Definition:Smooth Path/Closed/Complex Plane

Definition
Let $\gamma : \left[{ a \,.\,.\, b }\right] \to \C$ be a smooth path, where $\left[{ a \,.\,.\, b }\right]$ is a closed real interval.

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

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


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

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


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

$\gamma$ satisfies these conditions:


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


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

The smooth path $\gamma$ is a closed smooth path iff $\gamma \left({ a }\right) = \gamma \left({ b }\right)$.