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=-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
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
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)
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
Equals 4*A093766. - Omar E. Pol, Dec 30 2023
EXAMPLE
3.627598728468435701188156515284311464568132496185481151139769870776...
MATHEMATICA
RealDigits[2 Pi/Sqrt[3], 10, 111][[1]] (* 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
KEYWORD
cons,nonn
AUTHOR
Robert G. Wilson v, Feb 25 2011
STATUS
approved