Elefun

CHAPTER III. ELEMENTARY FUNCTIONS

SECTION 28. THE EXPONENTIAL FUNCTION

We consider various elementary functions studied in calculus and define corresponding functions of a complex variable that reduce to the elementary functions in calculus when z = x + i0. We start by defining the complex exponential function and then use it to develop the others.

The exponential function ez or exp z of a complex variable z=x+iy is defined as

e e z

x  iy

e e . x iy

Observe that ez reduces to the usual exponential function in calculus when y = 0.

When z = 1/n, then ez = e1/n. In calculus we take e1/n as the real positive nth root of e. Therefore here also we take ez as the real positive nth root of e if z = 1/n.

The exponential function has the following properties: z1 z2 z1  z2 (1) e e  e .

(2)

e  0 for any z. z

z1  z2

e e e . d z z (4) (e )  e . dz z (5) e is periodic with period 2 i. (3)

z1

(6)

z

z2

e can have negative values.

Problem 6/89. Show that 2

exp( z )  exp( z ). 2

Problem 9/90. Show that

exp(iz )  exp(i z ) if and only if z = nπ ; n integer.

SECTION 29. THE LOGARITHMIC FUNCTION

The (multiple-valued)logarithmic function of a nonzero complex variable z = reiΘ (- π < Θ ≤ π) is defined as log z = ln r + i (Θ+2nπ) (n integer) 0r log z = ln |z| + i arg z. Observe that exp(log z) = z (z ≠ 0) but log(exp z) is not simply z.

The principal value Log z of log z is defined as Log z = ln |z| + i Arg z. Note : (1) Log z is well defined and single valued when z ≠ 0 and log z = Log z + 2nπi ; n integer. (2) Log z reduces to usual logarithm in calculus when z is a positive real number. (3) log (-1) = (2n+1) πi and Log (-1) = πi.

Problem 5/94. Show that (a) The set of values of log(i½) is (n+¼)πi (n integer) and that the same is true of (½)log i. (b) The set of values of log(i2) is not the same as the set of values of 2 log i.

SECTION 30. BRANCHES AND DERIVATIVES OF LOGARITHMS

A branch of a multiple-valued function f is any single valued function F that is analytic in some domain D and F(z) is one of the values f(z) at each point of D.

For each fixed α, the single-valued function

log z  ln r  i (r  0,       2 ) is a branch of the multiple-valued function

log z  ln r  i (  2n ) (n  Z).

The function

Log z  ln r  i (r  0,      )

is called the principal branch.

A line or curve that is introduced in order to define a branch F of a multiple-valued function f is called a branch cut for F. Any point that is common to all branch cuts of f is called a branch point.

Problem 8/94. Suppose that the point z=x+iy lies in the horizontal strip α
log z  ln r  i (r  0,       2 ) of the logarithmic function is used, log(ez)=z.

SECTION 31. SOME IDENTITIES INVOLVING LOGARITHMS

(1) log( z1 z2 )  log z1  log z2 ( z1 , z2  0) (2) z  e n

(3) z

1n

n log z

( z  0, n  Z)

 exp( log z ) ( z  0, n  1, 2,3,K ) 1 n

Equation (1) is interpreted by saying that if two of the three logarithms are specified, then there is a value of the third such that the equation holds.

Note: Statement (1) is not, in general, valid when log is replaced everywhere by Log. Problem 1/96. Show that if real parts of z1 and z2 are positive, then

Log ( z1 z2 )  Log z1  Log z2 .

SECTION 32. COMPLEX EXPONENTS

When z≠0 and c is any complex number, the function zc is defined by the equation

z e c

c log z

,

where log z denotes the multiple valued logarithmic function. Note: Powers of z are, in general multiple valued.

If the branch

log z  ln r  i (r  0,       2 ) of logarithmic function is used, then zc is single valued and analytic in this domain. Within this domain

d c c 1 z  cz . dz

The principal value of zc is

P.V. z  e c

c log z

.

The principal branch of zc is defined on the domain

z  0,   Arg z   .

The exponential function with base c, where c is any nonzero complex constant, is defined as

c e z

z log c

.

When a value of log c is specified, cz is an entire function of z and

d z z c  c log c. dz

Problem 3/100. Show that

(1  3 i )

32

  2 2.

SECTION 33. TRIGONOMETRIC FUNCTIONS

The sine and cosine functions of a complex variable z are defined as

e e sin z  2i iz

 iz

 iz

e e , cos z  . 2 iz

It is easy to check that

d d (1) sin z  cos z , cos z   sin z. dz dz (2) sin( z )   sin z , cos( z )  cos z.

The following identities are easy to establish:

(1) sin( z1  z2 )  sin z1 cos z2  cos z1 sin z2 (2) cos( z1  z2 )  cos z1 cos z2  sin z1 sin z2 (3)sin(iy )  i sinh y, cos(iy )  cosh y (4)sin z  sin x cosh y  i cos x sinh y (5) cos z  cos x cosh y  i sin x sinh y

Both sine and cosine functions are periodic with period 2π and both are unbounded. Zeros of sin z are z = nπ and zeros of cos z are z = (π/2) + nπ, where n is an integer. Other trigonometric functions are defined as in real case.

Problem 16(a)/105. Show that

cos(iz )  cos(i z ) for all complex numbers z. Problem 17/105. Find all roots of the equation sin z = cosh 4 by equating the real parts and the imaginary parts of sin z and cosh 4.

SECTION 34. HYPERBOLIC FUNCTIONS

Hyperbolic sine and cosine functions of a complex variable are defined as z

z

e e e e sinh z  , cosh z  . 2 2 z

z

It is easy to check that d d (1) sinh z  cosh z, cosh z  sinh z. dz dz (2)sinh( z )   sinh z , cosh( z )  cosh z.

The following identities are easy to establish:

(1) sinh( z1  z2 )  sinh z1 cosh z2  cosh z1 sinh z2 (2) cosh( z1  z2 )  cosh z1 cosh z2  sinh z1 sinh z2 (3)  sinh(iz )  sin z, cosh(iz )  cos z (4)  i sin(iz )  sinh z, cos(iz )  cosh z

(5)sinh z  sinh x cos y  i cosh x sin y (6) cosh z  cosh x cos y  i sinh x sin y 2

2

2

2

(7) sinh z  sinh x  sin y 2

(8) cosh z  sinh x  cos y 2

Other hyperbolic functions are defined as in real case.

SECTION 35. INVERSE TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS

The inverse sine, cosine and tangent functions of a complex variable are defined as:

sin

1 1

z  i log[iz  (1  z ) ], 2 12

cos z  i log[ z  i (1  z ) ], i iz 1 tan z  log . 2 iz 2 12

The derivatives of these functions are as follows: d 1 1 sin z  , 2 12 dz (1  z ) d 1 1 cos z  , 2 12 dz (1  z ) d 1 1 tan z  . 2 dz 1 z

First two derivatives depend on the values chosen for the square roots.

The inverse hyperbolic sine, cosine and tangent functions of a complex variable are defined as:

sinh

1 1

z  log[ z  ( z  1) ], 2

12

z  log[ z  ( z  1) ], 1 1 z 1 tanh z  log . 2 1 z

cosh

2

12