PhD COMPREHENSIVE EXAM—COMPLEX ANALYSIS—August 1993

Notation: B = {z ∈ C: |z| < 1}, the open unit disc

1. Show that the real and imaginary parts of an analytic function are harmonic.

2. Show that the polynomial z^{5}+5z^{3}+z−2 has exactly three zeros (counting multiplicities) in B.

3. Let f be an entire function, and suppose there is a finite constant M so that

∫_{0}^{2π} |f(re^{iθ})| dθ ≤ M

for all r > 0. Show that f is constant by finding f^{(n)}(0) for all n ≥ 1.

4. Let F denote the family of functions f analytic on B so that f(0) = 0 and

|f′(z)| ≤ \frac{1}{1 − |z|}

for all z ∈ B. Show that F is a normal family.

5. a) Define, for |z| > 1,

f(z) = \frac{2}{z} + ∑_{n=2}^{∞} \frac{1}{z^{n}} .

Find an analytic continuation of f to the set D = {z: z ≠ 0 and z ≠ 1}; that is, find a function g analytic on D so that g(z) = f(z) if |z| > 1.

b) With notation as in a), how many such analytic continuations of f to D are there? Why?

c) If g is as in a), what is the residue of g at 0?

6. a) Does there exist a one-to-one analytic function f mapping the upper half plane H = {x + iy: −∞ < x < ∞ and y > 0} onto B? If so, can f be chosen to have an analytic continuation to a neighborhood of the closure of H in C? Explain your answers.

b) Repeat a) with B replaced by the square S = {x + iy: 0 < x < 1 and 0 < y < 1}.