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

Hence for $0 < x < 1$, $\map {\sech^{-1} } x$ has $2$ values.

For $x > 0$ and $x > 1$, $\map {\sech^{-1} } x$ is not defined.

Also see

 * Equivalence of Definitions of Real Area Hyperbolic Secant