# Talk:Inverse hyperbolic function

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## 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
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)

All of the references are about defending the use of the "ar" terminology which is not used by any professional mathematicians I've ever encountered. The notation for inverse function is ${\displaystyle f^{-1}}$, and is universally used. — Preceding unsigned comment added by 69.142.244.49 (talk) 18:50, 19 January 2013 (UTC)

Has been discussed before. See below. - DVdm (talk) 11:23, 20 January 2013 (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 ${\displaystyle \operatorname {arcosh} \,x=\ln(x+{\sqrt {x-1}}{\sqrt {x+1}})}$ be written as ${\displaystyle \operatorname {arcosh} \,x=\ln(x+{\sqrt {x^{2}-1}})}$? (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 ${\displaystyle x=-10-10i}$, ${\displaystyle {\sqrt {x^{2}-1}}\approx 10+10i}$, while ${\displaystyle {\sqrt {x-1}}{\sqrt {x+1}}\approx -10-10i}$, i.e. the argument shifts by ${\displaystyle \pi }$. Works with ${\displaystyle x\in \mathbb {R} }$, though.(212.247.11.156 (talk) 17:42, 29 May 2008 (UTC))

If principal values are intended, perhaps they should be capitalized (${\displaystyle \operatorname {Arcrcosh} }$ and ${\displaystyle \operatorname {Ln} }$)? --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 ${\displaystyle \mathrm {coth} (y)={\frac {1}{\mathrm {tanh} (y)}}}$ then ${\displaystyle \mathrm {acoth} (x)=\mathrm {atanh} \left({\frac {1}{x}}\right)}$ and
{\displaystyle {\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
${\displaystyle \tanh \left(x+{\frac {i\pi }{2}}\right)=\coth(x)}$
or more generally
${\displaystyle \tanh \left(x+(2n+1){\frac {i\pi }{2}}\right)=\coth(x)}$
for integer ${\displaystyle n}$. So
${\displaystyle \mathrm {arctanh} (x)=\mathrm {arccoth} (x)+(2n+1){\frac {i\pi }{2}}}$
or if you want to work with principle values
${\displaystyle \mathrm {Arctanh} (x)=\mathrm {Arccoth} (x)\pm {\frac {i\pi }{2}},\qquad \Im \{x\}\gtrless 0}$
so ${\displaystyle \mathrm {arctanh} (x)}$ and ${\displaystyle \mathrm {arccoth} (x)}$ differ by a constant term, and
${\displaystyle {\frac {d}{dx}}\mathrm {arctanh} (x)={\frac {d}{dx}}\mathrm {arccoth} (x)}$

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

Even restricting ourselves to real numbers,

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

The argument of the absolute value is negative when ${\displaystyle x\in [-1,1]}$, in which case ${\displaystyle \operatorname {arcosh} \,x}$ is not real, and positive otherwise, when ${\displaystyle \operatorname {arsinh} \,x}$ 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 ${\displaystyle \cosh ^{-1}z}$. 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.)-- 19:46, 20 January 2010 (UTC)

no that ugly
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 (talkcontribs) 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)

The whole "ar" business is quite dubious. It has been claimed that, for instance, the name of the inverse hyperbolic sine in Latin is "area sinus hyperbolicus", but there are no hits for this term in Google books prior to 1960, but loads of hits for "arcsinh" (e.g., Lambert). This makes this supposed etymology very questionable indeed. Since this seems to be the only argument supporting our usage, and since it is so clearly wrong, I propose that the article go back to the standard "arc" usage, that dominates nearly all mathematical sources. Sławomir Biały (talk) 14:19, 28 September 2011 (UTC)

Well I suggested the following over 18 months ago so I'll try again: why not avoid this whole controversy and use the sinh−1 notation instead? The impression I have is that arsinh or arcsinh are used only in America. As far as I can remember, all the textbooks, academic papers, schoolteachers and university lecturers I've ever seen always use the sinh−1 (and trig sin−1) notation. -- 19:09, 28 September 2011 (UTC)
Seems like a no-brainer if you ask me ;-) Sławomir Biały (talk) 13:39, 2 October 2011 (UTC)
Florian Cajori A History of Mathematical Notations: Vol. II pp. 175-179 [2] is worth reading on this question. --catslash (talk) 14:18, 2 October 2011 (UTC)

Regardless of historical claims, the prefix ar' is not used today by research mathematicians. I am a professional mathematician (admittedly American) who spends time in Europe collaborating with European mathematicians. I read journals from Europe and the States. I have never ever in 20 years of mathematics research seen ar' until today (except perhaps in Hardy's Introduction to mathematical analysis.) European mathematicians do not in general use ar' or arc'. American mathematicians do not use ar'. The claim that the inverse hyperbolic functions have no relevance for arclength is completely false. Arclength with respect to the Poincare (hyperbolic) metric is expressed in terms of inverse hyperbolic functions. Today, mathematicians understand that the trig functions and the hyperbolic functions are two sides of the an important duality between spherical and hyperbolic geometry (noncompact and compact real forms). This understanding has gradually replaced the narrow understanding of these functions from the previous century. Lost-n-translation (talk) 18:13, 14 March 2012 (UTC)

Today a reference (Busemann & Kelly) was placed to give an example of the use of "Area cosh" for arcosh in 1953. The usage occurs in a textbook in the series Pure and Applied Mathematics edited by Paul A. Smith and Samuel Eilenberg.Rgdboer (talk) 21:16, 14 March 2012 (UTC)

I propose that either the article be returned to "sinh^{-1}" with a section on notation and its (very interesting) history, or that every ar' be replaced with the arc'---used in every standard calculus textbook in use today---with a section on the (very interesting) history of the notation. Wikipedia is not the place to conduct a crusade on reverting to historically correct' notation. It is rather a place to record the facts on the ground. The facts on the ground are that though references to ar' can be found, arc' is the prefix that is in use today. Wikipedia should reflect the usage of today.Lost-n-translation (talk) 13:44, 21 March 2012 (UTC)

The "sinh^{-1}" notation is inferior as it can be easily confused with the reciprocal function (1 over sinh). Clearly arcsinh should be the default. Tkuvho (talk) 13:30, 31 May 2012 (UTC)

By ISO 8000-2-13.20 (and ISO 31-11-9.20) it’s arsinh x. By ISO 8000-2-13.21 (and ISO 31-11-9.21) it’s arcosh x. By ISO 8000-2-13.22 (and ISO 31-11-9.22) it’s artanh x. The ^(-1) notation is nowhere to be seen in ISO 80000-2 for inverse trig or hyp. — Preceding unsigned comment added by 83.223.9.101 (talk) 13:28, 14 June 2012 (UTC)

The ^(-1) notation is the absolute standard form to mean "the inverse function." I have never in 40 years of professional use of mathematics seen your "ar" notation, and I am English. When the ^(-1) notation is used to refer to inverse trig. functions either "arc" or "a" are used. This appears to be another typical pedantic wikiwar with no relationship to actual practice in the real world. — Preceding unsigned comment added by 173.251.79.58 (talk) 18:48, 30 November 2012 (UTC)

In my experience I have hardly ever seen the ^(-1) notation. The new statement is wp:unsourced. I have undone the change. Let's stick with the consensus over this. - DVdm (talk) 09:08, 1 December 2012 (UTC)
The edits of 173.251.79.58 seemed to me to violate the neutral point of view policy, and I was considering reverting them on these grounds. --catslash (talk) 18:50, 2 December 2012 (UTC)

## Notation

Fwiw, anonymous editor 170.170.59.138 (talk · contribs) and 170.170.59.139 (talk · contribs) made these edits: ([3], [4], [5]). 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)

Took into account ([6]) the remark following the overview. DVdm (talk) 10:19, 20 October 2010 (UTC)

## Figures

The figures under Inverse hyperbolic functions in the complex plane seem to be quite unhelpful without any general explanation, and lacking a specific indication of the meaning of the colours used. —DIV (138.194.11.244 (talk) 01:43, 15 August 2011 (UTC))

There is an explanation at Trigonometric_functions#Complex_graphs - though even with this, I don't find these plots very clear. --catslash (talk) 11:33, 15 August 2011 (UTC)
The neon figures should be deleted. Tkuvho (talk) 13:43, 31 May 2012 (UTC)
Then this link Trigonometric_functions#Complex_graphs should be given in the article and not only on the Talk page. — Preceding unsigned comment added by 132.187.40.102 (talk) 14:21, 5 December 2012 (UTC)
Good point. I've found the page domain colouring and linked to that. There is also a page color wheel graphs of complex functions which, judging by the appearance, may be the specific colour-mapping used here. I still don't find these plots particularly enlightening though. --catslash (talk) 00:25, 6 December 2012 (UTC)

## artanh

The definition of artanh should be changed to one that gives the same results as the one used in major programming languages (C, C++, etc.) and libraries (MFC, Boost, etc.) and motivated in W. Kahan's article "Branch Cuts for Complex Elementary Functions, or Much Ado About Nothing's Sign Bit". The definition

${\displaystyle \operatorname {artanh} \,z={\tfrac {1}{2}}\ln {\frac {1+z}{1-z}}}$

should be changed to the principal expression

${\displaystyle \operatorname {artanh} \,z={\tfrac {1}{2}}\left(\ln(1+z)-\ln(1-z)\right)}$

The two yield the same positions of branch cuts, the same values for z not on a branch cut, but different values right on the two branch cuts. If there are no objections I will change this in a couple of days.richy (talk) 08:43, 14 December 2011 (UTC)

I'm not seeing the benefit for the change. I think the first form is more familiar and I generally take the values on the branch cuts to be more or less undefined anyway. This function isn't built into C and C++ but is part of a library, and though the exist standards for C libraries variations are certainly possible. In any case if you're a programmer writing code that's going to succeed or fail based on what value the function takes on a branch cut then I'd suggest finding a different line of work.--RDBury (talk) 13:53, 14 December 2011 (UTC)
Regarding "benefit": The benefit is that algorithms become portable when all agree about one principal form. Much harm is done by implementations differing in the choice of branch cuts. Regarding values being "undefined": We must not confuse the liberty of making a choice with the values being undefined! The values of the square root offer the same liberty but everybody expects an implementation to return its branch cut running along the negative imaginary axis and (if signed zero is not supported) to be continuous with quadrant II. Regarding "familiarity": The statement that the first is more familiar is one of perspective. I recommend reading Kahan's paper. He discusses several different forms. Regarding "variations are possible": Well, let's agree to put bugs aside, those are not variations. Then, within C and c++, our choice is basically whether we support signed zero (as IEEE-754 does) or not. Even if we don't support signed zero, i.e. if we do not distinguish between +0 and -1, then the two expressions give different results on the branch cut along the positive real axis and the first one even violates the rule of CCC (counter-clockwise continuity).richy (talk) 16:25, 14 December 2011 (UTC)
If anything is just undefined unless you give a value to log 0 and infinity. I must admit I'm surprised those expressions give a different values if calculated using IEEE functions, are you sure - what exactly did you do? If you like you could put in the values from the IEEE-2008 standard since it has now been standardized there - that's good enough to cover all the different language versions. Dmcq (talk) 15:44, 14 December 2011 (UTC)
The IEEE-754 standard does not cover complex numbers. They give different values on the branch cuts. E.g. on z=(+2,+0), the expression we quote gives (+0.5493,-1.5708) whereas the other one gives (+0.5493,+1.5708). Similar for (-2,-0), due to symmetry.richy (talk) 16:01, 14 December 2011 (UTC)
I've now completed the change. (Let me note that it is also conforming to the LIA-3 standard.)richy (talk) 22:51, 18 December 2011 (UTC)

## Area

Corrections were made noting hyperbolic angle as the value of an inverse hyperbolic function, and the correct hyperbola as xy =1. The fact that areas correspond equally to angles for this hyperbola has frequently caused confusion when the hyperbola x2 − y2 = 1 is used, for then the areas are of half the size, just as in the circular case. The larger hyperbola became the standard of reference in the early days of transcendental functions when Gregoire de Saint Vincent found the area properties of that hyperbola, setting a grounding for the natural logarithm.Rgdboer (talk) 23:29, 11 July 2012 (UTC)

## Sections "Definition" and "Logarithmic representation in the complex plane"

I have recently edited and renamed the section that was formerly named "Logarithmic representation". In fact, previous section did not named the inverse hyperbolic and contained various mathematical errors, such as introducing a spurious apparent singularity at 0 for arcoth. With this edit section "Logarithmic representation in the complex plane" becomes redundant, except for the figures. I have thus repeated the definitions of the functions for explaining the figures. However the figures set problems. Firstly, it is not explained how colors represent the two dimensions of the complex plane. Secondly, the figures for arcoth and arcsch show a discontinuity over a segment, as if there were no monodromy when following a closed path around these segments. If this is true, this requires clarification. D.Lazard (talk) 16:06, 9 December 2015 (UTC)

Hi @D.Lazard:, I just noticed you changed that last section. I disagree with the edit, since the definitions you put are inequivalent to the ones on MathWorld, for example. See my last edit on the article where I added a disputed tag with extended reason. In addition, the branch cuts of your new definition of arcosh no longer correspond to the branch cuts of the plotted graph.
You might be interested additionally in the branch cuts shown in A&S: http://people.math.sfu.ca/~cbm/aands/page_86.htm . I believe these are identical to the branch cuts in the MathWorld definitions, and to those in the shown graphs. Unless you have a good reason I think we shouldn't deviate from this convention.
(I agree that the plots are not totally clear. How exactly the color corresponds to a complex number is not indicated.) --Nanite (talk) 17:49, 9 December 2015 (UTC)

## Assertion tagged as disputed

In section "Definition", the sentence has been tagged as "disputed" with the comment (the numbering or the lines have been added for making discussion easier)

This procedure does not give the true complex principal values for two out of six functions!;reason=
(1) The proper principal value definitions are listed in the article in a later section "Logarithmic representation in the complex plane". Note the difference in definitions compared to this section.
(2) Crucially, it is NOT true that sqrt(z+1)sqrt(z-1) = sqrt(z^2-1). As a counterexample try z = -1+1i .
(3) The main problem with using arcosh(z) = log(z + sqrt(z^2-1)) is that it yields four branch cuts instead of only the one cut that is necessary.
(4) Thus the definitions for arcosh and arsech in this section ONLY apply for real x and should not be reinterpreted to find the complex versions.

Assertion (1): The definition given there is not sourced. I agree that the present definition is not sourced as well, but as far as I know the standard way to get these logarithm expressions is to solve x = hyp-function(y) in terms of y, by using quadratic formula. This provides sqrt(z^2-1) and not sqrt(z+1)sqrt(z-1). Thus, the formulas given in the previous versions of the article are apparently WP:OR.

Assertion (2): True, but there is no reason for considering sqrt(z+1)sqrt(z-1) here.

Assertion (3): Using arcosh(z) = log(z + sqrt(z^2-1)) does not give 4 branch cuts: as the argument of the log cannot be zero, the function has only two singularities, at 1, and −1. Thus, one branch cut is sufficient, consisting of the real numbers ≤ 1. Writing this, I just remark the following: turning around a singularity of a square root changes the sign of the square root. Thus, turning around both singularities does not changes the square root. It follows that one may reduce the branch cut to the real interval [−1, 1]. I'll editing the article for improving the branch cuts. By the way, this answers my question in the preceding thread.

Assertion (4): No evidence that this assertion is correct. D.Lazard (talk) 18:01, 9 December 2015 (UTC)

Here is what I mean:
These figures are the imaginary parts of the indicated formula, with argument z = horizontal axis + i*vertical axis. I calculated them in python using the numpy library which in turn defines the arccosh(), sqrt(), and log() principal values all according to standard convention. The colors are not important but where you see a discontinuity there is a branch cut. The first two figures have the conventional branch cut for real numbers <= 1. The third panel, which is your formula, has two additional branch cuts along the imaginary axis. (this is what I meant by "four branch cuts"; they all meet at the origin)
Again, I stress that Mathworld gives the correct formulas with only one branch cut in the case of arcosh. It is not original research it is just the normal definitions. --Nanite (talk) 20:40, 9 December 2015 (UTC)
(P.S. in case you find a way to isolate the arcosh branch cut along the real interval (-1,1), do publish it! As far as I can tell it is impossible, I tried to do the same some time ago.) --Nanite (talk) 20:48, 9 December 2015 (UTC)
Sorry, I mixed two problems. The problem that I have in mind was to define a principal value for inverse hyperbolic functions by using their multi-valued logarithmic representation and analytic continuation. That is applying the general approach to these particular functions. The drawback of this general approach is that, for choosing a value at a point, one has to follow a path from a specific point in the real domain to the point of evaluation. Fortunately, here, the functions may be defined by mean of logarithms and square roots, and the principal values of these basic functions may be defined without using a continuation path: the principal value of the square root is the the square root of positive real part (not defined for real negative arguments), and the principal value of the logarithm is the determination of the logarithm that has an imaginary part of smallest absolute value (also not defined for real negative arguments). Thus, every expression of an inverse hyperbolic function in terms of principal values of the log and the square roots defines a function whose domain is the set of complex points such that none of the involved log and square root have a real negative argument. The game is thus to maximize the domain of definition, which is not done, when using ${\displaystyle {\sqrt {x^{2}-1}},}$ which, in terms of principal values excludes the imaginary line, and may not be continued into a function that is continuous on this line. That is this discontinuity that makes my formulation wrong.
I'll correct the section "Definition" by removing the mention of principal values. I'll also try to clarify the section about the complex log representation. In fact, this section is rather confusing (at least for readers who are more customized with multivalued functions than with principal values). Nevertheless, this clarification needs to not be too technical, and this is a challenge.
About your PS, this is indeed impossible, as, when following, by analytic continuation, the value of arcosh around a closed path that encloses the singularities, the value jumps of 2iπ after a complete turn. Thus, an unbounded branch cut is unavoidable. D.Lazard (talk) 11:38, 10 December 2015 (UTC)
No worries. ;) A year ago I was trying to use results from some scientific articles that had this error (e.g., this arxiv formula 25) and only realized later they were trying to talk about the arcosh function, which is how I came to this page. You might also be interested in the article mentioned earlier on the talk page, which I just noticed --- William Kahan's "Branch Cuts for Complex Elementary Functions, or Much Ado About Nothing's Sign Bit". His point is that if you care about the principal values evaluated exactly on the branch cuts, then special forms need to be observed, and if you allow for signed zero then the headache worsens. Observe:
In [20]: log(-(3+0j))
Out[20]: (1.0986122886681098-3.1415926535897931j)

In [21]: log(-3+0j)
Out[21]: (1.0986122886681098+3.1415926535897931j)

I am not convinced this is the best idea but I suppose it makes sense. I guess it is best if log(z*) = log(z)* and it's better to have a value on the branch cut than having a NaN. --Nanite (talk) 21:23, 10 December 2015 (UTC)
This makes sense, but, only from the restricted point of view of (reliable) numerical computation (this is clearly Bill Kahan's point of view). Thus, if things like this would be inserted in the article, care must be taken for clearly distinguish this point of view from the general mathematical considerations. By the way, you could be interested in the site Dynamic Dictionary of Mathematical Functions, that automatically implements numerical and non numerical computations (including arbitrary precision evaluation, asymptotic, plotting, Taylor expansion, antiderivative, ... ) for inverse hyperbolic functions and many other special functions. D.Lazard (talk) 08:33, 11 December 2015 (UTC)