PhD COMPREHENSIVE EXAM—COMPLEX ANALYSIS—August 1997

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

1. Let f be a function defined on an open set G in C.

a) Define in terms of difference quotients what it means for f to be analytic in G.

b) Give an equivalent definition of the analyticity of f in terms of power series.

c) In this part we take G = B. Suppose that ϕ is a continuous function on ∂B and define f on B by

f(z) = ∫_{γ} \frac{ϕ(w)}{w − z} dw ,

where γ(t) = e^{it} on [0, 2π]. Use the definition in part b) to show that f is analytic in B.

2. a) Show that the real and imaginary parts of an analytic function are harmonic. b) If u and v are real-valued harmonic functions, does it follow that u + iv is analytic? Explain your answer.

3. Fix a complex number λ so that λ^{3} = 1 but λ ≠ 1. Let f be an entire function so that f(λz) = f(z) for all z ∈ C. Prove using series that there exists an entire function g so that f(z) = g(z^{3}) for all z ∈ C.

4. Suppose f is analytic on a neighborhood of \overline{B} and |f(z)| ≤ 1 when |z| = 1. In parts a) and b) give the best upper bound you can for the given quantity.

a) |f(\frac{1}{2})|

b) |f′(\frac{1}{2})|

c) Do part a) again, but make the additional assumption that f(0) = 0.

5. Let G = {x + iy: x ∈ R, y > x^{2}}.

a) Show that for each n ∈ N there exists a one-to-one analytic map f_{n} of G onto B so that f_{n}(i) = 1 − \frac{1}{n}.

b) Does it follow that a sequence {f_{n}} as in part a) has a subsequence converging normally on G (i.e., uniformly on compact subsets of G)? If it does, what can you say about the limit function of such a subsequence? If it does not, explain why not.

6. a) State the Weierstrass theorem characterizing the zeros of entire functions. b) Show that every meromorphic function on C can be written as the quotient of two entire functions.