COMPLEX ANALYSIS GENERAL EXAM AUGUST 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

Using Cauchy's residue theorem, compute the definite integral

$$\int_{0}^{2 \pi} \frac{1}{5+4 \cos \theta} d \theta$$

Problem 2

Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be an entire function satisfying $|f(2 z)| \leq 2|f(z)|$ for all $z \in \mathbb{C}$. Show that either $f(z)=f(0)$ for all $z \in \mathbb{C}$ or $f(z)=f^{\prime}(0) z$ for all $z \in \mathbb{C}$.

Problem 3

Show that the only solution inside the unit disc $\mathbb{D}$ to the equation $e^{z}=2 z+1$ is $z=0$. (You may use without proof the fact that $e \approx 2.718<3$.)

Problem 4

Let $U:=\{z \in \mathbb{C}: 0<\operatorname{Re}(z)<1\}$. Let $f: \bar{U} \rightarrow \mathbb{C}$ be a continuous bounded function for which $\left.f\right|_{U}$ is holomorphic. Suppose that there exist constants $M_{1}, M_{2} \geq 0$ such that

$$\sup _{\operatorname{Re}(z)=0}|f(z)| \leq M_{0}, \quad \sup _{\operatorname{Re}(z)=1}|f(z)| \leq M_{1} .$$

Show that for $r \in[0,1]$,

$$\sup _{\operatorname{Re}(z)=r}|f(z)| \leq M_{0}^{1-r} M_{1}^{r} .$$

(Hint: for fixed $\varepsilon>0$, show that $f_{\varepsilon}(z):=f(z) M_{0}^{z-1} M_{1}^{-z} e^{\varepsilon\left(z^{2}-1\right)}$ satisfies $\sup _{z \in \bar{U}}\left|f_{\varepsilon}(z)\right| \leq$ 1.)

Problem 5

Let $\mathcal{F}$ denote the set of holomorphic functions $f$ on the disc $B_{2}(0)=\{z \in \mathbb{C}:|z|<2\}$ that satisfy

$$\int_{0}^{2 \pi}\left|f\left(e^{i \theta}\right)\right| d \theta \leq 1$$

Let

$$\mathcal{G}:=\left\{g: \mathbb{D} \rightarrow \mathbb{C}: g=\left.f\right|_{\mathbb{D}} \text { for some } f \in \mathcal{F}\right\} .$$

Show that $\mathcal{G}$ is a normal family.