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A173624
Decimal expansion of Pi^2*log(2)/8 - 7*zeta(3)/16, where zeta is the Riemann zeta function.
5
3, 2, 9, 2, 3, 6, 1, 6, 2, 8, 4, 9, 8, 1, 7, 0, 6, 8, 2, 4, 3, 5, 4, 9, 4, 4, 8, 5, 8, 3, 0, 0, 2, 6, 3, 7, 9, 5, 2, 7, 9, 0, 8, 7, 8, 1, 2, 4, 5, 2, 0, 9, 2, 8, 6, 3, 1, 3, 9, 7, 6, 7, 5, 6, 0, 2, 5, 8, 5, 4, 3, 9, 8, 3, 3, 8, 3, 4, 1, 1, 3, 8, 8, 1, 6, 6, 9, 3, 1, 8, 5, 3, 1, 5, 6, 4, 9, 9, 7, 2, 7, 8, 2, 2, 0
OFFSET
0,1
LINKS
Chenli Li, Wenchang Chu, Improper integrals involving powers of Inverse Trigonometric and hyperbolic Functions, Mathematics 10 (16) (2022) 2980
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (7.3.1)
Kazuhiro Onodera, Generalized log sine integrals and the Mordell-Tornheim zeta values, Trans. Am. Math. Soc. 363 (3) (2010), 1463-1485.
FORMULA
The absolute value of the Integral_{x=0..Pi/2} x*log(sin(x)) dx.
Equals A111003 * A002162 - 7*A002117/16. [typo corrected by R. J. Mathar, Nov 15 2010]
Equals Sum_{n>=1} (phi(-1,1,2n)/(2n-1)^2), where phi is the Lerch transcendent. - John Molokach, Jul 22 2013
Equals Sum_{n>=1} 4^n / (8*n^3*binomial(2*n,n)). - John Molokach, Aug 01 2013
Equals Integral_{y=0..1} Integral_{x=0..1} log(x*y+1)/(1-(x*y)^2) dx dy. - Amiram Eldar, Apr 17 2022
EXAMPLE
0.3292361628498170682435494485830026...
MAPLE
-7*Zeta(3)/16+Pi^2*log(2)/8 ; evalf(%) ;
MATHEMATICA
N[(1/8) (Pi^2 Log[2] - 7 Zeta[3]/2), 100] (* John Molokach, Aug 02 2013 *)
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
R. J. Mathar, Nov 08 2010
STATUS
approved