Definition:Inverse Hyperbolic Cosecant/Real

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Definition

Definition 1

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

$\forall x \in \R_{\ne 0}: \map \arcsch x := y \in \R: x = \map \csch y$

where $\map \csch y$ denotes the hyperbolic cosecant function of $y$.


Definition 2

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

$\forall x \in \R_{\ne 0}: \map \arcsch x := \map \ln {\dfrac 1 x + \dfrac {\sqrt {x^2 + 1} } {\size x} }$

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.


Graph of Inverse Hyperbolic Cosecant

The graph of the real inverse hyperbolic cosecant function appears as:


Arcsch.png


Also denoted as

The symbol used to denote the real area hyperbolic cosecant function is variously seen as:

  • $\arcsch$
  • $\operatorname {acosech}$
  • $\operatorname {acsch}$


Also known as

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

Note that as the real hyperbolic cosecant $\csch$ is injective, its inverse is properly a function on its domain.

Hence there is no need to make a separate distinction between branches in the same way as for real inverse hyperbolic cosine and real inverse hyperbolic secant.


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.


Also see

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


Sources

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