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%I #53 Dec 30 2023 23:06:48
%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=-oo..oo} (-1)^n*exp(-Pi*x*(6n-1)^2/12).
%C Now Integral_{x=0..oo} exp(-Pi*x*(6n-1)^2/12) dx = 12/(Pi*(6n-1)^2).
%C According to Maple, Sum_{n=-oo..oo} (-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)
%C Ratio of surface area of a sphere to the regular octahedron whose edge equals the radius of the sphere. - _Omar E. Pol_, Dec 30 2023
%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)
%F Equals Integral_{x=0..2*Pi} 1/(2 + sin(x)) dx; since for a>1: Integral_{x=0..2*Pi} 1/(a + sin(x)) dx = 2*Pi/sqrt(a^2-1). - _Bernard Schott_, Feb 18 2023
%F Equals 4*A093766. - _Omar E. Pol_, Dec 30 2023
%e 3.627598728468435701188156515284311464568132496185481151139769870776...
%t RealDigits[2 Pi/Sqrt[3], 10, 111][[1]] (* _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
%Y Cf. A073005, A073006, A093602, A093766,
%K cons,nonn
%O 1,1
%A _Robert G. Wilson v_, Feb 25 2011
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