Complex Analysis General Exam January 2024

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. Throughout $\mathbb{D}=\{z \in \mathbb{C}:|z|<1\}$. Don't use any of the Picard theorems.

Problem 1

Compute, for $b, \xi>0$ and $a \in \mathbb{R}$

$$\lim _{R \rightarrow \infty} \int_{-R}^{R} \frac{x e^{-i x \xi}}{(x-a)^{2}+b} d x$$

Show all estimates.

Problem 2

Define $g: \mathbb{C} \backslash\{1\} \rightarrow \mathbb{C}$ by $g(z)=\frac{z+1}{z-1}$, and let $f: \mathbb{C} \backslash\{1\} \rightarrow \mathbb{C}$ be given by $f(z)=e^{g(z)}$.
(i) Give an explicit description of $g(\mathbb{D})$.
(ii) Show that $f$ is bounded on $\mathbb{D}$.

Problem 3

Fix an integer $k \in \mathbb{N}$. Suppose that $a_{0}, \cdots, a_{k} \in \mathbb{C}$, set $p(z)=\sum_{j=0}^{k} a_{j} z^{j}$ and assume that $p(z)$ has no zeroes on $\partial \mathbb{D}$. Prove that there is an $\varepsilon>0$ so that if $b_{0}, \cdots, b_{k} \in \mathbb{C}$ with $\left(\sum_{j=0}^{k}\left|b_{j}-a_{j}\right|^{2}\right)^{1 / 2}<\varepsilon$, then setting $q(z)=\sum_{j=0}^{k} b_{j} z^{j}$ we have that $p, q$ have the same number of zeroes in $\mathbb{D}$, counted with multiplicity.

Problem 4

Let $A=\{z \in \mathbb{C}: 0<|z|<1\}$ and $B=\{z \in \mathbb{C}: 1<|z|<2\}$,
(i) Suppose that $\Omega \subseteq \mathbb{C}$ is a bounded open set and that $\phi: A \rightarrow \Omega$ is a holomorphic bijection. Show that $\phi$ extends to a holomorphic function $\widetilde{\phi}: \mathbb{D} \rightarrow \bar{\Omega}$ and that $\widetilde{\phi}(0) \in \partial \Omega$. (Suggestion: it may be helfpul to prove that if $z_{n}$ is a sequence in $A$ and $z_{n} \rightarrow 0$ then for every compact $K \subseteq \Omega$ we have that $\left\{n: \phi\left(z_{n}\right) \in K\right\}$ is finite).
(ii) Show that there is no holomorphic bijection from $A$ to $B$.

Problem 5

For $p \in \mathbb{C}$ and $s>0$, we use $B_{s}(p)=\{z \in \mathbb{C}:|z-p|<s\}$.
(i) Let $U \subseteq \mathbb{C}$ be open, $p \in U$ and $s>0$ with $\overline{B_{s}(p)} \subseteq U$. Show that if $f: U \rightarrow \mathbb{C}$ is holomoprhic, then

$$f(p)=\frac{1}{\pi s^{2}} \iint_{B_{s}(p)} f(x+i y) d x d y$$

(Hint: compute the integral in polar coordinates).
(ii) Let $\mathcal{F}$ be the family of all holomorphic functions $f: \mathbb{D} \rightarrow \mathbb{C}$ with

$$\sup _{0<s<1} \iint_{B_{s}(0)}|f(x+i y)| d x d y \leq 1$$

Show that $\mathcal{F}$ is normal meaning that if $\left(f_{n}\right)_{n=1}^{\infty}$ is a sequence in $\mathcal{F}$, then there is a subsequence $\left(f_{n_{k}}\right)_{k}$ in $\mathcal{F}$ which converges uniformly on compact subsets of $\mathbb{D}$.