Complex Analysis Exam
Part of the phd exam in analysis
2002 Nov 20
Do as many problems as you can.

Let $(X, d)$ be a metric topological space and $C \subset X$ be a closed subspace. Define a function $f\colon X \to \mathbb R$ by $f(a) = \inf\{d(x, y) : y\in C\}$.
(a) Show that $f(x)f(y) \leq d(x,y)$ for all $x, y \in X$.
(b) Suppose that $K \subset X$ is compact and disjoint from $C$. Prove that there is an $\epsilon >0$ so that $d(x,y) \geq \epsilon >0$ for all $x \in K$, $y\in C$. 
Define the sequence $\{c_n\}$ as follows: $$c_n = \left\{ \begin{array}{ll}(1)^n/4^{n+1}, & n = 0, 1, \dots,\\ (1)^n n, & n = 1, 2, \dots. \end{array}\right.$$
(a) Find the largest annulus $A(r_1, r_2) = \{z : r_1 < z < r_2\}$ so that the Laurent series $\sum_{n=\infty}^\infty c_n z^n$ converges to a holomorphic function on $A(r_1, r_2)$.
(b) Express this holomorphic function as a rational function of $z$. 
(a) State the Riemann Mapping Theorem and explain the meaning of the geometric statement.
(b) Find a holomorphic bijection between the unit disc $U$ and the semidisc $G = \{z \in U : \mathrm{Im}\ z > 0\}$. 
Suppose that $f(z)$ is an entire function and that $\mathrm{Im}\ f(z)= 0$ for all $0\leq x \leq 1$. Prove that $\mathrm{Im}\ f(x)= 0$ for all real $x$.

Suppose that $G\subset \mathbb C$ is a domain and that $f_n$ is a holomorphie function on $G$ for each $n=1,2,\dots$. Suppose further that $f(z) = \lim_{n\to \infty} f_n(z)$ exists for each $z\in G$ and that the convergence in uniform on every compact subset of $G$.
(a) Prove that $f$ is holomorphic on $G$.
(b) Prove that $f_n' \longrightarrow f'$ uniformly on compact subsets of $G$.
(c) Suppose that instead of uniform convergence on every compact subset of $G$ we only know that the sequence $\{f_n\}$ is bounded on every compact subset of $G$. Prove that $f$ is holomorphic on $G$. 
Let $G$ be the domain consisting of the entire complex plane with the integers removed, i.e., $G = \mathbb C \setminus \mathbb Z$. Prove that a bounded holomorphic function on $G$ must be constant.