COMPLEX ANALYSIS PRELIM – AUGUST 2006
1. Suppose fis a nonconstant entire function such that f◦f(z)=f(z) for all z. Prove
that fmust be the identity function.
2. Suppose fis entire, f(0) = 0 and
|f(z)|≤e1/|z|
for all z6= 0. Prove that fis identically 0.
3. Suppose for each nthat fnis a bounded continuous real-valued function on the unit
circle {z:|z|=1}. Suppose for each nthat unis a function that is continuous on the closed
unit disk {z:|z|≤1}, is harmonic in the open unit disk {z:|z|<1}, and agrees with fn
on the unit circle. Show that {fn}is an equicontinuous family on the unit circle if and only
if {un}is an equicontinuous family on the closed unit disk.
4. Use residues to evaluate the definite integral
Z∞
−∞
x2
(x2+1)
2dx.
5. Let D={z=x+iy :0<y<1,x > 0}. Find a conformal mapping of Donto the
open unit disk.
6. Suppose that for each nthe function fnis analytic in the open unit disk, |fn(0)|≤1,
and for each r<1 satisfies Z|z|=r
|fn(z)|2|dz|≤1.
Show that every subsequence of {fn}has a further subsequence which converges to a finite
analytic function uniformly on each compact subset of the open unit disk.
7. Suppose for each nthe function fnis analytic on the open unit disk Dand has exactly
one zero in D. Suppose the sequence {fn}converges to funiformly on each compact subset
of the unit disk.
(a) Show that either fis identically zero on Dor else has at most one zero in D.
(b) Give an example of a sequence {fn}where the limit function has no zeros in D.
Date: August 7, 2006.