Catalan's constant
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In mathematics, Catalan's constant K, which occasionally appears in estimates in combinatorics, is defined by
- <math>\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</math>
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.
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Integral identities
Some identities include
- <math>K = -\int_{0}^{1} \frac{\ln(t)}{1 + t^2} \mbox{ d} t</math>
along with
- <math> K = \frac{1}{2}\int_0^1 \mathrm{K}(x)\,dx</math>
where K(x) is a complete elliptic integral of the first kind, and
- <math> K = \int_0^1 \frac{\tan^{-1}x}{x}dx.</math>
Uses
K appears in combinatorics, as well as in values of the second polygamma function, also called the trigamma function, at fractional arguments:
- <math> \psi_{1}\left(\frac{1}{4}\right) = \pi^2 + 8K</math>
- <math> \psi_{1}\left(\frac{3}{4}\right) = \pi^2 - 8K</math>
Simon Plouffe gives an infinite collection of identities between the trigamma function, <math>\pi^2</math> and Catalan's constant; these are expressible as paths on a graph.
The probability that two randomly selected Gaussian integers are coprime is <math> \frac{6}{\pi^2 K} </math>.
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:
<math>K = \,</math> <math>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) -</math>
<math> 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)</math>
and
- <math>K = \frac{\pi}{8} \log(\sqrt{3} + 2) + \frac{3}{8} \sum_{n=0}^\infty \frac{(n!)^2}{(2n)!(2n+1)^2}.</math>
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)
- Template: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.).fr:Constante de Catalan