\begin{definition}[Definition:Inverse Hyperbolic Cosecant/Real/Definition 2]
The '''inverse hyperbolic cosecant''' $\arcsch: \R_{\ne 0} \to \R$ is a real function defined on the non-zero real numbers $\R_{\ne 0}$ as:
:$\forall x \in \R_{\ne 0}: \map \arcsch x := \map \ln {\dfrac 1 x + \dfrac {\sqrt {x^2 + 1} } {\size x} }$
where:
:$\sqrt {x^2 + 1}$ denotes the positive square root of $x^2 + 1$
:$\ln$ denotes the natural logarithm of a (strictly positive) real number.
\end{definition}