Complex Analysis 24 | Winding Number [dark version]

TL;DR

Winding number converts the geometric idea of “net turns around a point” into an integer computed by a contour integral.

Briefing Cornell Notes

Briefing

Winding number turns “how many times a curve loops around a point” into a precise integer you can compute with a contour integral. For a point z0 in the complex plane that does not lie on the curve, the winding number counts the net number of counterclockwise turns the curve makes around z0—so a single counterclockwise circle gives 1, two counterclockwise turns give 2, and a curve that doesn’t enclose the point yields 0. The key payoff is that this counting can be done algebraically, not just by visual inspection.

The definition is built from the classic contour integral of 1/z. For a circle centered at the origin, parameterized by γ(t)=e^{it} over t∈[0,2π], the integral ∮γ (1/z) dz evaluates to 2πi. Extending the parameter interval to [0,4π] makes the curve wind twice, and the integral scales to 4πi. More generally, if a curve lies entirely in a region that avoids the origin, Cauchy’s theorem implies ∮γ (1/z) dz = 0, matching the intuition that there are no turns around the origin.

To count turns around an arbitrary point z0, the integrand is shifted: the winding number is defined as

wind(γ, z0) = (1/(2πi)) ∮γ 1/(z − z0) dz,

with the only restriction that z0 must not lie on the image of γ (otherwise the integral—and the notion of winding—breaks down). While the definition works for general piecewise continuously differentiable curves, the most important case is closed curves, where the winding number becomes an integer.

The integer property is proved by rewriting a closed curve (after translating so the point of interest is at the origin) in polar form. Each point on the curve can be expressed as γ(t)=ρ(t)e^{iΦ(t)}, where ρ(t) and Φ(t) are real-valued, piecewise continuously differentiable functions. Substituting this into the integral ∮γ (1/z) dz and applying the product rule causes cancellations that reduce the integral to two simpler real integrals: one involving the derivative of log ρ(t) and another involving Φ′(t). The radius term contributes nothing over a closed loop because ρ(B)=ρ(A). The angle term contributes the net change in Φ over the loop, which must be a multiple of 2π, say 2πk, since the curve returns to the same point even though the polar angle may differ by full rotations.

After dividing by 2πi, the winding number equals k, an integer. That result matters because it converts geometric looping into a robust analytic invariant, setting up later contour-integral theorems where winding number will determine how many singularities a curve effectively “encircles.”

Cornell Notes

Winding number measures how many net counterclockwise turns a curve makes around a point z0, provided z0 is not on the curve. It is defined using a contour integral: wind(γ, z0) = (1/(2πi)) ∮γ 1/(z − z0) dz. For circles around the origin, the integral gives 2πi per full turn, so the winding number matches the number of rotations. For closed curves, the winding number is always an integer. The proof rewrites the curve in polar form γ(t)=ρ(t)e^{iΦ(t)}, reduces the integral to terms involving log ρ(t) and Φ(t), and uses the fact that returning to the same point forces the angle change to be a multiple of 2π, yielding an integer k.

Why does ∮γ (1/z) dz equal 2πi for a single counterclockwise circle around the origin?

For the circle γ(t)=e^{it} with t from 0 to 2π, the curve winds once around 0. The integral ∮γ (1/z) dz evaluates to 2πi, matching the geometric fact that the argument increases by 2π over one full rotation. Extending the parameter interval to 0 to 4π makes the argument increase by 4π, and the integral scales to 4πi, corresponding to two turns.

How does Cauchy’s theorem connect to winding number being 0 when the point is outside the curve?

If the curve stays in a region where the origin is not inside (so the integrand 1/z has no singularity on the region enclosed by the curve), Cauchy’s theorem gives ∮γ (1/z) dz = 0. Since the winding number is defined as (1/(2πi)) times this integral, the result becomes 0—consistent with the idea that the curve makes no net turns around the point.

What is the exact definition of wind(γ, z0), and what restriction is required?

The winding number around z0 is defined by wind(γ, z0) = (1/(2πi)) ∮γ 1/(z − z0) dz. The point z0 must not lie on the curve’s image (z0 ∉ γ([A,B])), because otherwise the integrand 1/(z − z0) becomes singular along the path and the winding notion loses meaning.

Why must the winding number of a closed curve be an integer?

After translating so z0 is at 0, write the closed curve as γ(t)=ρ(t)e^{iΦ(t)}. Substituting into ∮γ (1/z) dz and using the product rule reduces the integral to contributions from (d/dt)log ρ(t) and Φ′(t). Closure forces ρ(B)=ρ(A), so the log term cancels out. The angle term contributes Φ(B)−Φ(A), and because the curve returns to the same point, Φ(B)−Φ(A) must be 2πk for some integer k. Dividing by 2πi yields wind(γ,0)=k.

What role does the polar angle’s non-uniqueness (multiples of 2π) play in the proof?

The polar representation γ(t)=ρ(t)e^{iΦ(t)} is not unique because adding 2πk to Φ leaves e^{iΦ} unchanged. When the curve is closed, the endpoint in the plane matches, but Φ may differ by full rotations. That difference must be 2πk, and that k becomes the winding number.

Review Questions

How does shifting from 1/z to 1/(z − z0) change the geometric meaning of the contour integral?
In the polar-form proof, which term vanishes due to closure, and which term produces the integer multiple of 2π?
What condition on z0 relative to the curve is necessary for the winding number definition to make sense?

Key Points

1
Winding number converts the geometric idea of “net turns around a point” into an integer computed by a contour integral.
2
For a point z0 not on the curve, wind(γ, z0) is defined as (1/(2πi))∮γ 1/(z − z0) dz.
3
A circle around the origin gives wind = 1 per full counterclockwise rotation, because ∮ (1/z) dz = 2πi.
4
If the curve avoids the point so that the integrand has no enclosed singularity, Cauchy’s theorem forces the winding number to be 0.
5
For closed curves, wind(γ, z0) is always an integer because the polar angle change over a closed loop must be 2πk.
6
The proof relies on expressing γ(t)=ρ(t)e^{iΦ(t)} and showing the radius contribution cancels while the angle contribution yields k.

Highlights

The winding number is defined by wind(γ, z0) = (1/(2πi))∮γ 1/(z − z0) dz, turning “turn counting” into a computable invariant.

A single counterclockwise loop around the origin produces ∮γ (1/z) dz = 2πi, so the winding number equals 1.

For closed curves, the net change in polar angle over the loop must be 2πk, forcing the winding number to be an integer k.