\begin{definition}[Definition:Inverse Hyperbolic Secant/Real/Definition 2]
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.
\end{definition}