Definition:Inverse Hyperbolic Cosecant/Complex

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Definition

Definition 1

The inverse hyperbolic cosecant is a multifunction defined as:

$\forall z \in \C_{\ne 0}: \map {\csch^{-1} } z := \set {w \in \C: z = \map \csch w}$

where $\map \csch w$ is the hyperbolic cosecant function.


Definition 2

The inverse hyperbolic cosecant is a multifunction defined as:

$\forall z \in \C_{\ne 0}: \map {\csch^{-1} } z := \set {\map \ln {\dfrac {1 + \sqrt {\size {z^2 + 1} } e^{\paren {i / 2} \map \arg {z^2 + 1} } } z} + 2 k \pi i: k \in \Z}$

where:

$\sqrt {\size {z^2 + 1} }$ denotes the positive square root of the complex modulus of $z^2 + 1$
$\map \arg {z^2 + 1}$ denotes the argument of $z^2 + 1$
$\ln$ denotes the complex natural logarithm considered as a multifunction.


Principal Branch

The principal branch of the complex inverse hyperbolic cosecant function is defined as:

$\forall z \in \C_{\ne 0}: \map \Arcsch z := \map \Ln {\dfrac {1 + \sqrt {z^2 + 1} } z}$

where:

$\Ln$ denotes the principal branch of the complex natural logarithm
$\sqrt {z^2 + 1}$ denotes the principal square root of $z^2 + 1$.


Also known as

The principal branch of the inverse hyperbolic cosecant is also known as the area hyperbolic cosecant, as it can be used, among other things, for evaluating areas of regions bounded by hyperbolas.

Some sources refer to it as hyperbolic arccosecant, but this is strictly a misnomer, as there is nothing arc related about an inverse hyperbolic cosecant.


In the complex plane, $\mathsf{Pr} \infty \mathsf{fWiki}$ reserves the term area hyperbolic cosecant strictly for the principal branch.


Also see

  • Results about the inverse hyperbolic cosecant can be found here.


Sources

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