Complex Analysis/Chain

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

Definition - Chain

Let $G\subseteq \mathbb {C}$, let $n\in \mathbb {N}$, and let $\gamma _{i}\colon [a_{i},b_{i}]\to G$ 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 $\mathbb {C}$. 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 $\Gamma$ is the union of the traces of the individual curves $\gamma _{i}$, i.e.

$\mathrm {Trace} (\Gamma ):=\bigcup _{i=1}^{n}\mathrm {Trace} (\gamma _{i})$

Cycle

A chain $\Gamma =\sum _{i=1}^{n}n_{i}\gamma _{i}\in C(G)$ with $\gamma _{i}\colon [a_{i},b_{i}]\to G$ 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 $\Gamma$ be a cycle in $\mathbb {C}$, with the help of the winding number one can consider a decomposition of $\mathbb {C}$ into three parts determined by $\Gamma$, namely:

• The image set of $\mathrm {Trace} (\Gamma )$

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

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

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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