A396964 - OEIS

%I #36 Jun 18 2026 23:26:01

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

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

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

%N Decimal expansion of (-1)*Integral_{x=0..1} log(-log(x))/(1 + x^2 + 2*x*cos(4)) dx.

%H Robert Reynolds, <a href="https://arxiv.org/abs/2412.10395">Derivation of some definite integrals</a>, arXiv:2412.10395 [math.GM], 2024-2025, page 10, eq. 4.11.

%F Equals (1/2)*Pi*csc(4)*log((2*Pi)^(2 - 4/Pi)*Gamma(3/2 - 2/Pi)/Gamma(-1/2 + 2/Pi)).

%e 1.02756386551864675847502033400582086722448311493...

%t RealDigits[(-1)*Integrate[Log[-Log[x]]/(1 + x^2 + 2 x Cos[4]), {x, 0, 1}],10,105][[1]]

%t (* Alternative: *)

%t RealDigits[(1/2)*Pi Csc[4] Log[((2 Pi)^(2 - 4/Pi) Gamma[3/2 - 2/Pi])/Gamma[-1/2 + 2/Pi]],10,105][[1]]

%o (PARI) -intnum(x=0,1,log(-log(x))/(1+x^2+2*x*cos(4)))

%Y Cf. A396976.

%K nonn,cons

%O 1,3

%A _Artur Jasinski_, Jun 11 2026