A380206 - OEIS

%I #10 Feb 07 2025 16:48:17

%S 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,

%T 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,

%U 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

%N Decimal expansion of the generalized log-sine integral with k = 0, n = 3, m = 3, from {0 .. 5 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..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)).

%F 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.)

%e -4.841900132896448626653713755364830580644916393751353477278...

%t 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

%Y Cf. A379042, A379273, A380205.

%K nonn,cons

%O 1,1

%A _Detlef Meya_, Jan 16 2025