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$\begingroup$

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.

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    I think arc for arcus would be more plausibly misleading if more people actually knew about the word...2012-06-12
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    Another name for both is $\mathrm{ang\,sin}$ and $\mathrm{ang\,sinh}$.2012-06-12
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    @ArturoMagidin That's very interesting! Do you know what the "ang" prefix is for?2012-06-12
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    Please forgive me but I have always used $\operatorname{arcsinh}$ - I don't wish to be ignorant, but maybe I am.2012-06-12
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    I have seen $\sin^{-1}$ and $\sinh^{-1}$, which I think are a bit confusing (considering the meaning of $\sin^{2}$)2012-06-12
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    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.)2012-06-12
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    [In my old book](http://problemasteoremas.files.wordpress.com/2008/11/derivadas.jpg?w=500&h=660) the author used arg sh $x$, arg ch $x$ and arg th $x$, where arg stands for argument.2012-06-12
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    I think this notation came from French.2012-06-12
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    @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.2012-06-12
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    I was told that the prefix "arc" was for trigonometric functions because they get you an arc length, while the prefix "ar" was for hyperbolic functions because they get you an area. It makes sense to me that they're shorthand prefixes.2012-06-12
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    What is the confusion with sinh^-1 ? If you want to say 1/(sinh(x)) then you write (sinh(x))^-1. If you want to write the inverse sinh function then you write sinh^-1(x). Where is the confusion?2012-06-12
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    @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.2012-06-12
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    In some countries (such as Russia) $\sinh$, $\cosh$, and $\tanh$ are written as $\mathrm{sh}$, $\mathrm{ch}$, and $\mathrm{th}$... I used $\mathrm{th}$ in a paper where this function appeared a lot of times... and nobody complained because I *made clear what it means*. (Well, either that or nobody read the paper.) As long as the meaning of your notation is clear, the choice does not really matter.2012-06-12
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    @Arturo, does anybody write $\sin^{-1}x$ to mean $(\sin x)^{-1}$ (or even $\sin^{-2}x$ to mean $(\sin x)^{-2}$)? I do love the usual convention, but I feel it is restricted to positive exponents.2012-06-12
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    @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.2012-06-12
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    Some scientific calculators use "asin" and "asinh"; I guess that's where I picked up my preference. These have the advantage of being shorter than the alternatives ... and they can even be read informally as "anti-sine".2012-07-07
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    @DayLateDon That is also the name often used in programming.2012-07-07

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Which version should be preferred?

This is easy: the version you prefer. One of the most important lessons in math is to learn to become confident in your own use of the language. Math requires coming up with new symbolism all the time. Functions and variables need names, properties need names, operations need symbols; every new object you work with, you need to be confident enough to own it as your own and look at it however you want to. Start with knowing that you can work with whatever form makes most sense to you.

When you want to communicate a result, use some common term and don't purposely be obscure. But even then, it is extremely common to see that a paper summary uses a full, common term, that internally becomes something specific to the author and their own preference. There is usually an internal logic to such choices (and if there is, much the better). Which gives another reason why it's important to be flexible with one's choice of symbols: it will help you read other people's papers.