Talk:Inverse hyperbolic functions

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)

The hyperbola used should be either $2xy=1$ or $x^{2}-y^{2}=1$ for consistency with the unit circle interpretation of circular functions. An infinite number of possible other quadratic forms might be chosen instead, with arbitrary scaling. But the fraction of ${\textstyle {\tfrac {1}{2}}}$ is an important part of the area here (given the unit-length square as a conventional unit for area). –jacobolus (t) 05:36, 7 July 2022 (UTC)

The "ar-" prefix must go

Currently the article asserts that the "ar-" prefix is the most common usage. This is obviously false. Let's demonstrate using the inverse tanh function.

arctanh -- used by Wolfram Mathematica; Wolfram Alpha; Gradshteyn and Ryzhik; Abramowitz and Stegun;various other sources

atanh -- used by MATLAB, C math, Python math, IDL, and most other programming math packages

tanh^(-1) -- used basically everywhere in math and physics. I have little doubt that this is the dominant form in the literature.

artanh -- Bronshtein and Semendyayev, probably some other things

Now let's look at the citations. 1,3,4,5,9,10,11 support artanh. Except this is only 3 distinct authors. 1, 10, and 11 are all Bronshtein and Semendyayev, which isn't even in English; 3, 4, and 5 are all the same nobody; and 9 is a non-scientist and non-English. 7, 13, 14, and 15 use tan^(-1); 2 and 8 don't mention it; and 6 and 12 I couldn't access, though 12 apparently uses the bizarre argtanh. Meanwhile, the main sources for arctanh are each notable enough in their own right to have their own Wikipedia pages. Google search results verify that artanh is much less popular than other forms.

And yes, I read the article. "ar- is short for area" and something about Latin roots. Who cares? No one uses hyperbolic functions for actual hyperbolas. No one does math in Latin. This is just a historical artifact. Wikipedia should go with prevailing usage and not random prescriptivists. This also means the page needs to purge the whole "ar- prefix is totally correct, guys. It's everyone else who's wrong." thing.

As for whether to use arc- or ^(-1), the arc- prefix is probably better despite ^(-1)'s likely greater prevalence. It has better citations and is consistent with wiki's use of arc- for inverse trig functions.128.104.166.158 (talk) 00:10, 20 December 2017 (UTC)

I never heard of "ar" until I read this article. Its always arctanh or, in computer usage, atanh. PAR (talk) 02:51, 20 December 2017 (UTC)

I guess the article should only really say what's more common if there are reliable sources making these claims, but I don't know if such sources really even exist, or how you'd measure that. In any case, both notations are certainly in use, and the article should make note of that. I'd be okay with standardizing arc- instead of ar- here. I personally prefer a superscript −1, but it seems to be a lot less common on Wikipedia, so I wouldn't care that much. –Deacon Vorbis (carbon • videos) 03:41, 20 December 2017 (UTC)

I agree that "most common" is probably false, but the "ar-" prefix is really used in practice, e.g.: on the OEIS; in an exam (PDF); in Pure Mathematics 2 by Linda Bostock, Suzanne Chandler, F. S. Chandler; Handbook of Mathematics and Computational Science by John W. Harris, Horst Stöcker; other books that can be found on Google... Vincent Lefèvre (talk) 22:48, 20 December 2017 (UTC)

Yes, I recognize that there are people that use "ar-". My point is that it is much less common than both "arc-" and ^(-1), and major references used by millions of scientists and mathematicians use either "arc-" or ^(-1). I have no problem with the "ar-" prefix being mentioned in the notation section, but for the main body of the article either the "arc-" prefix or ^(-1) should be used.128.104.166.158 (talk) 04:47, 21 December 2017 (UTC)

NOTE that there is a parallel discussion of this on Talk:Hyperbolic function#Arc is not a misnomer PAR (talk) 05:13, 21 December 2017 (UTC)

From that talk page, FWIW, "arc" seems to win in Google Scholar, whereas "a" (not "ar") wins in Google Books, both with sinh and tanh:

	Google Scholar	Google Books	Sum
arcsinh	7440	5660	13100
asinh	5370	7890	13260
arsinh	2360	4710	7070
argsinh	479	838	1317

	Google Scholar	Google Books	Sum
arctanh	8710	5340	14050
atanh	4630	6350	10980
artanh	2070	3630	5700
argtanh	187	381	568

There could be some overlapping, but in the sums "asinh" just beats "arcsinh", but "arctanh" strongly beats "atanh". Clearly "ar" is significantly less common than both "arc" and "a", but not at all uncommon. - DVdm (talk) 07:28, 21 December 2017 (UTC)

Also note that some of them come from math software or programming languages (e.g., C), in particular those with the "a" prefix. Vincent Lefèvre (talk) 16:45, 21 December 2017 (UTC)

JSTOR search allows us to compare these to the ^(-1) construction (unfortunately Google doesn't because their algorithm is better).

	sinh	tanh
^(-1)	483	412
arc-	117	102
a-	82	60
ar-	23	26

In both cases we see that the ^(-1) construction is twice as common as all others combined, and I have no reason to think JSTOR is not representative of usage as a whole. This should probably be noted in the notation section. The main article should probably use the arc- prefix though, if only for consistency with the notation used in inverse trigonometric functions. 128.104.166.158 (talk) 00:10, 20 December 2017 (UTC)

Arc interpretation for inverse hyperbolic functions

"These are misnomers, since the prefix arc is the abbreviation for arcus, while the prefix ar stands for area; the hyperbolic functions are not directly related to arcs."

Actually, the arc interpretation works fine if you use the 1+1 Minkowski model for hyperbolic space (so that you use Minkowski arclength along the hyperbolas in the figure defining the inverse hyperbolic functions).

You can also define the inverse trigonometric functions using area, so "arsin", etc. would be appropriate from an "etymology" point of view.

This completes the symmetry between "hyperbolic" and "elliptic". Indeed, this symmetry is to be expected.

So arguing that ar or arc is more common, or is in reference books, is fine; but saying that it is a misnomer in its essence appears to be a misstatement.

2001:171B:2274:7C21:2D77:47E6:187A:F329 (talk) 20:41, 9 December 2021 (UTC)

This was also pointed out above by Lost-n-translation on 14 March 2012. Evidently this connection was missed in every cited source that has commented explicitly on "arc vs ar". Zeidler should have noticed it; he was a careful worker, and must have done many transformation where you "rotate by 90 degrees" to get from elliptic to hyperbolic.

2001:171B:2274:7C21:7834:C779:3E93:482B (talk) 16:43, 30 December 2021 (UTC)

Does anyone want to make a draft of a version of this article that discusses the "arc" interpretation in hyperbolic space? (And ideally compares with the "arclength" and "area" interpretations of trigonometric functions.) I am also not a fan of the area shown in hyperbolic angle being for the hyperbola ${\textstyle xy=1}$ instead of either ${\textstyle x^{2}-y^{2}=1}$ or ${\textstyle 2xy=1}$; area in the context of trigonometric-like functions is triangle-like (not rectangle-like) and should naturally involve halving [or if you like, the standard n-simplex can be used as a unit for n-volume instead of the unit-length n-cube]. (cf. "Keplerian Trigonometry".) You can see the ${\textstyle {\tfrac {1}{2}}}$ in e.g. shoelace formula.

If you want a source, try: Reynolds, William F. "Hyperbolic Geometry on a Hyperboloid." The American Mathematical Monthly 100.5 (1993): 442-455. –jacobolus (t) 04:54, 7 July 2022 (UTC)

There is some relevant discussion at Hyperbolic angle#Relation To The Minkowski Line Element. –jacobolus (t) 21:09, 13 July 2022 (UTC)

The problem is not whether ar(x) could be a reasonable abbreviation of "arc", "area" or "argument". The problem is what the inventors of the terminology had in mind when abbreviating. So, the only reliable sources are historical sources. D.Lazard (talk) 21:25, 13 July 2022 (UTC)

Can you clarify/elaborate? I don't think I quite understand your point. Historically, some people used the name (and abbreviation) arcsinh, arccosh, etc. (e.g. Gudermann (1833) wrote these as arc(sin=v), arc(tang=v), etc. but used roman letters for circular functions and fraktur letters for hyperbolic functions; Gudermann (1829) recommended those outside Germany who didn't want to use fraktur could use an initial capital letter Sin, etc. for hyperbolic functions; presumably following Gudermann, Ligowski (1879) writes ArcSin, etc. but in fractur). Later, other people used notation like ${\textstyle \sinh ^{-1}}$, etc. At some point, a few pedants declared that the arc notation was "incorrect" because under a Euclidean metric, the argument of hyperbolic functions does not represent arclength, and preferred calling these "argument" or "area" functions (because the the argument is half the sector area; or if you like the whole sector area using a standard simplex as unit for area). Later on, still other people pointed out that "arc" makes sense after all, because the argument is an arclength under the Minkowski metric (or if you use the split-complex plane, a.k.a. "hyperbolic number plane"), where the curve ${\textstyle x^{2}-y^{2}=1}$ is a kind of "circle" (locus of vectors with constant squared distance).

This article is currently misleading, about all three of (1) historical practice, (2) prevailing mathematical conventions, (3) mathematical correctness. It is over-relying on the claims of a few pedants whose arguments have not stood the test of time.

Personally I think it’s fine to continue using the arsinh, artanh, etc. names (as specified by ISO, etc.), but the article should not imply that these are universal (or even currently most common) or historically prior, or that the arc notation is incorrect or misleading. –jacobolus (t) 04:36, 14 July 2022 (UTC)

Apparently the arg notation comes from Houël (1878) who wrote Arg Sh, Arg Th, etc. Cf. Cajori. –jacobolus (t) 05:07, 14 July 2022 (UTC)

I just fixed up the notation section. I'll try to clean up the rest of the article at some future time, but this is enough for now; it was the section that was most directly incorrect/misleading. –jacobolus (t) 18:50, 2 May 2023 (UTC)

Opinion

The article states:

"[The terms "arcsinh", "arccosh", etc.] are misnomers, since the prefix arc is the abbreviation for arcus, while the prefix ar stands for area; the hyperbolic functions are not directly related to arcs"

This ridiculous comment displays a complete lack of understanding of how language evolves. It makes absolutely no difference whether "arc" derived from "arcus", which is not directly applicable to the inverse hyperbolic functions.

Furthermore and worse: To claim that it is a "misnomer" is inserting opinion into Wikipedia.

And no, it makes no difference that there are *citations* for this opinion. It's still an opinion. — Preceding unsigned comment added by 2601:200:c082:2ea0:4dfa:207a:64:d818 (talk) 04:16, 13 April 2023 (UTC)

I agree, and intend to fix this when I get the free time (it'll take me at least a few hours of work). If you don't want to wait indefinitely, feel free to do the research and writing required to demonstrate that the 'arc' language is widely adopted and explain why it makes sense (namely, it represents arclength in a pseudo-Euclidean plane with metric signature (1, 1)). ––jacobolus (t) 05:11, 13 April 2023 (UTC)

I cleaned this up just now. What do you think? –jacobolus (t) 18:51, 2 May 2023 (UTC)

Thanks! But I still see a number of instances of arsinh(x), arcosh(x), artanh(x). This will only sow confusion, no matter how "correct" it may be.

Bad caption

A caption below the last illustration reads as follows:

'Inverse hyperbolic functions in the complex z-plane: the colour at each point in the plane represents the complex value of the respective function at that point"

But this is useless without even the slightest attempt to tell the reader how complex values are represented by the colors.