Definition:Contour/Image/Complex Plane
< Definition:Contour | Image
Jump to navigation
Jump to search
Definition
Let $C$ be a contour in $\C$ defined by the (finite) sequence $\left\langle{C_1, \ldots, C_n}\right\rangle$ of directed smooth curves in $\C$.
Let $C_i$ be parameterized by the smooth path $\gamma_i: \left[{a_i\,.\,.\,b_i}\right] \to \C$ for all $i \in \left\{ {1, \ldots, n}\right\}$.
The image of $C$ is defined as:
- $\displaystyle \operatorname{Im} \left({C}\right) := \bigcup_{i \mathop = 1}^n \operatorname{Im} \left({\gamma_i}\right)$
where $\operatorname{Im} \left({\gamma_i}\right)$ denotes the image of $\gamma_i$.
If $\operatorname{Im} \left({C}\right) \subseteq D$, where $D$ is a subset of $\C$, we say that $C$ is a contour in $D$.
Also see
- Reparameterization of Directed Smooth Curve Preserves Image, from which it follows that this definition is independent of parameterizations of $C_1, \ldots, C_n$.
Sources
- 2001: Christian Berg: Kompleks funktionsteori: $\S 2.2$