\begin{definition}[Definition:Inverse Hyperbolic Secant/Real/Definition 1]
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:
:$\forall x \in S: \map {\sech^{-1} } x := y \in \R_{\ge 0}: x = \map \sech y$
where $\map \sech y$ denotes the hyperbolic secant function.
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}