Catalan's constant
In mathematics, Catalan's constant G, is defined by
where β is the Dirichlet beta function. Its numerical value[1] is approximately (sequence A006752 in the OEIS)
- G = 0.915965594177219015054603514932384110774…
It is not known whether G is irrational, let alone transcendental.[2] G has been called "arguably the most basic constant whose irrationality and transcendence (though strongly suspected) remain unproven".[3]
Catalan's constant was named after Eugène Charles Catalan, who found quickly-converging series for its calculation and published a memoir on it in 1865.[4][5]
Uses
In low-dimensional topology, Catalan's constant is 1/4 of the volume of an ideal hyperbolic octahedron, and therefore 1/4 of the hyperbolic volume of the complement of the Whitehead link.[6] It is 1/8 of the volume of the complement of the Borromean rings.[7]
In combinatorics and statistical mechanics, it arises in connection with counting domino tilings,[8] spanning trees,[9] and Hamiltonian cycles of grid graphs.[10]
In number theory, Catalan's constant appears in a conjectured formula for the asymptotic number of primes of the form according to Hardy and Littlewood's Conjecture F. However, it is an unsolved problem (one of Landau's problems) whether there are even infinitely many primes of this form.[11]
Catalan's constant also appears in the calculation of the mass distribution of spiral galaxies.[12][13]
Known digits
The number of known digits of Catalan's constant G has increased dramatically during the last decades. This is due both to the increase of performance of computers as well as to algorithmic improvements.[14]
Date | Decimal digits | Computation performed by |
---|---|---|
1832 | 16 | Thomas Clausen |
1858 | 19 | Carl Johan Danielsson Hill |
1864 | 14 | Eugène Charles Catalan |
1877 | 20 | James W. L. Glaisher |
1913 | 32 | James W. L. Glaisher |
1990 | 20000 | Greg J. Fee |
1996 | 50000 | Greg J. Fee |
August 14, 1996 | 100000 | Greg J. Fee & Simon Plouffe |
September 29, 1996 | 300000 | Thomas Papanikolaou |
1996 | 1500000 | Thomas Papanikolaou |
1997 | 3379957 | Patrick Demichel |
January 4, 1998 | 12500000 | Xavier Gourdon |
2001 | 100000500 | Xavier Gourdon & Pascal Sebah |
2002 | 201000000 | Xavier Gourdon & Pascal Sebah |
October 2006 | 5000000000 | Shigeru Kondo & Steve Pagliarulo[15] |
August 2008 | 10000000000 | Shigeru Kondo & Steve Pagliarulo[14] |
January 31, 2009 | 15510000000 | Alexander J. Yee & Raymond Chan[16] |
April 16, 2009 | 31026000000 | Alexander J. Yee & Raymond Chan[16] |
June 7, 2015 | 200000001100 | Robert J. Setti[17] |
April 12, 2016 | 250000000000 | Ron Watkins[17] |
February 16, 2019 | 300000000000 | Tizian Hanselmann[17] |
March 29, 2019 | 500000000000 | Mike A & Ian Cutress[17] |
July 16, 2019 | 600000000100 | Seungmin Kim[18][19] |
September 6, 2020 | 1000000001337 | Andrew Sun[20] |
March 9, 2022 | 1200000000100 | Seungmin Kim[20] |
Integral identities
As Seán Stewart writes, "There is a rich and seemingly endless source of definite integrals that can be equated to or expressed in terms of Catalan's constant."[21] Some of these expressions include:
where the last three formulas are related to Malmsten's integrals.[22]
If K(k) is the complete elliptic integral of the first kind, as a function of the elliptic modulus k, then
If E(k) is the complete elliptic integral of the second kind, as a function of the elliptic modulus k, then
With the gamma function Γ(x + 1) = x!
The integral is a known special function, called the inverse tangent integral, and was extensively studied by Srinivasa Ramanujan.
Relation to other special functions
G appears in values of the second polygamma function, also called the trigamma function, at fractional arguments:
Simon Plouffe gives an infinite collection of identities between the trigamma function, π2 and Catalan's constant; these are expressible as paths on a graph.
Catalan's constant occurs frequently in relation to the Clausen function, the inverse tangent integral, the inverse sine integral, the Barnes G-function, as well as integrals and series summable in terms of the aforementioned functions.
As a particular example, by first expressing the inverse tangent integral in its closed form – in terms of Clausen functions – and then expressing those Clausen functions in terms of the Barnes G-function, the following expression is obtained (see Clausen function for more):
If one defines the Lerch transcendent Φ(z,s,α) (related to the Lerch zeta function) by then
Quickly converging series
The following two formulas involve quickly converging series, and are thus appropriate for numerical computation: and
The theoretical foundations for such series are given by Broadhurst, for the first formula,[23] and Ramanujan, for the second formula.[24] The algorithms for fast evaluation of the Catalan constant were constructed by E. Karatsuba.[25][26] Using these series, calculating Catalan's constant is now about as fast as calculating Apery's constant, .[27]
Other quickly converging series, due to Guillera and Pilehrood and employed by the y-cruncher software, include:[27]
All of these series have time complexity .[27]
Continued fraction
G can be expressed in the following form[28]
- The simple continued fraction is given by[29]
- This continued fraction would have infinite terms if and only if is irrational, which is still unresolved.
See also
- Gieseking manifold
- List of mathematical constants
- Mathematical constant
- Particular values of Riemann zeta function
References
- ^ Papanikolaou, Thomas (March 1997). Catalan's Constant to 1,500,000 Places – via Gutenberg.org.
- ^ Nesterenko, Yu. V. (January 2016), "On Catalan's constant", Proceedings of the Steklov Institute of Mathematics, 292 (1): 153–170, doi:10.1134/s0081543816010107, S2CID 124903059.
- ^ Bailey, David H.; Borwein, Jonathan M.; Mattingly, Andrew; Wightwick, Glenn (2013), "The computation of previously inaccessible digits of and Catalan's constant", Notices of the American Mathematical Society, 60 (7): 844–854, doi:10.1090/noti1015, MR 3086394
- ^ Goldstein, Catherine (2015), "The mathematical achievements of Eugène Catalan", Bulletin de la Société Royale des Sciences de Liège, 84: 74–92, MR 3498215
- ^ Catalan, E. (1865), "Mémoire sur la transformation des séries et sur quelques intégrales définies", Ers, Publiés Par l'Académie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique. Collection in 4, Mémoires de l'Académie royale des sciences, des lettres et des beaux-arts de Belgique (in French), 33, Brussels, hdl:2268/193841
- ^ Agol, Ian (2010), "The minimal volume orientable hyperbolic 2-cusped 3-manifolds", Proceedings of the American Mathematical Society, 138 (10): 3723–3732, arXiv:0804.0043, doi:10.1090/S0002-9939-10-10364-5, MR 2661571, S2CID 2016662.
- ^ William Thurston (March 2002), "7. Computation of volume" (PDF), The Geometry and Topology of Three-Manifolds, p. 165, archived (PDF) from the original on 2011-01-25
- ^ Temperley, H. N. V.; Fisher, Michael E. (August 1961), "Dimer problem in statistical mechanics—an exact result", Philosophical Magazine, 6 (68): 1061–1063, Bibcode:1961PMag....6.1061T, doi:10.1080/14786436108243366
- ^ Wu, F. Y. (1977), "Number of spanning trees on a lattice", Journal of Physics, 10 (6): L113–L115, Bibcode:1977JPhA...10L.113W, doi:10.1088/0305-4470/10/6/004, MR 0489559
- ^ Kasteleyn, P. W. (1963), "A soluble self-avoiding walk problem", Physica, 29 (12): 1329–1337, Bibcode:1963Phy....29.1329K, doi:10.1016/S0031-8914(63)80241-4, MR 0159642
- ^ Shanks, Daniel (1959), "A sieve method for factoring numbers of the form ", Mathematical Tables and Other Aids to Computation, 13: 78–86, doi:10.2307/2001956, JSTOR 2001956, MR 0105784
- ^ Wyse, A. B.; Mayall, N. U. (January 1942), "Distribution of Mass in the Spiral Nebulae Messier 31 and Messier 33.", The Astrophysical Journal, 95: 24–47, Bibcode:1942ApJ....95...24W, doi:10.1086/144370
- ^ van der Kruit, P. C. (March 1988), "The three-dimensional distribution of light and mass in disks of spiral galaxies.", Astronomy & Astrophysics, 192: 117–127, Bibcode:1988A&A...192..117V
- ^ a b Gourdon, X.; Sebah, P. "Constants and Records of Computation". Retrieved 11 September 2007.
- ^ "Shigeru Kondo's website". Archived from the original on 2008-02-11. Retrieved 2008-01-31.
- ^ a b "Large Computations". Retrieved 31 January 2009.
- ^ a b c d "Catalan's constant records using YMP". Retrieved 14 May 2016.
- ^ "Catalan's constant records using YMP". Archived from the original on 22 July 2019. Retrieved 22 July 2019.
- ^ "Catalan's constant world record by Seungmin Kim". 23 July 2019. Retrieved 17 October 2020.
- ^ a b "Records set by y-cruncher". www.numberworld.org. Retrieved 2022-02-13.
- ^ Stewart, Seán M. (2020), "A Catalan constant inspired integral odyssey", The Mathematical Gazette, 104 (561): 449–459, doi:10.1017/mag.2020.99, MR 4163926, S2CID 225116026
- ^ Blagouchine, Iaroslav (2014). "Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results" (PDF). The Ramanujan Journal. 35: 21–110. doi:10.1007/s11139-013-9528-5. S2CID 120943474. Archived from the original (PDF) on 2018-10-02. Retrieved 2018-10-01.
- ^ Broadhurst, D. J. (1998). "Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5)". arXiv:math.CA/9803067.
- ^ Berndt, B. C. (1985). Ramanujan's Notebook, Part I. Springer Verlag. p. 289. ISBN 978-1-4612-1088-7.
- ^ Karatsuba, E. A. (1991). "Fast evaluation of transcendental functions". Probl. Inf. Transm. 27 (4): 339–360. MR 1156939. Zbl 0754.65021.
- ^ Karatsuba, E. A. (2001). "Fast computation of some special integrals of mathematical physics". In Krämer, W.; von Gudenberg, J. W. (eds.). Scientific Computing, Validated Numerics, Interval Methods. pp. 29–41. doi:10.1007/978-1-4757-6484-0_3.
- ^ a b c Alexander Yee (14 May 2019). "Formulas and Algorithms". Retrieved 5 December 2021.
- ^ Bowman, D. & Mc Laughlin, J. (2002). "Polynomial continued fractions" (PDF). Acta Arithmetica. 103 (4): 329–342. arXiv:1812.08251. Bibcode:2002AcAri.103..329B. doi:10.4064/aa103-4-3. S2CID 119137246. Archived (PDF) from the original on 2020-04-13.
- ^ "A014538 - OEIS". oeis.org. Retrieved 2022-10-27.
Further reading
- Adamchik, Victor (2002). "A certain series associated with Catalan's constant". Zeitschrift für Analysis und ihre Anwendungen. 21 (3): 1–10. doi:10.4171/ZAA/1110. MR 1929434. Archived from the original on 2010-03-16. Retrieved 2005-07-14.
- Fee, Gregory J. (1990). "Computation of Catalan's Constant Using Ramanujan's Formula". In Watanabe, Shunro; Nagata, Morio (eds.). Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC '90, Tokyo, Japan, August 20-24, 1990. ACM. pp. 157–160. doi:10.1145/96877.96917. ISBN 0201548925. S2CID 1949187.
- Bradley, David M. (1999). "A class of series acceleration formulae for Catalan's constant". The Ramanujan Journal. 3 (2): 159–173. arXiv:0706.0356. Bibcode:2007arXiv0706.0356B. doi:10.1023/A:1006945407723. MR 1703281. S2CID 5111792.
External links
- Adamchik, Victor. "33 representations for Catalan's constant". Archived from the original on 2016-08-07. Retrieved 14 July 2005.
- Plouffe, Simon (1993). "A few identities (III) with Catalan". Archived from the original on 2019-06-26. Retrieved 29 July 2005. (Provides over one hundred different identities).
- Plouffe, Simon (1999). "A few identities with Catalan constant and Pi^2". Archived from the original on 2019-06-26. Retrieved 29 July 2005. (Provides a graphical interpretation of the relations)
- Fee, Greg (1996). Catalan's Constant (Ramanujan's Formula). (Provides the first 300,000 digits of Catalan's constant)
- Bradley, David M. (2001). Representations of Catalan's constant. CiteSeerX 10.1.1.26.1879.
- Johansson, Fredrik. "0.915965594177219015054603514932". Ordner, a catalog of real numbers in Fungrim. Retrieved 21 April 2021.
- "Catalan's Constant". YouTube. Let's Learn, Nemo!. 10 August 2020. Retrieved 6 April 2021.
- Weisstein, Eric W. "Catalan's Constant". MathWorld.
- "Catalan constant: Series representations". Wolfram Functions Site.
- "Catalan constant". Encyclopedia of Mathematics. EMS Press. 2001 [1994].