## December 17, 2011

### MATHEMATICS 3 JNTU previous years question papers

Time: 3 hours Max Marks: 80

All Questions carry equal marks

1. (a) Prove that (2n+1)x Pn(x) = (n+1)Pn+1(x) + nPn−1(x).
(b) Show that x4 = 1
35 [8P4(x) + 20P2(x) + 7P0(x)]. [8+8]

2. (a) Find where the function
i. w = 1
z
ii. w = z
z −1
ceases (fails ) to be analytic.
(b) In a two dimensional fluid flow, the stream function ψ = tan−1 (y/x), then find velocity potential function φ [8+8]

3. (a) Separate into real and imaginary parts log cos ( x + iy )
(b) Determine all values of ( 1+ i )i [8+8]

4. (a) Evaluate RCRezdz, where C is the shortest path from 1+i to 3+2i.
(b) Use Cauchy’s integral formula to evaluate H
c
z−1
(z+1)2(z−2)dz where ‘c’ is the circle
|z − i|= 2 [8+8]

5. (a) Find the Taylor’s series expansion of sin z = π/2.
(b) Find the Taylor’s series expansion of ez about z = 3. [8+8]

6. (a) State and prove Cauchy’s Residue theorem.
(b) Find the residue at z = 0 of the function
f(z) = 1+ez sin z+z cos z [8+8]

7. Evaluate R
C
f1(z)
f(z) dz by using the Augument principle where C is a circle
|z| = 4 and f(z) = (z2+1)2
(z2+2z+2)3 

8. (a) If w = 1+iz
1−iz find the image of |z| < 1.
(b) Find the image of the unit circle |z| = 1 under the linear fractional transformation w(z) = 2iz−2−2i
(1−i)z−1 [8+8]

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