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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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11
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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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Use the definition of DedekindEta as a sum:
Eta(i*x) = Sum_{n=-oo..oo} (-1)^n*exp(-Pi*x*(6n-1)^2/12).
Now Integral_{x=0..oo} exp(-Pi*x*(6n-1)^2/12) dx = 12/(Pi*(6n-1)^2).
According to Maple, Sum_{n=-oo..oo} (-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)
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
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LINKS
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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
Equals Integral_{x=0..oo} log(1 + 1/x^3) dx.
Equals Integral_{x=-oo..oo} exp(x/3)/(exp(x) + 1) dx. (End)
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
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EXAMPLE
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3.627598728468435701188156515284311464568132496185481151139769870776...
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MATHEMATICA
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PROG
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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