Tamil Nadu Open University (TNOU) 2009-1st Year M.Sc Mathematics Tamilnadu open university Maths Real and complex analysis - Question Paper

M.Sc. DEGREE EXAMINATION JUNE 2009.

(AY 2005-06 and CY 2006 batches only)

First Year

Mathematics

REAL AND COMPLEX ANALYSIS

Time : 3 hours Maximum marks : 75

PART A (5 5 = 25 marks)

Answer any FIVE questions.

1.         If ; , then prove that there exists a such that .

2.         State and prove the root test for convergence of the series .

3.         Let f be a continuous real function on the interval . If , and c is a number satisfying show that there exist a point such that .

4.         If is a refinement of p, then prove that .

5.         Find the values of .

6.         Show that is an analytic function.

7.         Prove that every totally bounded set is bounded.

8.         State and prove Schwartz's lemma.

PART B (5 10 = 50 marks)

Answer any FIVE questions.

9.         In any metric space X , prove that every convergent sequence is a Cauchy sequence. If X is a compact metric space, prove that every Cauchy sequence converges to some point of X.

10.       Show that converges if , and diverges if .

11.       Show that a mapping f of a metric space X into a metric space Y is continuous on X iff is open in X for every open set V in Y.

12.       State and prove the fundamental theorem of calculus.

13.       If u and v have first-order partial derivatives that satisfy the Cauchy-Riemann equations, then show that is analytic.

14.       If is continuous, and X is compact, prove that f is uniformly continuous.

15.       State and prove Cauchys theorem for a rectangle.

16.       Let be the zeros of a function that is analytic in a disk and , each zero being counted as many times as its order indicates. For every closed curve r which does not pass through a zero, prove that .