LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – APRIL 2012

MT 3811 – COMPLEX ANALYSIS

 

 

Date : 24-04-2012             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

Answer all the questions.

 

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

OR

1. b) Let be a non-constant polynomial. Prove that there is a complex number  such that.                                                                                                       (5)
2. c) Let be an analytic function. Prove that  for  where  Hence prove that if f is analytic in an open disk  then prove that  for  where.

OR

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

 

2. a) State and prove Morera’s theorem.

OR

1. b) Prove that a differentiable function  on  is convex if and only if  is

increasing.                                                                                                             (5)

1. c) State and prove the Arzela-Ascoli theorem.

OR

1. d) State and prove the Riemann mapping theorem.

(15)

 

3. a) Show that  in the usual notation.

OR

1. b) If and  then prove that .

(5)

 

 

 

 

 

 

 

1. c) (i) Let be a compact metric space and let  be a sequence of continuous functions from X into  such that  converges absolutely and uniformly for x in X. Then prove that the product  converges absolutely and uniformly for x in X. Also prove that there is an integer  such that  if and only if  for some n, .

(ii) State and prove Weierstrass factorization theorem.                                     (7+8)

OR

1. d) Let , then prove that converges absolutely if and only if  converges absolutely.
2. e) State and prove Bohr-Mollerup theorem.    (7+8)

 

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

OR

1. b) If  is a metric space, then prove that  is also a metric on                                                                                                                                            (5)
2. c) State and prove Rung’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) Prove that an elliptic function without poles is a constant.

(5)

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

(ii) Derive Legendre’s relation                                                                           (7+8)

OR

1. d) (i) State and prove the addition theorem for the Weierstrass

(ii) Show that

(8+7)

 

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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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