Complex Analysis/Exercises/Sheet 5

Problem (Integrals, 5 points)

Calculate the following integrals using the Cauchy integral formula:

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

Problem (Cauchy, 10 points)

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

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

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

Problem (More Integrals, 5 points)

Use the Cauchy integral formula to determine

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

Hint: Consider ${\displaystyle z\mapsto {\frac {e^{kz}}{z}}}$ 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