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.
- (a) State and prove Rolle’s theorem.
(b) Justify Rolle’s theorem for the function in the interval.
- 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.
- (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
- (a) Evaluate (any two) (i) , (ii) ,
