# 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*: $\S 2.2$