# Definition:Complex Inverse Hyperbolic Function

## Definition

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:

### Inverse Hyperbolic Sine

The **inverse hyperbolic sine** is a multifunction defined as:

- $\forall z \in \C: \map {\sinh^{-1} } z := \set {w \in \C: z = \map \sinh w}$

where $\map \sinh w$ is the hyperbolic sine function.

### Inverse Hyperbolic Cosine

The **inverse hyperbolic cosine** is a multifunction defined as:

- $\forall z \in \C: \map {\cosh^{-1} } z := \set {w \in \C: z = \map \cosh w}$

where $\map \cosh w$ is the hyperbolic cosine function.

### Inverse Hyperbolic Tangent

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.

### Inverse Hyperbolic Cotangent

The **inverse hyperbolic cotangent** is a multifunction defined on $S$ as:

- $\forall z \in S: \map {\coth^{-1} } z := \set {w \in \C: z = \map \coth w}$

where $\map \coth w$ is the hyperbolic cotangent function.

### Inverse Hyperbolic Secant

The **inverse hyperbolic secant** is a multifunction defined as:

- $\forall z \in \C_{\ne 0}: \map {\sech^{-1} } z := \set {w \in \C: z = \map \sech w}$

where $\map \sech w$ is the hyperbolic secant function.

### Inverse Hyperbolic Cosecant

The **inverse hyperbolic cosecant** is a multifunction defined as:

- $\forall z \in \C_{\ne 0}: \map {\csch^{-1} } z := \set {w \in \C: z = \map \csch w}$

where $\map \csch w$ is the hyperbolic cosecant function.

## Also known as

The **inverse hyperbolic functions** are also known as the **area hyperbolic functions**, as they can be used, among other things, for evaluating areas of regions bounded by hyperbolas.

On $\mathsf{Pr} \infty \mathsf{fWiki}$, the term **area hyperbolic function** is specifically reserved for the principal branch of those **inverse hyperbolic functions** which are multifunctions.

Some sources refer to them as **hyperbolic arc functions**, but this is strictly a misnomer, as there is nothing **arc** related about an **inverse hyperbolic function**.

## Notation

In general, the inverse hyperbolic functions are multifunctions.

When used in their multifunction form, the notation of choice on $\mathsf{Pr} \infty \mathsf{fWiki}$ for the inverse of a hyperbolic function $\operatorname h$ is $\operatorname h^{-1}$.

When the **area** hyperbolic function is specifically required, the following prefixes are used:

- $\text {ar}$ for the real inverse hyperbolic functions
- $\text {Ar}$ for the complex inverse hyperbolic functions

where $\text{ar}$ is an abbreviation for **area**.

The prefix $\text {arc}$, borrowing from the notation for the inverse trigonometric functions, is often seen to mean the same thing, but this is erroneous.

$\text{ar}$ is an abbreviation for **area** hyperbolic function, which is another name for an inverse hyperbolic function.

The forms $\sinh^{-1}$ and $\Sinh^{-1}$, and so on, are often seen in the literature for the area hyperbolic form, that is, as a single-value function.

This can cause confusion, for the following reasons:

- $(1): \quad \sinh^{-1}$, for example, can be conflated with $\dfrac 1 {\sinh}$, as it conflicts with the similar notation $\sinh^2 x$ which means $\paren {\sinh x}^2$, and so on.

- $(2): \quad h^{-1}$ is strictly interpreted as the inverse of a mapping, and for such hyperbolic functions that are not bijective, such inverses are not actually mappings.

Hence $\mathsf{Pr} \infty \mathsf{fWiki}$ uses the notation $\text{ar-}$ or $\text{Ar-}$ for the **area** hyperbolic functions in preference to all others.

## Also see

- Results about
**inverse hyperbolic function**can be found here.

## Sources

- 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**