[seqfan] Re: simplify the constant A186706
israel at math.ubc.ca
israel at math.ubc.ca
Wed Jul 25 20:28:07 CEST 2012
On Jul 25 2012, Alexander P-sky wrote:
>WolframAlpha reports that
>
>Integrate[DedekindEta[x I], {x, 0, Infinity}] -
>sum(3^m/m/binomial(2*m,m),m=1..infinity)
>~= 1.53825*10^(-10)
>
That's only to 10 digits of accuracy.
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
It won't simplify the difference between this and 2 pi/sqrt(3) to 0, but
floating point evaluation at 1000 digits gives -.1e-998+0.*I. So it seems
very likely that these are equal.
Robert Israel
University of British Columbia
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