Definition:Concatenation of Contours

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.

• Definition:Concatenation of Paths

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.