A379273 Decimal expansion of the generalized log-sine integral with k = 0, n = 3, m = 3, from {0 .. 2*Pi/3} (negated).

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, 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, 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

Offset: 1

Author

Detlef Meya, Dec 19 2024

Keywords

• nonn
• cons

Examples

			-1.9403919820720596979364925591310637161191841878362545269432607629448...
		

Links

• Jonathan M. Borwein and Armin Straub, Special Values of Generalized Log-sine Integrals, ISSAC '11: Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation, 2011, pp. 43-50.
• Armin Straub, A Mathematica package for evaluating log-sine integrals

Crossrefs

Cf. A379042.

Programs

Mathematica

RealDigits[(1/162)*(-4*Pi^3 + 324*Im[PolyLog[3, 1 - (-1)^(2/3)]] -
   108*Pi*Log[3/2]^2 + 27*Pi*Log[3]^2 + 12*Sqrt[3]*Pi^2*Log[27/4] -
   18*Sqrt[3]*Log[27/4]*PolyGamma[1, 2/3])
, 10, 105] // First

Formula

-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))) -

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

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