\begin{definition}[Definition:Inverse Hyperbolic Cotangent/Real/Definition 1]
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:
:$\forall x \in S: \arcoth x := y \in \R: x = \coth y$
where $\coth y$ denotes the hyperbolic cotangent function.
\end{definition}