Let $U$ be an open set, $\gamma :[t_0,t_1]\to \mathbb{C}$ be a closed curve, and $z_0=0$ be a point not on the curve (otherwise just translate the curve)

Take a subdivision $t_0={\tau }_0<{\tau }_1<\cdots <{\tau }_n=t_1$ such that for any $k=1,\dots ,m$ we have that the segment $\gamma ([{\tau }_k,{\tau }_{k+1}])$ is contained in an open halfplane through $0$. We require this so that 1;;we have a well defined angle between the rays $[0,\gamma ({\tau }_k)]$ and $[0,\gamma ({\tau }_{k+1})]$. Find ${\theta }_k\in (-\pi ,\pi )$ satisfying: $\frac{\gamma ({\tau }_k)}{|\gamma ({\tau }_k)|}=e^{i{\theta }_k}\frac{\gamma ({\tau }_{k+1})}{|\gamma ({\tau }_{k+1})|}$

Finally, define the winding number: