Complex Analysis General Exam - August 2019

Problem 1

Let $\xi$ be a nonnegative real number. Compute $\int_{-\infty}^{\infty} \frac{e^{i x \xi}}{x^{2}+1} d x$.

Problem 2

Let $f$ be an entire function. Suppose that there is an $\alpha \in(0, \infty)$ and a $C>0$ so that $|f(z)| \leq C|z|^{\alpha}$ for all $|z| \geq 1$. Show that $f$ is a polynomial.

Problem 3

Let $f$ be an entire function. Suppose that $\lim _{z \rightarrow \infty} f(z)=\infty$. Show that $f$ is a polynomial.

Problem 4

Let $\Omega=\{z \in \mathbb{C}: \operatorname{Re}(z)>0\}$. Suppose that $f: \bar{\Omega} \rightarrow \mathbb{C}$ is continuous, and that $\left.f\right|_{\Omega}$ is holomorphic. Suppose that $|f(i y)| \leq 1$ for all $y \in \mathbb{R}$ and $|f(z)| \leq 2$ for all $z \in \Omega$. Show that in fact $|f(z)| \leq 1$ for all $z \in \Omega$.

Hint: For $\varepsilon>0$, consider $f_{\varepsilon}(z)=\frac{f(z)}{1+\varepsilon z}$. Show that $\left|f_{\varepsilon}\right| \leq 1$ for every $\varepsilon>0$.

Problem 5

Recall that if $U$ is a open subset of $\mathbb{C}$ and $\mathcal{G}$ is family of holomorphic functions on $U$ then we say that $\mathcal{G}$ is normal, if given any sequence $\left(f_{n}\right)_{n}$ in $\mathcal{G}$ there is a subsequence $\left(f_{n_{k}}\right)_{k}$ and a holomorphic function $g: U \rightarrow \mathbb{C}$ with $f_{n_{k}} \rightarrow_{k \rightarrow \infty} g$ uniformly on compact subsets of $U$.

Let $\mathbb{D}=\{z \in \mathbb{C}:|z|<1\}$. Suppose that $\mathcal{F}$ is a family of holmorphic functions on $\mathbb{D}$ and that $\sup _{f \in \mathcal{F}}|f(0)|<\infty$. Show that $\mathcal{F}$ is normal if and only if $\left\{f^{\prime}: f \in \mathcal{F}\right\}$ is normal.