LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – NOVEMBER 2012

MT 3811 – COMPLEX ANALYSIS

 

 

Date : 03/11/2012            Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

Answer all the questions:

 

1. a) Prove that  if  using Leibniz’s rule.

OR

1. b) State and prove Liouville’s theorem.        (5)
2. c) State and prove first version of Cauchy’s integral formula.

OR

1. d) State and prove the homotopic version of Cauchy’s theorem                  (15)

 

2. a) State and prove Hadamard’s three circles theorem.

OR

1. b) Define a convex function and prove that a function  is convex if and

only if the set  is a convex set.                           (5)

1. c) State and prove Goursat’s theorem.

OR

1. d) State and prove Arzela Ascoli theorem.      (15)

 

3. a) Let , for all . Then prove that  converges to a complex number different from zero if and only if  converges.

OR

1. b) Show that in the usual notation.                                                                                                                                            (5)

 

 

1. c) (i) If and  then prove that .

(ii) Prove that .

(iii) State and prove Gauss’s Formula.                                                                                                                                                                                              (5+5+5)

OR

1. d) (i) State and prove Bohr-Mollerup theorem.

(ii) Prove that (a)  converges to  in  and (b) if  then  for all .                                                                                                                                                                                          (8+7)

4. a) State and prove Jensen’s formula.

OR

1. b) Let  be a rectifiable curve and let K be a compact set such that .  If f is a continuous function on  and  then prove that there is a rational function  having all its poles on  and  such that  for all z in K.

(5)

1. c) State and prove Mittag-Leffler’s theorem.                                                          (15)

OR

1. d) State and prove Hadamard’s Factorization theorem.

(15)

5. a) Prove that any two bases of a same module are connected by a unimodular transformation.

OR

1. b) Show that and it is an odd function.                                                                                                                                                        (5)

 

1. c) (i) Prove that the zeros and poles  of an elliptic function satisfy .

(ii) Prove that   .                                  (7+8)

OR

1. d) (i) Show that

(ii) State and prove the addition theorem for the Weierstrass -function.                                                                                                                                                 (7+8)

 

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