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A186706
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Decimal expansion of the Integral of Dedekind Eta(x*I) from x = 0 ... Infinity.
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4
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3, 6, 2, 7, 5, 9, 8, 7, 2, 8, 4, 6, 8, 4, 3, 5, 7, 0, 1, 1, 8, 8, 1, 5, 6, 5, 1, 5, 2, 8, 4, 3, 1, 1, 4, 6, 4, 5, 6, 8, 1, 3, 2, 4, 9, 6, 1, 8, 5, 4, 8, 1, 1, 5, 1, 1, 3, 9, 7, 6, 9, 8, 7, 0, 7, 7, 6, 2, 4, 6, 3, 6, 2, 2, 5, 2, 7, 0, 7, 7, 6, 7, 3, 6, 8, 2, 4, 9, 9, 7, 6, 4, 2, 4, 1, 2, 0, 3, 3, 7, 7, 1, 2, 4, 4
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OFFSET
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1,1
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COMMENTS
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Reduction of the integral by Robert Israel, Jul 25 2012: (Start)
Use the definition of DedekindEta as a sum:
Eta(i x) = sum_{n=-infinity}^infinity (-1)^n exp(-pi x (6n-1)^2/12)
Now int_0^infinity exp(-pi x (6n-1)^2/12) dx = 12/(pi (6n-1)^2)
According to Maple, sum_{n=-infinity}^infinity (-1)^n 12/(pi (6n-1)^2) is
2*3^(1/2)*(dilog(1-1/2*I-1/2*3^(1/2))-dilog(1-1/2*I+1/2*3^(1/2))-dilog(1+1/2*I+1/2*3^(1/2))+dilog(1+1/2*I-1/2*3^(1/2)))/Pi
Jonquiere's inversion formula (see http://en.wikipedia.org/wiki/Polylogarithm)
but note that Maple's dilog(z) is L_2(1-z) in the notation there) gives
dilog(1-1/2*I-1/2*3^(1/2))+dilog(1+1/2*I-1/2*3^(1/2)) = 13/72*Pi^2
and
dilog(1-1/2*I+1/2*3^(1/2))+dilog(1+1/2*I+1/2*3^(1/2)) = -11*Pi^2/72
which give the desired multiple of Pi. (End)
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LINKS
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Table of n, a(n) for n=1..105.
D. H. Lehmer, Interesting series involving the central binomial coefficient, Am. Math. Monthly 92 (7) (1985) 449
Eric W. Weisstein's World of Mathematics, Dedekind Eta Function.
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FORMULA
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Equals 2*Pi/sqrt(3), 2 times A093602, and in consequence equal to sum_{m>=1} 3^m/(m*binomial(2m,m)) according to Lehmer . - R. J. Mathar, Jul 24 2012
Also equals Gamma(1/3)*Gamma(2/3) = A073005 * A073006. - Jean-François Alcover, Nov 24 2014
From Amiram Eldar, Aug 06 2020: (Start)
Equals Integral_{x=0..oo} log(1 + 1/x^3) dx.
Equals Integral_{x=-oo..oo} exp(x/3)/(exp(x) + 1) dx. (End)
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EXAMPLE
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3.627598728468435701188156515284311464568132496185481151139769870776...
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MATHEMATICA
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RealDigits[2 Pi/Sqrt[3], 10, 111][[1]] (* Robert G. Wilson v, Nov 18 2012 *)
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PROG
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(PARI) intnum(x=1e-7, 1e6, eta(x*I, 1)) \\ Charles R Greathouse IV, Feb 26 2011
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CROSSREFS
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Sequence in context: A169751 A105332 A274632 * A169749 A169750 A249558
Adjacent sequences: A186703 A186704 A186705 * A186707 A186708 A186709
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KEYWORD
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cons,nonn
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AUTHOR
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Robert G. Wilson v, Feb 25 2011
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STATUS
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approved
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