Definition:Contour/Endpoints/Complex Plane

Definition

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

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 $\sequence {C_1, \ldots, C_n}$.

The start point of $C$ is $\map {\gamma_1}{a_1}$.

The end point of $C$ is $\map {\gamma_n}{b_n}$.

Collectively, $\map {\gamma_1}{a_1}$ and $\map {\gamma_n}{b_n}$ 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 $C_1, \ldots, C_n$.

Sources

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