# $\operatorname{arsinh}$ vs $\operatorname{arcsinh}$

I note that some people like to write the inverse hyperbolic functions not with the prefix "arc" (like regular inverse trigonometric functions), but rather "ar". This is because the prefix "arc" (for arcus) is misleading, because unlike regular trigonometric functions, they are not used to find lengths (inverse trigonometric functions can be used to find arc length of ellipses like $x^2+y^2=1$) but rather find area of a sector of the unit hyperbola.

Which version should be preferred? $\operatorname{arsinh}$ initially confused me as I did not know why "ar" was used. However, some seem to prefer this notation as it is more of an accurate description of the function.

-
I think arc for arcus would be more plausibly misleading if more people actually knew about the word... – anon Jun 12 '12 at 20:05
My vote goes to $\operatorname{arsinh}$. I've never seen the $\operatorname{angsinh}$, but i have seen $\operatorname{argsinh}$ (arg as in argument, I suppose.) – mrf Jun 12 '12 at 20:25
@Argon: "ang" for "angle", as in "angle whose sine is...". As Americo writes, you also often see the prefix "arg" for argument. In Mexico, as I remember (which may be inaccurate), the common names were angsin and argsin; arcsin was not very common. – Arturo Magidin Jun 12 '12 at 20:33
@AdamRubinson: The confusion is that the convention breaks the usual convention for trignometric/hyperbolic functions that $\sin^n(x)$ means $\left(\sin(x)\right)^n$. So the convention would have to be stated as: $$\sin^n(x) = \left\{\begin{array}{ll}(\sin x)^n&\text{if }n\neq -1;\\ \arcsin(x)\text{ (or }\sin^{-1}(x)\text{)}&\text{if }n=-1.\end{array}\right.$$Granted, this is not an unsurmountable problem, but it is at least mildly annoying. – Arturo Magidin Jun 12 '12 at 21:37
@RahulNarain Here is an integration formula I copy from a book: "$\int \frac{\sin x}{\cos^n x}\,dx = \frac{1}{(n-1)\cos^{n-1} x}$ (if $n\ne 1$)". The values $n=0,-1,\dots$ are not excluded. – user31373 Jun 12 '12 at 23:55