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 if and only if:

$(1): \quad$ 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:

Let $C_1, \ldots, C_n$ be directed smooth curves in the complex plane $\C$.

Let $C_k$ be parameterized by the smooth path $\gamma_k: \closedint {a_k} {b_k} \to \C$ for all $k \in \set {1, \ldots, n}$.

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

$C$ is a simple contour if and only if:

$(1): \quad$ For all $j, k \in \set {1, \ldots, n}, t_1 \in \hointr {a_j} {b_j}, t_2 \in \hointr {a_k} {b_k}$ with $t_1 \ne t_2$, we have $\map {\gamma_j} {t_1} \ne \map {\gamma_j} {t_2}$.

$(2): \quad$ 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 $\map {\gamma_k} t \ne \map {\gamma_n} {b_n}$.

Also see

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

Notes

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Note that a simple contour may be a closed contour, so $\map {\rho_1} {a_1} = \map {\rho_n} {b_n}$.