Complex Analysis/Exercises/Sheet 4

Exercise on Complex Analysis

Problem (Integrals, 5 points)

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

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

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 $f:𝐂\to 𝐂$ such that $f^{\prime }=\mathrm{R}\mathrm{e}$.

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

$\underset{{\gamma }_R}{\int }\frac{1}{1+z^2}dz$

Problem (Length, 5 points)

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

Translation and Version Control

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• Source: Kurs:Funktionentheorie/Übungen/4._Zettel - URL:

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

• Date: 01/14/2024

de:Kurs:Funktionentheorie/Übungen/4._Zettel