Complex Analysis/Exercises/Sheet 3

Exercise on Complex Analysis

Task (Integrals, 5 Points)

Let$\gamma \colon [0,1]\to \mathbf {C}$,$t\mapsto \exp(2\pi it)$ ,be the standard parametrization of the unit circle. Determine for $n\in \mathbf {Z}$ the Integral

$\int _{\gamma }z^{n}\,dz$

Task (Quadratic Path, 5 Points)

Let $\gamma \colon [0,4]\to \mathbf {C}$ be parametrized by

$t\mapsto {\begin{cases}-(1+i)+2t&t\in [0,1]\\1-i+2i(t-1)&t\in [1,2]\\1+i-2(t-2)&t\in [2,3]\\-1+i-2i(t-3)&t\in [3,4]\end{cases}}$

given parameterization of the unit square.Calculate

$\int _{\gamma }z^{-1}\,dz$

Task (Estimation, 5 Points)

Let $f\colon [a,b]\to \mathbf {C}$ be integrable. Show that

$\left|\int _{a}^{b}\mathop {\rm {Re}} f(t)\,dt\right|\leq \left|\int _{a}^{b}f(t)\,dt\right|.$

Task (Inverse, 5 Points)

Let $\gamma \colon [a,b]\to \mathbf {C}$ be an integration path and $f\colon \mathbf {C} \to \mathbf {C}$ be continuous. Define $\gamma ^{-}\colon [a,b]\to \mathbf {C}$ durch $\gamma ^{-}(t):=\gamma (a+b-t)$. Man zeige

$\int _{\gamma ^{-}}f(z)\,dz=-\int _{\gamma }f(z)\,dz$

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/3._Zettel - URL:

https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Übungen/3._Zettel

• Date: 01/14/2024