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 A186706 Decimal expansion of the Integral of Dedekind Eta(x*I) from x = 0 ... Infinity. 2
 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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) LINKS 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. FORMULA 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) - Jean-François Alcover, Nov 24 2014 EXAMPLE 3.6275987284684357011881565152843114645681324961854811511397698728... MATHEMATICA RealDigits[2 Pi/Sqrt, 10, 111][] (* Robert G. Wilson v, Nov 18 2012 *) PROG (PARI) intnum(x=1e-7, 1e6, eta(x*I, 1)) \\ Charles R Greathouse IV, Feb 26 2011 CROSSREFS Sequence in context: A169751 A105332 A274632 * A169749 A169750 A249558 Adjacent sequences:  A186703 A186704 A186705 * A186707 A186708 A186709 KEYWORD cons,nonn AUTHOR Robert G. Wilson v, Feb 25 2011 STATUS approved

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Last modified November 12 05:52 EST 2019. Contains 329051 sequences. (Running on oeis4.)