Complex analysis qualifying exam, August 2009.
1. Give the statements of the following theorems:
(a) Jensen’s formula;
(b) Harnack’s principle.
2. A continuous function on the real line is called analytic if it can be continuously and analytically extended
to the upper half-plane. Similarly, a continuous function on the line is called antianalytic if it can be
continuously and analytically extended to the lower half-plane. Prove that fis analytic if and only if ¯
fis
antianalytic.
3. Give an example of a real-valued harmonic function in the unit disk Dthat cannot be represented as a
difference of two non-negative harmonic functions in D.
4. Prove that the equation
az3−z+b=e−z(z+ 2),
where a > 0 and b > 2, has two roots in the right half-plane <z≥0.
5. Let fbe an analytic function in the disk {|z|< R}satisfying |f(z)|< M. Suppose that f(z0) = 0 for
some z0,|z0|< R. Show that then
|f(z)| ≤ MR|z−z0|
|R2−z¯z0|
for any z, |z|< R, and
|f0(z0)| ≤ MR
R2− |z0|2.
6. Let fbe an entire function. Let a, b ∈R, a < b. Prove that the residue of the function
f(z) log z−b
z−a
at infinity is equal to
Zb
a
f(x)dx.
Here log z−b
z−adenotes any branch analytic in a neighborhood of infinity.
7. Let f(z) be a function analytic in a domain containing the segment [0,1] and satisfying
f(z+ 1) = azf(z) + p(z)
in that domain, where a∈Rand pis a polynomial. Show that fcan be analytically continued to a domain
{|=z|< ε}for some ε > 0.
8. Find the function w(z) that maps the domain Ω = {=z > 0} \ [0, i] conformally onto the unit disk and
satisfies
w5i
4= 0, w(i) = −i.
9. Let fbe analytic in D\ {0}. Suppose that 0 is an essential singularity of f. Denote
M(r) = max
|z|=r|f(z)|.
Prove that for any p > 0
lim
r→0+ rpM(r) = ∞.
10. Find a general formula for an entire function of finite order that has no zeros.
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