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.

1 Integral identities
2 Uses
3 Quickly converging series
4 References

[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, $π 2$ 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

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

[edit] References

Victor Adamchik, 33 representations for Catalan's constant (undated)
Victor Adamchik, A certain series associated with Catalan's constant, (2002) Zeitschrift fuer Analysis und ihre Anwendungen (ZAA), 21, pp.1-10.
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 at 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.).