Winding number of cycle around point

Definition

Definition in terms of complex integrals

Let $c$ be a cycle (a sum of loops) in $\mathbb{C}$ and $z_0\in \mathbb{C}$ be a point not lying on any of the loops comprising $c$. Then, the winding number of $c$ about $z_0$, denoted $n(c;z_0)$, is defined by:

$n(c;z_0):=\frac{1}{2\pi i}\int \frac{dz}{z-z_0}$

Definition in terms of homology classes

Let $c$ be a cycle (a sum of loops) in $\mathbb{C}$, and $z_0\in \mathbb{C}$ be a point not lying on any of the loos comprising $c$. The winding number of $c$ about $z_0$ is the homology class of $c$ viewed as a cycle in $\mathbb{C}\setminus \{z_0\}$.