Preliminary Examination Complex Analysis January 1998

1. Suppose that f is an entire function and f(C) ∩ {w : Re w = 0} = ∅. Prove that f is constant.

2. Evaluate ∫_{0}^{π} \frac{dθ}{2 + cos θ}.

3. Give an example of a function f which is holomorphic in C\{z_{0}} for some z_{0} ≠ 0, has an essential singularity at z_{0} and is continuous in {z : |z| ≤ |z_{0}|}. Show that the function given actually has these properties.

4. Find the maximum value of |g(z)| if g(z) = \frac{z}{4z^{2} − 1} and z varies over the region {z : |z| ≥ 1}.

5. A. State carefully the Riemann Mapping Theorem.

B. Let D = {z : |z| < 1}, Ω = {z : Re z > 0} and fix α ∈ Ω. Find all conformal maps g from Ω onto D such that g(α) = 0.

6. Suppose that f is a holomorphic function in an open disk D, f is continuous on \overline{D} and |f| is constant and nonzero on ∂D. Prove that f is a rational function.

7. Let P be a nonzero polynomial. Suppose that ∫_{|z|=r} \frac{1}{P (z)}dz ≠ 0 whenever r > 0 and the integral is defined. Show that deg P = 1.

8. Suppose that f is an entire function, and for r > 0 let M_{f}(r) = sup{|f(z)| : |z| ≤ r}.

Assume that 0 < α < 1 and let

L(α) = lim_{r→∞} \frac{M_{f}(αr)}{M_{f}(r)} .

(a) Determine L(α) in the case f is a polynomial.

(b) Show that L(α) = 0 if f is not a polynomial.