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 $\mathbb {C} \setminus \{z_{0}\}$.