Definition:Contour/Complex Plane

Definition
Let $C_1, \ldots, C_n$ be directed smooth curves in the complex plane $\C$.

For each $i \in \left\{ {1, \ldots, n}\right\}$, let $C_i$ be parameterized by the smooth path $\gamma_i: \left[{a_i \,.\,.\, b_i}\right] \to \C$.

For each $i \in \left\{ {1, \ldots, n - 1}\right\}$, let the endpoint of $\gamma_i$ equal the start point of $\gamma_{i + 1}$:


 * $\gamma_i \left({b_i}\right) = \gamma_{i + 1} \left({a_{i + 1} }\right)$

Then the finite sequence $\left\langle{C_1, \ldots, C_n}\right\rangle$ is a contour.

If $C_1, \ldots, C_n$ are defined only by their parameterizations $\gamma_1, \ldots, \gamma_n$, then the contour can be denoted by the same symbol $\gamma$.

Illustration

 * [[File:Contours.png]]

Images of four contours in the complex plane, showing from left to right:


 * A contour that is neither simple nor closed.


 * A simple contour that is not closed.


 * A closed contour that is not simple.


 * A simple closed contour, whose parameterization is a Jordan curve.

Also known as
A contour is called a directed contour, piecewise smooth path, or a piecewise smooth curve in many texts.

Some texts only use the name contour for a closed contour.

Also denoted as
Some texts write the sequence of directed smooth curves as:


 * $C_1 \cup C_2 \cup \ldots \cup C_n$

or with some other symbol denoting the concatenation of directed smooth curves.

Also see

 * Definition:Directed Smooth Curve (Complex Plane), the special case that $n = 1$.