Complex analysis qualifying exam, January 2009.
1. Give the statements of the following theorems:
(a) Runge’s theorem;
(b) the Mittag-Leffler theorem.
2. Let f(z) be analytic in {|Re z|<1}and continuous on the closure of that domain.
Suppose that f(z) is real on the lines x=±1. Prove that then f(z) can be analytically
continued to the whole plane and that the resulting entire function satisfies F(z+ 4) = F(z)
for all z∈C.
3. Let uand vbe non-constant harmonic functions on a complex domain. Prove that uv is
harmonic if and only if u+icv is analytic for some real c. (Hint: one of the possible ways to
prove the ”only if” part is to consider f/g with f=ux−iuy, g =vx−ivy.)
4. Prove that for any a∈Cand any integer n≥2 the polynomial 1 + z+aznhas at least
one root in the disk {|z| ≤ 2}. (Hint: use the Vieta theorem that says that the product of
the roots of a monic polynomial is equal to its constant term in absolute value.)
5. Let fbe an analytic function in the unit disk Dsatisfying 0 <|f(z)|<1. Show that
then for any z∈D
|f(z)| ≤ |f(0)|
1−|z|
1+|z|.
(Hint: estimate log |f|.)
6. Calculate the integral using residues:
Z∞
0
dx
(x2+ 4)x1/3.
7. (a) Let fbe a non-constant holomorphic function on a neighborhood of the closed unit
disk such that |f(z)|is constant on the unit circle. Prove that fhas at least one zero in the
unit disk.
(b) Find all entire fsuch that |f|is constant on the unit circle.
8. Let f(z) be analytic in the strip {|Re z|< π/4}and satisfy f(0) = 0,|f(z)|<1. Prove
that then |f(z)| ≤ | tan z|for all zfrom the strip.
9. Let fbe a holomorphic function in the unit disk Dthat is injective and satisfies f(0) = 0.
Prove that there exists a holomorphic function gin Dsuch that (g(z))2=f(z2) for all z∈D.
10. Let fbe analytic in a bounded connected domain Ω and continuous in the closure of Ω.
Suppose that the boundary of Ω consists of two disjoint smooth simple closed curves γ1and
γ2. Prove that fhas an analytic antiderivative in Ω if and only if Rγ1f(z)dz = 0 (note that
Rγ1f(z)dz =Rγ2f(z)dz).
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