Complex Analysis Preliminary Exam
June 4, 1999
1. Let f be a complex-valued harmonic function in a domain Ω ⊂ C. Prove that if
|f | = const in Ω, then f = const.
2. Let f be a holomorphic function in the unit disk which is continuous up to the bound-
ary of the disk T = ∂∆ = {z ∈ C : |z| = 1}. Prove that if |f(z)| = 1 for all z ∈ T,
then f is a rational function.
3. Let f be an entire function such that Re(f(z)) ≤ 0 for all z ∈ C. Prove that
f = const.
4. For each real t compute the integral ϕ(t) =
∫ ∞
−∞
eitx
1 + x2
dx.
5. Construct a conformal mapping of the unit disk onto the crescent
{z ∈ C : |z| < 1,
∣∣∣∣∣z − 12
∣∣∣∣∣ ≥ 12} .
6. How many complex solutions does the equation
z = cos z
have? Justify your answer.
Hint. Use the following fact:
If an entire function F (z) has no zeros and satisfies
|F (z)| ≤ C1eC2|z| (z ∈ C)
then F (z) = eaz+b.
7. Let f be a bounded analytic function in the right half-plane. Prove that if
f(n) = 0 for n = 1, 2, 3, . . . ,
then f ≡ 0.
8. Let f1, f2 be entire functions, and let J be the set of all combinations
A1f1 + A2f2,
where A1 and A2 are entire functions. Show that there exists an entire function f
such that J consist of all entire functions Af , where A is entire.
Hint: Use the result of Problem #9.
9. Let {an} be a sequence in C,
lim
n→∞
an = ∞. Prove that for any sequence {bn} of
complex numbers there exists an entire function f such that f(an) = bn.