Editing Catalan's constant (section)

==Series representations==
Catalan's constant appears in the evaluation of several rational series including:<ref name=":0">{{Cite web |last=Weisstein |first=Eric W. |title=Catalan's Constant |url=https://mathworld.wolfram.com/CatalansConstant.html |access-date=2024-10-02 |website=mathworld.wolfram.com |language=en}}</ref><math display="block">\frac{\pi^2}{16}+\frac G2 = \sum_{n=0}^\infty \frac{1}{(4n+1)^2}.</math><math display="block">\frac{\pi^2}{16}-\frac G2 = \sum_{n=0}^\infty \frac{1}{(4n+3)^2}.</math>
The following two formulas involve quickly converging series, and are thus appropriate for numerical computation:
<math display="block">\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 display="block">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|page=289|isbn=978-1-4612-1088-7}}</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| url=https://archive.org/details/scientificcomput00wals_919|url-access=limited|editor1-first=W.|editor1-last=Krämer| editor2-first=J. W.|editor2-last=von Gudenberg|pages=[https://archive.org/details/scientificcomput00wals_919/page/n35 29]–41|date=2001|doi=10.1007/978-1-4757-6484-0_3|isbn=978-1-4419-3376-8 }}</ref> Using these series, calculating Catalan's constant is now about as fast as calculating [[Apéry's constant]], <math>\zeta(3)</math>.<ref name="Yee_formulas" />

Other quickly converging series, due to Guillera and Pilehrood and employed by the [[y-cruncher]] software, include:<ref name="Yee_formulas">{{cite web|url=http://www.numberworld.org/y-cruncher/internals/formulas.html|title=Formulas and Algorithms|author=Alexander Yee|date=14 May 2019|access-date=5 December 2021}}</ref>

:<math>G = \frac{1}{2}\sum_{k=0}^{\infty }\frac{(-8)^{k}(3k+2)}{(2k+1)^{3}{\binom{2k}{k}}^{3}}</math>

:<math>G = \frac{1}{64}\sum_{k=1}^{\infty }\frac{256^{k}(580k^2-184k+15)}{k^3(2k-1)\binom{6k}{3k}\binom{6k}{4k}\binom{4k}{2k}}</math>

:<math>G = -\frac{1}{1024}\sum_{k=1}^{\infty }\frac{(-4096)^k(45136k^4-57184k^3+21240k^2-3160k+165)}{k^3(2k-1)^3}\left( \frac{(2k)!^6(3k)!^3}{k!^3(6k)!^3} \right)</math>

All of these series have [[time complexity]] <math>O(n\log(n)^3)</math>.<ref name="Yee_formulas"/>