WJD Open Notebook

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. $D(z_0, R) = \{z \in \mathbb C : |z-z_0| < R\}$, $R>0$. For an open set $G\subseteq \mathbb C$, $H(G)$ will denote the set of functions that are analytic in $G$.

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

2. (a) State the Casorati-Weierstrass Theorem.
   (b) Evaluate the integral $$I = \frac{1}{2\pi i}\int_{|z|=R} (z-3)\sin \left( \frac{1}{z+2}\right)\ dz,\quad \text{ where } R\geq 4.$$

3. 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 $z\in \mathbb C$.

4. 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 $C$.

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

6. (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.

7. 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.

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