Lemniscate constant explained

In mathematics, the lemniscate constant ^{[1] [2] [3] [4]} is a transcendental mathematical constant that is the ratio of the perimeter of Bernoulli's lemniscate to its diameter, analogous to the definition of for the circle. Equivalently, the perimeter of the lemniscate

(x ^{2+y} ^{2)} ^{2=x} ^{2-y} ^{2} is . The lemniscate constant is closely related to the lemniscate elliptic functions and approximately equal to 2.62205755.^{[5] [6]} It also appears in evaluation of the gamma and beta function at certain rational values. The symbol is a cursive variant of ; see Pi § Variant pi.

Sometimes the quantities or are referred to as the lemniscate constant.^{[7] [8]}

As of 2024 over 1.2 trillion digits of this constant have been calculated.^[9]

History

Gauss's constant, denoted by G, is equal to ^[10] and named after Carl Friedrich Gauss, who calculated it via the arithmetic–geometric mean as

1 /M l ( 1 ,\sqrt{ 2 } r ) .^[5] By 1799, Gauss had two proofs of the theorem that M l ( 1 ,\sqrt 2 r ) = \pi/\varpi where \varpi is the lemniscate constant.

John Todd named two more lemniscate constants, the first lemniscate constant and the second lemniscate constant .^{[11] [12] [13]}

The lemniscate constant

\varpi and Todd's first lemniscate constant A were proven transcendental by Theodor Schneider in 1937 and Todd's second lemniscate constant B and Gauss's constant G were proven transcendental by Theodor Schneider in 1941.^[14] In 1975, Gregory Chudnovsky proved that the set \{\pi,\varpi\} is algebraically independent over Q , which implies that A and B are algebraically independent as well.^{[15] [16]} But the set l \{\pi,M l ( 1 , 1 /\sqrt{ 2 } r ),M' l ( 1 , 1 /\sqrt{ 2 } r ) r \} (where the prime denotes the derivative with respect to the second variable) is not algebraically independent over Q . In fact,^[17]

$$\backslash pi=2\backslash sqrt\backslash frac=\backslash frac.$$

Forms

Usually,

\varpi is defined by the first equality below.^{[18] [19]}

where is the complete elliptic integral of the first kind with modulus, is the beta function, is the gamma function and is the Riemann zeta function.

M ,

$$\backslash varpi=\backslash frac.$$

Moreover,

$$e^=\backslash frac$$

which is analogous to

$$e^=\backslash frac$$

where

\beta is the Dirichlet beta function and \zeta is the Riemann zeta function.^[20]

The lemniscate constant is equal to

$$\backslash varpi\; =\; \backslash tfrac\backslash Beta\backslash bigl(\backslash tfrac14,\; \backslash tfrac12\backslash bigr)$$

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

$$\backslash varpi\; =\; \backslash pi\backslash vartheta\_^2\backslash left(e^\backslash right)$$

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

M l ( 1 , \sqrt 2 r ) published in 1800:$$G\; =\; \backslash frac$$John Todd's lemniscate constants may be given in terms of the beta function B:

Series

Viète's formula for can be written:

$$\backslash frac2\backslash pi\; =\; \backslash sqrt\backslash frac12\; \backslash cdot\; \backslash sqrt\; \backslash cdot\; \backslash sqrt\; \backslash cdots$$

An analogous formula for is:^[21]

$$\backslash frac2\backslash varpi\; =\; \backslash sqrt\backslash frac12\; \backslash cdot\; \backslash sqrt\; \backslash cdot\; \backslash sqrt\; \backslash cdots$$

The Wallis product for is:

$$\backslash frac\; =\; \backslash prod\_^\backslash infty\; \backslash left(1+\backslash frac\backslash right)^=\backslash prod\_^\; \backslash left(\backslash frac\; \backslash cdot\; \backslash frac\backslash right)=\; \backslash biggl(\backslash frac\; \backslash cdot\; \backslash frac\backslash biggr)\; \backslash biggl(\backslash frac\; \backslash cdot\; \backslash frac\backslash biggr)\; \backslash biggl(\backslash frac\; \backslash cdot\; \backslash frac\backslash biggr)\; \backslash cdots$$

An analogous formula for is:^[22]

$$\backslash frac\; =\; \backslash prod\_^\backslash infty\; \backslash left(1+\backslash frac\backslash right)^=\backslash prod\_^\; \backslash left(\backslash frac\; \backslash cdot\; \backslash frac\backslash right)=\; \backslash biggl(\backslash frac\; \backslash cdot\; \backslash frac\backslash biggr)\; \backslash biggl(\backslash frac\; \backslash cdot\; \backslash frac\backslash biggr)\; \backslash biggl(\backslash frac\; \backslash cdot\; \backslash frac\backslash biggr)\; \backslash cdots$$

A related result for Gauss's constant (

G = \varpi / \pi ) is:^[23]

$$\backslash frac\; =\; \backslash prod\_^\; \backslash left(\backslash frac\; \backslash cdot\; \backslash frac\backslash right)=\; \backslash biggl(\backslash frac\; \backslash cdot\; \backslash frac\backslash biggr)\; \backslash biggl(\backslash frac\; \backslash cdot\; \backslash frac\backslash biggr)\; \backslash biggl(\backslash frac\; \backslash cdot\; \backslash frac\backslash biggr)\; \backslash cdots$$

An infinite series discovered by Gauss is:^[24]

$$\backslash frac\; =\; \backslash sum\_^\backslash infty\; (-1)^n\; \backslash prod\_^n\; \backslash frac\; =\; 1\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash frac\; +\; \backslash cdots$$

The Machin formula for is $\backslash tfrac14\backslash pi\; =\; 4\; \backslash arctan\; \backslash tfrac15\; -\; \backslash arctan\; \backslash tfrac1,$ and several similar formulas for can be developed using trigonometric angle sum identities, e.g. Euler's formula $\backslash tfrac14\backslash pi\; =\; \backslash arctan\backslash tfrac12\; +\; \backslash arctan\backslash tfrac13$. Analogous formulas can be developed for, including the following found by Gauss:

\tfrac 12 \varpi = 2 \operatorname{arcsl} \tfrac 12 + \operatorname{arcsl} \tfrac 7 { 23 } , where \operatorname{arcsl} is the lemniscate arcsine.^[25]

The lemniscate constant can be rapidly computed by the series^{[26] [27]}

\varpi =2 ^{-1/2} \pi l (\sum _{n\inZ} e ^{- \pi n ^{2}} r ) ^{2=2} ^{1/4} \pi e ^{-\pi/12} l (\sum _{n\inZ} ( -1 ) ^{n} e ^{- \pi p _{n}} r ) ^{2}

where

p ^{2- n)} _{n = \tfrac 12 ( 3 n} (these are the generalized pentagonal numbers). Also^[28] \sum _{m,n\inZ} e ^{-2 \pi (m ^{2} + mn + n ^{2)}} = \sqrt{ 1+ \sqrt{ 3 }}\dfrac{\varpi}{ 12 ^{1/8} \pi}.

In a spirit similar to that of the Basel problem,

\sum _{z\inZ[i]\setminus\{0\} }\frac=G_4(i)=\fracwhere Z [i] are the Gaussian integers and G _{4} is the Eisenstein series of weight (see Lemniscate elliptic functions § Hurwitz numbers for a more general result).^[29]

A related result is

\sum ^{inf ty} _{n =1} \sigma ^{-2 \pi n} _{3 (n)e} = (\varpi ^{4})/(- 80 \pi ^{4}) (1)/(240) where \sigma _{3} is the sum of positive divisors function.^[30]

In 1842, Malmsten found

\beta'( 1 ) = \sum ^{inf ty} _{n =1} ( -1 ) ^{n+1} (log ( 2 n +1 ))/(= 2 n +1) (\pi)/(\left(\gamma +2log 4) (\pi)/(\varpi\sqrt{ 2) }\right)

where

\gamma is Euler's constant and \beta(s) is the Dirichlet-Beta function.

The lemniscate constant is given by the rapidly converging series

$$\backslash varpi\; =\; \backslash pi\backslash sqrt[4]e^\backslash biggl(\backslash sum\_^\backslash infty\; (-1)^n\; e^\; \backslash biggr)^2.$$

The constant is also given by the infinite product

\varpi = \pi\prod ^{inf ty} _{m =1} \tanh ^{2} \left( (\pi m)/(2) \right).

Continued fractions

A (generalized) continued fraction for is$$\backslash frac\backslash pi2=1\; +\; \backslash cfrac$$An analogous formula for is^[12] $$\backslash frac\backslash varpi2=\; 1\; +\; \backslash cfrac$$

Define Brouncker's continued fraction by^[31] $$b(s)=s\; +\; \backslash cfrac,\backslash quad\; s>0.$$Let

n\ge 0 except for the first equality where n\ge 1 . Then^{[32] [33]} For example,

Simple continued fractions

Simple continued fractions for the lemniscate constant and related constants include^{[34] [35]}

Integrals

The lemniscate constant is related to the area under the curve

x ^{4} + y ^{4} = 1 . Defining \pi _{n} l{:=} \Beta l (\tfrac 1 n, \tfrac 1 n r ) , twice the area in the positive quadrant under the curve x ^{n} + y ^{n} = 1 is $2\; \backslash int\_0^1\; \backslash sqrt[n]\backslash mathop\; =\; \backslash tfrac1n\; \backslash pi\_n.$ In the quartic case, \tfrac 14 \pi _{4} = \tfrac 1 \sqrt{ 2 } \varpi.

In 1842, Malmsten discovered that^[36]

$$\backslash int\_0^1\; \backslash frac\backslash ,\; dx=\backslash frac\backslash log\backslash frac.$$

Furthermore,$$\backslash int\_0^\backslash infty\; \backslash frace^\backslash ,\; dx=\backslash log\backslash frac$$

and^[37]

$$\backslash int\_0^\backslash infty\; e^\backslash ,\; dx=\backslash frac,\backslash quad\backslash text\backslash ,\backslash int\_0^\backslash infty\; e^\backslash ,\; dx=\backslash frac,$$a form of Gaussian integral.

The lemniscate constant appears in the evaluation of the integrals

$$=\; \backslash int\_0^\backslash sqrt\backslash ,dx=\backslash int\_0^\backslash sqrt\backslash ,dx$$

$$\backslash frac\; =\; \backslash int\_0^$$

John Todd's lemniscate constants are defined by integrals:^[11]

$$A\; =\; \backslash int\_0^1\backslash frac$$

$$B\; =\; \backslash int\_0^1\backslash frac$$

Circumference of an ellipse

The lemniscate constant satisfies the equation

$$\backslash frac\; =\; 2\; \backslash int\_0^1\backslash frac$$

Euler discovered in 1738 that for the rectangular elastica (first and second lemniscate constants)^[38]

$$\backslash textrm\backslash \; \backslash textrm\backslash cdot\backslash textrm\; =\; A\; \backslash cdot\; B\; =\; \backslash int\_0^1\; \backslash frac\; \backslash cdot\; \backslash int\_0^1\; \backslash frac\; =\; \backslash frac\backslash varpi2\; \backslash cdot\; \backslash frac\backslash pi\; =\; \backslash frac\backslash pi4$$

Now considering the circumference

C of the ellipse with axes \sqrt{ 2 } and 1 , satisfying 2 x ^{2} + 4 y ^{2} = 1 , Stirling noted that

$$\backslash frac\; =\; \backslash int\_0^1\backslash frac\; +\; \backslash int\_0^1\backslash frac$$

Hence the full circumference is

$$C\; =\; \backslash frac\; +\; \backslash varpi\; =3.820197789\backslash ldots$$

This is also the arc length of the sine curve on half a period:^[39]

$$C\; =\; \backslash int\_0^\backslash pi\; \backslash sqrt\backslash ,dx$$

Other limits

Analogously to$$2\backslash pi=\backslash lim\_\backslash left|\backslash frac\backslash right|^$$where

B _{n} are Bernoulli numbers, we have$$2\backslash varpi=\backslash lim\_\backslash left(\backslash frac\backslash right)^$$where H _{n} are Hurwitz numbers.

References

• • Sequences A014549, A053002, and A062539 in OEIS
• Cox . David A. . The Arithmetic-Geometric Mean of Gauss . L'Enseignement Mathématique . January 1984 . 30. 2 . 275–330 . 10.5169/seals-53831 . 25 June 2022.

External links

• Web site: 2021-10-17. Gauss's constant and where it occurs. www.johndcook.com. en-US.

Notes and References

1. Book: Gauss . C. F. . Werke (Band III). Herausgegeben der Königlichen Gesellschaft der Wissenschaften zu Göttingen . Latin, German. 1866. p. 404
2. Book: Eymard . Pierre . Lafon. Jean-Pierre . The Number Pi . American Mathematical Society . 2004 . 0-8218-3246-8. p. 199
3. Book: Bottazzini . Umberto . Umberto Bottazzini . Gray . Jeremy . Jeremy Gray . 2013 . Hidden Harmony – Geometric Fantasies: The Rise of Complex Function Theory . Springer . 10.1007/978-1-4614-5725-1 . 978-1-4614-5724-4 . p. 57
4. Book: Arakawa . Tsuneo . Ibukiyama. Tomoyoshi . Kaneko. Masanobu. Bernoulli Numbers and Zeta Functions . Springer . 2014 . 978-4-431-54918-5. p. 203
5. Book: Finch . Steven R. . Mathematical Constants . 18 August 2003 . Cambridge University Press . 978-0-521-81805-6 . 420 . en.
6. Web site: A062539 - Oeis .
7. Web site: A064853 - Oeis .
8. Web site: Lemniscate Constant.
9. Web site: Records set by y-cruncher . 2024-08-20 . numberworld.org.
10. Web site: A014549 - Oeis .
11. Todd . John . January 1975 . The lemniscate constants . . 18 . 14–19 . 10.1145/360569.360580 . 85873 . free . 1.
12. Web site: A085565 - Oeis .
13. Web site: A076390 - Oeis .
14. Theodor . Schneider . Zur Theorie der Abelschen Funktionen und Integrale . 1941 . Journal für die reine und angewandte Mathematik . 183 . 19 . 110–128 . 10.1515/crll.1941.183.110 . 118624331 .
15. G. V. Choodnovsky: Algebraic independence of constants connected with the functions of analysis, Notices of the AMS 22, 1975, p. A-486
16. G. V. Chudnovsky: Contributions to The Theory of Transcendental Numbers, American Mathematical Society, 1984, p. 6
17. Book: Borwein . Jonathan M. . Borwein. Peter B. . Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity . Wiley-Interscience . 1987 . First . 0-471-83138-7. p. 45
18. Book: Finch . Steven R. . Mathematical Constants . 18 August 2003 . Cambridge University Press . 978-0-521-81805-6 . 420–422 . en.
19. Book: Some milestones of lemniscatomy  . Schappacher . Norbert . Norbert Schappacher . 1997 . Sertöz . S. . Algebraic Geometry . Proceedings of Bilkent Summer School, August 7–19, 1995, Ankara, Turkey . Marcel Dekker . 257–290 . http://irma.math.unistra.fr/~schappa/NSch/Publications_files/1997_LemniscProvis.pdf.
20. Web site: A113847 - Oeis .
21. Levin (2006)
22. 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.
23. Hyde . Trevor . 2014 . A Wallis product on clovers . The American Mathematical Monthly . 121 . 3 . 237–243 . 10.4169/amer.math.monthly.121.03.237 . 34819500 .
24. Book: Bottazzini . Umberto . Umberto Bottazzini . Gray . Jeremy . Jeremy Gray . 2013 . Hidden Harmony – Geometric Fantasies: The Rise of Complex Function Theory . Springer . 10.1007/978-1-4614-5725-1 . 978-1-4614-5724-4 . p. 60
25. Todd (1975)
26. for the first equality. The second equality can be proved by using the pentagonal number theorem.
27. Book: Berndt . Bruce C. . Ramanujan's Notebooks Part V . Springer . 1998 . 978-1-4612-7221-2. p. 326
28. This formula can be proved by hypergeometric inversion: Let\operatorname{a}(q) = \sum _{m,n\inZ} q ^{m ^{2+mn+n} ^{2}} where q\in C with \left|q\right|< 1 . Then\operatorname{a}(q) = {} _{2F} , _{1 \left( (1)/(3)} (2)/(3) , 1 ,z\right) whereq = \exp\left( - (2 \pi)/(\sqrt{ 3) }\frac\right)where z\in C \setminus\{ 0 , 1 \} . The formula in question follows from setting $z=\backslash tfrac14\backslash bigl(3\backslash sqrt-5\backslash bigr)$.
29. Book: Eymard . Pierre . Lafon. Jean-Pierre . The Number Pi . American Mathematical Society . 2004 . 0-8218-3246-8. p. 232
30. Web site: Level-one elliptic modular forms . Garrett . Paul . University of Minnesota. p. 11—13
31. Book: Khrushchev . Sergey . Orthogonal Polynomials and Continued Fractions . Cambridge University Press . 2008 . First . 978-0-521-85419-1. p. 140 (eq. 3.34), p. 153. There's an error on p. 153: 4 [\Gamma( 3+ s/ 4 )/\Gamma( 1+ s/ 4 )] ^{2} should be 4 [\Gamma(( 3+ s)/ 4 )/\Gamma(( 1+ s)/ 4 )] ^{2} .
32. Book: Khrushchev . Sergey . Orthogonal Polynomials and Continued Fractions . Cambridge University Press . 2008 . First . 978-0-521-85419-1. p. 146, 155
33. Book: Perron . Oskar . German . Oskar Perron. Die Lehre von den Kettenbrüchen: Band II . B. G. Teubner . 1957 . Third. p. 36, eq. 24
34. Web site: A062540 - OEIS . 2022-09-14 . oeis.org.
35. Web site: A053002 - OEIS. oeis.org.
36. 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. 10.1007/s11139-013-9528-5 . 120943474 .
37. Web site: A068467 - Oeis .
38. Levien (2008)
39. Web site: An Eloquent Formula for the Perimeter of an Ellipse. Adlaj. Semjon. 2012. American Mathematical Society. 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 \sqrt{ 2 } l( 1 /\sqrt{ 2 }) = L + M .. In this paper M =1 /G = \pi/\varpi and L = \pi/M = G\pi = \varpi .