# Lemniscate constant

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

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

${\displaystyle \pi =2{\sqrt {2}}{\frac {M^{3}(1,1/{\sqrt {2}})}{M'(1,1/{\sqrt {2}})}}={\frac {1}{G^{3}M'(1,1/{\sqrt {2}})}}.}$

## Forms

Usually, ${\displaystyle \varpi }$ is defined by the first equality below.[2][21][22]

{\displaystyle {\begin{aligned}\varpi &=2\int _{0}^{1}{\frac {\mathrm {d} t}{\sqrt {1-t^{4}}}}={\sqrt {2}}\int _{0}^{\infty }{\frac {\mathrm {d} t}{\sqrt {1+t^{4}}}}=\int _{0}^{1}{\frac {\mathrm {d} t}{\sqrt {t-t^{3}}}}=\int _{1}^{\infty }{\frac {\mathrm {d} t}{\sqrt {t^{3}-t}}}\\[6mu]&=4\int _{0}^{\infty }{\Bigl (}{\sqrt[{4}]{1+t^{4}}}-t{\Bigr )}\,\mathrm {d} t=2{\sqrt {2}}\int _{0}^{1}{\sqrt[{4}]{1-t^{4}}}\mathop {\mathrm {d} t} =3\int _{0}^{1}{\sqrt {1-t^{4}}}\,\mathrm {d} t\\[2mu]&=2K(i)={\tfrac {1}{2}}\mathrm {B} {\bigl (}{\tfrac {1}{4}},{\tfrac {1}{2}}{\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}}}

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 ${\displaystyle M}$,

${\displaystyle \varpi ={\frac {\pi }{M(1,{\sqrt {2}})}}.}$

Moreover,

${\displaystyle e^{\beta '(0)}={\frac {\varpi }{\sqrt {\pi }}}}$

which is analogous to

${\displaystyle e^{\zeta '(0)}={\frac {1}{\sqrt {2\pi }}}}$

where ${\displaystyle \beta }$ is the Dirichlet beta function and ${\displaystyle \zeta }$ 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 ${\displaystyle M(1,{\sqrt {2}})}$ published in 1800:[24]

${\displaystyle G={\frac {1}{M(1,{\sqrt {2}})}}}$

Gauss's constant is equal to

${\displaystyle G={\frac {1}{2\pi }}\mathrm {B} {\bigl (}{\tfrac {1}{4}},{\tfrac {1}{2}}{\bigr )}}$

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

${\displaystyle G=\vartheta _{01}^{2}\left(e^{-\pi }\right)}$

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

${\displaystyle G={\frac {\Gamma {\bigl (}{\tfrac {1}{4}}{\bigr )}{}^{2}}{2{\sqrt {2\pi ^{3}}}}}}$

John Todd's lemniscate constants may be given in terms of the beta function B:

{\displaystyle {\begin{aligned}A&={\tfrac {1}{2}}\pi G={\tfrac {1}{2}}\varpi ={\tfrac {1}{4}}\mathrm {B} {\bigl (}{\tfrac {1}{4}},{\tfrac {1}{2}}{\bigr )},\\[3mu]B&={\frac {1}{2G}}={\tfrac {1}{4}}\mathrm {B} {\bigl (}{\tfrac {1}{2}},{\tfrac {3}{4}}{\bigr )}.\end{aligned}}}

## Series

Viète's formula for π can be written:

${\displaystyle {\frac {2}{\pi }}={\sqrt {\frac {1}{2}}}\cdot {\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {\frac {1}{2}}}}}\cdot {\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {\frac {1}{2}}}}}}}\cdots }$

An analogous formula for ϖ is:[25]

${\displaystyle {\frac {2}{\varpi }}={\sqrt {\frac {1}{2}}}\cdot {\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\bigg /}\!{\sqrt {\frac {1}{2}}}}}\cdot {\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\Bigg /}\!{\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\bigg /}\!{\sqrt {\frac {1}{2}}}}}}}\cdots }$

The Wallis product for π is:

${\displaystyle {\frac {\pi }{2}}=\prod _{n=1}^{\infty }\left(1+{\frac {1}{n}}\right)^{(-1)^{n+1}}=\prod _{n=1}^{\infty }\left({\frac {2n}{2n-1}}\cdot {\frac {2n}{2n+1}}\right)={\biggl (}{\frac {2}{1}}\cdot {\frac {2}{3}}{\biggr )}{\biggl (}{\frac {4}{3}}\cdot {\frac {4}{5}}{\biggr )}{\biggl (}{\frac {6}{5}}\cdot {\frac {6}{7}}{\biggr )}\cdots }$

An analogous formula for ϖ is:[26]

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

A related result for Gauss's constant (${\displaystyle G=\varpi /\pi }$) is:[27]

${\displaystyle G=\prod _{n=1}^{\infty }\left({\frac {4n-1}{4n}}\cdot {\frac {4n+2}{4n+1}}\right)={\biggl (}{\frac {3}{4}}\cdot {\frac {6}{5}}{\biggr )}{\biggl (}{\frac {7}{8}}\cdot {\frac {10}{9}}{\biggr )}{\biggl (}{\frac {11}{12}}\cdot {\frac {14}{13}}{\biggr )}\cdots }$

An infinite series of Gauss's constant discovered by Gauss is:[28]

${\displaystyle G=\sum _{n=0}^{\infty }(-1)^{n}\prod _{k=1}^{n}{\frac {(2k-1)^{2}}{(2k)^{2}}}=1-{\frac {1^{2}}{2^{2}}}+{\frac {1^{2}\cdot 3^{2}}{2^{2}\cdot 4^{2}}}-{\frac {1^{2}\cdot 3^{2}\cdot 5^{2}}{2^{2}\cdot 4^{2}\cdot 6^{2}}}+\cdots }$

The Machin formula for π is ${\textstyle {\tfrac {1}{4}}\pi =4\arctan {\tfrac {1}{5}}-\arctan {\tfrac {1}{239}},}$ and several similar formulas for π can be developed using trigonometric angle sum identities, e.g. Euler's formula ${\textstyle {\tfrac {1}{4}}\pi =\arctan {\tfrac {1}{2}}+\arctan {\tfrac {1}{3}}}$. Analogous formulas can be developed for ϖ, including the following found by Gauss: ${\displaystyle {\tfrac {1}{2}}\varpi =2\operatorname {arcsl} {\tfrac {1}{2}}+\operatorname {arcsl} {\tfrac {7}{23}}}$, where ${\displaystyle \operatorname {arcsl} }$ is the lemniscate arcsine.[29]

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

${\displaystyle \varpi =2^{-1/2}\pi \left(\sum _{n\in \mathbb {Z} }e^{-\pi n^{2}}\right)^{2}=2^{1/4}\pi e^{-\pi /12}\left(\sum _{n\in \mathbb {Z} }(-1)^{n}e^{-\pi p_{n}}\right)^{2}}$

where ${\displaystyle p_{n}=(3n^{2}-n)/2}$ (these are the generalized pentagonal numbers).

In a spirit similar to that of the Basel problem,

${\displaystyle \sum _{z\in \mathbb {Z} [i]\setminus \{0\}}{\frac {1}{z^{4}}}=G_{4}(i)={\frac {\varpi ^{4}}{15}}}$

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

A related result is

${\displaystyle \sum _{n=1}^{\infty }\sigma _{3}(n)e^{-2\pi n}={\frac {\varpi ^{4}}{80\pi ^{4}}}-{\frac {1}{240}}}$

where ${\displaystyle \sigma _{3}}$ is the sum of positive divisors function.[33]

In 1842, Malmsten found

${\displaystyle \sum _{n=1}^{\infty }(-1)^{n+1}{\frac {\log(2n+1)}{2n+1}}={\frac {\pi }{4}}\left(\gamma +2\log {\frac {\pi }{\varpi {\sqrt {2}}}}\right)}$

where ${\displaystyle \gamma }$ is Euler's constant.

Gauss's constant is given by the rapidly converging series

${\displaystyle G={\sqrt[{4}]{32}}e^{-{\frac {\pi }{3}}}\left(\sum _{n=-\infty }^{\infty }(-1)^{n}e^{-2n\pi (3n+1)}\right)^{2}.}$

The constant is also given by the infinite product

${\displaystyle G=\prod _{m=1}^{\infty }\tanh ^{2}\left({\frac {\pi m}{2}}\right).}$

## Continued fractions

A (generalized) continued fraction for π is

${\displaystyle {\frac {\pi }{2}}=1+{\cfrac {1}{1+{\cfrac {1\cdot 2}{1+{\cfrac {2\cdot 3}{1+{\cfrac {3\cdot 4}{1+\ddots }}}}}}}}}$
An analogous formula for ϖ is[12]
${\displaystyle {\frac {\varpi }{2}}=1+{\cfrac {1}{2+{\cfrac {2\cdot 3}{2+{\cfrac {4\cdot 5}{2+{\cfrac {6\cdot 7}{2+\ddots }}}}}}}}}$

Define Brouncker's continued fraction by[34]

${\displaystyle b(s)=s+{\cfrac {1^{2}}{2s+{\cfrac {3^{2}}{2s+{\cfrac {5^{2}}{2s+\ddots }}}}}},\quad s>0.}$
Let ${\displaystyle n\geq 0}$ except for the first equality where ${\displaystyle n\geq 1}$. Then[35][36]
{\displaystyle {\begin{aligned}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{aligned}}}
For example,
{\displaystyle {\begin{aligned}b(1)&={\frac {4}{\pi }}\\b(2)&={\frac {\varpi ^{2}}{\pi }}\\b(3)&=\pi \\b(4)&={\frac {9\pi }{\varpi ^{2}}}.\end{aligned}}}

### Simple continued fractions[37][38]

{\displaystyle {\begin{aligned}\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{aligned}}}

## Integrals

A geometric representation of ${\displaystyle \varpi /2}$ and ${\displaystyle \varpi /{\sqrt {2}}}$

ϖ is related to the area under the curve ${\displaystyle x^{4}+y^{4}=1}$. Defining ${\displaystyle \pi _{n}\mathrel {:=} \mathrm {B} {\bigl (}{\tfrac {1}{n}},{\tfrac {1}{n}}{\bigr )}}$, twice the area in the positive quadrant under the curve ${\displaystyle x^{n}+y^{n}=1}$ is ${\textstyle 2\int _{0}^{1}{\sqrt[{n}]{1-x^{n}}}\mathop {\mathrm {d} x} ={\tfrac {1}{n}}\pi _{n}.}$ In the quartic case, ${\displaystyle {\tfrac {1}{4}}\pi _{4}={\tfrac {1}{\sqrt {2}}}\varpi .}$

In 1842, Malmsten discovered that[39]

${\displaystyle \int _{0}^{1}{\frac {\log(-\log x)}{1+x^{2}}}\,dx={\frac {\pi }{2}}\log {\frac {\pi }{\varpi {\sqrt {2}}}}.}$

Furthermore,

${\displaystyle \int _{0}^{\infty }{\frac {\tanh x}{x}}e^{-x}\,dx=\log {\frac {\varpi ^{2}}{\pi }}}$

and[40]

${\displaystyle \int _{0}^{\infty }e^{-x^{4}}\,dx={\frac {\sqrt {2\varpi {\sqrt {2\pi }}}}{4}},\quad {\text{analogous to}}\,\int _{0}^{\infty }e^{-x^{2}}\,dx={\frac {\sqrt {\pi }}{2}},}$
a form of Gaussian integral.

Gauss's constant appears in the evaluation of the integrals

${\displaystyle {\frac {1}{G}}=\int _{0}^{\frac {\pi }{2}}{\sqrt {\sin(x)}}\,dx=\int _{0}^{\frac {\pi }{2}}{\sqrt {\cos(x)}}\,dx}$

${\displaystyle G=\int _{0}^{\infty }{\frac {dx}{\sqrt {\cosh(\pi x)}}}}$

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

${\displaystyle A=\int _{0}^{1}{\frac {dx}{\sqrt {1-x^{4}}}}}$

${\displaystyle B=\int _{0}^{1}{\frac {x^{2}\,dx}{\sqrt {1-x^{4}}}}}$

### Circumference of an ellipse

Gauss's constant satisfies the equation[41]

${\displaystyle {\frac {1}{G}}=2\int _{0}^{1}{\frac {x^{2}\,dx}{\sqrt {1-x^{4}}}}}$

Euler discovered in 1738 that for the rectangular elastica (first and second lemniscate constants)[42][41]

${\displaystyle {\textrm {arc}}\ {\textrm {length}}\cdot {\textrm {height}}=A\cdot B=\int _{0}^{1}{\frac {\mathrm {d} x}{\sqrt {1-x^{4}}}}\cdot \int _{0}^{1}{\frac {x^{2}\mathop {\mathrm {d} x} }{\sqrt {1-x^{4}}}}={\frac {\varpi }{2}}\cdot {\frac {\pi }{2\varpi }}={\frac {\pi }{4}}}$

Now considering the circumference ${\displaystyle C}$ of the ellipse with axes ${\displaystyle {\sqrt {2}}}$ and ${\displaystyle 1}$, satisfying ${\displaystyle 2x^{2}+4y^{2}=1}$, Stirling noted that[43]

${\displaystyle {\frac {C}{2}}=\int _{0}^{1}{\frac {dx}{\sqrt {1-x^{4}}}}+\int _{0}^{1}{\frac {x^{2}\,dx}{\sqrt {1-x^{4}}}}}$

Hence the full circumference is

${\displaystyle C={\frac {1}{G}}+G\pi \approx 3.820197789\ldots }$

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

${\displaystyle C=\int _{0}^{\pi }{\sqrt {1+\cos ^{2}(x)}}\,dx}$

## Notes

1. ^ although neither of these proofs was rigorous from the modern point of view.
2. ^ In particular, he proved that the beta function ${\displaystyle \mathrm {B} (a,b)}$ is transcendental for all ${\displaystyle a,b\in \mathbb {Q} \setminus \mathbb {Z} }$ such that ${\displaystyle a+b\notin \mathbb {Z} _{0}^{-}}$. The fact that ${\displaystyle \varpi }$ is transcendental follows from ${\displaystyle \varpi ={\tfrac {1}{2}}\mathrm {B} \left({\tfrac {1}{4}},{\tfrac {1}{2}}\right)}$ and similarly for B and G from ${\displaystyle \mathrm {B} \left({\tfrac {1}{2}},{\tfrac {3}{4}}\right).}$

## References

1. ^ Gauss, C. F. (1866). Werke (Band III) (in Latin and German). Herausgegeben der Königlichen Gesellschaft der Wissenschaften zu Göttingen. p. 404
2. ^ a b c 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. ^ a b Finch, Steven R. (18 August 2003). Mathematical Constants. Cambridge University Press. p. 420. ISBN 978-0-521-81805-6.
7. ^ Kobayashi, Hiroyuki; Takeuchi, Shingo (2019), "Applications of generalized trigonometric functions with two parameters", Communications on Pure & Applied Analysis, 18 (3): 1509–1521, arXiv:1903.07407, doi:10.3934/cpaa.2019072, S2CID 102487670
8. ^ Asai, Tetsuya (2007), Elliptic Gauss Sums and Hecke L-values at s=1, arXiv:0707.3711
9. ^
10. ^
11. ^ a b c Todd, John (January 1975). "The lemniscate constants". Communications of the ACM. 18 (1): 14–19. doi:10.1145/360569.360580. S2CID 85873.
12. ^ a b
13. ^
14. ^ Carlson, B. C. (2010), "Elliptic Integrals", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248
15. ^
16. ^
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. S2CID 118624331.
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). Mathematical Constants. Cambridge University Press. pp. 420–422. ISBN 978-0-521-81805-6.
22. ^ Schappacher, Norbert (1997). "Some milestones of lemniscatomy" (PDF). In Sertöz, S. (ed.). Algebraic Geometry (Proceedings of Bilkent Summer School, August 7–19, 1995, Ankara, Turkey). Marcel Dekker. pp. 257–290.
23. ^
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" (PDF). The American Mathematical Monthly. 121 (3): 237–243. doi:10.4169/amer.math.monthly.121.03.237. S2CID 34819500.
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" (PDF). University of Minnesota. 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: ${\displaystyle 4[\Gamma (3+s/4)/\Gamma (1+s/4)]^{2}}$ should be ${\displaystyle 4[\Gamma ((3+s)/4)/\Gamma ((1+s)/4)]^{2}}$.
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). Die Lehre von den Kettenbrüchen: Band II (in German) (Third ed.). B. G. Teubner. p. 36, eq. 24
37. ^ "A062540 - OEIS". oeis.org. Retrieved 2022-09-14.
38. ^ "A053002 - OEIS". oeis.org.
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. S2CID 120943474.
40. ^
41. ^ a b 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" (PDF). American Mathematical Society. p. 1097. 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 ${\displaystyle {\sqrt {2}}l(1/{\sqrt {2}})=L+M}$. In this paper ${\displaystyle M=1/G=\pi /\varpi }$ and ${\displaystyle L=\pi /M=G\pi =\varpi }$.