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.

It is also common to see the notation $\sinh^{-1}$, $\cosh^{-1}$, and so on, for the inverse hyperbolic functions, but this is discouraged for various reasons cited in the literature, some more compelling than others:


 * $(1): \quad \sinh^{-1}$, 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.