Definition:Directed Smooth Curve/Parameterization/Complex Plane/Reparameterization

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

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