Catalan's constant

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 [http://www.gutenberg.org/etext/812] is approximately

:"G" = 0.915 965 594 177 219 015 054 603 514 932 384 110 774 …

It is not known whether "G" is rational or irrational.

Catalan's constant was named after Eugène Charles Catalan.

Integral identities

Some identities include

:$G\; =\; -int\_\{0\}^\{1\}\; frac\{ln(t)\}\{1\; +\; t^2\}\; dt$

:$G\; =\; int\_0^1\; int\_0^1\; frac\{1\}\{1+x^2\; y^2\}\; dx\; dy$

:$G\; =\; int\_\{0\}^\{pi/4\}\; frac\{t\}\{sin(t)\; cos(t)\}\; dt$

along with

:$G\; =\; frac12int\_0^1\; mathrm\{K\}(x),dx$

where K("x") is a complete elliptic integral of the first kind, and

:$G\; =\; int\_0^1\; frac\{arctan\; x\}\{x\}dx.$

Uses

"G" appears in combinatorics, as well as in values of the second polygamma function, also called the trigamma function, at fractional arguments:

:$psi\_\{1\}left(frac\{1\}\{4\}\; ight)\; =\; pi^2\; +\; 8G$

:$psi\_\{1\}left(frac\{3\}\{4\}\; ight)\; =\; pi^2\; -\; 8G$

Simon Plouffe gives an infinite collection of identities between the trigamma function, $pi^2$ 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

:$G\; =\; frac\{pi\}\{8\}\; log(sqrt\{3\}\; +\; 2)\; +\; frac38\; sum\_\{n=0\}^infty\; frac\{(n!)^2\}\{(2n)!(2n+1)^2\}.$

The theoretical foundations for such series is given by Broadhurst. [D.J. Broadhurst, " [http://arxiv.org/abs/math.CA/9803067 Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5)] ", (1998) "arXiv" math.CA/9803067]

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. [Gourdon, X., Sebah, P; [http://numbers.computation.free.fr/Constants/constants.html Constants and Records of Computation] ]

ee also

* Zeta constant

References

* Victor Adamchik, " [http://www-2.cs.cmu.edu/~adamchik/articles/catalan/catalan.htm 33 representations for Catalan's constant] " (undated)
* Victor Adamchik, " [http://www-2.cs.cmu.edu/~adamchik/articles/csum.html A certain series associated with Catalan's constant] ", (2002) Zeitschrift fuer Analysis und ihre Anwendungen (ZAA), 21, pp.1-10.
* Simon Plouffe, " [http://www.lacim.uqam.ca/~plouffe/IntegerRelations/identities3a.html A few identities (III) with Catalan] ", (1993) "(Provides over one hundred different identities)".
* Simon Plouffe, " [http://www.lacim.uqam.ca/~plouffe/IntegerRelations/identities3.html A few identities with Catalan constant and Pi^2] ", (1999) "(Provides a graphical interpretation of the relations)"
*
* [http://functions.wolfram.com/Constants/Catalan/06/01/ Catalan constant: Generalized power series] at the Wolfram Functions Site
* Greg Fee, " [http://www.gutenberg.org/etext/682 Catalan's Constant (Ramanujan's Formula)] " (1996) "(Provides the first 300,000 digits of Catalan's constant.)".