Winding numbers and contour topology

Try

A circle can go around a point twice even though its geometric trace looks like a circle drawn once. The parametrization γ(t)=exp(2it), for 0≤t≤2π, records two counterclockwise turns. Its reciprocal integral is 4πi rather than 2πi.

Winding number keeps track of that multiplicity and orientation. It is defined for a closed piecewise smooth contour and a point a not on the contour. The excluded-point condition is essential because the defining integrand has denominator z−a.

Predict

Predict the signs for the boundary of a ring-shaped region. Traversing the outer boundary counterclockwise keeps the region on your left. Traversing the inner boundary must therefore be clockwise.

A theorem about the whole boundary may imply equality between two counterclockwise circle integrals. It need not imply that either one vanishes separately. Test that distinction using 1/z.

Experiment

Let the circles be centered at zero.

Oriented contour	Winding about zero
One counterclockwise circle	1
One clockwise circle	−1
Two counterclockwise traversals	2
Outer counterclockwise plus inner clockwise	0

The last row is a sum of contour components, often called a cycle. It measures cancellation of the hole’s contributions, not the absence of a geometric hole.

Understand

Define Ind(γ,a)=(1/(2πi))∫γ dz/(z−a). To see why this is an integer, choose a continuous argument along the parameter interval, using successive local branches. Write γ(t)−a=r(t)exp(iθ(t)), piecewise smoothly. Integrating its logarithmic derivative gives the change in log r plus i times the change in θ. Closure cancels the radial change, while equality of endpoint directions forces the angular change to be 2πk. This argument chooses an angle along a path; it does not assume a global angle on the punctured plane.

The index varies continuously with a away from the compact trace: locally the denominator has a uniform positive lower bound. An integer-valued continuous function is locally constant, so the index is constant on each complementary component. Far from the contour, ML bounds its modulus by contour length divided by 2π times distance. Hence the index is zero in the unbounded component.

A useful general Cauchy theorem says that if f is holomorphic on a domain D and a closed cycle in D has index zero about every point outside D, its integral is zero. This is the homological form; a full planar subdivision proof is beyond the triangle proof already given. For a positively oriented boundary of a finitely connected region on whose closure f is holomorphic, it gives the outer-boundary integral minus the counterclockwise inner-boundary integrals equal to zero.

Simply connected domains make every closed curve contractible. Local primitives then give zero integrals by patchwise cancellation along a contraction.

Apply

For f=1/z on 1≤|z|≤3, compute both counterclockwise circle integrals. Each equals 2πi, while their difference is zero. Explain why replacing the difference by a sum would correspond to the wrong inner orientation.

Repair this claim: “A closed contour has winding number one.” It can have winding zero, negative winding, or multiple winding, and its index depends on the chosen point. Draw a point outside a simple circle and compare it with the center.

Continue

The Cauchy formula next reconstructs a value from a contour that winds once around it. Keep the more general integer index in mind for repeated traversals. Optional depth: prove index invariance under a homotopy avoiding a by continuity and integrality, using compactness of the parameter square to keep the denominator away from zero.