# Definition:Inverse Hyperbolic Secant/Real/Definition 2

## Definition

Let $S$ denote the half-open real interval:

$S := \hointl 0 1$

The inverse hyperbolic secant $\sech^{-1}: S \to \R$ is a real function defined on $S$ as:

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

where:

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

## Also known as

The inverse hyperbolic secant function is also known as the hyperbolic arcsecant function.