Definition:Smooth Path/Complex

Definition
Let $a, b \in \R : a < b$.

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

That is, $\gamma$ is a continuous mapping on $\C$.

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 a smooth path iff:


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


 * $(2)$: Their derivates $x'$ and $y'$ are both continuous.


 * $(3)$: For all $t \in \left[ a \,.\,.\, b \right]$, either $x' \left( { t } \right) \ne 0$ or $y' \left( { t } \right) \ne 0$.

If $\gamma$ is a smooth path, its derivate $\gamma ' : \left[ a \,.\,.\, b \right] \to \C$ is defined by:


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

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