Definition:Derivative of Smooth Path/Complex Plane

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


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

where:
 * $\operatorname{Re} \left({\gamma \left({t}\right)}\right)$ denotes the real part of the complex number $\gamma \left({t}\right)$


 * $\operatorname{Im} \left({\gamma \left({t}\right)}\right)$ denotes the imaginary part of $\gamma \left({t}\right)$.

It follows from the definition of a smooth path that both $x$ and $y$ are continuously differentiable.

Let $x' \left({t}\right)$ and $y' \left({t}\right)$ denote the derivative of $x$ and $y$ $t$.

The derivative of $\gamma$ is the continuous complex function $\gamma': \left[{a \,.\,.\, b}\right] \to \C$ defined by:


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