Catalan's constant
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Catalan's constant K, which occasionally appears in estimates in combinatorics, is defined by
- <math>\Kappa = \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} + ...<math>
or equivalently
- <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>
- <math> K = \int_0^1 \frac{\tan^{-1}x}{x}dx<math>
where K(x) is a complete elliptic integral of the first kind, and has nothing to do with the constant itself.
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>
Its numerical value is approximately
- K = .915 965 594 177 219 015 054 603 514 932 384 110 774 ...
It also appears in connection with the hyperbolic secant distribution.
It is not known whether K is rational or irrational.
See Also
External links
Catalan's Constant -- from MathWorld (http://mathworld.wolfram.com/CatalansConstant.html|)pl:Stała Catalana