Catalan's constant

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

\Kappa = \beta(2) = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)^2} = \frac{1}{1^2} - \frac{1}{3^2} + \frac{1}{5^2} - \frac{1}{7^2} + \cdots

where β is the Dirichlet beta function. Its numerical value 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.

Contents

[edit] Integral identities

Some identities include

K = -\int_{0}^{1} \frac{\ln(t)}{1 + t^2} \mbox{ d} t
K = \int_0^1 \int_0^1 \frac{1}{1+x^2 y^2} dx dy

along with

K = \frac{1}{2}\int_0^1 \mathrm{K}(x)\,dx

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

K = \int_0^1 \frac{\tan^{-1}x}{x}dx.

[edit] Uses

K 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 + 8K
\psi_{1}\left(\frac{3}{4}\right) = \pi^2 - 8K

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.

The probability that two randomly selected Gaussian integers are coprime is \frac{6}{\pi^2 K}.

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:

K = \, 3 \sum_{n=0}^\infty \frac{1}{2^{4n}} \left( -\frac{1}{2(8n+2)^2} +\frac{1}{2^2(8n+3)^2} -\frac{1}{2^3(8n+5)^2} +\frac{1}{2^3(8n+6)^2} -\frac{1}{2^4(8n+7)^2} +\frac{1}{2(8n+1)^2} \right) -
2 \sum_{n=0}^\infty \frac{1}{2^{12n}} \left( \frac{1}{2^4(8n+2)^2} +\frac{1}{2^6(8n+3)^2} -\frac{1}{2^9(8n+5)^2} -\frac{1}{2^{10} (8n+6)^2} -\frac{1}{2^{12} (8n+7)^2} +\frac{1}{2^3(8n+1)^2} \right)

and

K = \frac{\pi}{8} \log(\sqrt{3} + 2) + \frac{3}{8} \sum_{n=0}^\infty \frac{(n!)^2}{(2n)!(2n+1)^2}.

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.

Number of known decimal digits of Catalan's constant K
Date Decimal digits Computation performed by
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

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