The winding number vanishes on the unbounded component of the complement of the trace

Given: A closed complex contour $\gamma :[a,b]\to \mathbb{C}$.

For a compact $K\subseteq \mathbb{C}$, the complement $\mathbb{C}\setminus K$ has exactly one unbounded connected component $U_{\mathrm{\infty }}$, every other component is bounded, and $\{z:\mathrm{\mid }z\mathrm{\mid }>R\}\subseteq U_{\mathrm{\infty }}$ whenever $R>0$ satisfies $K\subseteq \{z:\mathrm{\mid }z\mathrm{\mid }\le R\}$ (The complement of a compact plane set has exactly one unbounded connected component).

The index $n(\gamma ,\cdot )$ is constant on every connected component of $\mathbb{C}\setminus {\gamma }^{\ast }$ (The winding number is constant on each connected component of the complement of the trace).

The winding number of a closed complex contour about a point off its trace is an integer (The winding number of a closed contour is an integer).

If $\mathrm{\mid }f(z)\mathrm{\mid }\le M$ on the trace of a rectifiable contour $\gamma$, with $M\ge 0$, then $\mathrm{\mid }{\int }_{\gamma }f(z)dz\mathrm{\mid }\le ML(\gamma )$ (ML estimate: a contour integral is bounded by a supremum bound times path length).

$n(\gamma ,p)=(2\pi i{)}^{-1}{\int }_{\gamma }dz\mathrm{/}(z-p)$ (The winding number of a closed contour about a point off its trace).

A compact subset of a metric space is closed and bounded (A compact subset of a metric space is closed and bounded); the continuous image of a compact subset is compact (The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset); a closed bounded interval is compact (Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

The connected component $C(x)$ is the union of all connected subsets containing $x$ (Connected components, quasicomponents, and totally disconnected spaces) and contains every connected subset containing $x$ (The components of a space are its maximal connected subsets, they partition it, and each of them is closed).

A subset of a metric space is bounded when it is empty or contained in some ball (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space).

$\mathrm{\mid }z+w\mathrm{\mid }\le \mathrm{\mid }z\mathrm{\mid }+\mathrm{\mid }w\mathrm{\mid }$ for complex $z,w$ (Conjugation is an involutive real-field automorphism, $z\bar{z}=\mathrm{\mid }z{\mathrm{\mid }}^2$, and modulus is definite, multiplicative, and subadditive).

The integers form an ordered commutative ring and their canonical image in $\mathbb{R}$ is discrete; in particular the only integer of modulus below $1$ is $0$ (The integers form a commutative ring, The integers form a totally ordered ring, Integer part: for every real $x$ there is exactly one integer $m$ with $m\le x<m+1$).