Complex analysis qualifying exam, January 2012.
1. Give the statements of the following:
(a) The Mittag-Leffler theorem;
(b) Harnack’s lemma.
2. Consider an infinite product
πz
∞
Y
n=1 1−z2
n2.
(a) Prove that the product converges normally in C.
(b) Find the elementary entire function that the product converges to (prove your answer).
(c) Use the previous parts to prove Wallis’ formula:
π
2=2
1·2
3·4
3·4
5·6
5·...
3. Let F={fa}a∈Abe a family of functions holomorphic in a neighborhood of the closed unit disk
¯
D={|z| ≤ 1}. Suppose that
Z2π
0
|fa(eiφ)|1/2dφ ≤1
for any a∈A. Prove that Fis a normal family in the unit disk D={|z|<1}.
4. Let γbe a closed curve in the right half-plane that has index Nwith respect to the point 1. Find
Zγ
e
1
z2
−1sin(πz)dz.
5. Let Ω 6=Cbe a simply-connected complex domain containing a point c. Let φ: Ω →Dbe a conformal
mapping such that φ(c) = 0. The function gc(z) = log |φ(z)|is called the Green function of Ω corresponding
to c. Prove that ga(b) = gb(a) for any a,b ∈Ω.
6. Write a formula for a conformal map from the upper half-plane to {z| <z > 0,|=z|<1}.
7. Let Fbe an entire function such that
|F(z)| ≤ e|z|λ
for some λ > 0 and large enough |z|. Let F(z) = P∞
0anznfor all z∈C. Prove that then
|an| ≤ eλ
nn/λ
for large enough n.
8. Let Fbe a function holomorphic and bounded in the upper half-plane C+. Suppose that Fhas period 1
(F(z+ 1) = F(z) for all z∈C+). Prove that F(z) has a finite limit as =z→+∞.
9. Let u(z) be a bounded harmonic function in Dsuch that the limit
lim
r→1−u(reiφ)
is equal to 1 for 0 < φ < π and to 0 for π < φ < 2π. Find u(1/2).
10. Let Fbe an entire function. We say that a∈C∪ {∞} is an asymptotic value for Fif there exists a
continuous curve going from a finite point to infinity such that Ftends to aalong that curve. Prove that
for any non-constant entire function ∞is an asymptotic value.
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