PhD COMPREHENSIVE EXAM—COMPLEX ANALYSIS—June 1998

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

1. Suppose that f, g, and \overline{f}g are all analytic in B. Show that either f is constant or g is identically 0.

2. (a) Suppose that f is analytic in B. Show that f = F′ for some F analytic in B. (b) Let B′ = B \ \{0\} = \{z ∈ C: 0 < |z| < 1\}. Define f(z) = \frac{1}{z}. Show that there does not exist a function F analytic in B′ with F′ = f.

(c) Explain the difference between B and B′ here, by filling in the blank in the following theorem: Suppose G is a connected open subset of the plane. Every analytic function in G is the derivative of an analytic function if and only if G is . (No proof required here, just a correct statement).

3. (a) Suppose that u is a (real-valued) harmonic function in B. Show that u is the real part of some function f analytic in B.

(b) Define B′ = B \ {0} as in the previous problem. Let u(z) = log(|z|). Show that u is not the real part of any function analytic in B′. (One way to do part (a) would be to show somehow that the function u_{x} − iu_{y} was analytic, and then use that fact somehow.)

4. Suppose that f is an entire function with period 1 and period i. (That is, f is analytic in the entire plane and f(z + 1) = f(z + i) = f(z) for all z.) Show that f is constant.

5. Suppose that f is analytic and one-to-one in a neighborhood of the closed disc \overline{B}. Let γ = ∂B, oriented counterclockwise. Show that the inverse function f^{−1} is given by the formula

f^{−1}(α) = \frac{1}{2πi} ∫_{γ} \frac{zf′(z)}{f(z) − α} dz

for any α ∈ f(B).

6. Let F denote the set of all f analytic on B such that

∫∫_{B} |f(x + iy)|^{2} dxdy ≤ 1.

Show that F is a normal family (that is, F is uniformly bounded on compact subsets of B). Hint: Fix p ∈ B. Show first that

|f(p)|^{2} ≤ \frac{1}{2π} ∫_{0}^{2π} |f(p + re^{it})|^{2} dt

for 0 < r < 1 − |p|. It follows from this (how?) that

|f(p)|^{2} ≤ \frac{1}{π(1 − |p|)^{2}} ∫∫_{|x+iy−p|<1−|p|} |f(x + iy)|^{2} dxdy,

and the result follows.