\begin{definition}[Definition:Complex Inverse Hyperbolic Function]
Let $h: \C \to \C$ be one of the hyperbolic functions on the set of complex numbers.
The '''inverse hyperbolic function''' $h^{-1} \subseteq \C \times \C$ is actually a multifunction, as in general for a given $y \in \C$ there is more than one $x \in \C$ such that $y = \map h x$.
As with the inverse trigonometric functions, it is usual to restrict the codomain of the multifunction so as to allow $h^{-1}$ to be single-valued. 
There are six basic hyperbolic functions, so each of these has its inverse functions:
\end{definition}