OFFSET
0,2
COMMENTS
The absolute value of the integral {x=0..Pi/2} x^3*log(sin(x )) dx or (d^3/da^3 (integral {x=0..Pi/2} sin(ax )*log(sin(x )) dx)) at a=0. The absolute value of (sum {n=1..infinity} (limit { a -> 0} (d^3/da^3 ((1-cos((a+2n)*Pi/2))/n/(a+2n)))))-(Pi/2)^4*log(2)/4. [Seiichi Kirikami and Peter J. C. Moses]
REFERENCES
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, series and Products, 4th edition, 1.441.2, log(sin(x))=-(sum {1..infinity} cos(2nx)/n)-log(2).
LINKS
S. Koyama and N. Kurokawa, Euler’s integrals and multiple sine functions, Proc. Amer. Math. Soc. 133(2005), 1257-1265.
EXAMPLE
-0.14002410170685231710...
MATHEMATICA
RealDigits[N[(2 Pi^4 Log[2] - 18 Pi^2 Zeta[3] + 93 Zeta[5]) / 128, 105]][[1]]
PROG
(PARI) Pi^4*log(2)/64 - 9*Pi^2*zeta(3)/64 + 93*zeta(5)/128 \\ Michel Marcus, Oct 25 2017
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Seiichi Kirikami, Aug 03 2011
STATUS
approved