Re: ArcTanh[x,y] & Wikipedia.

"Roger Beresford" <mail@xxxxxxxxxxxxxxxxxxxxxxxxxxx> writes:

> When I added the definition ArcTanh[x,y]:=Log[(x+y)/SQRT[x^2-y^2]] (due

By the way, the names of the inverse hyperbolic functions should be
arsinh, arcosh, artanh etc. and not arc*. The latin names of the functions
are 'area sinus hyperbolicus' etc. where 'area' refers to the area of a
sector bounded by the unit hyperbola. In the trigonometric case, 'arc' is
correct because the value of the function represents the length of an arc.
(It could also be considered as area of a sector and therefore, 'ar' would
in principle be correct also here, but it has never been used.) In the
hyperbolic case, there is no arc, and the use of 'arc' should be considered
as a mistake.

In the older litterature and good encylopedias the names are correct.
See e.g. Courant & John, Introduction to Calculus and Analysis, 1965;
Wolff & Gloor & Richard, Analysis Alive, 1998; Kluwer Encyclopedia of
Mathematics. In the computer algebra systems like Mathematica or Maple
the 'arc' form is used. I don't know why. A pure mistake?

SKK
--
Simo K. Kivelä Tel. + 358 9 451 3032
Helsinki University of Technology Fax + 358 9 451 3016
Institute of Mathematics E-mail Simo.Kivela@xxxxxx
P.O.Box 1100, FIN-02015 TKK, Finland http://math.tkk.fi/~kivela/
Street address: Otakaari 1, Otaniemi, Espoo http://matta.hut.fi/matta/
.

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