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[[File:Lemniscate of Bernoulli.svg|thumb|Lemniscate of Bernoulli]]
In [[Mathematics|mathematics]], the '''lemniscate constant''' {{math|''ϖ''}} is a [[Transcendental number|transcendental]] mathematical constant that is the ratio of the [[Perimeter|perimeter]] of [[Lemniscate of Bernoulli|Bernoulli's lemniscate]] to its [[Diameter#Generalizations|diameter]], analogous to the definition of [[Pi|{{pi}}]] for the circle. It is approximately equal to 2.62205755.<ref name=Finch>{{cite book |last1=Finch |first1=Steven R. |title=Mathematical Constants |date=18 August 2003 |publisher=Cambridge University Press |isbn=978-0-521-81805-6 |page=420 |url=https://www.google.com/books/edition/Mathematical_Constants/Pl5I2ZSI6uAC?hl=en |language=en}}</ref><ref>{{citation|title=Applications of generalized trigonometric functions with two parameters|last1=Kobayashi|first1=Hiroyuki|last2=Takeuchi|first2=Shingo|journal=Communications on Pure & Applied Analysis|arxiv=1903.07407|year=2019|volume=18|issue=3|pages=1509–1521|doi=10.3934/cpaa.2019072|s2cid=102487670}}</ref><ref>{{citation|title=Elliptic Gauss Sums and Hecke L-values at s=1|last1=Asai|first1=Tetsuya|arxiv=0707.3711|year=2007}}</ref>

'''Gauss's constant''', denoted by ''G'', is equal to {{math|''ϖ'' /{{pi}} ≈ 0.8346268}}.

John Todd named two more lemniscate constants, the ''first lemniscate constant'' {{math|1=''L'' = ''ϖ''/2 ≈ 1.3110287771}} and the ''second lemniscate constant'' {{math|1=''M'' = {{pi}}/(2''ϖ'') ≈ 0.5990701173}}.<ref name="todd">{{Cite journal|url=https://dl.acm.org/doi/10.1145/360569.360580|title=The lemniscate constants|last=Todd|first=John|date=January 1975|journal=[[Communications of the ACM]]|volume=18|number=1|pages=14–19|doi=10.1145/360569.360580|s2cid=85873}}</ref><ref>{{Cite web|url=http://oeis.org/A085565|title = A085565 - Oeis}}</ref><ref>{{dlmf|first=B. C.|last=Carlson|id=19.20.E2|title=Elliptic Integrals}}</ref>

Sometimes the quantities {{math|2''ϖ''}} or {{math|''L''}} are referred to as ''the'' lemniscate constant.<ref>{{Cite web | url=http://oeis.org/A064853 | title=A064853 - Oeis }}</ref><ref>{{Cite web|url=http://www.numberworld.org/digits/Lemniscate/|title=Lemniscate Constant}}</ref> Throughout this article, we strictly follow Gauss' definition for the lemniscate constant. 

== History ==

Gauss's constant is named after [[Biography:Carl Friedrich Gauss|Carl Friedrich Gauss]], who calculated it via the [[Arithmetic–geometric mean|arithmetic–geometric mean]] as <math>1/M(1,\sqrt{2})</math>.<ref name=Finch/> By 1799, Gauss had two proofs of the theorem that <math>M(1,\sqrt{2})=\frac{\pi}{\varpi}</math> where <math>\varpi</math> is the lemniscate constant.{{efn|although neither of these proofs was rigorous from the modern point of view.}}

The lemniscate constant and first lemniscate constant <math>L</math> were proven [[Transcendental number|transcendental]] by [[Biography:Theodor Schneider|Theodor Schneider]] in 1937 and the second lemniscate constant <math>M</math>and Gauss's constant <math>G</math> were proven transcendental by Theodor Schneider in 1941.<ref name="todd" /><ref>{{cite journal |first=Theodor |last=Schneider |url=https://www.deepdyve.com/lp/de-gruyter/zur-theorie-der-abelschen-funktionen-und-integrale-mn0U50bvkB |title=Zur Theorie der Abelschen Funktionen und Integrale |year=1941 |journal=Journal für die reine und angewandte Mathematik |volume=183 |number=19 |pages=110–128 |doi=10.1515/crll.1941.183.110 |s2cid=118624331 }}</ref>{{efn|In particular, he proved that the [[Beta function|beta function]] <math>\Beta (a,b)</math> is transcendental for all <math>a,b\in\mathbb{Q}\setminus\mathbb{Z}</math> such that <math>a+b\notin \mathbb{Z}_0^-</math>. The fact that <math>\varpi</math> is transcendental follows from <math>\varpi=\tfrac{1}{2}\Beta \left(\tfrac{1}{4},\tfrac{1}{2}\right)</math> and similarly for {{math|''M''}} and {{math|''G''}} from <math>\Beta \left(\tfrac{1}{2},\tfrac{3}{4}\right)</math> }} In 1975, Gregory Chudnovsky proved that the set <math>\{\pi,\varpi\}</math> is [[Algebraic independence|algebraically independent]] over <math>\mathbb{Q}</math>, which implies that <math>L</math> and <math>M</math> are algebraically independent as well.<ref>G. V. Choodnovsky: ''Algebraic independence of constants connected with the functions of analysis'', Notices of the AMS 22, 1975, p. A-486</ref><ref>G. V. Chudnovsky: ''Contributions to The Theory of Transcendental Numbers'', American Mathematical Society, 1984, p. 6</ref> But the set <math>\{\pi,M(1,1/\sqrt{2}),M'(1,1/\sqrt{2})\}</math> (where the prime denotes the [[Derivative|derivative]] with respect to the second variable) is not algebraically independent over <math>\mathbb{Q}</math>. In fact,<ref>{{Cite book |last1=Borwein |first1=Jonathan M. |last2=Borwein| first2=Peter B. |title=Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity |publisher=Wiley-Interscience |year=1987 |edition=First |isbn=0-471-83138-7}} p. 45</ref>

<math display="block">\pi=2\sqrt{2}\frac{M^3(1,1/\sqrt{2})}{M'(1,1/\sqrt{2})}=\frac{1}{G^3 M'(1,1/\sqrt{2})}.</math>

== Forms ==

Usually, <math>\varpi</math> is defined by the first equality below.

<math display=block>\begin{aligned}
\varpi
&= 2\int_0^1\frac{\mathrm{d}t}{\sqrt{1-t^4}}
 = \sqrt2\int_0^\infty\frac{\mathrm{d}t}{\sqrt{1+t^4}}
 = \int_0^1\frac{\mathrm dt}{\sqrt{t-t^3}} \\[6mu]
&= 4\int_0^\infty\Bigl(\sqrt[4]{1+t^{4}}-t\Bigr)\,\mathrm{d}t
 = 2\sqrt2\int_0^1 \sqrt[4]{1-t^{4}}\mathop{\mathrm{d}t} =3\int_0^1 \sqrt{1-t^4}\,\mathrm dt\\[2mu]
&= 2K(i)
 = \tfrac{1}{2}\Beta\bigl( \tfrac14, \tfrac12\bigr)
 = \frac{\Gamma (1/4)^2}{2\sqrt{2\pi}}
 = \sqrt{\pi}e^{\beta '(0)}
 = \frac{2-\sqrt{2}}{4}\frac{\zeta(3/4)^2}{\zeta(1/4)^2}\\[5mu]
&= 2.62205\;75542\;92119\;81046\;48395\;89891\;11941\ldots,
\end{aligned}</math>

where {{math|''K''}} is the [[Elliptic integral#Complete elliptic integral of the first kind|complete elliptic integral of the first kind]] with modulus {{math|''k''}}, {{math|&Beta;}} is the [[Beta function|beta function]], {{math|&Gamma;}} is the [[Gamma function|gamma function]], {{math|β'}} is the derivative of the [[Dirichlet beta function]] and {{math|ζ}} is the [[Riemann zeta function]]. 

The lemniscate constant can also be computed by the [[Arithmetic–geometric mean|arithmetic–geometric mean]] <math>\operatorname{M}</math>,

<math display=block>\varpi=\frac{\pi}{\operatorname{M}(1,\sqrt{2})}.</math>

Gauss's constant is typically defined as the [[Multiplicative inverse|reciprocal]] of the [[Arithmetic–geometric mean|arithmetic–geometric mean]] of 1 and the [[Square root of 2|square root of 2]], after his calculation of <math>\operatorname{M}\left(1, \sqrt{2}\right)</math> published in 1800:{{sfn|Cox|1984|p=277}}

<math display=block>G = \frac{1}{\operatorname{M}\left(1, \sqrt{2}\right)}</math>

Gauss's constant is equal to

<math display=block> G = \frac{1}{2\pi}\Beta\bigl( \tfrac14, \tfrac12\bigr)</math>

where Β denotes the [[Beta function|beta function]]. A formula for ''G'' in terms of  [[Theta function|Jacobi theta function]]s is given by

<math display=block>G = \vartheta_{01}^2\left(e^{-\pi}\right) </math>

Gauss's constant may be computed from the [[Particular values of the gamma function#General rational argument|gamma function at argument&nbsp;]]{{sfrac|1|4}}:

<math display=block> G = \frac{\Gamma\bigl( \tfrac{1}{4}\bigr){}^2}{2\sqrt{ 2\pi^3}} </math>

John Todd's lemniscate constants may be given in terms of the [[Beta function|beta function]] B:
<math display=block>\begin{aligned}
L &= \tfrac12\pi G = \tfrac12\varpi = \tfrac14 \Beta \bigl(\tfrac14,\tfrac12\bigr), \\[3mu]
M &= \frac{1}{2G} =\tfrac14\Beta \bigl(\tfrac12,\tfrac34\bigr).
\end{aligned}</math>

== Series ==
[[Viète's formula]] for {{math|''π''}} can be written:

<math display=block>
\frac2\pi = \sqrt\frac12 \cdot \sqrt{\frac12 + \frac12\sqrt\frac12} \cdot \sqrt{\frac12 + \frac12\sqrt{\frac12 + \frac12\sqrt\frac12}} \cdots
</math>

An analogous formula for {{math|''ϖ''}} is:<ref>Levin (2006)</ref>

<math display=block>
\frac2\varpi = \sqrt\frac12 \cdot \sqrt{\frac12 +  \frac12 \bigg/ \!\sqrt\frac12} \cdot \sqrt{\frac12 +  \frac12 \Bigg/ \!\sqrt{\frac12 +  \frac12 \bigg/ \!\sqrt\frac12}} \cdots
</math>

The [[Wallis product]] for {{math|''π''}} is:

<math display=block>
\frac{\pi}{2} = \prod_{n=1}^\infty \left(1+\frac{1}{n}\right)^{(-1)^{n+1}}=\prod_{n=1}^{\infty} \left(\frac{2n}{2n-1} \cdot \frac{2n}{2n+1}\right)
= \biggl(\frac{2}{1} \cdot \frac{2}{3}\biggr) \biggl(\frac{4}{3} \cdot \frac{4}{5}\biggr) \biggl(\frac{6}{5} \cdot \frac{6}{7}\biggr) \cdots
</math>

An analogous formula for {{math|''ϖ''}} is:<ref>Hyde (2014) proves the validity of a more general Wallis-like formula for clover curves; here the special case of the lemniscate is slightly transformed, for clarity.</ref>

<math display=block>
\frac{\varpi}{2} = \prod_{n=1}^\infty \left(1+\frac{1}{2n}\right)^{(-1)^{n+1}}=\prod_{n=1}^{\infty} \left(\frac{4n-1}{4n-2} \cdot \frac{4n}{4n+1}\right)
= \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
</math>

A related result for Gauss's constant (<math>G=\varpi / \pi</math>) is:

<math display=block>
G = \prod_{n=1}^{\infty} \left(\frac{4n-1}{4n} \cdot \frac{4n+2}{4n+1}\right)
= \biggl(\frac{3}{4} \cdot \frac{6}{5}\biggr) \biggl(\frac{7}{8} \cdot \frac{10}{9}\biggr) \biggl(\frac{11}{12} \cdot \frac{14}{13}\biggr) \cdots
</math>

An infinite series of Gauss's constant discovered by Gauss is:<ref>Bottazzini & Gray (2013), p. 60</ref>

<math display=block>
G = \sum_{n=0}^\infty (-1)^n \prod_{k=1}^n \frac{(2k-1)^2}{(2k)^2}  
= 1 - \frac{1^2}{2^2} + \frac{1^2\cdot3^2}{2^2\cdot4^2} - \frac{1^2\cdot3^2\cdot5^2}{2^2\cdot4^2\cdot6^2} + \cdots
</math>

The [[Machin-like formula|Machin formula]] for {{math|''π''}} is <math display=inline>\tfrac14\pi = 4 \arctan \tfrac15 - \arctan \tfrac1{239},</math> and several similar formulas for {{math|''π''}} can be developed using trigonometric angle sum identities, e.g. Euler's formula <math display=inline>\tfrac14\pi = \arctan\tfrac12 + \arctan\tfrac13</math>. Analogous formulas can be developed for {{math|''ϖ''}}, including the following found by Gauss: <math>\tfrac12\varpi = 2 \operatorname{arcsl} \tfrac12 + \operatorname{arcsl} \tfrac7{23}</math>, where <math>\operatorname{arcsl}</math> is the [[Lemniscate elliptic functions#Inverse functions|lemniscate arcsine]].<ref>Todd (1975)</ref>

The lemniscate constant can be rapidly computed by the series<ref>{{harvnb|Cox|1984|p=307|loc=eq. 2.21}} for the first equality.</ref><ref>{{Cite book |last1=Berndt |first1=Bruce C. |title=Ramanujan's Notebooks Part V |publisher=Springer |year=1998 |isbn=978-1-4612-7221-2}} p. 326</ref>

<math>\varpi=2^{-1/2}\pi\left(\sum_{n\in\mathbb{Z}}e^{-\pi n^2}\right)^2=2^{1/4}\pi e^{-\pi/12} \left(\sum_{n\in\mathbb{Z}}(-1)^n e^{-\pi p_n}\right)^2</math>

where <math>p_n=(3n^2-n)/2</math> (for <math>n\ge 1</math>, these are the pentagonal numbers)

In a spirit similar to that of the [[Basel problem]],
:<math>\sum_{z\in\mathbb{Z}[i]\setminus\{0\}}\frac{1}{z^4}=G_4(i)=\frac{\varpi ^4}{15}</math>
where <math>\mathbb{Z}[i]</math> are the Gaussian integers and <math>G_4(\tau)</math> is the [[Eisenstein series]] of weight <math>4</math>.<ref>{{Cite book |last1=Eymard |first1=Pierre |last2=Lafon| first2=Jean-Pierre |title=Autour du nombre Pi |language=French|publisher=HERMANN |year=1999 |isbn=2705614435}} p. 224</ref>

Gauss's constant is given by the rapidly converging series

<math display=block>G = \sqrt[4]{32}e^{-\frac{\pi}{3}}\left (\sum_{n = -\infty}^\infty (-1)^n e^{-2n\pi(3n+1)} \right )^2.</math>

The constant is also given by the [[Infinite product|infinite product]]

:<math>G = \prod_{m = 1}^\infty \tanh^2 \left( \frac{\pi m}{2}\right).</math>

An analog of the [[Wallis product]] is:<ref>{{cite journal |last1=Hyde |first1=Trevor |year=2014 |title=A Wallis product on clovers |journal=The American Mathematical Monthly |volume=121 |issue=3 |pages=237–243 |url=https://math.uchicago.edu/~tghyde/Hyde%20--%20A%20Wallis%20product%20on%20clovers.pdf |doi=10.4169/amer.math.monthly.121.03.237 |s2cid=34819500 }}</ref>

:<math>G = \prod_{n=1}^{\infty} \left(\frac{4n-1}{4n} \cdot \frac{4n+2}{4n+1}\right)
= \biggl(\frac{3}{4} \cdot \frac{6}{5}\biggr) \biggl(\frac{7}{8} \cdot \frac{10}{9}\biggr) \biggl(\frac{11}{12} \cdot \frac{14}{13}\biggr) \cdots
</math>

Gauss' constant as a [[Continued fraction|continued fraction]] is [0,&nbsp;1,&nbsp;5,&nbsp;21,&nbsp;3,&nbsp;4,&nbsp;14,&nbsp;...]. {{OEIS|id=A053002}}

== Integrals ==

[[File:Lemniscate constant as an integral.png|thumb|upright=1.3|A geometric representation of <math>\varpi/2</math> and <math>\varpi/\sqrt{2}</math>]]
{{math|''ϖ''}} is related to the area under the curve <math>x^4 + y^4 = 1</math>. Defining <math>\pi_n \mathrel{:=} \Beta\bigl(\tfrac1n, \tfrac1n \bigr)</math>, twice the area in the positive quadrant under the curve <math>x^n + y^n = 1</math> is <math display="inline">2 \int_0^1 \sqrt[n]{1 - x^n}\mathop{\mathrm{d}x} = \tfrac1n \pi_n.</math> In the quartic case, <math>\tfrac14 \pi_4 = \tfrac1\sqrt{2} \varpi.</math>

Gauss's constant appears in the evaluation of the integrals

<math display=block> {\frac{1}{G}} = \int_0^{\frac{\pi}{2}}\sqrt{\sin(x)}\,dx=\int_0^{\frac{\pi}{2}}\sqrt{\cos(x)}\,dx </math>

<math display=block> G = \int_0^{\infty}{\frac{dx}{\sqrt{\cosh(\pi x)}}} </math>

The first and second lemniscate constants are defined by integrals:

<math display=block>L = \int_0^1\frac{dx}{\sqrt{1 - x^4}}</math>

<math display=block>M = \int_0^1\frac{x^2 dx}{\sqrt{1 - x^4}} </math>

=== Circumference of an ellipse ===

Gauss's constant satisfies the equation{{sfn|Cox|1984|p=313}}

<math display="block">\frac{1}{G} = 2 \int_0^1\frac{x^2\, dx}{\sqrt{1 - x^4}} </math>

Euler discovered in 1738 that for the rectangular elastica (first and second lemniscate constants)<ref>Levien (2008)</ref>{{sfn|Cox|1984|p=313}}

<math display=block>\textrm{arc}\ \textrm{length}\cdot\textrm{height} = L \cdot M = \int_0^1 \frac{\mathrm{d}x}{\sqrt{1 - x^4}} \cdot \int_0^1 \frac{x^2 \mathop{\mathrm{d}x}}{\sqrt{1 - x^4}} = \frac\varpi2 \cdot \frac\pi{2\varpi} = \frac\pi4</math>

Now considering the circumference <math>C</math> of the ellipse with axes <math>\sqrt{2}</math> and <math>1</math>, satisfying <math>2x^2 + 4y^2 = 1</math>, Stirling noted that{{sfn|Cox|1984|p=312}}

<math display="block"> \frac{C}{2} = \int_0^1\frac{dx}{\sqrt{1 - x^4}} + \int_0^1\frac{x^2\,dx}{\sqrt{1 - x^4}}</math>

Hence the full circumference is

<math display="block"> C = \frac{1}{G} + G \pi \approx 3.820197789\ldots </math>

This is also the arc length of the [[Sine and cosine|sine]] curve on half a period:<ref name="ams">{{Cite web|url=https://www.ams.org/notices/201208/rtx120801094p.pdf|title=An Eloquent Formula for the Perimeter of an Ellipse|last=Adlaj|first=Semjon|date=2012|website=American Mathematical Society|page=1097|quote=One might also observe that the length of the “sine” curve over half a period, that is, the length of the graph of the function sin(t) from the point where t = 0 to the point where t = π , is <math>\sqrt{2} l(1/\sqrt{2}) = L + M</math>.}}</ref>

<math display="block"> C = \int_0^\pi \sqrt{1+\cos^2(x)}\,dx </math>

==See also==
*Lemniscate elliptic function

==Notes==
{{notelist}}

==References==
{{Reflist}}
*{{mathworld|urlname=GausssConstant|title=Gauss's Constant}}
* Sequences A014549, A053002, and [https://oeis.org/A062539 A062539] in OEIS
* {{cite journal |last1=Cox |first1=David A. |title=The Arithmetic-Geometric Mean of Gauss |journal=L'Enseignement Mathématique |date=January 1984 |volume=30|issue=2 |pages=275-330 |doi=10.5169/seals-53831 |url=https://webspace.science.uu.nl/~wepst101/elliptic/cox_agm.pdf |access-date=25 June 2022}}

== External links ==
* {{Cite web|date=2021-10-17|title=Gauss's constant and where it occurs|url=https://www.johndcook.com/blog/2021/10/17/gauss-constant/|website=www.johndcook.com|language=en-US}}

{{Irrational number}}

[[Category:Mathematical constants]]
[[Category:Real transcendental numbers]]

{{Sourceattribution|Gauss's constant|1}}