PhD COMPREHENSIVE EXAM—COMPLEX ANALYSIS—January 1998

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

1. (a) State Cauchy’s Theorem (in the most general form you know).

(b) Give an example of an open set Ω ⊂ C, a function f analytic on Ω, and a smooth closed curve γ in Ω such that ∫_{γ} f(z) dz ≠ 0. Explain why this example does not contradict Cauchy’s Theorem.

2. 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.

3. Let D be a connected open set in C and f a function analytic on D.

a) Assuming that f has an analytic logarithm in D, show that f has analytic mth roots for all m. Explicitly: Assume that there exists h analytic on D so that f = e^{h}, and show that for each m ∈ N there exists a function g_{m} analytic on D so that f = (g_{m})^{m}.

b) Assume now that f ≢ 0 and f has analytic mth roots for all m. Show that f(z) ≠ 0 for all z ∈ D. Hint: Look at multiplicities.

4. Suppose f: B → B is analytic, |f(z)| → 1 as |z| → 1, and f has exactly one zero in B (counting multiplicities). Show that f is one-to-one and onto. Hint: For a given λ ∈ B, show that f and f − λ have the same number of zeros in B.

5. 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.

6. Let E be a set with no limit point in C. Show that there exists a meromorphic function f on C so that f has a pole of order two at each point of E and f has no zero in C.