Complex Analysis/Exercises/Sheet 3

Exercise on Complex Analysis

Task (Integrals, 5 Points)

Let

$\gamma :[0,1]\to 𝐂$

,

$t\mapsto \exp (2\pi it)$

,be the standard parametrization of the unit circle. Determine for

$n\in 𝐙$

the Integral

$\underset{\gamma }{\int }{z}^{n}dz$

Task (Quadratic Path, 5 Points)

Let $\gamma :[0,4]\to 𝐂$ be parametrized by

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

given parameterization of the unit square.Calculate

$\underset{\gamma }{\int }{z}^{-1}dz$

Task (Estimation, 5 Points)

Let $f:[a,b]\to 𝐂$ be integrable. Show that

$|\underset{a}{\overset{b}{\int }}\mathrm{R}\mathrm{e}f(t)dt|\le |\underset{a}{\overset{b}{\int }}f(t)dt|.$

Task (Inverse, 5 Points)

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

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

Translation and Version Control

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• Source: Kurs:Funktionentheorie/Übungen/3._Zettel - URL:

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

• Date: 01/14/2024

de:Kurs:Funktionentheorie/Übungen/3._Zettel