Definition:Complex Inverse Hyperbolic Function

Definition
Let $f: \C \to \C$ be one of the hyperbolic functions on the set of complex numbers.

The inverse hyperbolic function $f^{-1} \subseteq \C \times \C$ is actually a multifunction, as in general for a given $y \in \C$ there is more than one $x \in \C$ such that $y = f \left({x}\right)$.

As with the inverse trigonometric functions, it is usual to restrict the codomain of the multifunction so as to allow $f^{-1}$ to be single-valued.

There are six basic hyperbolic functions, so each of these has its inverse functions: