- G = 0.915 965 594 177 219 015 054 603 514 932 384 110 774 …
|Unsolved problem in mathematics:
Is Catalan's constant irrational? Is it transcendental?
(more unsolved problems in mathematics)
Catalan's constant was named after Eugène Charles Catalan (1814-94).
Some identities include
- If K(t) is a complete elliptic integral of the first kind, then
- With Gamma function
- The integral
- is a known special function, called the Inverse tangent integral, and was extensively studied by Ramanujan.
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 functions
Catalan's constant occurs frequently in relation to the Clausen function, the Inverse tangent integral, the Inverse sine integral, 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 easily obtained (N.B. all the relevant relations for this derivation have been added to the page for the Clausen function):
If one defines the Lerch transcendent, , (related to the Lerch zeta function) by,
then it is clear that
Quickly converging series
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||20,000||Greg J. Fee|
|1996||50,000||Greg J. Fee|
|August 14, 1996||100,000||Greg J. Fee & Simon Plouffe|
|September 29, 1996||300,000||Thomas Papanikolaou|
|January 4, 1998||12,500,000||Xavier Gourdon|
|2001||100,000,500||Xavier Gourdon & Pascal Sebah|
|2002||201,000,000||Xavier Gourdon & Pascal Sebah|
|October 2006||5,000,000,000||Shigeru Kondo & Steve Pagliarulo|
|August 2008||10,000,000,000||Shigeru Kondo & Steve Pagliarulo|
|January 31, 2009||15,510,000,000||Alexander J. Yee & Raymond Chan|
|April 16, 2009||31,026,000,000||Alexander J. Yee & Raymond Chan|
|April 6, 2013||100,000,000,000||Robert J. Setti|
|June 7, 2015||200,000,001,100||Robert J. Setti|
- Nesterenko, Yu. V. (January 2016), "On Catalan's constant", Proceedings of the Steklov Institute of Mathematics, 292 (1): 153–170, doi:10.1134/s0081543816010107.
- Agol, Ian (2010), "The minimal volume orientable hyperbolic 2-cusped 3-manifolds", Proceedings of the American Mathematical Society, 138 (10): 3723–3732, doi:10.1090/S0002-9939-10-10364-5, MR 2661571.
- Broadhurst, D.J. (1998). "Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5)". arXiv: .
- B.C. Berndt, Ramanujan's Notebook, Part I., Springer Verlag (1985)
- E.A. Karatsuba, Fast evaluation of transcendental functions, Probl. Inf. Transm. Vol.27, No.4, pp. 339–360 (1991)
- E.A. Karatsuba, Fast computation of some special integrals of mathematical physics. Scientific Computing, Validated Numerics, Interval Methods, W.Krämer, J.W.von Gudenberg, eds.; pp. 29–41, (2001)
- Gourdon, X., Sebah, P; Constants and Records of Computation
- Shigeru Kondo's website
- Constants and Records of Computation
- Large Computations
- Catalan's constant records using YMP
- 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. 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 ISSAC '90, pp. 157–160, doi:10.1145/96877.96917
- Bradley, David M. (1999). "A class of series acceleration formulae for Catalan's constant". The Ramanujan Journal. 3 (2): 159–173. doi:10.1023/A:1006945407723. MR 1703281.
- Bradley, David M. (2007). "A class of series acceleration formulae for Catalan's constant". arXiv: .
- Bradley, David M. (2001), Representations of Catalan's constant, CiteSeerX