COMPLEX ANALYSIS GENERAL EXAM JANUARY 2025

Solve as many problems as you can. Full solutions on a smaller number of problems will be worth more than partial solutions on several problems. Don’t use any of the Picard theorems. Throughout, $\mathbb{D}$ denotes the unit disc $\{z \in \mathbb{C}:|z|<1\}$.

Problem 1

Let $f: \overline{\mathbb{D}} \rightarrow \mathbb{C}$ be a continuous function for which $\left.f\right|_{\mathbb{D}}$ is holomorphic and $|f(z)|=1$ for all $z \in \partial \mathbb{D}$. Show that that there exists a finite collection of points $z_{1}, \ldots, z_{n} \in \mathbb{D}$ and some $\theta \in \mathbb{R}$ such that

$$f(z)=e^{i \theta} \prod_{j=1}^{n} \frac{z_{j}-z}{1-\overline{z_{j}} z}$$

Problem 2

Let $f: \mathbb{C} \backslash\{2,3\} \rightarrow \mathbb{C}$ be the holomorphic function

$$f(z):=\frac{z}{(z-2)(z-3)}$$

For each of the following regions, determine the Laurent series of $f$ that converges on the given region.
(a) $\{z \in \mathbb{C}:|z|<2\}$
(b) $\{z \in \mathbb{C}: 2<|z|<3\}$
(c) $\{z \in \mathbb{C}:|z|>3\}$

Problem 3

Let $u: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a harmonic function, and suppose that $u(x, y) \leq x$ for all $(x, y) \in \mathbb{R}^{2}$. Show that there exists a constant $c \in(-\infty, 0]$ such that $u(x, y)=x+c$ for all $(x, y) \in \mathbb{R}^{2}$.

Problem 4

Using Cauchy's residue theorem, show that for $a, b \in \mathbb{R}$ with $0<b<a$,

$$\int_{0}^{2 \pi} \frac{1}{(a+b \sin \theta)^{2}} d \theta=\frac{2 \pi a}{\left(a^{2}-b^{2}\right)^{3 / 2}}$$

Problem 5

(a) Let $f: \mathbb{D} \rightarrow \mathbb{D}$ be a holomorphic function satisfying $f(0)=\frac{2}{3}$. Show that $\left|f^{\prime}(0)\right| \leq \frac{5}{9}$.
(b) Determine all holomorphic functions $f: \mathbb{D} \rightarrow \mathbb{D}$ that simultaneously satisfy both $f(0)=\frac{2}{3}$ and $\left|f^{\prime}(0)\right|=\frac{5}{9}$.