Catalan's constant

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

${\displaystyle G=\beta (2)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{2}}}={\frac {1}{1^{2}}}-{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}-{\frac {1}{7^{2}}}+{\frac {1}{9^{2}}}-\cdots }$

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

G = 0.915965594177219015054603514932384110774…

Unsolved problem in mathematics: Is Catalan's constant irrational? If so, is it transcendental? (more unsolved problems in mathematics)

It is not known whether G is irrational, let alone transcendental.[2]

Catalan's constant was named after Eugène Charles Catalan.

The similar but apparently more complicated series

${\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{3}}}={\frac {1}{1^{3}}}-{\frac {1}{3^{3}}}+{\frac {1}{5^{3}}}-{\frac {1}{7^{3}}}+{\frac {1}{9^{3}}}-\cdots }$

can be evaluated exactly and is equal to π³/32.

Integral identities

Some identities involving definite integrals include

${\displaystyle {\begin{aligned}G&=-{\frac {1}{\pi i}}\int _{0}^{\frac {\pi }{2}}\ln \ln \tan x\ln \tan x\,dx\\[3pt]G&=\iint _{[0,1]^{2}}\!{\frac {1}{1+x^{2}y^{2}}}\,dx\,dy\\[3pt]G&=\int _{0}^{1}\int _{0}^{1-x}{\frac {1}{1-x^{2}-y^{2}}}\,dy\,dx\\[3pt]G&=\int _{1}^{\infty }{\frac {\ln t}{1+t^{2}}}\,dt\\[3pt]G&=-\int _{0}^{1}{\frac {\ln t}{1+t^{2}}}\,dt\\[3pt]G&=\int _{0}^{\frac {\pi }{4}}{\frac {t}{\sin t\cos t}}\,dt\\[3pt]G&={\frac {1}{4}}\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}{\frac {t}{\sin t}}\,dt\\[3pt]G&=\int _{0}^{\frac {\pi }{4}}\ln \cot t\,dt\\[3pt]G&=-\int _{0}^{\frac {\pi }{4}}\ln \tan t\,dt\\[3pt]G&={\frac {1}{2}}\int _{0}^{\frac {\pi }{2}}\ln \left(\sec t+\tan t\right)\,dt\\[3pt]G&=\int _{0}^{1}{\frac {\arccos t}{\sqrt {1+t^{2}}}}\,dt\\[3pt]G&=\int _{0}^{1}{\frac {\operatorname {arsinh} t}{\sqrt {1-t^{2}}}}\,dt\\[3pt]G&=\int _{0}^{\infty }\operatorname {arccot} e^{t}\,dt\\[3pt]G&={\frac {1}{4}}\int _{0}^{\frac {\pi ^{2}}{4}}\csc {\sqrt {t}}\,dt\\[3pt]G&={\frac {1}{16}}\left(\pi ^{2}+4\int _{1}^{\infty }\operatorname {arccsc} ^{2}t\,dt\right)\\[3pt]G&={\frac {1}{2}}\int _{0}^{\infty }{\frac {t}{\cosh t}}\,dt\\[3pt]G&={\frac {\pi }{2}}\int _{1}^{\infty }{\frac {\left(t^{4}-6t^{2}+1\right)\ln \ln t}{\left(1+t^{2}\right)^{3}}}\,dt\\[3pt]G&=1+\lim _{\alpha \to {1^{-}}}\!\left\{\int _{0}^{\alpha }\!{\frac {\left(1+6t^{2}+t^{4}\right)\arctan {t}}{t\left(1-t^{2}\right)^{2}}}\,dt+2\operatorname {arctanh} {\alpha }-{\frac {\pi \alpha }{1-\alpha ^{2}}}\right\}\\[3pt]G&=1-{\frac {1}{8}}\iint _{\mathbb {R} ^{2}}\!\!{\frac {x\sin \left(2xy/\pi \right)}{\,\left(x^{2}+\pi ^{2}\right)\cosh x\sinh y\,}}\,dxdy\end{aligned}}}$

where the last three formulas are related to Malmsten's integrals.[3]

If K(k) is the complete elliptic integral of the first kind, as a function of the elliptic modulus k, then

${\displaystyle G={\tfrac {1}{2}}\int _{0}^{1}\mathrm {K} (k)\,dk}$

With the gamma function Γ(x + 1) = x!

${\displaystyle {\begin{aligned}G&={\frac {\pi }{4}}\int _{0}^{1}\Gamma \left(1+{\frac {x}{2}}\right)\Gamma \left(1-{\frac {x}{2}}\right)\,dx\\&={\frac {\pi }{2}}\int _{0}^{\frac {1}{2}}\Gamma (1+y)\Gamma (1-y)\,dy\end{aligned}}}$

The integral

${\displaystyle G=\operatorname {Ti} _{2}(1)=\int _{0}^{1}{\frac {\arctan t}{t}}\,dt}$

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

Uses

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

${\displaystyle {\begin{aligned}\psi _{1}\left({\tfrac {1}{4}}\right)&=\pi ^{2}+8G\\\psi _{1}\left({\tfrac {3}{4}}\right)&=\pi ^{2}-8G.\end{aligned}}}$

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.[4]

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 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 obtained (see Clausen function for more):

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

If one defines the Lerch transcendent Φ(z,s,α) (related to the Lerch zeta function) by

${\displaystyle \Phi (z,s,\alpha )=\sum _{n=0}^{\infty }{\frac {z^{n}}{(n+\alpha )^{s}}},}$

then

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

Quickly converging series

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

${\displaystyle {\begin{aligned}G&=3\sum _{n=0}^{\infty }{\frac {1}{2^{4n}}}\left(-{\frac {1}{2(8n+2)^{2}}}+{\frac {1}{2^{2}(8n+3)^{2}}}-{\frac {1}{2^{3}(8n+5)^{2}}}+{\frac {1}{2^{3}(8n+6)^{2}}}-{\frac {1}{2^{4}(8n+7)^{2}}}+{\frac {1}{2(8n+1)^{2}}}\right)-\\&\qquad -2\sum _{n=0}^{\infty }{\frac {1}{2^{12n}}}\left({\frac {1}{2^{4}(8n+2)^{2}}}+{\frac {1}{2^{6}(8n+3)^{2}}}-{\frac {1}{2^{9}(8n+5)^{2}}}-{\frac {1}{2^{10}(8n+6)^{2}}}-{\frac {1}{2^{12}(8n+7)^{2}}}+{\frac {1}{2^{3}(8n+1)^{2}}}\right)\end{aligned}}}$

and

${\displaystyle G={\frac {\pi }{8}}\log \left(2+{\sqrt {3}}\right)+{\frac {3}{8}}\sum _{n=0}^{\infty }{\frac {1}{(2n+1)^{2}{\binom {2n}{n}}}}.}$

The theoretical foundations for such series are given by Broadhurst, for the first formula,[5] and Ramanujan, for the second formula.[6] The algorithms for fast evaluation of the Catalan constant were constructed by E. Karatsuba.[7][8]

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.[9]

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	20000	Greg J. Fee
1996	50000	Greg J. Fee
August 14, 1996	100000	Greg J. Fee & Simon Plouffe
September 29, 1996	300000	Thomas Papanikolaou
1996	1500000	Thomas Papanikolaou
1997	3379957	Patrick Demichel
January 4, 1998	12500000	Xavier Gourdon
2001	100000500	Xavier Gourdon & Pascal Sebah
2002	201000000	Xavier Gourdon & Pascal Sebah
October 2006	5000000000	Shigeru Kondo & Steve Pagliarulo[10]
August 2008	10000000000	Shigeru Kondo & Steve Pagliarulo[11]
January 31, 2009	15510000000	Alexander J. Yee & Raymond Chan[12]
April 16, 2009	31026000000	Alexander J. Yee & Raymond Chan[12]
June 7, 2015	200000001100	Robert J. Setti[13]
April 12, 2016	250000000000	Ron Watkins[13]
February 16, 2019	300000000000	Tizian Hanselmann[13]
March 29, 2019	500000000000	Mike A & Ian Cutress[13]
July 16, 2019	600000000100	Seungmin Kim[14][15]
July 16, 2019	600000000100	Robert Reynolds[16]

See also

• Particular values of Riemann zeta function
• Mathematical constant

References

1. Papanikolaou, Thomas (March 1997). "Catalan's Constant to 1,500,000 Places". Gutenberg.org.
2. Nesterenko, Yu. V. (January 2016), "On Catalan's constant", Proceedings of the Steklov Institute of Mathematics, 292 (1): 153–170, doi:10.1134/s0081543816010107, S2CID 124903059.
3. Blagouchine, Iaroslav (2014). "Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results" (PDF). The Ramanujan Journal. 35: 21–110. doi:10.1007/s11139-013-9528-5. S2CID 120943474. Archived from the original (PDF) on 2018-10-02. Retrieved 2018-10-01.
4. Agol, Ian (2010), "The minimal volume orientable hyperbolic 2-cusped 3-manifolds", Proceedings of the American Mathematical Society, 138 (10): 3723–3732, arXiv:0804.0043, doi:10.1090/S0002-9939-10-10364-5, MR 2661571, S2CID 2016662.
5. Broadhurst, D. J. (1998). "Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5)". arXiv:math.CA/9803067.
6. Berndt, B. C. (1985). Ramanujan's Notebook, Part I. Springer Verlag. p. 289.
7. Karatsuba, E. A. (1991). "Fast evaluation of transcendental functions". Probl. Inf. Transm. 27 (4): 339–360. MR 1156939. Zbl 0754.65021.
8. Karatsuba, E. A. (2001). "Fast computation of some special integrals of mathematical physics". In Krämer, W.; von Gudenberg, J. W. (eds.). Scientific Computing, Validated Numerics, Interval Methods. pp. 29–41.
9. Gourdon, X.; Sebah, P. "Constants and Records of Computation".
10. "Shigeru Kondo's website". Archived from the original on 2008-02-11. Retrieved 2008-01-31.
11. Constants and Records of Computation
12. Large Computations
13. Catalan's constant records using YMP
14. Catalan's constant records using YMP
15. Catalan's constant world record by Seungmin Kim
16. A Definite Integral Involving the Logarithmic Function in Terms of the Lerch Function by Robert Reynolds and Allan Stauffer

External links

• 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. doi:10.4171/ZAA/1110. 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 international symposium on Symbolic and algebraic computation - ISSAC '90, Proceedings of the ISSAC '90, pp. 157–160, doi:10.1145/96877.96917, ISBN 0201548925, S2CID 1949187
• Bradley, David M. (1999). "A class of series acceleration formulae for Catalan's constant". The Ramanujan Journal. 3 (2): 159–173. arXiv:0706.0356. doi:10.1023/A:1006945407723. MR 1703281. S2CID 5111792.
• Bradley, David M. (2007). "A class of series acceleration formulae for Catalan's constant". The Ramanujan Journal. 3 (2): 159–173. arXiv:0706.0356. Bibcode:2007arXiv0706.0356B. doi:10.1023/A:1006945407723. S2CID 5111792.
• Bradley, David M. (2001), Representations of Catalan's constant, CiteSeerX 10.1.1.26.1879

• "Catalan constant", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Catalan's Constant — from Wolfram MathWorld
• Catalan's Constant (Ramanujan's Formula)
• catalan's constant — www.cs.cmu.edu