Complex Analysis/chain

Definition - Chain

Let ${\displaystyle G\subseteq \mathbb {C} }$ be a region, ${\displaystyle n\in \mathbb {N} }$ and let ${\displaystyle \gamma _{i}\colon [a_{i},b_{i}]\to G}$ be curves in ${\displaystyle G}$ and ${\displaystyle n_{i}\in \mathbb {Z} }$. Then the formal linear combination ${\displaystyle \sum _{i=1}^{n}n_{i}\gamma _{i}}$ is called a chain in ${\displaystyle \mathbb {C} }$. The set of all chains in ${\displaystyle G}$ that form an abelian group in a natural way is denoted by ${\displaystyle C(G)}$.

Definition - Trace of a chain

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

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

Cycle

A chain ${\displaystyle \Gamma =\sum _{i=1}^{n}n_{i}\gamma _{i}\in C(G)}$ with ${\displaystyle \gamma _{i}\colon [a_{i},b_{i}]\to G}$ is called a cycle if every point in ${\displaystyle G}$ appears the same number of times as the starting and ending point of curves in ${\displaystyle G}$, i.e. if

${\displaystyle \sum _{i=1}^{n}n_{i}|\{i:\gamma _{i}(a_{i})=z\}|=\sum _{i=1}^{n}n_{i}|\{i:\gamma _{i}(b_{i})=z\}|}$

holds for every ${\displaystyle z\in G}$.

Inner and outer regions

Let ${\displaystyle \Gamma }$ be a cycle in ${\displaystyle \mathbb {C} }$, using theWinding number, we can consider a decomposition of ${\displaystyle \mathbb {C} }$ determined by ${\displaystyle \Gamma }$ into three parts, namely:

• The image of the trace of ${\displaystyle \Gamma }$
• The outer region, the points that are not traversed by ${\displaystyle \Gamma }$, i.e.

${\displaystyle A_{\Gamma }:=\{z\in \mathbb {C} \setminus \mathrm {Spur} (\Gamma ):n(\Gamma ,z)=0\}}$

• The inner region are the points that are traversed by ${\displaystyle \Gamma }$, i.e.

${\displaystyle I_{\Gamma }:=\{z\in \mathbb {C} \setminus \mathrm {Spur} (\Gamma ):n(\Gamma ,z)\neq 0\}}$

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