Definition:Contour/Image

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Definition

Let $\R^n$ be a real cartesian space of $n$ dimensions.

Let $C$ be a contour in $\R^n$ defined by the (finite) sequence $\left\langle{C_1, \ldots, C_n}\right\rangle$ of directed smooth curves in $\R^n$.

Let $C_i$ be parameterized by the smooth path $\rho_i: \left[{a_i \,.\,.\, b_i}\right] \to \R^n$ for all $i \in \left\{ {1, \ldots, n}\right\}$.


The image of $C$ is defined as:

$\displaystyle \operatorname{Im} \left({C}\right) := \bigcup_{i \mathop = 1}^n \operatorname{Im} \left({\rho_i}\right)$

where $\operatorname{Im} \left({\rho_i}\right)$ denotes the image of $\rho_i$.


Complex Plane

The definition carries over to the complex plane, in which context it is usually applied:


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