COMPLEX ANALYSIS GENERAL EXAM SPRING 2022

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 $0<\alpha<1$

$$\int_{0}^{\infty} \frac{1}{x^{\alpha}(1+x)} d x$$

Show all estimates.

Problem 2

Suppose that $f$ is an entire function, and that there are constants $a, b>0$ so that

$$|f(z)| \leq a+b|z| \text { for all } z \in \mathbb{C}$$

Show that $f$ is a polynomial of degree at most one.

Problem 3

Give an explicit example of an unbounded harmonic function $u: \mathbb{D} \rightarrow(0,+\infty)$ with the property that

$$\lim _{z \rightarrow \zeta} u(z)=0$$

for all $\zeta \in \partial \mathbb{D}$ with $\zeta \neq 1$. It is acceptable to leave your answer as the real (or imaginary) part of an explicit holomorphic function.

Problem 4

Let $S=\{z \in \mathbb{C}: 0<\operatorname{Re}(z)<1\}$. Suppose that $f: \bar{S} \rightarrow \mathbb{C}$ is bounded, continuous, and that $\left.f\right|_{S}$ is analytic. If

$$\sup _{t \in \mathbb{R}} \max (|f(i t)|,|f(1+i t)|) \leq 1,$$

show that $|f(z)| \leq 1$ for all $z \in S$.
Suggestion: consider, for $\varepsilon>0$, the function $f_{\varepsilon}(z)=\frac{f(z)}{1+\varepsilon z}$. Show that $\left|f_{\varepsilon}\right| \leq 1$ for all $\varepsilon>0$, and use this to conclude that $|f| \leq 1$.

Problem 5

Let $\left(M_{n}\right)_{n=0}^{\infty}$ be a sequence of positive real numbers. Assume that the series $\sum_{n=0}^{\infty} M_{n} z^{n}$ has radius of convergence at least 1 . Let $\mathcal{F}$ be the set of holomorphic functions on $\mathbb{D}$ which satisfy

$$\left|\frac{f^{(n)}(0)}{n!}\right| \leq M_{n} \text { for all } n \in \mathbb{N} \cup\{0\} .$$

Show that $\mathcal{F}$ is normal (i.e. the closure of $\mathcal{F}$ is compact for the topology of uniform convergence on compact subsets of $\mathbb{D}$ ).