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