Definition:Contour/Closed/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 $C_1, \ldots, C_n$.

A contour $C$ is a closed contour  the start point of the contour is equal to the endpoint of the contour.

That is, $\gamma_1 \left({a_1}\right) = \gamma_n \left({b_n}\right)$.

Also see

 * Reparameterization of Directed Smooth Curve Maps Endpoints To Endpoints, from which it follows that this definition is independent of the parameterizations of $C_1$ and $C_n$.

Also known as
A closed contour is called a loop in some texts.

Some texts define a contour to be what refers to as a closed contour.