Catalan's constant

In mathematics, Catalan's constant $G$ , which appears 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}}}+{\frac {1}{9^{2}}}-\cdots

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

G = 6999915965594177219♠ 0.915 965594177219015054603514932384110774 \dots

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

\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 π³/32.

Integral identities

Some identities involving definite integrals include

{\begin{aligned}G&=\iint _{[0,1]^{2}}\!{\frac {1}{1+x^{2}y^{2}}}\,dx\,dy\\[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&={\tfrac {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&={\tfrac {1}{2}}\int _{0}^{\frac {\pi }{2}}\ln(\sec t+\tan t)\,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 }\arctan e^{-t}\,dt\\[3pt]G&={\tfrac {1}{2}}\int _{0}^{\infty }{\frac {t}{\cosh t}}\,dt\\[3pt]G&={\frac {\pi }{2}}\int _{1}^{\infty }{\frac {(t^{4}-6t^{2}+1)\ln \ln t}{(1+t^{2})^{3}}}\,dt\\[3pt]G&=1+\lim _{\alpha \to {1^{-}}}\!\left\{\int _{0}^{\alpha }\!{\frac {(1+6t^{2}+t^{4})\arctan {t}}{t(1-t^{2})^{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(2xy/\pi )}{\,(x^{2}+\pi ^{2})\cosh x\sinh y\,}}\,dxdy\end{aligned}}

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

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

G={\tfrac {1}{2}}\int _{0}^{1}\mathrm {K} (t)\,dt

With the gamma function $Γ(x + 1) = x!$

{\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

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:

{\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, $π$ 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.^[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):

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

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

then

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:

{\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

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	7004200000000000000♠20000	Greg J. Fee
1996	7004500000000000000♠50000	Greg J. Fee
August 14, 1996	7005100000000000000♠100000	Greg J. Fee & Simon Plouffe
September 29, 1996	7005300000000000000♠300000	Thomas Papanikolaou
1996	7006150000000000000♠1500000	Thomas Papanikolaou
1997	7006337995700000000♠3379957	Patrick Demichel
January 4, 1998	7007125000000000000♠12500000	Xavier Gourdon
2001	7008100000500000000♠100000500	Xavier Gourdon & Pascal Sebah
2002	7008201000000000000♠201000000	Xavier Gourdon & Pascal Sebah
October 2006	7009500000000000000♠5000000000	Shigeru Kondo & Steve Pagliarulo^[10]
August 2008	7010100000000000000♠10000000000	Shigeru Kondo & Steve Pagliarulo^[11]
January 31, 2009	7010155100000000000♠15510000000	Alexander J. Yee & Raymond Chan^[12]
April 16, 2009	7010310260000000000♠31026000000	Alexander J. Yee & Raymond Chan^[12]
April 6, 2013	7011100000000000000♠100000000000	Robert J. Setti
June 7, 2015	7011200000001100000♠200000001100	Robert J. Setti^[13]

References

↑ Papanikolaou, Thomas (March 1997). "Catalan's Constant to 1,500,000 Places". Gutenberg.org.
↑ Nesterenko, Yu. V. (January 2016), "On Catalan's constant", Proceedings of the Steklov Institute of Mathematics, 292 (1): 153–170, doi:10.1134/s0081543816010107 .
↑ 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.
↑ 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 .
↑ Broadhurst, D. J. (1998). "Polylogarithmic ladders, hypergeometric series and the ten millionth digits of $ζ (3)$ and $ζ (5)$ ". arXiv:math.CA/9803067.
↑ Berndt, B. C. (1985). Ramanujan's Notebook, Part I. Springer Verlag.
↑ Karatsuba, E. A. (1991). "Fast evaluation of transcendental functions". Probl. Inf. Transm. 27 (4): 339–360. MR 1156939. Zbl 0754.65021.
↑ Karatsuba, E. A. (2001). "Fast computation of some special integrals of mathematical physics". In Krämer, W.; von Gudenberg, J. W. Scientific Computing, Validated Numerics, Interval Methods. pp. 29–41.
↑ Gourdon, X.; Sebah, P. "Constants and Records of Computation".
↑ Shigeru Kondo's website
↑ Constants and Records of Computation
1 2 Large Computations
↑ Catalan's constant records using YMP

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 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. arXiv:0706.0356. doi:10.1023/A:1006945407723. MR 1703281.
Bradley, David M. (2007). "A class of series acceleration formulae for Catalan's constant". The Ramanujan Journal. 3: 159–173. arXiv:0706.0356. doi:10.1023/A:1006945407723.
Bradley, David M. (2001), Representations of Catalan's constant, CiteSeerX 10.1.1.26.1879

Hazewinkel, Michiel, ed. (2001) [1994], "Catalan constant", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, 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

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[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 .

[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.

[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 .

[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.

[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. Scientific Computing, Validated Numerics, Interval Methods. pp. 29–41.

[9] Gourdon, X.; Sebah, P. "Constants and Records of Computation".

[10] Shigeru Kondo's website

[11] Constants and Records of Computation

[yee_chan-12] 1 2 Large Computations

[setti-13] Catalan's constant records using YMP