Definition:Smooth Path/Complex

Definition
Let $\closedint a b$ be a closed real interval.

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

That is, $\gamma$ is a continuous complex-valued function from $\closedint a b$ to $\C$.

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 $\map \gamma t$.

Then $\gamma$ is a smooth path (in $\C$) :


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


 * $(2): \quad$ For all $t \in \closedint a b$, either $\map {x'} t \ne 0$ or $\map {y'} t \ne 0$.

Also see

 * Definition:Directed Smooth Curve (Complex Plane)
 * Definition:Derivative of Smooth Path in Complex Plane
 * Definition:Smooth Real Function, which defines real functions of differentiability class $C^\infty$.
 * Definition:Complex Contour Integral