Definition:Contour/Simple

Definition
Let $\R^n$ be a real cartesian space of $n$ dimensions. Let $C_1, \ldots, C_n$ be directed smooth curves in $\R^n$.

Let $C_i$ be parameterized by the smooth path $\rho_i: \closedint {a_i} {b_i} \to \R^n$ for all $i \in \set {1, 2, \ldots, n}$.

Let $C$ be the contour in $\R^n$ defined by the finite sequence $C_1, \ldots, C_n$.

$C$ is a simple contour :


 * $(1): \quad$ For all $i, j \in \set {1, \ldots, n}, t_1 \in \hointr {a_i} {b_i}, t_2 \in \hointr {a_j} {b_j}$ with $t_1 \ne t_2$, we have $\map {\rho_i} {t_1} \ne \map {\rho_j} {t_2}$


 * $(2): \quad$ For all $k \in \set {1, \ldots, n}, t \in \hointr {a_k} {b_k}$ where either $k \ne 1$ or $t \ne a_1$, we have $\map {\rho_k} t \ne \map {\rho_n} {b_n}$.

Thus a simple contour is a contour that does not intersect itself.

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

Also see

 * Reparameterization of Directed Smooth Curve Preserves Image, from which it follows that this definition is independent of the parameterizations of $C_1, \ldots, C_n$.