Complex Analysis/Exercises/Sheet 5

Exercise on Complex Analysis

Problem (Integrals, 5 points)

Calculate the following integrals using the Cauchy integral formula:

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

Problem (Cauchy, 10 points)

Let $S^{1}:=\{z\in \mathbf {C} :|z|=1\}$. On $\mathbf {C} \setminus S^{1}$, consider the function

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

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

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

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