Definition:Concatenation of Contours/Complex Plane

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Definition

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 see


Linguistic Note

The word concatenation derives from the Latin word catena for chain.

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


Sources