Definition:Contour/Image
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 $\sequence {C_1, \ldots, C_n}$ of directed smooth curves in $\R^n$.
Let $C_i$ be parameterized by the smooth path $\rho_i: \closedint {a_i} {b_i} \to \R^n$ for all $i \in \set {1, \ldots, n}$.
The image of $C$ is defined as:
- $\ds \Img C := \bigcup_{i \mathop = 1}^n \Img {\rho_i}$
where $\Img {\rho_i}$ 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 $\sequence {C_1, \ldots, C_n}$ of directed smooth curves in $\C$.
Let $C_k$ be parameterized by the smooth path $\gamma_k: \closedint {a_k} {b_k} \to \C$ for all $k \in \set {1, \ldots, n}$.
The image of $C$ is defined as:
- $\ds \Img C := \bigcup_{k \mathop = 1}^n \Img {\gamma_k}$
where $\Img {\gamma_k}$ denotes the image of $\gamma_k$.
If $\Img C \subseteq D$, 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 $C_1, \ldots, C_n$.