Definition:Contour/Simple/Complex Plane

Definition
Let $C_1, \ldots, C_n$ be directed smooth curves.

Let $C_i$ be parameterized by the smooth path $\gamma_i: \left[{a_i\,.\,.\,b_i}\right] \to \C$ for all $i \in \left\{ {1, \ldots, n}\right\}$.

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

A contour $C$ is a simple contour iff:


 * $(1): \quad$ For all $i,j \in \left\{ {1, \ldots, n}\right\}, t_1 \in \left[{a_i\,.\,.\,b_i}\right), t_2 \in \left[{a_j\,.\,.\,b_j}\right)$ with $t_1 \ne t_2$, we have $\gamma_i \left({t_1}\right) \ne \gamma_j \left({t_2}\right)$.


 * $(2): \quad$ For all $k \in \left\{ {1, \ldots, n}\right\}, t \in \left[{a_k\,.\,.\,b_k}\right)$ with $t \ne \gamma_1 \left({a_1}\right)$, we have $\gamma_k \left({t}\right) \ne \gamma_n \left({b_n}\right)$.

Also see

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