Catalan's constant
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In mathematics, Catalan's constant K, which occasionally appears in estimates in combinatorics, is defined by
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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[edit] Integral identities
Some identities include
along with
where K(x) is a complete elliptic integral of the first kind, and
[edit] Uses
K appears in combinatorics, as well as in values of the second polygamma function, also called the trigamma function, at fractional arguments:
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 .
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:
and
[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.).