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Mathematical Description of Winding Numbers

The mathematical underpinings of the winding number formula consist of Cauchy's Theorem and a calculation. The calculation is encapsulated in the following lemma.

Lemma 1   Let $ \gamma(t) = p + r e^{it}$ where $ t \in \left[0,2 \pi n\right]$ . Then $ \int_\gamma \frac{dz}{z - p}= 2 \pi i n$ .

Proof. By the Chain rule, $ \gamma'(t) = ir e^{it}$ . Thus

$\displaystyle \int_\gamma \frac{dz}{z - p} = \int_0^{2\pi n} \frac{i r e^{it}}{(p + re^{it}) - p} \, dt = \int_0^{2\pi n} i dt = 2 \pi i n. $

$ \qedsymbol$

We also have:

Theorem 1 (Cauchy)   Suppose $ f(z)$ is a holomorphic function on and inside a simple closed curve $ \gamma$ . Then $ \int_\gamma f(z) \,dz = 0$ .

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 $ \gamma$ and $ \sigma$ are homotopic in $ A$. Then

$\displaystyle \int_\gamma f(z) \, dz = \int_\sigma f(z) \, dz $

Using Theorem 2 and our lemma, the motivation for the winding number formula is clear. If $ \gamma(t) = p + r e^{it}$ where $ t \in [0,2 \pi n]$ . and $ \sigma$ is homotopic to $ \gamma$ on $ \mathbf{C} \setminus p$ , then

$\displaystyle \int_\sigma \frac{dz}{z - p} = \int_\gamma \frac{dz}{z - p} = 2 \pi i n $

since $ \frac{1}{z-p}$ is holomorphic away from $ p$.

next up previous
Up: Winding Number Illustrator Previous: Homotopy
Ross Moore
1998-07-22