# 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.

## Sources

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- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 8$: Hyperbolic Functions: Inverse Hyperbolic Functions - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**inverse hyperbolic function** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**inverse hyperbolic function** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**inverse hyperbolic function**