Definition:Smooth Path/Complex

Definition
Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval.

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

That is, $\gamma$ is a continuous function from $\left[{a \,.\,.\, b}\right]$ to $\C$.

Define the real function $x : \left[{a \,.\,.\, b}\right] \to \R$ by:


 * $\forall t \in \left[{a \,.\,.\, b}\right]: x \left({t}\right) = \operatorname{Re} \left({\gamma \left({t}\right)}\right)$

Define the real function $y: \left[{a \,.\,.\, b}\right] \to \R$ by:


 * $\forall t \in \left[{a \,.\,.\, b}\right]: y \left({t}\right) = \operatorname{Im} \left({\gamma \left({t}\right)}\right)$

Here, $\operatorname{Re} \left({\gamma \left({t}\right)}\right)$ denotes the real part of the complex number $\gamma \left({t}\right)$, and $\operatorname{Im} \left({\gamma \left({t}\right)}\right)$ denotes the imaginary part of $\gamma \left({t}\right)$.

Then $\gamma$ is said to be a smooth path iff:


 * $(1): \quad$ Both $x$ and $y$ are continuously differentiable.


 * $(2): \quad$ For all $t \in \left[{a \,.\,.\, b}\right]$, either $x' \left({t}\right) \ne 0$ or $y' \left({t}\right) \ne 0$.

Derivative of Smooth Path
If $\gamma$ is a smooth path, its derivative $\gamma' : \left[{a \,.\,.\, b}\right] \to \C$ is defined by:


 * $\forall t \in \left[{a \,.\,.\, b}\right] : \gamma' \left({t}\right) := x' \left({t}\right) + i y' \left({t}\right)$

Also see

 * Directed Smooth Curve