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).

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Complex numbers object
Contour integral definition
Winding number definition

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Tag	`0X7SBS8E0F`
URI	`m2://0X7SBS8E0F`
Aliases	`index-of-a-curve`
Kind	definition
Writing	W1 stated
MSC	55M25
Defines	Winding number
Uses	Contour integral

Winding number

Definition

Remarks

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