Definition:Contour/Endpoints/Complex Plane

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

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 $C_1, \ldots, C_n$.

The start point of the contour $C$ is $\gamma_1 \left({a_1}\right)$.

The end point of the contour $C$ is $\gamma_n \left({b_n}\right)$.

Collectively, $\gamma_1 \left({a_1}\right)$ and $\gamma_n \left({b_n}\right)$ are referred to as the endpoints of $C$.

Also see
From Reparameterization of Directed Smooth Curve Maps Endpoints To Endpoints, it follows that this definition is independent of the parameterizations of $C_1, \ldots, C_n$.