Catalan's constant
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In mathematics, Catalan's constant K, which occasionally appears in estimates in combinatorics, is defined by
where β is the Dirichlet beta function. Its numerical value [1] is approximately
- K = 0.915 965 594 177 219 015 054 603 514 932 384 110 774 …
It is not known whether K is rational or irrational.
Catalan's constant was named after Eugène Charles Catalan.
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[edit] Integral identities
Some identities include
along with
where K(x) is a complete elliptic integral of the first kind, and
[edit] Uses
K appears in combinatorics, as well as 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.
It also appears in connection with the hyperbolic secant distribution.
[edit] 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 is given by Broadhurst.[1]
[edit] Known digits
The number of known digits of Catalan's constant K has increased dramatically during the last decades. This is due both to the increase of performance of computers as well as to algorithmic improvements.[2]
Date | Decimal digits | Computation performed by |
---|---|---|
October 2006 | 5,000,000,000 | Shigeru Kondo[3] |
2002 | 201,000,000 | Xavier Gourdon & Pascal Sebah |
2001 | 100,000,500 | Xavier Gourdon & Pascal Sebah |
January 4, 1998 | 12,500,000 | Xavier Gourdon |
1997 | 3,379,957 | Patrick Demichel |
1996 | 1,500,000 | Thomas Papanikolaou |
September 29, 1996 | 300,000 | Thomas Papanikolaou |
August 14, 1996 | 100,000 | Greg J. Fee & Simon Plouffe |
1996 | 50,000 | Greg J. Fee |
1990 | 20,000 | Greg J. Fee |
1913 | 32 | James W. L. Glaisher |
1877 | 20 | James W. L. Glaisher |
[edit] See also
[edit] References
- ^ D.J. Broadhurst, "Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5)", (1998) arXiv math.CA/9803067
- ^ Gourdon, X., Sebah, P; Constants and Records of Computation
- ^ Shigeru Kondo's website
- Victor Adamchik, 33 representations for Catalan's constant (undated)
- Victor Adamchik, A certain series associated with Catalan's constant, (2002) Zeitschrift fuer Analysis und ihre Anwendungen (ZAA), 21, pp.1-10.
- Simon Plouffe, A few identities (III) with Catalan, (1993) (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)
- Eric W. Weisstein, Catalan's Constant at 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.).