06 - Paths

Path

Path in the Complex Plane is continuous function

$$\gamma :[a,b]\to \mathbb{C}$$

Closed Path

Let $\gamma :[a,b]\to \mathbb{C}$ be a path then

$\gamma$ is a closed path if

$$\gamma (a)=\gamma (b)$$

Image of a Path

Let $\gamma :[a,b]\to \mathbb{C}$ be a path then

Image of path ${\gamma }^{\ast }$ is defined as

$$\{z\in C:z=\gamma (t)\text{ for some }t\in [a,b]\}$$

Note that by abuse of notation the image of $\gamma$ is also denoted by $\gamma$

Alternative Definition (Personal)

$$\{\gamma (t):t\in [a,b]\}$$

Differentiable Path

Let $\gamma :[a,b]\to \mathbb{C}$ be a path then

$\gamma$ is differentiable if it’s real and imaginery parts are differentiable as real-valued functions

Equivalently, $\gamma$ is differentiable at $t_0\in [a,b]$ if

$$\underset{t\to t_0}{\lim }\frac{\gamma (t)-\gamma (t_0)}{t-t_0}\text{ exists}$$

in which cases the limit is denoted as ${\gamma }^{\prime }(t_0)$

Path is $C^1$

Let $\gamma :[a,b]\to \mathbb{C}$ be a path then

$\gamma$ is $C^1$ if it is differentiable and it’s derivative ${\gamma }^{\prime }(t)$ is continuous

Piecewise $C^1$ Path

Let $\gamma :[a,b]\to \mathbb{C}$ be a path

$\gamma$ is piecewise $C^1$ if

1. Continuous on $[a,b]$
2. Interval $[a,b]$ can be divided into subintervals on each of which $\gamma$ is $C^1$
   So there is a finite sequence $a=a_0<a_1<\cdots a_m=b$ such that

$$\gamma {\mid }_{[a_i,a_{i+1}]}\text{ is }{C}^1$$

Note that in particular it is not necessary that the left-hand and right-hand derivatives of $\gamma$ are equal at $a_i$

Equivalent Paths

Let ${\gamma }_1:[a,b]\to \mathbb{C}$ and ${\gamma }_2:[c,d]\to \mathbb{C}$ be two parametrised paths

${\gamma }_1$ and ${\gamma }_2$ are equivalent if there exists continuously differentiable bijective function

$$s:[a,b]\to [c,d]$$

such that

1. $s^{\prime }(t)>0$ for all $t\in [a,b]$
2. ${\gamma }_1={\gamma }_2\circ s$

Note that ${\gamma }_1$ is piecewise $C^1$ if and only if ${\gamma }_2$ is piecewise $C^1$

Oriented Curves

As equivalence between paths is a equivalence relation then

The equivalence class of $\gamma$ is

$$[\gamma ]$$

where condition $s^{\prime }(t)>0$ ensures the paths are traversed in the same direction

Simply Connected

Let $U$ be a domain in $C$

$U$ is simply connected if for every $a,b\in U$ any two paths ${\gamma }_1,{\gamma }_2$ from $a$ to $b$ then

$${\gamma }_1\text{ and }{\gamma }_2\text{ are homotopic}$$

Equivalently, any closed curve is null homotopic