Definition:Inverse Hyperbolic Sine/Real/Definition 2
Definition
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 real inverse hyperbolic sine is also known as the (real) area hyperbolic sine, as it can be used, among other things, for evaluating areas of regions bounded by hyperbolas.
Note that as the real hyperbolic sine $\sinh$ is injective, its inverse is properly a function on its domain.
Hence there is no need to make a separate distinction between branches in the same way as for real inverse hyperbolic cosine and real inverse hyperbolic secant.
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
- 1972: Frank Ayres, Jr. and J.C. Ault: Theory and Problems of Differential and Integral Calculus (SI ed.) ... (previous) ... (next): Chapter $15$: Differentiation of Hyperbolic Functions: Definitions of Inverse Hyperbolic Functions
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): inverse hyperbolic function
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): inverse hyperbolic function
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): inverse hyperbolic function