Talk:Dixon elliptic functions

Citation problem
I'm relatively new editor, so please forgive me for not including a citation for multiplication formulas. Since I derived them myself (Most probably somebody else derived it earlier, but I have not recearched on who has done it earlier) I can't provide a citation. Is a written proof in talk section, a good citation source, or should I delete that information until somebody publishes a proof on another website?

Proofs:

cm duplication formula (cm(u)not=0):

at first we insert u, -u in difference formula, (cm^2(u)cm(-u)-sm(u)sm^2(-u))/(cm(u)cm^2(-u)-sm^2(u)sm(-u))

then use identities cm(-u)=1/cm(u), sm(-u)=-sm(u)/(cm(u):

(cm^2(u)/cm(u)-sm(u)sm^2(u)/cm^2(u))/(cm(u)/cm^2(u)+sm^2(u)sm(u)/cm(u))

then we multiply both sides by cm^2(u):

(cm^3(u)-sm^3(u))/(cm(u)+sm^3(u)cm(u))

then we use identity cm^3(u)=1-cm^3(u)

(2cm^3(u)-1)/(cm(u)+cm(u)(1-cm^3(u)))

And by opening brackets we get:

(2cm^3(u)-1)/(2cm(u)-cm^4(u))

sm duplication formula (cm(0)not=0):

at first we insert u, -u in difference formula, (sm(u)cm(u)-sm(-u)cm^(-u))/(cm(u)cm^2(-u)-sm^2(u)sm(-u))

then use identities cm(-u)=1/cm(u), sm(-u)=-sm(u)/(cm(u):

(cm(u)sm(u)+sm(u)/cm^2(u))/(cm(u)/cm^2(u)+sm^2(u)sm(u)/cm(u))

then we multiply both sides by cm^2(u)

(cm^3(u)sm(u)+sm(u))/(cm(u)+sm^3(u)cm(u))

we use cm^3(u)+sm^3(u)=1:

((1-sm^3(u))sm(u)+sm(u))/(cm(u)+(1-cm^3(u))cm(u))

By opening brackets we get:

(2sm(u)-sm^4(u))/(2cm(u)-cm^4(u))

For general n, I substituted (u, (n-1)u) into sum formula for cm, and sm.

Proof for triplication I will post in this thread tommorow (standart European time) if this post won't be deleted. Great Cosine (talk) 19:16, 7 January 2023 (UTC)


 * Please review WP:OR, especially WP:CALC. Short derivations like this are kind of a borderline case, but if there's no source for the formula, it makes me wonder if the formula is important enough to bother including. Apocheir (talk) 01:28, 8 January 2023 (UTC)
 * Dixon (1890) suggests: $$\begin{align}

\operatorname{sm} 2u &= \frac { \operatorname{sm} u (1 + \operatorname{cm}^3 u)} { \operatorname{cm} u (1 + \operatorname{sm}^3 u) } \\ \operatorname{cm} 2u &= \frac { \operatorname{cm}^3 u - \operatorname{sm}^3 u} { \operatorname{cm} u (1 + \operatorname{sm}^3 u) } \vphantom{\frac{}{\big|}} \end{align}$$ And also has triplication formulas on the following page. –jacobolus (t) 02:54, 8 January 2023 (UTC)
 * Dixon's duplication and triplication formulas can also be found in Robinson (2019). –jacobolus (t) 02:59, 8 January 2023 (UTC)
 * Thank you for sources, I included citation. Also can you cite a calculator work (If I want to add more specific values, can I cite calculators like WolphramAlpha in doing work, and explaining inputs. Will it count as simple calculation?)? Great Cosine (talk) 08:27, 8 January 2023 (UTC)
 * Which specific values are you hoping to add? I don’t think these necessarily need a citation (it’s more or less a routine calculation), but they also probably aren’t that valuable for readers. I put a few particular values in so that e.g. people can double-check if they have some code evaluating these functions. A table with more than maybe 10 or 12 entries might start to feel out of scope for the article. You can see that e.g. trigonometric functions includes 7 rows in its table of values, gamma function has 10 specific values listed, and Gudermannian function also has 10. –jacobolus (t) 08:37, 8 January 2023 (UTC)
 * After reading your reply, I chose not to (maybe creating page specific values, could be a good idea). Also to avoid edit war, I checked that if your cm triplication formula is correct, my cm triplication formula is also correct and it is a bit more elegant than yours.
 * Proof: let cm^3(u)=c, and sm^3(u)=s,
 * numerator: cc-s-3cs-ssc (I avoid ^2 notation in some parts to not confuse ^2c with ^(2c))
 * using identity: s+c=1
 * cc-(1-c)-3c(1-c)-c(1-c)^2,
 * cc-1+c-3c+3cc-c(1-2c+cc),
 * -1-2c+4cc-c+2cc-c^3,
 * -(c^3)+6cc-3c-1
 * denominator: c-ss+3cs+ccs,
 * using s+c=1:
 * c-((1-c)^2)+3c(1-c)+cc(1-c),
 * c-(1-2c+cc)+3c-3cc+cc-c^3,
 * -(c^3)-2cc+4c-1+2c-cc,
 * -(c^3)-3cc+6c-1.
 * Then we multiply numerator and denominator by -1, and we get: (ccc-6cc+3c+1)/(ccc+3cc-6c+1)
 * Finally we replace c with cm^3(u) and s with sm^3(u) to get original formula. (I think it is more elegant because all terms are powers of cm(u), and has same amount of terms), with sn, I suggest to change denominator only (it is the same as cm denominator), and multiply numerator and denominator by -1 (to get rid of a lot of minuses)
 * For duplication formulas I suggest writing mine nearby, like they did in list of trig identities because some people may prefer to have expressions without brackets. And also because it is useful to have cm duplication formula in cm only. Great Cosine (talk) 10:40, 8 January 2023 (UTC)
 * Sorry, I wasn't trying to clobber your formulas before, only putting more directly what was in the published sources. I don't really care one way or another about rewriting the expression for cm 3z entirely in terms of cm z; seems fine. I fixed up your sm formula (previously had a typo). I think it should be okay for readers to have it in factored form, and not worth the space to write again expanded out. –jacobolus (t) 04:49, 10 January 2023 (UTC)
 * Could it be that the values for ±pi3/12 are incorrect? They do not seem to fit well with an implementation of the Taylor series I'm writing. I'm a bit suspicious that maybe somewhere a formatting error has occured. Thanks for all the great work though! KeithWM (talk) 22:46, 13 January 2023 (UTC)
 * I didn't check the values at 1/12ths. @Great Cosine did you try sanity checking your expressions against a numerical estimate? Aside: out of curiosity, @KeithWM what are you using these functions for? –jacobolus (t) 08:02, 14 January 2023 (UTC)
 * Thank you for notifying, pi3/12 was correct, but -pi3/12 had a mistake which I already fixed. To be completely sure that these radicals are right, I plugged them in Wolphram Alpha duplication formula. Sorry for misspelling. Great Cosine (talk) 15:21, 14 January 2023 (UTC)
 * I'm using them to try to create the Lee conformal world in a tetrahedron map projections in Julia. But also just for plaing around and learning some aspects of the language. All pass with the new values! KeithWM (talk) 10:23, 18 January 2023 (UTC)
 * @KeithWM You may find https://observablehq.com/@jrus/conformal-octahedron useful –jacobolus (t) 13:57, 18 January 2023 (UTC)
 * Thanks for that! I was wondering if maybe including some imaginary (or complex) default values would be useful too? 2001:1C03:430E:1E00:24B0:FF80:B539:76F8 (talk) 11:09, 25 January 2023 (UTC)
 * Just in case, I have some calculated complex values. But in that case, I think creating another page for specific values will be useful. Should I create another page for specific values? Great Cosine (talk) 13:08, 25 January 2023 (UTC)
 * User:Great Cosine/sandbox here is how I imagine this page (work in progress) Great Cosine (talk) 14:21, 25 January 2023 (UTC)
 * If you don't have a link to a published source for these, this will all look suspiciously like original research, and some editor is likely to come delete your page, citing WP:OR or WP:N. –jacobolus (t) 17:11, 25 January 2023 (UTC)
 * A better option could be to publish it somewhere other than Wikipedia, e.g. in a paper on the arXiv, or your own website. We could provide a link at the bottom of the page. –jacobolus (t) 17:12, 25 January 2023 (UTC)

should I include half-argument formula for cm? Because it is pretty simple, and can be used in most cases. I checked it for x=pi3 and pi3/2, and it was correct. If so, which variant is better? Great Cosine (talk) 20:57, 21 January 2023 (UTC)
 * $$\alpha = 1-\sqrt[3]{4}\operatorname{cm}(x)\operatorname{sm}(x)$$
 * $$\operatorname{cm}(\frac{x}{2}) = \frac{-1 \pm_p \sqrt{\alpha} \pm_q \sqrt{3-\alpha \pm_p \frac{4\operatorname{cm^3}(x)-2}{\sqrt{\alpha}}}}{2\operatorname{cm}(x)}$$
 * $$\operatorname{cm}(\frac{x}{2}) = \frac{-1 \pm_p \sqrt{1-\sqrt[3]{4}\operatorname{cm}(x)\operatorname{sm}(x)} \pm_q \sqrt{2+\sqrt[3]{4}\operatorname{cm}(x)\operatorname{sm}(x) \pm_p \frac{4\operatorname{cm^3}(x)-2}{\sqrt{1-\sqrt[3]{4}\operatorname{cm}(x)\operatorname{sm}(x)}}}}{2\operatorname{cm}(x)}$$
 * $$\operatorname{cm}(\frac{x}{2}) = \frac{-1 \pm_p \sqrt{1-\sqrt[3]{4}\operatorname{cm}(x)\operatorname{sm}(x)} \pm_q \sqrt{2+\sqrt[3]{4}\operatorname{cm}(x)\operatorname{sm}(x) \pm_p \frac{4\operatorname{cm^3}(x)-2}{\sqrt{1-\sqrt[3]{4}\operatorname{cm}(x)\operatorname{sm}(x)}}}}{2\operatorname{cm}(x)}$$
 * $$\operatorname{cm}(\frac{x}{2}) = \frac{-1 \pm_p \sqrt{1-\sqrt[3]{4}\operatorname{cm}(x)\operatorname{sm}(x)} \pm_q \sqrt{2+\sqrt[3]{4}\operatorname{cm}(x)\operatorname{sm}(x) \pm_p \frac{4\operatorname{cm^3}(x)-2}{\sqrt{1-\sqrt[3]{4}\operatorname{cm}(x)\operatorname{sm}(x)}}}}{2\operatorname{cm}(x)}$$
 * Can you find a published source? If not, it starts to get at least a bit into a gray area about whether this kind of thing counts as "original research" or not. –jacobolus (t) 22:22, 21 January 2023 (UTC)

Some questions

 * 1) I found out and did some routine calculations that adding almost tangent equivalent: $$ \operatorname{tm} z = -\frac{\operatorname{cm} z}{\operatorname{sm} z}$$ will yield some elegant results, and will make some formulas more elegant. like $$\begin{align}

\operatorname{sm} 2u &= \frac { 2\operatorname{tm}^3 u - 1} { 2\operatorname{tm} u - \operatorname{tm}^4 u }, \\[8mu] \operatorname{tm} 2u &= \frac { 2\operatorname{sm}^3 u - 1} { 2\operatorname{sm} u - \operatorname{sm}^4 u }, \end{align}$$and

Just like sm, tm also satisfies a nice formula with cm. $$\frac{1}{\operatorname{cm}^3 z} + \frac{1}{\operatorname{tm}^3 z} = 1$$Maybe it could be included in this page?

2. Are there hyperbolic Dixon elliptic functions? Like which are related to this curve $$x^3 + \omega y^3 = 1$$? ($$x^3 - y^3 = 1$$ is just rotated $$x^3 + y^3 = 1$$, so not that intresting)

3.Should I create another page for specific values? Because somebody requested complex specific values, and I feel this is too much to include in this page, but since some people need them, so creating another page can be a good option. It's draft is here: User:Great Cosine/sandbox. Great Cosine (talk) 17:11, 25 January 2023 (UTC)


 * User:Great Cosine: These are all good questions worthy of research (and I agree with you that having 3 or 6 named functions instead of 2 is more symmetrical; though trigonometric "sine", "cosine", and "tangent" aren’t ideal analogs). But instead of just doing a bunch of your own computations, for at least some of which you are unlikely to find a source and start to get into borderline "original research" territory, you might instead try to hit the academic literature and see what kinds of sources you can dig up. For example, inre content about origami constructibles, you might start by looking at:
 * https://people.reed.edu/~jerry/Clover/cloverexcerpt.pdf
 * https://case.edu/artsci/math/langer/jlpreprints/Trefoil.pdf
 * https://link.springer.com/article/10.1007/s00009-017-1037-0
 * https://www.hindawi.com/journals/geometry/2013/897320/
 * https://www.mdpi.com/2227-7390/10/9/1512/htm
 * –jacobolus (t) 17:27, 25 January 2023 (UTC)
 * If you want to invent new names for new functions, research their properties, and so on, then Wikipedia is (for better or worse) probably not the ideal venue for publication, even if you can write up full proofs for everything. Instead you could try to publish your findings in a journal or conference, and then Wikipedia could cite those. Or you could put them on arXiv or your own website (citing non-peer-reviewed self-published sources from here gets a bit dicier, see WP:RS). –jacobolus (t) 17:32, 25 January 2023 (UTC)

generalized trigonometry
Hi User:Great Cosine. I hope it doesn't seem like I'm stepping on your toes. I cut the section about the quartic case down to just one sentence, because to put more here seems a bit out of scope for this article, unless we are going to more specifically discuss the relation to the cubic case. If we want to discuss this broader topic in more generality, it might instead be better to create a new article (or greatly expand the article generalized trigonometry), where the first several cases (n = 2, 3, 4, 5, ...) could be compared directly, alongside formulas for arbitrary n or maybe even other curves (cf. Gambini, Nicoletti & Ritelli (2021) "Keplerian Trigonometry"). –jacobolus (t) 19:24, 25 January 2023 (UTC)


 * Do you think I am allowed to create a new page for Keplerian trigonometry? And there rewiew special cases like regular Trigonometric functions, and Dixonian ones, Quartic case, etc... If so will anybody else improve that article with me? Like adding cos_4 domain colouring, etc... Great Cosine (talk) 12:58, 26 January 2023 (UTC)
 * I don't think "Keplerian trigonometry" per se is a well enough established idea to satisfy the Notability guidelines since is discussed in only one research paper. It would probably be deleted.
 * But I cited from this page a bunch of papers about a more specific kind of generalized trigonometry that results from looking at just the Fermat curves. I think an article about that one could be defended. Such an article could probably support a short section about "Keplerian trigonometry". –jacobolus (t) 13:23, 26 January 2023 (UTC)