Winding number ($\mathrm{wind}(\gamma ,w)$)

For a closed piecewise smooth curve $\gamma$ avoiding $w$,It is the net change of a continuous argument divided by $2\pi$ and is an integer.

If $\gamma :[0,1]\to \mathbb{C}\setminus \{0\}$ is continuous and closed, choose a continuous lift $\theta$ such thatThen $\mathrm{wind}(\gamma ,0)=\theta (1)-\theta (0)\in \mathbb{Z}$. This agrees with the contour-integral definition when the path is piecewise smooth.