\begin{definition}[Definition:Inverse Hyperbolic Cotangent/Real/Definition 2]
Let $S$ denote the union of the unbounded open real intervals:
:$S := \openint \gets {-1} \cup \openint 1 \to$
The '''inverse hyperbolic cotangent''' $\arcoth: S \to \R$ is a real function defined on $S$ as:
:$\forall x \in S: \arcoth x := \dfrac 1 2 \map \ln {\dfrac {x + 1} {x - 1} }$
where $\ln$ denotes the natural logarithm of a (strictly positive) real number.
\end{definition}