Catalan's constant

In mathematics, Catalan's constant G, which occasionally appears in estimates in combinatorics, is defined by

$G = \beta(2) = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n%2B1)^2} = \frac{1}{1^2} - \frac{1}{3^2} %2B \frac{1}{5^2} - \frac{1}{7^2} %2B \cdots \!$

where β is the Dirichlet beta function. Its numerical value [1] is approximately (sequence A006752 in OEIS)

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

It is not known whether G is irrational, let alone transcendental.

Catalan's constant was named after Eugène Charles Catalan.

1 Integral identities
2 Uses
3 Quickly converging series
4 Known digits
5 See also
6 Notes
7 References

Integral identities

Some identities include

$G = -\int_{0}^{1} \frac{\ln(t)}{1 %2B t^2} \,dt \!$

$G = \int_0^1 \int_0^1 \frac{1}{1%2Bx^2 y^2} \,dx\, dy \!$

$G = \int_{0}^{\pi/4} \frac{t}{\sin(t) \cos(t)} \;dt \!$

$G = \tfrac14 \int_{-\pi/2}^{\pi/2} \frac{t}{\sin(t)} \;dt \!$

$G = \int_{0}^{\pi/4} \ln ( \cot(t) ) \,dt \!$

along with

$G = \tfrac12\int_0^1 \mathrm{K}(x)\,dx \!$

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:

$\psi_{1}\left(\frac{1}{4}\right) = \pi^2 %2B 8G$

$\psi_{1}\left(\frac{3}{4}\right) = \pi^2 - 8G.$

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:

$G = \,$	$3 \sum_{n=0}^\infty \frac{1}{2^{4n}} \left( -\frac{1}{2(8n%2B2)^2} %2B\frac{1}{2^2(8n%2B3)^2} -\frac{1}{2^3(8n%2B5)^2} %2B\frac{1}{2^3(8n%2B6)^2} -\frac{1}{2^4(8n%2B7)^2} %2B\frac{1}{2(8n%2B1)^2} \right) -$
	$2 \sum_{n=0}^\infty \frac{1}{2^{12n}} \left( \frac{1}{2^4(8n%2B2)^2} %2B\frac{1}{2^6(8n%2B3)^2} -\frac{1}{2^9(8n%2B5)^2} -\frac{1}{2^{10} (8n%2B6)^2} -\frac{1}{2^{12} (8n%2B7)^2} %2B\frac{1}{2^3(8n%2B1)^2} \right)$

and

$G = \frac{\pi}{8} \log(\sqrt{3} %2B 2) %2B \tfrac38 \sum_{n=0}^\infty \frac{(n!)^2}{(2n)!(2n%2B1)^2}.$

The theoretical foundations for such series is given by Broadhurst (the first formula)^[1] and Ramanujan (the second formula)^[2] The algorithms for fast evaluation of the Catalan constant is constructed by E. Karatsuba^[3]^[4].

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.^[5]

**Number of known decimal digits of Catalan's constant G**
Date	Decimal digits	Computation performed by
1832	16	Thomas Clausen
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
1996	1,500,000	Thomas Papanikolaou
1997	3,379,957	Patrick Demichel
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^[6]
August 2008	10,000,000,000	Shigeru Kondo & Steve Pagliarulo^[7]
January 31, 2009	15,510,000,000	Alexander J. Yee & Raymond Chan^[8]
April 16, 2009	31,026,000,000	Alexander J. Yee & Raymond Chan^[8]

Notes

^ D.J. Broadhurst, "Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5)", (1998) arXiv math.CA/9803067
^ 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
^ ^a ^b Large Computations

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. MR 1929434. http://www-2.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. MR 1703281.
Bradley, David M. (2007). "A class of series acceleration formulae for Catalan's constant". arXiv:0706.0356.
Bradley, David M. (2001), Representations of Catalan's constant, http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.26.1879
Carella, N. A. (2011). "Note On the Irrationality of the L-Function Constants L(s, X)". arXiv:1105.2042v1.