Winding number

Definition

Let $\mathrm{\Gamma }$ be a cycle in $ℂ$, and let $z\in ℂ$ be a point that $\mathrm{\Gamma }$ does not intersect. Then

$n(\mathrm{\Gamma },z):=\frac{1}{2\pi i}\underset{\mathrm{\Gamma }}{\int }\frac{1}{w-z}dw$

is called the *winding number* of $\mathrm{\Gamma }$ around $z$.

Motivation

First, consider the case where $\mathrm{\Gamma }=\gamma$ consists of a single closed curve. Then $\gamma$ is homologous in $ℂ\setminus z$ to an $n$-fold (for some $n\in \mathbb{Z}$) traversed circle $\partial D_r(z)$ around $z$ with $r>0$. Now,

$\underset{\gamma }{\int }\frac{1}{w-z},dw=\underset{n\cdot \partial B(z)}{\int }\frac{1}{w-z},dw=n\underset{\partial B(z)}{\int }\frac{1}{w-z},dw=2\pi i\cdot n.$

Thus, this integral counts how many times the curve $\gamma$ winds around the point $z$.

Task

Let the closed integration path $\gamma :[-\pi ,+\pi ]\to ℂ$ be defined as:

$\gamma (t):=(2+\cos (t))\cdot e^{2it}.$

1. Plot the trace of the integration path.

2. Determine the winding number $n(\gamma ,1+i)$.

3. Determine the winding number $n(\gamma ,0)$.

4. Determine the winding number $n(\gamma ,1)$.

Additivity of the Integral

For a cycle $\mathrm{\Gamma }=\sum _{i=1}^k{n}_i\cdot {\gamma }_i$ with closed ${\gamma }_i$, due to the additivity of the integral, we have

$n(\mathrm{\Gamma },z)=\sum _{i=1}^k{n}_i\cdot n({\gamma }_i,z).$

Thus, the winding number also counts how many times the point $z$ is encircled.

Length of the Cycle

For a cycle $\mathrm{\Gamma }=\sum _{i=1}^k{n}_i{\gamma }_i$ with closed ${\gamma }_i$, the length of the cycle is defined additively over the lengths of the individual integration paths:

$L(\mathrm{\Gamma }):=\sum _{i=1}^k{n}_i\cdot L({\gamma }_i).$

See Also

• Cauchy's Integral Theorem for Disks

Page Information

You can display this page as Wiki2Reveal slides

Wiki2Reveal

The Wiki2Reveal slides were created for the Complex Analysis' and the Link for the Wiki2Reveal Slides was created with the link generator.

• This page is designed as a PanDocElectron-SLIDE document type.
• Source: Wikiversity https://en.wikiversity.org/wiki/Winding%20number
• see Wiki2Reveal for the functionality of Wiki2Reveal.

Translation and Version Control

This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Umlaufzahl Wikiversity source page] and uses the concept of Translation and Version Control for a transparent language fork in a Wikiversity:

• Source: Kurs:Funktionentheorie/Umlaufzahl - URL:

https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Umlaufzahl

• Date: 12/17/2024

de:Kurs:Funktionentheorie/Umlaufzahl