Complex Analysis/chain

Definition - Chain

Let $G\subseteq ℂ$ be a region, $n\in \mathbb{N}$ 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 $G$ that form an abelian group in a natural way 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.

$\mathrm{S}\mathrm{p}\mathrm{u}\mathrm{r}(\mathrm{\Gamma }):=\bigcup _{i=1}^n\mathrm{S}\mathrm{p}\mathrm{u}\mathrm{r}({\gamma }_i)$

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 every point in

$G$

appears the same number of times 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

$z\in G$

.

Inner and outer regions

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

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

$A_{\mathrm{\Gamma }}:=\{z\in ℂ\setminus \mathrm{S}\mathrm{p}\mathrm{u}\mathrm{r}(\mathrm{\Gamma }):n(\mathrm{\Gamma },z)=0\}$

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

$I_{\mathrm{\Gamma }}:=\{z\in ℂ\setminus \mathrm{S}\mathrm{p}\mathrm{u}\mathrm{r}(\mathrm{\Gamma }):n(\mathrm{\Gamma },z)\ne 0\}$

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