View source for Catalan's constant

In [[mathematics]], '''Catalan's constant''' {{mvar|G}}, which appears in [[combinatorics]], is defined by

:<math>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 </math>

where {{mvar|β}} is the [[Dirichlet beta function]]. Its numerical value<ref>{{cite web|last1=Papanikolaou|first1=Thomas|title=Catalan's Constant to 1,500,000 Places|url=http://www.gutenberg.org/etext/812|website=Gutenberg.org|date=March 1997}}</ref> is approximately {{OEIS|A006752}}

:{{math|''G'' {{=}} {{val|0.915965594177219015054603514932384110774}}…}}

{{unsolved|mathematics|Is Catalan's constant irrational? If so, is it transcendental?}}
It is not known whether {{mvar|G}} is [[irrational number|irrational]], let alone [[transcendental number|transcendental]].<ref>{{citation
 | last = Nesterenko | first = Yu. V.
 | date = January 2016
 | doi = 10.1134/s0081543816010107
 | issue = 1
 | journal = Proceedings of the Steklov Institute of Mathematics
 | pages = 153–170
 | title = On Catalan's constant
 | volume = 292}}.</ref>

Catalan's constant was named after [[Eugène Charles Catalan]].

The similar but apparently more complicated series
:<math> \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 </math>

can be evaluated exactly and is equal to π<sup>3</sup>/32.

==Integral identities==
Some identities involving [[definite integral]]s include

:<math>\begin{align}
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 &= \tfrac14 \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 &= \tfrac12 \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 &= \tfrac12 \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 - \frac18 \iint_{\mathbb{R}^2}\!\!\frac{x\sin(2xy/\pi)}{\,(x^2+\pi^2)\cosh x\sinh y\,} \,dxdy
\end{align}</math>

where the last three formulas are related to Malmsten's integrals <ref>
{{Cite journal
|first1=Iaroslav 
|last1=Blagouchine
|title=Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results
|year=2014
|doi=10.1007/s11139-013-9528-5
|url=https://iblagouchine.perso.centrale-marseille.fr/publications/Blagouchine-Malmsten-integrals-and-their-evaluation-by-contour-integration-methods-(Ramanujan-J-2014).pdf
|volume=35
|journal=The Ramanujan Journal
|pages=21–110
}}
</ref>.

If {{math|K(''k'')}} is the [[Elliptic_integral#Complete_elliptic_integral_of_the_first_kind|complete elliptic integral of the first kind]], as a function of the elliptic modulus {{math|''k''}}, then
:<math> G = \tfrac12 \int_0^1 \mathrm{K}(k)\,dk </math>

With the [[gamma function]] {{math|Γ(''x'' + 1) {{=}} ''x''!}}
:<math>\begin{align}
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^\frac12\Gamma(1+y)\Gamma(1-y)\,dy \end{align}</math>

The integral
:<math> G = \operatorname{Ti}_2(1)=\int_0^1 \frac{\arctan t}{t}\,dt </math>

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

==Uses==
{{mvar|G}} appears in [[combinatorics]], as well as in values of the second [[polygamma function]], also called the [[trigamma function]], at fractional arguments:

:<math>\begin{align} \psi_1 \left(\tfrac14\right) &= \pi^2 + 8G \\
\psi_1 \left(\tfrac34\right) &= \pi^2 - 8G. \end{align}</math>

[[Simon Plouffe]] gives an infinite collection of identities between the trigamma function, {{pi}}<sup>2</sup> 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]].<ref>{{citation
 | last = Agol | first = Ian | authorlink = Ian Agol
 | doi = 10.1090/S0002-9939-10-10364-5
 | issue = 10
 | journal = [[Proceedings of the American Mathematical Society]]
 | mr = 2661571
 | pages = 3723–3732
 | title = The minimal volume orientable hyperbolic 2-cusped 3-manifolds
 | volume = 138
 | year = 2010| arxiv = 0804.0043}}.</ref>

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|Barnes {{mvar|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 {{mvar|G}}-function, the following expression is obtained (see [[Clausen function]] for more):

:<math>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)</math>.

If one defines the '''Lerch transcendent''' {{math|Φ(''z'',''s'',''α'')}} (related to the [[Lerch zeta function]]) by

:<math>\Phi(z, s, \alpha) = \sum_{n=0}^\infty \frac { z^n} {(n+\alpha)^s},</math>

then

:<math> G = \tfrac{1}{4}\Phi\left(-1, 2, \tfrac{1}{2}\right).</math>

==Quickly converging series==
The following two formulas involve quickly converging series, and are thus appropriate for numerical computation:
: <math>\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)- \\
& \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{align}</math>

and

:<math>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}}.</math>

The theoretical foundations for such series are given by Broadhurst, for the first formula,<ref>{{cite arXiv|first1=D. J. |last1=Broadhurst|eprint=math.CA/9803067 |title=Polylogarithmic ladders, hypergeometric series and the ten millionth digits of {{math|''ζ''(3)}} and {{math|''ζ''(5)}}| year=1998}}</ref> and Ramanujan, for the second formula.<ref>{{cite book|first=B. C.|last=Berndt|title=Ramanujan's Notebook, Part I|publisher=Springer Verlag|date=1985}} {{ISBN missing}}</ref> The algorithms for fast evaluation of the Catalan constant were constructed by E. Karatsuba.<ref>{{cite journal|first=E. A.|last=Karatsuba|title=Fast evaluation of transcendental functions|journal=Probl. Inf. Transm.|volume=27|issue=4|pages=339–360|date=1991|zbl=0754.65021|mr=1156939}}</ref><ref>{{cite book|first=E. A.|last=Karatsuba|contribution=Fast computation of some special integrals of mathematical physics|title=Scientific Computing, Validated Numerics, Interval Methods|editor1-first=W.|editor1-last=Krämer|editor2-first=J. W.|editor2-last=von Gudenberg|pages=29–41|date=2001}} {{ISBN missing}}</ref>

==Known digits==
The number of known digits of Catalan's constant {{mvar|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.<ref>{{cite web|last1=Gourdon|first1=X.|last2=Sebah|first2=P.|url=http://numbers.computation.free.fr/Constants/constants.html|title=Constants and Records of Computation}}</ref>

{| class="wikitable" style="margin: 1em auto 1em auto"
|+ Number of known decimal digits of Catalan's constant {{mvar|G}}
! Date || Decimal digits || Computation performed by
|-
| 1832 ||align="right"| 16 || [[Thomas Clausen (mathematician)|Thomas Clausen]]
|-
| 1858 ||align="right"| 19 || Carl Johan Danielsson Hill
|-
| 1864 ||align="right"| 14 || [[Eugène Charles Catalan]]
|-
| 1877 ||align="right"| 20 || [[James Whitbread Lee Glaisher|James W. L. Glaisher]]
|-
| 1913 ||align="right"| 32 || [[James Whitbread Lee Glaisher|James W. L. Glaisher]]
|-
| 1990 ||align="right"| {{val|20000}} || Greg J. Fee
|-
| 1996 ||align="right"| {{val|50000}} || Greg J. Fee
|-
| August 14, 1996 ||align="right"| {{val|100000}} || Greg J. Fee & [[Simon Plouffe]]
|-
| September 29, 1996 ||align="right"| {{val|300000}} || Thomas Papanikolaou
|-
| 1996 ||align="right"| {{val|1500000}} || Thomas Papanikolaou
|-
| 1997 ||align="right"| {{val|3379957}} || Patrick Demichel
|-
| January 4, 1998 ||align="right"| {{val|12500000}} || Xavier Gourdon
|-
| 2001 ||align="right"| {{val|100000500}} || Xavier Gourdon & Pascal Sebah
|-
| 2002 ||align="right"| {{val|201000000}} || Xavier Gourdon & Pascal Sebah
|-
| October 2006 ||align="right"| {{val|5000000000}} || Shigeru Kondo & Steve Pagliarulo<ref>{{Cite web |url=http://ja0hxv.calico.jp/pai/ecatalan.html# |title=Shigeru Kondo's website |access-date=2008-01-31 |archive-url=https://web.archive.org/web/20080211185703/http://ja0hxv.calico.jp/pai/ecatalan.html# |archive-date=2008-02-11 |url-status=dead }}</ref>
|-
| August 2008 ||align="right"| {{val|10000000000}} || Shigeru Kondo & Steve Pagliarulo<ref>[http://numbers.computation.free.fr/Constants/constants.html Constants and Records of Computation]</ref>
|-
| January 31, 2009 ||align="right"| {{val|15510000000}} || Alexander J. Yee & Raymond Chan<ref name=yee_chan>[http://www.numberworld.org/nagisa_runs/computations.html Large Computations]</ref>
|-
| April 16, 2009 ||align="right"| {{val|31026000000}} || Alexander J. Yee & Raymond Chan<ref name=yee_chan/>
|-
| June 7, 2015 ||align="right"| {{val|200000001100}} || Robert J. Setti<ref name=setti>[http://www.numberworld.org/digits/Catalan/ Catalan's constant records using YMP]</ref>
|-
| April 12, 2016 ||align="right"| {{val|250000000000}} || Ron Watkins<ref name=setti>[http://www.numberworld.org/digits/Catalan/ Catalan's constant records using YMP]</ref>
|-
| February 16, 2019  ||align="right"| {{val|300000000000}} || Tizian Hanselmann<ref name=setti>[http://www.numberworld.org/digits/Catalan/ Catalan's constant records using YMP]</ref>
|-
| March 29, 2019  ||align="right"| {{val|500000000000}} || Mike A & Ian Cutress<ref name=setti>[http://www.numberworld.org/digits/Catalan/ Catalan's constant records using YMP]</ref>
|-
| July 16, 2019  ||align="right"| {{val|600000000100}} || Seungmin Kim<ref name=yee>[https://web.archive.org/web/20190722034426/http://www.numberworld.org/y-cruncher/ Catalan's constant records using YMP]</ref><ref name=kim>[https://smkim.sourceforge.io/2019/07/23/catalans-constant/ Catalan's constant world record by Seungmin Kim]</ref>
|-
|}

==See also ==
* [[Particular values of Riemann zeta function]]
* [[Mathematical constant]]

==References==
{{reflist}}

==External links==
* Victor Adamchik, ''[http://www-2.cs.cmu.edu/~adamchik/articles/catalan/catalan.htm 33 representations for Catalan's constant]'' (undated)
* {{cite journal
|first1=Victor
|last1= Adamchik,
|year=2002
|journal=Zeitschrift für Analysis und ihre Anwendungen
|volume=21
|issue=3
|pages=1–10
|url=http://www-2.cs.cmu.edu/~adamchik/articles/csum.html
|title=A certain series associated with Catalan's constant
|mr=1929434
|doi=10.4171/ZAA/1110
}}
* {{cite web
|first1=Simon
|last1=Plouffe
|url=http://www.lacim.uqam.ca/~plouffe/IntegerRelations/identities3a.html
|title= A few identities (III) with Catalan
|year=1993 }} (Provides over one hundred different identities)''.
* Simon Plouffe, ''[http://www.lacim.uqam.ca/~plouffe/IntegerRelations/identities3.html A few identities with Catalan constant and Pi^2]'', (1999) ''(Provides a graphical interpretation of the relations)''
* {{MathWorld|title=Catalan's Constant|urlname=CatalansConstant}}
* [http://functions.wolfram.com/Constants/Catalan/06/01/ Catalan constant: Generalized power series] at the Wolfram Functions Site
* Greg Fee, ''[http://www.gutenberg.org/etext/682 Catalan's Constant (Ramanujan's Formula)]'' (1996) ''(Provides the first 300,000 digits of Catalan's constant.)''.
* {{citation
|first1=Greg
|last1=Fee
|year=1990
|title=Computation of Catalan's constant using Ramanujan's Formula
|pages=157–160
|series=Proceedings of the ISSAC '90
|doi=10.1145/96877.96917
}}
* {{Cite journal
|first1=David M.
|last1=Bradley
|title=A class of series acceleration formulae for Catalan's constant
|doi=10.1023/A:1006945407723
|year=1999
|journal=The Ramanujan Journal
|volume=3
|issue=2
|pages=159–173
|mr=1703281
|arxiv=0706.0356
}}
* {{Cite journal
|first1=David M.
|last1=Bradley
|title=A class of series acceleration formulae for Catalan's constant
|arxiv=0706.0356
|year=2007
|doi=10.1023/A:1006945407723
|volume=3
|journal=The Ramanujan Journal
|pages=159–173
}}
* {{Citation
|first1=David M.
|last1=Bradley
|title = Representations of Catalan's constant
|citeseerx = 10.1.1.26.1879
|year=2001
}}

*{{springer|title=Catalan constant|id=p/c130040}}
*[http://mathworld.wolfram.com/CatalansConstant.html Catalan's Constant &mdash; from Wolfram MathWorld]
*[https://archive.org/details/ctcst10a Catalan's Constant (Ramanujan's Formula)]
*[https://www.cs.cmu.edu/~adamchik/articles/catalan/catalan.htm catalan's constant] &mdash; www.cs.cmu.edu

[[Category:Combinatorics]]
[[Category:Mathematical constants]]