Definition:Contour/Simple/Complex Plane

Definition

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 $\map {\gamma_k} t \ne \map {\gamma_n} {b_n}$.

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

Also see

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

• Definition:Closed Contour (Complex Plane)

• Definition:Jordan Arc and Definition:Jordan Curve, which are the corresponding definitions for a path in the Euclidean space $\R^2$.

Sources

• 2001: Christian Berg: Kompleks funktionsteori: $\S 2.2$