# Definition:Area Hyperbolic Cosine

## Definition

### Complex

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

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

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 principal branch of the real inverse hyperbolic cosine function is defined as:

$\forall x \in S: \map \arcosh x := \map \ln {x + \sqrt {x^2 - 1} }$

where:

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

That is, where $\map \arcosh x \ge 0$.

## Symbol

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

arcosh

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

acosh

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