\begin{definition}[Definition:Inverse Hyperbolic Tangent/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 tangent''' is a multifunction defined on $S$ as:
:$\forall z \in S: \map {\tanh^{-1} } z := \set {w \in \C: z = \map \tanh w}$
where $\map \tanh w$ is the hyperbolic tangent function.
\end{definition}