Complex Analysis/Chain

A chain is a formal linear combination ofTrace of Curve, we have

Definition - Chain

Let $G\subseteq ℂ$, let $n\in \mathbb{N}$, and let ${\gamma }_i:[a_i,b_i]\to G$ be curves in $G$ and $n_i\in \mathbb{Z}$. Then the formal linear combination $\sum _{i=1}^n{n}_i{\gamma }_i$ is called a chain in $ℂ$. The set of all chains in $G$, which is naturally an abelian group, is denoted by $C(G)$.

Definition - Trace of a Chain

The trace of a chain $\mathrm{\Gamma }$ is the union of the traces of the individual curves ${\gamma }_i$, i.e.

$\mathrm{T}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(\mathrm{\Gamma }):=\bigcup _{i=1}^n\mathrm{T}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}({\gamma }_i)$

Cycle

A chain $\mathrm{\Gamma }=\sum _{i=1}^n{n}_i{\gamma }_i\in C(G)$ with ${\gamma }_i:[a_i,b_i]\to G$ is called a cycle if each point of $G$ occurs equally often as the starting and ending point of curves in $G$, i.e., if

$\sum _{i=1}^n{n}_i|\{i:{\gamma }_i(a_i)=z\}|=\sum _{i=1}^n{n}_i|\{i:{\gamma }_i(b_i)=z\}|$

holds for every $z\in G$.

Interior and Exterior Region

Let $\mathrm{\Gamma }$ be a cycle in $ℂ$, with the help of the winding number one can consider a decomposition of $ℂ$ into three parts determined by $\mathrm{\Gamma }$, namely:

• The image set of $\mathrm{T}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(\mathrm{\Gamma })$

• The exterior region, those points that are not traversed by $\mathrm{\Gamma }$, i.e. $A_{\mathrm{\Gamma }}:=z\in ℂ\setminus \mathrm{T}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(\mathrm{\Gamma }):n(\mathrm{\Gamma },z)=0$

• The interior region consists of those points that are traversed by $\mathrm{\Gamma }$, i.e. $I_{\mathrm{\Gamma }}:=z\in ℂ\setminus \mathrm{T}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(\mathrm{\Gamma }):n(\mathrm{\Gamma },z)\ne 0$

Page Information

You can display this page as Wiki2Reveal slides

Wiki2Reveal

TheWiki2Reveal slides were created for the Complex Analysis' and the Link for the Wiki2Reveal Slides was created with the link generator.

• This page is designed as a PanDocElectron-SLIDE document type.

Translation and Version Control

This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Kette Wikiversity source page] and uses the concept of Translation and Version Control for a transparent language fork in a Wikiversity:

• Source: Kurs:Funktionentheorie/Kette - URL:

https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Kette

• Date: 12/17/2024

de:Kurs:Funktionentheorie/Kette