Talk:Hyperbolic functions

Use of exponents on function names

Hi, I have seen that there is some use of the notation $\cosh ^{2}()$ to indicate $\cosh(\cosh())$. This should not be done as in some countries $\cosh ^{2}()$ actually means $[\cosh()]^{2}$, and this may be a source of confusion. It's evil. Orzetto 09:14, 16 March 2005 (UTC)

Is this actually true of non-kiddie level math in those countries: Function composition TomJF 04:09, 12 April 2006 (UTC)

For trigonometric functions, $function^{2}()$ is synonymous with $function()\cdot function()$. For example, $\cos ^{2}(t)+\sin ^{2}(t)=1$. Out of curiosity, when would you ever nest trigonometric functions? You'd run into unit problems, wouldn't you? Your basic trigonometric functions (cosine, sine, and tangent) have domains in radians, degrees, or gradians, yet have ranges in unit lengths. The cosine of the cosine of an angle would be meaningless. Sobeita (talk) 02:36, 16 December 2010 (UTC)

Almost nine years late, but why not? You might want to find the fixed point of cos(x). Double sharp (talk) 04:24, 9 December 2019 (UTC)

@Sobeita: I have replaced the multiplication symbol in your entry with a mid-dot. We often use the star glyph for a multiplication symbol in a plain text, because ASCII did not contain a mid-dot · or a times symbol ×. However, in some contexts the star symbol $*$ can be confused with the convolution operator, so it's better to avoid the star whenever more apropriate symbols are available (like in MathJax/LaTeX). --CiaPan (talk) 18:46, 13 December 2019 (UTC)

I have fixed symbols like $cosh()$ to render as $\cosh().$ --CiaPan (talk) 18:30, 13 December 2019 (UTC)

The Imaginary Unit

The article states that i is defined as the square root of -1, and that's incorrect. It's defined by i^2 = -1.

A fine case of tetrapilotomy. Yes, I guess we should distinguish between the positive and negative square roots of -1 (i.e. ±i). Urhixidur 03:15, 25 September 2006 (UTC)

Ah, that beautiful word! But it was originally tetrapyloctomy in Umberto Eco's Foucault's Pendulum, combining Greek tetra "four", Latin pilus "hair", and Greek -tomy "cutting"; thus it means "[the art of] splitting a hair four ways". Now that makes it a hybrid word, so linguistic purists may prefer to use Greek tricho- for "hair", creating tetratrichotomy (although, come to think of it, that's confusing with trichotomy). And yes, I'm aware that this comment is approaching icosapyloctomy... Double sharp (talk) 22:03, 16 March 2015 (UTC)

Wow, an interesting lesson in Greek word compounds . I see that the original comment has been addressed in the article. —Quondum 23:37, 16 March 2015 (UTC)

@Quondum: Well, of course it has been addressed; it was posted eight and a half years ago! But I couldn't resist giving this lesson .

Technically, the definition i² = −1 is still ambiguous, as both i and −i are solutions to the equation x² = −1. But this is moot because they behave exactly the same way algebraically, despite being quantitatively different. If you wanted to really nail it down, we could construct C as ordered pairs of real numbers (R²) and explicitly choose (0, 1) to be the imaginary unit and not (0, −1), but not only would that approach ogdoëcontapyloctomy (we've just split each piece of hair in quarters, haven't we?), it is also rather irrelevant for this article.

P.S. Argh, I can't help myself; one would expect *octaconta- as a Greek prefix for 80, following the more well-known triaconta- for 30 and hexeconta- for 60, but it turns out that the Ancient Greek word for 80 was actually ὁγδοήκοντα ogdoëconta, presumably related to ordinal ὄγδοος ogdoös (eighth). (I couldn't find anyone using the diaeresis on that, incidentally, but it seemed prudent here as there's no audio.) Ancient numbering systems are not remotely the most consistent in the world. Double sharp (talk) 20:49, 18 March 2015 (UTC)

I'm not sure that I'd agree with you about the ambiguity. And no, your suggestion does not nail it down any better, even by a sub-hair's breadth. But I disagree with the article saying "defined as": this is not a definition. I'll tweak it. —Quondum 04:21, 19 March 2015 (UTC)

Whatever; it's not like it's going to make any difference, as i and −i are going to behave exactly the same way in this context. Probably Imaginary unit#i and −i is the place for this. (Wait, that section says that the ambiguity can be solved in the way I mentioned? Am I missing something?) Double sharp (talk) 04:32, 19 March 2015 (UTC)

P.S. But do make it a proper definition, please. If we say "define", that's what we'd better give the reader. Double sharp (talk) 05:02, 19 March 2015 (UTC)

I already replaced the phrase "defined by". It cannot be defined adequately in half a sentence. I think we should only say "... where i is the imaginary unit." My reasoning is that there are many structures that contain elements that conform to his characterization (an infinite number of them in in the quaternions, for example). Once one has pinned it down to the complex numbers, though, this final ambiguity between two points is resolved by the equivalence: until you choose one, you cannot distinguish which you are working with – just as there is no special point on a given sphere without reference to some other points. Do you agree that we should strip off the half-hearted "definition" and leave it to the link Imaginary unit? —Quondum 14:32, 19 March 2015 (UTC)

What's wrong with defining i as the square root of -1? It seems the objection is that it is better defined as z such that z^2 =-1, where z will have two possible values. But if you define it as the square root of -1, don't you get the implication that it must be that i = -i anyway? (maybe not; I'm asking) If that is true, then the simpler definition is equivalent.-- editeur24 (talk) 01:23, 30 December 2020 (UTC)

Arc is not a misnomer

Contrary to the unreferenced assertions made above, the terms arcsinh, arccosh, etc., are not misnomers. The fact that the hyperbolic angle is equal to twice the area described in a unit hyberbola does not mean or even imply that it is not an arc. After all, even a circular angle is equal to twice the the area described in a unit circle, and nobody says that therefore it is not an arc. Nobody says "arsin" or "arcos" or "area sine" or "area cosine", but one rather says arcsin, arccos, arc sine, and arc cosine.

In fact, in this sense, the circular angle and the hyperbolic angle are quite analagous. Both circular and hyperbolic angles are equal to twice the area of their paradigmatic unit shapes. Saying "arcsin" but not saying "arcsinh" obscures this analogy to no good effect.

More importantly for Wikipedia, no good references are produced to defend the ar- usage. Instead, baseless assertions are made by individuals not known for their expertise in etymology nor Latin, who imply that the Latin word arcus cannot refer to an area. A glance at the Lewis and Short dictionary is sufficient to falsify that argument.

I propose that use of the ar- prefix be banished from this article. If good references are produced for it, then it may perhaps be profitable to make a mention of it while noting that its use is far from widespread.

Rwflammang (talk) 02:11, 16 October 2016 (UTC)

You do not provide any WP:reliable source supporting your assertion, while the article on inverse hyperbolic functions provide 3 reliable sources (each has at least an author having a WP article) asserting that arc- is a misnomer. D.Lazard (talk) 08:58, 16 October 2016 (UTC)

You misunderstand me. I do not propose putting text amounting to "Arc- is not a misnomer" into the article, so I need no source for this statement. I merely point out that there are reliable sources which use arc- and that such sources can be multiplied ad nauseum. You aren't going to make me list them all, are you? I have yet to see any reliable sources saying otherwise. Rwflammang (talk) 16:32, 16 October 2016 (UTC)

Look at notes 1, 2 and 3 in Inverse hyperbolic function § References. D.Lazard (talk) 17:09, 16 October 2016 (UTC)

This is a nonsense. The arguments of cos(z1) and arccos(z2) are any complex number. Actually, they are anything that renders the infinite series 1/0!-x^2/2!+... and its inversion well defined. They only have "arcs" and "areas" when the arguments happen to be real numbers with 1<=z1<=-1. Ok, historically they were named based on what is now seen as a narrow interpretation, and yes, in this ancient history mindset, the argument of Cosh[z] is not a normalized arc length, but twice a normalized area. So what? If arccos[z] is the length of an arc swept out by angle z on the unit circle, what is arccos[2]? Its perfectly well defined and imaginary, and cosh(-I arccos(2))=2. So do we use ARCcos[z] when -1<z<1, but "ARcos" when not? What is the "arc" associated with arccos of a 5x5 matrix of quaternions? Again, the value of the function is perfectly well defined. This is just stupid. The "arc" prefix has been redefined, in light of the extension to complex numbers, to mean "inverse" when applied to circular or hyperbolic trig functions. The decision to replace "arc" with "ar" should not be based on obsolete ancient history, it should be based on A) Modern accepted usage and B) Do we want to destroy, by our notation, the connection, via complex numbers, between the circular and hyperbolic trig functions? I vote to ignore the obsolete ancient history lesson, and go with common usage, and to not destroy the notational connection. PAR (talk) 16:53, 20 December 2017 (UTC)

This argument is WP:OR, and is thus of little value for WP. Moreover, an abbreviated name (that is what is discussed here) cannot be wrong; it may only be frequently or rarely used. Thus the above argument, based on mathematical meaning, has no more value than the etymological argument (area). Thus, there are only two things that have to be considered: The common usage(s), which is not easy to measure. The content of reliable secondary sources. We have sources that assert that "arc" is a misnomer, and, apparently, we have no reliable sources asserting the contrary (the usage, by mathematicians who have not really thought on this, is not a secondary source). So, I suggest to replace "arc is a misnomer" by "some notable authors assert that arc is a misnomer". Otherwise, Wikipedia has to make a choice for coherency across articles. This is "ar". It is not my personal preference, but as, firstly, "ar" and "arc" are both commonly used, and, secondly, some sources says that "arc" is a misnomer, there is no reason to change Wikipedia's convention. D.Lazard (talk) 18:02, 20 December 2017 (UTC)

• "Moreover, an abbreviated name (that is what is discussed here) cannot be wrong" - I agree completely.
• "Thus the above argument, based on mathematical meaning, has no more value than the etymological argument (area)." - both having value zero, I agree completely.
• "the usage, by mathematicians who have not really thought on this, is not a secondary source" - We aren't looking for a secondary source (see #1 above). If the literature were to be dominated by such mathematicians, we go with their usage.
• "ar" is not Wikipedia's convention. It's what various Wikipedia authors decide to use. I can go thru all pages changing arcos to arccos, and then, by your definition, it is Wikipedia's convention. Which it still won't be. Searching for "arcos cosine" (to eliminate non-mathematical arcos) gives 17 hits on Wikipedia, searching for "arccos cosine" gives 127. There is an "arccos" Wikipedia page, but no "arcos" Wikipedia page. etc. etc. All of which is irrelevant. Wikipedia should use one or the other consistently. The question is "which?".
• "ar" and/or "arc" do not actually mean area or arc any more, they mean "inverse of a circular or hyperbolic trig function". To imply otherwise is counterproductive.
• The bottom line is that we want to help the reader understand Wikipedia articles by using a consistent notation, and use the notation that is most commonly found in the literature, but noting other common notations.
• It is my opinion that "arc" should be used for both circular and hyperbolic functions throughout Wikipedia, with "arc" signifying inverse, nothing else, also noting other notations found in the literature. From my experience, "arc" is the most common usage. Any discussion of the origins of the different notations can be relegated to a "history" section.PAR (talk) 19:53, 20 December 2017 (UTC)

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". So indeed the literature seems to accept "arc", even if it is or looks like a misnomer. - DVdm (talk) 22:35, 20 December 2017 (UTC)

NOTE that there is a parallel discussion of this on Talk:Inverse hyperbolic functions#The "ar-" prefix must go PAR (talk) 05:18, 21 December 2017 (UTC)

I do not have the slightest doubt that the historic/linguistic roots in Latin (arcus (bow) and area (ground), perseus.tufts.edu) pale besides the prevalent prefixes (a-, arc-, ar-). Nevertheless, the use of arc- in connection with inverse hyperbolics is a misnomer. It is not the first, and will not be the last misnomer that becomes a general habit in referring to notions. I do not agree to encyclopedias having the task to avoid hurt feelings for using such made explicit misnomers, but rather to have the noble task of passing on the evidenced true roots of naming conventions. Reporting the updates in contemporary prevalence is a newly acquired advantage of electronic encyclopedias. My preference for the prefix "a-", for both inverses of "circular" and "hyperbolic" trigs, may be obvious from the above, but is no guidance for WP ("arg-" would be tedious, and ^(-1) is too mathy). Please, do not conceal that "arc-" for inverse hyperbolics results from a misnomer. Purgy (talk) 08:51, 21 December 2017 (UTC)

It is not a misnomer because inverse hyperbolic functions do represent an arc, which is imaginary. Think of hyperbolic functions as trigonometric functions with imaginary arguments and you'll understand: θ = arccosh x means that the corresponding arc length is iθ. Flora Canou (talk) 06:56, 12 December 2018 (UTC)

As an old IT boy, I tell you that the abbreviation of the scientific names of trigonometric and hyperbolic functions was determined by the fact that the names of the functions in Fortran IV could be at most 8 characters and two characters were reserved for the type of function (integer, real, double), respectively the type of argument. Function names longer than 6 characters have been truncated. Many programmers wrote the names of the functions in mathematical texts as they knew them from programming. --Turbojet (talk) 08:43, 29 March 2021 (UTC)

It is not just an imaginary arclength, but an actual one if you look at 1+1 Minkowski space with the "unit circle" (actually a hyperbola) embedded in it. You have to use the Minkowski metric to get the arclength along the "unit circle". See my comment at Talk:Inverse hyperbolic functions, section "Arc interpretation for inverse hyperbolic functions". 2001:171B:2274:7C21:59C0:D11E:8871:EC52 (talk) 22:22, 10 May 2022 (UTC)

Short description

Before January 27, the short description (imported from Wikidata) was analog of the ordinary trigonometric function. Because "analog" is confusing here, I have changed if to Mathematical function related with trigonometric functions. Macrakis changed it to Mathematical functions on hyperbolas similar to trigonometric functions on circles with edit summary "also better grammar".

I disagree with this new description for two reasons. Firstly, it is wrong or at least confusing, since "on" after "function" generally specifies the domain, being an abbreviation of "defined on" (function on a curve, function on a manifold, function on an algebraic variety, ...). Secondly, the relationship between hyperbolic functions and hyperbolas is unclear for many readers, as most applications are not related to geometry. As short descriptions are aimed for easier navigation and searching, this seem a bad idea to mention in the short description some relatively minor facts that are ignored by most readers.

About the grammar: As the article title is singular, it should refered as such in the short description. So, the plural in Makrakis' version may be confusing. IMO, this article title and Trigonometric function should be moved to plural per WP:PLURAL#Exceptions, but this is another question. (This move has been requested and done. D.Lazard (talk) 14:39, 28 January 2020 (UTC))

Do someone have a (short) formulation that is fine for everybody? D.Lazard (talk) 10:59, 28 January 2020 (UTC)

Happy to work with you on a better short description!

"related with" is rare and unidiomatic; "to" is the usual construction.

"Related to trigonometric functions" seems rather vague. They are both also related to the exponential function.

How about "The hyperbolic functions are to hyperbolas what the trigonometric functions are to circles." (Avoiding the word analog -- not quite sure why you object to it.)

Alternatively, we can emphasize their role in DE's: "The hyperbolic functions are solutions to many important differential equations."

It is, after all, a short description, so can't mention all the important characteristics.

Thoughts? --Macrakis (talk) 15:17, 28 January 2020 (UTC)

What about "Main solutions of the differential equation y″ = y "? (Apparently italics are impossible in short description.) D.Lazard (talk) 16:31, 28 January 2020 (UTC)

Too technical for a short description. --Macrakis (talk) 17:35, 28 January 2020 (UTC)

'Osborn's rule'

The 'Useful relations' section mentions 'Osborn's rule', citing a 'mnemonic' in a 1902 paper which neither proves it nor states in the way the article does. It even mentions it fails for fourth terms (?). This needs a better reference or removal. Kranix (talk | contribs) 17:03, 27 May 2020 (UTC)

@Kranix: Fixed: special:diff/959218432. --CiaPan (talk) 17:58, 27 May 2020 (UTC)

The error about '2, 6, 10, 14... sinh's' has been introduced in this edit special:diff/253404134 in 2008. The error about 'any identity' was even older. --CiaPan (talk) 18:06, 27 May 2020 (UTC)

@CiaPan: As it is now, I don't see what $\theta ,2\theta ,\varphi ,$ etc. could refer to. The exponent bit was more or less right, but wasn't explained well. As far as sourcing, it's fine to include the original, but we should include something else, especially for establishing that this is actually referred to as "Osborne's rule". I'll take a look a bit later if it's still hanging around. –Deacon Vorbis (carbon • videos) 18:21, 27 May 2020 (UTC)

@Deacon Vorbis: Hopefully some of these would qualify as RS?

• https://undergroundmathematics.org/glossary/osborns-rule
• https://archive.uea.ac.uk/jtm/4/dg4p1.pdf
• https://www.cambridge.org/core/journals/mathematical-gazette/article/9739-fibonometry/F708D0F6A464669A928835FC16FE856D

This one contains a clear proof of the rule

• https://math.stackexchange.com/questions/138842/proof-of-osbornes-rule

alas, as a user-generated content, it's not reliable enough. :( --CiaPan (talk) 19:43, 27 May 2020 (UTC)

@Deacon Vorbis: I think this would be more readable if $\theta$ and $\varphi$ were changed to x and y, respectively. Then they would correspond to the identities which follow.—Anita5192 (talk) 20:17, 27 May 2020 (UTC)

The 1902 article seems like a fine reference. The section does need amending, though, because (a) Osborne's method needs an example to be easily understood, and (b) this section goes on to things unrelated to Osborne's method, without any transition. I don't know trig functions well enough to do it myself. Also, Osborne's example in his 1902 article is too hard to understand-- pick something simple. --editeur24 (talk) 02:05, 30 December 2020 (UTC)

"Hypersine" listed at Redirects for discussion

A discussion is taking place to address the redirect Hypersine. The discussion will occur at Wikipedia:Redirects for discussion/Log/2020 July 3#Hypersine until a consensus is reached, and readers of this page are welcome to contribute to the discussion. 1234qwer1234qwer4 (talk) 17:48, 3 July 2020 (UTC)

Editing for concision.

The article begins, “In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle.” Is “ordinary” redundant? Later in the same paragraf, it says, “Also, similarly to how the derivatives…” Should “similarly to” be shortened to “just as”? Solomonfromfinland (talk) 15:52, 29 September 2024 (UTC)

No, ordinary is not redundant. The derivatives should not be mentioned in the lead section at all. –jacobolus (t) 03:08, 30 September 2024 (UTC)

Inequalities

I would like to add some useful inequalities for cosh/sinh to the page, since these are often useful when analyzing these functions, and generally hard to find online.

I tried to make an edit here: special:diff/1264119498

But it was reverted. If anyone has suggestions for improvement, I'd be happy to take them. Thomasda (talk) 22:24, 21 December 2024 (UTC)

I think D.Lazard may have been concerned that you might be trying to promote the work of the cited author, Zhu Ling, which might involve a conflict of interest (hence the edit summary "... Apparent COI and self-promoting"). The material added here also consisted entirely of decontextualized symbolic formulas whose purpose and significance is not made clear. If these formulas are "often useful", is there some secondary survey source explaining when and how, and how they fit in with other scholarship? If you can convey some of that, an addition might be easier to defend.

Part of the problem is a broader one: this article is currently extremely incomplete, not particularly well organized, not very well sourced or illustrated, etc. So there might also be a bit of an "undue weight" concern. There are at least many dozens of more important and relevant formulas to include in a generic article about the hyperbolic functions, and these inequalities seem like a bit of a recent niche topic. Making them prominent here may imply more significance than justified. Personally I don't think this is usually enough reason to remove material (unless it's way out of scope / clearly belongs at a different page), but the only real way to address such concerns is to put a whole lot more work into improving the core parts of an article, which nobody seems to be in a hurry to do here. –jacobolus (t) 22:56, 21 December 2024 (UTC)

Logistic function

The article on the logistic function discusses the hyperbolic tangent. It doesn't have to be much in this article, but I am thinking that we should at least let the reader know that there is a relationship between the hyperbolic functions and the (standard) logistic function. What do you say? —Quantling (talk | contribs) 21:04, 1 May 2025 (UTC)

Figure in Definitions section is misleading

The figure in the Definitions section makes it seem that "hyperbolic angle u" is the Euclidian angle between OA and OM, which is not true. PatmaxDaddy (talk) 15:18, 4 September 2025 (UTC)

@PatmaxDaddy: You're right, the image File:Cartesian hyperbolic rhombus.svg is misleading, it should rather be File:Cartesian hyperbolic triangle.svg. --CiaPan (talk) 15:40, 4 September 2025 (UTC)

I'm not convinced that the proposed alternative is much better. The placement of the "u", and the white arc that it's next to, are the usual way Euclidian angles are drawn. It would be better if the while arc were removed, and the "u" placed in the center of the red sector to suggest area rather than angle. — Preceding unsigned comment added by PatmaxDaddy (talk • contribs) 16:56, 4 September 2025 (UTC)

This article would benefit from a substantial rewrite with better explanation of the context. It should be made clear that, if a diagram depicts a pseudo-Euclidean plane of indefinite signature (1, 1), the hyperbolic angle is precisely analogous to a Euclidean angle. (Obviously our own eyes and environment is Euclidean, so this interpretation takes some amount of imagination.) –jacobolus (t) 18:29, 4 September 2025 (UTC)

One possibility for a drawing convention is to use a portion of a small hyperbola near the vertex, rather than a portion of a small circle. Another possibility is, as you say, to focus on areas, e.g. as this image of mine from Gudermannian function does:

–jacobolus (t) 18:38, 4 September 2025 (UTC)