A379042 Decimal expansion of the generalized log-sine integral with k = 0, n = 3, m = 3, from {0 .. Pi/3} (negated).

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

Offset

1,2

Links

Table of n, a(n) for n=1..101.

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; alternative link.

Armin Straub, A Mathematica package for evaluating log-sine integrals.

Formula

Equals -Integral_{0..Pi/3} log(3*sin(x/2))^2 dx.

Equals -((7*Pi^3)/108) - (2*Pi^2*log(3/2))/(3*sqrt(3)) - (1/3)*Pi*log(2)^2 + (2/3)*Pi*log(2)*log(3) - (1/3)*Pi*log(3)^2 + (log(3/2)*PolyGamma(1, 1/3))/sqrt(3). (This formula was suggested by Mathematica.)

Example

-1.3587805883266742456054317574967336704631556699546811878188991347065...

Maple

Digits:= 106: evalf(Int(log(3*sin(x/2))^2, x = 0..Pi/3)); # Peter Luschny, Dec 16 2024

Mathematica

RealDigits[-((7*Pi^3)/108) - (2*Pi^2*Log[3/2])/(3*Sqrt[3]) - (1/3)*Pi* Log[2]^2 + (2/3)*Pi*Log[2]*Log[3] - (1/3)*Pi*Log[3]^2 + (Log[3/2]*PolyGamma[1, 1/3])/Sqrt[3], 10, 105] // First

Prog

(PARI) -intnum(x = 0, Pi/3, log(3*sin(x/2))^2) \\ Amiram Eldar, Jun 29 2026

Crossrefs

Cf. A258759 (Ls_3(Pi/3)).

Sequence in context: A388447 A304026 A110641 * A121729 A072105 A350693

Adjacent sequences: A379039 A379040 A379041 * A379043 A379044 A379045

Keyword

nonn,cons,changed

Author

Detlef Meya, Dec 14 2024

Status

approved