A379273 - OEIS

%I #4 Dec 30 2024 17:21:36

%S 1,9,4,0,3,9,1,9,8,2,0,7,2,0,5,9,6,9,7,9,3,6,4,9,2,5,5,9,1,3,1,0,6,3,

%T 7,1,6,1,1,9,1,8,4,1,8,7,8,3,6,2,5,4,5,2,6,9,4,3,2,6,0,7,6,2,9,4,4,8,

%U 5,7,1,3,2,3,5,9,3,4,5,8,6,7,4,5,8,9,4,9,5,4,5,5,7,2,3,2,4,8,7,3

%N Decimal expansion of the generalized log-sine integral with k = 0, n = 3, m = 3, from {0 .. 2*Pi/3} (negated).

%H Jonathan M. Borwein and Armin Straub, <a href="https://carmamaths.org/resources/jon/logsin3.pdf">Special Values of Generalized Log-sine Integrals</a>, ISSAC '11: Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation, 2011, pp. 43-50.

%H Armin Straub, <a href="https://arminstraub.com/software/lstoli">A Mathematica package for evaluating log-sine integrals</a>

%F -Integral_{0..2*Pi/3} log(3*sin(x/2))^2 dx = (1/162)*(-4*Pi^3 + 324*Im(PolyLog(3, 1 - (-1)^(2/3))) -

%F    108*Pi*Log(3/2)^2 + 27*Pi*Log(3)^2 + 12*Sqrt(3)*Pi^2*Log(27/4) -

%F    18*Sqrt(3)*Log(27/4)*PolyGamma(1, 2/3)). (This formula was suggested by Mathematica.)

%e -1.9403919820720596979364925591310637161191841878362545269432607629448...

%t RealDigits[(1/162)*(-4*Pi^3 + 324*Im[PolyLog[3, 1 - (-1)^(2/3)]] -

%t    108*Pi*Log[3/2]^2 + 27*Pi*Log[3]^2 + 12*Sqrt[3]*Pi^2*Log[27/4] -

%t    18*Sqrt[3]*Log[27/4]*PolyGamma[1, 2/3])

%t , 10, 105] // First

%Y Cf. A379042.

%K nonn,cons

%O 1,2

%A _Detlef Meya_, Dec 19 2024