Definition:Contour

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

Let $C_1, \ldots, C_n$ be directed smooth curves in $\R^n$.

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

For each $i \in \left\{ {1, \ldots, n-1}\right\}$, let the end point of $\rho_i$ equal the start point of $\rho_{i + 1}$:


 * $\rho_i \left({b_i}\right) = \rho_{i + 1} \left({a_{i + 1} }\right)$

Then the finite sequence $\left\langle{C_1, \ldots, C_n}\right\rangle$ is called a contour (in $\R^n$).

If $C_1, \ldots, C_n$ are defined only by their parameterizations $\rho_1, \ldots, \rho_n$, then the contour can be denoted by the same symbol $\rho$.

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

Also see

 * Directed Smooth Curve, the special case that $n = 1$.