Catalan's constant

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

$G = \beta(2) = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)^2} = \frac{1}{1^2} - \frac{1}{3^2} + \frac{1}{5^2} - \frac{1}{7^2} + \cdots \!$

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

G = 0.915 965 594 177 219 015 054 603 514 932 384 110 774 …

It is not known whether G is irrational, let alone transcendental.

Catalan's constant was named after Eugène Charles Catalan.

Integral identities

Some identities include

$G = \int_0^1 \int_0^1 \frac{1}{1+x^2 y^2} \,dx\, dy \!$

$G = -\int_0^1 \frac{\ln t}{1 + t^2} \,dt \!$

$G = \int_{0}^{\pi/4} \frac{t}{\sin t \cos t} \;dt \!$

$G = \frac{1}{4} \int_{-\pi/2}^{\pi/2} \frac{t}{\sin t} \;dt \!$

$G = \int_0^{\pi/4} \ln ( \cot(t) ) \,dt \!$

$G = \int_0^\infty \arctan (e^{-t}) \,dt \!$

$G = \int_1^\infty \frac{\ln t}{1 + t^2} \,dt \!$

• If K(t) is a complete elliptic integral of the first kind, then

$G = \frac{1}{2} \int_0^1 \mathrm{K}(t)\,dt \!$

• With Gamma function $\Gamma(x+1) = x!$

$G = \frac{\pi}{4} \int_0^1 \Gamma(1+\tfrac{x}{2})\Gamma(1-\tfrac{x}{2})\,dx = \frac{\pi}{2} \int_0^\tfrac12\Gamma(1+y)\Gamma(1-y)\,dy$

• The integral

$G = \text{Ti}_2(1)=\int_0^1 \frac{\arctan t}{t}\,dt. \!$

is a known special function, called the Inverse tangent integral, and was extensively studied by Ramanujan.

Uses

G appears in combinatorics, as well as in values of the second polygamma function, also called the trigamma function, at fractional arguments:

$\psi_1 \left(\tfrac14\right) = \pi^2 + 8G$

$\psi_1 \left(\tfrac34\right) = \pi^2 - 8G.$

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.

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.^[1]

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, Barnes G-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 G-function, the following expression is easily obtained (N.B. all the relevant relations for this derivation have been added to the page for the Clausen function):

$G=4\pi \log\left( \frac{ G(\tfrac{3}{8}) G(\tfrac{7}{8}) }{ G(\tfrac{1}{8}) G(\tfrac{5}{8}) } \right) +4 \pi \log \left( \frac{ \Gamma(\tfrac{3}{8}) }{ \Gamma(\tfrac{1}{8}) } \right) +\frac{\pi}{2} \log \left( \frac{1+\sqrt{2} }{2 \, (2-\sqrt{2})} \right)$.

If one defines the Lerch transcendent, $\Phi(z, s, \alpha)$, (related to the Lerch zeta function) by,

$\Phi(z, s, \alpha) = \sum_{n=0}^\infty \frac { z^n} {(n+\alpha)^s}.$,

then it is clear that

$G = \tfrac{1}{4}\,\Phi(-1, 2, \tfrac{1}{2}).$

Quickly converging series

The following two formulas involve quickly converging series, and are thus appropriate for numerical computation:

$\begin{align} G & = 3 \sum_{n=0}^\infty \frac{1}{2^{4n}} \left( -\frac{1}{2(8n+2)^2} +\frac{1}{2^2(8n+3)^2} -\frac{1}{2^3(8n+5)^2} +\frac{1}{2^3(8n+6)^2} -\frac{1}{2^4(8n+7)^2} +\frac{1}{2(8n+1)^2} \right) \\ & {}\quad -2 \sum_{n=0}^\infty \frac{1}{2^{12n}} \left( \frac{1}{2^4(8n+2)^2} +\frac{1}{2^6(8n+3)^2} -\frac{1}{2^9(8n+5)^2} -\frac{1}{2^{10} (8n+6)^2} -\frac{1}{2^{12} (8n+7)^2} +\frac{1}{2^3(8n+1)^2} \right) \end{align}$

and

$G = \tfrac18\pi \log(2 + \sqrt{3}) + \tfrac38 \sum_{n=0}^\infty \frac{(n!)^2}{(2n)!(2n+1)^2}.$

The theoretical foundations for such series are given by Broadhurst, for the first formula,^[2] and Ramanujan, for the second formula.^[3] The algorithms for fast evaluation of the Catalan constant were constructed by E. Karatsuba.^[4][5]

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.^[6]

Number of known decimal digits of Catalan's constant G

Date	Decimal digits	Computation performed by
1832	16	Thomas Clausen
1858	19	Carl Johan Danielsson Hill
1864	14	Eugène Charles Catalan
1877	20	James W. L. Glaisher
1913	32	James W. L. Glaisher
1990	20,000	Greg J. Fee
1996	50,000	Greg J. Fee
August 14, 1996	100,000	Greg J. Fee & Simon Plouffe
September 29, 1996	300,000	Thomas Papanikolaou
1996	1,500,000	Thomas Papanikolaou
1997	3,379,957	Patrick Demichel
January 4, 1998	12,500,000	Xavier Gourdon
2001	100,000,500	Xavier Gourdon & Pascal Sebah
2002	201,000,000	Xavier Gourdon & Pascal Sebah
October 2006	5,000,000,000	Shigeru Kondo & Steve Pagliarulo[7]
August 2008	10,000,000,000	Shigeru Kondo & Steve Pagliarulo[8]
January 31, 2009	15,510,000,000	Alexander J. Yee & Raymond Chan[9]
April 16, 2009	31,026,000,000	Alexander J. Yee & Raymond Chan[9]
April 6, 2013	100,000,000,000	Robert J. Setti
June 7, 2015	200,000,001,100	Robert J. Setti

See also

• Zeta constant
• Mathematical constant

Notes

1. Agol, Ian (2010), "The minimal volume orientable hyperbolic 2-cusped 3-manifolds", Proceedings of the American Mathematical Society 138 (10): 3723–3732, doi:10.1090/S0002-9939-10-10364-5, MR 2661571.
2. Broadhurst, D.J. (1998). "Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5)". arXiv:math.CA/9803067.
3. B.C. Berndt, Ramanujan's Notebook, Part I., Springer Verlag (1985)
4. E.A. Karatsuba, Fast evaluation of transcendental functions, Probl. Inf. Transm. Vol.27, No.4, pp. 339–360 (1991)
5. E.A. Karatsuba, Fast computation of some special integrals of mathematical physics. Scientific Computing, Validated Numerics, Interval Methods, W.Krämer, J.W.von Gudenberg, eds.; pp. 29–41, (2001)
6. Gourdon, X., Sebah, P; Constants and Records of Computation
7. Shigeru Kondo's website
8. Constants and Records of Computation
9. Large Computations

References

• Victor Adamchik, 33 representations for Catalan's constant (undated)
• Adamchik,, Victor (2002). "A certain series associated with Catalan's constant". Zeitschrift für Analysis und ihre Anwendungen 21 (3): 1–10. MR 1929434.
• Plouffe, Simon (1993). "A few identities (III) with Catalan". (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", 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.).
• Fee, Greg (1990), Computation of Catalan's constant using Ramanujan's Formula, Proceedings of the ISSAC '90, pp. 157–160, doi:10.1145/96877.96917
• Bradley, David M. (1999). "A class of series acceleration formulae for Catalan's constant". The Ramanujan Journal 3 (2): 159–173. doi:10.1023/A:1006945407723. MR 1703281.
• Bradley, David M. (2007). "A class of series acceleration formulae for Catalan's constant". arXiv:0706.0356.
• Bradley, David M. (2001), Representations of Catalan's constant, CiteSeerX: 10.1.1.26.1879

External links

• Hazewinkel, Michiel, ed. (2001), "Catalan constant", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Catalan's Constant — from Wolfram MathWorld
• Catalan's Constant (Ramanujan's Formula)
• catalan's constant — www.cs.cmu.edu