Winding number

The winding number of a closed contour $C$ around a point $z_0\notin C$ is the integer number of times $C$ wraps around $z_0$ counterclockwise. Notation: $n(C,z_0)$.

Computed by

$n(C,z_0)=\frac{1}{2\pi i}{\oint }_C\frac{dz}{z-z_0}.$

This integral is always an integer for a closed contour not passing through $z_0$.

Why integer

Parameterize $z-z_0=\rho (t)e^{i\theta (t)}$ along $C$. Then $dz/(z-z_0)=d\rho /\rho +id\theta$. The real part $d\rho /\rho$ is exact: $\rho$ is continuous and $\oint d(\ln \rho )=0$. The imaginary part contributes $i\oint d\theta$. For a closed contour, $\theta$ returns to its starting value modulo $2\pi$, so $\oint d\theta =2\pi n$ for some integer $n$. Dividing by $2\pi i$ gives $n(C,z_0)=n$ — the net number of counterclockwise wraps.

Examples

• $C=$ unit circle counterclockwise, $z_0=0$: $n=1$.
• Same $C$, $z_0=2$ (outside): $n=0$.
• $C$ traverses the unit circle twice counterclockwise: $n=2$.
• $C$ traverses the unit circle clockwise: $n=-1$.
• $C$ is a figure-eight around two points $z_0=1$ and $z_0=-1$: depending on how the loops are oriented, $n=1,-1,0,$ etc., at each point.

Application to contour integrals

For a function with an isolated singularity at $z_0$:

${\oint }_C\frac{dz}{z-z_0}=2\pi in(C,z_0).$

So contour integrals “count” how many times the path encircles each singularity, weighted by orientation.

The general residue theorem in full form:

${\oint }_{C}f(z)dz=2\pi i{\sum }_{k}n(C,z_k)\mathrm{Res}(f,z_k).$

For the simple closed contours that come up in practice (and that Vector Calculus and Complex Analysis emphasizes), all winding numbers are $0$ or $\pm 1$, so this reduces to $2\pi i\sum \mathrm{Res}$ over enclosed singularities with the natural sign convention.

In context

The winding number formalizes “how many times a path encircles a singularity,” which is the natural way to think about Residue theorem applications. For most calculations the winding number is just $\pm 1$ or $0$ and gets absorbed into the orientation convention, but the concept is essential when contours are non-trivial.

It’s also a foundational concept in algebraic topology: the fundamental group of $\mathbb{C}\setminus \{0\}$ is $\mathbb{Z}$, generated by loops around the missing point with the winding number as the homomorphism. This is the same topological obstruction that makes the Complex logarithm multi-valued and the 2D rotational field non-conservative.