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/