\begin{definition}[Definition:Inverse Hyperbolic Secant/Real/Principal Branch]
Let $S$ denote the subset of the real numbers:
:$S := \hointl 0 1$
The principal branch of the real inverse hyperbolic secant function is defined as:
:$\forall x \in S: \map \arsech x := \map \ln {\dfrac {1 + \sqrt {1 - x^2} } x}$
where:
:$\ln$ denotes the natural logarithm of a (strictly positive) real number.
:$\sqrt {1 - x^2}$ specifically denotes the positive square root of $x^2 - 1$
That is, where $\map \arsech x \ge 0$.
\end{definition}