Complex Analysis Exam
Part of the phd exam in analysis
2004 April 19
Instructions. Use a separate sheet of paper for each new problem. Do as many problems as you can. Complete solutions to five problems will be considered as an excellent performance. Be advised that a few complete and well written solutions will count more than several partial solutions.
Notation. $D(z_0, R) = \{z \in \mathbb C : zz_0 < R\}$, $R>0$. For an open set $G\subseteq \mathbb C$, $H(G)$ will denote the set of functions that are analytic in $G$.

Let $\gamma$ be a rectifiable curve and let $\phi \in C(\gamma^\ast)$. (That is, $\phi$ is a continuous complexvalued function defined on the trace, $\gamma^\ast$, of $\gamma$.) $$\text{Let } \ F(z) = \int_\gamma \frac{\phi(w)}{wz}\ dw,\ z \in \mathbb C  \gamma^\ast. \quad \text{ Prove that } \ F'(z) = \int_\gamma \frac{\phi(w)}{(wz)^2}\ dw,\ z \in \mathbb C  \gamma^\ast,$$ without using Leibniz's Rule.

(a) State the CasoratiWeierstrass Theorem.
(b) Evaluate the integral $$I = \frac{1}{2\pi i}\int_{z=R} (z3)\sin \left( \frac{1}{z+2}\right)\ dz,\quad \text{ where } R\geq 4.$$ 
Let $f(z)$ be an entire function such that $f(0) = 1$, $f'(0)=0$ and $0< f(z) \leq e^{z},$ for all $z\in \mathbb C$. Prove that $f(z) = 1$ for all $z\in \mathbb C$.

Let $C$ be an arbitrary circle through $1$ and $1$. Suppose that $z_1$ and $z_2$ are two points that satisfy $z_1z_2=1$ and do not lie on the circle $C$. Show that one of these points lies inside $C$ and the other lies outside $C$.

Show that there is no onetoone analytic function that maps $G = \{z : 0 < z < 1\}$ onto the annulus $\Omega = \{z : r < z < R \}$, where $r > 0$.

(a) State a theorem that gives a sufficient condition for a family $\mathcal F$ of analytic functions to be normal in a domain $G$.
(b) Let $\mathcal F \subseteq H(D)$ be a family of analytic functions in the open unit disk $D=D(0,1)$. Let $\{M_n\}$ be a sequence of positive real numbers such that $\limsup_{n\to \infty} (M_n)^{1/2} < 1$. If for each $f(z) = \sum_{n=0}^\infty a_n z^n \in \mathcal F$, $a_n \leq M_n$ for all $n$, prove that $\mathcal F$ is a normal family. 
Is there a harmonic function $u(z)$ defined on the open unit disk, $D(0, 1)$, such that $u(z_n)\to \infty$ whenever $z_n \to 1^$? Prove your answer.

Let $G$ be a simply connected domain with at least $2$ boundary points. Let $S = \{\psi \in H(G)\ \ \psi \colon G \to D(0, 1),\ \psi \text{ is onetoone}\}$. Prove, without using the Riemann Mapping Theorem, that the set $S$ is nonempty.