Complex Analysis/Exercises/Sheet 4

Exercise on Complex Analysis

Problem (Integrals, 5 points)

Let $\gamma \colon [0,1]\to \mathbf {C}$, $t\mapsto \exp(2\pi it)$ denote the standard parametrization of the unit circle. Determine the integral

$\int _{\gamma }{\frac {1}{z+{\frac {1}{2}}}}\,dz$

Hint: Use the geometric series to express ${\frac {1}{z+{\frac {1}{2}}}}$ in the form $\sum _{n}a_{n}z^{-n}$. Then integrate term by term, using the first problem from the previous exercise sheet.

Problem (Antiderivatives, 5 points)

Prove that there is no function $f\colon \mathbf {C} \to \mathbf {C}$ such that $f'=\mathop {\rm {Re}}$.

Problem (More Integrals, 5 points)

Let $R>1$, and let $\gamma _{R}$ denote the semicircle consisting of $[-R,R]$, followed by $t\mapsto R\exp(it),t\in [0,\pi ]$. Determine

$\int _{\gamma _{R}}{\frac {1}{1+z^{2}}}\,dz$

Problem (Length, 5 points)

Determine the length of the unit circle, $\gamma \colon [0,1]\to \mathbf {C}$, $t\mapsto \exp(2\pi it)$.

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/4._Zettel - URL:

https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Übungen/4._Zettel

• Date: 01/14/2024