# Definition:Contour/Closed/Complex Plane

## Definition

Let $C$ be a contour in $\C$ defined by the (finite) sequence $\left\langle{C_1, \ldots, C_n}\right\rangle$ of 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 $i \in \left\{ {1, \ldots, n}\right\}$.

$C$ is a closed contour if and only if the start point of $C$ is equal to the end point of $C$:

$\gamma_1 \left({a_1}\right) = \gamma_n \left({b_n}\right)$

## Also known as

A closed contour is called a loop in some texts.

Some texts define a contour to be what $\mathsf{Pr} \infty \mathsf{fWiki}$ refers to as a closed contour.