10 - Winding Numbers

Existence of a Continuous Argument

Let $\gamma :[0,1]⟹\mathbb{C}\setminus \{0\}$ be a path then

There exists continuous function $a:[0,1]\to \mathbb{R}$ such that

$$\gamma (t)=|\gamma (t)|e^{2\pi ia(t)}$$

If $a,b$ are two such functions that satisfy the above then there exists $n\in \mathbb{Z}$ such that

$$a(t)=b(t)+n\text{ for all }t\in [0,1]$$

So $a(t_0)$ at any $t_0$ uniquely determines $a(t)$ for all $t$

Proof

Replacing $\gamma (t)$ with $\frac{\gamma (t)}{|\gamma (t)|}$, assume that $|\gamma (t)|=1$

Since $\gamma$ is continuous on a compact set, then $\gamma$ is uniformly continuous
So there is $\delta >0$ such that

$$|\gamma (s)-\gamma (t)|<\sqrt{3}\text{ for all }s,t\text{ with }|s-t|<\delta$$

Choose integer $n>0$ such that $n>\frac{1}{\delta }$ so on subinterval $\left[\frac{j}{n},\frac{j+1}{n}\right]$ then

$$|\gamma (s)-\gamma (t)|<\frac{\sqrt{3}}{2}$$

So for any half-plane in $\mathbb{C}$ then continuous argument function can be defined

If $|z_1|=|z_2|=1$ and $|z_1-z_3|<\sqrt{3}$ then angle between $z_1$ and $z_2$ is at most $\frac{2\pi }{3}$

So there exists continuous function $a_i:\left[\frac{j}{n},\frac{j+1}{n}\right]\to \mathbb{R}$ such that

$$\gamma (t)=e^{2\pi i{a}_j(t)}\text{ for }t\in \left[\frac{j}{n},\frac{j+1}{n}\right]$$

as $\gamma \left(\left[\frac{j}{n},\frac{j+1}{n}\right]\right)$ must lie in an arc of length at most $\frac{2\pi }{3}$

As $e^{2\pi i{a}_j(j/n)}=e^{2\pi i{a}_{j-1}(j/n)}$ and $a_{j-1}\left(\frac{j}{n}\right)$ and $a_i\left(\frac{j}{n}\right)$ differ by an integer then
Successively adjust $a_j$ for $j>1$ by an integer to get continuous function $a:[0,1]\to \mathbb{C}$
so that

$$\gamma (t)=e^{2\pi ia(t)}$$

Uniqueness statement follows from as

$$e^{2\pi i(a(t)-b(t))}=1$$

with $a(t)-b(t)\in \mathbb{Z}$ so since $[0,1]$ is connected then $a(t)-b(t)$ is constant

Winding Number

Let $\gamma :[0,1]\to \mathbb{C}\setminus \{0\}$ be a closed path and

$$\gamma (t)=|\gamma (t)|e^{2\pi ia(t)}$$

As $\gamma (0)=\gamma (1)$ then

$$I(\gamma ,0):=a(1)-a(0)\in \mathbb{Z}\text{ is known as the winding number}$$

where $I(\gamma ,0)$ is the winding number of $\gamma$ around $0$

If $z_0$ is not in the image of $\gamma$, it is still defined in the same fashion where
Let $t:\mathbb{C}\to \mathbb{C}$ be defined by $t(z)=z-z_0$ so

$$I(\gamma ,z_0)=I(t\circ \gamma ,0)$$

where the quantity is called the index of $z_0$ with respect to $\gamma$

Uniqueness

Uniquely determined by path $\gamma$ as function $a$ is unique up to an integer

Positively Oriented

Let $U$ be an open set in $\mathbb{C}$
Let $\gamma :[0,1]\to U$ be a closed curve

$\gamma$ is positively oriented then if
Winding number around any point from bounded component is $1$

Otherwise $\gamma$ is negatively oriented

Equivalent Definition $w$ be a point from the bounded component then

Let

$\gamma$ is positively oriented if

$$\frac{1}{2\pi i}{\int }_{\gamma }\frac{1}{z-w}dz=1$$

Homotopy between Interior Closed Paths and Interior Circle

Let $U$ be a domain
Let $\gamma$ be a simply positively oriented curve such that $\gamma$ and its interior are inside $U$

Let $r>0$ such that $B(w,r)$ is inside $\gamma$
Denote the positively oriented circle of radius $r$ around $w$ by ${\gamma }_r$ then

$$\gamma \text{ is homotopic to }{\gamma }_r\text{ inside }U\setminus \{w\}$$

Winding Number of a Opposite Path

Let $\gamma$ be a closed path

Let $f(z)$ be a continuous function on ${\gamma }^{\ast }$ then

$$I(f,{\gamma }^{-1})=I(f,-\gamma )$$