Definition:Contour/Closed

Definition

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

Let $C$ be the 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 $i \in \set {1, \ldots, n}$.

$C$ is a closed contour if and only if the start point of $C$ is equal to the end point of $C$:

$\map {\rho_1} {a_1} = \map {\rho_n} {b_n}$

Complex Plane

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

$C$ is a closed contour if and only if the start point of $C$ is equal to the end point of $C$:

$\map {\gamma_1} {a_1} = \map {\gamma_n} {b_n}$

Also see

• Reparameterization of Directed Smooth Curve Maps Endpoints To Endpoints, from which it follows that this definition is independent of the parameterizations of $C_1$ and $C_n$.