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A186706 Decimal expansion of the Integral of Dedekind Eta(x*I) from x = 0 ... Infinity. 1
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 (list; constant; graph; refs; listen; history; text; internal format)
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=-infinity}^infinity (-1)^n exp(-pi x (6n-1)^2/12)

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

According to Maple, sum_{n=-infinity}^infinity (-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)

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

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

Also equals Gamma(1/3)*Gamma(2/3) - Jean-François Alcover, Nov 24 2014

EXAMPLE

3.6275987284684357011881565152843114645681324961854811511397698728...

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

Sequence in context: A046901 A169751 A105332 * A169749 A169750 A249558

Adjacent sequences:  A186703 A186704 A186705 * A186707 A186708 A186709

KEYWORD

cons,nonn

AUTHOR

Robert G. Wilson v, Feb 25 2011

STATUS

approved

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Last modified December 22 07:03 EST 2014. Contains 252328 sequences.