Complex Analysis General Exam August 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 $0<\alpha<1$,

$$\int_{0}^{\infty} \frac{x^{\alpha}}{x^{2}+4} d x$$

You must show all estimates.

Problem 2

Suppose that $U \subseteq \mathbb{C}$ is open and $f: U \rightarrow \mathbb{C} \backslash\{0\}$ is holomorphic. Prove that $u(z)=\log |f(z)|$ is harmonic.

Problem 3

Let $\mathbb{H}=\{z \in \mathbb{C}: \operatorname{Im}(z)>0\}$. Suppose that $f: \overline{\mathbb{H}} \rightarrow \mathbb{C}$ is continuous and that $\left.f\right|_{\mathbb{H}}$ is holomorphic. Suppose that $f(\mathbb{R}) \subseteq \mathbb{R}$ and that $f(\mathbb{H}) \subseteq\{z \in \mathbb{C}: \operatorname{Im}(z) \geq 0, \operatorname{Re}(z) \geq 0\}$. Show that $f$ is constant.

Problem 4

Suppose that $f$ and $g$ are analytic in an open subset $U$ of $\mathbb{C}$ containing $\{z \in \mathbb{C}:|z| \leq 1\}$. Suppose that $f$ has a zero of order 1 at $z=0$, and no other zero in $U$.
(i) For $\zeta \in \mathbb{C}$, set $f_{\zeta}=f(z)+\zeta g(z)$. Show that if $|\zeta|$ is small enough (depending upon $g$ ), then $f_{\zeta}$ has no zeroes on $\partial \mathbb{D}$, has a unique zero in $\mathbb{D}$, and that the order of this zero is 1 .
(ii) Show that for all sufficiently small $|\zeta|$, if $z_{\zeta}$ is the unique zero of $f_{\zeta}$ in $\mathbb{D}$, then

$$z_{\zeta}=\frac{1}{2 \pi i} \int_{\partial \mathbb{D}} w \frac{f_{\zeta}^{\prime}(w)}{f_{\zeta}(w)} d w$$

Hint: evaluate the integral by the residue theorem.

Problem 5

Let $U$ be a connected open subset of $\mathbb{C}$. Suppose that $\left(f_{n}\right)_{n=1}^{\infty}$ is a sequence of holomorphic functions on $U$, and that there is a continuous $g: U \rightarrow[0,+\infty)$ with $\left|f_{n}(z)\right| \leq g(z)$ for all $n \in \mathbb{N}$ and all $z \in U$. Assume that $p \in U$ and that $r>0$ has $B_{r}(p) \subseteq U$ and that $\lim _{n \rightarrow \infty} f_{n}(z)=0$ for all $z \in B_{r}(p)$.
(i) Suppose that $h: U \rightarrow \mathbb{C}$, and that there is a subsequence $\left(f_{n_{k}}\right)_{k}$ of $\left(f_{n}\right)_{n}$ with $f_{n_{k}} \rightarrow_{k \rightarrow \infty} h$ uniformly on compact sets. Show that necessarily $h=0$.
(ii) Show that $f_{n} \rightarrow 0$ uniformly on compact subsets of $U$.