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 A186706 Decimal expansion of the Integral of Dedekind Eta(x*I) from x = 0 ... Infinity. 4

%I

%S 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,

%T 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,

%U 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

%N Decimal expansion of the Integral of Dedekind Eta(x*I) from x = 0 ... Infinity.

%C Reduction of the integral by _Robert Israel_, Jul 25 2012: (Start)

%C Use the definition of DedekindEta as a sum:

%C Eta(i x) = sum_{n=-infinity}^infinity (-1)^n exp(-pi x (6n-1)^2/12)

%C Now int_0^infinity exp(-pi x (6n-1)^2/12) dx = 12/(pi (6n-1)^2)

%C According to Maple, sum_{n=-infinity}^infinity (-1)^n 12/(pi (6n-1)^2) is

%C 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

%C Jonquiere's inversion formula (see http://en.wikipedia.org/wiki/Polylogarithm)

%C but note that Maple's dilog(z) is L_2(1-z) in the notation there) gives

%C dilog(1-1/2*I-1/2*3^(1/2))+dilog(1+1/2*I-1/2*3^(1/2)) = 13/72*Pi^2

%C and

%C dilog(1-1/2*I+1/2*3^(1/2))+dilog(1+1/2*I+1/2*3^(1/2)) = -11*Pi^2/72

%C which give the desired multiple of Pi. (End)

%H D. H. Lehmer, <a href="http://www.jstor.org/stable/2322496">Interesting series involving the central binomial coefficient</a>, Am. Math. Monthly 92 (7) (1985) 449

%H Eric W. Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DedekindEtaFunction.html">Dedekind Eta Function</a>.

%F 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

%F Also equals Gamma(1/3)*Gamma(2/3) = A073005 * A073006. - _Jean-François Alcover_, Nov 24 2014

%F From _Amiram Eldar_, Aug 06 2020: (Start)

%F Equals Integral_{x=0..oo} log(1 + 1/x^3) dx.

%F Equals Integral_{x=-oo..oo} exp(x/3)/(exp(x) + 1) dx. (End)

%e 3.6275987284684357011881565152843114645681324961854811511397698728...

%t RealDigits[2 Pi/Sqrt, 10, 111][] (* _Robert G. Wilson v_, Nov 18 2012 *)

%o (PARI) intnum(x=1e-7,1e6,eta(x*I,1)) \\ _Charles R Greathouse IV_, Feb 26 2011

%K cons,nonn

%O 1,1

%A _Robert G. Wilson v_, Feb 25 2011

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Last modified April 13 06:02 EDT 2021. Contains 342935 sequences. (Running on oeis4.)