Definition:Directed Smooth Curve/Complex Plane

Definition
Let $\gamma : \left[{ a \,.\,.\, b }\right] \to \C$ 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 $\sigma: \left[{ a \,.\,.\, b }\right] \to \C$ belongs to the equivalence class of $\gamma$ :
 * there exists a bijective differentiable strictly increasing real function:
 * $\phi: \left[{c \,.\,.\, d}\right] \to \left[{a \,.\,.\, b}\right]$
 * 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 $\gamma$.

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.