Some complex analysis prelim questions
January 21, 2009
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 fnon 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.
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