# Definition:Area Hyperbolic Sine

## Definition

### Complex

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

$\forall z \in \C: \map \Arsinh z := \map \Ln {z + \sqrt {z^2 + 1} }$

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 sine $\arsinh: \R \to \R$ is a real function defined on $\R$ as:

$\forall x \in \R: \map \arsinh x := \map \ln {x + \sqrt {x^2 + 1} }$

where:

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

## Symbol

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

arsinh

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

asinh

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