In mathematics, Catalan's constant G, which occasionally appears in estimates in combinatorics, is defined by
where β is the Dirichlet beta function. Its numerical value [1] is approximately (sequence A006752 in OEIS)
It is not known whether G is rational or irrational.
Catalan's constant was named after Eugène Charles Catalan.
Contents
Integral identities
Some identities include
along with
where K(x) is a complete elliptic integral of the first kind, and
Uses
G 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, π^{2} and Catalan's constant; these are expressible as paths on a graph.
It also appears in connection with the hyperbolic secant distribution.
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]}
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.^{[2]}
See also
Notes
References
 Victor Adamchik, 33 representations for Catalan's constant (undated)
 Adamchik,, Victor (2002). "A certain series associated with Catalan's constant". Zeitschr. f. Analysis und ihre Anwendungen (ZAA) 21 (3): 1–10. MR1929434. http://www2.cs.cmu.edu/~adamchik/articles/csum.html.
 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)
 Weisstein, Eric W., "Catalan's Constant" from 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.).
 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. MR1703281.
 Bradley, David M. (2007). "A class of series acceleration formulae for Catalan's constant". arΧiv:0706.0356.
 Bradley, David M. (2001), Representations of Catalan's constant
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