Definition:Inverse Hyperbolic Cosecant

From ProofWiki
Jump to navigation Jump to search

Definition

Complex Plane

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.


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.