Integration over a complex chain and the index of a chain

Let $\mathrm{\Gamma }={\sum }_{k<r}{m}_k{\gamma }_k$ be a complex chain (Complex chains, their traces, and cycles) with trace ${\mathrm{\Gamma }}^{\ast }$, and let $f$ be continuous on ${\mathrm{\Gamma }}^{\ast }$. The integral of $f$ over $\mathrm{\Gamma }$ is

${\int }_{\mathrm{\Gamma }}f(z)dz:={\sum }_{\begin{array}{c}k<r\\ m_k\mathrm{\ne }0\end{array}}{m}_k{\int }_{{\gamma }_k}f(z)dz,$

a finite sum (A finite sum in a commutative monoid indexed by an arbitrary finite set) of the complex line integrals of The complex line integral over a rectifiable path as a componentwise Riemann–Stieltjes integral. Each summand exists: for $k$ with $m_k\mathrm{\ne }0$ the trace ${\gamma }_k^{\ast }$ is contained in ${\mathrm{\Gamma }}^{\ast }$, so $f$ is continuous on it, and ${\gamma }_k$ is rectifiable, so Continuous integrands have complex and absolute line integrals along every rectifiable path applies. Terms with $m_k=0$ are omitted, so no integral of $f$ over a contour outside the trace is required. The empty chain, and any chain all of whose coefficients vanish, give ${\int }_{\mathrm{\Gamma }}fdz=0$.

For $p\in \mathbb{C}\setminus {\mathrm{\Gamma }}^{\ast }$, the index of $\mathrm{\Gamma }$ about $p$ is

$n(\mathrm{\Gamma },p):=\frac{1}{2\pi i}{\int }_{\mathrm{\Gamma }}\frac{dz}{z-p}\mathrm{.}$

This is defined: $z\mapsto 1\mathrm{/}(z-p)$ is complex differentiable, hence continuous, on $\mathbb{C}\setminus \{p\}\supseteq {\mathrm{\Gamma }}^{\ast }$ by Linearity, product, reciprocal, and quotient rules for complex derivatives and Complex differentiability at a point implies continuity there.

The notation is consistent with the single-contour case. If $r=1$, $m_0=1$ and ${\gamma }_0$ is closed, then ${\mathrm{\Gamma }}^{\ast }={\gamma }_0^{\ast }$, the sum has the one term ${\int }_{{\gamma }_0}fdz$, and $n(\mathrm{\Gamma },p)$ is the winding number $n({\gamma }_0,p)$ of The winding number of a closed contour about a point off its trace for every $p\mathrm{\notin }{\gamma }_0^{\ast }$. So writing $n$ for both costs no ambiguity.

Linearity in the integrand is inherited termwise from Complex line integrals are linear in the integrand, finite sums in the additive commutative monoid of $\mathbb{C}$ (A finite sum in a commutative monoid indexed by an arbitrary finite set), and distributivity in the complex field ($\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)\mathrm{/}(a^2+b^2)$): for $f,g$ continuous on ${\mathrm{\Gamma }}^{\ast }$ and $\alpha ,\beta \in \mathbb{C}$, ${\int }_{\mathrm{\Gamma }}(\alpha f+\beta g)dz=\alpha {\int }_{\mathrm{\Gamma }}fdz+\beta {\int }_{\mathrm{\Gamma }}gdz$.

The index is not defined on the trace. For $p\in {\mathrm{\Gamma }}^{\ast }$ the integrand is undefined at $z=p$, and no value is assigned; every statement about $n(\mathrm{\Gamma },\cdot )$ below carries the hypothesis $p\mathrm{\notin }{\mathrm{\Gamma }}^{\ast }$.