Definition:Contour/Closed/Complex Plane

Definition

Let $C$ be a contour in $\C$ defined by the (finite) sequence $\sequence {C_1, \ldots, C_n}$ of 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 $k \in \set {1, \ldots, n}$.

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

$\map {\gamma_1} {a_1} = \map {\gamma_n} {b_n}$

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 $C_n$.

• Definition:Interior of Simple Closed Contour

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.

Sources

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