Complex Analysis/Exercises/Sheet 3

Task (Integrals, 5 Points)

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

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

Task (Quadratic Path, 5 Points)

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

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

given parameterization of the unit square.Calculate

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

Task (Estimation, 5 Points)

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

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

Task (Inverse, 5 Points)

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

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

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