Complex Analysis - May 2023

Note: all statements require proofs. You can make references to standard theorems from the course; however, you need state the relevant part of the theorem in your own words, unless it’s a well known named theorem. For example, ”we had a theorem in the class that said that any continuous function on a compact subset of $\mathbb R^n$ is uniformly continuous” is a good reference, and ”by the uniqueness theorem from the class, $f$ is unique” is not a good reference.

Problem 1

Suppose that $f:\mathbb C\to\mathbb C$ is totally differentiable at $z_0$ and the limit

$$\lim_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0}$$

exists. Show that either $f$ or $\overline f$ is complex differentiable at $z_0$.

Problem 2

Let $\gamma:[0,1]\to\mathbb C\setminus\{i,-i\}$ be a continuous path with $\gamma(0)=0$, $\gamma(1)=1$. Show that

$$\int_\gamma\frac{dz}{1+z^2}\in\frac\pi4+2\pi i\mathbb Z.$$

Problem 3

For $R>0$, find a conformal isomorphism between

$$\mathbb C\setminus\big((-\infty,-R)\cup(R,+\infty)\big)$$

and the upper half plane $\mathbb C^+=\{z\in\mathbb C:\operatorname{Im}z>0\}$.

Problem 4

Calculate the following integral for all values of $N\in\mathbb N$:

$$\oint_{|z|=\pi N+1/2}\frac{dz}{\sin z},$$

where the circle $|z|=\pi N+1/2$ is parametrized counter-clockwise.

Problem 5

Show that for every $\rho\in(0,1)$ there exists $N=N(\rho)$ such that the function

$$f_n(z)=1+z+2z+3z^2+4z^3+\ldots$$

does not have any zeros in $\{z\in\mathbb C:|z|<\rho\}$ for $n>N(\rho)$.