A380206 Decimal expansion of the generalized log-sine integral with k = 0, n = 3, m = 3, from {0 .. 5 Pi/3} (negated).
4, 8, 4, 1, 9, 0, 0, 1, 3, 2, 8, 9, 6, 4, 4, 8, 6, 2, 6, 6, 5, 3, 7, 1, 3, 7, 5, 5, 3, 6, 4, 8, 3, 0, 5, 8, 0, 6, 4, 4, 9, 1, 6, 3, 9, 3, 7, 5, 1, 3, 5, 3, 4, 7, 7, 2, 7, 8, 2, 7, 7, 8, 8, 5, 9, 6, 5, 4, 7, 4, 8, 7, 9, 4, 5, 5, 8, 6, 1, 0, 0, 9, 5, 9, 1, 7, 4, 1, 6, 3, 5, 3, 4, 7, 5, 9, 2, 3, 1, 0
Offset: 1
Examples
-4.841900132896448626653713755364830580644916393751353477278...
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
Programs
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Mathematica
RealDigits[(1/108)*(-11*Pi^3 + 24*Sqrt[3]*Pi^2*Log[3/2] - 180*Pi*Log[3/2]^2 - 36*Sqrt[3]*Log[3/2]*PolyGamma[1, 1/3]), 10, 100] // First
Formula
-Integral_{0..5*Pi/3} log(3*sin(x/2))^2 dx = (1/108)*(-11*Pi^3 + 24*Sqrt(3)*Pi^2*Log(3/2) - 180*Pi*Log(3/2)^2 - 36*Sqrt(3)*Log(3/2)*PolyGamma(1, 1/3)).
Equals (-Integral_{0..2 Pi} log(3*sin(x/2))^2 dx) - (-Integral_{0..Pi/3} log(3*sin(x/2))^2 dx). (This formula was suggested by Mathematica.)