Mathematical Description of Winding Numbers

Lemma. Let where Then .

Proof. By the Chain rule, Thus

We also have:

Theorem 1. (Cauchy) Suppose f(z) is a holomorphic function on and inside a simple closed curve . Then

The standard proof of this theorem amounts to an application of Green's theorem; the definition of holomorphic implies the integrand vanishes identically on the interior of the curve.

Using Cauchy's theorem, it is not too difficult to prove the following theorem.

Theorem 2. Suppose f(z) is holomorphic on a region A, and suppose further that two closed curves and are homotopic in A. Then

Using Theorem 2 and our lemma, the motivation for the winding number formula is clear. If where and is homotopic to on , then