# 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 $\gamma$.

### 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 $\C$.

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

Illustration of the 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**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 $\mathsf{Pr} \infty \mathsf{fWiki}$ 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 $\mathsf{Pr} \infty \mathsf{fWiki}$ 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 $n = 1$.

## Sources

- 2001: Christian Berg:
*Kompleks funktionsteori*: $\S 2.2$