Math 5120
Preliminary Exam in Complex Analysis
August 2010
1. Characterize all those analytic functions defined in the unit disc ∆ with the property
that, for all a, b ∈∆, f(ab) = f(a)f(b).
2. Prove that there does not exist a 1-1 analytic function mapping an annulus onto a punc-
tured disc.
3. Evaluate Z|z|=1
z11
12z12 −4z9+ 2z6−4z3+ 1 dz and justify all steps. Hint: one of the
ways to approach this problem is to make the change of variable w=1
z.
4. Suppose the sequence {fn}of 1-1 analytic functions converges uniformly on compact sub-
sets of a region Ω to a function f. Show that fis analytic, and is either constant or is
also 1-1.
5. Let Ω be a bounded, simply connected domain in the plane. Suppose g: Ω →Ω is
holomorphic and not the identity. Show that gcan have at most one fixed point.
(a) First show it when Ω is the unit disc. Then
(b) Show it when Ω is a bounded, simply connected region in the plane.
6. Evaluate the integral Z∞
0
log x
x2+a2dx where ais real and positive.
7. Prove that if fis a non-constant entire function then f(C) is dense in C. (You cannot
just quote Picard’s Theorem.)
