Definition:Contour/Closed/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 closed contour iff $\gamma_1 \left({a_1}\right) = \gamma_n \left({b_n}\right)$.

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

Some texts define a contour to be what we call closed contours.