Lemniscate constant

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Short description: Ratio of the perimeter of Bernoulli's lemniscate to its diameter
Lemniscate of Bernoulli

In mathematics, the lemniscate constant ϖ[1][2][3][4][5] is a transcendental mathematical constant that is the ratio of the perimeter of Bernoulli's lemniscate to its diameter, analogous to the definition of π for the circle. Equivalently, the perimeter of the lemniscate [math]\displaystyle{ (x^2+y^2)^2=x^2-y^2 }[/math] is 2ϖ. The lemniscate constant is closely related to the lemniscate elliptic functions and approximately equal to 2.62205755.[6][7][8][9] The symbol ϖ is a cursive variant of π; see Pi § Variant pi.

Gauss's constant, denoted by G, is equal to ϖ /π ≈ 0.8346268.[10]

John Todd named two more lemniscate constants, the first lemniscate constant A = ϖ/2 ≈ 1.3110287771 and the second lemniscate constant B = π/(2ϖ) ≈ 0.5990701173.[11][12][13][14]

Sometimes the quantities 2ϖ or A are referred to as the lemniscate constant.[15][16]

History

Gauss's constant [math]\displaystyle{ G }[/math] is named after Carl Friedrich Gauss, who calculated it via the arithmetic–geometric mean as [math]\displaystyle{ 1/M(1,\sqrt{2}) }[/math].[6] By 1799, Gauss had two proofs of the theorem that [math]\displaystyle{ M(1,\sqrt{2})=\pi/\varpi }[/math] where [math]\displaystyle{ \varpi }[/math] is the lemniscate constant.[2][lower-alpha 1]

The lemniscate constant [math]\displaystyle{ \varpi }[/math] and first lemniscate constant [math]\displaystyle{ A }[/math] were proven transcendental by Theodor Schneider in 1937 and the second lemniscate constant [math]\displaystyle{ B }[/math] and Gauss's constant [math]\displaystyle{ G }[/math] were proven transcendental by Theodor Schneider in 1941.[11][17][lower-alpha 2] In 1975, Gregory Chudnovsky proved that the set [math]\displaystyle{ \{\pi,\varpi\} }[/math] is algebraically independent over [math]\displaystyle{ \mathbb{Q} }[/math], which implies that [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math] are algebraically independent as well.[18][19] But the set [math]\displaystyle{ \{\pi,M(1,1/\sqrt{2}),M'(1,1/\sqrt{2})\} }[/math] (where the prime denotes the derivative with respect to the second variable) is not algebraically independent over [math]\displaystyle{ \mathbb{Q} }[/math]. In fact,[20]

[math]\displaystyle{ \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]\displaystyle{ \varpi }[/math] is defined by the first equality below.[2][21][22]

[math]\displaystyle{ \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}} = \int_1^\infty \frac{\mathrm dt}{\sqrt{t^3-t}}\\[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}} = \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 K is the complete elliptic integral of the first kind with modulus k, Β is the beta function, Γ is the gamma function and ζ is the Riemann zeta function.

The lemniscate constant can also be computed by the arithmetic–geometric mean [math]\displaystyle{ M }[/math],

[math]\displaystyle{ \varpi=\frac{\pi}{M(1,\sqrt{2})}. }[/math]

Moreover, [math]\displaystyle{ e^{\beta'(0)}=\frac{\varpi}{\sqrt{\pi}} }[/math]

which is analogous to

[math]\displaystyle{ e^{\zeta'(0)}=\frac{1}{\sqrt{2\pi}} }[/math]

where [math]\displaystyle{ \beta }[/math] is the Dirichlet beta function and [math]\displaystyle{ \zeta }[/math] is the Riemann zeta function.[23]

Gauss's constant is typically defined as the reciprocal of the arithmetic–geometric mean of 1 and the square root of 2, after his calculation of [math]\displaystyle{ M(1, \sqrt{2}) }[/math] published in 1800:[24]

[math]\displaystyle{ G = \frac{1}{M(1, \sqrt{2})} }[/math]

Gauss's constant is equal to

[math]\displaystyle{ G = \frac{1}{2\pi}\Beta\bigl( \tfrac14, \tfrac12\bigr) }[/math]

where Β denotes the beta function. A formula for G in terms of Jacobi theta functions is given by

[math]\displaystyle{ G = \vartheta_{01}^2\left(e^{-\pi}\right) }[/math]

Gauss's constant may be computed from the gamma function at argument 1/4:

[math]\displaystyle{ 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 B: [math]\displaystyle{ \begin{aligned} A &= \tfrac12\pi G = \tfrac12\varpi = \tfrac14 \Beta \bigl(\tfrac14,\tfrac12\bigr), \\[3mu] B &= \frac{1}{2G} =\tfrac14\Beta \bigl(\tfrac12,\tfrac34\bigr). \end{aligned} }[/math]

Series

Viète's formula for π can be written:

[math]\displaystyle{ \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 ϖ is:[25]

[math]\displaystyle{ \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 π is:

[math]\displaystyle{ \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 ϖ is:[26]

[math]\displaystyle{ \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]\displaystyle{ G=\varpi / \pi }[/math]) is:[27]

[math]\displaystyle{ 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:[28]

[math]\displaystyle{ 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 formula for π is [math]\displaystyle{ \tfrac14\pi = 4 \arctan \tfrac15 - \arctan \tfrac1{239}, }[/math] and several similar formulas for π can be developed using trigonometric angle sum identities, e.g. Euler's formula [math]\displaystyle{ \tfrac14\pi = \arctan\tfrac12 + \arctan\tfrac13 }[/math]. Analogous formulas can be developed for ϖ, including the following found by Gauss: [math]\displaystyle{ \tfrac12\varpi = 2 \operatorname{arcsl} \tfrac12 + \operatorname{arcsl} \tfrac7{23} }[/math], where [math]\displaystyle{ \operatorname{arcsl} }[/math] is the lemniscate arcsine.[29]

The lemniscate constant can be rapidly computed by the series[30][31]

[math]\displaystyle{ \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]\displaystyle{ p_n=(3n^2-n)/2 }[/math] (these are the generalized pentagonal numbers).

In a spirit similar to that of the Basel problem,

[math]\displaystyle{ \sum_{z\in\mathbb{Z}[i]\setminus\{0\}}\frac{1}{z^4}=G_4(i)=\frac{\varpi ^4}{15} }[/math]

where [math]\displaystyle{ \mathbb{Z}[i] }[/math] are the Gaussian integers and [math]\displaystyle{ G_4 }[/math] is the Eisenstein series of weight [math]\displaystyle{ 4 }[/math] (see Lemniscate elliptic functions § Hurwitz numbers for a more general result).[32]

A related result is

[math]\displaystyle{ \sum_{n=1}^\infty \sigma_3(n)e^{-2\pi n}=\frac{\varpi^4}{80 \pi^4}-\frac{1}{240} }[/math]

where [math]\displaystyle{ \sigma_3 }[/math] is the sum of positive divisors function.[33]

In 1842, Malmsten found

[math]\displaystyle{ \sum_{n=1}^\infty (-1)^{n+1}\frac{\log (2n+1)}{2n+1}=\frac{\pi}{4}\left(\gamma+2\log\frac{\pi}{\varpi\sqrt{2}}\right) }[/math]

where [math]\displaystyle{ \gamma }[/math] is Euler's constant.

Gauss's constant is given by the rapidly converging series

[math]\displaystyle{ 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

[math]\displaystyle{ G = \prod_{m = 1}^\infty \tanh^2 \left( \frac{\pi m}{2}\right). }[/math]

Continued fractions

A (generalized) continued fraction for π is [math]\displaystyle{ \frac\pi2=1 + \cfrac{1}{1 + \cfrac{1\cdot 2}{1 + \cfrac{2\cdot 3}{1 + \cfrac{3\cdot 4}{1+\ddots}}}} }[/math] An analogous formula for ϖ is[12] [math]\displaystyle{ \frac\varpi2= 1 + \cfrac{1}{2 + \cfrac{2\cdot 3}{2 + \cfrac{4\cdot 5}{2 + \cfrac{6\cdot 7}{2+\ddots}}}} }[/math]

Define Brouncker's continued fraction by[34] [math]\displaystyle{ b(s)=s + \cfrac{1^2}{2s + \cfrac{3^2}{2s + \cfrac{5^2}{2s+\ddots}}},\quad s\gt 0. }[/math] Let [math]\displaystyle{ n\ge 0 }[/math] except for the first equality where [math]\displaystyle{ n\ge 1 }[/math]. Then[35][36] [math]\displaystyle{ \begin{align}b(4n)&=(4n+1)\prod_{k=1}^n \frac{(4k-1)^2}{(4k-3)(4k+1)}\frac{\pi}{\varpi^2}\\ b(4n+1)&=(2n+1)\prod_{k=1}^n \frac{(2k)^2}{(2k-1)(2k+1)}\frac{4}{\pi}\\ b(4n+2)&=(4n+1)\prod_{k=1}^n \frac{(4k-3)(4k+1)}{(4k-1)^2}\frac{\varpi^2}{\pi}\\ b(4n+3)&=(2n+1)\prod_{k=1}^n \frac{(2k-1)(2k+1)}{(2k)^2}\,\pi.\end{align} }[/math] For example, [math]\displaystyle{ \begin{align}b(1)&=\frac{4}{\pi}\\ b(2)&=\frac{\varpi^2}{\pi}\\ b(3)&=\pi\\ b(4)&=\frac{9\pi}{\varpi^2}.\end{align} }[/math]

Simple continued fractions[37][38]

[math]\displaystyle{ \begin{align}\varpi&=[2,1,1,1,1,1,4,1,2,\ldots]\\ 2\varpi&=[5,4,10,2,1,2,3,29,\ldots]\\ \frac{\varpi}{2}&=[1,3,4,1,1,1,5,2,\ldots]\\ G&=[0,1,5,21,3,4,14,\ldots]\end{align} }[/math]

Integrals

A geometric representation of [math]\displaystyle{ \varpi/2 }[/math] and [math]\displaystyle{ \varpi/\sqrt{2} }[/math]

ϖ is related to the area under the curve [math]\displaystyle{ x^4 + y^4 = 1 }[/math]. Defining [math]\displaystyle{ \pi_n \mathrel{:=} \Beta\bigl(\tfrac1n, \tfrac1n \bigr) }[/math], twice the area in the positive quadrant under the curve [math]\displaystyle{ x^n + y^n = 1 }[/math] is [math]\displaystyle{ 2 \int_0^1 \sqrt[n]{1 - x^n}\mathop{\mathrm{d}x} = \tfrac1n \pi_n. }[/math] In the quartic case, [math]\displaystyle{ \tfrac14 \pi_4 = \tfrac1\sqrt{2} \varpi. }[/math]

In 1842, Malmsten discovered that[39]

[math]\displaystyle{ \int_0^1 \frac{\log (-\log x)}{1+x^2}\, dx=\frac{\pi}{2}\log\frac{\pi}{\varpi\sqrt{2}}. }[/math]

Furthermore, [math]\displaystyle{ \int_0^\infty \frac{\tanh x}{x}e^{-x}\, dx=\log\frac{\varpi^2}{\pi} }[/math]

and[40]

[math]\displaystyle{ \int_0^\infty e^{-x^4}\, dx=\frac{\sqrt{2\varpi\sqrt{2\pi}}}{4},\quad\text{analogous to}\,\int_0^\infty e^{-x^2}\, dx=\frac{\sqrt{\pi}}{2}, }[/math] a form of Gaussian integral.

Gauss's constant appears in the evaluation of the integrals

[math]\displaystyle{ {\frac{1}{G}} = \int_0^{\frac{\pi}{2}}\sqrt{\sin(x)}\,dx=\int_0^{\frac{\pi}{2}}\sqrt{\cos(x)}\,dx }[/math]

[math]\displaystyle{ G = \int_0^{\infty}{\frac{dx}{\sqrt{\cosh(\pi x)}}} }[/math]

The first and second lemniscate constants are defined by integrals:[11]

[math]\displaystyle{ A = \int_0^1\frac{dx}{\sqrt{1 - x^4}} }[/math]

[math]\displaystyle{ B = \int_0^1\frac{x^2\, dx}{\sqrt{1 - x^4}} }[/math]

Circumference of an ellipse

Gauss's constant satisfies the equation[41]

[math]\displaystyle{ \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)[42][41]

[math]\displaystyle{ \textrm{arc}\ \textrm{length}\cdot\textrm{height} = A \cdot B = \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]\displaystyle{ C }[/math] of the ellipse with axes [math]\displaystyle{ \sqrt{2} }[/math] and [math]\displaystyle{ 1 }[/math], satisfying [math]\displaystyle{ 2x^2 + 4y^2 = 1 }[/math], Stirling noted that[43]

[math]\displaystyle{ \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]\displaystyle{ C = \frac{1}{G} + G \pi \approx 3.820197789\ldots }[/math]

This is also the arc length of the sine curve on half a period:[44]

[math]\displaystyle{ C = \int_0^\pi \sqrt{1+\cos^2(x)}\,dx }[/math]

Notes

  1. ↑ although neither of these proofs was rigorous from the modern point of view.
  2. ↑ In particular, he proved that the beta function [math]\displaystyle{ \Beta (a,b) }[/math] is transcendental for all [math]\displaystyle{ a,b\in\mathbb{Q}\setminus\mathbb{Z} }[/math] such that [math]\displaystyle{ a+b\notin \mathbb{Z}_0^- }[/math]. The fact that [math]\displaystyle{ \varpi }[/math] is transcendental follows from [math]\displaystyle{ \varpi=\tfrac{1}{2}\Beta \left(\tfrac{1}{4},\tfrac{1}{2}\right) }[/math] and similarly for B and G from [math]\displaystyle{ \Beta \left(\tfrac{1}{2},\tfrac{3}{4}\right). }[/math]

References

  1. ↑ Gauss, C. F. (1866) (in Latin, German). Werke (Band III). Herausgegeben der Königlichen Gesellschaft der Wissenschaften zu Göttingen. https://gdz.sub.uni-goettingen.de/id/PPN235999628.  p. 404
  2. ↑ 2.0 2.1 2.2 Cox 1984, p. 281.
  3. ↑ Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi. American Mathematical Society. ISBN 0-8218-3246-8.  p. 199
  4. ↑ Bottazzini, Umberto; Gray, Jeremy (2013). Hidden Harmony – Geometric Fantasies: The Rise of Complex Function Theory. Springer. doi:10.1007/978-1-4614-5725-1. ISBN 978-1-4614-5724-4.  p. 57
  5. ↑ Arakawa, Tsuneo; Ibukiyama, Tomoyoshi; Kaneko, Masanobu (2014). Bernoulli Numbers and Zeta Functions. Springer. ISBN 978-4-431-54918-5.  p. 203
  6. ↑ 6.0 6.1 Finch, Steven R. (18 August 2003) (in en). Mathematical Constants. Cambridge University Press. p. 420. ISBN 978-0-521-81805-6. https://books.google.com/books?id=Pl5I2ZSI6uAC. 
  7. ↑ Kobayashi, Hiroyuki; Takeuchi, Shingo (2019), "Applications of generalized trigonometric functions with two parameters", Communications on Pure & Applied Analysis 18 (3): 1509–1521, doi:10.3934/cpaa.2019072 
  8. ↑ Asai, Tetsuya (2007), Elliptic Gauss Sums and Hecke L-values at s=1 
  9. ↑ "A062539 - Oeis". http://oeis.org/A062539. 
  10. ↑ "A014549 - Oeis". http://oeis.org/A014549. 
  11. ↑ 11.0 11.1 11.2 Todd, John (January 1975). "The lemniscate constants". Communications of the ACM 18 (1): 14–19. doi:10.1145/360569.360580. https://dl.acm.org/doi/10.1145/360569.360580. 
  12. ↑ 12.0 12.1 "A085565 - Oeis". http://oeis.org/A085565. 
  13. ↑ "A076390 - Oeis". http://oeis.org/A076390. 
  14. ↑ Carlson, B. C. (2010), "Elliptic Integrals", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, http://dlmf.nist.gov/19.20.E2 
  15. ↑ "A064853 - Oeis". http://oeis.org/A064853. 
  16. ↑ "Lemniscate Constant". http://www.numberworld.org/digits/Lemniscate/. 
  17. ↑ Schneider, Theodor (1941). "Zur Theorie der Abelschen Funktionen und Integrale". Journal für die reine und angewandte Mathematik 183 (19): 110–128. doi:10.1515/crll.1941.183.110. https://www.deepdyve.com/lp/de-gruyter/zur-theorie-der-abelschen-funktionen-und-integrale-mn0U50bvkB. 
  18. ↑ G. V. Choodnovsky: Algebraic independence of constants connected with the functions of analysis, Notices of the AMS 22, 1975, p. A-486
  19. ↑ G. V. Chudnovsky: Contributions to The Theory of Transcendental Numbers, American Mathematical Society, 1984, p. 6
  20. ↑ Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience. ISBN 0-471-83138-7.  p. 45
  21. ↑ Finch, Steven R. (18 August 2003) (in en). Mathematical Constants. Cambridge University Press. pp. 420–422. ISBN 978-0-521-81805-6. https://books.google.com/books?id=Pl5I2ZSI6uAC. 
  22. ↑ Schappacher, Norbert (1997). "Some milestones of lemniscatomy ". in Sertöz, S.. Algebraic Geometry (Proceedings of Bilkent Summer School, August 7–19, 1995, Ankara, Turkey). Marcel Dekker. pp. 257–290. http://irma.math.unistra.fr/~schappa/NSch/Publications_files/1997_LemniscProvis.pdf. 
  23. ↑ "A113847 - Oeis". http://oeis.org/A113847. 
  24. ↑ Cox 1984, p. 277.
  25. ↑ Levin (2006)
  26. ↑ 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.
  27. ↑ Hyde, Trevor (2014). "A Wallis product on clovers". The American Mathematical Monthly 121 (3): 237–243. doi:10.4169/amer.math.monthly.121.03.237. https://math.uchicago.edu/~tghyde/Hyde%20--%20A%20Wallis%20product%20on%20clovers.pdf. 
  28. ↑ Bottazzini, Umberto; Gray, Jeremy (2013). Hidden Harmony – Geometric Fantasies: The Rise of Complex Function Theory. Springer. doi:10.1007/978-1-4614-5725-1. ISBN 978-1-4614-5724-4.  p. 60
  29. ↑ Todd (1975)
  30. ↑ Cox 1984, p. 307, eq. 2.21 for the first equality. The second equality can be proved by using the pentagonal number theorem.
  31. ↑ Berndt, Bruce C. (1998). Ramanujan's Notebooks Part V. Springer. ISBN 978-1-4612-7221-2.  p. 326
  32. ↑ Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi. American Mathematical Society. ISBN 0-8218-3246-8.  p. 232
  33. ↑ Garrett, Paul. "Level-one elliptic modular forms". http://www-users.math.umn.edu/~garrett/m/mfms/notes_2015-16/10_level_one.pdf.  p. 11—13
  34. ↑ Khrushchev, Sergey (2008). Orthogonal Polynomials and Continued Fractions (First ed.). Cambridge University Press. ISBN 978-0-521-85419-1.  p. 140 (eq. 3.34), p. 153. There's an error on p. 153: [math]\displaystyle{ 4 [\Gamma(3+s/4)/\Gamma(1+s/4)]^2 }[/math] should be [math]\displaystyle{ 4[\Gamma((3+s)/4)/\Gamma((1+s)/4)]^2 }[/math].
  35. ↑ Khrushchev, Sergey (2008). Orthogonal Polynomials and Continued Fractions (First ed.). Cambridge University Press. ISBN 978-0-521-85419-1.  p. 146, 155
  36. ↑ Perron, Oskar (1957) (in German). Die Lehre von den Kettenbrüchen: Band II (Third ed.). B. G. Teubner.  p. 36, eq. 24
  37. ↑ "A062540 - OEIS". http://oeis.org/A062540. 
  38. ↑ "A053002 - OEIS". http://oeis.org/A053002. 
  39. ↑ Blagouchine, Iaroslav V. (2014). "Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results". The Ramanujan Journal 35 (1): 21–110. doi:10.1007/s11139-013-9528-5. https://www.researchgate.net/publication/257381156. 
  40. ↑ "A068467 - Oeis". https://oeis.org/A068467. 
  41. ↑ 41.0 41.1 Cox 1984, p. 313.
  42. ↑ Levien (2008)
  43. ↑ Cox 1984, p. 312.
  44. ↑ Adlaj, Semjon (2012). "An Eloquent Formula for the Perimeter of an Ellipse". p. 1097. https://www.ams.org/notices/201208/rtx120801094p.pdf. "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]\displaystyle{ \sqrt{2} l(1/\sqrt{2}) = L + M }[/math]."  In this paper [math]\displaystyle{ M=1/G=\pi/\varpi }[/math] and [math]\displaystyle{ L = \pi/M=G\pi=\varpi }[/math].

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