Catalan's constant

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In mathematics, Catalan's constant G, which occasionally appears in estimates in combinatorics, is defined by

where β is the Dirichlet beta function. Its numerical value [1] is approximately (sequence A006752 in OEIS)

It is not known whether G is rational or irrational.

Catalan's constant was named after Eugène Charles Catalan.

Contents

Integral identities

Some identities include

along with

where K(x) is a complete elliptic integral of the first kind, and

Uses

G 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.

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:

and

The theoretical foundations for such series is given by Broadhurst.[1]

Known digits

The number of known digits of Catalan's constant G 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]

See also

Notes

References

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