Definition:Inverse Hyperbolic Secant/Real/Definition 2

Definition
Let $S$ denote the half-open real interval:
 * $S := \left({0 \,.\,.\, 1}\right]$

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


 * $\forall x \in S: \operatorname{sech}^{-1} \left({x}\right) := \ln \left({\dfrac {1 + \sqrt{1 - x^2} } x}\right)$

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.

Also see

 * Equivalence of Definitions of Real Inverse Hyperbolic Secant