Talk:Lemniscate constant

General Formula

The general formula that ties together Gauss's constant with Gaussian integrals, the factorial or gamma function, and the beta function:

$\int _{0}^{1}{\sqrt[{m}]{1-x^{n}}}dx=\int _{0}^{1}{\sqrt[{n}]{1-x^{m}}}dx={\frac {{\tfrac {1}{m}}!{\tfrac {1}{n}}!}{({\tfrac {1}{m}}+{\tfrac {1}{n}})!}}={\frac {\int _{0}^{\infty }{e^{-x^{m}}}dx\int _{0}^{\infty }{e^{-x^{n}}}dx}{\int _{0}^{\infty }{e^{-x^{mn \over m+n}}}dx}}=(1+{\tfrac {1}{m}}+{\tfrac {1}{n}})\cdot \mathrm {B} (1+{\tfrac {1}{m}},1+{\tfrac {1}{n}})$

— 79.118.171.165 (talk) 15:48, 30 May 2013 (UTC)

More digits

as the formula mentioned in the text (section 1.0), using windows calculator's factorial button (scientific mode); it gives the value for
G = 0.83462684167407318628142973279905
knew that: 0.25 Γ(0.25)= 0.25!
Tabascofernandez (talk) 02:43, 12 July 2017 (UTC)

Requested move 25 June 2022

The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this discussion.

The result of the move request was: moved. Unopposed request made on proven common-name grounds. (closed by non-admin page mover) — Ceso femmuin mbolgaig mbung, mellohi! (投稿) 22:33, 2 July 2022 (UTC)

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Gauss's constant → Lemniscate constant – Lemniscate constant is more widely known/used and also doesn't overlap with Gauss's gravitational constant.

• Google pages of results: 12 for "Lemniscate constant", 11 for "Gauss's constant"
• Google hits with quotes: 3920 vs 1490
• Google scholar: 149 vs 43
• Google books, pages of results with direct quotes: 6 vs 12, but Gauss's results are mostly for the gravitational constant.
• Material: "Mathematical Constants" by Finch calls his section "Gauss's lemniscate constant". Also [1] uses the term "Gauss's constant" to refer to the lemniscate constant.
• Overlap: Since the two constants differ by a factor of pi almost all formulas/properties can be adjusted from one to the other Mathnerd314159 (talk) 20:04, 25 June 2022 (UTC)

Seems like a fine idea. When I was working on the Lemniscate elliptic functions article many months ago, there wasn’t such a large amount of material there about these constants. After moving material here, some of the content over there can be summarized or removed, and that section can have a link here, following WP:SS. –jacobolus (t) 02:40, 26 June 2022 (UTC)

Yeah, my plan is to shrink the section there down to {{Main|Lemniscate constant}} plus the few sentences on the role the constant plays for the functions. But I don't want to modify all the redirects twice, so first we should figure out the title. Mathnerd314159 (talk) 16:40, 26 June 2022 (UTC)

I think it’s worth leaving a few extra lines of “highlights” there, showing analogies to π. I find the Viète formula, Manchin formula, and Basel problem analogs all quite surprising. But yes as that section grows it starts to be a distraction there; copying/moving material over here gives it space to grow and some chance for its own narrative structure with more organization so it’s a bit less of a list of miscellaneous facts. –jacobolus (t) 17:09, 26 June 2022 (UTC)

Move to Lemniscate constants Since several versions of the lemniscate constant are defined in the article, I would recommend using a plural title covering all of them. –LaundryPizza03 (dc̄) 09:03, 26 June 2022 (UTC)

I think it would be safe to just stick to writing this article in terms of ϖ and π and √2 instead of a bunch of other named constants which are simple combinations of those. –jacobolus (t) 17:09, 26 June 2022 (UTC)

We could move the other constants to a new first section. Since the sources define and use all of these constants in their formulas I hesitate to rewrite the formulas to just ϖ. Mathnerd314159 (talk) 05:22, 28 June 2022 (UTC)

But as far as the plural for the title, I think it's a bad idea, because there is a single constant known as "the lemniscate constant", and references such as Transcendental number § Numbers proven to be transcendental would have to use piping [[lemniscate constants|lemniscate constant]] to reference it . Mathnerd314159 (talk) 05:31, 28 June 2022 (UTC)

Although in this case all the lemniscate constants are transcendental, so it's a bad example. But e.g. on Arithmetic–geometric mean § Related concepts ϖ is specifically referenced. Mathnerd314159 (talk) 05:35, 28 June 2022 (UTC)

Move to Lemniscate constant We still need to choose just one lemniscate constant, the article would be messy otherwise. Multiplying the constant by $2$ or $1/2$ is a triviality which is not of mathematical interest. Of course, we can mention that some people use different conventions. A1E6 (talk) 12:19, 26 June 2022 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Notation

@Mathnerd314159: Why did you use $L$ and $M$ for the first and the second lemniscate constant? They are usually denoted by $A$ and $B$ instead. A1E6 (talk) 13:57, 3 July 2022 (UTC)

Well, I thought Adlaj's L and M were the first and second lemniscate constants. But now I see that's wrong, L is ϖ. So you're right, A and B are Todd's notation and seem more standard, so I have changed it. Mathnerd314159 (talk) 16:19, 3 July 2022 (UTC)

Inline citations template

Do we need more inline citations? Please specify where exactly. A1E6 (talk) 19:25, 3 July 2022 (UTC)

Well, ideally each formula would have a citation. The forms section has no citations, and the integrals section before the circumference subsection has no citations. Also Gauss's constant in the lead has no citations. I think those are the most egregious. Mathnerd314159 (talk) 16:51, 4 July 2022 (UTC)

I have lots of citations I can add, but it's really a mess – quite often, the results in the sources are written in terms of $G$, $\Gamma (1/4)$ and other constants, but not $\varpi$. I know these are trivialities from a mathematical standpoint, but I'm not sure if that would be acceptable as "citations". A1E6 (talk) 17:23, 4 July 2022 (UTC)

Well there is WP:CALC, which as I understand it means that converting formulas to use different constants is allowed. With G vs ϖ sometimes you want the factor of pi and sometimes you don't, so in the article I would use whichever constant makes the formula simpler but cite whatever form appeared in the literature. $\Gamma (1/4)$ is a bit more estranged but I would be comfortable including it in the article if the formula used $\Gamma (1/4)^{2}$, again rewritten in terms of G or ϖ. Mathnerd314159 (talk) 15:58, 5 July 2022 (UTC)

I am not concerned about rewriting formulas from literature by changing from G to ϖ or whatever; such transformations are clearly not “original research” from the perspective of WP:OR. Still please add citations. Readers can figure out the details if they want to if they have a citation to a source. For that matter, please add referecnes to the best of the relevant literature to even if there aren’t yet specific places where it is being cited yet. It’s also possible to add proofs/justifications of otherwise mysterious statements if the referenced literature is hard to obtain or if proofs there are confusing or pitched at too advanced an audience; if proofs break up the narrative flow of the article they can be moved to a footnote, put inside Template:Collapse or the like, etc.

As an example of trivial rewriting that I think seems justified, the Wallis-like formula for ϖ in Hyde (2014) was in the form:

$$\varpi _{m}={\frac {2(m+2)}{m}}\prod _{n=1}^{\infty }{\frac {n(2(mn+1)+m)}{(mn+1)(2n+1)}}$$

Since this is more general and not as obviously analogous to the conventional expression for π, I rewrote it to the cuter version

$${\begin{aligned}{\frac {\varpi }{2}}&=\prod _{n=1}^{\infty }\left({\frac {4n-1}{4n-2}}\cdot {\frac {4n}{4n+1}}\right)\\[10mu]&={\biggl (}{\frac {3}{2}}\cdot {\frac {4}{5}}{\biggr )}{\biggl (}{\frac {7}{6}}\cdot {\frac {8}{9}}{\biggr )}{\biggl (}{\frac {11}{10}}\cdot {\frac {12}{13}}{\biggr )}\cdots \end{aligned}}$$

I don’t think this is “original research” per WP:OR; Hyde did the hardest part here of having the idea and writing the proof. –jacobolus (t) 18:21, 5 July 2022 (UTC)

Alright, I'll add the citations. A1E6 (talk) 19:28, 5 July 2022 (UTC)

Pronunciation

I removed the "(pronounced 'pi script')" from the first sentence, as I can't find any other sources for this, so it’s hard to tell if anyone other than the authors of one book call it this; to me that sounds very awkward as a name (ordinarily in English I'd expect an adjective first: "script pi"). Hunting around I see some people pronounce this letter (in other contexts) as "pomega" or "omega pi" or (following the LaTeX command) "varpi" but none of the sources I can find about this seem especially reliable – just random websites and forum comments etc. I’m not sure this symbol is common enough to have an accepted name. Maybe it would be fine to mention the "pi script" name if the statement is qualified that this name is not widely known/used. But I think it’s also fine to leave out. What do other people think? –jacobolus (t) 02:11, 8 October 2022 (UTC)

I think linking the symbol to Pi (letter)#Variant pi like you suggested in your edit is a good idea. The different pronunciations can be discussed there. Mathnerd314159 (talk) 05:32, 8 October 2022 (UTC)

Rename to 'lemniscate constants'?

I realized that this article maybe violates WP:UCRN. The only source (as far as I know) which uses 2.622... as the lemniscate constant is this: http://oeis.org/A062539. I agree with LaundryPizza03 – perhaps the name 'lemniscate constants' instead of 'lemniscate constant' is more suitable for this article. A1E6 (talk) 06:28, 15 December 2022 (UTC)

I think it's better as-is. Todd mucked things up with his weird choice of constants and naming, but most of the more recent sources either call 2.62... the "lemniscate constant", for instance:

• Hoffmann (2003) "Pi and the Arithmetic-Geometric Mean"
• Levin (2006) "A Geometric Interpretation of an Infinite Product for the Lemniscate Constant"
• Borwein "Also Eisenstein"
• Fidytek (2010) "Some Properties of the Curves $x^{n}+y^{n}=1$ with Even Exponents"
• Boyd (2011) "New series for the cosine lemniscate function and the polynomialization of the lemniscate integral"
• Hyde (2012) "A Wallis Product on Clovers"
• Clemons (2012) "Connectivity of Julia Sets for Weierstrass Elliptic Functions on Square Lattices"
• McInnes (2012) "Universality of the holographic angular momentum cutoff"
• Moree (2013) "Counting Numbers in Multiplicative Sets: Landau Versus Ramanujan"
• Pulov, Hadzhilazova, Mladenov (2014) "The Mylar Balloon"
• Fong (2014) "Analytical Methods for Squaring the Disc"
• Sherbon (2014) "Fundamental Nature of the Fine-Structure Constant"
• Takeuchi (2016) "Multiple-angle formulas of generalized trigonometric functions with two parameters"
• Osler (2016) "Iterations for the Lemniscate Constant Resembling the Archimedean Algorithm for Pi"
• Hawkins, Rocha (2018) "Dynamics and Julia sets of iterated elliptic functions"
• Lütken (2019) "Elliptic mirror of the quantum Hall effect"
• Frauenfelder (2020) "Helium and Hamiltonian delay equations"
• Skopenkov, Ustinov (2022) "Feynman checkers: towards algorithmic quantum theory"
• Finch (2022) "Errata and Addenda to Mathematical Constants"

... or introduce $\varpi$ (or another symbol e.g. $L$) as an analog of $\pi$ without ever calling it the "lemniscate constant" per se, but clearly privilege it over alternatives based on the analogy with $\pi$. Looking at all of the available sources $\varpi$ is clearly the most common out of $\varpi ,$ $2\varpi ,$ ${\tfrac {1}{2}}\varpi ,$ ${\sqrt {2}}\varpi ,$ ${\tfrac {1}{\sqrt {2}}}\varpi ,$ ${\tfrac {1+i}{2}}\varpi ,$ etc. I didn’t look too hard but I can’t find any recent sources naming those other constants.

This would be sort of like hosting pi at circle constants instead and discussing $\pi$ equally with $2\pi$, ${\tfrac {1}{2}}\pi ,$ ${\sqrt {2\pi }},$ etc.; there’s nothing inherently wrong with that as an idea, but it would be a bit cumbersome and doesn’t match the prevailing convention.

Finally, if people want to wiki-link a plural they can just add an s at the end – [[lemniscate constant]]s produces lemniscate constants – but not vice-versa. So if the choice seems like it could go either way I would generally prefer the singular form. –jacobolus (t) 17:12, 15 December 2022 (UTC)

Thanks for the reply :) A1E6 (talk) 19:12, 15 December 2022 (UTC)

Infinite Series

The german wikipedia article (see here) gives two infinite series:

$\varpi ={\frac {\pi }{\sqrt {2}}}\sum _{n=0}^{\infty }{\binom {2n}{n}}^{2}{\frac {1}{2^{5n}}}$ and $\varpi =2\sum _{n=0}^{\infty }{\binom {2n}{n}}{\frac {1}{(4n+1)2^{2n}}}$

Are there any references for that? MuCephei (talk) 12:37, 20 August 2024 (UTC)

The first one comes from the power series of $\operatorname {K}$, the complete elliptic integral of the first kind, evaluated at $1/2$ (or $1/{\sqrt {2}}$, depending on whether you use the parameter or modulus); while the second one comes from the power series of $\operatorname {sl} ^{-1}$ evaluated at $1$. The first result is known: https://functions.wolfram.com/EllipticIntegrals/EllipticK/06/01/03/01/0003/. Unfortunately, I can't seem to find a reference for the second result but it can be proved as well: $1$ is outside of the disk of convergence of the power series of $\operatorname {sl} ^{-1}$, so use

${\dbinom {2n}{n}}{\frac {1}{(4n+1)2^{2n}}}\sim {\frac {1}{{\sqrt {\pi n}}(4n+1)}}=\operatorname {O} \left({\frac {1}{n^{3/2}}}\right)$

and Abel's theorem. A1E6 (talk) 13:13, 21 August 2024 (UTC)

Thank you! MuCephei (talk) 16:36, 3 September 2024 (UTC)