Definition:Reversed Directed Smooth Curve/Complex Plane
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Definition
Let $C$ be a directed smooth curve in the complex plane $\C$.
Let $C$ be parameterized by the smooth path $\gamma: \closedint a b \to \C$.
The reversed directed smooth curve of $C$ is denoted $-C$ and is defined as the directed smooth curve that is parameterized by:
- $\rho: \closedint a b \to \C: \map \rho t = \map \gamma {a + b - t}$
From Reversed Directed Smooth Curve is Directed Smooth Curve, it follows that $-C$ is a directed smooth curve.
Also known as
A reversed directed smooth curve is called a reversed curve or a reciprocal curve in some texts.
Also denoted as
The reversed directed smooth curve of $C$ is denotes as $C^-$ in some texts.
Sources
- 2001: Christian Berg: Kompleks funktionsteori: $\S 2.2$