Definition:Inverse Hyperbolic Cosecant

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Definition

Complex Plane

Definition 1

The inverse hyperbolic cosecant is a multifunction defined as:

$\forall z \in \C_{\ne 0}: \operatorname{csch}^{-1} \left({z}\right) := \left\{{w \in \C: z = \operatorname{csch} \left({w}\right)}\right\}$

where $\operatorname{csch} \left({w}\right)$ is the hyperbolic cosecant function.


Definition 2

The inverse hyperbolic cosecant is a multifunction defined as:

$\forall z \in \C_{\ne 0}: \operatorname{csch}^{-1} \left({z}\right) := \left\{{\ln \left({\dfrac {1 + \sqrt{\left|{z^2 + 1}\right|} e^{\left({i / 2}\right) \arg \left({z^2 + 1}\right)}} z}\right) + 2 k \pi i: k \in \Z}\right\}$

where:

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


Hyperbolic Arccosecant

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

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

where:

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


Real Numbers

Definition 1

The inverse hyperbolic cosecant $\operatorname{csch}^{-1}: \R_{\ne 0} \to \R$ is a real function defined on the non-zero real numbers $\R_{\ne 0}$ as:

$\forall x \in \R_{\ne 0}: \operatorname{csch}^{-1} \left({x}\right) := y \in \R: x = \operatorname{csch} \left({y}\right)$

where $\operatorname{csch} \left({y}\right)$ denotes the hyperbolic cosecant function.


Definition 2

The inverse hyperbolic cosecant $\operatorname{csch}^{-1}: \R_{\ne 0} \to \R$ is a real function defined on the non-zero real numbers $\R_{\ne 0}$ as:

$\forall x \in \R_{\ne 0}: \operatorname{csch}^{-1} \left({x}\right) := \ln \left({\dfrac {1 + \sqrt{x^2 + 1} } x}\right)$

where:

$\sqrt{x^2 + 1}$ denotes the positive square root of $x^2 + 1$
$\ln$ denotes the natural logarithm of a (strictly positive) real number.


Also see

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