[seqfan] Re: simplify the constant A186706

[israel at math.ubc.ca]

Ah: 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 result.

Robert Israel
University of British Columbia


On Jul 25 2012, israel at math.ubc.ca wrote:

>
>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
>
>
>_______________________________________________
>
>Seqfan Mailing list - http://list.seqfan.eu/
>