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 mapping 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:


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


 * $(2)$: 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:


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