# Definition:Real Inverse Hyperbolic Function

## Definition

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.

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

### Real Inverse Hyperbolic Sine

The **inverse hyperbolic sine** $\arsinh: \R \to \R$ is a real function defined on $\R$ as:

- $\forall x \in \R: \map \arsinh x := \map \ln {x + \sqrt {x^2 + 1} }$

where:

- $\ln$ denotes the natural logarithm of a (strictly positive) real number
- $\sqrt {x^2 + 1}$ denotes the positive square root of $x^2 + 1$.

### Real Inverse Hyperbolic Cosine

Let $S$ denote the subset of the real numbers:

- $S = \set {x \in \R: x \ge 1}$

The **inverse hyperbolic cosine** $\cosh^{-1}: S \to \R$ is a real multifunction defined on $S$ as:

- $\forall x \in S: \map {\cosh^{-1} } x := \map \ln {x \pm \sqrt {x^2 - 1} }$

where:

- $\ln$ denotes the natural logarithm of a (strictly positive) real number.
- $\sqrt {x^2 - 1}$ denotes the square root of $x^2 - 1$

### Real Inverse Hyperbolic Tangent

Let $S$ denote the open real interval:

- $S := \openint {-1} 1$

The **inverse hyperbolic tangent** $\artanh: S \to \R$ is a real function defined on $S$ as:

- $\forall x \in S: \map \artanh x := \dfrac 1 2 \map \ln {\dfrac {1 + x} {1 - x} }$

where $\ln$ denotes the natural logarithm of a (strictly positive) real number.

### Real Inverse Hyperbolic Cotangent

Let $S$ denote the union of the unbounded open real intervals:

- $S := \openint \gets {-1} \cup \openint 1 \to$

The **inverse hyperbolic cotangent** $\arcoth: S \to \R$ is a real function defined on $S$ as:

- $\forall x \in S: \arcoth x := \dfrac 1 2 \map \ln {\dfrac {x + 1} {x - 1} }$

where $\ln$ denotes the natural logarithm of a (strictly positive) real number.

### Real Inverse Hyperbolic Secant

Let $S$ denote the half-open real interval:

- $S := \hointl 0 1$

The **inverse hyperbolic secant** $\sech^{-1}: S \to \R$ is a real function defined on $S$ as:

- $\forall x \in S: \map {\sech^{-1} } x := \map \ln {\dfrac {1 + \sqrt {1 - x^2} } x}$

where:

- $\ln$ denotes the natural logarithm of a (strictly positive) real number.
- $\sqrt {1 - x^2}$ denotes the positive square root of $1 - x^2$

### Real Inverse Hyperbolic Cosecant

The **inverse hyperbolic cosecant** $\arcsch: \R_{\ne 0} \to \R$ is a real function defined on the non-zero real numbers $\R_{\ne 0}$ as:

- $\forall x \in \R_{\ne 0}: \map \arcsch x := \map \ln {\dfrac 1 x + \dfrac {\sqrt {x^2 + 1} } {\size x} }$

where:

- $\sqrt {x^2 + 1}$ denotes the positive square root of $x^2 + 1$
- $\ln$ denotes the natural logarithm of a (strictly positive) real number.

## 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

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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**