# Definition:Inverse Hyperbolic Sine

## Definition

## Complex Plane

### Definition 1

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.

### Definition 2

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

- $\forall z \in \C: \map {\sinh^{-1} } z := \set {\map \ln {z + \sqrt {\size {z^2 + 1} } e^{\paren {i / 2} \map \arg {z^2 + 1} } } + 2 k \pi i: k \in \Z}$

where:

- $\sqrt {\size {z^2 + 1} }$ denotes the positive square root of the complex modulus of $z^2 + 1$
- $\map \arg {z^2 + 1}$ denotes the argument of $z^2 + 1$
- $\ln$ denotes the complex natural logarithm considered as a multifunction.

## Real Numbers

### Definition 1

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

- $\forall x \in \R: \map \arsinh x := y \in \R: x = \map \sinh y$

where $\map \sinh y$ denotes the hyperbolic sine function.

### Definition 2

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 $x^2 + 1$.

## Also known as

The principal branch of the **inverse hyperbolic sine** is also known as the **area hyperbolic sine**, as it can be used, among other things, for evaluating areas of regions bounded by hyperbolas.

Some sources refer to it as **hyperbolic arcsine**, but this is strictly a misnomer, as there is nothing **arc** related about an **inverse hyperbolic sine**.

## 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

- Definition:Inverse Hyperbolic Cosine
- Definition:Inverse Hyperbolic Tangent
- Definition:Inverse Hyperbolic Cotangent
- Definition:Inverse Hyperbolic Secant
- Definition:Inverse Hyperbolic Cosecant

- Results about
**the inverse hyperbolic sine**can be found here.