Mathematics Iii May2003 Or 220556

Code No. 220556 II-B.Tech. II-Semester Examinations April/May, 2003

OR

MATHEMATICS-III (Common to Electrical and Electronics Engineering, Electronics and Communication Engineering, Electronics and Instrumentation Engineering, Bio – Medical Engineering, Electronics and Control Engineering, Metallurgical and Material Technology, Electronics and Telematics) Time: 3 hours

Max. Marks: 70 Answer any Five questions All questions carry equal marks --1

∫ ( log 1 / y )

1. a) Show that t Γn =

n−1

dy

0

1

b) Prove that

∫ 0

dx 1− x

n

=

π Γ (1 / n) n Γ (1 / n + 1 / 2)

2. a) Write down the power series expansion for J n ( x ) and hence show that

2 sin x. πx d n x J n ( x ) = x n J n −1 ( x ) b) Show that dx J 1/ 2 ( x ) =

[

3. a) b) 4. a) b)

]

Define analyticity of a complex function at a point P and in a domain D. Prove that the real and imaginary parts of an analytic function satisfy Cauchy – Riemann Equations. Show that w = zn (n , a positive integer) is analytic and find it’s derivative. Find an analytic function f(z) such that Real [f ' (z)] = 3x2 – 4y – 3y2 and f(1+i) = 0. Determine p such that the function  px  1   2 2 –1  y  2 f(z) = log (x +y ) + i tan be an analytic function e

Contd…2

Code No. 220556

.2..

OR

5. a) Show that

∫

x dx – y dy √ ( x2 + y2 )

C

Is independent of any path of integration which does not pass through the origin. b) Evaluate

∫

dz

.

( z- 2i )2 ( z + 2i )2

c

c being the circumference of the ellipse x2 + 4( y - 2 )2 = 4

6. a) b) 7. a)

b)

n When n is an integer, show that J − n ( x ) = ( −1) J n ( x )

Prove that

d  J n ( x)  −n = − x J n +1 ( x ) n   dx  x 

Find the residue of Z 2 − 2Z f(z) = at each pole. ( Z + 1) 2 ( Z 2 + 1) 4 − 3z 3 Evaluate ∫ z ( z − 1)( z − 2) dz Where c is the circle | Z | = 2 c

8. a)

Find the image of the rectangle R: -π< x <π; 1/2 < y< 1, under the transformation w = sin z. b) Find the bilinear transformation that maps the points ,(∞ i, 0), into the points (0, i, ∞) . ###