Difference Between arsinh and arcsinh Functions
Date: 05/13/2004 at 13:18:17 From: Matt Subject: The missing c in arsinh Why isn't there a "c" in arsinh, arcosh, and artanh? These are the equivalents of arcsin, arccos and arctan but they are hyperbolic. I was just curious as to why there is no "c". Is there a reason? - Matt (from England)
Date: 05/13/2004 at 23:04:56 From: Doctor Peterson Subject: Re: The missing c in arsinh Hi, Matt. None of us seem to have heard of this; we all use arcsinh, arccosh, and arctanh for the inverse functions. Can you tell me where you see your versions--in a math text, a programming language, or what? Is it the standard British usage? Searching the web for the terms, I see indications that some programming languages use your forms. Perhaps because they were limited to 6 characters in a name. I also seem to see them used in German and Finnish pages, and some in the UK. I also find a few references to "arsinh" as meaning "area hyperbolic sine"; this seems to be the reason for using "ar" rather than "arc" in some languages. The idea evidently is that whereas the trig functions take an angle (or arc) as their argument, so that the inverse function returns an angle (or arc), the hyperbolic functions actually take an area (which in the case of circular functions is proportional to the arc length, so we don't see the difference). So it really does make more sense to use "area" rather than "arc"; the inverse function returns the "area" whose hyperbolic sine is such and such. I suppose we use "arc" here just because we use it for trig functions without thinking of its meaning, and therefore think of "arc" as if it just meant "inverse". I never thought about this before! Here is a reference that gives both names and the reason; it doesn't indicate just how common "arsinh" is, or where it is used: http://mathworld.wolfram.com/InverseHyperbolicSine.html Here are some other interesting references: http://planetmath.org/encyclopedia/InverseHyperbolicFunctions.html http://en.wikipedia.org/wiki/Hyperbolic_function The latter uses the more usual (at least in my part of the world) "arc" names, though it earlier emphasized that the argument is not an angle but an area. Your "ar" names turn out to be better. I'd love to know the history of this, and how some parts of the world use one set of names while the rest use one that is less meaningful. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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