Winding Number (Contour)

Denoted $n(\gamma,z_0)$ and defined as the number of times a path $\gamma$ curve passes around a point.

$$n(\gamma,a) = {1\over 2\pi i}\int_\gamma {dz\over z-a}.$$

The contour winding number was part of the inspiration for the idea of the Degree of a Map between two Compact, oriented Manifolds of the same Dimension. In the language of the Degree of a Map, if $\gamma: [0,1] \to \Bbb{C}$ is a closed curve (i.e., $\gamma(0) = \gamma(1)$), then it can be considered as a Function from $\Bbb{S}^1$ to $\Bbb{C}$. In that context, the winding number of $\gamma$ around a point $p$ in $\Bbb{C}$ is given by the degree of the Map

$${\gamma-p\over \vert\gamma-p\vert}$$

from the Circle to the Circle.