Talk:List of formulae involving π

Article created
I have created the article. Regarding the name: Wiktionary says "formulae in scientific use, formulas in informal use." - but if anyone objects (or to any other aspect of the name), we can move. In any case, we can place here all the forumlae that are interesting, but not important enough to be included in Pi. I've placed a few formulae for starters. Of course, comments and expansions are welcome. -- Meni Rosenfeld (talk) 09:24, 7 April 2006 (UTC)

Article name
Apart from the burning question whether it should be "formulas", "formulae", or "formulæ", involving "Pi" or "π", isn't the notion of "involving" somewhat broad? What about sin n π = 0 and arctan 1 = π/4? Does that belong here or not?
 * Yes, I also think that the title of the page is too broad and the word 'involving' should be removed from the title to be replaced by some more appropriate word. Deepak.sachan 22:44, 8 April 2006 (UTC)


 * I'm open to suggestions. -- Meni Rosenfeld (talk) 06:10, 9 April 2006 (UTC)

With very few exceptions, the formulas listed could be used to define pi. Should we make that the distinguishing feature? If not, is there any discerning criterion as to what can go here? (Remember: Wikipedia is NOT a collection of indiscriminate information). Lambiam Talk 14:15, 8 April 2006 (UTC)
 * I guess every formula involving pi can be be used to define it, with sufficient conditions. For example, for sin &pi; = 0 you can define: &pi; is the smallest positive number such that sin x = 0. So I don't know how much can be worked out from such a criterion. The criterion should probably be how interesting and elegant the formula is, while trying to avoid too many formulae around the same theme. So it's okay to write &pi;/4 = arctan 1 and give a series expansion, but we wouldn't write it as the arctan of every possible number. Although we probably would include several efficient Machin-like formulae. -- Meni Rosenfeld (talk) 17:35, 8 April 2006 (UTC)

New approximation
Is the new formula by Fredrik really notable? It is basically just a discretization of $$\int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}$$. -- Meni Rosenfeld (talk) 12:55, 4 May 2006 (UTC)
 * It's not mine; it's the Borweins'. Regardless of whether it is trivial (I think not), the fact that such approximations can be constructed at all is certainly noteworthy. Fredrik Johansson 13:57, 4 May 2006 (UTC)

I don't see why; It's just a brute force calculation of the integral, and an impractical one as well. If one is serious about calculating Pi by evaluating the integral, he is much more likely to use Gaussian quadrature or the like. I have a mind to remove the formula, unless you strongly object. -- Meni Rosenfeld (talk) 14:20, 4 May 2006 (UTC)
 * That's a bit like omitting 355/133 because it's just a truncated continued fraction, or for that matter, log(6403203+744)/1631/2 because it's easily extracted from the series expansion of the j-function. It may be obvious to you, but that's because you already know the math behind it. Most readers won't. It'd be more helpful to amend the section with an explanation. Fredrik Johansson 14:39, 4 May 2006 (UTC)

Can't say that I agree, but I understand - so I'll leave it there. If the idea of removing it surfaces again, though, I will support the removal. -- Meni Rosenfeld (talk) 15:36, 4 May 2006 (UTC)

Name
Would the page titles Formulae involving π or Formulae using π be more appropriate for this article? Deathbob 07:28, 15 January 2007 (UTC)


 * I think not. The prefix "list of..." is common for pages which are, in fact, lists. -- Meni Rosenfeld (talk) 17:54, 15 January 2007 (UTC)

a suggestion
after reading the AFD discussion (Articles_for_deletion/List_of_formulae_involving_π), i was thinking it'd be good to have a prose description at the top of the article of what types of formulas are in the article. that is, explain that only certain formulas of historical significance or intellectual significance appear here. (or whatever criteria editors want to set.)

links to related articles are nice, too, and many of the formulas already have them. on a quick glance, i noticed the borwein formula for hexadecimal digits of pi doesn't have a link. hasn't some mad pi fanatic already created a wiki article on that? ;)

cheers, Lunch 17:45, 25 January 2007 (UTC)
 * It probably should have been deleted. There's still nothing important here. (But don't waste time discussing it.) -A876 (talk) 22:05, 14 May 2014 (UTC)

sin(1/5555)
Does anyone have a source for the sin(1/5555) limit? 70.116.1.27 (talk) 07:01, 2 December 2010 (UTC)
 * I don't have a source, but it follows from the following:
 * $$\frac{1}{180}=\frac{5}{900}=0.0055555 \cdots$$
 * $$\Rightarrow \lim_{x \rightarrow \infty} \sin \left( \frac{1}{x}^o \right) = \frac{\pi}{180x} = (0.005555 \cdots)\frac{\pi}{x}$$ Gandalf61 (talk) 13:46, 2 December 2010 (UTC)

Explanation of move from "$\pi$" to "pi"
It is unconventional to use a math symbol in a title even when that symbol is universally understood. It's Three Blind Mice, not "3 Blind Mice", a Dirac delta function, not a δ function, and so forth. In this case, "pi" is a far more common usage than "π". On Google books, the formula pi -wikipedia gets vastly more hits than formula π -wikipedia. CNN uses "pi", the BBC uses "pi", and the New York Times uses "pi". The entry in Britannica is entitled "pi," as is the entry in Merriam-Webster. Math journals can go either way. Check here and here for journal articles with "pi" in their titles. Kauffner (talk) 04:10, 19 April 2011 (UTC)
 * It was NOT moved (apparently not officially proposed). It remains at "List of formulae involving π". -A876 (talk) 22:05, 14 May 2014 (UTC)

Move discussion in progress
There is a move discussion in progress on Talk:Liu Hui's pi algorithm which affects this page. Please participate on that page and not in this talk page section. Thank you. —RM bot 07:16, 27 April 2011 (UTC)
 * It was moved to "Liu Hui's π algorithm". It remains there. -A876 (talk) 22:05, 14 May 2014 (UTC)

Move discussion in progress
There is a move discussion in progress on Talk:Pi which affects this page. Please participate on that page and not in this talk page section. Thank you. —RM bot 13:15, 2 June 2011 (UTC)
 * It was NOT moved. It remains at "Pi". -A876 (talk) 22:05, 14 May 2014 (UTC)

Is there a source for this?
Under Other infinite series and with $$\zeta(2n)$$, is there a source for this equation? What does the $$B_{2n}$$ mean? John W. Nicholson (talk) 02:29, 22 September 2013 (UTC)

Recent edits (July 5, 2017)
According to the goal stated in the article's lead, these recent edits shouldn't be added. Moreover, they appear to have been added by the original author (with just a primary source), and I'm not so sure that's okay. --Deacon Vorbis (talk) 17:02, 5 July 2017 (UTC)


 * Which goal do you mean?


 * I do not know if you refer to my editions: the new section "Other Ramanujan-like series". I have only tried to add a new section with the new formulas I got since 2002. See, for example https://projecteuclid.org/euclid.em/1087568026


 * I have written many papers using latex. I send papers to journals, to the arXiv, etc, and I have never had problems. But Wikipedia is getting me crazy. Please, tell me what is wrong and how to solve it, or otherwise, please cancel my account because I do not want to be a contributor. J314G (talk) 16:52, 7 July 2017 (UTC)


 * First of all, the article lead says, in full, "The following is a list of significant formulae involving the mathematical constant π. The list contains only formulae whose significance is established either in the article on the formula itself, the article Pi, or the article Approximations of π.". Thus, your additions would not appear to qualify.  Second, the conjectures don't belong either way.
 * But also more generally, see WP:NOR. Wikipedia isn't the place to publish original research.  That being said, the information might have a place in a more specialized article, but (as I understand things), it should probably be added by someone else that's gathered the information from a secondary source.  --Deacon Vorbis (talk) 17:30, 6 July 2017 (UTC)


 * The formulas I tried to publish in wikipedia have been cited by Jonathan Borwein, Bruce Berndt, Doron Zeilberger, etc, and all them have the opinion that they are very significant (the formulas I could prove and also the conjectured ones). In the contribution I tried to make, I cited a paper written by me, but with references to other authors (Borwein, Berndt, etc) who cite my formulas, also I gave a reference of a paper by the author Wadim Zudilin. However I agree that probably is better that other mathematicians write about my formulas in wikipedia.
 * I have been visiting several pages of wikipedia on the number pi. In the page "Demostración de la irracionalidad de pi", the first proof is FALSE (great error) and should be corrected (the english "version" of it is correct). In the page I tried to edit (list of formulas involving pi) one can read "is the Pochhammer symbol for the falling factorial". It should say "the rising factorial" instead of "falling". J314G (talk) 16:52, 7 July 2017 (UTC)


 * I am afraid that you are not understanding the nature of our objection to these additions. It has nothing to do with the quality of the mathematics you are presenting which I assume is significant and valuable. The problem has to do with the nature of the article that you are trying to add these formulae to. Because there are so many possible formulae that involve π we need to restrict what we put on this page to keep it manageable. The restriction that we are using is that the only formulae that should appear here have already appeared in other articles on Wikipedia (either an article about the formulae themselves, or one of two very specific articles mentioned above). Since your work does not fall under this umbrella, we do not want it to appear on this page. The way you can get your formulae to be appropriate for this page is to write an independent article about them first, and then they could be listed here. I caution you however that this is not as easy as it sounds. To create an independent page you need to establish the notability of the topic which requires citations to secondary sources where the topic is specifically appraised by reliable researchers. I hope this helps you understand our position. --Bill Cherowitzo (talk) 03:06, 7 July 2017 (UTC)


 * I understand what you mean.
 * I consult the wikipedia mathematical pages frequently and is a useful source of information. As there is in wikipedia a page with title "list of formulas involving pi" I thought that some of the formulas in the new family that I discovered should be in the list in order that it was updated. In addition, in my plans were to improve the page correcting mistakes, for example, it reads "falling factorial" instead of "rising factorial", replacing the title of the section "Infinite series" (all the series in the page are infinite) with "Ramanujan-type series" (the formulas in the table are the 17 series by Ramanujan), etc, but you did not give the chance. Probably I should have edit the page "Approximations of Pi" instead, which is much better in my opinion, but when I looked for the terms "Pi formulas" with google, the page I found was "list of formulas involving Pi".
 * As I have written before, the page "Demostración de la irracionalidad de pi" (written in Spanish) contains a fatal mistake: the first proof of the irrationality of Pi is FALSE from the beginning to the end (the english "version" of it is correct). J314G (talk) 16:52, 7 July 2017 (UTC)


 * If you see an error on the page, you can always just be WP:BOLD and correct it. Corrections are generally very welcome.  --Deacon Vorbis (talk) 13:12, 7 July 2017 (UTC)


 * The first proof in the page "Demostración de la irracionalidad de pi" (Spanish) is FALSE and should be deleted completely. The correct proof using a continued fraction is due to LAMBERT and a brief outline of the proof can be seen in the English "version" of this page "Proof that pi is irrational".


 * Please tell me how to change my username. J314G (talk) 16:52, 7 July 2017 (UTC)


 * The different language versions of Wikipedia are not in general maintained by the same editors and they are independent of one another. To discuss a change on the Spanish page, you need to bring it up on the talk page of that article as bringing it up here will have no effect. As to changing your username, while I do know that it can be done (I can think of two cases that I am familiar with) I do not know the particulars of what is involved outside of the fact that it does require some assistance from an administrator.--Bill Cherowitzo (talk) 14:55, 7 July 2017 (UTC)


 * Thank you. J314G (talk) 16:52, 7 July 2017 (UTC)


 * The formulas 26, 27 and 28 in section 15 of https://carma.newcastle.edu.au/jon/pi-2012.pdf are three of the 8 formulas I tried to write in wikipedia but that were removed. I ask for your approval to write them in "Approximations to Pi". — Preceding unsigned comment added by J314G (talk • contribs) 07:44, 8 July 2017 (UTC)


 * A few comments:
 * You don't need my (or anyone's) approval per se. We're all just editors here and you can always be WP:BOLD and make the additions.
 * However, it's certainly okay to discuss additions first. But you should really do so at the article in question, not here.
 * But I took a look at Approximations of π, and I'll say that I don't think those formulas are appropriate for that article. It seems to mainly be concerned with historically significant attempts to approximate π or to compute it to large numbers of digits.
 * --Deacon Vorbis (talk) 12:59, 8 July 2017 (UTC)


 * Is there no a good page in Wikipedia for very significant formulas for Pi of any nature? It seems odd!. J314G (talk) 13:51, 8 July 2017 (UTC)


 * Probably https://www.wikiwand.com/en/Pi is the most suitable place for significant formulas of any nature. I can try to write the formulas there after commenting the idea in the corresponding talk page. J314G (talk) 14:11, 8 July 2017 (UTC)


 * I have just tried to correct a mistake: at the end of the section "Infinite Series" it says "falling factorial" instead of "rising factorial". I do not know why my edit was undone. J314G (talk) 13:44, 13 July 2017 (UTC)


 * Please leave your signature at the same indent level as the rest of your comment; I've gone ahead and fixed the last few. I only happened to see this comment because I'm watching your talk page because I was involved in a conversation with you recently.  First, if you want to ask about a specific revert, ask the person who actually did the revert, or ask on the talk page of the article in question.  Second, that change was made by an IP address.  So if it was you, you weren't logged in when you made it.  Third, that edit wasn't undone; it's still current.  --Deacon Vorbis (talk) 14:24, 13 July 2017 (UTC)

Some things that should be corrected
The second formula (wrongly written) and the last one of the table of the section "Infinite Series" are the same formula. The formulas in the table are the 17 series given by Ramanujan, however one is lacking: the one with 1103 (which is in the Section "Efficient Infinite Series"). All the formulas in the table should be written in the same way: using the rising factorial. A better title for the section "Infinite Series" is "Ramanujan Series". The first one and the last one of the formulas in "Efficient Infinite Series" are not very efficient because there exist formulas much more efficient than those.

J314G (talk) 09:48, 14 July 2017 (UTC)


 * If you notice any errors, you can always just go ahead and fix them. As far as "efficient" goes, yes, it's probably a bit vague anyway.  Normal factorials should be used in place of rising factorials if appropriate (I'm assuming you mean writing $$(1)_n$$ in place of $$n!$$).  This is how they're given in the source anyway.  And yes, the section titling and organization here is definitely kind of wonky.  --Deacon Vorbis (talk) 14:09, 14 July 2017 (UTC)


 * I mean that inside the table the formulas are written sometimes with factorials and other times with rising factorials. There is an equivalence, for example (4n)!/(n!)^4 is the same than (1/2)_n(1/4)_n(3/4)_n / (n!)^3 * 256^n. I think that all the formulas in the table should be written with the same symbols, either all with ordinary factorial or either all with rising factorial which  the exceptional case of (1)_n which is more frequently written as n!. The writing with Pochhammer's symbols has some advantages, however ordinary factorial are wider known.   J314G (talk) 15:13, 14 July 2017 (UTC)

Demonstration required: Sums of powers of reciprocals
Would it be possible to add a reference with a demonstration for the nice alternating sums of reciprocals of the odd numbers? It would be nice to understand the deep meaning under the reported fact that $$\sum \left(\frac{-1}{2k+1}\right)^k \propto \pi^k$$. Giacomo Lanza (talk) 21:06, 30 April 2021 (UTC)


 * I've just added a reference. A1E6 (talk) 20:56, 9 October 2022 (UTC)

Integrals yielding π
When f(x) is a piecewise continuous function having zeros at $$x_1$$ < 0 and $$x_2$$ > 0, but f(0) ≠ 0, $$\int_{x_1}^{x_2}\frac{f-xf'}{x^2+f^2}\,dx = \pi$$ is the integral of $$\frac{d}{dx}atan\left(\frac{f}{x}\right)$$, and always yields $$\pi$$. The proof is obtained by confirming that the integrand is indeed the derivative pointed.

Hitsuji Santos(talk) 02:10, 7 September 2021 (UTC)

functional equation of the Riemann zeta function (under Miscellaneous)
This form of the functional equation is quite tortuous (and maybe torturous) so that the two sides are equivalent exchanging $$s$$ and $$1-s$$. I would suggest the much simpler form at Riemann's functional equation in the article on the zeta function. Natkuhn (talk) 22:29, 14 March 2023 (UTC)

Last two Jacobi function pi equivalencies are inaccurate
Title, both are not equal to pi. The second one is close, but I don't know why the first one is there. If someone could prove me wrong, that would be great. 131.106.140.129 (talk) 05:44, 26 June 2023 (UTC)