# Definition:Area Hyperbolic Cosecant

## Definition

### Complex

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 $z^2 + 1$.

### Real

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.

## Symbol

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

arcsch

The usual symbol used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to denote the area hyperbolic cosecant function is $\arcsch$.

acsch

A variant symbol used to denote the area hyperbolic cosecant function is $\operatorname {acsch}$.

acosech

A variant symbol used to denote the area hyperbolic cosecant function is $\operatorname {acosech}$.