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| In [[mathematics]], '''Catalan's constant''' ''G'', which occasionally appears in estimates in [[combinatorics]], is defined by
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| :<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} + \cdots \!</math>
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| where ''β'' is the [[Dirichlet beta function]].
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| Its numerical value [http://www.gutenberg.org/etext/812] is approximately {{OEIS|A006752}}
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| :''G'' = 0.915 965 594 177 219 015 054 603 514 932 384 110 774 …
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| It is not known whether ''G'' is [[irrational number|irrational]], let alone [[transcendental number|transcendental]].
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| Catalan's constant was named after [[Eugène Charles Catalan]].
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| ==Integral identities==
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| Some identities include
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| :<math>G = \int_0^1 \int_0^1 \frac{1}{1+x^2 y^2} \,dx\, dy \!</math>
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| :<math>G = -\int_0^1 \frac{\ln t}{1 + t^2} \,dt \!</math>
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| :<math>G = \int_{0}^{\pi/4} \frac{t}{\sin t \cos t} \;dt \!</math>
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| :<math>G = \frac{1}{4} \int_{-\pi/2}^{\pi/2} \frac{t}{\sin t} \;dt \!</math>
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| :<math>G = \int_0^{\pi/4} \ln ( \cot(t) ) \,dt \!</math>
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| :<math>G = \int_0^\infty \arctan (e^{-t}) \,dt \!</math>
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| along with
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| :<math> G = \frac{1}{2} \int_0^1 \mathrm{K}(t)\,dt \!</math>
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| where K(''t'') is a complete [[elliptic integral]] of the first kind.
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| And with [[Gamma function]] Γ(x+1) = x!
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| :<math> G = \frac{\pi}{2} \int_0^\tfrac12\Gamma(1+x)\Gamma(1-x)\,dx</math>
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| The integral:
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| :<math> G = \text{Ti}_2(1)=\int_0^1 \frac{\arctan t}{t}\,dt. \!</math>
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| is a known special function, called the [[Inverse tangent integral]], and was extensively studied by [[Ramanujan]]
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| ==Uses==
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| ''G'' appears in [[combinatorics]], as well as in values of the second [[polygamma function]], also called the [[trigamma function]], at fractional arguments:
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| :<math> \psi_1 \left(\tfrac14\right) = \pi^2 + 8G</math>
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| :<math> \psi_1 \left(\tfrac34\right) = \pi^2 - 8G.</math>
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| [[Simon Plouffe]] gives an infinite collection of identities between the trigamma function, π<sup>2</sup> and Catalan's constant; these are expressible as paths on a graph.
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| It also appears in connection with the [[hyperbolic secant distribution]].
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| ==Relation to other special function==
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| 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.
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| As a particular example, by first expressing the [[Inverse tangent integral]] in it's 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]])''':
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| :<math>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)</math>
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| ==Quickly converging series==
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| The following two formulas involve quickly converging series, and are thus appropriate for numerical computation:
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| : <math>
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| \begin{align}
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| G & =
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| 3 \sum_{n=0}^\infty \frac{1}{2^{4n}}
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| \left(
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| -\frac{1}{2(8n+2)^2}
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| +\frac{1}{2^2(8n+3)^2}
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| -\frac{1}{2^3(8n+5)^2}
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| +\frac{1}{2^3(8n+6)^2}
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| -\frac{1}{2^4(8n+7)^2}
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| +\frac{1}{2(8n+1)^2}
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| \right) \\
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| & {}\quad -2 \sum_{n=0}^\infty \frac{1}{2^{12n}}
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| \left(
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| \frac{1}{2^4(8n+2)^2}
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| +\frac{1}{2^6(8n+3)^2}
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| -\frac{1}{2^9(8n+5)^2}
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| -\frac{1}{2^{10} (8n+6)^2}
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| -\frac{1}{2^{12} (8n+7)^2}
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| +\frac{1}{2^3(8n+1)^2}
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| \right)
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| \end{align}
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| </math>
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| and
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| :<math>G = \tfrac18\pi \log(2 + \sqrt{3}) + \tfrac38 \sum_{n=0}^\infty \frac{(n!)^2}{(2n)!(2n+1)^2}.</math>
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| The theoretical foundations for such series is given by Broadhurst (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 ζ(3) and ζ(5)| year=1998}}</ref> and Ramanujan (the second formula).<ref>B.C. Berndt, Ramanujan's Notebook, Part I., Springer Verlag (1985)</ref> The algorithms for fast evaluation of the Catalan constant is constructed by E. Karatsuba.<ref>E.A. Karatsuba, Fast evaluation of transcendental functions, Probl. Inf. Transm. Vol.27, No.4, pp. 339–360 (1991)</ref><ref>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)</ref>
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| ==Known digits==
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| 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.<ref>Gourdon, X., Sebah, P; [http://numbers.computation.free.fr/Constants/constants.html Constants and Records of Computation]</ref>
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| {| class="wikitable" style="margin: 1em auto 1em auto"
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| |+ '''Number of known decimal digits of Catalan's constant ''G'' '''
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| ! Date || Decimal digits || Computation performed by
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| |-
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| | 1832 || 16 || [[Thomas Clausen (mathematician)|Thomas Clausen]]
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| |-
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| | 1858 || 19 || Carl Johan Danielsson Hill
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| |-
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| | 1864 || 14 || [[Eugène Charles Catalan]]
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| |-
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| | 1877 || 20 || [[James Whitbread Lee Glaisher|James W. L. Glaisher]]
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| |-
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| | 1913 || 32 || [[James Whitbread Lee Glaisher|James W. L. Glaisher]]
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| |-
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| | 1990 || 20,000 || Greg J. Fee
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| |-
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| | 1996 || 50,000 || Greg J. Fee
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| |-
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| | August 14, 1996 || 100,000 || Greg J. Fee & [[Simon Plouffe]]
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| |-
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| | September 29, 1996 || 300,000 || Thomas Papanikolaou
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| | 1996 || 1,500,000 || Thomas Papanikolaou
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| | 1997 || 3,379,957 || Patrick Demichel
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| | January 4, 1998 || 12,500,000 || Xavier Gourdon
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| | 2001 || 100,000,500 || Xavier Gourdon & Pascal Sebah
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| | 2002 || 201,000,000 || Xavier Gourdon & Pascal Sebah
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| | October 2006 || 5,000,000,000 || Shigeru Kondo & Steve Pagliarulo<ref>[http://ja0hxv.calico.jp/pai/ecatalan.html Shigeru Kondo's website]</ref>
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| | August 2008 || 10,000,000,000 || Shigeru Kondo & Steve Pagliarulo<ref>[http://numbers.computation.free.fr/Constants/constants.html Constants and Records of Computation]</ref>
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| | January 31, 2009 || 15,510,000,000 || Alexander J. Yee & Raymond Chan<ref name=yee_chan>[http://www.numberworld.org/nagisa_runs/computations.html Large Computations]</ref>
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| | April 16, 2009 || 31,026,000,000 || Alexander J. Yee & Raymond Chan<ref name=yee_chan/>
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| | April 6, 2013 || 100,000,000,000 || Robert J. Setti<ref>[http://settifinancial.com/0441-world-record-catalans-constant-calculation-complete/ 100 Billion Digits Catalan's Constant Complete]</ref>
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| |-
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| |}
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| ==See also ==
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| * [[Zeta constant]]
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| * [[Mathematical constant]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * Victor Adamchik, ''[http://www-2.cs.cmu.edu/~adamchik/articles/catalan/catalan.htm 33 representations for Catalan's constant]'' (undated)
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| * {{cite journal
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| |first1=Victor
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| |last1= Adamchik,
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| |year=2002
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| |journal=Zeitschr. f. Analysis und ihre Anwendungen (ZAA)
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| |volume=21
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| |issue=3
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| |pages=1–10
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| |url=http://www-2.cs.cmu.edu/~adamchik/articles/csum.html
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| |title=A certain series associated with Catalan's constant
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| |mr=1929434
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| }}
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| * {{cite web
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| |first1=Simon
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| |last1=Plouffe
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| |url=http://www.lacim.uqam.ca/~plouffe/IntegerRelations/identities3a.html
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| |title= A few identities (III) with Catalan
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| |year=1993 }} (Provides over one hundred different identities)''.
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| * 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)''
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| * {{MathWorld|title=Catalan's Constant|urlname=CatalansConstant}}
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| * [http://functions.wolfram.com/Constants/Catalan/06/01/ Catalan constant: Generalized power series] at the Wolfram Functions Site
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| * 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.)''.
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| * {{citation
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| |first1=Greg
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| |last1=Fee
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| |year=1990
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| |title=Computation of Catalan's constant using Ramanujan's Formula
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| |pages=157–160
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| |series=Proceedings of the ISSAC '90
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| |doi=10.1145/96877.96917
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| }} | |
| * {{Cite journal
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| |first1=David M.
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| |last1=Bradley
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| |title=A class of series acceleration formulae for Catalan's constant
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| |doi=10.1023/A:1006945407723
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| |year=1999
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| |journal=The Ramanujan Journal
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| |volume=3
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| |issue=2
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| |pages=159–173
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| |mr=1703281
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| }}
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| * {{Cite arxiv
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| |first1=David M.
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| |last1=Bradley
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| |title=A class of series acceleration formulae for Catalan's constant
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| |eprint=0706.0356
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| |year=2007
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| }}
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| * {{Citation
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| |first1=David M.
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| |last1=Bradley
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| |title = Representations of Catalan's constant
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| |url = http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.26.1879
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| |year=2001
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| }}
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| ==External links==
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| *{{springer|title=Catalan constant|id=p/c130040}}
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| *[http://mathworld.wolfram.com/CatalansConstant.html Catalan's Constant — from Wolfram MathWorld]
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| *[http://archive.org/details/ctcst10a Catalan's Constant (Ramanujan's Formula)]
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| *[http://www.cs.cmu.edu/~adamchik/articles/catalan/catalan.htm catalan's constant] — www.cs.cmu.edu
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| [[Category:Combinatorics]]
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| [[Category:Mathematical constants]]
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