0.45363 it 8%: 0: 33799 0.30409 0. 27264 0.24363 0: 17074

Et %ii: 0: 05681 0.04809 0.04968

0.08968

0.03438 0.02903 0.02449 0.02065 0.01739

t %::

[ (-44)3] (27.41

H.

L. Johnston, butions

to

L. Savedoff the

oscillator

NAVEXOS

search,

p.

Department

in 646,

of the

D.C.

(1949).

Values

--In

(1-e-z)

and

x=0(.001)3(.01)

J. Belzer,

thermodynamic

Planck-Einstein dom,

and

one Office

degree

of free

Naval

Navy,

x(e*--l)-I-ln 5D

a Re-

Washington,

first

in Terms

27.4.2 e e-z mc $&#)q s0

27.4.1

Function

see chapter 5.)

to the

27.4.3 c ’ e-za*o+&=K&

S 0

(For Ki,(z),

o<e<;)

1.3.5...(2&-1) 2.4.6...(2k)

CG=l’ak=

deCKi,

(For erf, see chapter 7.)

(se) (520,

Integral

Error

~>~+‘&+a

diierencee.

Relation

to the

Integrals

for

(For Ean+&),

Relation

of Exponential

e-z !3ec4(j+

-~&OS

x(e=-1)-i, (1-e-s)

with

Represent&on

by

of

[ ‘-;I”]

[ (-;)“I

Contri-

functions

of zae*(e=--1)-z,

14.99,

27.4. Sievert

[ (-:)“I

Integral

of the

Bessel

Function

m

(z) =

S z

- (+T)+

{

K,(t)&

where

l-tz+$f$

301035 -- 2655 1024&%i%?4see chapter 11.)

--

Ko@)

***

MISCELLANEOUS [27.5]

National integral, Printing

[27.6]

Bureau of Standards, Table of the Sievert Applied Math. Series(U.S. Government Office, Washington, D.C. In press).

z=0(.01)2(.02)5(.05)10,

e=0”(1”)90”,

1001

FUNCTIONS

9D.

R.

M. Sievert, Die v-Strahlungsintensitiit an der Oberfliiche und in der niichsten Umgebung von Radiumnadeln, Acta Radiologica 11, 239-301 (1930).

s

’ 0-r 0

Sievert

Integral

0 S

,-zssc*

#*a *d+, ~=30°(lo)900,

A=O(.01).5,

3D.

Table

&J

27.4

0

x\e

2o”

4o”

5o”

60”

174533 157843 142749 129099 116754

0.349066 0.315187 0. 284598 0.256978 0.232040

0.424515 0.382255 0.344209

0. 698132 0.625886 0.561159 0.503165 0.451198

0.872665 0.777323 0. 692565 0. 617194 0.550154

1.047198 0.923778 0.815477 0.720366 0. 636769

1.308997 1. 123611 0. 968414 x: EE::

1. 570796 1.228632 1.023680 0.868832 0. 745203

0. 105589 0.095492 0.086361 0.078103 0.070634

0.209522 0. 189191 0. 170833 0. 154256 0. 139289

0.309957 0.279118 0.251353 0. 226354 0.203845

t 0: 0. 0.

0.490508 0.437428 0.390178 0.348109 0.310642

0.563236 0.498504 0.441478 0.391204 0.346851

,O. 632830 0.552287 0.483134 0.423535 0.371996

0.643694 0.558890 0.487198 0.426062 0.373579

0.063880 0.052247 0.042733 0.034951 0.028587

0. 125775 0. 102553

0. 183579 0. 148899 0. 120780 0.097979 0.079488

0.234956 0. 189138 z Et; 0: 098829

0.277267 0.221027 0. 176336 0. 140792 0. 112497

0.307694 0.242523 0.191533 0. 151541 0. 120105

0.327288 0.254485 0. 198885 0. 156087 0. 122932

0.328286 0.254889 0. 199051 0. 156156 0. 122961

0.023381 0.019123 0. 015641 0.012793 0. 010463

0.045335

t o”Ei 0: 024582 0.020045

0. 064492 0.052329 0. 042463 0.034460 0.027968

0.079644 0. 064201 0.051766 0.041750 0.033680

0.089954 0.071979 0.057635 0.046179 0. 037024

t X%? 0: 060342 0.048100 0.038387

0.097108 0.076905 0.061040 0.048541 0.038667

0.097121 0.0769 11 0.061043 0.048542 0.038668

0.016347 0.009817 0.005896 0.003542 0.002127

0.022700 0.013477 0.008005 0.004756 0. 002828

0.027177 0.015912 0.009330 0.005478 0.003221

t EEz 0: 009951 0.005787 0.003374

0.030670 0.017576 0.010128 0.005862 0.003407

0.030848 0.017634 0. 010147 0.005869 0.003409

0.030848 0.017634 0.010147 0.005869 0.003409

0: 000461 0.000277 0.000167

0.001682 0.001001 0.000596 0. 000355 0.000211

0.001896 0.001117 0. 000659 0. 000389 0.000230

0.001972 0.001155 0.060678 0.000399 0. 000235

0.001986 0. 001162 0.000681 0.000400 0.000235

0.001987 0.001162 0.000681 0.000400 0.000235

0.001987 0.001162 0.000681 0.000400 0.000235

0.000100 0.000060 0.000036 0.000022 0.000013

0.000126 0.000075 0.000045 0.000027 0.000016

0.000136 0.000081 0.000048 0.000028 0.000017

0.000139 0. 000082 0.000048 0.000029 0.000017

0.000139 0.000082 0.000048 0.000029 0.000017

0.000139 0.000082 0.000048 0.000029 0.000017

0.000139 0. 000082 0.000048 0.000029 0.000017

[ (-;I51

[(-y]

[‘-y]

[(-f”]

[ Ff’“]

[ (-;)“I

[(1?2]

loo 0. 0. 0. 0. 0.

:: i Ki 0:9 1. 0 1. 2 i-t 1: 8 Ei 2: 4 i:t

i EE d. 055597

0.008558 E 4: 0 4. 5 5. 0 5. 5 E P:X 8. 0 :X 9: 5 10. 0

t xxz

0.000694 0.000420 0. 000254 0.000154 0.000093 0.000056 _..__ 0.000034 0.000021 0.000012 0.000008

-

[ 921

27.5. f,,,(z)=

S

OD t”‘e-‘*-$&

fi%i 325486 291957 261901

Power

and

Series

Representations

0

Related

m=o,

Integrals

1,2..

Differential

j;=

-j,,,ml

(m=l,

Recurrence

27.5.3

-2akea

‘“=k(k-1) %=a*=0

zjy+P-l,j~‘+2jm=0

27.5.1

27.5.2

.

Equations

2, . . .)

Relation

2j,,,= (m-l)jm-a+zjm-a

(m23)

--

I

(k-2)

b,=l

] (For y, see chapter 6.)

b ---2b,-,(3ka-6k+2)an ‘k(k-1) (k-2) e=-b, b,=-,G

bs=; (1-y)

1002

MISCELLANEOUS

FUNCTIONS Asymptotic

27.5.5

Representation

27.5.11 -.01968;cb+.00324x6+.000188x7

gI(x)=Gyz

. . .

-x2 ln x(1-.0833322+.001389ti-.00000835cs+.

. .)

i exp [ --i(iy’]

(A sin e+B

27.5.12 g2(x)=-~~2fexp[-~(~~3](A

27.5.6 2j2(x)=g-x+$

co9 0)

co8 &-Bsin

0)

x2-.3225~-.1477x*+.03195x’ +.00328x’-

.000491x’-

.0000235ti

+x3 ln x(+.01667;F2+.000198x4-

... . . .)

27.5.7 2j~(x)=l-q

x+;-.295421+.10142’+.02954x’ -.00578x8-

.00047x’+

.000064x” . . .

-x4 ln x(.0833-.00278~+.000025x*Asymptotic

. . .) uQ=l

Representation

aa=-.

27.5.8

fm(z)-di

3-;vy

em0(ao+:+$+

. . . +$+

. . .)

b-+-1 v=3

: ,2f3 02

[27.7] M. s

+3(m--2k)(2k+3--m)(2k+3+2m)u~

lw

(k=O, m(x) +ig,(x)

=~mt3e-ta+i

fdt

27.5.10 91(x> = 9i%(;z>

s2(4

13, l-57

a,==.148534

us=-.000762

Evaluation

of

the

integral

J. Math. Phys. 32,188-192

(1953).

in series of the integral Ark.

Mat., Astr., Fys.

(1921).

[27.9] J. E. Kilpatrick and M. F. Kilpatrick, Discrete energy levels associated with the LennardJones potential, J. Chem. Phys. 19, 7, 930-933 (1951). [27.10] U. E. Kruse and N. F. Ramsey, The integral

(3m2+3m-l)

12(k+2)uk+2=-(12k2+36k-3m2-3m+25)u~+I

27.5.9

Abramowitz, m e-u’-Judu,

[27.8] H. ‘Fax&r, Expansion (D exp [-z(tft-“)]tadt, s 1;,

c&)=1, a*=+2

a1 = .972222 al= .004594

= --J%(~l

1,2 . . .)

ya exp (-@+i

i)

dy, J. Math.

Phys. 30,

40 (1951). [27.11] 0. Laporte, Absorption coefficients for thermal neutrons, Phys. Rev. 52, 72-74 (1937). [27.12] H. C. Torrey, Notes on intensities of radio fre. quency spectra, Phya. Rev. 59, 293 (1941). (27.131 C. T. Zahn, Absorption coefficients for thermal neutrons, Phys. Rev. 52, 67-71 (1937). I- 0

m~~e-z-zl~dy

for n=O, 4, 1; z=O(.Ol).l(.l)l.

MISCELLANEOUS

x

fib)

fib)

A(z)

0. 0. 0. 0. 0. 0.

0. 5000 0.4956 0. 4912 0. 4869

1003

FUNCTIONS

--

f3W

---

I 0.

00

0. 01 0. 02 0.03 0. 04 0. 05

::

:Ei

0. 4832 0.4753 0. 4676 0. 4602

4431 4382 4333 4285 4238 4191

0. 4263 0. 3697 0. 3238 0.2855 0.2531

0. 0. 0. 0. 0.

0. 3970 8: E 0. 2923 0. 2654

4580 4204 3864 3557 3278

I

i: ES [‘-s5]

X

~f3(4

0.00000 0.08764 0. 16933 0. 24139 0. 30136

6

0: 8

::X :.t

-0. 2626 -0.2552 -0. 2441 -0. 2299 -0. 2132

0. 34805

1: 8

0.0430

0. 0. 0. 0.

iii! Et:

2; ;;y; -0: 0490

-0.0734

10: 0

-0.1374 -0. 1455

10. 11. 11. 12. 12.

5 0 5 0 5

0. 10288 +O. 03892

t E?::; 0: 40910 0.40592

1; :;g -0: 1536 -0.1322 -0. 1108

1: -0: -0. -0.

;;04;2 1221 1629 1966

it fE” 0: 3448 0. 3122 0.2759

-0.0896 -0.0691 -0.0493 -0.0307 -0.0132

1; :g -0: 1535 -0.1515 -0. 1476

:tx 14: 0 14. 5 15. 0

-0. -0. -0. -0.

2233 2432 2565 2639 2657

it % 0: 1569 0. 1173 0.0792

+; ye;

-0.

14211

0: 03061 0. 04220 0. 05224

1; -0: -0.

:;;g 11805 10830

16. 17. 18. 19. 20.

it0: ;%i 16972

2255 2015 1807 1626 1466

2415 2202 2011 1839 1685

:: ;%z 0. 2584 0.2392 0. 2215

[1(-s”]

[ (-;)4]

0.06078 0.07562 0.08221 0.08191 0.07626

-0.09808 -0.07131 -0.04496 -0.02082 -0.00010

it XE 0.06684

+;: 0.03707 ;;W$

8X

4f3W

-0. 50000 0. 49019 46229 0. 41950 0. 36543

0. 0. 0. 0. 0.

f&

fd4

fib)

-(-$61’

=

8:“z tt

Table 27.5

0 0 0 0 0

0: 02937 0.01727

: . 84;::

4-O. 00650 -0.00259 zoo. gg;

0.04109 0.03758 0.03268 0.02696

-0:

0.02089

01872

-0.02118 -0.01906 -0.01435 -0. 00879 -0.00360

+O. 00921 -0.00022 -0.00650 -0.00965 -0.01021

[ q1-J

[ (-;)7]

[ (q3P-j Compiled

[ ‘-921

[ (-$51

[ (--:)4]

from U. E. Kruse and N. F. Ramsey, The integral

du, J. Math.

Phys. 30,40 (1951) (with permission) Asymptotic

Power

27.6.1

f(z)=-e-z21n

Series

27.6.4

Representation

z+emz2[&

&

Representation

k,Gyil)

-2

kg&l

27.6.2 z-e

-9 In ,&+i .& 2

(For y and the digamma ter 6.) Relation

to the

(--lMk+lwk

k=O

function

Exponential

k!

yS(x)),see chap-

[27.14]

A. ErdBlyi, Note on the paper gral” by R. H. Ritchie, Comp. 4, 31, 179 (1950).

“On Math.

a definite Tables

[27.15]

E. T. Goodwin

Table

of

Quart. J. Mech. x=0(.02)2(.05)3(.1)10. z=O(.Ol)l.

Integral

emz2Ei (2”) +&ems JO’ e12dt z (For Ei (CC)see chapter 5; emz2 ef2 dt, see chapter s0 78

and J. Staton,

27.6.3 f(x) =-5

[27.16]

R. H. Ritchie, Aids Comp.

Appl.

Math. Auxiliary

On a definite integral, 4, 30, 75 (1950).

inteAids

0m $ du, s 1, 319 (1948). function for

Math.

Tables

1004 Table

MISCELLANEOUS

FUNCTIONS

27.6

= x

f(z)

+ In z

--

II---

0. 00

0. 0. 0. 0.

05 10 15 20

0. 0. 0. 0. 0.

25 30 35 40 45

-0. 2886 -0. 2081 -0. 1375 - 0.0735 -0.0146 ‘Z:

0. 50

x

f(z) + In z

x

II

----I

= --

0. 0. 0. 0. 0.

50 55 60 65 70

0. 2704 0.3100 0.3479 0.3842 0. 4192

E 0. 1398 0. 1856 0.2290

0. 0. 0. 0. 0.

75 80 85 90 95

0.4529 0. 4854 0.5168 0. 5472 0.5766

0.2704

1. 00

1. 0

::i ::;: 1. 9

0.6051

2. 0

from

[(-$2-j

E. T . Goodwin

and J. St&on,

27.7.

27.7.1

Integral

f(x)=-1

f(x) =kg

2: 2 2. 3 2. 4

0.4460 0.4239 0.4040 0.3860 0.3695

2. 5

0.2944

4: F 2. 8 2. 9

0. 2848 2758 0. 2673 0.2594

t:: 7. 5

0.3543

3. 0

0.2519

8. 0

Functional

for

of

s

0

ag c u+z

du, Quart.

J. Me&.

+f(l+x)

27.7.5 f(x>+f

Math.

1,319

(1948)

to

[ (-yJ]

(with

permission).

Debye

Functions

(l-z)+;

[27.18]

K. Mitchell,

[27.19]

E. 0. Powell, integrals,

(12x20)

(12

=;f(l-x?>

(In s)*

mew

functions

of the function

An integral Phil. Mag.

related 7, 34,

x=0(.01)2(.02)6,

to the 600-607

radiation (1943).

7D.

A. van Wijngaarden, Polylogarithms, by the Staff of the Computation Department, Report R24, Mathematisch Centrum, Amsterdam, Holland (1954).

5 In (x+1) -g-;j(x*)

and associated England, 1958).

* --log Il-Yl dy s0 Y ’ with an account of some properties of this and related functions, Phil. Mag. 40, 351-368 (1949). z=-l(.Ol)l; 2=0(.001).5, 9D. Tables

:zdy,

[27.20]

-!&

Lewin, Dilogarithms (Macdonald, London,

s

x>O)

(O_
f(~-‘)=-f(e’)-;=So’ L.

(21x10)

27.7.6 f(s+l)-f(z)=-In

Appl.

[27.17]

Relationships

(k)=-k

1468 1356 1259 1175 1102

0. 1037

[‘-y-j

27.7.7

dt

27.7.4 f(l-4

0. 0. 0. 0. 0.

fl = 2)

@$

2 In

2 i 4. 5 5. 0

-

27.7.3 f(z)+f(l--2)=-ln

0.2519 0. 2203 0. 1958 0. 1762 0. 1602

:: Fi

[ (-i)7]

Expansion

(-1)’

.-

0.3543 0.3404 0.3276 0.3157 0.3046

Relation

gl

Series

27.7.2

Table

f(z)

X

3. 0

;:

Dilogaritbm

@pence’s

f(z)

-

[(-y] Compiled

x

f(z) 0.6051 0.5644 0.5291 0.4980 0.4705

:. 4 1: 3 1. 4

-

=

z=ti, 10D.

P,,(z)=~& for

z=O(.Ol)l;

h- nzh for

z=z=

z=eira/*

for

- l(.Ol)l; a=0(.01)2,

MISCELLANEOUS

1005

FUNCTIONS

Table

Dilogarithm

27.7

f(x)=-S,y+it I

x

f(z)

2

x

f(z)

x

f(z)

_-

Fixi 0: 34

40 41 42 43 44

0. 0. 0. 0. 0.

72758 71239 69736 68247 66774

6308 5042 1058 9725 6644

9393 6675 7798 0654 4053

0. 0. 0. 0. 0.

0.80608 0.79002 0. 77415 0. 75846 0.74293

2689 6024 3992 0483 9737

0. 0. 0. 0. 0.

45 46 47 48 49

0.65315 0. 63870 0. 62439 0. 61021 0.59616

7631 8705 6071 6108 5361

7624

0. 40

0.72758

6308

0. 50

0. 58224

0526

20 21 22 23 24

1. 07479 1.05485 1. 03527 1. 01603 0.99709

4600 9830 7934 0062 9088

0. 30 0. 31

0. 15 0. 16 0. 17 0.18 0. 19

1. 18058 1. 15851 1. 13693 1.11580 1.09510

1124 6487 6560 8451 3088

0. 0. 0. 0. 0.

25 26 27 28 29

0.97846 0.96012 0. 94205 0.92425 0.90669

0.20

1.07479

4600

0.30

0.88937

4067 5448 9712 9041 5860

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

05 06 07 08 09

1.44063 1.40992 1.38068 1. 35267 1. 32572

3797 8300 5041 5161 8728

0. 10

1. 29971

4723

10 11 12 13 14

1. 1. 1. 1. 1.

-0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

1.64493 1.58862 1. 54579 1.50789 1.47312

35 36 37 38 39

-

[l--3)2] K. Mitchell,

From (with

[ (-!)5]

[(-4’11 Tables

dy, with

of the function

f(z)

X

7624 1733 7404 6261 0471

4723 9160 7584 0101 7961

01 02 03 04

f(x) 0. 88937 0.87229 0. 85542 0.83877 0. 82233

29971 27452 25008 22632 20316

0. 0. 0. 0.

0. 00

II

an account

[‘-p-j

[ (-;)“I

of Some properties

of this and related

functions,

Phil.

Meg. 40,351~363

(1949)

permi.?a1on).

27.8.

Clausen’s

Integral Summations

and

Summahle

Related

27.8.6

27.8.1

c

cos

c n=l

Series

cn4

(w
on-l Cw

‘%+’

&(2=-l)

2k(2k+l)

!

bdKe<4 Functional

to Spence’s

where g(z)=

-

~08

(o<e<27r)

(osel27)

no r4 a282 n4 =iG-12+12--G

s!322!=~ n

c n=~

s3-7r;

c n=1

!!Epg!T+g-go

[27.21]

(o<e
[27.22]

[27.23]

S

,“$lnJl+i!l

[27.24]

84

71+

(T-e)

A. Ashour

(o_<esar)

0w<w

77;2

I g

n3

c&9 =ng

27.8.5

ij(e)=&P)+~

e 2 sin 2

cz!+f-$+f

m c n=~

Relationship

27.8.4 f&--e)=f(e)--$(28) Relation

‘rJ

Representation

27.8.2

f(?r-e>=e ln 2-g

no ( >

-=--In

n=l

Series

(o<e<2r) (o<e52n)

and y

A. Sabri, t Math.

Tabulation Tables

of the function Aids

Comp.

10,

54, 57-65 (1956). T. Clausen, uber die Zerlegung reeller gebrochener Funktionen, J. Reine Angew. Math. 8, 298-300 (1832). z=O”(10)1800, 16D. L. B. W. Jolley, Summation of series (Chapman Publishing Co., London, England, 1925). A. D. Wheelon, A short table of summable series, Report No. SM-14642, Douglas Aircraft Co., Inc., Santa Monica, Calif. (1953).

1006

MISCELLANEOUS

Table 27.8

FUNCTIONS

Clausen’s f(0)

= -I”

Integral In (2 sin f) dt

-P

f@)+e In 0 0.000000 0. 017453 0. 034908 0. 052362 0.069818 0.087276

0.612906 0.635781 0. 657571 0. 678341 0. 698149 0.717047

0. 104735 0. 122199 0. 139664 0.157133 0. 174607

0.735080 0.752292 0. 768719 0.784398 0. 799360

0. 192084 0. 209567 0. 227055 0. 244549 0.262049

0.813635 0.827249 0.840230 0.852599 0.864379

80

f@)

80

f@)

80

f(e)

f(e)

--

0.864379 0. 886253 0.906001 0.923755 0.939633 0.953741

1.014942 1.014421 1.012886 1. 010376 1.006928 1. 002576

1;: 105 110 115

0.915966 0.883872 0.848287 0.809505 0. 767800 0.723427

2 50

0.966174 0.977020 0.986357 0.994258 1.000791

0.997355 0.991294 0.984425 0. 976776 0. 968375

120 125 130 135 140

: 0: 0. 0.

x2 56 58 60

1.006016 1. 009992 1. 012773 1.014407 1. 014942

0.959247 0.949419 0.938914 0.927755 0.915966

145 150 160 170 180

0.413831 0.356908 0. 240176 0. 120755 0.000000

3x xi 38 40 2

90

W%E 576647 523889 469554

[ 7’83 Compiled

from A. Ashour

[t-y]

and A. Sabri,

[(-y]

[t-213]

Tabulation

of the function

sin +(0) = c d, n.9 ?&=I

27.9. Vector-Addition

(Wigner

Math.

Tables

Aida

Comp.

[ q4]

10. bl. 67435 (19561 (with

permission).

Coefficients

coefficients or Clebsch-Gordan

coefficients)

Definition

27.9.1 (j,j2m,m*lj1j2jm)=~(m,

m1+m2)

*

<j,+j,-~~!0’+j*-j,>!(j+j,-j,,!(aj+l> J

o’+jl+j2+1>!

(-1>“JO;+~l>~(jl-~l>~(j2+~2>~(j2-m2>!O’+m>~(j-m>! * T

k!(j,+j,-j-~>!(j,-m,-k>!(j,+m~--k)!(j-jz+m,+k)!0’-j1-m~+~)!

Conditions

27.9.2 27.9.3

jl, j2, j=+n

or+:

27.9.8

(n=integer)

27.9.9

ImlIj1,

Imzlljz,

(jlj2mlm21j~j2jm)=0

14 Ij mli-m,Zm

jl+j2+j=n Special

Values

27.9.4 27.9.5

27.9.10

(jlOmlOIjlOjm)=~(jl, j)s(m,, m)

27.9.11

(jlj2001j&jO) =0

27.9.12

(j,jlmlmlIj,jljm)=O

27.9.6 27.9.7

j,+j,+j=Sn+l 2jI+j=2n+l

MISCELLANEOUS Symmetry

27.9.17

Relations

27.9.13 (j,j2mmlihjm>

27.9.18

=(-1)11+‘~-‘(j,jz-m,-m2~jIj2j-m) 27.9.14

=t.hjl-ma-mlIj&j-m)

27.9.15

=(-1)1~+t~-f(j2jlm,m21

1007

FUNCTIONS

= gE$ (-l)ll-~l+l-~(jjam-ma J Ijjkh) =

J

E

(-l)l-"+ll-"l(jljma-m

lj2jj~--md

27.9.19

j2jljm)

zz

$J+

(-l)Jl-“ll(jljml-m

J Ijljh-m2>

27.9.16

27.9.20 =

J

g

(-l)jp+%(jj,-mm,

=

E

(-l)jl-“l(

jj,m-m,

J

Ijj2jl -4

I j&h>

(jl% ml ma IA %j ml

j=

m=H

I

Table 27.9.1

<

ma=-%

Table 27.9.2 m=-1

h+ 1

~r+m)~l-t-m+l) J

(2ji+ 1)(2j1+2)

(j,-m+1)0'l+m+l)

J

(2jl+l)l.,h+l)

(j,-m)($-m+l) -\I (2jl+l)@j1+2)

1008

MISCELLANEOUS

Table 27.9.3

(A % ml maIA 74j 4

j=

A--%

ma=%

ma=% (j,+m-f/2)Cj~+mf~/z)(jl+m+~) (2jt+1)(2j1+2)(2j1+3)

A+%

jl+!h

FUNCTIONS

_ J

3(j,+m-M)(j1+m+%)(j~--m+%) 2jr(2jl+ 1)(%+3)

J

a(j,+m+~)(j,+m+h)(j,-m+S/2) (2jl+l)(2j1+2)(2j1+3)

-lir3m+4~;

30'lf

(2j,-1fi,~~,';2j1+2)

(2j,-1)(2jl+l)(2jl+2)

J

O',-m-~)(j,-m+Yz)(jl-m+3/1) 2ji(2jl-11)(2jl+l)

j,-#

30'1+m-~)O'I-m--Im,~,-m+Yz) 2j,(2jl-1)(2jl+l) J

ms=-$4

J

30'l+m+~)(j,-m+y,)O'I-m+~) (2jl+1)(2jr+2)(2ji+3)

(jl+3m+K)Jii) -(jl-3m-%) -

J

j,+m+% (2j,-1)(2jl+l)(2j1+2)

30'l+m-~)(j,+m+~)Cil-m-~) 2j1(2jl-11)(2jl+l)

ma=-%

J

(j,-m-%)(j,-m+~)(jl-m+~) (2jl+l)(2jt+2)(2j1+3)

J

30',+m+~)(j,-m-y,)O'l-m+~) 2j1(2jl+ 1) C&+3) 30',+m+~)(j,+m+K)O'l-m-Yz) (2jl-l)(2jl+1)(2j1+2) O',+rn-~)(j,+m+M)O'l+m+3/2) 2j,(2j,- 1) (2jl+ 1)

-

-

Table

27.9.4

(5 2 ml ma IA 2 j ml

-

m9=2

j=

iI+2

(j,+m-l)(j,+m)(j,+m+l)O',+m+2) (2jl+1)(2j,+2)(2j,+3)(2j,+4

J _

A+1

J

A

J

J

jl-2

J

j,+l

5

O',+m-l!O'1-m)O',-m+1)0'1-m+2) 2(~,-1)~10',+1)(2j,+l)

J

J

0',+2m-1) 1)

-

J

(j,-m+ 1) i&-m) ($-l)j,(2jl+1)(2j,+2)

,-m+l)O',-m)O',-m-1)0'l+m-l) (jl-1)(2j,-l)jt(2j,+l)

J

J

(j,-m-l)(j,+m+l)(j,+m)O',+m-l) 6,-l) (2j,- l)j,@j,+

(2j,+3)

J

(j,-m-l)(j,+m)(j,+m+l)O',+m+2) (j,-l)j,(2j,+l)(2j,+2) ~,+m-l)~,+m)~,+m+l)(jl+m+2)

1)

J

(2j,+4)

(j,-m-l)(j,-m)(j,-m+l)O',+m+2) jl(2jl+ 1) O'I+ 1) (%+4 (j,-m-l)(j,-m)(j,+m+l)O'l+m+2) (2j,-l)j,(2j,+2)(2j,+3)

1)

(j,+m+l)O'l+m) (.j,-l)j,(2jl+1)(2j,+2) -

(2j,+ 1) (2j,+2)

(2j,-2)(2j,-1)2j,(2j,+l)

(j,-m+l)(j,+m+l) j1(2jl+l)C.j~+l)o'1+2) 3m*-jlC$+l) 11j16'1+ 1) (%+3) 3Cj,-m)ti,+m) ,--l)j,(Zj,+l)&+l) 3(j,-m)(j,-m-l)(j,+m)ti,+m-1) (2jl-2)(2jrl)j,(2j,+

~,-m-i)~,-m)~,-m+l)(jl-m+2) J

(j,-m+l)O',-m) J j,(2jl+ 1) C&+2)&+2)

3($-m) ti,+m+ (2m+1)-/(2j,-l)j,(2j,+2)(2j,+3)

3(j,-m+2)(j,-m+l)(j,+m+2)0',+m+l) (2j,+1)(2j,+2)(2j,+3)0',+2)

d(2jl-

mz=-2

(j,-m+2)(j,-m+l)(j,-m)ti,+m+2) (2j,+l)(j,+l)@j,+3)0',+2)

t.j1+2m+2)

J

3(j,-m+l)til+m) (2j,-l)j,@j,+2)(2j,+3)

ma=-1

j,-1 h-2

O',-m+2)O',+m+2)O't+m+l)O'l+m) (2j,+l)(j,+l)@j,+3)0',+2)

(l--m)

(.j,-m-l)($-m)(j,-m+l)(j,-m+2) (2jl-2) (2jl- 1)2j,(2j,+

ml=0

-0.,-2m+2)J2j,(~~~~~~~~~2~

O',+m-1)O',+m)O',+m+1)0'1-m+2) Zj,(j,+1)0',+2)(2j,+l)

j=

5+2

J

3~,+m-l)~l+m)~~-m+1)O'~-m+2) (2j,-l)2jl(j,+l)(2j,+3)

_

ji-1

m=l

1)

1010

MISCELLANEOUS

FUNCTIONS

27.9.5 [By use of symmetry relations, coefficients may be put in standard form j,<j,< j and m>O]

Table

m2 1 m 1 A 1 j

1

(jlj2mmzl

j, j2jm)

j2=!4

-E H

$2

81

it1: 00000 E::

-

46

-1

-5E -2

0.70711 0.00000 -t y;;; -0:0.81650 70711

x4-i

0. 57735 1.00000

5#

0.81650 0.40825

5; 4

0.40825 0.70711 0.70711 1.00000

*

-

h=K

0.73030 -0.25820 -0. 63246 0. 63246 -0. 77460 0. 70711 0.70711 0.86603 0.50000 1.00000 0.50000 0.50000 -0.50000 - 0.50000 0. 70711 0.00000 -0.70711 0. 70711 -0.70711 0.54772 0.77460 0.31623 0. 77460 0. 63246 1.00000

Compiled cients, Oak @54) (with

from A. Simon, Ridge National permission).

Numerical Laboratory

tables of the Clebsch-Oordan Report 1718, Oak Ridge,

co&lTerm.

[27.25] E. U. Condon and 0. A. Shortley, Theor atomic s ectra (Cambridge Univ. Press, Sal2 bridge, 2 ngland, 1935). 127.26) M. E. Rose, Elementary theory of an ular momemtum (John Wiley & Sons, Inc., If ew York, N.Y., 1955). 127.271 A. Simon, Numerical tables of the Clebsch-Gordan coefficients, Oak Ridge National Laboratory Report 1718, Oak Ridge, Tenn. (1954). ~o~j~p3; mimzm) for all angu%r moments <#,

*see

page

n.

28. Scales of Notation A. SCHOPF~

S. PEAVP,’

Contents Page Representation

. . . . . . . . . . . . . . . . . .

1012

. . . . . . . . . . . . . . . . . . . . . .

1013

References.

. . . . . . . . . . . . . . . . . . . . . . . . . .

1015

Table 28.1.

2f” in Decimal,

n=0(1)50,

. . . . . . . . . . .

1016

Table 28.2.

2” in Decimal,

zc=.001(.001) .01(.01).1(.1).9,

15D . . .

1017

Table 28.3.

lo*” in Octal,

Exact or 20D . . . . . . .

1017

Table 28.4.

n log,, 2, n log, 10 in Decimal,

10D . . . .

1017

Table 28.5.

Addition and Multiplication Tables, Binary and Octal Scales. . . . . . . . . . . . . . . . . . , . .

1017

Mathematical

1017

Numerical

Table28.6.

of Numbers.

Methods

The authors acknowledge checking of the tables.

n=0(1)18,

Constants

Exact

n=l(l)lO,

in Octal

Scale . . . . . . . .

the assistance of David S. Liepman

1 National Bureau of Standards. * Guest worker, National Bureau (deceased).

of Standards,

from

The

in the preparation

American

and

University 1011

28. Scales of Notation Representation Any positive real number x can be uniquely represented in the scale of some integer b>l as x=(A,

. . . AIA,,.a-la-2

. . .)(b),

of Numbers Integers

X=(A,

. . .

(I) b-scale arithmetic. and define

AlAo)

Convert ?; to the b-scale

where every A, and am5 is one of the integers 0, 1 not all Ai, aWj are zero, and * *, b-l, Am)0 if z 11. There is a one-to-one correspondence between the number and the sequence x=A,b”+

. . . SAlb+%+~

a- jb-5

where the infinite series converges. The integer b is called the base or radix of the scale. The sequence for x in the scale of b may terminate, i.e., a-.-l=a-n-2= . . . =0 for some n>l so that x=(A,,,

. . . A1Ao.a-la-2

where 2, xi, . . ., xi are the remainders and Xl, x2, * . ., X; the quotients (in the b-scale) where X, X1, . . ., X;.+ respectively are divided by z in the b-scale. Then convert the remainders to the z-scale,

. . . a-,Jca,;

then x is said to be a finite b-adic number. A sequence which does not terminate may have the property that the infinite sequence aml, a-,, . . . becomes periodic from a certain digit a-,(n>l) on; according as n=l or n>l the sequence is then said to be pure or mixed recurring. A sequence which neither terminates nor recurs represents an irrational number.

(z&,=&

(z-4&;)=&

. . ., (Ti&;)=Ji;

and obtain x=

(24; . . . ZJ&).

(II) &scale arithmetic. Convert b and Ao, A,, . . ., A,,, to the g-scale and define, using arithmetic operations in the J-scale, Xm_1=A,b+A,-1, X,-2=X,-1b+A,-,,

Names of Scales

X,=-&b+A,, Base

Scale

2 3 4

Binary Ternary Qu?ternary

2 7

2eiEiy Septenary

I

Base 8 9 10 11 12 16

I

Scale

X=X,b+A,. Octal Nonary Decimal Undenary Duodenary Hexadecimal

General Conversion Methods

Any number can be converted from the scale of b to the scale of some integer z# b, z>l, by using arithmetic operations in either the b-scale or the $-scale. Accordingly, there are four methods of conversion, depending on whether the number to be converted is an integer or a proper fraction. 1012

then

Proper

fractions

2=

(0.~3~la-,

. . . )(a)

To convert a proper fraction x, given to n digits in the b-scale, to the scale of ii Z b such that inverse conversion from the J-scale may yield the same n rounded digits in the b-scale, the representation of x in the z-scale must be obtained to n rounded digits where n satisfies i?>b*. (III) b-scale arithmetic. Convert a to the b-scale and define

SCALES

OF

where &, SiL, . . ., ii-;; are the integral parts and Xl, 22, * - *, G the fractional parts $n the b-scale) of the products & z$, . . ., ~-lb, respectively. Then convert the integral parts to the kcale, (&)(B)=ii-l,

1013

NOTATION

Convert 6 and a-,, (IV) z-scale arithmetic. to the J-scale and define, using arithmetic operations in the &scale,

a-2, . . .) a-,

. . ., E&,=L,

(z.2)(;,=ii-2,

x+2=x-.+db+a-.+a x-1=xJbSa-1;

and obtain

then x= (OLIii-2

Numerical

Methods

The examples are restricted to the scales of 2, 8, 10 because of their importance to electronic computers. Note that the octal scale is a power of the binary scale. In fact, an octal digit corresponds to a triplet of binary digits. Then, binary arithmetic may be used whenever a number either is to be converted to the octal scale or is given in the octal scale and is to be converted to some other scale. Decimal

1 2 3

4

5

6

7

8

9

10

Octal

123

4

5

6

7

10

11

12

Binary

1 10 11 100 101 110 111 1000 1001 1010

Convert X= (1369)c10j to the octal scale. By (I) we have b=lO, z=8c0, and so, using decimal arithmetic, Example

x=xsel/b.

. . . &J(6).

1.

Convert X=-(2531) (*) to the decimal scale. By (I) we have b= lO= (12) (*) and hence, using octal arithmetic, Example

2.

2531/12=210+11/12 210/12=15+6/12 15/12=1+3/12 l/12=0$1/12

Thus, converting &=

(11)(B)=9,

to the decimal &=6ce,=6,

scale,

x2=3(*)=3,

&

1,

and so X= (1369)(10,. By (II) we have b= 10, and the octal digits of X are unchanged in the decimal scale. Hence, using decimal arithmetic,

1369/8=171+1/8, 171/8=21+3/8,

21/8=2+5/8, 2/a=o+218;

Using binary arithmetic we have, by b=8=(lOOO)o, and A,=l,A1=(11)(2),A2=(101)(2), Aa= (10)c2,. Then,

then X=

(2531),,.

By (II) we have b= (12)(*, and As= lc8), A2 =3(*), A1=6(,=,), Ao=(ll)(g,. Hence, using octal arithmetic, x,=1-12+3=

(15)(*),

X~=10~1000+101=(10

x,=10 x=10

(II),

lOl),,,

101~1000+11=(10

101 Oil)@,,

101 011*1000+1=(10

101 011 001)(2),

xr=15~12+6=(210)~8,, x=210-

12+11=

(2531)(*,.

Using binary arithmetic we have, by (II), ~=(1010)~2~ and &=l~, A2=(ll)~2,, A=(l10)c2j, AO(lOO1)o,. Thus x,=1*1010+11=(1101)(2,, x,=1101*1010+110=(10

x=10

001 OOO)@,,

001 ooo.lolo+lool=(lo

whence, on converting

101011 OOl),,,

to the octal scale,

X=(2531)(8,.

whence, on converting x=

to the decimal scale, (1369)clo,.

Observe that in both examples above, octal arithmetic is used as an intermediate step to convert. according to (II), the given number to the binary scale. If, instead, the given number is first converted to the binary scale, then binary arithmetic may be applied directly to convert, according to (I), the given number from the binary scale to the scale desired.

1014

SCALES

OF

For example, in converting X= (2531),8, to the decimal scale, we find fbst X=(10101011001)~~, and then obtain, using (I) with g=lO=(lO1O)ca,, 10 101 011 001/1010=10

NOTATION

Alternatively, we can apply (III) using binary arithmetic: (0.010 110 11).1010=11+(0.100

(0.100 011 1).1010=101+(0.100

001 ooo+lool~lolo,

(0.100 011)*1010=101+(0.011

10 001 000/1010=1101+110/1010,

(0.011 11) ~1010=100+(0.101

1101/1010=1+11/1010, 1/1010=0+1/1010.

Thus, on converting Ao=(1001)ca,=9,

to the decimal scale,

Al=

(110)cs,=6,

Aa=l,

~&=(11)~~,=3,

whence Convert

3.

z= (0.355)c10j to the bi-

(0.355) .8=2+0.840,

(0.080) .8=0+0.640

(0.840) .8=6+0.720,

(0.640) .8=5+0.120

(0.720) .8=5+0.760,

(0.120) .8=0+0.960

(0.760) .8=6+0.080,

(0.960) -8=7+0.680

whence x=(0.26560507 . . .)ca,. Thus, verting to the binary scale,

on con-

In order that inverse conversion of x from the binary to the decimal scale yield again x to the given number n of decimal digits, we must round x in the binary scale to at least Z digits where 5 is chosen such that 2g>10n. As a working rule, we n.

Hence, to obtain x= (0.355) clgj

by inverse conversion, binary scale to Z 2:

x must be rounded

in the

3= 10 digits.

Thus, 2=(0.010

with

b=8,

and k are to be chosen so as to satisfy d-k

2y.

15

=50.

From Table 28.1 we find

Thus, we must take k=29 and, consequently, choose n>21. The conversion on a desk calculator thus proceeds as follows. First, we obtain by use of Table 28.1

Then, for convenience’s sake, we convert this number to the octal scale, using the method of Example 3 and rounding as required, to at least 7 octal (=2l binary) digits. We find

using decimal

27?z= (1.537 4337)@,.

Hence

x-,=6/8+6=6.75,

x= (1.537 433 7)(8) * 2 -2s

and, consequently,

x-1=6.75/8$2=2.84375, x=2.84375/8=0.355

. . . a-,),a,*2-k

2mx= (1.686 629 899) tloj

x= (0.266)(s),

(IV)

1).

where n and k are such that inverse conversion from the binary scale to the decimal scale will produce x to the same given 15 decimal digits. Accordingly, by the rule stated in Example 3, n

110 110 O),,.

TO carry out the inverse conversion we can first convert to the octal scale,

and then apply arithmetic:

ll),

Note that the fractional part in any step is the unconverted remainder. Thus, to round at any step, it is only necessary to ascertain whether the unconverted portion to be neglected is greater or less than 4; i.e., whether, in the binary scale, the first neglected digit is 1 or 0. Example 4. Convert z= (3.141593)c10j~ 10m9 to the binary scale. The desired representation is x=(l.u-la-a

110 101 110 000 101 000 111 . . .)@,.

may take ?i >$

Oil),

x= (0.3554)ClO,

nary scale. We first convert to the octal scale, using decimal arithmetic. By (III), we find with b=8

2=(0.010

011 l),

Converting the integral parts to the decimal scale, we find z-,=(11),,=3, ^a-a=Z-3=(101)(2,=5, &=(100),=4, and thus

X= (.1369) (10). Example

with g= (1010) (aj,

46875.

x=(1.

101 011 111 100 011 011 111)@,*2-“.

SCALES

To convert x back to the decimal scale we only need to obtain from Table 28.1 the various powers of 2 which appear in the above representation and sum them. However, since 2-ln=2-m+1-2-m for any real constant m, it is more convenient to reduce first the binary representation of x to the form x,2-24 ,2-31_2-33_2-39+2-42_2-46_2-w

and then sum these powers of 2. (Note that the number of summands is thereby decreased from 16 to 7.) From Table 28.1 we have +2-“=+3.725 -2-31=-2-33=-2-3Q=+2-“=-I-2-4(1=-2-W=x=

.465 .116

290 661 415

298 287 322

*lo-* .lO+’ .lO+’

.OOl

818

989

000 227 :OOO 028 .ooo 000

374 422 888

.lO+’ -10-Q .lO+’ -10-Q

764

.lO+

3.141

592

Nine decimal digits are used for sufficient accuracy reserve. Hence, rounding to seven significant figures, we find x= (3.141593)(1lJ,*lo

-9

1015

OF NOTATION

We first compute, using 4.1.19 and Table 4.1, .05764 log,, x 83.44295 log, x=-= .30103 =277+mt kl0 2 and find from Hence log, x=277+

Table

4.1, .05764=loglo

log,, 1.1419 log10

=277+1og,

Conversion back to the decimal as follows, we write log,, x=log,, 2 log2 x =log,,2{265+logz

To convert a number such as

=log,,

2

265+lOg,o

=265 log,, 2 +lO&,

10

scale proceeds

(11105)

,3,}

(11105)(8) log10

log, x=&==k++2

1.1419

2

and so x=(1.1419)(10, *2 277. , Now we apply the methods of Example 3 to obtain (1.1419)~,o,=(1.110516),8, where octal notation is used for the sake of convenience. To round such that inverse conversion will yield the same decimal digits of x, observe that the last non-zero decimal digit of x is 3. 1080. Table 28.4 shows tha’t 2286<10*<2266. Hence, in the binary scale, x must be a binary integer times 22aa; i.e., (1.110516)(8, must be rounded to 4 octal (= 12 binary) digits. As a result, x=(1.1105)~g~~2277=(11105)~g~~2286 =(l 001 001 000 lOl)n22@

.

to the binary scale, where k is a positive integer so large that Table 28.1 cannot be used, apply the following device: Compute

1.1419.

2

(11105),8,.

Hence, converting (11105)(8J to the decimal scale by any of the methods of Example 2, we obtain log,, x=265 log,, 2+logl,, 4677

10

which yields, using Table 4.1 where k is the quotient and xl the remainder, the division being carried out in the decimal scale. Then find r]=lO’l, i.e., xl=loglo 7, so that . log, x=k+B=k+log,

rl

1

log,0 x=83.44292 Thus, by Table 4.1, we find, rounded significant figures, x= (2.773)cIo,. lOa.

to four

whence x=

h)

References

(10,2’.

Now convert (I),~~, to the binary scale by any of the methods described above. A similar device may be used to convert to the decimal scale a binary number that is outside the range of Table 28.1. Example 5. Convert x= (2.773)(ro,.1083 to the binary scale.

J. Malengreau, fitude dea Bcritures bin&es, Bibliothhque Sci. 32 Mathkmatique. Edition Griffon, NeuchBtel, Suisse (1958). [28.2] D. D. McCracken, Digital computer programming (John Wiley & Sons, Inc., New York, N.Y., 1957). [28.3] R. K. Richards, Arithmetic operation in digital computers (D. Van Nostrand Co., Inc., New York, N.Y., 1955). [2&l]

SCALES OF NOTATION

1016 Table 28.1

2*”

2n

IN DECIMAL

n

I

2 4 1: 32

i

2 3 4 5

0.125

64 128 256 512 1024 2048

1;: 11

4096 8192 16384

12

32768 65536 1 31072

15 16 17

2 62144 5 24288 10 48576

18 19 20

46 97265 625 73 48632 8125 36 74316 40625

20 97152 41 94304 83 88608

21 22 23

68 37158 20312 5 84 18579 10156 25 92 09289 55078 125

167 77216 335 54432 671 08864

zz 26

96 04644 77539 0625 98 02322 38769 53125 49 01161 19384 76562 5

:i

0.000s952587 89062 5 0.00000,76293 94531 25

1342 17728 2684 35456 5368 70912

27 St

50580 59692 38281 25 25290 29846 19140 625 62645 14923 09570 3125

10737 41824 21474 83648 42949 67296

30 31 32

31322 57461 54785 15625 65661 28730 77392 57812 5 32830 64365 38696 28906 25

85899 34592 1 71798 69184 3 43597 38368

33

6 87194 76736 13 74389 53472 27 48779 06944

36

32182 69348 14453 125 66091 34674 07226 5625 83045 67337 03613 28125

;z

;i

0.00000 00000 0.00000 00000

51 91522 83668 51806 64062 5 75 95761 41834 25903 32031 25 37 97880 70917 12951 66015 625

54 97558 13888 109 95116 27776 219 90232 55552

39 40 41

0.00000 00000 0.00000 00000

18 98940 35458 56475 83007 8125 09 49470 17729 28237 91503 90625 54 74735 08864 64118 95751 95312 5

439 80465 11104 879 60930 22208 1759 21860 44416

42

0.00000 00000 00%27 37367 54432 32059 47875 97656 25 0.00000 00000 00 3 68683 77216 16029 73937 98828 125 0.00000 00000 000 6 84341 88608 08014 86968 99414 0625

3518 43720 88832 7036 87441 77664 14073 74883 55328 28147 49767 10656 56294 99534 21312 112589 99068 42624

ti

0.00000

00000

0.00000 00000 000 0.00000 00000 000 0.00000 00000 000

42170 94304 04007 43484 49707 03125 21085 47152 02003 71742 24853 51562 5 10542 73576 01001 85871 12426 75781 25

00000 000 t 55271 36788 00500 92935 56213 37890 625 0.00000 00000 000 77635 68394 00250 46467 78106 68945 3125 0.00000 00000 00000 8817 84197 00125 23233 89053 34472 65625 0.00000

SCALES

1017

OF NOTATION

2” IN DECIMAL x

2”

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009

1.00069 1.00138 1.00208 1.00277 1.00347 1.00416 1.00486 1.00556 1.00625

33874 72557 16050 64359 17485 75432 38204 05803 78234

x 62581 11335 79633 01078 09503 38973 23785 96468 97782

1.00695 1.01395 1.02101 1.02811 1.03526 1.04246 1.04971 1.05701 1.06437

0. 01 0. 02 0.03 0.04 0. 05 0.06 0.07 0. 08 0. 09

lo*11

lo”

3 46 575 1 346

0 :

1.000 0.063 0.005

000 000 000 146 314 075 341 631 217

000 270 463

1 750 23 420

3 4

0.000 0.000

406 111 032 155

570 530

303 641 113 360 545

240 100 200 400 000

564 613

:

00'000 000

000 476 002 206 157 132

364 610

055 706

37 64

0.000 000 001 0:OOO 015 327 257

143 745

561 152

06 75

9

0.000

560

276

41

104

1 16 221 2 657 34 434 5 432 67 405

n log10

n loml 2

n

0. 6 o"G 0: 9

327 157 127 553

n

n log* 10

:

0.60205 0.30102

99913 99957

3.32192 6.64385

61898 80949

:

1.20411 0.90308

99827 99870

13.28771 9.96576

23795 42847

5

1.50514

99763

16.60964

04744

ADDITION

28.3

10 I1

762 564 210 520 440

000 000 000 000 000

10 11 12 13 14

0.000 0.000 0.000 0.000 0.000

000 000 000 000 000 000 000 000 000 000

006 000 000 000 000

676 537 043 003 000

337 657 136 411 264

66 77 32 35 11

724 461 500 115 760 200 413 542 400 164 731 000

000 000 000 000

15 16 17 18

0.000 0.000 0.000 0.000

000 000 000 000

000 000 000 000

000 000 000 000

022 001 000 000

01 63 14 01

112 351 432 411 142

402 035 451 634 036

n log2 10 IN DECIMAL

2,

Talh!

34625 83549 44133 79107 35623 65665 47927 11265 59830

IN OCTAL

000 00 146 31 243 66 651 77 704 15

ll

000 000

1.07177 1.14869 1.23114 1.31950 1.41421 1.51571 1.62450 1.74110 1.86606

10”

1::

28.2

2”

56719 90029 07193 56067 41377 41121 23067 613.90 53360

lo-”

n 1

2”

55500 94797 21257 38266 49238 57608 66836 80405 01824

‘I’alde

n

Ta,lr

; i 10

-4ND MULTIPLICATION

99740 99696 99653 99610 99566

19.93156 23.25349 26.57542 29.89735 33.21928

TABLES

Addition

28.-l

72log2 10

72log,0 2 1.80617 2.10720 2.40823 2.70926 3.01029

000 000 000 000

85693 66642 47591 28540 09489

TaI)lc 28.5

Multiplication Binary Scale

O+l=l+O=

o+o=

0

1

0x1=1x0=0

l+l=lO

0x0=0 1x1=1

Octal Scale 01

02

03

04

05

06

07

1

02

03

04

05

06

07

02

03

04

05

06

07

10

2

04

06

10

12

14

16

11

14

17

22

25

03

04

05

06

07

10

11

3

06

04

05

06

07

10

11

12

4

10

14

20

24

30

34

05

06

07

10

11

12

13

5

12

17

24

31

36

43

06

07

10

11

12

13

14

6

14

22

30

36

44

52

07

10

11

12

13

14

15

7

16

25

34

43

52

61

10

11

12

13

14

15

16

MATHEMATICAL *=

(3.11037

552421)c8j

* -l=

(0.24276

3O1556)c8j

ii=

(1.61337

611O67)c8)

CONSTANTS e= e-l= y'i=

IN OCTAL

211.6

521335)cB)

(0.27426

53O661)c8j

hl

I=

-(0.43127

2336O2)c8)

(1.51411

23O7O4)c8)

lo&r2

.,=

do.62573

O3O645)c8)

T= (1.11206

404435)(*)

log10

e=

(0.33626

754251)cB)

lo@

T= (1.51544

163223)(,)

log2

e=

(1.34252

166245)c8)

(3.12305

407267)(*)

1(-j=

(3.24464

741136)(*)

logi

Table

(2.55760

h

dlo=

SCALE Y=

(0.44742

~‘2= (1.32404

hl

1477O7)cB,

74632O)c8j

In 2=

(0.54271

027760)c81

lo=

(2.23273

O67355)c8)

29. Laplace

Transforms

Contents 29.1. Definition

of the Laplace

29.2. Operations

for the Laplace

29.3. Table of Laplace

..........................

.............

Transform

Transforms

29.4. Table of Laplace-Stieltjes References

Transform

............

................

Transforms

Page 1020 1020 1021

............

1029 1030

29. Laplace 29.1.

Definition

of the

One-dimensional

29.1.1

f(s)=.Y{F(t)]

Laplace

Transforms function of s in the half-plane B?s>s,.

Transform

Laplace

Transform

=lrn

e-““F(t)dt

Two-dimensional

j(u, v)=Y{F(z,

B-t-

for

the

Laplace

Function

(t=O) u(t) (t
Transform’ Image

m

F(t)

S

e-“‘F(t)dt

&W

+&Cd

Function j(s)

0

Formula

S

c_‘im c Irn

Linearity

Step

y)dm!y

In the following tables the factor is to be understood as multiplying the original function F(t).

Function F(t)

Inversion

Unit

+ 1 1

AB e--ElotF(t)dt

29.2.1

29.2.3

u(t)=

29.1.3

29.2. Operations

-2’,,

of the

e-uz-ouF(x,

0

exists, then it converges for all s with %?s>sO,and the image function is a single valued analytic

29.2.2

y) 1=lmS,-

Definition

S

Original

Transform

29.1.2

F(t) is a function of the real variable t and s is a F(t) is called the original funccomplex variable. If the tion andf(s) is called the image function. integral in 29.1.1 converges for a real s=so, i.e., l&

Laplaee

e'"j(s)ds

Property

AF(t)+BG(t) Differentiation

29.2.4

F’(t)

29.2.5

F’“‘(t)

V(S)---F(+O)

s”f(s)-s”-1F(+0)-s”-2F’(+O)-

. . . --F'"-l'(+O)

Integration

29.2.6

S Ir

’ F(T)ds

ff(4

0

29.2.7

;A4

F(X)dAdr ss 0 0 Convolution

29.2.8

(Faltung)

S

’ F,(t--)Fz(7)dr=Fl*Fz

fib) j*(s)

0

29.2.9 29.2.10

Theorem

Differentiation

f’(s)

-tF(t) (-l)vF(t)

1Adapted by permission from IL V. Churchill, York, N.Y., 1958. 1020

P’ (8) Operational

mathelhatics,

2d ed., McGraw-Hill

Book Co., Inc., New

LAPLACE Original

Function

1021

TRANSFORMS

F(t)

Zmage

Function

f(s)

Integration

29.2.11

f F(t)

29.2.12

e”‘F‘(t)

29.2.13

;F

0

f

Transformation

fc-4

fbs)

(c>O)

1 (Slc)tF i ce 0

29.2.14

S

8m f(x)dx

Linear

f (cs- b)

(c>O)

Translation

29.2.15

F(t-b)u(t-6) Periodic

29.2.16

e-""j(s)

(bX-0

a S a S

Functions

e-“‘F(t)dt

F(t+a)=F(t)

0

l-e-as

29.2.17

F(t+a)

e-“F(t)dt

= - 6’(t)

0

l+e+

Half-Wave

29.2.18 Full-Wave

Rectification

of F(t)

in

29.2.17

f(s) 1-e-s

F(t) .$ (-l)%(t--na) 9 Rectification

29.2.19

of

F(t)

in 29.2.17

j(s) coth 2 2

IF(t) I Heaviside

Expansion

Theorem

PW n(s>’ q(s)=(s-a,)(s-az)

. . . (s-a,,)

p(s) a polynomial of degree<m 29.2.21

r per-n,(u) eatz

(r-n)!

p-1 (n-l)! p(s)

a polynomial of degree
29.3. Table of Laplace Transforms*, 3 For a comprehensive table of Laplace a.nd other integral t8ransforms SW [29.9]. dimensional Laplace transforms see [29.11]. f(s)

F(t)

29.3.1

1 s

1

29.3.2

1 3

t

For a tnblc of tn-o-

2 The numbers in bold type in the f(s) and F(t) columns indicate the chapters in which the propcrtics of tlw rcq(xtivc: higher mathematical functions are given. 3 Adapted by permission from R. V. Churchill, Operational mathematics, 2d. ed., McGraw-Hill Book Co., IIIC., SW York, N. Y., 1958.

1022

LAF’LACE

TRANSFORMS

f(s) 29.3.3

1 F

F(t)

(n=l,

tn-I

2,3, . . .)

(n-l)!

kt

29.3.4

$i 29.3.5 29.3.6

s-3/2

&,- (n+))

(?a=1,2,3,.

r(k) ts

29.3.7

24F

. .)

(k>o)

terna

(SL2

1

(n=l,

(sfd”

29.3.11

(a#b)

(s+utb+b) s @+a)

29.3.16

29.3.17

29.3.18

29.3.19

29.3.20

29.3.21

(n-l)!

e

ae

(s+b)

1 (s+d(s+b)

(s+c)

-at-be-b1

a-b ~(b-c)e-a’+(c--a)e-b’+(a-b)e-c’ (u-b) (b-c) (c-a)

constants)

1 s2+a2 S

-al-e-“’

b-a

b#b)

(a, b, c distinct 29.3.15

. . .)

tk-l,-at

(k>o)

29.3.12

29.3.14

p-le-ar

2,3,

I- (k) (s+aY

29.3.13

. (2n-l)&

e-“l

29.3.9 29.3.10

1.3.5.. ik-1

I sfa

29.3.8

yy”-,

1 ; sin at

s2+a2

cos at

1 s2-a2

a

S

s2-a2

1

1

sinh at

cash at

s(s2+a2)

$ (I-cos

1 s”(s’+a’)

f (at-sin

(s2+&

&

at)

at)

(sin at---at cos at)

LAPLACE

TRANSFORMS

1023

f(8)

F(t)

29.3.22 29.3.23 29.3.24

4 sin at 2a $ (sin at+&

(s&)2

t co9 at

s2-a2 Cs2+a2)2

29.3.25

s (~“+a”) (s2+b2)

29.3.26

1 b+4’+b2

1 - em” sin bt b

29.3.27

s+a (S+d2+b2

e-ar cos bt

29.3.28

3a2 s3+a3

29.3.29

4a3 s4+4a4

29.3.30 29.3.31

S

s4+4a4 1 s4--a4

co9 at)

(a2# b2)

cos at-cos bt b2- a2

e-at -eW sin at cash at-cos - 1 sin at sinh at 2a2 ,$ (sinh at-sin

29.3.32

&

29.3.33

(1+a2t2) sin at-at

co8 at

22

29.3.35 29.3.36

L(e-b’-e-~‘) 2 J*t”

d&a

29.3.38

fi s-a2

29.3.39

G s+a2

29.3.4Q

at)

(cash at-cos at)

29.3.34

29.3.37

at sinh at

1 &qs-d)

1024

LAPLACE

TRANSFORMS

f(s)

F(t)

-2 f-G

1

29.3.41

fib+&

29.3.42

(s-a”)

b2-a2 @+&I

s0

1

erfc buG

b2--a2 &s-4

(6+

29.3.46

(l-s)* p-+-t

29.3.47

(1 -sjn an+:

Js+2a -6

ea% z erf (u&)-l]+ebzf [

b)

($&t

1

7 7

-at erf (46-u&)

(s+h’s+b

29.3.48

7

eaZi erfc a&

&+d

29.3.45

eA2dX

eaQ[b--a erf afl--bbebQ

29.3.43

29.3.44

4

e-a2t

Hzn+~

ue-“‘[II

+lo(ut)]

7

erfc bJi

22

HPn(~)

n! (2n+l)!&

7

(4)

22

9

29.3.49

9

r (k) (s+a)k(s+b>k

29.3.50

(k>O) 6

10

29.3.51

9 Js+zaJP&&

29.3.52

29.3.53

29.3.54

(JiG+

(a-b)” Jsfby

c&G+ ma” dWG

29.3.55

29.3.56

29.3.57

JS

(k>o)

9

k e-#(a+b):I k t

9

(v>-l)

9

J&2 WV-g &q2

1 t e’“‘I,(ut)

(y>-

1)

Jo@4

9

u”J,(ut)

9

--fi I’(k)

t ‘-‘J&t) 0 2a

6,lO

LAPLACE

f(s) 29.3.58

29.3.59

29.3.60

F(t)

( Jzw-s)k

(k>o)

(s-&=a” JF2

(v>-l)

(k>o)

(s&k

kGJ,(at)

9

U”l&)

9

J;; 4 k-fI&*(ut)

1 s e-ks

u(t-k)

29.3.62

1 2 emks

(t-k)u(t-k)

1 -g em’”

1 -e-kd

29.3.64

29.3.65

(t-k)r-’ r(p)

WO)

1

l+coth

s(l-emk8)=

G--k)

I I

2s

$ks

1

L-L-

u(t)--u(t-k)

S

6,lO

r(k) 0 2a

29.3.61

29.3.63

1025

TRANSFORMS

go u(t--nk)

1

:

L+

29.3.66

1 s(eka-u)

g

an-‘u(t--nk)

29.3.67

i tanh ks

u(t)+2

29.3.68

1 s(l+emk”)

go(--l)“u(t--nk) km--2131

29.3.69

-$ tanh ks

h(t)+2

gl (-l)%(t-2nk) z

5

(-l)n(t-22nk)u(t-22nk)

It=1

29.3.70

1 s sinh ks

29.3.71

1 s cash ks

2 go u[t-(2n+l)kl

1026

LAPLACE

TRANSFORMS

F(t)

f(s) 1

29.3.72

i

coth ks

Lx!5

u(t) +2 5 u(t-2nk) lb=1

2

2

0

2k . 41 -

k

29.3.73

coth c s2+k2 2k

Isinkt’ IzmL0 IL

(s’+l)(Le-“)

29.3.74

2

(-l)“u(t-nlr)

sin

kwl 0

n-0

29.3.75

2” L t I

277 3r

9

Joh’~)

29.3.76

-L

cos 24z

’ Jz

cash 24%

- ’ a

sinh 2m

fi

29.3.77

*

-

29.3.78 29.3.79 29.3.80

$ es1

(p>O)

t’-’ 0E

29.3.81

1 5 sre’

(PX)

0

2i.3.82

em”&

(k>O)

29.3.83

’ -‘& se

(k>O)

,g9.3.84

1

-k&

6”

29.3.85 29.3.86 29.3.87

1 ae

1 pie n-l

~‘3 e

-k&

-k&

-k&

;

2 J,+&&d ‘+ 1#-1(2~>

7

erfc k 24

(k20)

(k>

k erfc -&=24

0)

(n=O, 1,2, . . .; k>O)

(4t))”

i erfc k 24

7

7

i” erfc k 24

22

(n=O, 1,2, . . .; k>O)

1

G *Bee page xx. .

exp

k2 ( .-Ft

>

-aeaea2’ erfc (a&+-$)

7

LAF’LACE

1027

TRANSFORMS

29.3.89 (k> 0)

-eakeaQ erfc (ay%+-$)+erfc

(k>O)

eakeaZterfc (aJz+-$)

29.3.90

7

-&

7

29.3.91 29.3.92

9 e--k~r+al Js2+a2

(k20)

9

(kko)

9

29.3.93 29.3.94

,-k(w-8)

(k20)

J,(u@i?%i)

9

.-A&-

J1 (a#=i@u(t-k)

9

-!+Jt2_k2

I (a+2-k2)u(t-k)

9

“Jv(a,@?>u(t-k)

9

Jiq2

29.3.95 29.3.96

e -ka-

e

-km

e-n&a

(k>o)

-e-ka

(k>O)

Js7F

l

awe --tJll+at

.

29*3*97

-Js2+a2( Js2+a2+s)’

29.3.98 29.3.99 29.3.100

b>-l,k20)

--r-ln

fins $ln

s

In s s-a

t(r=.57721

(k>o)

s

@>O>

ea’[ln a+E,(at)]

In 8 &?a+1

co9 t Si (i!)-sin

29.3.102

s In s s2+1

-sin

f In (l+ks)

29.3.104

In -s+a s+b

29.3.105

f In (1 +k2s2)

29.3.106

i In (s2+a2)

(k>o)

G

0

t Si (t)-cos

constant)

6

W(k)-In tl

29.3.101

29.3.103

56649 . . . Euler’s

5 t Ci (t) t Ci (t)

5 5 5

;

f (e-“‘-e-“~) (k>O)

(G-0)

;

5

2 In a-2 Ci (at)

5

-2Ci

0

~

1028

LAPLACE

TRANSFORMS

F(t)

f(8) 29.3.107

$ln

(s2+a2)

29.3.108

s*+d hgp

29.3.109

‘“7

29.3.110

arctan ks

29.3.111

f arctan ;

29.3.112 29.3.113

29.3.114

29.3.115

29.3.116

s [at In a+sin at-m!

~~>o)

; (1 -cm

s2-a2

; (l-cash

1 se

kW

erfc ks

ekr erfc fi erfc &

$

at)

f sin kt k

ek2r2erfc ks

at)

Si (kt) (k>o) (k>O) (k>O)

(k 20)

e” erfc &

7

&

7

erf 2

7

JE d(t+lc)

7

t2

exp

(

t

-!-

u(t-k)

sin 2kG

Jz

(k10)

29.3.117

7

i

29.3.118

7

T w

(k>o)

9

&2N”-k)

(kX-0

9

& exp

(k>O)

9

@G-m

(k>o)

9

EexP

9

-?- Ko(2JEt) Jz

29.3.119 29.3.120 29.3.121 29.3.122 29.3.123

Ko (W Ko W) f e%

(ks)

$ & (M) II k - es K O0 s G

>

-43

@X-N

29.3.124

ne-“Io(ks)

@X0

29.3.125

e+ll(ks>

(k>O)

1 e-2k&

(

(

-g

-t

>

k2 >

[u(t)---u(t-2k)]

Ci (at)]

LAPLACE

F(t)

f(8)

29.3.126

e"E,

29.3.127

+m,(as)

(as)

1029

TRANSFORMS

5

(a>@

5-

(a>@

29.3.128

a'-"emdEn

b>O;n=O,

29.3.129

b-Si(B)]

cos s+Ci(s) sin s

1,2,. . J

29.4. Table

-!t+a (t-ta)’

5

ft:a)n

5

&

of Laplace-Stieltjes

Transforms w

ds) m

29.4.1

4

S

e-*[d@(t)

emb

(k>o)

w>

0

29.4.2

1

29.4.3

u(t-k)

l-e-”

(k>o)

go dt-nk)

29.4.4

1 1+e-k”

(k>O)

go (-l)‘W--nk)

29.4.5

1 sinh

(k>O)

2 go

cash ks

(k>o)

2 F. (-l)“u[t-(2n+l)k]

tanh ks

(k>o)

u(t)+2

1

29.4.6 29.4.1

1

29.4.8 29.4.9 29.4.10

e+ sinh (ks+a) sinh (hs+b) sinh (ks+a)

RX,

%cI (-l)“u(t-2nk)

2 5 e-‘2n+1%[t-(2n+l)k] n-o

(k>o)

sinh (ks +a)

4t-@n+l)kl

2 2 e-(2n+1%[t-h-(2n+l)k] n=O

h>O)

(o
n%e-

t2n+1’o{ebu[t+h-(2n+l)k] -eebu[t-h-(2n+

29.4.11

2 a,emkns 7k=O

@
. . .>

OkI 1

nTow(t--k,)

For the definition of the Laplace-Stieltjes transform see[29.7]. In practice, Laplace-Stieltjes transforms are often written as ordinary Laplace transforms involving Dirac’s delta function This “function” may formally be considered as

the derivative of the unit step function, du(t) =6(t) dt, so t,hstJ’

6(t).

du(t)=J’

6(t)dt={

F fz’>ii.

29.22, for instance, then m e-“16(t-k)dt. assumesthe form eVks= The correspo;dence

S 0

4 Adapted by permission from P. M. Morse and H. Feshbach, Hill Book Co., Inc., New York, N.Y., 1953.

Methods

of theoretical

physics, ~01s. 1, 2, McGraw-

1030

LAPLACE

TRANSFORMS

References Texts

[29.1] H. S. Carslaw and J. C. Jaeger, Operational methods in applied mathematics, 2d ed. (Oxford Univ. Press, London, England, 1948). [29.2] R. V. Churchill, Operational mathematics, 2d ed. (McGraw-Hill Book CO., Inc., New York, N.Y., Toronto, Canada, London, England, 1958). [29.3] G. Doetsch, Handbuch der Laplace-Transformation, ~01s. I-III (Birkhauser, Basel, Switzerland, 1950; Basel, Switzerland, Stuttgart, Germany, 1955, 1956). [29.4] G. Doetsch, Einfiihrung in Theorie und Anwendung der Laplace-Transformation (Birkhauser, Base& Switzerland, Stuttgart, Germany, 1958). [29.5] P. M. Morse and H. Feshbach, Methods of theoretical physics, ~01s. I, II (McGraw-Hill Book Co., Inc., New York, N.Y., Toronto, Canada, London, England, 1953). [29.6] B. van der Pol and H. Bremmer, Operational

calculus, 2d ed. (Cambridge Univ. Press, Cambridge, England, 1955). [29.7] D. V. Widder, The Laplace transform (Princeton Univ. Press, Princeton, N.J., 1941). Tables

[29.8] G. Doetsch, Guide to the applications of Laplace transforms (D. Van Nostrand, London, England; Toronto, Canada; New York, N.Y.; Princeton, N.J., 1961). [29.9] A. Erdelyi et al., Tables of integral transforms, ~01s. I, II (McGraw-Hill Book Co., Inc., New York, N.Y., Toronto, Canada, London, England, 1954). [29.10] W. Magnus and F. Oberhettinger, Formulas and theorems for the special functions of mathematical physics (Chelsea Publishing Co., New York, N.Y., 1949). [29.11] D. Voelker and G. Doetsch, Die zweidimensionale Laplace-Transformation (Birkhauser, Base& Switzerland, 1950).

Subject A Page Adam’s formulas- _______ __ __ __ _- ______ - _______ 896 Airy functions-- ________-_______ 367,446,510,540, 689 ascending series for-- _____________ - ______ -_-_ 446 ascending series for integrals of---- _ _ - _- __ __ _ - _ 447 asymptotic expansions of ______ - .___ _--___-_ __ 448 asymptotic expansions of modulus and phase--449 asymptotic forms of integrals of-- _ ____ _- ____ _449 asymptotic forms of related functions---------450 complex zeros and associated values of Bi(r) and Bil(z)_-_-______________________________-478 computation of--- _________ - ____ - ____________ 454 computation of zeros of- _ _ _ _-_ __ _ _ ___-_ _-___ _ 454 definitions---___________________ -_- ____ -___ 446 differential equations- __ __ ___ __ ______ ______ 446,448 graphsof_~~~_--~~~~~_-------_-------_---~~~~~~ 446 integral representations of ____________________ 447 integrals involving ____ -_ __ ___ __ __ _-__ _ ___ __.__ 448 modulus and phase __________ - ______ - ____ -___ 449 related functions- _ ___ _ _ __ ______ _______ _ ___-_ 448 relations between--- __ _-_ ____ __ __ -_ __ _ __ _____ 446 representations in terms of Bessel functions-- _ _ _ 447 tables of _____________ - ______ - _______________ 475 tablesofintegralsof--______ - ____ - ______ -_-_ 478 Wronskian for products of ______________ -_ _-_ 448 Wronskian relations- _ _ _ __ __ __ ___ __ _ __ __ ___ __ 446 zeros and their asymptotic expansions-------..-450 Airy functions and their derivatives- _ _ __ _-_- _____ 446 zeros and associated values of _ _ _ _____________ 478 Airy integrals-__-_-_-_-_-_______-___________-447 Aitken’s@-process ___________ - ____ -__-_-___-___ 18 Aitken’s iteration method- _ _ __ _ __ ____ _ _-_-___ __ 879 Anger’s function. _ _ _-_ __ __ __ _ _ __ _-_ _-___ __ __ 498 relation to Weber’s function- _- - _ _ _ __ _ - _ _ _ _ _ 498 Antilogarithm _______ - ____ -__-_- ____ - ______ -___ 89 Approximate values- ____ __ _ __ _-___ ________ -_ __ _ 14 Approximation methods _____ ___-___ ___ __ _- ____ 18 Aitken’s @-process _____L___ -_ .__- ____ - _______ 18 Newton’s method _____ ____ -__ ___ ____ ___ _ __-_ _ 18 regula falsi- _ _ _ _____ _ __ _____ -_ __ _ __ ___ ____ __ 18 successive substitution___ __ ___ ____ ____ - _____ 18 Argument---_____-___-___________________---16 Arithmeticfunctions _________ - ____ -_-__-___-_-_ 826 tableof~~~~_-_~~_~__-_~~_~_-___~_~___-_~___ 840 Arithmetic-geometric mean- _ _ 571, 577, 578, 580, 598, 601 Arithmetic mean- _________________ - ______ -__-_ 10 Arithmetic progression- _ _ _ _ _ _ _- - _ - _ _ _ _ _ _ __ - _ _ _ _ 10 Associated Legendre functions- _ _ ___ _-_ __ __ _-_ __ 331 (see Legendre functions) Asymptotic expansions- ____ - _____________ -__-_ 15

Index PtW3 B Basic numbers- ___________ - _____________ ______ 822 Bateman function- _ ____ _-_- ______ _____________ 510 Bernoulli numbers-- ____ __ _-__ _______-_________ 804 table of- _ _____ -- ___________________________ 810 Bernoulli polynomials ____ __ __ __ __ ____ __ __ ___ ___ 803 as sums of powers- ________________-_-_______ 804 coefficients of _ ___ _ __ ____ __ __ ____ __ -___ ______ 809 derivativesof-_________ - ______________ - ____ 804 differences of ______ ____ _________ - ______ ______ 804 expansionsof___-----------------------,---804 Fourier expansions of _ _ _ - - _ _ _ _ _ _ _ __ - _ _ _ _ _ _ _ _ _ 805 generating function for _______________________ 804 inequalities for---- _ _-_-_- _____ __ __ __ ____ ___ _ 805 integrals involving _________________-_-_ ----805 multiplication theorem for __________.._-_______ 804 relations with Euler polynomials- _ _ ___________ 806 special values of ___________________-_________ 805 symbolic operations ____ _-_ _______ __-_____ _ __ _ 806 symmetry relations-- __ _______ __ ________ _____ 804 Bessel functions as parabolic cylinder functions- _____________ 692,697 definite integrals- _ _ _ ___ _____ ____ -_ __________ 485 modified _______________ - __________________ 374, 509 notation for ______ _ __ ____ _____ _-_ ____ __ ______ 358 of fractional order _________________-_________ 437 of the first kind ___________ -__-__- ______ --_ 358,509 of the second kind- ________________________ 358,509 of the third kind ______ - ______ -__- ______ -__ 358,510 orthogonality properties of _____ _ __-_ ________ __ 485 representations in terms of Airy functions _ _ _ __ _ 447 spherical- _ _ ____ __ _ _ __ __ _- _ ___ _______ _____ 437,309 Bessel functions of half-integer order--- _ _ _ _ _ _ _ _ 437,497 zeros and associated values- __________________ 467 zeros of the derivative and associated values---468 Bessel functions, integrals- _ _ __ _-__ _- _____ _-__ 479,485 asymptotic expansions- _ _ _ __ __-_ _-___ _-__ _ _ 480,482 computation of __________ - ______ - _______ - ____ 488 convolution type---------------------------485 Hankel-Nicholson type--- ___ _ -_ _- _____ _-_ _ __ _ 488 involving products of- _ _ _ ___ _ _ __ _ ___ __ __ _-_ __ 484 polynominal approximations_______________ 481,482 recurrence relations ________________________ 480,483 reduction formulas- _ _ _ _ _ _ _ _ __ __ - - _ _ _ _ _ _ _ _ _ _ _ 483 repeated_-__--_---_______---_--__-----_-_--482 simple- _ _ _ ____ -_ __ __ _ __ ____ _ __ __ -_ __ _ ______ 480 tables of _______________________________ -_-__ 492 Weber-Schafheitlin type ______________________ 487 Bessel functions Jl(z),Y.(z)--________ 358, 379, 381, 385 addition theorems for--- ____________ - ________ 363 1031

1032

INDEX

Page Bessel functions JP(Z), Yv(z)-Continued analytic continuation of- ___ _ ___ _ ____ _-__ _ _ _- _ 361 ascending series for- ___ ____________ __________ 360 asymptotic expansion for large arguments- _ ____ 364 asymptotic expansions for large orders- _ _ __ _ _ _ _ 365 asymptotic expansions for zeros- ______________ 371 asymptotic expansions in the transition region for large orders---- ____ -_- _________________ 367 asymptotic expansions of modulus and phase for large arguments-________-_-_______________ 365 connection with Legendre functions ____ ____ -_ _ _ 362 continued fractions for- -_-_ ___ __ _ _ _ __ _ __ _ __ __ 363 derivatives with respect to order- _ _ _ _ __ _-_ _-_ _ 362 differential equation- _ _ ______________________ 358 differential equations for products- _ ___________ 362 formulas for derivatives ______________________ 361 generating function and associated series- - _ _ _ __ 361 graphs of_-_-_-_-___________-_-_-_-_-_-_-_ 359,373 in terms of hypergeometric functions-. ________ 362 integral representations of- _ _ _ __ __ _ ___ _ ___ __ __ 360 limiting forms for small arguments ____ -_ _ __ _-__ 360 modulus and phase- _ __ __ _ __ __ __ _ __ __ _ _ __-__ _ 365 multiplication theorem for--- _____ _ ____ __ _ ____ 363 Neumann’s expansion of an arbitrary function- _ 363 notation ____ _-_ ______ __ __ __ __ ____ ____ _______ 358 other differential equations- _ _ ________________ 362 polynomial approximations_ ___ ___________ ___ 369 recurrence relations ____ -_ ___ _ _-_-_-_ _________ 361 recurrence relations for cross-products---_ _ _ _ _ _ 361 relations between..-- __ __ __ _ __ ______ -_-_ ____ __ 358 tables of ______ _ __ __ _ __ ___ _________ ____ __ ____ 390 uniform asymptotic expansions for large orders..358 upperbounds-~~~~~~_-~-~__-________________ 362 Wronskian relations- _ _ _ _ _______ _____ ___ _-___ 360 zeros of ______ - _______________________ -_-_-_ 370 zeros, complex- _ _ _______________ _ __ __ _ _-_-__ 372 zeros, infinite products for ________ -_-_ -:- _-_-_ 370 zeros, McMahon’s expansions for-- __ __ -_-_ __ __ 371 zeros of cross products of ____________ _________ 374 zeros, tables of ________________________ 371,409,414 zeros, uniform expansions of- - _-_ __- _ _-_ _- ____ 371 Bessel’s interpolation formula- _ - - - _ - _ - _ - _ _ _ _ __ _ _ 881 betafunction_-_-L-__-___--------------------258 Biharmonic operator-----___ ___- __ __ __________ 885 Binaryscale-~~___~_~_~_~_____________________ 1017 Binomial coefficients- ____ _-_-_ ____ __ _____ _ 10,256,822 table of~~-~__~_~_~~~______________________ 10,828 Binomial distribution_ __ _ __ ___ __ _ ____ __ _-_ ____ 960 Binomialseries--________ -_- ____________ -_-___ 14 Binomial theorem- _ _ _ _______ ___ _-_-_ __ _______ _ 10 Bivariate normal probability function-----------936 computation of ________ - ________ - _______ -_-_ 955 graphs of-_______-_-______-_________________ 937 special values of--- _ __ _ __ __ __ ___ _-_ __ __ __ ____ 937 Bode’s rule-----_-_-_-_-_-_-_-----------------886 C Cartesian form-. ______________________________ Catalan’s constant __-__________________________ Cauchy-Riemann equation ________________ - ____ Cauchy’s inequality--___ _ ________ __ _-_-_-_ __ _Characteristic function--- _ _ _ _ __ _____ __ __ __ __ _ __

16 807 17 11 928

%X0 Chebyshev integration- _ -____ ________________ __ 887 abscissas for- _ -_-___ _____ __ ___ ________ __ __ __ 920 Chebyshev polynomials- - -_______________ 486,561,774 (see orthogonal polynomials) coefficients for and zn in terms of _________ _ __ 795 graphs of-_-_-_-_------------------------778 values of_--_____----___-----------------795 Chebyshev’s inequality _________________________ 11 Chi-square distribution function computation of-------______________________ 958 940 C&i-square probability function- _ _ _-___ _ ______ __ 941 approximations to- ___________ __________-____ asymptotic expansion- _ _______________ _ ____ __ 941 941 continued fraction for--- __ ____ __ __-___ __ _____ cumulants for--- ____ ____ __ ____ ____ __ ___ _____ 940 942 non-central--------------------------------recurrence and differential properties of _ _ _ _ _ _ - _ 941 940 relation to the normal distribution _______ -_ ____ 941 relation to other functions ____________________ series expansions for. _ _________ ______ __ ____ __ 941 943 statistical properties of--- __ ______ __ ____ ______ 785 Christoffel-Darboux formula ___-________________ Circular functions _____________________________ 71,91 72 addition and subtraction of ________ _________ __ 72 addition formulas for- _ _ _______________ I _____ 76 Chebyshev approximations_ _ _ _____ -___ __ ____ 75 continued fractions for- ______________________ 78 definite integrals- _ _ ___ ________ - _______ ______ 74 DeMoivre’s formula- _ _ - _____ __-_ _________ ___ 77 differentiation formulas- - _ _________ -__ _ ____ __ 74 Euler’s formulas ____________________ _________ 75 expansion in partial fractions- _ _ _ ___ __ __ __ ____ 72 graphs of-___------------------------------72 half-angle formulas- _ _- ________________ ______ 77 indefinite integrals _____ ______ ________ __,______ 75 inequalities for _____ ____________ __ ___________ 75 infinite products __________ ____________ I__ __ __ 75 limiting values- __________ - _____ -_- ___, _______ 74 modulus and phase- _ __ ____ ___________ _-_ _ ___ 72 multiple angle formulas- _ _ _ _ ____ - ____ _ __ _ ___ _ 72 negative angle formulas- _____________________ 72 periodic properties of _ _ _ __ __ _ ___ __ __ __ __ _____ 76 polynomial approximations_ _ __ _ _ _ _ _ _ _ _ _ _ __ __ 72 products of_-------------------------------74 realandimaginaryparts _______ -_-___-_- _____ 73 reduction to first quadrant __--_________ _______ 72 relations between-_ _ __ __ _ _ _ _ _ __ _ _ _ _ - _ _ _ _ - _ - 74 relation to hyperbolic functions- _ _ _ ___. __ ___ -74 series expansions for _ _ __ __ _- _ _ _ _ _ __ _ _ ___ _ __ _ _ _ 73 signs of ______________ - _____________.._______ 142 tablesof _________ - ______ - ___________.___-___ Circular normal distribution_ _ _ _ _ _ _ _ __ _ _ __ _ _ _ _ _ 936 calculation over an offset circle-- _ _ _ __ _ __ _ __ _ _ _ 957 Clausen’s integral- _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ - _ - _ _ _ _ _ - - - - 1005 Clebsch-Gordan coefficients-- __ __ _ _ _ _ _ - _ _ _ _ _ - - - - 1006 89 Cologarithm------------..--------------------822 Combinatorial analysis- _ _ _ _ __ __ _ _ _ _ _ - _ _ _ _ _ __ _ 16 Complex numbers- _ ____ __- _________ ______ ____logarithm of- _____________________ __ ____---_ 67,90 16 multiplication and division of _ _ _ _ _ _ __ _._ _ _ - - - _ -

1033 powers of __________________ - ________________ 16 rootsof_________---______~~~~~~~~~~---~~~~~ 17 Confluent hypergeometric functions---_ __ _ __ _ _ _ - 262, 298,300,362,377,486,503,686,691,695,780 alternative notations for -------- ---- -- ------504 analytic continuation of- _- ____-- --___-----___ 504 asymptotic expansions and limiting forms- _ _ _ _ _ 508 Barnes-type contour integrals __---- - -__------_ 506 calculation of zeros and turning points of-- _ _ _ _ _ 513 computation of ____ _____-. ____--_ __ __________ 511 connections with Bessel functions -_______- _____ 506 differential properties of-- - _ _ _ _- - - - _ _ _ _ - - _ __ __ 506 expansions in series of Bessel functions- _ _ _ __ _ _ _ 506 general confluent equation_ __ - - _ __ __ _ _ _ __ _ _ _ 505 graphingof_~-~~~.__..~~-~~~~~--~~~~~~~~~~_~_ 513 graph of zeros of- _ _ ________ ______ __ __ __ _____ 513 graphsof___~-~~~~~~_~~----~~----~~~~---~~~~ 514 integral representations of _ __ ______ __ ____ __ ___ 505 Kummer’s equation-.. _- ______ __ ____ _________ 504 Kummer’s functions- ____ ______ _ ___________ __ 504 Kummer’s transformations_ _ _ _-_ ____-_ ____ __ 505 recurrence relations-- _ _ _ _ __ _ _ _ _ _ _ _ _ _ __ _ __ _ _ _ _ 506 special cases of ______ ___-_____- __ ____c____ __ _ 509 tableof_____~~~~____~~~~~~~~--~~~~~_~~~~_~_ 516 table of zeros of __________ ___________________ 535 Whittaker’s equation- _ _ ______--__ ____ ____ ___ 505 Whittaker’s functions _______________ __ __ _____ 505 Wronskian relations- _ _ _ __ _ _ _ _ __ __ _ _ _ _ _ _ _ _ _ _ _ 505 zeros and turning values--- __ ____ ________ ___- _ 510 Conformal mapping ______---______ ______ _______ 642 Conical functions- ______- __ ______ __ __ _L_______ 337 Constants mathematical- _ _ _ _ _ _ _ _ _ _ ___ _ _ _ _ _ _ __ _ __ _ _ _ _ _ _ 1 physical, miscellaneous-- __ _ _ _ _ _ _ __ __ __ _ __ __ _ _ 5 Continued fractions ______________________ 19,22,68,70, 75, 81, 85, 88, 229, 258, 263, 298,.363, 932, 941, 944 Conversion factors mathematical _ _ _ _ __ _ _ __ _ _ _ _ _ __ _ __ _ _ _ _ _ _ _ _ __ _ 1 physical-____------__-----------------------5 Cornish-Fisher asymptotic expansions--- _ _ _ _ __ __ _ 935 Correlation- _ _ _ _ _ _ _ _ __ __ _ _ _ _ _ _ _ _ _ _ _ __ __ _ _ _ _ _ _ _ 936 Cosine integral ______________________________ 231,510 asymptotic expansions of--- __ _____ _ _ _ _ ___ _ _ _ _ 233 computation of--- _ _ _ _ _ __ __ __ _ _ _ _ _ _- _ _ __ __ _ __ 233 definitions- _ __ _ _ _ _ _ _ _ _ __ __ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ 231 graphsof-__,~~~~____~~--~~~~--~~~~~~~~~~~~~ 232 integral representation of _______ ______________ 232 integrals involving--__ __ ____ _- _ _ _ __ _ __ _ _ _ __ _ 232 rational approximations_ __ _ _ _ _ _ _ _ _ _ _ - __ __ _ - _ 233 relation to the exponential integral- _ _ _________ 232 series expansions for- _ _ _- - __ _ _ - _ + _ _ _ __ _ _ _ _ _ _ 232 symmetry relations- _________________________ 232 tables of _______ _____ __ __ __ _____ __ ___ __-_ __ 238,243 Coulomb wave functions- _ _ __________________ 509,537 asymptotic behavior of--- _____ ___________ -___ 542 asymptotic expansions of- _ _ _ _ _ __ _ __ _ __ __ _ _ __ _ 540 computationof ____ -___- ___________ - _____ -___ 543 differential equation- _ _ _ _ _ _ __ __ _ _ _ _ _ _ _ _ __ __ __ 538 expansions in terms of Airy functions- _________ 540 expansions in terms of Bessel-Clifford functions539 expansions in terms of spherical Bessel functions540

general solution- _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ __ _ _ _ 538 graphs of_-_---_---__------_-------------_ 539,541 integral representations of _________ _________ __ 539 recurrence relations ______ __-_- ____ _______ ____ 539 series expansions for- _ - _ _ _ _ _ _ - _ _ _ _ _ __ _ __ _- __ _ 538 special values of----- ________________________ 542 tables of--- _ _ _- __ __ _ _ _ _ __ __ _ _ _ __- _ _ __ __ _ _ _ _ _ 546 Wronskian relations- _ _ _ - _ _ _ _ - _ _ __ __ _ _ _ _ _- _ _ _ 539 Cubic equation, solution of- _ ___ __ ______________ 17,20 Cumulants.. L _ _ _ _ __ _ _ ___ __ _ _ _ _ _ __ __ _ _ _ _ _ __ _ __ _ 928 Cumulative distribution function multivariate_ _ __ _ __ _ _ __ _ _- _ _ _ _ __ _ __ _ __ _ _ _ _ _ 927 univariate- _ _ _ _ __ _ __ _ __ _ _ _ _ _ _ _ _ _ _ __ __ _ _ _ _ _ _ _ 927 Cunningham function___ _____ _______ _______ _ 510 Cylinder functions-- _ __ _ _ _ __ _ _ _ _ _ _ __ __ _ _ _ _ _ _ _ _ _ 361 D Dawson’s integral- _ _________________ 262,298,305,692 graph of__________________-----------------297 tableof____-________---_-___---_---_------319 Debye functions ___-___ ___ _______ ___ __ _______ __ 998 DeMoivre’s theorem_- - _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 74,84 Derivatives- __ _ _ _ __ _ _ _ _ _ _ __ _ __ _ _ __ _ _ - _ _ _ _ _ _ _ _ _ 11 of algebraic functions- _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _- __ _ __ 11 of circular functions- _ _ _ _- _ __ _ _ __- _ __ _ _ _ _ _ __ _ 77 of hyperbolic functions ____ _ ____ _______ __ _____ 85 of inverse circular functions __________ _ ________ 82 of inverse hyperbolic functions- _ __ _ _ _ _ _ __ _ _ _ _ _ 88 of logarithmio functions ____ __________ _____ ___ 69 partial---_.._-----__------___-------_-----..883 Differences____-____-------------------------877 central__-___---_____-______--------------------877 divided______-______--_____-------------------877 forward_____---_____----------------------877 in terms of derivatives- _ __ _ ___ _ _ _ _-__- __ __ _ __ 883 mean---_---------_-----------------------877 reciprocal--- _ _ _ _ _ _ _ _ _ _ _ __ __ _- _ _ _ _ _ _ __ _ _ _ __ __ 878 Differential equations- _ ___ _ _ _ __ _ _ _ __ _ _ ___ _ -_ _ _ _ 896 of second order with turning points ____ _____ ___ 450 ordinary first order.. _____________________ ____ 896 solution by Adam’s formulas- _ _ _ _ _ _ _ _ _ _ _ _ __ _ __ 896 by Gill’s method- _ _ __ _ _ __ _ __ __ _ _- _ _ _ _ _ _ __ _ 896 by Mime’s method- _ _ __ _______ __________ __ 896 by point-slope formula- ___ __ _ _ __ _ _ __ __ _ _ _ _ _ 896 by predictor-corrector methods--- _ _ _ _ __ _ _ _ _ _ 896 by Runge-Kutta method- _ _______ __ _______ _ 896 solution by trapezoidal formula- _ _. _ __ _ _ _ __ __ 896 systemof-____-_________________________-__ 897 Differentiation_ _ _ _ _ _ _ _ _ _ _ __ _ __ _ __ _ _ _ _ _ _ _ _ __ __ 882 Everett’s formula ____-_______________________ 883 Lagrange’s formula- _ __ _ _- ____ _ ______ __ ______ 882 Markoff’s formula _____________________ - _____ 883 Differentiation coefficients--- ___ __________ ______ 882 tableof-_-___________________-_____________ 914 Digamma function- _________ __ __ ______________ 258 (see psi function) Dilogarithm function- _ _ _______________________ 1004 Distribution functions __________________________ 927 asymptotic expansions of-- _____ -_-___ _ ____ __ _ 935 characteristics of _ _ _ _ _ _ - - _ _ _ _ _ _ _ _ __ _ __ _ __ _ _ _ _ 928 continuous-----____-_________________-_----927

,

INDEX

Distribution functions-Continued discrete- _ _ ___ __- __________ ---______--____-inequalities for ______________--_______ ______lattice ________ __-__ ___ _-____________________ one-dimensional continuous--_____ _- _____ -_-_ one-dimensional discrete --__ _ __ _ _ _ _ _ __ _ __ _ - _ __ Divisor functions--- _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ __ tableof______________----_-------_-----_----Double precision operations __-___ _ ____________ __

Page 927 931 927 930 929 827 840 21

E

Economization of series- _ ______________________ 791 Edgeworth asymptotic expansion ______ - _____ -_ 935, 955 Einstein functions _____________________________ 999 Elliptic functions Jacobian__--________-_-____--------------------567 (see Jacobian elliptic functions) Weierstrass- _ __ _ _ _ _ _ __ __ _ _ _ _ __ _ _ _ _ _ _ _ _ __ _ _ _ . 627 (see Weierstrass elliptic functions) Elliptic integrals- _ _ _ ____-_____ _-______________ 587 amplitude- ______ - _______________________ -__ 590 canonical forms- _ _ _________________L________ 589 characteristic_ _ _ ___ ___ _____ ___ ____ - ____ ___ __ 590 definition____________-_-____c___________---589 graphs of the complete __--____---____________ 592 graphs of the first kind ________-____________ 592,593 graphs of the incomplete- _ _ -__---____-_____ 593,594 graphs of the second kind- _ _ __-____________ 592,594 graphs of the third kind ______________________ 600 modular angle- _ _ _ _ __ ___ __ _-__ _ _ ___ __ ___ __ __ 590 modulus __________________________________ __ 590 of thefirst kind __________________________ -__ 589 of the second kind....-- __-_____-______________ 589 590,599 of the third kind -___________ c ---__________ parameter ___________________-____________ 590, 602 reduction formulas _________________________ 589, 597 reduction to canonical form -___ r _ _ _ _____ _ ___ _ _ 600 relation to Weierstrass elliptic functions-- __ _ _ _ _ 649 tables of complete- _____--___________________ 608 tables of the incomplete ______________________ 613 tables of the third kind _---___________________ 625 Elliptic integrals, complete- - - . __--_____________ 590 computation of ________ ___ ___ __ ___-__ ___ ____ _ 601 infinite series for-. _ _ ___-_____-______________ 591 Legendre’s relation- _ _ _ _____ ___ __ __- ______ __ _ 591 limiting values- _ __ _ ___ ___ __ ___ _______ ___ ____ 591 of the first kind -_____________________ -- _____ 590 ofthesecondkind___________________________ 590 of the third kind- _ ________________________ 599,605 polynomial approximations_ ___-_ __ __________ 591 q-series for- _ _ _ ___- __ ___ __-__ ___ ___ ___ __ ____ 591 relation to hypergeometric functions-----..-....-591 Elliptic integrals, incomplete ____________________ 592 amplitude of any magnitude- _________________ 592 amplitude near n/2- _ _____________________. _ _ 593 complex amplitude.. _ _ __-____________________ 592 computations involving- - _ _____________ 595,602,605 imaginary amplitude _________________________ 594 Jacobi’s imaginary transformation _____________ 592 negative amplitude- _________________________ 592 negative parameter- _ _ __ _ _ _ _ _ __ _ _ _ _ _ _ _ _- _ _ _ _ _ 593 numerical evaluation of- _ ____________________ 595

Page

593 parameter greater than unity- _ _______________ special cases of-- _ _ __ __- _ _ _ _ _ _ __ __ _ _ _ _ _ _ _ _ __ _ 594 752 Elliptical coordinates- _ _ _- ____ - ___ ______ ___.____ 652 Equianharmonic case- _ _ __ ______ __________ _-___ Error 14 absolute _______________--_--__ --- ----- - -.---_ percentage________-_-----------------------14 14 relative- _ _ _ _-_- ______________-______ _-_-___ Error function- _ ____________ 262,297,301,304,306,509 298 altitude chart in the complex plane ________-___ 298 asymptotic expansion of--- _ _ _ _ _ _ _ _ __ ___ _ _ _ _ _ 329 complex zerosof _____ -_- ______ - __________..___ 298 continued fraction for--- ____________ __ __ _.. ___ definite and indefinite integrals related to-- _ _ _ _ 302 298 derivatives of __ ______________ ___________.____ 297 graphs of_----__---------------------------inequalities for _________ -_-__--___-_____. __-_ 298 infinite series approximation for the complex function--- ______ --_- ______ - ______________ 299 297 integral representation of _____________ __ __ . _ __ 299 rational approximations--___________ ___ _.._ __ 298 relation to the confluent hypergeometric functionrepeatedintegralsof---____ -_- _____ - ____.. -_299 297 seriesexpansionsfor------______________.. -__ 297 symmetry relations- __ _____ ___ ____ __-_ ___ ..___ table for complex’arguments---__ __ _ _ __ __ ..-_ _ 325 table of repeated integrals of------___- ______-_ 317 tables of _____________________________ 310,312,316 value at infinity _____ - ____ - _____ -_-_- _________ 298 577 Eta functions----_-----_-----_-_-----_----..Euler function- _ _ ____________________ _____ _____ 826 840 table of ______________ - ____ - _____ - ______ _._-_ Euler-Maclaurinformulas ____ - _____ -_-_-_- __.___ 806 Euler-Maclaurin summation formula--- _ _ 16, 22, 806, 886 Euler numbers-- _____ -- _____________ -_- _______ 804 810 table of _____________ -- ____ -- ______ - _____.___ 803 Euler polynomials __________ --_- _-___ - ______.___ as sums of powers- _______________________ ___ 804 809 coefficientsof....- _____ - _____ -___- ____________ 804 derivativesof ______ - _____ -__-__- _________._ -_ 804 differences of- _______________________________ 804 expansions of- _ _______ - _____ - __________ _____ Fourier expansions of- _ ___-_- _______________ i 805 804 generating function for ________ -__ _______ __ ___ inequalities for ______ -_-___-_- _______________ 805 805 integrals involving---____________________ ___ 804 multiplication theorem for--- _____ -_-___ -__ ___ relations with Bernoulli polynomials- ____ -___ __ _ 806 special values of ______________ - ______ -__- ____ 805 806 symbolic operations _____________ -__- _________ 804 symmetry relations---____ - _____ - _____ -___-_ 806 Euler summation formula- _ _ _____________ -_-_ __ 826 Euler-Totient function- ___ _____ -__-__ __- _______ Euler’s constant_-------___-_____________-_-___ 255 Euler’s formula- _ _ ___________________________ 74,255 Euler’sintegral _____________ -_- _____ - __________ 255 Euler’s transformation of series- _ _ _ _____________ 16, 21 Everett interpolation coefficients- _ _-_ _-___ ______ 880 relation to Lagrange coefficients ____ - ____ -_ __ __ 880 Everett’s formula- _ _ _ _______________________ 880,883 Excess--------__________________________----928

INDEX Pam Expected value operator ______________________ __ 928 Exponential function ________ __-__ __ __-_-_ _ _ 69,90, 509 Chebyshev approximation__ __-__ _- ____ -_ __-_ 71 continued fractions for-- __ __ __ __-__ __ __ __-_-_ 70 differentiation formulas.. _ _ _.___ __ __ __ _____ -_-_ 71 Euler’s formula- _ _ __ __ __- ____ _____ __________ 74 graph of ____ - ___________________ - ___________ 70 identities__---____-___-_______________ ___-_ 70 indefinite integrals ___________________________ 71 inequalitiesfor ____ -_-___- ______________ - ____ 70 limiting values----________ ______ __ _________ 70 periodic property of _ - _ _ __ __ __ __ __ __-_ ____ -__ 70 polynomial approximations__________________ 71 series expansions for- _ __ ____ __ __ __ __ __ __ ____ 69 tables of ______________________________ 116,140,219 Exponential integral---_________________ 227,262, 510 asymptotic expansion of--- ___ _ __ __ __ __ ____ -__ 231 computation of ______ -_- ___________________ -_ 233 continued fraction for--- __ _ _ __ __ __ __ __ _. __ __229 definite integrals- _ _ _________________________ 230 derivatives of- __________________ -___-___-___ 230 graphs of_-___-________----_-_______________ 228 indefinite integrals----._____ __ _____ ___ .._____ 230 inequalities for- __ __ __ ____ ____ __ ______ __ _____ 229 interrelations_ _ _ _ _ _ _ __ __ _ _ _ _ _ _ _ _ __ __ _ _ _ _ _ _ _ 228 polynomial approximations_ ____ ___ __-_-_-_-_ 231 rational approximations_____ __ __ ____ ______ __ 231 recurrence relations--- __ ___ _ ___ __ __ __ __-_ ____ 229 relation to incomplete gamma function-- _ _ _ _ _ _ _ 230 relation to spherical Bessel functions-..--------230 series expansions for- ________________________ 229 tables of __________________ 238,243, 245, 248, 249,251 F F-distribution function-- __ ____ ______ ____ __ _____ 946 approximations to- _-___ __ __ __ ____ __ ___ _ __ _ 947,948 computation of ____ _ __-_ ____ ____________ _____ 961 limiting forms _____________________________ 947,948 non-central _ _ _ __ _ __ _ _ _ _ _ _ _ _ _ _ __ _ __ _ _ _ _ _ _ _ _ _ _ 947 reflexive 1elation _ _ _ _ - _ - _ _ _ _ _ _ _ _ _ _ __ __ _ _ _ _ _ _ _ 946 relation to the x*-distribution____ _ _ ____ __ ____ 946 relation to Student’s &distribution-_______ _ ___ 947 series expansions for- ____________ .--_-_ ____ 946,948 statistical properties of-- ______ __ _______ ______ 946 Factorial function- _ _-_ _ __ __ ____ __ _ __ _____ __ ___ 255 (see gamma function) Factorization table of- _ __ ______________________ ____ ____ 844,864 Faltung theorem- _ ______ - _______ ._____________ 1020 Filon’s quadrature formula- _ _ ___ -_-_--_ ___ _ -_-_ 890 coefficients for- _ _ ___________________________ 924 Floquet’s theorem _____________________________ 727 Fresnel integrals ____________----___-_____ 262,300,440 asymptotic expansions of _____________________ 302 auxiliary functions---- _ _ _ _ _- - - _ - - - - - - -_ - _ _ _ 300,323 complex zeros of ___________________-_________ 329 definition_____.________________________----300 derivatives of _ _____ _ __-___ __________________ 301 graph of________________-__--------------,__ 301 integrals involving-_- _ __ ___ __________ _ __ __ ___ 303 interrelations _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ __ _ __ _ _ 300

1035 Pam

maxima and minima of ____ - _______ ____ _ _ _ __ _ 329 rational approximations____ __ ____ _-__ _-__ ___ 302 relation to spherical Bessel functions--- _ __ __ __ _ 301 relation to the confluent hypergeometric function301 relation to the error function __________________ 301 series expansion for----- _________________ -___ 301 symmetry relations-- __-_-_ __-_ __ ____ __ __ __ __ 301 table of--- _______ - ___________ - ___________ 321,323 value at infinity _____________________________ 301 Fundamental period parallelogram ________________ _ 629 G Gamma function- _ ____ - _____________________ 255,263 asymptotic formulas _________________________ 257 binomial coefficient- _- _ _ _ _ _ _ __ __ _ _ _ _ _ _ _ _ __ __ _ 256 continued fraction for ________________________ 258 definite integrals ______ ___ __ _ __ __ __ _ _ _ _ __ __ __ _ 258 duplication formula-----_ _ _ _ _ __ __ ___ _ _ __ _ _ __ 256 Euler’s formula ______________________________ 255 Euler’s infinite product--- _ _ _ _ __ __ __ __ _ _ _ __ __ _ 255 Euler’s integral ____ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ _ __ __ __ _ __ 255 fractional values of- _ ___________ _____________ 255 Gauss’ multiplication formula-- __ _ _ _ __ ____ _ _ _ _ 256 graph of_-_____-_-_________-----------------255 Hankel’s contour integral _____________________ 255 in the complex plane _______ __________________ 256 integer values of ____ _ _ _- _ _ _ _ _ __ __ _ _ _ __ __ __ __ _ 255 PO&hammer’s symbol _____ _ _ _ _ - _ __ _ _ _ _ _ _ __ __ _ 256 polynomial approximations_ ________ ____ __ ___ 257 power series for _____ _- __ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ __ 256 recurrence formulas _____________________ _____ 256 reflection formula ____________________________ 256 series expansion for l/r(z) _ ________________-__ 256 Stirling’s formula--- _ _-_ __ _ __ _ __ __ __ _ _ __ ____ _ 257 tables of __________________________ 267,272,274,276 triplication formulas- ____________________ ____ 256 Wallis’ formula--- _ _ _ _ _ _ _ __ __ _ __ _ _ _ _ _ ___ _ _ -il _ 258 Gauss series _____ _ ___ __ __ _ _ __ _ ___ _ _ _ _ _ _ _ _ _ __ _ _ _ 656 Gaussian integration-.. _ __ _ _ __ _ __ __ _ _ _ _ _ _ __ _ ___ _ 887 abscissas an.d weight factors for ________________ 916 for integrands with a logarithmic singularity- _ _ _ 920 of moments-- __ __ _ _ _ _ _ _ _- _ _ __ _ __ _ __ __ _ _ __ _ __ 921 Gaussian probability function-- __ _ _ _ _ _ _ _ _ _ _ ___ __ 931 Gauss’ transformation--_ _ _ _ _ __ __ __ __ __ _ _ ____ __ 573 Gegenbauer polynomials ____ _ _ _ _ _ _ _.._ _ _ _ _ _ __ __ _561 (see orthogonal polynomials) coefficients for and z’ in terms of ____________ 794 graphs of----------------------------.---776 Generalized hypergeometric function----_ _ 362,377,556 Generalized Laguerre polynomials- _ __ _ _ __ _ __ _ _ __ 771 (see orthogonal polynomials) Generalized mean _____ _ _ __ ___ __ __ _ _ _ _ _ _- - _ - -- -10 Geometric mean _______________________________ 10 Geometric progression--- __ __ _ _ __ _ _ _ _ _ _ ___ _ _ _ __ _ 10 Gill’s method- _ __ __ _ _ _ _ __ ___ _ _ _ _ _ _ _ - - - - - -- - - - 896 Gudermannian-_ __ _- _ _ __ __ __ __ __ _ _ _ _ __ _ _ _ _ _ _ _ 77 H Hankel functions ________________________ 358,379,510 Hankel’s contour integral _______________________ 255 Harmonic analysis ______ ____L_________ ___---_ 202,881

1036 page

Harmonic mean- _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _- _ _ _ _ _ _ _ _ __. 10 Haversine____~~~~~________~~_~~_______~~~~~~~ 78 Heaviside expansion theorem ____ _ _ __ _ _ _ _ _ _ - - _- _ _ 1021 Hermite functions ______________ _____________ 509,691 Hermite integration _________ __ _________________ 890 abscissas and weight factors for- _..____________ 924 Hermite polynomials--______________ 300,510, 691, 775 (see orthogonal polynomials) coefficients for and z” in terms of _________ ___ 801 graph of.-- _ _ _. _ _ _ _ __ _ __ _ _ __ _ _ _ _ _ _ _ __ ___ __ 780 values of______------_____________________ 802 Heuman’s lambda function_ _ __ __________ __ ____ 595 graph of~-~~_~_~~~~~~~~~~~__________________ 595 table of _.._ _ _ _ _ - - __ _ _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ 622 Hh function-___ __ _____________ ___________ 300, 691 Holder’s inequality for integrals- _ _ _ _ _ _ _ _ _ _ _ _ __ - 11 for surns-~--~-----~___-_--------~----------11 Horner’s scheme ________ ____ ___ __ __ ____________ 788 Hyperbolic functions_..-- __ __ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ 83 addition and subtraction of- _ _ _ _ _ _ _ __ __-__ _ _ - _ 84 addition formulas for- _ _ _ _ _ _ _ _ _ _ _ __ _ __ _ _ _ _ _ - 83 continued fraction for..-- _ _________________ -__ 85 84 DeMoivre’s theorem _______-___________-_____ differentiation formulas- _ _ ___________________ 85 graph of_____---____________________________ 83 83 half-angle formulas- _ ________________________ indefinite integrals--- _ _ _ _ _ __ __ _ _ _ _ __ _ _ __ __ __ _ 86 infinite products ____ _ _ _ _ _ _ _ _ __ __ __ __ __ __ __ ___ 85 modulus and phase- _ _ __ _ _ __ __ _ _ __ __ __ __ __ __ _ 84 multiple angle formulas- _ _ __ _ _ _ _ __ _ _ _ __ ___ __ _ 84 negative angle formulas- __________________ -_83 periodic properties of- _ _ _____________________ 83 productsof__--___________---_______________ 84 real and imaginary parts.. _ ___________________ 84 relations between--- __ __ __ _ _ _ _ _ __ _ _ _ __ _ _ _ __ _ _ 83 relation to circular functions- _ _ __ __ _ _ _ _ _ _ __ __ _ 83 series expansions for- _ __ _ _ _ __ _ _ _ __ __ __ __ __ __ _ 85 tables of __________________________________ 213,219 Hypergeometric differential equation ____ __ __ __ __ _ 562 solution of--_ __ _ _ __ _- _ _ _ __ __ __ _ _ __ _ _ _ __ __ __ _ 563 Hypergeometric functions- - - -_ __ __ _-_ _-_ __ _-__ _ 332, 335,336,362,377,487,555,779 as Legendre functions ________________________ 561 aspolynomials_~~~~~____-_______________-___ 561 as reductions of Riemann’s P-function_---_ _ _ _ _ 565 asymptotic expansions of _____________ - _______ 565 differentiation formulas_ _ _ _ _ __ _ - _ _ __ __ _ _ _ _ _ __ 557 Gauss series- _ _ _ _ _ _ _ _ __ _ __ __ _ _ _ _ _ _ _ __ _ _ _- __ _ 556 Gauss’ relations for contiguous functions- _ _ - _ _ _ 557 integral representations_ _ _ _ _ _- _ _ _ _ __ __ __ _ _ _ _ _ 558 special cases of _____________ i ____________ -___ 561 special elementary cases ____________________ __ 556 special values of the argument- _ _ _______ - _____ 556 transformation formulas ______________________ 559 I Incomplete beta function _____________________ 263,944 approximations to- - - _ __ __ _ _ _ _ _____ __ _ _ __ __ __ 945 asymptotic expansions of..- __ __ __ __ _-_ _ ____ _ __ 945 computation of _--_- - _ ____ __ _ __ __ __ __ _ ____ __ _ 959 continued fraction for--- __ __ __ __ __ _ _ __ __ _ __ __ 944

P8ITe

recurrence formulas ________-----_-_____---263,944 relation to other functions ______----__________ 945 relation to the binomial expansion_ ___ _ __.. __ _ 263 relation to the xz-distribution_ __ __ __ __ __ _ _ _ _ _ 944 relation to the hypergeometric function _ _ _ _ _ _ _ _ 263 series expansion for- _ _ _ _ _ __ _ __ __ _ _ _ _ __ _ ___ __ _ 944 symmetry relation-- _ _ __ _ __ _ _ _ _ _ _ _ _ _ _- _ _ _ _-__ 263 Incomplete gamma function- _ ____ ____ 230,260,486,509 as a confluent hypergeometric function-- _ _ _ _ _ _ _ 262 asymptotic expansions of-- _ _ _ _ _ _ __ _ _ __ _ _ _ _ _-_ 263 computation of ____ __ _ __ __ _ _ _ _ __ __ _ _ __ _ _ __ __ _ 959 continued fraction for ________________________ 263 definite integrals- _ _ _ _ _ _ _ _ __ _ _ __ __ __ _ _ _ _ ___ _ _ 263 derivatives and differential equations- _ _ _ _ _- _ _ _ 262 graph of-----------------------------------261 Pearson’s form of ____________________________ 262 recurrence formulas ____________-__ ___________ 262 series developments for _______________________ 262 special values of _____________________________ 262 table of-- ______________________________ -___ 978 Indeterminate forms (L’Hospital’s rule) _ _ _ _ _ _ _ _ _ _ 13 Inequality, Cauchy’s ___________________________ 11 Chebyshev’s_---_________________________--11 Holder’s for integrals- _ _ _ _ _ - _ - _ - _ __ _ __ _ - - - _ _ _ 11 Holder’s for sums __________-_---_____________ 11 11 Minkowski’s for integrals--- _ _ __ _ _ _ _ _ _ _ _ - _ _ _ __ Minkowski’s for sums-- _ _ _ _ - - - __ _ _ _ _ _ _ _ - - _ __ _ 11 Schwa&s _____- _ _ _ _ _ _ _ _ _ __ __ _ _ _ _ _ _ _ _ _ _ _ __ __ _ 11 triangle- _ _ __ __ __ __ __ _ _ _ __ __ _ _ ___ _ _ _ _ ___ _ __ _ 11 Integral of a bivariate normal distribution over a polygon-~--~~~-----~~~~~~~~~_~~~~~~~_______ 956 Integrals of circular functions- _ _ _ __ _____________ - _____ 77 of exponential functions ______________________ 71 of hyperbolic functions _______________________ 86 of inverse circular functions ____ _ _ _ _ _ __ __ _-__ __ 82 of inverse hyperbolic functions_ _ _ _ _ ___ - _ _ - _ _ 88 of irrational algebraic functions- _ _ _ ________ -__ 12 of logarithmic functions- _____ - _______________ 69 of rational algebraic functions ____________ __ -__ 12 Integration~~~~~-~~~____________________~~~~~~ 885 Bode’srule-~~~---~~~~~_______~_______-_____ 886 byp~~-------------____________________--12 Chebyshev’s equal weight formula--- _ __ __ __ __ _ 887 Euler-Maclaurin summation formula ___________ 886 Filon’s formula ____________________ - _________ 890 five-point rule for analytic functions_ _ __ ___ _ _ _ _ 887 Gaussian type formulas- _ _ _ _ _ _ _ _ _ - :_ _ _ _ _ _ _ _ _ _ 887 iterated integrals--_ __ ___ __ __ __ _ _ __ __ __ __ __ _ 891 Lagrange formula- - _ _ __ __ __ __ __ _ __ __ __ _ __ __ _ 886 Lobatto’s integration formula _________________ 888 multidimensional _____ _ _ _ _ _ _ _ _ - _ _ __ _ _ _ _ _ _ ___- _ 891 Newton-Cotes formula- ______________________ 886 Radau’s integration formula- _ ________________ 888 Simpson’s rule- _ _ __ __ _ __ _ _ __ _ __-_ __ __ _ __ __ _ 886 trapezoidal rule _--_ __ ___ __ __ __ _ __ _ _ __ _- _____ _ 885 Interpolation_ _ _ _____ __ __ __ __ __ _ __ __-_ __ __ __ _ 878 Aitken’s iteration method-- __ __ __ __ _ _ _ _ _ _ ____ _ 879 Bessel’s formula-- _ ___ __ __ __ __ ____ _ _ _-_ __ __ _ _ 881 bivariate~~~~~~--~~~~~_____________-________ 882 Everett’s formula _____ _ __ _ _ __ _ __ __ __ _ __ _ __ __ _ 880

Page harmonic analysis- _ _ _ _ _ _____ _ __ _ _ _- __ -_ __ _ __ 881 inverse________~~_____________~~~~~~~~~~~~~~ 881 Lagrange formula- _ _ ________________________ 878 Newton’s divided difference formula- _ _- _ _ _ _ _ _ _ 880 Newton’s forward difference formula ___________ 880 Taylor’s expansion _____. __ _ _ __ __ _ _ __ __ _ _ __ _ _ _ 880 Thiele’s formula ___________ -.___-_-_--________ 881 throwback formulas __________________________ 880 trigonometric_ _ _ _ _ _ _ _ _ _ ___ - _ __ _ _ _ _ __ __ __ __881 Invariants~~~~__~~~~~~~~~______~~~~~__________ 629 tables of _____ _ _ _ _ __ __ __ __ _ _ __ __ __ __ __ _ _ _ _ _ _ _ 680 Inverse circular functions ____ _ _ _ _ _ __ _ ___ _ _ _ _ _ 22, 79, 92 addition and subtraction of-- ______ ___________ 80 Chebyshev approximations_ _ _ _ _ _ _ __ _ _ _ _ _ _ _ __ 82 continued fractions for- _ __ __________ ____ _____ 81 differentiation formulas- _ _ _ __________________ 82 graph of_____---____________________________ 79 indefinite integrals ____ _ _ _ _ _ __ _ __ __ _ _ _ __ _ _ _ _ _ _ 82 logarithmic representation ______________---____ 80 negative arguments __________________________ 80 polynomial approximations_ __ __ __ __ __ _ _ _ __ __ 81 real and imaginary parts- _ _ _ _ __-_ __ _ _ _ _ _ _ __ __ 80 relation to inverse hyperbolic functions- ___ _ _ - _ _ 80 series expansions for- _ ___ _ __ __ __ __ __ __ _-__ __ _ 81 table of- _ _ _ ___ __ __ __ __ __ __ __ __ __ __ _ ___ __ _ __ 203 Inverse hyperbolic functions- _ _ _ _ _ __ __ __ _ _ _ _ _ __- 86,93 addition and subtraction of-- __ _ _ _ __ __ __ _ _ _ _ _87 continued fractions for _______________________ 88 differentiation formulas- _ _ _ _ _ _ _ __ __ __- __ _ _ __ _ 88 graphsof~~~_~~~~~~~______~_________________ 86 indefinite integrals _____ _- ___ -_ __ __ _ _ __ ___-_ __ 88 logarithmic representations- - __-__ _ __ __ __ _ ___ _ 87 negative arguments--- _ _ __ __ __ __ __ _ _ __ __ __ __ _ 87 relation to inverse circular functions- _ __ _-_ _ _ _ _ 87 series expansions for- _ __ _ __ _ _ - _ _ __- __ _ _ __ __ __ 88 tables of--.. _ _ _ __ _ __ _ __ __ _ _ _ _ __ __ _ _ _ _ _ _ _ _ __ __ 221 J Jacobian elliptic functions- _____ - ______________ _ 567 addition theorems for- _ _ _ _ _ _ _ __ _ _ _ _ _ __ __ _ __ __ 574 approximations in terms of circular functions- _ _ 573 approximations in terms of hyperbolic functions574 calculation by use of the arithmetic-geometric rnean__~~~~~~_~~~~~~~____________________ 571 calculation of _ _ __ _ _ _- _ _ _ _ _ _ _ ___ _ _ _ _ __ _ _ __ _ 579,581 change of argument_-- _- _- __ _ __ _ _ _ _ _ _ _ _ _ _- _ __ 572 change of parameter- ________ __-______ __ ____ _ 573 classification of __-_ __ __ _ _ _ _ _ _ _ __ __ _ _ _ _ _ _ _ A_ __ 570 complex arguments- __ _ _ _ _ _ _ _ _ __ __ ___ _ _ _ _ _ _ _ _ 575 definitions-------_______________________---569 derivatives of _ _ _ __ _ _ _ _ _ _ _ _ __ __ __ __ _ _ _ _ _ __ __ _ 574 double arguments- _ _ _ _ _ _ _ _ _ _ _ __ __ _ _ _ __ _ _ _ _ __ 574 graphsof______---__________________________ 570 half-arguments ______________ -- _________ ----_ 574 integrals- _ _ __ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ 575 integrals of the squares _____ - __ _ ______ - _______ 576 Jacobi’s imaginary transformation-___________ 574 Landen transformation______________________ 573 leading terms of series in powers of u- _ __ ____ ___ 575 p~arneter~~~~~----,~~~~~~~___~~~~~_________ 569 principal terms-- - -_ __ _ _ _ ___ __ __ __ _ _ _ _ __ __ ___ 572

Page quarter periods--- _ _ _ __ _ _ __ __ __ _ __ __ __ __ _ __ __ 569 reciprocal parameter-- __ _ _ _ _ __ _ _ _ _ _ _ _ _ _ __ __ _ _ 573 relations between squares of the functions- _ _ _ _ _ 573 relation to the copolar trio ______________ ______ 570 relation with Weierstrass functions.. _ _ _ - - - - - - __ 649 series expansions in terms of the nome q-- ______ 575 special arguments- _ _ _____ ___-_______________ 571 Jacobi’s eta function ____________ ____ ___________ 577 Jacobi’s polynomials ___________-_-___________ 561,773 (see orthogonal polynomials) coefficients for- _ _ _ ____-_ -_-___-__ _________ 793 graphs of_-_________-____--------------773,776 Jacobi’s theta function (see theta functions) _ _ _ _ _ _ 576 Jacobi’s zeta function ________________________ 578, 595 addition theorem for-- _ _ _ __ ___ _ _ _ _ _ __ _ _ _ __ __ _ 595 calculation by use of the arithmetic-geometric rnean__-____-~~~--_______________________ 578 graph of__-____________---__________________ 595 Jacobi’s imaginary transformation____ ________ 595 q-seriesfor ______ -_--_-- _____________________ 595 relation to theta functions-- ________________ 578,595 special v~ues_~~____-_~~~~~~~~~~~~~~~~~_-___ 595 table of _______________-----___ ______ ____ ____ 619 K Kelvin functions- _ _ _ _ ____ _______________ 379,387,509 ascending series for- __ ___ _ _ _ _ _ ____ __ __ __ _ _-_ _ 379 ascending series for products of--- ___ _ __ _ _ __ _ _ _ 381 asymptotic expansions for large arguments- _ _ _ 381 asymptotie expansions for large zeros- _ ________ 383 asymptotic expansions of modulus and phase- - _ 383 asymptotic expansions of products--- _ _ _ __ __ _-_ 383 definitions__-------_____________________---379 differential equations ____ __ _ __ __ _-_ _ _ __ __ __ _ __ 379 expansions in series of Bessel functions _ _ _ - _ _ _ _ _ 381 graphs of___~~~~~_____~~~~~_________________ 382 indefinite integrals- _ _ __ __ __ _ __ __ __ __ __- __ __ _ 380 modulus and phase- _ _ _ _ __ __ __ __ __ _ _ ___ __ ____ 382 of negative argument- _ _ __ __ _ _ __ ___ _ __ _ ___ _ __ 380 polynomial approximations_ __ _ __ _ __ _ _ _ _ _ -_ __ 384 recurrence relations--- _ _ __ __ _ ___ _ _ _ _ _ __ _ _ __ __ 380 recurrence relations for products _____ __________ 380 relations between ____ _ _ __ -_ _ __ __ _ _ _ _ __ _ __ __ _ _ 379 tables of ____________________________________ 430 uniform asymptotic expansions for large orders-384 zeros of functions of order zero ________________ 381 Kroneckerdelta-__-____--_-___________________ 822 Kummer functions ____ _ _ _ ___ ____ __ __ _ _ _ _ _ __ __ __ 504 Kummer’s transformation of series ____ __ __ __ _ _ _ _16 L Lagrange differentiation formula ____ __ __ __ _ __ __ __ Lagrange integration coefficients ____ _ _ _ _ __ __ __ __ _ tableof____-~~~~~~~~~~~-~~~~~~~~~__________ Lagrange interpolation coefficients-- __ _ _ _ __ __ ____ table of _ _ ___ - -- - _ _ __ __ _ __ ___ ___ __ __ __ _ -L--Lagrange interpolation formula ____ _ __ _ _ __ __ __ _ _ _ Lagrange’s expansion- _ _ __ __ __ ____ __ __ __ __ __ __ _ Laguerre integration ______________ __ __ __ __ __ __ _ abscissae and weight factors for- _ _____________

882 886 915 878 900 878 14 890 923

1038

INDEX Page

PSiW

Laguerre polynomials.. ___________________ 509,510,773 (see orthogonal polynomials) coefficients for and x” in terms of __---------799 graph of___---___------------------------780 valuesof__------------------------------800 Lame’s equation----_ _ __ __ _ _ _ _ __ _- - _ _ _ __ ___ _ _ _ 641 Landen’s transformation ascending- - - _ _ _______________________ 573,598,604 descending- _ _ _ _ ____ __________________ 573,597,604 Laplace transforms- ___________________________ 1019 definition-_--_____-------------------------1020 operations-------_-------------------------1020 tables of----- _______________________________ 1021 Laplace-Stieltjes transforms ____ _ _ _ _ __ _ __ _ _ _ __ _ __ 1029 tables of__-__------~~-----~----------------1029 Laplace’s equation- ____________________________ 17 Laplacian _ _ _ _ _ _ _ _ __ _ __ _ _ _ _ _r _ _ _ _ _ _ _ __ _ _ _ _ - - - 885 in spherical coordinates- _ _ _____________ Y_-___ 752 Laurent series ____ ___-_ _______-______----__---635 Least square approximations _______ __ ___________ 790 Legendre functions_-- ___________ 332,362,377,561,774 asymptotic expansions of - _ _ _ _ __ - - _ _ _ _ __ _ _ _ _ _ _ 335 computation of ____ - ___________ ______________ 339 explicit expressions for- _ _- _ _ _ _ __ _ - _ _ _ _ _ _ _ __ __ 333 graphs of____-______-______--------------338,780 integral representations of ___________ _________ 335 integrals involving _______ __F_________________ 337 notation for- ______ - _________________________ 332 of negative argument- _ _ _ __ __________________ 333 of negative degree--- ___ __ _F__________ _______ 333 ofnegativeorder-__------_-----------------333 recurrence relations ______________ - ____ _______ 333 relation to elliptic integrals- _ ______ _________ __ 337 relations between _______ _____ _______________ _ 333 Rodrigues’ formula- ___ ___ ___ ____ __ __________ 334 special values of_- _ _ _ _ _ __ _ _ _ _ _ _ _ _ __ _ _ _ _- _ __ __ 334 summation formulas- _ __ _______ __ ____________ 335 tables of-.- __ _-_ ___ __ _. _____________________ 342 trigonometric expansions of-- _ _ __ _ _ _ _ _ - __ ___ __ 335 values on the cut-__-----_-----___----------333 Wronskian relation- _ ___ _____ ___ ____ _________ 333 Legendre polynomials ____________________ 332,486, 773 (see orthogonal polynomials) coefficients for and zn in terms of--- _ ________ 798 graph of___------_-----_---------------338,780 valuesof-_--_____-..-_..---.._-__-.._____.._-342 Legendre’s differential equation- _ _ ____ __ ____ __. _ 332 solutions of_~--~~~~_~~~~~__~~~~~~~~~~~~~~~~~ 332 Leibnia’ theorem for differentiation of a product- __________________ 12 differentiation of an integral- _ __ _ _ _ _ _ _ _ __ __ _ _ _ 11 Lemnisoate constant _____ ______ ________ - ______ 658 Lemniscatic case- _ _ ___ ___ ___ __ _____ __ ______ 658 L’Hospital’s rule- _ ____________________________ 13 Lobattointegration______________________-----888 abscissas and weight factors for- _ _ -__ ___ ___-__ 920 Logarithmic function __________________________ 67,89 chan~eofbase______________________________ 67,89 characteristic- _ _ _______ _ ___ __ _-__ ______ ____ _ 89 Chebyshev approximations_ _ _________________ 69 common__________________________________-68,89

continued fractions for- _ _ _ _ _ __ __ __- - -- -- - --- 68 definite integrals- _ _ _ ____________----------69 differentiation formula- ____________-_-___..--69 graph of__---------------------------------67 indefinite integrals ________________---------69 inequalities for ______________________________ 68 68 limiting values _ _ _______________------------89 mantissa---_______-___-______------------polynomial approximations_____________-.-___ 68 series expansions for- ___ ________________--_-_ 68 tables of __________ - ____________-----_-- .---95 Logarithmic integral _____________________ 228,231,510 231 graph of ____ -_- _________________L_-_________ M Markoff’s differentiation formula ____________-___ 883 Mathematical constants _____________--_____---1 in octal scale __________ - _____________________ 1017 Mathieu functions-- ______________________^ ____ 721 asymptotic representations of _________________ 740 comparative notation for-- _______ ________ _ ___ 744 expansions for small Q_____________-______---730 expansions in terms of parabolic cylinder functions------------------------------------742 graphs of___-----------------------------725,734 integral equations for- _______________________ 735 integral representations of ________________-___ 736 joining factors, table of ___________________ ____ 748 normalization of-- ___________ ____________ _ ___ 732 orthogonality properties of ____________________ 732 other properties of _______________________ __ 735,738 power series in q for periodic solutions ______ ____ 725 proportionality factors- ______________________ 735 recurrence relations among the coefficients-, __ __ 723 special cases of ______________________________ 728 special values of ________________________^ ____ 740 table of coefficients for _______________________ 760 table of critical values of- __________-_____ ____ 748 zeros of _ _ ______________ __ ______________ ____ 739 Mathieu’s equation ________________________ ____ 722 characteristic exponent _______________________ 727 generation of _______ _______________________ 727 graphs of--------------,------,------,-----728 characteristic values- _ ____________-______ __ 722,748 asymptotic expansions of ___________________ 726 determination of _ _ _ _ ___ ___________________ 722 graph of__------_--------------------,----724 power series for- __________________________ 724 Floquet solutions-- ______________________ __ __ 727 other solutions-,--------------_----________ 730 relation to spheroidal wave equation _______ ____ 722 solutions involving products of Bessel functions731 stability regions of ___________________________ 728 Mathieu’s modified equation- ____ -__ ___________ 722 radial solutions of- _ _________________________ 732 Maxima-_---_-----__-____________________~--14 Mean__-----_---_-_-_____________________---928 arithmetic _______ __ ___ _____ ___ __ ________ ____ 10 generalized- _ _ __ ___________ _- ___________ __ __ 10 geometric _______________ - _______________ ____ 10 harmonic _______________________________ ____ 10

Page Milne’s method- _ _ _ _____ -_- ____ __ ______ -- ____896 14 Minima_________-____--___--------------------Minkowski’s inequality for integrals- _ _ ______ ____ 11 for sums---- ________________ -_- -----_------11 Miscellaneous functions- _______________________ 997 Mobius function _______________________________ 826 Modified Bessel functions, L(z), K.(z) _ _______ -___ 374 analytic continuation of-- _____ __ __ __ __ _ _-__ __ 376 ascending series for- _ _____ __ __ __ ___ __ _ __ __ _-_ 375 asymptotic expansions for large arguments-..---377 connection with Legendre functions-_-- _ _ _- ____ 377 derivatives with respect to order- _ _ _ __________ 377 differential equation- _ _ _ _ _ _ _ _ _- - _ _ _ _ _ - _ _ _ - _ __ 374 formulas for derivatives- _____________________ 376 generating function and associated series- _ _ - _ - _ 376 graphs of-_________-____--_____-_____-_____374 in terms of hypergeometric functions ___________ 377 integral representations of- __-___ _ _ ___ _ _ ___ _-_ 376 limiting form5 for small arguments _____________ 375 multiplication theorem5 for- _ _ _ _ _ _ ___- ____ ___ _ 377 Neumann series for K,(z)-- _ .___ __ __ _ __-__ ____ _ 377 other differential equations- _ _ __ - ____ _________ 377 polynomial approximations_ __ _- _ __ __ _ __ _-_-_ 378 recurrence relations-- _-__ __ _ _ __ __ _ ___ _ __ _ __ __ 376 relations between- __________ --- ____ -___-_-_-_ 375 tables of ____- _________ -_-___- _______________ 416 uniform asymptotic expansions for large orders- _ 378 Wronskian relations- _ _ _ __ __ _ _ __ __ _ _ _ _-__ _-__ 375 zerosof ____ - ____ --___-_- ________________ -__ 377 Modified Mathieu functions- _ - _ __- _______ - _____ 722 graphs of____-_____-_____-_____--------__734 Modified spherical Bessel functions- _ _ _ __ _ _ 443, 45 3,498 addition theorems for- _______________________ 445 ascending series for- _________________________ 443 computation of ____ _ ____ _-__ __ __ _ _ __ __ __ __ ___ 453 definitions_______________-____--_____-__-___ 443 degenerate forms- _ _ _ __-_ ___ __ __ __ __ __ __ _-___ 445 derivatives with respect to order- ______________ 445 differential equation- _ -__- _____ - _____________ 443 differentiation formulas- _ _ _ ____ - _____ ________ 444 duplication formula---_____ _________________ 445 formulas of Rayleigh’s type--- _ _ _ _ __ _ _ _ __ __ ___ 445 generating function5 for- _______ -_- ___________ 445 graphs of________-_______-_________________444 recurrence relations---_ __ _ _ _ __ __ _ __ _ __ _ _ __-_ 444 representations by elementary functions-- _ _ - _ _ _ 443 tables of--- ___________________ -___- _________ 469 Wronskian relations- _ _ _ _______ _______ _______ 443 Modified Struve functions- _____________________ 498 asymptotic expansion for large 121____ _ __ _ _ _ _ _ _ _ 498 computation of_-- _____ - ____ --- ________ - _____ 499 graphof __________ - _________ -- ________ - _____ 498 integral representations of- _ __________________ 498 integrals--__________ - ____ -- _________ - _____ 498 power series expansion for- _____ - _________ -___ 498 recurrence relations __________________________ 498 relation to modified spherical Bessel functions- _ 498 tables of _________ - ____________________ - _____ 501 Modulus---____-_________________________---16 Moments--__-_--_________________________---928 Multidimensional integration- _ _ _ _____ __ ________ 891

Page

Multinomial coefficients---_ __ ___ _-_-__ ___ ____ _ table of __________ -___-_-___- _____ - _________

823 831

N

Neumann’s polynomial _________ - _________ ______ 363 Neville’snotation _________ -- _____ --- ______ -___ 578 Neville’s theta functions ____ - - _ - _ __ _ _- - _ _ _ _ _ _ _ _ _ 578 expression as infinite product5 _________________ 579 expression as infinite series- --__ _ _ _ _ _ - _ _ __ _ __ _ _ 579 graphs of___----------_--------------------578 tables of___--------------------------------582 Newton coefficients ____ -___---- _____ -- _________ 880 relation to Lagrange coefficient5 _______-_ __ --- 880 Newton interpolation formula....- _______._____ 880,883 Newton’s method of approximation _______ __ _ ____ 18 Newton-Cotes formula- -___- _____ - __ _- ___ __ _ ___ 886 Nome_------------------------------------591,602 tableof- ____ -- __F________ --- _________ 608,610,612 Normal probability density function derivativesof________ -___-_- ____________--_ 933 Normal probability function_ _- - _ - _ _ _ _ _ _ _ __ _ _ - _ _ 931 asymptotic expansion5 of _____ __ __ _ __ -- -___- - 932 bound5 for-_----_____________ -- ____________ 933 computation of ____-_____ ---- ________________ 953 continued fraction for--- ____ __ __ ____________ _ 932 error curves for _________ ---- ________________ 933 polynomial and rational approximations____- - 932 power series for _______ - _____ - ____________--932 relation to other functions- _- __ _____ -__ __ __ __ _ 934 values of z for extreme values of P(z) and Q(z) - 977 values of 2 in terms of P(z) and Q(z) ________--976 values of Z(z) in terms of P(z) and Q(z) ---__--975 Normal probability functions and derivatives - _ _ _ _ 933 tables of____--___---_-_--------------------966 Normal probability integral repeated integrals of _____ -__ _____ - _______ ____ 934 Number theoretic functions--- _ _ _ _ - _ - - - _ _ __ _ __ _ _ 826 0

Oblate spheroidal coordinates- - _ _ _ _ _ _ _ _ __ __ _; _ _ _ 752 Octal scale___--------------------------------1017 Octal tables_--~____-_--_----~~-~-------------1017 Operations with series--- __ __ __ __ _ _ _ _ _ __ _ _ _ __ __ _ 15 Orthogonal polynomials- ____ - _____ - ______ ______ 771 as confluent hypergeometric functions __________ 780 as hypergeometric functions.. _ _ _______ ________ 779 as Legendre functions_-- _ _ _ - - __ _ _ - _ - _ _ _ _ _ _ __ _ 780 as parabolic cylinder functions- _ __ - - _ _ _ _ _ _ _ _ _ _ 780 change of interval of orthogonrtlity.. _ _ _________ 790 coefficients for- _ _ ______ _____ - __________ _____ 793 definition_-______-_------------------------773 differential equations----- _ _ __ _ __ _ _ _ _ __ _ _ _ _ 773, 781 differential relations- _ _ _ _- - _ - _ _ - _- _ _ _ _ _ __ _ r _ _ 783 evaluation of ____ - __ _ _ _ _ _ _ _ _ _ _ _ _ - _ _ ___ _ _ _ _ _ _ _ 788 explicit expressions for- _ ___ _-__-__-___ __ _____ 775 generating functions for--- __ -__- _- ___________ 783 graphs of-_--_-_--____--___-~~---~~-~--773,776-780 inequalities for __________ -- ____ -__- __________ 786 integral representation5 of_ _- _______ -_ ________ 784 integral5 involving ____ -_ __ __ _ ________ -__- ____ 785 interrelations_ _ _ __ __ __ _ -__- _ _ ____ - __ ____ ___ 777

-

INDEX

1040

Page Orthogonal polynomials-Continued limit relations ____ __ ____- __ __--____ _____ __-__ 787 of a discrete variable-_-_ _ _ _ - _ _ __ _- - _ _ _ _ _ _ __ _ _ 788 774 orthogonality relations- ____ - __ __ _- _ _________ _ powers of z in terms of-_. __ -__ ___--_ ___ 793,794-801 recurrence relations, miscellaneous-- _ _ - _ - _ __ - 773,782 782 recurrence relations with respect to degree n- _ _ _ Rodrigues’ formula- - __ ___ __ _ _ ___ _______. __ 773,785 special values of ____ -_-_ __ _-___ __ ___________ _ 777 sumformulas _____ -_-- __________ -- ______ -___ 785 tablesof-.--_____ -_-- _______.____ 795,796,800,802 787 zeros of- _ _ __ ___ _____ __- ___ __. _- __ _-_ __ __- --

P U(a, z), V(a, cc)- __ _ 300, 509,685, 780 asymptotic expansions of _____________________ 689 computation of ______________________________ 697 connection with Bessel functions-- _ _________ 692, 697 connection with confluent hypergeometric functions_________--____-______--___________-691 connection with Hermite polynomials and functions___-___________-_____--____________-691 connection with probability integrals and Dawson’s integral- _ _ ______________________ 691 689 Darwin’s expansions- __ ___ _ ___ __ _ _ _ __ _ ___ __ __ differential equation- _ _ _--____-______________ 686 expansions for a large, z moderate- ____________ 689 expansions for x large, a moderate- ________ ____ 689 689 expansions in terms of Airy functions __________ integral representations of- _________ __________ 687 modulus and phase- _ ___---______ ____________ 690 power series in x for-- _______________________ 686 recurrence relations _____-____________________ 688 standard solutions--- ____ ___________ _________ 687 table of- _ -_________________________________ 700 Wronskian and other relations- _ _ ____ ___ ___-_ _ 687 zeros of- _ _ __ ___ _ ___ ___ _ _ _ ___ __ _ _ __ __ _ __ __ __ 696 Parabolic cylinder functions W(a, x) ______ ________ 692 asymptotic expansions of _______________ ______ 693 complex solutions ____________________________ 693 computation of ______________________________ 699 connection with Bessel functions- _ _ _ __________ 695 connection with confluent hypergeometric functions.-_______________________________ 695 Darwin’s expansions--- ______________________ 694 differential equation- _ _ ____________________ 686,692 expansions for o large, x moderate- ____________ 694 expansions for x large, a moderate _____________ 693 expression in terms of Airy functions _____ __ ___ _ 693 integral representations of- __ _ _ ___ __ ___ _ ___ ___ 693 rnodulusandphase~~___-~~~~_~~__~~~~~~~~~~~ 695 power series in x for _____ ___ _ ___ __ ___ __ __ _ __ _692 standard solutions ___________________________ 692 table of- _ _ __ ___ ____ __ _ -__ ___ __ _ _ __ __ __ ___ __ 712 Wronskian and other relations- _ ____________ __ 693 zeros of ____--_________-________________ -___ 696 Parameter m ____________________ __ __________ 569,602 table of------------___---__________________ 612 Partitions- _ _ _ _ __ _ _ _ _ _ _ _ _ __ _ _ _ __ _ _ _ _ _ _ _ _ __ _ _ _ _ 825 into distinct parts- _ ___ _-_ __ __ _ _ _ __-_ __ __-___ 825 tables of__________________________________ 831,836

Parabolic

cylinder

functions

Pago

825 unrestricted ________ ____ ___-__ ______ __ _____- 262 Pearson’s form of the incomplete gamma funct,ion-260 Pentagamma function __________________________ (see polygamma functions) Percentage points of the x*-distribution 984 values of ~2 in terms of & and Y____-_-__----Percentage points of the F-distribution 986 values of F in terms of Q, Y,, YS--- ___L_____--__ Percentage points of the t-distribution 990 values of t in terms of A and Y_________-_-_-_999 Planck’s radiation function- ____ ________________ Plane triangles, solution of ______ ________ _ _______ 78, 92 256 Pochhammer’s symbol- _ _______________________ 896 Point-slope formula __________ ______ _________-__ 509 Poisson-Charlier function ____________ ___________ 959 Poisson distribution ________________________ ____ 978 table of cumulative sums of ________________--16 Polar form ____________ - _______________________ 260 Polygamma functions __________________________ 260 asymptotic formulas- ________________________ fractional values of- _ ______ - _________________ 260 integer values of _________ ____________________ 260 260 multiplication formula for- _ ____ ______________ 260 recurrence formula ____ _ _ __ _ _ _ _ _ _ _ __ _ __ _ __ _ _ _& reflection formula ____________________________ 260 series expansions for- _ __ _ __ __ ___ __ _ _ __ __ _ __ __ 260 tables of _________________________ - ________ 267, 271 788 Polynomial evaluation- _ _ _ ___ __ ______ _ __ __ _ ____ Powers computation of ______________________________ 19,90 general_,,-----_-----_-----_---______-__---69 graphof__..----_c----_---_-----------___---_ 19 16 of complex numbers- _ _ ______________________ of two- - - - ---- -- -- -- -- - -- -- - --- -_ -- -_ __-- -- 1016 of Z”/?z!- - -----_---___--____________________ 818 24 tables of ____________________________________ Predictor-corrector method- ____________________ 896 231 Primes..-__-..----------_----__-----_---___-_-_ tableof__----__--_____-____________________ 870 Primitive roots ________________________________ 827 tableof____-_______________________________ 864 Probability density function- ___________________ 931 asymptotic expansion of ______ __ __ _____ _______ 935 Probability functions- _ _ _ _ __ _ _____ _-__ _________ 927 Probability integral __________________________ 262,691 of the ,+distribution, table of _________________ 978 Progressions arithmetic- __ ____ ___ _-_ ___ __ ___ ________ __ __ _ 10 geometric _____ __ __ __ __ _____ ________ __ _______ 10 Prolate spheroidal coordinates __________ __ __ _ _ ___ 752 Pseudo-lemniscatic case- _ _ _ ___ ___- ____ __ __ _ _ ___ 662 Psi function- - _ _ __ __ __ ___ __ _- ______ ____ __ _ _ _ 258,264 asymptotic formulas-- __ __ _ _ ___ __ ____ __-_ ____ 259 definite integrals- _ _ _ _ __ __ _ _ __ __ __ __ ___ __ __ __ 259 duplication formula--- __ __ _____ ____ __ ____ ____ 259 fractional values of- _____________________ ____ 258 graph of ___--_--_--_________ - _______________ 258 in the complex plane--- __ __-__ __ __ __ __ ___ ____ 259 integer values of __________ __ _- ____ __ __ ___ __ __ 258 recurrence formulas--- _ _ _ _ _ _ _- _ _ _ _ _ _ _ __ _ _ __ _ _ 258 reflection formula ______ __ ___ __ __ _-__ ___ _ _ ____ 259

INDEX series expansions for- _ __ __ __ __ __ ____ ___-_ ___ _ 259 t,ables of __________________________ 267,272,276,288 zeros of ______________ - _______ - _____________ 259

Q Quadratic equation, solution of ____ -__ _ __- _______ Quartic equation, resolution into quadratic factors--

17,19 17, 20

R Radau’s integration formula- ___________________ 888 Random deviates, generation of _____________ ____ 949 Random numbers- _ _ __________________________ 949 ‘methods of generation of- ____________________ 949 tableof______-___-_-____--_-_---___----___991 Repeated integrals of the error function __________ 299 as a single integral--________________________ 299 asymptotic expansion of ______________________ 300 definition___---___-------------------------299 derivatives of _____________________ __________ 300 differential equation- _ _ _ _-__ _ __________ _____ _ 299 graph of _____ _______________________________ 300 power series for- _ - __________________________ 299 recurrence relations--- _______________________ 299 relation to Hermite polynomials ___-_____-_____ 300 relation %o parabolic cylinder functions-- _ - _ _ _ _ _ 300 relation to the confluent hypergeometric function_-__-_-----_-------------------------300 relation to the Hh function ________ _____-_____ 300 table of____-_-___________---------------..-.. 317 value at zero ________________________________ 300 Representation of numbers- _ _--__-_---_-_-_---_ 1012 Reversion of series _________________-___________ 16, 882 Riccati-Bessel functions- __ __ ___ __ _ _ _ _ ___ _- _____ 445 definitions-_-------------------------------445 differential equation- _ ____--------_---- ----_ 445 Wronskian relations- _ _ _ ___ __ __ __ - _ _ __ _-_-___ 445 Riemann zeta function ___________________-___ 256,807 special values of _________________-_-______ ___ 807 Riemann’s differential equation- _ _ _ ______---_-__ 564 solutions of- ____ ___ __ _____ _____ __ _____ - _____ 564 Riemann’s P function ______ ________ ___ ___- -____ 564 transformation formulas _____ ____-_-_-__----__ 565 Ring functions..---_-_------------------------336 Rodrigues’ formula- _ __ ___ __ __ __ ____ _____ 334,773,785 Roots computation of ________ ______________________ 19,89 graph of____--_____------------------------19 of complex numbers- _ _ _ ________- __ ____----__ 17,20 tables of--_-- _- __ ____________________ _-____ 24,223 Runge-Kutta methods- _ __ ____ __ __ -_______-__ __ 896 S Scales of notation- _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ __ - - _ _ _ _ 1011 general conversion methods ____ __ __ __ __ ____ __ _ 1012 Schwarz’s inequality _______ - ___________________ 11 Sectoral harmonics--_ __ ____-_____-----_-----332 Series___--____-_-__~~----------~------------14 binomial ______________ -_--- ________________ 14 Euler-Maclaurin summation formula ___________ 16,22 Euler’s transformation of------______________ 16, 21 exponential_____-__-----~-------~----------69

Kummer’s transformation of-- __ _ _ _ _ _ - - -_ _ _ _ _ _ 16 Lagrange’s expansion- _ _ _ _ - _ _ __ _ _ - - _ _ __ _ __ _ _ _ 14 logarithmic_-------------------------------68 operations with _____________ ---- ____________ 15 reversionof- __________ - _________ -- _____ ---__ 16 Taylor’s_____-~~---~------------~---~-~~---14 trigonometric-----_________ --- ________ - ____ 74 Shifted Chebyshev polynomials- _ _ _ _______ __ ____ 774 (see orthogonal polynomials) Shifted Legendre polynomials- - - _ _ _ _ _ __ _ _ _ _ - _ _ _ _ 774 (see orthogonal polynomials) Sievertintegral_______------------------------1000 886 Simpson’srule ______ -__--- ____________ -- ______ Sine integral __________ -_-- __________________ 231,510 asymptotic expansions of-- - _ _ _ _ _ _ _ _ __ _ __ _ _ _ _ _ 233 computation of ___________________ ---- _______ 233 definitions_- __ _- ____ __ ______________ __ ______ 231 232 graphs of______-_-_------------------------232 integral representation of-- __________ ____ __ ___ integrals-_--__ _ _ _ _ _ _ _ - __ __ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ 232 rational approximations_- __ __. __ __ __ __ ___ ___ 233 relation to exponential integral--- ___ __ __ __ ____ 232 series expansions for- _ _ __ _ _ _ _ _ _ _ __ __ - - _ _ _ _ _ _ _ 232 232 symmetry relations- ___ __-__ __ __ __ __ ______ ___ tablesof __________________ -__- ____________ 238,243 928 Skewness.-----------------------------------Spence’s integral- _ _ ______ ___________ _________ _ 1004 Spherical Bessel functions.. __ _________ 230, 301, 435, 540 addition theorems for- ________ _ __ __ ___- ______ 440 439 analytic continuation of ______ _______ ____ _ _ ___ ascending series for- _ _ _ _ _ _ _ _ _ _ _ __ _ __ _ _ _ _ _ _ _ __ 437 complex zeros of h’:‘(z), h:‘(z)-- _ _ _ __ _ _ _ _ _ _ _ __ 441 computation of _____ __ __ ____ __ __ _____ ______ __ 452 439 cross products of- ______ __ __ __ __ ____ __ __ _- --definitions___------------------------------437 degenerate forms- ______________ __ __ __ __ __ __ _ 440 440 derivatives with respect to order- _ _ _______---differential equation- _ __ __ __ __ __ -_ __ ____ __ __ _ 437 439. differentiation formulas- _ _ _ ____ ____ _ __ __ - -- -duplication formula ___________ ______ __ __ __ __ _ 440 Gegenbauer’s generalization for- _ _ _ ______ ____ _ 438 generating functions for- ________ ____________ _ 439 438 graphs of____------------------------------infinite series involving j:(z) _ _________________ 440 limiting values as z-0 ____ ______________ __ __ _ 437 439 modulus and phase- _______ ____________ ___ __Poisson’s integral for ____________ ____________ _ 438 Rayleigh’s formula for- _ __ _________ _____--_ -439 439 recurrence relations ____ _-_______________-..-relation to Fresnel integrals _____ __________ __ __ 440 representations by elementary functions- _ _ _ _ _ _ _ 437 457 tables of_____------------------------------Wronskian relations_ _ _ _ _ _ ____ __ -___ ____ ___ __ 437 440 zeros and their asymptotic expansions .____ -- --Spherical polynomials (Legendre) ____ _ _ - _ - - _ _ _ _ _ _ 332 (see orthogonal polynomials) Spherical triangles, solution of- _ _ ____-_____-____ 79 751 Spheroidal wave functions- __-__ ____- __ _______ - 756 asymptotic behavior of __________________----755 asymptotic expansions of- _ ____ _______ __ -- -- --

.

1042

INDEX

Page Spheroidal wave functionsContinued characteristic values for ____________________ 753,756 differential equations----_ _ _ _ _ - _ _ - _ _ _ __ __ __ _ _ 753 evaluation of coefficients for- _ ________________ 755 expansions for _____ ___ _- __ ___ __ _________ _ __ __ 755 joining factors for- _ _____-__-__-_-__-________ 757 normalisation of ____ - __ _ _ _ __ ______ __ _ ___-__ __ 755 tables of _____ _ ___ __ __ _ _- - - - -- - - - -- -- ------_ 766 table of eigenvalues of- _ _ _ _____-__-__________ 760 table of prolate joining factors- _ _ _____________ 769 Stirling numbers- _ _ - _ _-_ _ _ __ _ __ __ __ ____ -_ ____ _ 824 table of the first kind- ____________-__________ 833 table of the second kind-.. ___ __ __ __ __ __-_-___ _ 835 Stirling’s formula ________--_--_-_----__________ 257 Struve’s functions- _ ___________________________ 495 asymptotic expansions for large orders _ _ - _ _ __ _ _ 498 asymptotic expansions for large (z( ___________ 497,498 computation of ______________________________ 499 differential equation- _ - ______________________ 496 graphs of_____-________--_-_________________ 496 integral representations of- _______________ -_ 496,498 integrals----_ __ __ _ _ _ _ __ _-_ _ _ __ _ ___ ___ __ __ _ 497,498 modified ______ - _____________________________ 498 power series expansion for- _ _ __ _ ____ __ _ ____ - 496,498 recurrence relations ____ - _____________ - _____ 496,498 relation to Weber’s function- _ _ __ _- ____ __ __ __ _ 498 special properties of __________________________ 497 tables of _____ ___ __ _ _ _ __ __ __ __ _ ___ __ _____ _-__ 501 Student’s t-distribution _________________________ 948 approximations to ___________________________ 949 asymptotic expansion of ______________________ 949 limiting distribution_ _ _ _ ___ __ __-__ _ _ _-_ ___ __ 949 non-central---______________ - __________ -_-_ 949 series expansions for- __-_____________________ 948 statistical properties of-- _ _ __ __ __ _ __ __ _-__-__ _ 948 Subtabulation ____ __ __ __ ___ __ ___ _-_ - __ __ __ ____ _ 881 Summable series __________ - ________ -_ __________ 1005 Summation of rational series- _ - __ __ __ ____ _ __ ___ 264 Sums of positive powers-.. _ __ __ _ _ __ __ ___ _ _-__ __ _ 813 Sums of powers-__-____________________ - _____ 804 Sums of reciprocal powers- _ __ ____ _ ___-__ __ __ _ 807,811 Systems of differential equations of first order----897

T Taylor expansion________---___________________ 880 Taylor’s formula- _ _ ___________________________ 14 Tesseral harmonics- _ _ - __ __ _ __ _ _ _ __ ___ _ __ __ _ _ __ 332 Tetrachoric functions- ____________________ - ____ 934 Tetragamma function ___-_____ _________________ 260 (see polygamma functions) Theta functions- _ __ __-_ __ _ _ __ __ __ __ ___ _ _ _ __ __ _ 576 addition of quarter-periods_ _ _ __ __ __ _ __ __ _ _ __ 577 calculation by use of the arithmetic-geometric mean-_________________________________ 577,580 expansions in terms of the nome p _____________ 576 Jacobi’s notation for __________________ - ____ -_ 577 logarithmic derivatives of _____________________ 576 logarithms of sum and difference_ _ ____________ 577 Neville’s notation for- _ ____ - _______________ 578,582 relations between squares of the functions _ _ _ _ __ 576 relation to Jacobi’s zeta function-- ____________ 578 relation with Weierstrass elhptic functions-- _ __ _ 650

Page

Thiele’s interpolation formula-- _ _ _ __ _- _ __ _ _ _ ___ _ 881 Toroidal functions _____________________________ 336 Toronto function-_ _ _ ____ ___ ___- ____ _______ _ ___ 509 Trapezoidal rule ____ __-_ __ _____ -_ __ ____ ___ ___ __ 885 Triangle inequality _ _ _ _ _ _ _ _ _ _ __ - _ _ _ _ _ __ _ __ _ _ _ _ _ 11 Trigamma function ____________________________ 260 (see polygamma functions) Trigonometric functions ________________________ 71 (see circular functions) Truncated exponential function- _ __ ____________ 70,262 U Ultraspherical polynomials __________ ____ ________ (see orthogonal polynomials) coefficients for and P in terms of ____-___ ____ graphsof_____~-__-_--~--_-----..--~--..---Unit step function _____________________________

774 794 776 1020

V Variance- _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ __ _ __ _ _ 928 Variance-ratio distribution function ________ _ ___ _ _ 946 (see F-distribution function) Vector-addition coefficients- _ ___________________ 1006 W Wallis’ formula _____ - __________________________ 258 Wave equation in prolate and oblate spheroidal coordinates----752 Weber’s function ______________________________ 498 relation to Anger’s function-..- ________________ 498 relation to Struve’s function- _________________ 498 Weierstrass elliptic functions ______ _____ _ __ __ ____ 627 addition formulas for- _ _ - _-__ _ __ __ __ ___ __ __ __ 635 case A=O-_-------------_-------_-----------651 computation of_-..-- ___ __ _ __ __ __ _____ ________ 663 conformal mapping of--- _ _ __ __ __ _-_ __ __ 642,654,659 definitions______-______________________----629 derivatives of- ______________________________ 640 determination of periods from given invariants- 665 determination of values at half-periods, etc., from givenperiods ____________ - ___-_ -_- ________ 664 differential equation- _ _____________________ 629,640 discriminant____ __ _ _____ ___ _ _______ __-_ -___ 629 equianharmonic case _________________________ 652 expressing any elliptic function in terms of pand p’- _____________________________________ 651 fundamental period parallelogram------------629 fundamental rectangle- _ _ _- ____________ ______ 630 homogeneity relations-_-- _ __ __ __ _ __ _ _ _-___ __ _ 631 integrals- ___ __ _ _ _ _ _ _ _ _ _ _ _ __ _ __ _ _ _ _ _ _ _ - _ _ __ _ 641 invariants- _ _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ 629 Legendre’s relation- _ _ _____ - _________________ 634 lemniscatic case- _ __ _ _ _ ___ _ _ __ _ __-__ __ __ ___ __ 658 maps of- _ _ __ __ __ __ __ _ _ _ _ _ __ __ __ __ _ _ _ _ 642,654,659 multiplication formulas ______ _ _ _ ___ __ __ __ _ _ _ __ 635 other series involving 9, 8’, 5 __________ _ ____ 639 pseudo-lemniscatic case- _ _ _______ ____________ 662 reduction formulas _______ ____________________ 631 relation with complete elliptic integrals _________ 649 relations with Jacobi’s elliptic functions- __ _ _ _ _ _ 649 relations with theta functions _________________ 650

Page

reversedseriesforlarge 191, 19’1, 1(1--------reversed series for small (VI--. ______ ___-______ series expansions for- _ _______________________ special values and relations- ___- _______ _______ symbolism ______ -- ________ -- ______ -- ______ -_ tables of___---.. __ __ _ __ _ _ _ _ ___ _ __ _ _ _ __ __ __ __ _

638 640 635 633 629 673

Page

Whittaker functions-----_____ - _______ - _______ 505 Wigner coefficients ____ _- - - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 1006 z Zeta function Jacobi’s_-----__---------------------------578 Riemann’s ____________ - ___________ --- _____ 256,807

Index

of Notations Page

=p(a+n)/l”(a) (Pochhammer’s symbol) _ _ _ _ 256 a,(g) characteristic value of Mathieu’s equation- _ 722 = 2P(z) - 1 normal probability function- _ _ _ 931 A(4 Ai Airy function---- ________ --- ____________ 446 A.G.M. arithmetic-geometric mean-. _ _ _ _ _ _ _ __ _ _ 571 am z amplitude of the complex number z____--__ 16 antilog antilogarithm (log-r)-L- - - - _ _ __ _ _ _ _ _ _ __ _ 89 arcsin 2, arccos 2 inverse circular functions- ____ _ 79 arctan 2, arccot 2 arcsec 2, arccsc 2 arcsinh z, arccosh z inverse hyperbolic functions.86 arctanh z, arccoth z arcsech z, arccsch z arg z argument of z- - _ _ ___ __ _____- -_ __ __ ---__ 16 b,(p) characteristic value of Mathieu’s equation- 722 B, Bernoulli number.. _ _ -___-----_-----------804 B.(z) Bernoulli polynomial-- __ _ _ _ _ _ _ __ . _ _ _ _ _ __ 804 ber,.z, bei*, Kelvin functions- _________________ 379 Bi(z) Airy function ___________________________ 446 cd, sd, nd Jacobian elliptic functions ____________ 570 c.d.f. cumulative distribution function- _ _ __ _ ____ 927 ce,(z, g) Mathieu function- ____________________ 725 cn Jacobian elliptic function__ _ - - _ _ _ _ _ _ __ ___ _ 569 Cn, Dn, Sn integrals of the squares of Jacobian elliptic functions- _ _ _. __ __________________ _ 576 cs, ds, ns Jacobian elliptic functions.. _ __________ 570 C(z) Fresnel integral __________________________ 300 C,(z) Chebyshev polynomial of the second kind-774 C(z, a) generalized Fresnel integral _____ ________ 262 Ce,(z, 9) modified Mathieu function_ _ _________ _ 732 C,(z), Cz(z) Fresnel integrals _______-___________ 300 C,(“(z) ultraspherical (Gegenbauer) polynomialL_ 774 Chi(z) hyperbolic cosine integral- _ _ _ _ __ _ _ _ _ _ _ __ 231 Ci(z) cosine integral- _ _ _ _ _ __ _ _ _ _ _ _ _ _ ___ _ _ _ _ _ __ 231 Gin(2) cosine integral----_--_____ -- __________ 231 Cinh(z) hyperbolic cosine integral------__-----231 colog cologarithm--_____ - ____ -_- ____ -- _____. 89 covers A, coversine A---_-------____ - _______ 78 dc, nc, SC Jacobian elliptic functions- _ _________. 570 dn = A( q) delta amplitude (Jacobian elliptic function)___--_____--___---~--~-----~--------569 D,(z) parabolic cylinder function (Whittaker’s forrn)__________-____--~~-~--------------687 er, el, c~ roots of a polynomial (Weierstrass form)-629 e* exponential function_--- - _ ___ _ - - _ _ _ __ - _ _ _ __ _ 69 e,(z) truncated exponential function ____ - _______ 262 E(~\or) elliptic integral of the second kind- _ _ _ - 589 E(a,z) parabolic cylinder function- _ _ - - _ __ __ - __ _ 693 E.(z) Weber’s function ____.__ --___-- _____ - ____ 498 E,(m)(z) Weber parabolic cylinder function---- _ _ _ 509 K(m) complete elliptic integral of the second kind---___-___-_---_____________________590 (4

I

1044

page

Ei(z) exponential integral ___________________-228 E,(z) exponential integral- _ _____ ____________-228 E[g(X)] expected value operator for the function s(x)_-___-____---------------------------928 Ein(z) modified exponential integral __________-_ 228 B, Euler number _____ _ _ _ _ _ _- _- _ _ __ __ __ _ _ _ _ _ _804 E,(z) Euler polynomial- _ _ _ _ - _ __ _ __ _ _ _ _ _ _ __ _ _ _ 804 E,(z) exponential integral- _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ 228 erf z error function ___________________________ 297 erfc 2 complementary error function __________-297 exp z=ez exponential function- ________ ________ 69 exsec A, exsecant A ____________________________ 78 f*,,, f.,, joining factors for Mathieu functions-- _ _ 735 F(a, b; c; z) hypergeometric function- _ _________ 556 F(p\a) elliptic integral of the first kind- ________ 589 F~(q,p) Coulomb wave function (regular)---_ _- 538 FPP fundamental period parallelogram_ _ ______ 629 hyper.,Fmh, . . ., a,; bl, . . ., b,; z) generalised geometric function _________________________ 556 g2, g3 invariants of Weierstrass elliptic functions- _ 629 740 Be.., B0.r joining factors for Mathieu functions--g(z, y, p) bivariate normal probability function--936 Gi(z) related Airy function- __________ _ ________ 448 GL(Q p) Coulomb wave functiou (irregular or logarithmic)----------------------..----------538 G,(p, q, z) Jacobi polynomial- _________________ 774 gd(z) Gudermannian ______ _ ___ __ ___ __ __ __ __ __ _ 77 Q(z) spherical Bessel function of the third kind437 hav A haversine A- ______________________ -___ 78 H,(z) Struve’s function- _ _ _ _ _ ___ __ _ _-_ __ __ __ __ 496 Hi(z) related Airy function _____-______________ 448 He,(z) Hermite polynomial ____ __ _ _ __ __ __ _ _-___ 775 H?(z) Bessel function of the third kind (Hankel) _ 358 HA,(z) Hh (probability) function ________-____ 300,691 H,(z) Hermite polynomial- _ _ _________________ 775 H(m, n, z) confluent hypergeometric function---695 Z.(z) modified Bessel function _____ __ _ _ _ _ _ _ - _ __ _ 374 modified spherical Bessel function dGmn+$&) of the first kind- _ ___-____________________ 443 modified spherical Bessel function dim--44 of the second kind _____ - _______ - ___________ 443 Z(rL, p) incomplete gamma function (Pearson’s form)_-----__----__---------------------262 Z,(a, b) incomplete beta function _______________ 263 92 imaginary part of z(=y)- ___________________ 16 in erfc z repeated integral of the error function--299 j,(z) spherical Bessel function of the first kind--437 J,(z) Anger’s function ____ ____________________ 498 J,(z) Bessel function of the first kind ___________ 358 k modulus of Jacobian elliptic functions-.. __ __ _ _ _ 590 k’ complementary modulus ____________________ 590 k,(z) Bateman’s function-- __ _ _ _ _ __ _ __ _ _ _ _ _ _ _ _ _ 510

INDEX

OF NOTATIONS Page

Page

483 repeated integrals of Z&J(Z)- _ _. -__------_ modified spherical Bessel funcKnfH(Z) 443 tion of the third kind- _ ______-______-._____ 374 modified Bessel function_. _. __ _- __ __ ___ K,(z) 590 K(m) complete elliptic integral of the first kind-ker,z, kei,z Kelvinfunctions----__._ -_-___---379 228 ii(z) logarithmic integral _____ ____ __. _ ___- -- -- -lim limit_..-____----_-_-.--------.----.----13 log,& common (Briggs) logarithm_ _ _ _ - -__- __ - __ 68 log,2 logarithm of z to base ~n_~--~~-----~-~~~-67 In 2 (=log,z) natural, Naperian or hyperbolic logarithrn_______-----..~~~---~-----------flF(t)]=f(s) Laplace transform._---____---___ 10:: L(h, k, p) cumulative hivariate normal probability function ____ _ _ ______________________ 936 L.(s) Laguerre polynomial- _ _ _ - - _ _ _ _ _ _ _ . _ _ __ - _ 775 L,(Q)(Z) generalized Laguerre polynomial-- __ . _ _ _ 775 modified Struve function-. . _. . _ _ -. _ _ _ __ 498 L(z) m=p,’ mean~~~~.~~-----~.~----.~.~-~-~~~---928 m parameter (elliptic functions) -- __ . . . _ _ _ _ _ . _ __ 569 _. _ 569 ml complementary parameter-. ._.-._______. M(o, b, z) Kummer’s confluent hypergeometric function--__-___.___--.-.-----.-.~-------504 733 Mc,“‘(z, qj modified Mathieu function_-_ __ ____ 733 itfsMs,(i)(z,q) modified Mathieu function __________ M,,,(z) Whittaker function- - - _ _ _ __ __ __ __ __ _ - 505 n characteristic of the elliptic integral of the third kind__~------.~~-~~-~.---~-~-.----~-----590 O(u,j =un, un is of the order of vn (U./U, is bounded) _ 15

Ki.(z) Am

lim %=O __________________ ____ ____ 259 o(v,) =% Ta+- vn O,,(z) Neumann’s polynomial----~-~---_--~---363 p(n) number of partitions _.______.-__.____._._ 825 q(z) Weierstrass elliptic function--- __ __ _______ 629 ph z phase of t.he complex number z----..------16 P(a,z) incomplete gamma function- _ - - _ - _ _ _ _ __ _ 260 P(x*]u) probability of the x2-distribution- - _ _ 262,940 P:(z) associated Legendre function of the first kind_______-_______-____________________332 P(z) normal probability function--- _ __ _ _ _- _ - - _ _ 931 P,(z) Legendre function (spherical polynomials) -333, 774 P:(z) shifted Legendre polynomial ____ _-_ __ __-_ 774 P,,(a*fl)(z) Jacobi polynomial- __ __ __ _ _ _ _ _ _ __ _ - _ _ _ 774 Pr(X<=s) probability of the event X
number of partitions into distinct integer summands ______________ -_--- _____ - __._.__ Q:(z) associated Legendre function of the second kind---_._.___---_..-~.~~~~-~.--.-~~-.-.$(z) Legendre function of the second kind_-- ._. z real part of z(=z) _____.__ -_- _._.._.______ P(c, -9 radial spheroidal wave function_ ___. __ _ ,!a) Stirling number of the first kind--- _ _-_ ___ _ s),(“+) Stirling number of the second kind _ _ _ - - - - se,(t, q) Mathieu function ____. _ _ _ __. _. . _ _ __ __ _ sn Jacobian elliptic function-- _- _ -_. __ __ _ _ _ _ _ __ S(z) Fresnel integral _____ - ________._._______._ S,(z), &(z) Fresnel integrals-_---_____________ Se,(z, q) modified Mathieu function. _ _ _________ S(z, a) generalized Fresnel integral _______ - _____ Shi(z) hyptrbolic sine integral- _ _ _ _ _ _ __. . ___ ___ Si(z) sine integral- _.___ -___-_--- _______ - _.___ S,(z) Chebyshev polynomial of the first kind- _ _ _ Sib(z) hyperbolic sine integral- _ _ _ _ - _ _ _ - - _ _ __ __ S$i (c, 7) angular spheroidal wave function- _. . __ si(r) sine integral ______. - __ _ _ _ _- _. . _. _ __ _ _ _ __ sin z, cos z, tan 2 circular functions ___. . . .______ cot z, set 2, csc 2 __________ --- _______ - _.___ sinh z, cash z, tanh z hyperbolic functions _______ coth z, sech z, csch z____ --- ______ - _______ -_ T(m, n,r) Toronto function ______________.____ T.(z) Chebyshev polynomial of the first kind-- - T:(z) shifted Chebyshev polynomial of the first kind_____~____---_-_-------~~-----------U(a, b, z) Kummer’s confluent hypergeometric function_______--------------------------U,(z) Chebyshev polynomial of the second kindV”(z) shifted Chebyshev polynomial of the second kind__________-___-_------------------.-U(a, z) Weber parabolic cylinder function--- - - - vers A, versine A _______ __ _________ __ __ _-- -__ V(a, z) Weber parabolic cylinder function _ _ _ - _ w(z) error function- _ ______ __ ____ - __ _ ________ W(a, z) Weber parabolic cylinder function _ _ _ _ _ _ Wr.a(z) Whittaker function ___________________ IV{ f(z), g(z) ) (=f(z)g’(z) -f’(z)g(z)) Wronskian reIation___-__________---------~--~-----.--divided difference _______ __ _- -- -ko, 211 * - . , zb] y,(z) spherical Bessel function of the second kindY,(z) Bessel function of the second kind- _ __---Y:(e, p) surface harmonic of the first kind _ _ __ _ _ Z(z) normal probability .density function- _ _ _ __ __ q(n)

825 332 334 16 753 824 824 725 569 300 300 733 262 231 231 774 231 753 232 71 72 83 83 509 774 774 504 774 774 687 78 687 297 692 505 505 877 437 358 332 931

Notation

-

Greek Letters

Page

.

a

modular

angle (elliptic

44

= lmtne-sldt s

B.(z)

=c

function) _ _ _ _ --___-___

___________

t*e-grdt_

-_-___--

- ____________

________

-_ ____ -- ____

590 228

228

Page

@(u]m) Jacobi’s theta function _________________ I,, nth curnulant--~---~~~~-__________________ ~$4 joining factor for spheroidal wave functions- _

577 928 757

X(n)&Sk+l)-.

807

----______-____

- ____________

k-0

/3(n)

=-&-1)h(2k+l)-”

_____________ -_- _____

807

B,(n, b) incomplete beta function- _ _ _ _ _ - - _ __ _ _ _ B(z, w) betafunction--______ -- ____ - _________ y Euler’s constant ______________ -- ____ -- ______ ~(a, 2) incomplete gamma function (normalired) _

263 258 255 260

?I=:

928

k=O

coefficient of skewness __________________

y~=&-3 coefficient of excess ______ - ___________ d p(z) gamma function- _ __ _- - _ _ _ _ _ _ _ __ __ _ _ _ _ _ _ _ p(a, 2) incomplete gamma function _____________ 8ii Kronecker delta (=0 if i#k; =l if i=k)----e(L) central difference- _ _ _ __________________ _ A difference operator __________________________ A discriminant of Weierstrass’ canonical form--- _ difference- _ _ _ _ __ _ _ _ _ _ - _ _ _ _ _ __ __ A( f,,) forward AX absolute error _______________ -- _________ -__ r(s) Riemann zeta function ____ -__ __ __-_- _____ l(z) Weierstrass zeta function _______ __ _____ - __ _ Z(u]m) Jacobi’s zeta function __________________

928 255 260 822 877 822 629 877 14 807 629 578

s(n)

807

-2

(-l)r-%-”

_.__----______---_-------

Weierstrasselliptic function-------‘lo =r(Z.{ H(u), H,(u) Jacobi’s eta function- _ _ ___________ 9,(z) theta function __________________________ ti,(e\\a),aa(e\aQ, Nevilb’s notation _____________ for theta functions

631 577 576 578

Miscellaneous

x*II characteristic

value of the spheroidal wave equation-___--_________________________ A”(p\a) Heuman’s lambda function-- _ _ __ __.__-_ &fJ mean difference- _ _-_____________________ c(n) Mobius function ____________________. ____ fin nth central moment ____ __________________ -_ B’,, nth moment about the origin ___________ ____ r(Z) number of primes 5% ________ - ______. ____ T=(X) = (Z-20) (x-q) . . . (z-2,) ____---_---.---II@; ~\a) elliptic integral of the third kind- _ _ _ __ II(z) factorial function ____________________ _-__ p correlation coefficient- _______________ ___ ____ Pn(%~1, . . .,z.) reciprocal difference ________ ____ P”(v, z) Poisson-Charlier function- _____ __ _ __ __ __ * standard deviation ______________________ ____ 2 variance----____--______________________-u(z) Weierstrass sigma function _---____________ Q k(n) divisor function ___- __ _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ __ T,(Z) tetrachoric function- _ _ _ _________________ a=am p1, amplitude- _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ --__-----a(n) Euler-Totient function- _ __--_____________ characteristic function of X- _ _ _ __ _ a(t) = E(e”*) *(a; b; z) confluent hypergeometric function-- _ __ $(z) logarithmic derivative of the gamma function__----------_---------------------*‘(a; c; z) confluent hypergeometric function___ W, period of Weierstrass elliptic functions- _ _ _ _ _ _ W.,,(Z) Cunningham function _____ _ _ __ __ ___ __ __ _

19 19 752 877

& partial

883

derivative _______________

- ________ -__

i (= V-l)- ___--______ - _----- - _______________ (;) binomial coefhcient __________ -_-___- _______ n! factorial function- _ _ _ _---__________________ (2n)!! =2*4.6 . . . (272) = 2%!-- _ _ _ __ __ _ _ _ _ _ _ _ _ (m,n) greatest common divisor--- _ _ _ _ _ _ _-_ __ _ _ _

70 10 255 258 822

(n,k) =wi+n+w krp(s+n-k)

437

(n; 781, nz; . . ., n,) [z]

largest integer 1046

(Hankel’s

symbol) _ _ _____

mu!tinomial coefficient-- _ _ _ _ Is-- ___- _______ _______ ___-_

823 66

268 SO4 629 510

Notations

PW0

[oia] determinant _____________________________ [ai] column matrix ______ ______________________ Vn Laplacian operator _ - _ __-___ _____--_-_ __ ___ AL forward difference operator __---__-----__-__

753 595 877 826 928 928 231 878 590 255 936 878 509 298 928 629 827 934 669 826 928 so4

<x> nearest integer to 5 _____________________ Z complex conjugate of 2 (=?:-iy)__________ -_ z=z+iy complex number (Cartesian form)-----= rei” (polar form) ____ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ Izj absolute value or modulus of z __________ __ __ I: overall summation _____________________ I ____ Z’ restricted summation ______________________ I: II sum or product taken over all prime numbers p-

sD

PM3 222 16 16 16 16 822 75s 807

Z IId,” sum or product overall positRe divisors d of A-6,”

826

f Cauchy’s principal value of the integral _ _ - ___ = approximately equal _______________________ N asymptotically equal-- - _ _ _ _ _ _ _ _ _ _ _ - __ __ _- __ <, >, I L inequality, inclusion _ _ _ _ - _ _ ___ _ - - _ # unequal----____--_______________________-

228 14 1s 10 12

,

A CATALOGUE OF SELECTED DOVER IN ALL FIELDS OF INTEREST

BOOKS

A CATALOGUE OF SELECTED DOVER IN ALL FIELDS OF INTEREST

BOOKS

AMERICA’S OLD MASTERS, James T. Flexner. Four men emerged unexpectedly from provincial 18th century America to leadership in European art: Benjamin West, J. S. Copley, C. R. Peale, Gilbert Stuart. Brilliant coverage of lives and contributions. Revised, 1967 edition. 69 plates. 365~~. of text. 21806-6 Paperbound $3.00 FLOWERS OF OUR WILDERNESS: AMERICAN PAINTING, THE COLONIAL PERIOD, James T. Flexner. Painters, and regional painting traditions from earliest Colonial times up to the emergence of Copley, West and Peale Sr., Foster, Gustavus Hesselius, Feke, John Smibert and many anonymous painters in the primitive manner. Engaging presentation, with 162 illustrations. xxii + 368~~. 22180-6 Paperbound $3.50 FIRST

THE LIGHT OF DISTANT SKIES: AMERICAN PAINTING, .1760-1835, James T. Flexner. The great generation of early American painters goes to Europe to learn and to teach: West, Copley, Gilbert Stuart and others. Allston, Trumbull, Morse; also contemporary American painters-primitives, derivatives, academics-who remained in America. 102 illustrations. xiii + 306~~. 22179-2 Paperbound $3.00 A HISTORY OF THE RISE STATES, William Dunlap.

AND

PROGRESS

OF THE ARTS

OF DESIGN

IN THE UNITED

Much the richest mine of information on early American painters, sculptors, architects, engravers, miniaturists, etc. The only source of information for scores of artists, the major primary source for many others. Unabridged reprint of rare original 1834 edition, with new introduction by James T. Flexner, and 394 new illustrations. Edited by Rita Weiss. 65/a x 95/a. 21695-0, 21696-9, 21697-7 Three volumes, Paperbound $13.50 EPOCHS OF CHINESE AND JAPANESE ART, Ernest F. Fenollosa. From primitive Chinese art to the 20th century, thorough history, explanation of every important art period and form, including Japanese woodcuts ; main stress on China and Japan, but Tibet, Korea also included. Still unexcelled for its detailed, rich coverage of cultural background, aesthetic elements, diffusion studies, particularly of the historical period. 2nd, 1913 edition. 242 illustrations. lii + 439~~. of text. 20364-6, 20365-4 Two volumes, Paperbound $6.00

THE GENTLE ART OF MAKING ENEMIES, James A. M. Whistler. Greatest wit of his day deflates Oscar Wilde, Ruskin, Swinburne; strikes back at inane critics, exhibitions, art journalism; aesthetics of impressionist revolution in most striking form. Highly readable classic by great painter. Reproduction of edition designed by Whistler. Introduction by Alfred Werner. xxxvi + 334~~. 21875-9 Paperbound $2.50

CATALOGUE

OF DOVER

BOOKS

VISUAL ILLUSIONS: THEIR CAUSES, CHARACTERISTICS, AND APPLICATIONS, Matthew Luckiesh. Thorough description and discussion of optical illusion, geometric and perspective, particularly; size and shape distortions, illusions of color, of motion; natural illusions; use of illusion in art and magic, industry, etc. Most useful today with op art, also for classical art. Scores of effects illustrated. Introduction by William H. Ittleson. 100 illustrations. xxi + 252~~. 21530-X Paperbound $2.00 A HANDBOOK OF ANATOMY FOR ART STUDENTS, Arthur Thomson. Thorough, virtually exhaustive coverage of skeletal structure, musculature, etc. Full text, supplemented by anatomical diagrams and drawings and by photographs of undraped figures. Unique in its comparison of male and female forms, pointing out differences of contour, texture, form. 211 figures, 40 drawings, 86 photographs. xx f 459~~. 21163-O Paperbound $3.50 53/s x S$‘a. 150 MASTERPIECES OF DRAWING, Selected by Anthony Toney. Full page reproductions of drawings from the early 16th to the end of the 18th century, all beautifully reproduced: Rembrandt, Michelangelo, Diirer, Fragonard, Urs, Graf, Wouwerman, many others. First-rate browsing book, model book for artists. xviii + 150~~. 21032-4 Paperbound $2.50 83/8 X 111/4. THE LATER WORK OF AUBREY BEARDSLEY, Aubrey Beardsley. Exotic, erotic, ironic masterpieces in full maturity: Comedy Ballet, Venus and Tannhauser, Pierrot, Lysistrata, Rape of the Lock, Savoy material, Ali Baba, Volpone, etc. This material revolutionized the art world, and is still powerful, fresh, brilliant. With The Early Ii+‘ork, all Beardsley’s finest work. 174 plates, 2 in color. xiv + 176~~. 81/8 x 11. 21817-1 Paperbound $3.00 DRAWINGS OF REMBRANDT, Rembrandt van Rijn. Complete reproduction of fabulously rare edition by Lippmann and Hofstede de Groot, completely reedited, updated, improved by Prof. Seymour Slive, Fogg Museum. Portraits, Biblical sketches, landscapes, Oriental types, nudes, episodes from classical mythology-All Rembrandt’s fertile genius. Also selection of drawings by his pupils and followers. “Stunning volumes,” Satunz’u~ Review. 550 illustrations. lxxviii + 552~~. 21485-0, 21486-9 Two volumes, Paperbound $10.00 9?/a x 121/. THE DISASTERS OF WAR, Francisco Goya. One of lization-83 etchings that record Goya’s shattering, war that swept through Spain after the insurrection Reprint of the first edition, with three additional Fine Arts. All plates facsimile size. Introduction v + 97pp. 93/a x Sr/&

the masterpieces of Western civibitter reaction to the Napoleonic of 1808 and to war in general. plates from Boston’s Museum of by Philip Hofer, Fogg Museum. 21872-4 Paperbound $2.00

GRAPHIC WORKS OF ODILON REDON. Largest collection of Redon’s graphic works ever assembled: 172 lithographs, 28 etchings and engravings, 9 drawings. These include some of his most famous works. All the plates from Odilon Redon: oeuvre graphique romplet, plus additional plates. New introduction and caption translations by Alfred Werner. 209 illustrations. xxvii -1 209~~. 9$$ x 121/4. 21966-8 Paperbound $4.00

CATALOGUE

OF DOVER BOOKS

JOHANN SEBASTIAN BACH, Philipp Spitta. One of the great classics of musicology, this definitive analysis of Bach’s music (and life) has never been surpassed. Lucid, nontechnical analyses of hundreds of pieces (30 pages devoted to St. Matthew Passion, 26 to B Minor Mass). Also includes major analysis of 18th-century music. 450 musical examples. 40-page musical supplement. Total of xx + 1799~~. (EUK) 22278-0, 22279-9 Two volumes, Clothbound $15.00 MOZART AND HIS PIANO CONCERTOS, Cuthbert Girdlestone. The only full-length study of an important area of Mozart’s creativity. Provides detailed analyses of all 23 concertos, traces inspirational sources. 417 musical examples. Second edition. (USO) 21271-S Paperbound $3.50 509pp. THE

PERFEIZT

WAGNERITE:

A COMMENTARY

ON THE

NIBLUNG’S

RING,

George

Bernard Shaw. Brilliant and still relevant criticism in remarkable essays on Wagner’s Ring cycle, Shaw’s ideas on political and social ideology behind the plots, role of Leitmotifs, vocal requisites, etc. Prefaces. xxi + 136~~. 21707-S Paperbound $1.50 GIOVANNI, W. A. Mozart. Complete libretto, modern English translation; biographies of composer and librettist; accounts of early performances and critical reaction. Lavishly illustrated. All the material you need to understand and appreciate this great work. Dover Opera Guide and Libretto Series; translated and introduced by Ellen Bleiler. 92 illustrations. 209~~. 21134-7 Paperbound $1.50 DON

HIGH FIDELITY SYSTEMS: A LAYMAN’S GUIDE, Rof F. Allison. All the basic information you need for setting up your own audio system: high fidelity and stereo record players, tape records, F.M. Connections, adjusting tone arm, cartridge, checking needle alignment, positioning speakers, phasing speakers, adjusting hums, trouble-shooting, maintenance, and similar topics. Enlarged 1965 edition. More than 50 charts, diagrams, photos. iv + 9lpp. 21514-8 Paperbound $1.25 REPRODUCTION

high fidelity loudspeakers, technicalities iv f 92pp. HEAR MADE

OF SOUND, Edgar Villchur. Thorough coverage for laymen of systems, reproducing systems in general, needles, amplifiers, preamps, feedback, explaining physical background. “A rare talent for making vividly comprehensible,” R. Darrell, High Fidelity. 69 figures. 21515-6 Paperbound $1.00

ME TALKIN’ TO YA: THE STORY OF JAZZ AS TOLD BY THE MEN WHO IT, Nat Shapiro and Nat Hentoff. Louis Armstrong, Fats Wailer, Jo Jones,

Clarence Williams, Billy Holiday, Duke Ellington, Jelly Roll Morton and dozens of other jazz greats tell how it was in Chicago’s South Side, New Orleans, depression Harlem and the modern West Coast as jazz was born and grew. xvi + 429~~. 21726-4 Paperbound $2.50 OF AESOP, translated by Sir Roger LEstrange. A reproduction of the very rare 1931 Paris edition; a selection of the most interesting fables, together with 50 imaginative drawings by Alexander Calder. v + 128~~. 61/2x91/4. 21780-9 Paperbound $1.25 FAEILES

i --

.-

CATALOGUE

OF

DOVER

BOOKS

ALPHABETS AND ORNAMENTS, Ernst Lehner. Well-known pictorial source for decorative alphabets, script examples, cartouches, frames, decorative title pages, calligraphic initials, borders, similar material. 14th to 19th century, mostly European. Useful in almost any graphic arts designing, varied styles. 750 illustrations. 256~~. 7 x 10. 21905-4 Paperbound $4.00 PAINTING: A CREATIVE APPROACH, Norman Colquhoun. For the beginner simple guide provides an instructive approach to painting: major stumbling blocks for beginner; overcoming them, technical points; paints and pigments; oil painting; watercolor and other media and color. New section on “plastic” paints. Glossary. Formerly Painf Your Own Pictures. 221~~. 22000-1 Paperbound $1.75 THE ENJOYMENT AND USE OF COLO.R, Walter Sargent. Explanation of the relations between colors themselves and between colors in nature and art, including hundreds of little-known facts about color values, intensities, effects of high and low illumination, complementary colors. Many practical hints for painters, references to great masters. 7 color plates, 29 illustrations. x + 274~~. 20944-X Paperbound $2.75 THE NOTEBOOKS OF LEONARDO DA VINCI, compiled and edited by Jean Paul Richter. 1566 extracts from original manuscripts reveal the full range of Leonardo’s versatile genius: all his writings on painting, sculpture, architecture, anatomy, astronomy, peography, topography, physiology, mining, music, etc., in both Italian and English, with 186 plates of manuscript pages and more than 500 additional drawings. Includes studies for the Last Supper, the lost Sforza monument, and other works. Total of xlvii + 866~~. 7ya x 103,4. 22572-0, 22573-9 Two volumes, Paperbound $10.00 MONTGOMERY WARD CATALOGUE OF 1895. Tea gowns, yards of flannel and pillow-case lace, stereoscopes, books of gospel hymns, the New Improved Singer Sewing Machine, side saddles, milk skimmers, straight-edged razors, high-button shoes, spittoons, and on and on . . listing some 25,000 items, practically all illustrated. Essential to the shoppers of the 1890’s, it is our truest record of the spirit of the period. Unaltered reprint of Issue No. 57, Spring and Summer 1895. Introduction by Boris Emmet. Innumerable illustrations. xiii + 624~~. 81/2 x 115/8. 22377-9 Paperbound $6.95 THE CRYSTAL PALACE EXHIBITION ILLUSTRATED CATALOGUE (LONDON, 185 1). One of the wonders of the modern world-the Crystal Palace Exhibition in which all the nations of the civilized world exhibited their achievements in the arts and sciences-presented in an equally important illustrated catalogue. More than 1700 items pictured with accompanying text-ceramics, textiles, cast-iron work, carpets, pianos, sleds, razors, wall-papers, billiard tables, beehives, silverware and hundreds of other artifacts-represent the focal point of Victorian culture in the Western World. Probably the largest collection of Victorian decorative art ever assembledindispensable for antiquarians and designers. Unabridged republication of the Art-Journal Catalogue of the Great Exhibition of 1851, with all terminal essays. New introduction by John Gloag, F.S.A. xxxiv + 426~~. 9 x 12. 22503-8 Paperbound $4.50

:

CATALOGUE

OF DOVER

BOOKS

DESIGN BY ACCIDENT; A BOOK OF “ACCIDENTAL EFFECTS” FOR ARTISTS AND DESIGNERS, James F. O’Brien. Create your own unique, striking, imaginative effects by “controlled accident” interaction of materials: paints and lacquers, oil and water based paints, splatter, crackling materials, shatter, similar items. Everything you do will be different; first book on this limitless art, so useful to both fine artist and commercial artist. Full instructions. 192 plates showing “accidents,” 8 in color. viii + 215~~. 83/a x 111/q. 21942-9 Paperbound $3.50 THE BOOK OF SIGNS, Rudolf Koch. Famed German type designer draws 493 beautiful symbols: religious, mystical, alchemical, imperial, property marks, runes, etc. Remarkable fusion of traditional and modern. Good for suggestions of timelessness, smartness, modernity. Text. vi + 104~~. 61/s x 91/L. 20162-7 Paperbound $1.25 HISTORY OF INDIAN AND INDONESIAN ART, Ananda K. Coomaraswamy. An unabridged republication of one of the finest books by a great scholar in Eastern art. social backgrounds; Sunga reliefs, Rajput Rich in descriptive material, history, paintings, Gupta temples, Burmese frescoes, textiles, jewelry, sculpture, etc. 400 21436-2 Paperbound $4.00 photos. viii + 423~~. 63/ x 93/4. PRIMITIVE ART, Franz Boas. America’s foremost anthropologist surveys textiles, ceramics, woodcarving, basketry, metalwork, etc.; patterns, technology, creation of symbols, style origins. All areas of world, but very full on Northwest Coast Indians. More than 350 illustrations of baskets, boxes, totem poles, weapons, etc. 378 pp. 20025-6 Paperbound $3.00 THE GENTLEMAN AND CABINET reprint (third edition, 1762) of master cabinetmaker. 200 plates, plus 24 photographs of surviving stock. vi f 249~~. 97’ x 123/.

MAKER’S DIRECTOR, Thomas Chippendale. Full most influential furniture book of all time, by illustrating chairs, sofas, mirrors, tables, cabinets, pieces. Biographical introduction by N. Bienen21601-2 Paperbound $4.00

AMERICAN ANTIQUE FURNITURE, Edga*. G. Miller, Jr. The basic coverage of all American furniture before 1840. Individual chapters cover type of furnitureclocks, tables, sideboards, etc.-chronologically, with inexhaustible wealth of data. More than 2100 photographs, all identified, commented on. Essential to all early American collectors. Introduction by H. E. Keyes. vi + 1106~~. 778 x 103/4. 21599-7, 21600-4 Two volumes, Paperbound $11.00 PENNSYLVANIA DUTCH AMERICAN FOLK ART, Henry J. Kauffman. 279 photos, 28 drawings of tulipware, Fraktur script, painted tinware, toys, flowered furniture, quilts, samplers, hex signs, house interiors, etc. Full descriptive text. Excellent for tourist, rewarding for designer, collector. Map. 146~~. 77/8 x 103/. 21205-X Paperbound $2.50 EARLY NEW ENGLAND GRAVESTONE RUBBINGS, Edmund V. Gillon, Jr. 43 photographs, 226 carefully reproduced rubbings show heavily symbolic, sometimes macabre early gravestones, up to early 19th century. Remarkable early American primitive art, occasionally strikingly beautiful; always powerful. Text. xxvi + 207~~. 83/s x 111/,. 21380-3 Paperbound $3.50