OFFSET
0,2
COMMENTS
The absolute value of the integral {x=0..Pi/2} x^2*log(sin(x )) dx or (d^2/da^2 (integral {x=0..Pi/2} cos(ax)*log(sin(x )) dx)) at a=0. The absolute value of (sum {n=1..infinity} (limit { a -> 0} (d^2/da^2 (sin((a+2n)*Pi/2)/n/(a+2n)))))-(Pi/2)^3*log(2)/3. [Seiichi Kirikami and Peter J. C. Moses]
REFERENCES
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, series and Products, 1.441.2, 4th edition, log(sin(x))=-(sum {1..infinity} cos(2nx)/n)-log(2).
LINKS
R. E. Crandall, J. P. Buhler, On the evaluation of Euler sums, Exper. Math. 3 (4) (1994) 275 (discuss int_{0..1} x^n*cot(x) dx which is obtained by partial integration).
S. Koyama and N. Kurokawa, Euler’s integrals and multiple sine functions, Proc. Amer. Math. Soc. 133(2005), 1257-1265.
EXAMPLE
0.18742642282823108026...
MATHEMATICA
RealDigits[ N[Pi (2 Pi^2 Log[2] - 9 Zeta[3]) / 48, 105] ][[1]]
PROG
(PARI) Pi^3*log(2)/24 - 3*Pi*zeta(3)/16 \\ Michel Marcus, Oct 25 2017
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Seiichi Kirikami, Aug 03 2011
STATUS
approved