Complex analysis qualifying exam, January 2010.
1. Give the statements of the following theorems:
(a) Montel’s theorem;
(b) The Weierstrass factorization theorem.
2. Suppose that a function fis holomorphic in {0<|z−a|< r}for some r > 0, a ∈Cand
that f′/f has a pole of order one at a. Prove that then fhas a pole or a zero at a.
3. Prove that all zeros of the function tanz−zare real.
4. Let {fn}be a sequence of holomorphic functions in a complex domain Ω. Suppose that
fn(a) converges for some a∈Ω and that the functions ℜfnconverge normally in Ω. Prove
that then fnconverge normally in Ω.
5. Let f1, f2, ..., fnbe holomorphic in a bounded complex domain Ω and continuous in the
closure of Ω. Let g=|f1|+|f2|+... +|fn|.
a) Prove that the maximum of gis attained on the boundary of Ω.
b) Prove that if g≡const in Ω then all fkare constants.
6. Let fbe a function holomorphic in the unit disk Dand continuous in the closure ¯
D.
a) Show that if ℜf= 0 on ∂D then fis a constant.
b) Show that the previous statement becomes false if ∂Dis replaced with a proper subarc
of ∂D.
7. Let entire functions fand gsatisfy ef+eg≡1. Prove that then both are constants.
8. Find a general formula for all functions w(z) that map the domain Ω = {|z|<1}\ [1/2,1]
conformally onto the domain {|ℑz|<1}.
9. Let ube a real-valued harmonic function in C\ {0}. Show that then
u(z) = clog |z|+ℜf(z)
for some real constant cand a function fholomorphic in C\ {0}.
10. Prove that for a function f:C→ˆ
Cit holds that
resz=af=−resz=−af
if fis even and that
resz=af=resz=−af
if fis odd. We assume that all the residues are correctly defined, i.e. fis holomorphic in a
punctured neighborhood of a.
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