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 $\gamma$.
Parameterization
Let $C_1, \ldots, C_n$ be directed smooth curves in the complex plane $\C$.
Let $C_i$ be parameterized by the smooth path $\gamma_i: \closedint {a_i} {b_i} \to \C$ for all $i \in \set {1, \ldots, n}$.
Let $C$ be the contour defined by the finite sequence $C_1, \ldots, C_n$.
The parameterization of $C$ is defined as the function $\gamma: \closedint {a_1} {c_n} \to \C$ with:
- $\map {\gamma \restriction_{\closedint {c_i} {c_{i + 1} } } } t = \map {\gamma_i} t$
where $\ds c_i = a_1 + \sum_{j \mathop = 1}^i b_j - \sum_{j \mathop = 1}^i a_j$ for $i \in \set {0, \ldots, n}$.
Here, $\gamma \restriction_{\closedint {c_i} {c_{i + 1} } }$ denotes the restriction of $\gamma$ to $\closedint {c_i} {c_{i + 1} }$.
Closed Contour
$C$ is a closed contour if and only if the start point of $C$ is equal to the end point of $C$:
- $\gamma_1 \left({a_1}\right) = \gamma_n \left({b_n}\right)$
Simple Contour
Let $C_1, \ldots, C_n$ be directed smooth curves in the complex plane $\C$.
Let $C_i$ be parameterized by the smooth path $\gamma_i: \closedint {a_i} {b_i} \to \C$ for all $i \in \set {1, 2, \ldots, n}$.
Let $C$ be the contour defined by the finite sequence $C_1, \ldots, C_n$.
$C$ is a simple contour if and only if:
- $(1): \quad$ For all $i, j \in \set {1, \ldots, n}, t_1 \in \hointr {a_i} {b_i}, t_2 \in \hointr {a_j} {b_j}$ with $t_1 \ne t_2$, we have $\map {\gamma_i} {t_1} \ne \map {\gamma_j} {t_2}$.
- $(2): \quad$ For all $k \in \set {1, \ldots, n}, t \in \hointr {a_k} {b_k}$ where either $k \ne 1$ or $t \ne a_1$, we have $\map {\gamma_k} t \ne \map {\gamma_n} {b_n}$.
Length
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 $C_i$ be parameterized by the smooth path $\gamma_i: \closedint {a_i} {b_i} \to \C$ for all $i \in \set {1, \ldots, n}$.
The length of $C$ is defined as:
- $\ds \map L C := \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} \size {\map {\gamma_i'} t} \rd t$
Image
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 $C_i$ be parameterized by the smooth path $\gamma_i: \closedint {a_i} {b_i} \to \C$ for all $i \in \set {1, \ldots, n}$.
The image of $C$ is defined as:
- $\ds \Img C := \bigcup_{i \mathop = 1}^n \Img {\gamma_i}$
where $\Img {\gamma_i}$ denotes the image of $\gamma_i$.
If $\Img C \subseteq D$, where $D$ is a subset of $\C$, we say that $C$ is a contour in $D$.
Endpoints
Let $C_1, \ldots, C_n$ be 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 $i \in \left\{ {1, \ldots, n}\right\}$.
Let $C$ be the contour defined by the finite sequence $C_1, \ldots, C_n$.
The start point of $C$ is $\gamma_1 \left({a_1}\right)$.
The end point of $C$ is $\gamma_n \left({b_n}\right)$.
Collectively, $\gamma_1 \left({a_1}\right)$ and $\gamma_n \left({b_n}\right)$ are referred to as the endpoints of $C$.
Illustration
Images of four contours in the complex plane, showing from left to right:
- 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 $n = 1$.
Sources
- 2001: Christian Berg: Kompleks funktionsteori: $\S 2.2$