Winding number

Definition

Let $\gamma : [a, b] \to \mathbb{C}$ be a closed path that is continuous and piecewise continuously differentiable. Let $w \in \mathbb{C}$ be a point that is not on the image of $\gamma$. The winding number of $\gamma$ around $w$ is $$n(\gamma, w) = \frac{1}{2 \pi i} \oint_\gamma \frac{dz}{z - w},$$ where the integral is a contour integral.

Remarks

The number $n(\gamma, w)$ is an integer. It counts how many times $\gamma$ goes around $w$ counterclockwise, minus the times it goes around $w$ clockwise. As a function of $w$, it is constant on each connected component of the complement of the image of $\gamma$. It is $0$ on the unbounded component. For the circle $|z - c| = R$, traveled once counterclockwise, $n = 1$ for $|w - c| < R$ and $n = 0$ for $|w - c| > R$, by the integral of 1/(z - w).

Explanations

L1 human

Stand at the point $w$ and watch a point that moves along the closed curve. The winding number is the number of full turns that you make. A circle around you gives $1$, the same circle traveled twice gives $2$, and a circle that does not go around you gives $0$.