Definition:Contour/Complex Plane

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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 $\gamma$.


Parameterization

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

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

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:

$\map {\gamma \restriction_{\closedint {c_k} {c_{k + 1} } } } t = \map {\gamma_k} t$

where $\ds c_k = a_1 + \sum_{j \mathop = 1}^k b_j - \sum_{j \mathop = 1}^k a_j$ for $k \in \set {0, \ldots, n}$.

Here, $\gamma \restriction_{\closedint {c_k} {c_{k + 1} } }$ denotes the restriction of $\gamma$ to $\closedint {c_k} {c_{k + 1} }$.


Closed Contour

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

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


Simple Contour

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

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

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 $j, k \in \set {1, \ldots, n}, t_1 \in \hointr {a_j} {b_j}, t_2 \in \hointr {a_k} {b_k}$ with $t_1 \ne t_2$, we have $\map {\gamma_j} {t_1} \ne \map {\gamma_j} {t_2}$.
$(2): \quad$ 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 $C_k$ be parameterized by the smooth path $\gamma_k: \closedint {a_k} {b_k} \to \C$ for all $k \in \set {1, \ldots, n}$.


The length of $C$ is defined as:

$\ds \map L C := \sum_{k \mathop = 1}^n \int_{a_k}^{b_k} \size {\map {\gamma_k'} t} \rd t$


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


The image of $C$ is defined as:

$\ds \Img C := \bigcup_{k \mathop = 1}^n \Img {\gamma_k}$

where $\Img {\gamma_k}$ denotes the image of $\gamma_k$.


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

Let $C$ be the contour defined by the finite sequence $\sequence {C_1, \ldots, C_n}$.


The start point of $C$ is $\map {\gamma_1}{a_1}$.

The end point of $C$ is $\map {\gamma_n}{b_n}$.


Collectively, $\map {\gamma_1}{a_1}$ and $\map {\gamma_n}{b_n}$ are referred to as the endpoints of $C$.


Illustration

Contours.png

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


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


Sources