Definition:Inverse Hyperbolic Function

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.

Real Inverse Hyperbolic Function

Let $f: \R \to \R$ be one of the hyperbolic functions on the set of real numbers.

Certain of the inverse hyperbolic function $f^{-1} \subseteq \R \times \R$ are actually multifunctions, such that for a given $y \in \R$ there may be more than one $x \in \R$ such that $y = \map f x$.

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.