The winding number of a closed contour about a point off its trace

Let $\gamma :[a,b]\to \mathbb{C}$ be a closed complex contour, that is a rectifiable path with $\gamma (a)=\gamma (b)$ (Rectifiable complex contours, reversal, concatenation, closedness, and orientation), with trace ${\gamma }^{\ast }$, and let $p\in \mathbb{C}$ with $p\mathrm{\notin }{\gamma }^{\ast }$. The winding number, or index, of $\gamma$ about $p$ is

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

the complex line integral of The complex line integral over a rectifiable path as a componentwise Riemann–Stieltjes integral.

The integral exists: $z\mapsto 1\mathrm{/}(z-p)$ is complex differentiable, hence continuous, on $\mathbb{C}\setminus \{p\}$ by Linearity, product, reciprocal, and quotient rules for complex derivatives and Complex differentiability at a point implies continuity there, the trace ${\gamma }^{\ast }$ is contained in that set, and $\gamma$ is rectifiable, so Continuous integrands have complex and absolute line integrals along every rectifiable path applies.

The index is attached to the parametrised contour and not to its trace. Two closed contours with the same trace can have different indices about the same point, because the parametrisation records how many times, and in which direction, the trace is traversed; the definition above reads $\gamma$ as a map and the integral depends on that map.

The point $p$ is required to lie off the trace. On the trace the integrand $1\mathrm{/}(z-p)$ is undefined at $z=p$, so no value $n(\gamma ,p)$ is defined there and none is asserted anywhere below.

No connectedness is assumed of the set $\mathbb{C}\setminus {\gamma }^{\ast }$ where the index lives; when a complex domain (A complex domain is a nonempty connected open subset of $\mathbb{C}$) is wanted it is said so explicitly.