# Definition:Inverse Hyperbolic Sine/Complex/Definition 2

## Definition

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.

## Also defined as

In expositions of the inverse hyperbolic functions, it is frequently the case that the $2 k \pi i$ constant is ignored, in order to simplify the presentation.

It is also commonplace to gloss over the multifunctional nature of the complex square root, and report this definition as:

- $\forall z \in \C: \map {\sinh^{-1} } z := \map \ln {z + \sqrt {z^2 + 1} }$

## 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

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.55$: Inverse Hyperbolic Functions

- Weisstein, Eric W. "Inverse Hyperbolic Sine." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/InverseHyperbolicSine.html