ArcTanh[x,y] & Wikipedia.
Roger Beresford
to KLUELESS, this group, June 2001) to Wikipedia "Hyperbolic-function",
it was deleted as being "not standard" by MACRAKIS, who accused me of
"original research"! Just as ArcTan[-x,-y] differs by Pi/2 from
ArcTan[(-x)/(-y)], the hyperbolic functions differ by i Pi/2, as the
definitions take the quadrant into account. Making trigonometric and
hyperbolic functions of z, rather than adjacent, opposite, &
hypotenuse, seems to be an oversimplification.
(Note that x & y can be complex, and are used as [y,x] in some
conventions.)
Can anyone provide me with evidence (published applications, original
source of the definition, etc.) that the use of two-argument hyperbolic
functions is well-established?
Roger Beresford.
"Veritas vincet fortius." (The greater truth must prevail?)
Julian V. Noble
In a career spanning more than 4 decades I have never come across one.
You will find definitions suitable for complex arithmetic, but not, AFIK,
a two-argument version for real values.
--
Julian V. Noble
Professor Emeritus of Physics
j...@lessspamformother.virginia.edu
^^^^^^^^^^^^^^^^^^
http://galileo.phys.virginia.edu/~jvn/
"For there was never yet philosopher that could endure the
toothache patiently."
-- Wm. Shakespeare, Much Ado about Nothing. Act v. Sc. 1.
Simo K Kivelä
> 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....@tkk.fi
P.O.Box 1100, FIN-02015 TKK, Finland http://math.tkk.fi/~kivela/
Street address: Otakaari 1, Otaniemi, Espoo http://matta.hut.fi/matta/
David N. Williams
> "Roger Beresford" <ma...@beresford22.freeserve.co.uk> 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.
A mistake in that sense, but "arc" has very wide usage. And
it's standardized in the open math standard, e.g.,
http://www.openmath.org/cocoon/openmath/cd/transc3.html#arctanh
I believe their case for standardizing such things, including
the definitions of principal branches, is compelling.
-- David
Richard J. Fateman
I would not expect the openmath standard to be held up
as an example to be followed; rather the openmath people
seem to try to follow someone else's standard. e.g. what
does "cos" mean? Well, what it usually means.
I also would not want Mathematica to become a standard
setter because it sometimes differs from the previous
convention for no reason I can see. For example,
I think reversing the arguments of arctan2 from convention.
[not the arctanh2 of the subject line]
RJF
Jean-Michel Collard
argsh,argch,argth while the inverse trigonometric functions are arcsin,
arccos,arctg.( For example Bourbaki "Fonctions d'une variable réelle",
chapter III)
(arg is for argument)
(this is not the notations in MMA or Maple)
David W. Cantrell
> Macsyma / Maxima uses asin and asinh (etc.)
>
> I would not expect the openmath standard to be held up
> as an example to be followed; rather the openmath people
> seem to try to follow someone else's standard. e.g. what
> does "cos" mean? Well, what it usually means.
>
> I also would not want Mathematica to become a standard
> setter because it sometimes differs from the previous
> convention for no reason I can see. For example,
> I think reversing the arguments of arctan2 from convention.
> [not the arctanh2 of the subject line]
>
> RJF
FWIW, since this concerns Wikipedia:
At the end of last Dec., in the Talk section, I wrote
"Arc{hyperbolic function} is a misnomer". See
<http://en.wikipedia.org/wiki/Talk:Hyperbolic_function>
if interested.
David Cantrell
David N. Williams
> Macsyma / Maxima uses asin and asinh (etc.)
Actually the language I use most (Forth) also uses asin and
asinh, for example, in the form fasinh and zasinh, for the real
and complex floating point functions.
> I would not expect the openmath standard to be held up
> as an example to be followed; rather the openmath people
> seem to try to follow someone else's standard. e.g. what
> does "cos" mean? Well, what it usually means.
Well, "what it usually means" isn't so clear for the complex
inverse functions. :-) I believe you're an expert in this
general area. I think having a standard helps here, and I have
bought into the openmath version, which I understand to be the
same as Kahan's. Is it the same as Macsyma/Maxima?
-- David