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 boundary of the disk T = ∂Δ = {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) = ∫_{−∞}^{∞} \frac{e^{itx}}{1 + x^{2}} dx.

5. Construct a conformal mapping of the unit disk onto the crescent

\{z ∈ C : |z| < 1, |z − \frac{1}{2}| ≥ \frac{1}{2}\} .

6. How many complex solutions does the equation

z = cos z

have? Justify your answer. \underline{Hint}. Use the following fact: If an entire function F (z) has no zeros and satisfies

|F (z)| ≤ C_{1}e^{C_{2}|z|} (z ∈ C)

then F (z) = e^{az+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 f_{1}, f_{2} be entire functions, and let J be the set of all combinations

A_{1}f_{1} + A_{2}f_{2},

where A_{1} and A_{2} are entire functions. Show that there exists an entire function f such that J consist of all entire functions Af, where A is entire. \underline{Hint}: Use the result of Problem #9.

9. Let {a_{n}} be a sequence in C, lim_{n→∞} a_{n} = ∞. Prove that for any sequence {b_{n}} of complex numbers there exists an entire function f such that f(a_{n}) = b_{n}.