Definition:Derivative of Smooth Path/Complex Plane

Definition

Let $\gamma: \closedint a b \to \C$ be a smooth path in $\C$.

Define the real function $x : \closedint a b \to \R$ by:

$\forall t \in \closedint a b: \map x t = \map \Re {\map \gamma t}$

Define the real function $y: \closedint a b \to \R$ by:

$\forall t \in \closedint a b: \map y t = \map \Im {\map \gamma t}$

where:

$\map \Re {\map \gamma t}$ denotes the real part of the complex number $\map \gamma t$

$\map \Im {\map \gamma t}$ denotes the imaginary part of $\map \gamma t$.

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

Let $\map {x'} t$ and $\map {y'} t$ denote the derivative of $x$ and $y$ with respect to $t$.

The derivative of $\gamma$ is the continuous complex function $\gamma': \closedint a b \to \C$ defined by:

$\forall t \in \closedint a b: \map {\gamma'} t = \map {x'} t + i \map {y'} t$