PhD COMPREHENSIVE EXAM—COMPLEX ANALYSIS—August 2000

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

1. 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.

2. (a) Prove that if a, b, c and d are real numbers satisfying ad − bc = 1, then

M(z) = \frac{az + b}{cz + d}

maps the upper half-plane H = {x + iy: y > 0} conformally onto itself.

(b) Let f: B → C be such that the imaginary part of f is non-negative for all z in B. Assume that f is non-constant and is analytic in B. Show that

(i) the imaginary part of f is strictly positive for all z in B; (ii) if f(0) = i, then

|f(z)| ≤ \frac{1 + |z|}{1 − |z|}.

Hint: Use the Schwarz Lemma.

3. (a) Suppose f is an entire function such that f^{2} is a polynomial. Does it follow that f is a polynomial? (b) Suppose f is analytic on B and f^{2} is a polynomial on B. Does it follow that f is a polynomial on B? (Of course, you have to explain your answers).

4. Use residue theory to calculate with justification that

∫_{0}^{∞} \frac{x sin x}{x^{2} + 4} dx = \frac{π}{2e^{2}} .

5. (a) Let G denote an open set in the complex plane containing \overline{B}. Suppose f is analytic on G and is non-zero on ∂B. Show that, with the counterclockwise orientation for ∂B,

\frac{1}{2πi} ∫_{∂B} \frac{f′(z)}{f(z)} dz = N(f, B),

where N(f, B) counts the number of zeros of f in B with multiplicity.

(b) Suppose f_{n} is a sequence of functions analytic on B such that f_{n} → f uniformly on B. Suppose f has exactly R zeros in the set D = \{z: |z| < \frac{1}{2}\} and no zeros on ∂D. Show that there exists an N > 0 such that, for all n > N, the function f_{n} has exactly R zeros in D.

6. By definition, an automorphism of a connected open set G is a one-to-one analytic function mapping G onto G.

(a) Let f and g be automorphisms of B. Suppose there are two distinct points P and Q in B such that f(P) = g(P) and f(Q) = g(Q). Show that f = g on B.

(b) Let D be a simply connected domain in C which is not equal to C, and let f and g be automorphisms of D. Suppose there are two distinct points P and Q in D such that f(P) = g(P) and f(Q) = g(Q). Show that f = g on D.