\begin{definition}[Definition:Inverse Hyperbolic Cotangent/Complex/Definition 1]
Let $S$ be the subset of the complex plane:
:$S = \C \setminus \set {-1 + 0 i, 1 + 0 i}$
The '''inverse hyperbolic cotangent''' is a multifunction defined on $S$ as:
:$\forall z \in S: \map {\coth^{-1} } z := \set {w \in \C: z = \map \coth w}$
where $\map \coth w$ is the hyperbolic cotangent function.
\end{definition}