A186706 - OEIS

A186706

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

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

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 https://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

REFERENCES

Joel L. Schiff, The Laplace Transform: Theory and Applications, Springer-Verlag New York, Inc. (1999). See p. 149.

LINKS

Table of n, a(n) for n=1..105.

D. H. Lehmer, Interesting series involving the central binomial coefficient, Am. Math. Monthly 92 (7) (1985) 449.

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 511.

Eric W. Weisstein's World of Mathematics, Dedekind Eta Function.

Index entries for transcendental numbers.

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) = A073005 * A073006. - Jean-François Alcover, Nov 24 2014

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

From Stefano Spezia, Jun 05 2025: (Start)

Equals Beta(1/3,2/3).

Equals Integral_{x=-oo..oo} 1/(x^2 + x + 1) dx.

Equals 2*Integral_{x=0..oo} log(1 + x^3)/x^3 dx.

Equals Integral_{x=0..oo} log(1 + 4/(x*(x + 2))) dx. (End)

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