Definition:Derivative of Smooth Path/Complex Plane

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

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 $\gamma \left({t}\right)$.

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

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


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