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 $i \in \left\{ {1, \ldots, n}\right\}$, 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 $\gamma_i: \left[{a_i \,.\,.\, b_i}\right] \to \C$ 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 parameterization of $C$ is defined as the function $\gamma: \left[{a_1 \,.\,.\, c_n}\right] \to \C$ with:

$\gamma \restriction_{\left[{c_i \,.\,.\, c_{i + 1} }\right] } \left({t}\right) = \gamma_i \left({t}\right)$

where $\displaystyle c_i = a_1 + \sum_{j \mathop = 1}^i b_j - \sum_{j \mathop = 1}^i a_j$ for $i \in \left\{ {0, \ldots, n}\right\}$.

Here, $\gamma \restriction_{\left[{c_i \,.\,.\, c_{i + 1} }\right] }$ denotes the restriction of $\gamma$ to $\left[{c_i \,.\,.\, c_{i + 1} }\right]$.


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 $\gamma_i: \left[{a_i \,.\,.\, b_i}\right] \to \C$ for all $i \in \left\{ {1, \ldots, n}\right\}$.

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


$C$ is a simple contour if and only if:

$(1): \quad$ For all $i,j \in \left\{ {1, \ldots, n}\right\}, t_1 \in \left[{a_i \,.\,.\, b_i}\right), t_2 \in \left[{a_j \,.\,.\, b_j}\right)$ with $t_1 \ne t_2$, we have $\gamma_i \left({t_1}\right) \ne \gamma_j \left({t_2}\right)$.
$(2): \quad$ For all $k \in \left\{ {1, \ldots, n}\right\}, t \in \left[{a_k \,.\,.\, b_k}\right)$ where either $k \ne 1$ or $t \ne a_1$, we have $\gamma_k \left({t}\right) \ne \gamma_n \left({b_n}\right)$.


Length

Let $C$ be a contour in $C$ defined by the (finite) sequence $\left\langle{C_1, \ldots, C_n}\right\rangle$ of 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 $i \in \left\{ {1, \ldots, n}\right\}$.


The length of $C$ is defined as:

$\displaystyle L \left({C}\right) := \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} \left\vert{\gamma_i' \left({t}\right) }\right\vert \rd t$


Image

Let $C$ be a contour in $\C$ defined by the (finite) sequence $\left\langle{C_1, \ldots, C_n}\right\rangle$ of 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 $i \in \left\{ {1, \ldots, n}\right\}$.


The image of $C$ is defined as:

$\displaystyle \operatorname{Im} \left({C}\right) := \bigcup_{i \mathop = 1}^n \operatorname{Im} \left({\gamma_i}\right)$

where $\operatorname{Im} \left({\gamma_i}\right)$ denotes the image of $\gamma_i$.


If $\operatorname{Im} \left({C}\right) \subseteq D$, 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

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