# Definition:Concatenation of Contours

## Contents

## Definition

Let $\R^n$ be a real cartesian space of $n$ dimensions.

Let $C$ and $D$ be contours in $\R^n$.

Thus:

- $C$ is a (finite) sequence of directed smooth curves $\sequence {C_1, \ldots, C_n}$

- $D$ is a (finite) sequence of directed smooth curves $\sequence {D_1, \ldots, D_m}$.

Let $C_i$ be parameterized by the smooth path:

- $\gamma_i: \closedint {a_i} {b_i} \to \R^n$ for all $i \in \set {1, \ldots, n}$

Let $D_i$ be parameterized by the smooth path:

- $\sigma_i: \closedint {c_i} {d_i} \to \R^n$ for all $i \in \set {1, \ldots, m}$

Let $\map {\gamma_n} {b_n} = \map {\sigma_1} {c_1}$.

Then the **concatenation of the contours** $C$ and $D$, denoted $C \cup D$, is the contour defined by the (finite) sequence:

- $\sequence {C_1, \ldots, C_n, D_1, \ldots, D_m}$

### Complex Plane

The definition carries over to the complex plane, in which context it is usually applied:

Let $C$ and $D$ be contours in the complex plane $\C$.

Thus:

- $C$ is a (finite) sequence of directed smooth curves $\sequence {C_1, \ldots, C_n}$

- $D$ is a (finite) sequence of directed smooth curves $\sequence {D_1, \ldots, D_m}$.

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 $D_i$ be parameterized by the smooth path:

- $\sigma_i: \closedint {c_i} {d_i} \to \C$ for all $i \in \set {1, \ldots, m}$

Let $\map {\gamma_n} {b_n} = \map {\sigma_1} {c_1}$.

Then the **concatenation of the contours** $C$ and $D$, denoted $C \cup D$, is the contour defined by the (finite) sequence:

- $\sequence {C_1, \ldots, C_n, D_1, \ldots, D_m}$

## Also denoted as

**Concatenation of contours** $C$ and $D$ can also be seen denoted as:

- $C D$
- $C + D$
- $C * D$

None of these notations, including $C \cup D$, fully comply with standard notation.

## Also known as

**Concatenation of contours** can also be referred to as **join of contours**, but that usage is deprecated on $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Also see

- Concatenation of Contours is Contour: demonstration that $C \cup D$ is also a contour.

## Linguistic Note

The word **concatenation** derives from the Latin **com-** for **with/together** and the Latin word **catena** for **chain**.

However, the end result of such an operation is not to be confused with a (set theoretical) chain.