Category:Smooth Paths (Complex Analysis)

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This category contains results about Smooth Paths (Complex Analysis).
Definitions specific to this category can be found in Definitions/Smooth Paths (Complex Analysis).

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 $\map {y'} t \ne 0$.

Pages in category "Smooth Paths (Complex Analysis)"

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