Real & Complex Analysis Code –3701 M Sc. Preliminary Test Examination-2010

Govt. M M College, Jessore

Department of Mathematics

M Sc. Preliminary Test Examination-2010

Subject – Real & Complex Analysis Code –3701

     Time – 4 hours             Full Marks – 100

(Answer any six questions)

 

Group-A: (Real Analysis)

(Answer Any Four)

1(a) Define supremum and infimum of a set of real numbers, state and prove the supremum and infimum principles for R and show that they are equivalent.

(b) Prove that if , then there exists a positive integer n such that

. If A= then find sup A, inf A, max A and min A.

  • (a) Define open set and closed set of real number. Show that the union of two open

sets is open.

(b) Define limit point of a set of real number. State and prove the Bolzano–weierstrass theorem. If A ={(-1)n} then show that A has no limit point .

3 (a) Define countable set and uncountable set. Show that the interval [0, 1] is not countable.

(b) Define interior, exterior and boundary points of a set of real numbers. Prove that a subset is closed if it contains all of its limit points.

4.(a) Define continuity and differentiability of a function. Show that the function

is continuous at

(b) If then show that f(x) is not differentiable at x=0.

  1. (a) State and prove Rolle’s theorem.

(b) Justify Rolle’s theorem for the function in the interval.

  1. Justify the Mean-value theorem theorem for the function in the interval.

Group-B: (Complex Analysis)

(Answer Any Two)

7 (a) If be two complex numbers then prove that . (b) Sketch the region described by .

8.(a) If is analytic then find it interms of z when .

(b) State and prove Rouchy’s theorem.

  1. (a) State and prove Cauchy’s integral formula the derivative of an analytic function.

(b) Use Cauchy’s integral formula to evaluate
where c is the circle

  1. (a) Evaluate (any two) (i) , (ii) ,

 

 




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