# Definition:Area Hyperbolic Secant

## Definition

### Complex

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

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

where:

$\Ln$ denotes the principal branch of the complex natural logarithm
$\sqrt {1 - z^2}$ denotes the principal square root of $1 - z^2$.

### Real

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

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

where:

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

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

## Symbol

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

arsech

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

asech

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