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Where is the 'c' in the spelling of the inverse hyperbolic function of sine? Isn't it just a hyperbolic version of $\operatorname{arcsin}(x)$? That is, why is it written $\operatorname{arsinh}(x)$ and not $\operatorname{arcsinh}(x)$?

The same question applies to the other functions as well.

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    $\begingroup$ arc=Arcus ($\approx$ circle) , ar=area . Never seen "arcsinh" (it doesn't make sense). $\endgroup$ – user90369 Mar 8 '18 at 11:05
  • $\begingroup$ Another notation is asinh ... but user is right. The inverse hyperbolic sine has nothing to do with arc-length, so arcsinh is nonsense. $\endgroup$ – GEdgar Mar 8 '18 at 11:14
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    $\begingroup$ @user90369, you should post as an answer and maybe expand a little $\endgroup$ – Mark S. Mar 8 '18 at 11:29
  • $\begingroup$ @MarkS.: Sorry, I don't want to waste more time on it. It's a question of ChubbyChoc and I hope he has now the answer which he has looked for. $\endgroup$ – user90369 Mar 8 '18 at 11:41
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    $\begingroup$ @Lorenzo the notazione section of the same article suggests that maybe all three of "arcsinh" "arsinh" and "settsinh" are in use. $\endgroup$ – Mark S. Mar 8 '18 at 22:44
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As user90369 pointed out, the "arc" in "$\arcsin$" comes from Latin arcus, which is directly related to the English word "arc". This makes sense because functions like $\arcsin$ give you the length of the corresponding arc of the unit circle (which also happens to be twice the area of the corresponding sector).

In contrast, "ar" is short for "area", since the hyperbolic functions are related to areas bounded by the unit hyperbola. Specifically, $\mathrm{arsinh}$ gives you twice the area of the corresponding sector and isn't related to the arclength of the piece of the hyperbola in a simple way.

Wikipedia confirms this story with references to Mathematics: From the Birth of Numbers, Oxford Users' Guide to Mathematics, and Handbook of Mathematics.

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    $\begingroup$ Nice answer. (+1) :-) $\endgroup$ – user90369 Mar 8 '18 at 13:08

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