UNIVERSITY OF MALTA UNIVERSITY OF MALTA

FACULTY OF SCIENCE

Department of Mathematics

B.Sc. Part II FINAL EXAMINATION

June Session 1995


MA223 Complex Analysis 8 June 1995

0900 - 1130 hrs


Answer THREE questions

1. Suppose g is a triangular contour in \Bbb C, and let the complex function f be analytic inside and on g. Prove that
ó
õ

g 
 f = 0.
[You may assume, if required, that the intersection of an infinite nested sequence of compact subsets of \Bbb C contains exactly one point.]

Indicate briefly how this result can be used to show that if f is analytic on a convex set U then the integral of f round any closed contour in U is equal to zero.



2. Suppose f:\Bbb C®\Bbb C is analytic for all z in \Bbb C and that there exists a constant M such that |f(z)| £ M for all z Î \Bbb C. Let a, b Î \Bbb C, a ¹ b and let R be a real number such that a and b lie inside the circle |z| = R. By resolving 1/(z-a)(z-b) into partial fractions and applying Cauchy's Integral Formula evaluate

I = ó
õ

g 
 f(z)
(z-a)(z-b)
 dz,
where g is the circle centre the origin and radius R.

Find the limit of I as R ® ¥. Deduce that f is constant.

[Liouville's Theorem should not be used in this problem.]



3. State and prove Cauchy's Integral Formula.

Let f:U®\Bbb C be analytic on the open region U Ì \Bbb C. Let z0 Î U and let R be a positive real number such that the closed disc [`(D)] with centre z0 and radius R is included in U. Prove that, in D, the function f can be written as a power series centre z0.



4. (a) Prove that, for all finite z except 0,

cosh(z+ 1
z
 ) = c0+ ¥
å
 n = 1 
 cn(zn+ 1
zn
 ),
where
cn = 1
2p
 ó
õ 		2p

0 
 cosnqcosh(2cosq)dq.

(b) Classify the singularities of the function

f(z) = (z-2)
z2
 sin( 1
1-z
 )
and find the residue at each one of them.



5. (a) The only singularities in [`(\Bbb C)] (that is, including the point at infinity) are simple poles at z = 1 and z = 2, with residues at these poles equal to -3 and 7, respectively. If f(0) = 1/2, determine the function f.

(b) Evaluate, by means of contour integration, the integral

ó
õ 		¥

0 
 cosx
x2+1
 dx.

File translated from TEX by TTH, version 2.00.
On 23 Dec 1999, 14:26.