Winding number

Definition

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

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

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

Motivation

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

\int _{\gamma }{\frac {1}{w-z}},dw=\int _{n\cdot \partial B(z)}{\frac {1}{w-z}},dw=n\int _{\partial B(z)}{\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 ${\displaystyle \gamma$ be defined as:

\gamma (t):=\left(2+\cos(t)\right)\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 $\Gamma =\sum _{i=1}^{k}n_{i}\cdot \gamma _{i}$ with closed $\gamma _{i}$ , due to the additivity of the integral, we have

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

L(\Gamma ):=\sum _{i=1}^{k}n_{i}\cdot L(\gamma _{i}).

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Date: 12/17/2024

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