Definition:Directed Smooth Curve/Complex Plane

Definition

Let $\gamma : \closedint a b \to \C$ be a smooth path in $\C$.

The directed smooth curve with parameterization $\gamma$ is defined as an equivalence class of smooth paths as follows:

A smooth path $\sigma: \closedint c d \to \C$ belongs to the equivalence class of $\gamma$ if and only if:

there exists a bijective differentiable strictly increasing real function:

$\phi: \closedint c d \to \closedint a b$

such that $\sigma = \gamma \circ \phi$.

It follows from Directed Smooth Curve Relation is Equivalence and Fundamental Theorem on Equivalence Relations that this does in fact define an equivalence class.

If a directed smooth curve is only defined by a smooth path $\gamma$, then it is often denoted with the same symbol $\gamma$.

Parameterization

Let $C$ be a directed smooth curve in the complex plane $\C$.

Let $\gamma: \closedint a b \to \C$ be a smooth path in $\C$.

Then $\gamma$ is a parameterization of $C$ if and only if $\gamma$ is a representative of the equivalence class that constitutes $C$.

Reparameterization

Let $\phi: \closedint c d \to \closedint a b$ be a bijective differentiable strictly increasing real function.

Let $\sigma : \closedint c d \to \C$ be defined by:

$\sigma = \gamma \circ \phi$

Then $\sigma$ is called a reparameterization of $C$.

Endpoints

Let $C$ be a directed smooth curve in the complex plane $\C$.

Let $C$ be parameterized by a smooth path $\gamma: \left[{a \,.\,.\, b}\right] \to \C$.

Then:

$\gamma \left({a}\right)$ is the start point of $C$

$\gamma \left({b}\right)$ is the end point of $C$.

Collectively, $\gamma \left({a}\right)$ and $\gamma \left({b}\right)$ are known as the endpoints of $\rho$.

Also known as

A directed smooth curve is called an oriented smooth curve, a smooth curve with orientation or simply a curve in many texts.

Sources

• 2001: Christian Berg: Kompleks funktionsteori $\S 2.2$