Complex Analysis/Exercises/Sheet 5

Exercise on Complex Analysis

Problem (Integrals, 5 points)

Calculate the following integrals using the Cauchy integral formula:

1. $\underset{|z+1|=1}{\int }\frac{dz}{(z+1)(z-1{)}^3}$
2. $\underset{|z|=2}{\int }\frac{\sin z}{z+i}dz$
3. $\underset{|z|=r}{\int }\frac{dz}{(z-a{)}^n(z-b{)}^m}$ mit $|a|<r<|b|$, $n,m\ge 1$.

Problem (Cauchy, 10 points)

Let

$S^1:=\{z\in 𝐂:|z|=1\}$

. On

$𝐂\setminus S^1$

, consider the function

$f(z):=\underset{S^1}{\int }\frac{d\zeta }{\zeta (\zeta -z)}$

Determine $f$. At which points of $S^1$ does $f$ have a limit?

Problem (More Integrals, 5 points)

Use the Cauchy integral formula to determine

$\underset{0}{\overset{2\pi }{\int }}{e}^{k\sin t}\sin (k\sin t)dt$

Hint: Consider

$z\mapsto \frac{e^{kz}}{z}$

on the unit circle.

Translation and Version Control

This page was translated based on the following Wikiversity source page and uses the concept of Translation and Version Control for a transparent language fork in a Wikiversity:

• Source: Kurs:Funktionentheorie/Übungen/5._Zettel - URL:

https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Übungen/5._Zettel

• Date: 01/14/2024

de:Kurs:Funktionentheorie/Übungen/5._Zettel