Definition:Inverse Hyperbolic Function/Notation

Notation for Inverse Hyperbolic Function
For a given hyperbolic function, the generally accepted way to denote the corresponding inverse hyperbolic function is by prepending ar to its name.

For example:


 * for the hyperbolic sine $\sinh$, the inverse hyperbolic sine is denoted $\arsinh$


 * for the hyperbolic cosine $\cosh$, the inverse hyperbolic sine is denoted $\arcosh$

and so on.

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 notation $\sinh^{-1}$, $\cosh^{-1}$, and so on, are generally used for the inverse hyperbolic functions when they are used in their multifunction form.

However, they are often seen in the literature when used in 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.


 * $(2): \quad f^{-1}$ 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 recommends the use of the area hyperbolic function notation $\text{ar-}$ in preference to all others.