Catalan's constant


In mathematics, Catalan's constant, which appears in combinatorics, is defined by
where is the Dirichlet beta function. Its numerical value is approximately
It is not known whether is irrational, let alone transcendental.
Catalan's constant was named after Eugène Charles Catalan.
The similar but apparently more complicated series
can be evaluated exactly and is equal to π3/32.

Integral identities

Some identities involving definite integrals include
where the last three formulas are related to Malmsten's integrals.
If is the complete elliptic integral of the first kind, as a function of the elliptic modulus, then
With the gamma function
The integral
is a known special function, called the inverse tangent integral, and was extensively studied by Srinivasa Ramanujan.

Uses

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, 2 and Catalan's constant; these are expressible as paths on a graph.
In low-dimensional topology, Catalan's constant is a rational multiple of the volume of an ideal hyperbolic octahedron, and therefore of the hyperbolic volume of the complement of the Whitehead link.
It also appears in connection with the hyperbolic secant distribution.

Relation to other special functions

Catalan's constant occurs frequently in relation to the Clausen function, the inverse tangent integral, the inverse sine integral, the Barnes -function, as well as integrals and series summable in terms of the aforementioned functions.
As a particular example, by first expressing the inverse tangent integral in its closed form – in terms of Clausen functions – and then expressing those Clausen functions in terms of the Barnes -function, the following expression is obtained :
If one defines the Lerch transcendent by
then

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 are given by Broadhurst, for the first formula, and Ramanujan, for the second formula. The algorithms for fast evaluation of the Catalan constant were constructed by E. Karatsuba.

Known digits

The number of known digits of Catalan's constant has increased dramatically during the last decades. This is due both to the increase of performance of computers as well as to algorithmic improvements.
DateDecimal digitsComputation performed by
183216Thomas Clausen
185819Carl Johan Danielsson Hill
186414Eugène Charles Catalan
187720James W. L. Glaisher
191332James W. L. Glaisher
1990Greg J. Fee
1996Greg J. Fee
August 14, 1996Greg J. Fee & Simon Plouffe
September 29, 1996Thomas Papanikolaou
1996Thomas Papanikolaou
1997Patrick Demichel
January 4, 1998Xavier Gourdon
2001Xavier Gourdon & Pascal Sebah
2002Xavier Gourdon & Pascal Sebah
October 2006Shigeru Kondo & Steve Pagliarulo
August 2008Shigeru Kondo & Steve Pagliarulo
January 31, 2009Alexander J. Yee & Raymond Chan
April 16, 2009Alexander J. Yee & Raymond Chan
June 7, 2015Robert J. Setti
April 12, 2016Ron Watkins
February 16, 2019Tizian Hanselmann
March 29, 2019Mike A & Ian Cutress
July 16, 2019Seungmin Kim