In mathematics, the lemniscate constant ϖ 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.[1] Equivalently, the perimeter of the lemniscate is 2ϖ. The lemniscate constant is closely related to the lemniscate elliptic functions and approximately equal to 2.62205755.[2] It also appears in evaluation of the gamma and beta function at certain rational values. The symbol ϖ is a cursive variant of π; see Pi § Variant pi.

Lemniscate of Bernoulli

Sometimes the quantities 2ϖ or ϖ/2 are referred to as the lemniscate constant.[3][4]

As of 2024 over 1.2 trillion digits of this constant have been calculated.[5]

History

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Gauss's constant, denoted by G, is equal to ϖ /π ≈ 0.8346268[6] and named after Carl Friedrich Gauss, who calculated it via the arithmetic–geometric mean as  .[7] By 1799, Gauss had two proofs of the theorem that   where   is the lemniscate constant.[8]

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.[9][10][11]

The lemniscate constant   and Todd's first lemniscate constant   were proven transcendental by Carl Ludwig Siegel in 1932 and later by Theodor Schneider in 1937 and Todd's second lemniscate constant   and Gauss's constant   were proven transcendental by Theodor Schneider in 1941.[9][12][13] In 1975, Gregory Chudnovsky proved that the set   is algebraically independent over  , which implies that   and   are algebraically independent as well.[14][15] But the set   (where the prime denotes the derivative with respect to the second variable) is not algebraically independent over  .[16] In 1996, Yuri Nesterenko proved that the set   is algebraically independent over  .[17]

Forms

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Usually,   is defined by the first equality below, but it has many equivalent forms:[18]

 

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  ,

 

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   published in 1800:[19] John Todd's lemniscate constants may be given in terms of the beta function B:  

As a special value of L-functions

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which is analogous to

 

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

Analogously to the Leibniz formula for π,   we have[21][22][23][24][25]   where   is the L-function of the elliptic curve   over  ; this means that   is the multiplicative function given by   where   is the number of solutions of the congruence   in variables   that are non-negative integers (  is the set of all primes). Equivalently,   is given by   where   such that   and   is the eta function.[26][27][28] The above result can be equivalently written as   (the number   is the conductor of  ) and also tells us that the BSD conjecture is true for the above  .[29] The first few values of   are given by the following table; if   such that   doesn't appear in the table, then  :  

As a special value of other functions

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Let   be the minimal weight level   new form. Then[30]   The  -coefficient of   is the Ramanujan tau function.

Series

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Viète's formula for π can be written:

 

An analogous formula for ϖ is:[31]

 

The Wallis product for π is:

 

An analogous formula for ϖ is:[32]

 

A related result for Gauss's constant ( ) is:[33]

 

An infinite series discovered by Gauss is:[34]

 

The Machin formula for π is   and several similar formulas for π can be developed using trigonometric angle sum identities, e.g. Euler's formula  . Analogous formulas can be developed for ϖ, including the following found by Gauss:  , where   is the lemniscate arcsine.[35]

The lemniscate constant can be rapidly computed by the series[36][37]

 

where   (these are the generalized pentagonal numbers). Also[38]

 

In a spirit similar to that of the Basel problem,

 

where   are the Gaussian integers and   is the Eisenstein series of weight   (see Lemniscate elliptic functions § Hurwitz numbers for a more general result).[39]

A related result is

 

where   is the sum of positive divisors function.[40]

In 1842, Malmsten found

 

where   is Euler's constant and   is the Dirichlet-Beta function.

The lemniscate constant is given by the rapidly converging series

 

The constant is also given by the infinite product

 

Also[41]

 

Continued fractions

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A (generalized) continued fraction for π is   An analogous formula for ϖ is[10]  

Define Brouncker's continued fraction by[42]   Let   except for the first equality where  . Then[43][44]   For example,  

In fact, the values of   and  , coupled with the functional equation   determine the values of   for all  .

Simple continued fractions

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Simple continued fractions for the lemniscate constant and related constants include[45][46]  

Integrals

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A geometric representation of   and  

The lemniscate constant ϖ is related to the area under the curve  . Defining  , twice the area in the positive quadrant under the curve   is   In the quartic case,  

In 1842, Malmsten discovered that[47]

 

Furthermore,  

and[48]

  a form of Gaussian integral.

The lemniscate constant appears in the evaluation of the integrals

 

 

John Todd's lemniscate constants are defined by integrals:[9]

 

 

Circumference of an ellipse

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The lemniscate constant satisfies the equation[49]

 

Euler discovered in 1738 that for the rectangular elastica (first and second lemniscate constants)[50][49]

 

Now considering the circumference   of the ellipse with axes   and  , satisfying  , Stirling noted that[51]

 

Hence the full circumference is

 

This is also the arc length of the sine curve on half a period:[52]

 

Other limits

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Analogously to   where   are Bernoulli numbers, we have   where   are Hurwitz numbers.

Notes

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  1. ^ See:
    • Gauss, C. F. (1866). Werke (Band III) (in Latin and German). Herausgegeben der Königlichen Gesellschaft der Wissenschaften zu Göttingen. p. 404
    • Cox 1984, p. 281
    • Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi. American Mathematical Society. ISBN 0-8218-3246-8. p. 199
    • 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
    • Arakawa, Tsuneo; Ibukiyama, Tomoyoshi; Kaneko, Masanobu (2014). Bernoulli Numbers and Zeta Functions. Springer. ISBN 978-4-431-54918-5. p. 203
  2. ^ See:
  3. ^ "A064853 - Oeis".
  4. ^ "Lemniscate Constant".
  5. ^ "Records set by y-cruncher". numberworld.org. Retrieved 2024-08-20.
  6. ^ "A014549 - Oeis".
  7. ^ Finch 2003, p. 420.
  8. ^ Neither of these proofs was rigorous from the modern point of view. See Cox 1984, p. 281
  9. ^ 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.
  10. ^ a b "A085565 - Oeis". and "A076390 - Oeis".
  11. ^ 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.
  12. ^ In particular, Siegel proved that if   and   with   are algebraic, then   or   is transcendental. Here,   and   are Eisenstein series. The fact that   is transcendental follows from   and  
    Apostol, T. M. (1990). Modular Functions and Dirichlet Series in Number Theory (Second ed.). Springer. p. 12. ISBN 0-387-97127-0.
    Siegel, C. L. (1932). "Über die Perioden elliptischer Funktionen". Journal für die reine und angewandte Mathematik (in German). 167: 62–69.
  13. ^ In particular, Schneider proved that the beta function   is transcendental for all   such that  . The fact that   is transcendental follows from   and similarly for B and G from  
    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.
  14. ^ G. V. Choodnovsky: Algebraic independence of constants connected with the functions of analysis, Notices of the AMS 22, 1975, p. A-486
  15. ^ G. V. Chudnovsky: Contributions to The Theory of Transcendental Numbers, American Mathematical Society, 1984, p. 6
  16. ^ In fact,  
    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
  17. ^ Nesterenko, Y. V.; Philippon, P. (2001). Introduction to Algebraic Independence Theory. Springer. p. 27. ISBN 3-540-41496-7.
  18. ^ See:
  19. ^ Cox 1984, p. 277.
  20. ^ "A113847 - Oeis".
  21. ^ Cremona, J. E. (1997). Algorithms for Modular Elliptic Curves (2nd ed.). Cambridge University Press. ISBN 0521598206. p. 31, formula (2.8.10)
  22. ^ In fact, the series   converges for  .
  23. ^ Murty, Vijaya Kumar (1995). Seminar on Fermat's Last Theorem. American Mathematical Society. p. 16. ISBN 9780821803134.
  24. ^ Cohen, Henri (1993). A Course in Computational Algebraic Number Theory. Springer-Verlag. pp. 382–406. ISBN 978-3-642-08142-2.
  25. ^ "Elliptic curve with LMFDB label 32.a3 (Cremona label 32a2)". The L-functions and modular forms database.
  26. ^ The function   is the unique weight   level   new form and it satisfies the functional equation
     
  27. ^ The   function is closely related to the   function which is the multiplicative function defined by
     
    where   is the number of solutions of the equation
     
    in variables   that are non-negative integers (see Fermat's theorem on sums of two squares) and   is the Dirichlet character from the Leibniz formula for π; also
     
    for any positive integer   where the sum extends only over positive divisors; the relation between   and   is
     
    where   is any non-negative integer.
  28. ^ The   function also appears in
     
    where   is any positive integer and   is the set of all Gaussian integers of the form
     
    where   is odd and   is even. The   function from the previous note satisfies
     
    where   is positive odd.
  29. ^ Rubin, Karl (1987). "Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication". Inventiones Mathematicae. 89: 528.
  30. ^ "Newform orbit 1.12.a.a". The L-functions and modular forms database.
  31. ^ Levin (2006)
  32. ^ 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.
  33. ^ 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.
  34. ^ 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
  35. ^ Todd (1975)
  36. ^ Cox 1984, p. 307, eq. 2.21 for the first equality. The second equality can be proved by using the pentagonal number theorem.
  37. ^ Berndt, Bruce C. (1998). Ramanujan's Notebooks Part V. Springer. ISBN 978-1-4612-7221-2. p. 326
  38. ^ This formula can be proved by hypergeometric inversion: Let
     
    where   with  . Then
     
    where
     
    where  . The formula in question follows from setting  .
  39. ^ Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi. American Mathematical Society. ISBN 0-8218-3246-8. p. 232
  40. ^ Garrett, Paul. "Level-one elliptic modular forms" (PDF). University of Minnesota. p. 11—13
  41. ^ The formula follows from the hypergeometric transformation
     
    where   and   is the modular lambda function.
  42. ^ 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:   should be  .
  43. ^ Khrushchev, Sergey (2008). Orthogonal Polynomials and Continued Fractions (First ed.). Cambridge University Press. ISBN 978-0-521-85419-1. p. 146, 155
  44. ^ Perron, Oskar (1957). Die Lehre von den Kettenbrüchen: Band II (in German) (Third ed.). B. G. Teubner. p. 36, eq. 24
  45. ^ "A062540 - OEIS". oeis.org. Retrieved 2022-09-14.
  46. ^ "A053002 - OEIS". oeis.org.
  47. ^ 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.
  48. ^ "A068467 - Oeis".
  49. ^ a b Cox 1984, p. 313.
  50. ^ Levien (2008)
  51. ^ Cox 1984, p. 312.
  52. ^ 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  . In this paper   and  .

References

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