A173624 Decimal expansion of Pi^2*log(2)/8 - 7*zeta(3)/16, where zeta is the Riemann zeta function.
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
Examples
0.3292361628498170682435494485830026...
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.
Programs
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Maple
-7*Zeta(3)/16+Pi^2*log(2)/8 ; evalf(%) ;
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Mathematica
N[(1/8) (Pi^2 Log[2] - 7 Zeta[3]/2), 100] (* John Molokach, Aug 02 2013 *)
Formula
The absolute value of the Integral_{x=0..Pi/2} x*log(sin(x)) dx.
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