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 [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.

Contents

[edit] Integral identities

Some identities include

K = -\int_{0}^{1} \frac{\ln(t)}{1 + t^2} dt
K = \int_0^1 \int_0^1 \frac{1}{1+x^2 y^2} dx dy
K = \int_{0}^{\pi/4} \frac{t}{\sin(t) \cos(t)} dt

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.

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.[2]

Number of known decimal digits of Catalan's constant K
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