login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A186706 Decimal expansion of the Integral of Dedekind Eta(x*I) from x = 0 ... Infinity. 5
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) = 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)

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

Sequence in context: A169751 A105332 A274632 * A169749 A169750 A249558

Adjacent sequences: A186703 A186704 A186705 * A186707 A186708 A186709

KEYWORD

cons,nonn

AUTHOR

Robert G. Wilson v, Feb 25 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 5 02:30 EST 2022. Contains 358572 sequences. (Running on oeis4.)