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Re: [Patch, Fortran, Doc] Inverse hyperbolic functions

Hi All,

With my remark to the info page for gfortran-4.5, I didn't want to cause so 
much trouble ;-)). However, I'm pretty sure that also in the US the correct 
name is area (co)sine/tangent hyperbolicus. I think this happened only quite 
recently that the functions got wrong names. E.g., in Mathematica they are 
also called ArcTanh etc. :-(. If you really want to change the info pages for 
gfortran, I'd also say, take the neutral name, "inverse hyperbolic 

The geometric meaning is clear. If you look at the unit hyperbola, given by 
the parametrization

\vec{r}(t)=(cosh t,sinh t); t \in |R,

then t/2 is the area of a sector of the right branch of this hyperbola.

That's pretty much the same as for the usual trigonometric functions. If you 
take the parametrization

\vec{r}(t)=(cos t,sin t), t \in [0,2 pi)

then t/2 is the area of the corresponding sector of the circle. 

In the case of the circle the angle, t, at the same time is the length of the 
arc of this sector.

This latter is not true in the analoguous parametrization of the hyperbola 
given above! That's why the inverse hyperbolic functions are named area 
functions and not arc functions.


PS: See also Wikipedia about this issue:

On Sunday 08 August 2010 19:55:08 Janus Weil wrote:
> Hi Dominique,
> > I agree that "Inverse hyperbolic *" is better than "Hyperbolic arc*"
> > (when I was taught hyperbolic functions and their inverses some 45 years
> > ago, they were denoted ch, sh, and th and the inverses argch, argsh,
> > and argth, I have no idea about the today fashion!-).
> > 
> > However the inverse hyperbolic functions appear in many places and
> > not in non-Euclidian geometry only, so I am not very fond
> > (to say the least) of "hyperbolic area". Why not
> > "computes the inverse @var{X} of the hyperbolic * (@code{*(X)})."
> > (or any suitable translation from Frenglish to native English!).
> well, I'm starting to think that there may be regional differences in
> the naming of the inverse hyperbolics (though I'm not sure about
> this)?
> The common naming convention in Germany, I think, is that the inverse
> trigonometrics are called ARCTAN etc, while in contrast the inverse
> hyperbolics are labeled ATANH or ARTANH, where the 'A' or 'AR' in
> front stands for 'area' (and not for 'arc'). This is confirmed by my
> copy of the Mathematical Handbook by Bronstein et al. (which I think
> is Russian by origin), and which even goes into some depth to explain
> where this naming comes from.
> However, I just had a look into Abramowitz & Stegun, which has
> ARCTANH, cf.
> So maybe it is indeed common to talk about an 'hyperbolic arctangent"
> in the US (and France?). What is the ultimate mathematical instance
> than we should follow? Maybe it is best to stick to the neutral
> "inverse hyperbolic tangent"? Or should we rather mention both naming
> conventions in the manual?
> Cheers,
> Janus
Hendrik van Hees
Justus-Liebig Universität Gießen
D-35392 Gießen

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