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 $\C$) if and only if:

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

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

Let $\gamma$ be a smooth path in $\C$.

Then $\gamma$ is a closed smooth path if and only if $\gamma$ is a closed path.

That is, if and only if $\map \gamma a = \map \gamma b$.

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

$\gamma$ is a simple smooth path if and only if:

$(1): \quad \gamma$ is injective on the half-open interval $\hointr a b$

$(2): \quad \forall t \in \openint a b: \map \gamma t \ne \map \gamma b$

That is, if $t_1, t_2 \in \openint a b$ with $t_1 \ne t_2$, then $\map \gamma a \ne \map \gamma {t_1} \ne \map \gamma {t_2} \ne \map \gamma b$.