|
|
A193717
|
|
Decimal expansion of Pi^4*log(2)/64 - 9*Pi^2*zeta(3)/64 + 93*zeta(5)/128.
|
|
3
|
|
|
1, 4, 0, 0, 2, 4, 1, 0, 1, 7, 0, 6, 8, 5, 2, 3, 1, 7, 1, 0, 0, 2, 7, 0, 5, 7, 8, 8, 7, 5, 5, 3, 5, 0, 7, 5, 3, 2, 2, 4, 2, 8, 2, 1, 2, 7, 8, 5, 7, 7, 0, 5, 0, 8, 9, 8, 8, 1, 8, 5, 9, 6, 3, 1, 4, 1, 1, 6, 2, 7, 7, 1, 4, 6, 3, 7, 0, 5, 9, 7, 0, 2, 3, 0, 4, 9, 0, 7, 6, 1, 1, 0, 2, 6, 6, 3, 0, 9, 0, 5
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
FORMULA
|
|
|
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
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|