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 <IMG WIDTH="105" HEIGHT="34" ALIGN="MIDDLE" BORDER="0" SRC="img20.gif" ALT="$ \gamma(t) = p + r e^{it}$"> where <IMG WIDTH="83" HEIGHT="31" ALIGN="MIDDLE" BORDER="0" SRC="img21.gif" ALT="$ t \in \left[0,2 \pi n\right]$"> . Then <IMG WIDTH="103" HEIGHT="35" ALIGN="MIDDLE" BORDER="0" SRC="img22.gif" ALT="$ \int_\gamma \frac{dz}{z - p}= 2 \pi i n$"> .

Proof. By the Chain rule, <IMG WIDTH="88" HEIGHT="34" ALIGN="MIDDLE" BORDER="0" SRC="img24.gif" ALT="$ \gamma'(t) = ir e^{it}$"> . Thus

<IMG WIDTH="455" HEIGHT="69" ALIGN="MIDDLE" BORDER="0" SRC="img25.gif" ALT="$\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 <IMG WIDTH="13" HEIGHT="28" ALIGN="MIDDLE" BORDER="0" SRC="img8.gif" ALT="$ \gamma$"> . Then <IMG WIDTH="101" HEIGHT="33" ALIGN="MIDDLE" BORDER="0" SRC="img27.gif" ALT="$ \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 <IMG WIDTH="13" HEIGHT="28" ALIGN="MIDDLE" BORDER="0" SRC="img8.gif" ALT="$ \gamma$"> and <IMG WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0" SRC="img9.gif" ALT="$ \sigma$"> are homotopic in $A$. Then

<IMG WIDTH="196" HEIGHT="61" ALIGN="MIDDLE" BORDER="0" SRC="img29.gif" ALT="$\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 <IMG WIDTH="105" HEIGHT="34" ALIGN="MIDDLE" BORDER="0" SRC="img20.gif" ALT="$ \gamma(t) = p + r e^{it}$"> where <IMG WIDTH="80" HEIGHT="31" ALIGN="MIDDLE" BORDER="0" SRC="img30.gif" ALT="$ t \in [0,2 \pi n]$"> . and <IMG WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0" SRC="img9.gif" ALT="$ \sigma$"> is homotopic to <IMG WIDTH="13" HEIGHT="28" ALIGN="MIDDLE" BORDER="0" SRC="img8.gif" ALT="$ \gamma$"> on <IMG WIDTH="40" HEIGHT="31" ALIGN="MIDDLE" BORDER="0" SRC="img31.gif" ALT="$ \mathbf{C} \setminus p$"> , then

<IMG WIDTH="237" HEIGHT="62" ALIGN="MIDDLE" BORDER="0" SRC="img32.gif" ALT="$\displaystyle \int_\sigma \frac{dz}{z - p} = \int_\gamma \frac{dz}{z - p} = 2 \pi i n $">

since <IMG WIDTH="31" HEIGHT="34" ALIGN="MIDDLE" BORDER="0" SRC="img33.gif" ALT="$ \frac{1}{z-p}$"> is holomorphic away from $p$.