- G = 0.915965594177219015054603514932384110774…
|Unsolved problem in mathematics:|
Is Catalan's constant irrational? If so, is it transcendental?(more unsolved problems in mathematics)
Catalan's constant was named after Eugène Charles Catalan.
The similar but apparently more complicated series
can be evaluated exactly and is equal to π3/32.
Some identities involving definite integrals include
where the last three formulas are related to Malmsten's integrals.
If K(k) is the complete elliptic integral of the first kind, as a function of the elliptic modulus k, then
With the gamma function Γ(x + 1) = x!
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.
In low-dimensional topology, Catalan's constant is a rational multiple of the volume of an ideal hyperbolic octahedron, and therefore of the hyperbolic volume of the complement of the Whitehead link.
It also appears in connection with the hyperbolic secant distribution.
Relation to other special functionsEdit
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
Quickly converging seriesEdit
The following two formulas involve quickly converging series, and are thus appropriate for numerical computation:
The theoretical foundations for such series are given by Broadhurst, for the first formula, and Ramanujan, for the second formula. The algorithms for fast evaluation of the Catalan constant were constructed by E. Karatsuba.
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.
|Date||Decimal digits||Computation performed by|
|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|
|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|
|August 2008||10000000000||Shigeru Kondo & Steve Pagliarulo|
|January 31, 2009||15510000000||Alexander J. Yee & Raymond Chan|
|April 16, 2009||31026000000||Alexander J. Yee & Raymond Chan|
|June 7, 2015||200000001100||Robert J. Setti|
|April 12, 2016||250000000000||Ron Watkins|
|February 16, 2019||300000000000||Tizian Hanselmann|
|March 29, 2019||500000000000||Mike A & Ian Cutress|
|July 16, 2019||600000000100||Seungmin Kim|
- Papanikolaou, Thomas (March 1997). "Catalan's Constant to 1,500,000 Places". 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.
- 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.
- 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.
- 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 missing]
- 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.[ISBN missing]
- Gourdon, X.; Sebah, P. "Constants and Records of Computation".
- "Shigeru Kondo's website". Archived from the original on 2008-02-11. Retrieved 2008-01-31.
- Constants and Records of Computation
- Large Computations
- Catalan's constant records using YMP
- Catalan's constant records using YMP
- Catalan's constant world record by Seungmin Kim
- Victor Adamchik, 33 representations for Catalan's constant (undated)
- 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.
- Plouffe, Simon (1993). "A few identities (III) with Catalan". (Provides over one hundred different identities).
- Simon Plouffe, A few identities with Catalan constant and Pi^2, (1999) (Provides a graphical interpretation of the relations)
- Weisstein, Eric W. "Catalan's Constant". MathWorld.
- Catalan constant: Generalized power series at the Wolfram Functions Site
- Greg Fee, Catalan's Constant (Ramanujan's Formula) (1996) (Provides the first 300,000 digits of Catalan's constant.).
- Fee, Greg (1990), "Computation of Catalan's constant using Ramanujan's Formula", Proceedings of the international symposium on Symbolic and algebraic computation - ISSAC '90, Proceedings of the ISSAC '90, 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. doi:10.1023/A:1006945407723. MR 1703281. S2CID 5111792.
- Bradley, David M. (2007). "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. S2CID 5111792.
- Bradley, David M. (2001), Representations of Catalan's constant, CiteSeerX 10.1.1.26.1879