Talk:Inverse hyperbolic functions

arc --> area

Hi, Wikipedia staff: In accordance with the explanation given in the article "Inverse_hyperbolic_function", the URL of the related article "List_of_integrals_of_arc_hyperbolic_functions" should be renamed to "List_of_integrals_of_area_hyperbolic_functions", shouldn't it? Thanks for your attention, 62.180.184.8 (talk) 04:20, 7 January 2009 (UTC).

It was (briefly), but then I changed it to List of integrals of inverse hyperbolic functions on the grounds that:

• the 'ar'/'area' thing is not commonly known, where as 'inverse' is widely recognized
• it's consistent with with this page

There are still redirect pages List of integrals of area hyperbolic functions and List of integrals of arc hyperbolic functions, which take you to List of integrals of inverse hyperbolic functions --catslash (talk) 10:45, 7 January 2009 (UTC)

x --> z?

Should we change x to z since all of this holds for the inverse hyperbolic functions of a complex variable? futurebird (talk) 04:12, 30 November 2007 (UTC)

acosh

Shouldn't $\operatorname {arcosh} \,x=\ln(x+{\sqrt {x-1}}{\sqrt {x+1}})$ be written as $\operatorname {arcosh} \,x=\ln(x+{\sqrt {x^{2}-1}})$? (Asech too.) MagiMaster (talk) 05:07, 16 February 2008 (UTC)

No. That gives wrong branch cuts. Fredrik Johansson 15:45, 20 March 2008 (UTC)

For example, with $x=-10-10i$, ${\sqrt {x^{2}-1}}\approx 10+10i$, while ${\sqrt {x-1}}{\sqrt {x+1}}\approx -10-10i$, i.e. the argument shifts by $\pi$. Works with $x\in \mathbb {R}$, though.(212.247.11.156 (talk) 17:42, 29 May 2008 (UTC))

If principal values are intended, perhaps they should be capitalized ($\operatorname {Arcrcosh}$ and $\operatorname {Ln}$)? --catslash (talk) 22:07, 10 July 2008 (UTC)

arcoth

The artile states the derivative of both artanh and arcoth as 1/(1-x**2). Is that a typo? —Preceding unsigned comment added by 88.112.61.116 (talk) 09:49, 30 May 2008 (UTC)

It is correct. Since $\mathrm {coth} (y)={\frac {1}{\mathrm {tanh} (y)}}$ then $\mathrm {acoth} (x)=\mathrm {atanh} \left({\frac {1}{x}}\right)$ and

${\begin{aligned}{\frac {d}{dx}}\mathrm {acoth} (x)&={\frac {d}{dx}}\mathrm {atanh} \left({\frac {1}{x}}\right)\\&={\frac {1}{1-{\frac {1}{x^{2}}}}}\left(-{\frac {1}{x^{2}}}\right)\\&={\frac {1}{1-x^{2}}}\end{aligned}}$

It is a bit surprising though, so it would be good to think of a nice way to make it clear in the article that this isn't a mistake. --catslash (talk) 15:01, 30 May 2008 (UTC)

Another way of looking at it: from the exponential definitions, it follows immediately that

$\tanh \left(x+{\frac {i\pi }{2}}\right)=\coth(x)$

or more generally

$\tanh \left(x+(2n+1){\frac {i\pi }{2}}\right)=\coth(x)$

for integer $n$. So

$\mathrm {arctanh} (x)=\mathrm {arccoth} (x)+(2n+1){\frac {i\pi }{2}}$

or if you want to work with principle values

$\mathrm {Arctanh} (x)=\mathrm {Arccoth} (x)\pm {\frac {i\pi }{2}},\qquad \Im \{x\}\gtrless 0$

so $\mathrm {arctanh} (x)$ and $\mathrm {arccoth} (x)$ differ by a constant term, and

${\frac {d}{dx}}\mathrm {arctanh} (x)={\frac {d}{dx}}\mathrm {arccoth} (x)$

--catslash (talk) 16:59, 5 July 2008 (UTC)

Even restricting ourselves to real numbers,

$\int {\frac {1}{1-x^{2}}}\,\mathrm {d} x=\int {\frac {1}{2}}\left({\frac {1}{x+1}}-{\frac {1}{x-1}}\right)\,\mathrm {d} x={\frac {1}{2}}\ln \left|{\frac {x+1}{x-1}}\right|+{\text{const.}}$

The argument of the absolute value is negative when $x\in [-1,1]$, in which case $\operatorname {arcosh} \,x$ is not real, and positive otherwise, when $\operatorname {arsinh} \,x$ isn't real.

Notation challenge

square root for artanh is wrong surely, it should be over the whole quotient

No, the square root is correct. You are forgetting that the square root is of 1-x^2, not 1+x. Since 1-x^2 factorizes, it works out: ln sqrt(1-x^2)/(1-x)=ln sqrt(1-x)sqrt(1+x)/(1-x)=ln sqrt(1+x)/sqrt(1-x)=1/2 ln (1+x)/(1-x). However, an actual issue: the 'arsinh', etc. names are much less common the arc ones. I found a mathematician on the web stating that he'd never encountered this notation. 'ar' may be more correct etymologically, but math notation doesn't work that way, it works based on what people actually use. A place like wikipedia records mathematical usage of these terms: it should use the common notation rather than spreading more confusion by having people looking up the functions be confused about whether they're thinking of the same thing. —Preceding unsigned comment added by 71.182.182.215 (talk) 09:39, 2 December 2007 (UTC)

Absolutely, it's ridiculous to come up with your own notation because you think that the common usage isn't proper. Something like that should

be left to the literature. Wikipedia should reflect the literature, not some personal conceptions about proper usage. (see below) The notation

should be changed back. Perhaps inverse hyperbolic functions are preferred, as in $\cosh ^{-1}z$. HowiAuckland (talk) 22:30, 12 May 2009 (UTC)

WP:Naming conventions says the most recognizable name to the English reader must be used, except in this case the page is not being named, rather the inverse function is being identified with a character string, possibly with a −1 superscript. Another part of the Manual of Style, Wikipedia:Manual_of_Style_(dates_and_numbers)#Unit_names_and_symbols, shows some of the issues that have arisen elsewhere. On the positive side of the arcosh and arsinh notation is the reference in Wolfram Research given, and the book reference found there. Furthermore, the explained motivation interpreting the function value as an area, as illustrated in the hyperbolic function article. Nevertheless, the scarcity of references is evident, and mathematical readers will see some innovation in this notation. Since keyboard work leading to superscripts is messy, and since there is ambiguity between reciprocal and inverse function, there is good reason to stick with arsinh and arcosh. The big question is whether the recognizability criterion is fulfilled; if not bring on the minus one.Rgdboer (talk) 04:53, 14 May 2009 (UTC)

This was a serious issue when the area/arc thing was in the article titles (List of integrals of area hyperbolic functions / List of integrals of arc hyperbolic functions), but now we have inverse (as in List of integrals of inverse hyperbolic functions), the only occurrence of ar is in the actual formulae. I'd prefer arc, on the grounds that this is what is used in most texts, but I reckon ar (or even a) is tolerable because the difference is barely noticeable. However objectionable the ar, it's surely not as bad as a −1 superscript. This argument is likely to run forever. --catslash (talk) 09:45, 14 May 2009 (UTC)

Just to respond to Rgdboer, Wolfram Research and Mathematica use ArcCosh etc. MathWorld uses ^-1 notation. I did turn up some references for the ar notation but none that I was familiar with. There is a forum discussion of this at [1].--RDBury (talk) 18:26, 2 August 2009 (UTC)

ar vs. arc (Notation challenge continued)

I just wanted to point out that the ar (as in arsinh vs. arcsinh) notation is non-standard.

Only in the US, I think. I grew up in Europe on "ar" and all textbooks and math dictionaries I saw there had an "ar", and it was pronounced "area" instead of "arc" or "arcus". I consider "arc" to be an (obvious) misnomer in this context. JanBielawski (talk) 23:08, 5 January 2010 (UTC)

Abramowitz and Stegun, Springer's Encyclopaedia of Mathematics, and Mathematica/MathWorld all use either arc or the ^-1 notation and none mentions the ar notation as an alternative. The ar notation may be more rational but it's confusing and makes Wikipedia less authoritative when it's the only one using it.--RDBury (talk) 18:04, 2 August 2009 (UTC)

You may be right if you assume American readers as the main audience. I think "ar" is standard around the world though. JanBielawski (talk) 23:08, 5 January 2010 (UTC)

Sorry, I missed that this came up in the previous section. I realize that the ar notation now has tremendous momentum in Wikipedia, so how much effort would it take to convert to the standard?--RDBury (talk) 18:10, 2 August 2009 (UTC)

Nobody appears to be defending the use of ar. Perhaps we should just change to arc if no objections are forthcoming in the next week or so? Of course ar should still be mentioned, as it appears to be backed by references. --catslash (talk) 22:10, 2 August 2009 (UTC)

It does not make sense to write 'arc' as it does not describe an arc, 'ar' makes a lot more sense, of course with trigometric functions such as sine it makes sense to write arcsin as that has reference to an arc. Besides it does not contradicts the Hyperbolic function page which states "The inverse hyperbolic functions are the area hyperbolic sine "arsinh" (also called "asinh", or sometimes by the misnomer of "arcsinh") and so on." --92.1.243.179 (talk) 15:16, 29 October 2009 (UTC)

...and this is explained in the lead section. --catslash (talk) 15:25, 29 October 2009 (UTC)

Well, there still doesn't seem to be any consensus one way or the other over this. And the only reference in this article supports "arsinh". Wouldn't it be better to avoid the issue all together and use "sinh⁻¹"? (Which, incidentally, is the only notation in the books I've ever seen.)-- Dr Greg  talk  19:46, 20 January 2010 (UTC)

no that ugly

how about the bible

http://www.math.ucla.edu/~cbm/aands/page_86.htm —Preceding unsigned comment added by 170.170.59.138 (talk) 23:06, 6 March 2010 (UTC)

Well we must infer that Dr Greg has not read Abramowitz and Stegun. Please do go out and buy a copy Dr Greg! I think/hope you will consider it money well spent. Of course you can see it at a number of places on the web, but it's nice to have a paper copy (I have two; one at work and one at home). --catslash (talk) 00:27, 7 March 2010 (UTC)

The "arcsinh" type of notation for the inverse hyperbolic functions is a pure misnomer and semantic error that came about due to a false grammatical analogy with the inverse trigonometric functions (which are arc functions), which was made possible via ignorance of these functions' historical names and inattention to detail (particularly the detail that using "arc" for the inverse hyperbolic functions makes no sense). I don't know who first started using the "arcsinh" type of notation, but what must have occured is that someone likely sometime in the early to mid-20th century noticed the correct notations of arsinh, etc., and thoughtlessly attempted to regularize the notation in conformation with arcsin, etc., without understanding why the inverse hyperbolic functions used the "ar" notation; thereby giving the false backronym of arc hyperbolic sine, in analogy with the inverse trigonometric functions. But this is a false etymology for the "ar" functions, as well as making no sense.

This incorrect usage then (in the United States, at any rate) propagated without people considering what the correct historical names for these functions were and without them considering whether or not such terminology made sense. It would be an interesting task for lexicographers to track down where this incorrect terminology originated.

In addition to the citation given on Wikipedia's "Inverse hyperbolic function" article of Jan Gullberg, Mathematics: From the Birth of Numbers (New York: W. W. Norton & Company, 1997) which points out the correct historical names of these functions and points out why the "arc" terminology is simply a mistake, see also Eberhard Zeidler, W. Hackbusch and Hans Rudolf Schwarz (editors), Bruce Hunt (translator), Oxford User's Guide to Mathematics (Oxford: Oxford University Press, 2004), Section 0.2.13: "The inverse hyperbolic functions", p. 68. Under that section heading, there is a subheading which reads "Arcsinh: The equation", but on the same page in footnote 41 is written the following:

The Latin names for the inverse hyperbolic functions are area sinus hyperbolicus, area cosinus hyperbolicus, area tangens hyperbolicus and area cotangens hyperbolicus (of x). The notation used here is from the fact that these functions give values which are the arguments of the hyperbolic functions.

Yet besides the subheading of "Arcsinh: The equation", from that point on, the section exclusively uses the "ar" notation: specifically, arsinh, arcosh, artanh, and arcoth. Perhaps the "Arcsinh" subheading was used because the authors figured a number of their U.S. readership would be familiar with it. But again, that's the only usage of it, whereupon it uses the historically and semantically correct "ar" notation.

See also Simo K Kivelä, "Re: ArcTanh[x,y] & Wikipedia.", sci.math.symbolic, October 13, 2005, Message-ID: <w4vy84xloyp.fsf@bessel.hut.fi> http://groups.google.com/group/sci.math.symbolic/msg/9b7f5a10c05f69e3 , wherein Kivelä (Senior Lecturer Emeritus of the Helsinki University of Technology, Institute of Mathematics) writes that

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. ...

--71.0.146.150 (talk) 16:37, 16 March 2010 (UTC)

nonsense

you can parse it as "h" of "arctan" if you like

anyway it *is* arc length, you just use ds^2 = dx^2 - dy^2

consistent with everything else about hyperbolic functions

in spec. rel. this come up all the time —Preceding unsigned comment added by 208.2.172.2 (talk) 20:37, 2 April 2010 (UTC)

I might also point out the usage of atanh, etc. as in programming languages. Asmeurer (talk ♬ contribs) 02:31, 3 June 2010 (UTC)

At the risk of beating a dead horse, I would like to point out that on page 127 of the NIST Handbook of Mathematical Functions recently published by the National Institute of Standards and Technology the inverse hyperbolic functions are given as "arcsinh", "arccosh", "arctanh", "arccoth", "arcsech" and "arccsch". The Preface says the Handbook was authored by "subject experts from around the world" and then proceeds to list dozens of them.

Wikipedia should consistently use throughout what has become, for better or worse, the standard names for these functions. But in the Inverse Hyperbolic Function section, explain why "arc" is a misnomer, and that the "ar" prefix is preferable. If in the future, the "ar" prefix is adopted, then Wikipedia can adopt it too. Aloha from Hawaii, Albert D. Rich (talk) 04:58, 14 August 2010 (UTC)

The problem with the NIST Handbook of Mathematical Functions vis-à-vis this matter is that it's not a reference pertaining to this subject. That is, it displays no familiarity that there even exists an issue concerning the notation. It doesn't address the matter one way or another. Every reference that I've come across which does display awareness of the issue and which does offer a usage recommendation is quite clear that the "area" form is the correct form.--Jamie Michelle (talk) 04:29, 31 August 2010 (UTC)

Notation

Fwiw, anonymous editor 170.170.59.138 (talk · contribs) and 170.170.59.139 (talk · contribs) made these edits: ([2], [3], [4]). I took the liberty to improve (and partly undo) the edits with this and this. See the edit summaries for explanation. I think it looks much better this way. DVdm (talk) 21:42, 19 October 2010 (UTC)