Definition:Contour/Complex Plane

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

For each $k \in \set{ 1, \ldots, n}$, let $C_k$ be parameterized by the smooth path $\gamma_k: \closedint {a_k}{b_k} \to \C$.

For each $k \in \set{ 1, \ldots, n-1}$, let the endpoint of $\gamma_k$ equal the start point of $\gamma_{k + 1}$:


 * $\map {\gamma_k}{b_k} = \map {\gamma_{k + 1} }{a_{k + 1} }$

Then the finite sequence $\sequence{C_1, \ldots, C_n}$ 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:ContoursComplexPlane.png]]

Illustration of the 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 with positive orientation, where its interior is drawn as shaded.

Their endpoints are marked as dots.

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 defined as
Some texts define a contour as a complex function $\gamma: \closedint a b \to \C$ that is piecewise continuously differentiable on the closed real interval $\closedint a b$.

This is what refers to as a parameterization of a contour.

Some texts define a contour $C$ as the image of a function $\gamma: \closedint a b \to \C$, defined as above.

This is what refers to as the image of a 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 $n = 1$.