No. of Printed Pages : 3

MMT-005

M.Sc. (MATHEMATICS WITH APPLICATIONS IN COMPUTER SCIENCE) M.Sc. (MACS) Term-End Examination June, 2015

00898 MMT-005 : COMPLEX ANALYSIS Time : 1-1- hours

Maximum Marks : 25

2

Note : Question no. 1 is compulsory. Attempt any three questions from questions no. 2 to 5. Use of calculator is not allowed.

1.

State giving reasons whether the following statements are true or false : 5x 2=10 (a)

The function f(z) (x3 -y3)+i(x3 +y3) X

2 +y 2

z 0,

ff0) = 0, is differentiable at 0. (b)

MMT-005

Product of two harmonic functions is harmonic. 1

P.T.O.

(c)

The function f(z) =

sin z z # 0 , has a pole z3

of order 3 at z = 0. (d)

sin z

If C: Izi = 1 and f(z) =

(z -

then

(z - 2)

f(z) dz = 2rci.

(e) sin (1) has a simple pole at z = 0. 2.

(a) State Cauchy Riemann equations in Polar form and use them to find the harmonic conjugate of u(r, 0) = in r, r > 0.

3

(b) Find Laurent's expansion of 1 . f(z) = m the region 1 < I z < 2. z` — 3z + 2 + 3.

Evaluate

dx J.(X2 + 1)3

using contour integration.

- 00

4.

(a)

Use Residue theorem to evaluate

dz eiz sin z C

where C is a positively oriented quadrilateral with vertices ± 2 ± 3i. MMT-005

2

5

(b) Find the radius of convergence of the series n! zn Also find the domain of n2 n=0

convergence of the series. 5. (a) If fz) is an entire function such that Z V z, show that f(z) = ae-iz f(z) where I a I 1.

2

(b) Prove that Mobius transformation w

MMT-005

2z -11 maps unit disc to itself. 2-z

3

3

1,000

June 2015

IN COMPUTER SCIENCE). M.Sc. (MACS). Term-End Examination. June, 2015. 00898. MMT-005 : COMPLEX ANALYSIS. Time : 1-1- hours. Maximum Marks : ...

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