A339135 - OEIS

%I #52 Jan 15 2021 21:21:13

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

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

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

%N Decimal expansion of J = 2*log(2)/3 - Re(Psi(1/2 + i*sqrt(3)/2)), where Psi is the digamma function and i=sqrt(-1).

%C Generally in the literature there is no explicit formula for the real part of the function Psi(x + i*y) when y != 0.

%C Up to now there is no explicit formula expressing the constant J in terms of other mathematical constants.

%F J = -log(2)/3 - (1/2)*Pi/cosh(Pi*sqrt(3)/2) - Re(Psi(1/4 + i*sqrt(3)/4)).

%F J = -log(2)/3 + (1/2)*Pi/cosh(Pi*sqrt(3)/2) - Re(Psi(3/4 + i*sqrt(3)/4)).

%F J = 3 + gamma + (2/3)*log(2) - (1/2)* sqrt(3)*Pi*tanh(Pi*sqrt(3)/2) - 3*(Sum_{n>=1} zeta(3*n)-1), where zeta is Riemann zeta function and gamma is Euler gamma constant see A001620.

%F J = -(1/2) + gamma + (2/3)*log(2) + (3/2)*(Sum_{n>=1} zeta(3*n+1)-1).

%F J = -1 + gamma + (2/3)*log(2) + (1/2)*sqrt(3)*Pi*tanh(Pi*sqrt(3)/2) - 3*(Sum_{n>=0} zeta(3*n+2)-1).

%F J = -(3/8) + gamma + (2/3)*log(2) + (3/2)*(Sum_{n>=1} zeta(6*n+1)-1).

%F J = 1/4 + gamma + (2/3)*log(2) - 3*(Sum_{n>=0} zeta(6*n+3)-1).

%F J = -(1/4) + gamma + (2/3)*log(2) - 3 (Sum_{n>=0} zeta(6*n+5)-1).

%F J = (11/12 - (1/4)*i*sqrt(3))*Psi(1/2 + i*sqrt(3)/2) + (-(5/4) + (1/4)*i*sqrt(3))*Psi(-(1/2) - i*sqrt(3)/2) + (-(17/24) + (1/8)*i*sqrt(3))* Psi(1/4 + i*sqrt(3)/4) + (3/8 - (1/8)*i*sqrt(3))*Psi(-(1/4) - i*sqrt(3)/4) + (-(17/24) + (1/8)*i*sqrt(3))*Psi(3/4 + i*sqrt(3)/4) + (3/8 - (1/8)*i*sqrt(3))*Psi(-(3/4) - i*sqrt(3)/4).

%F J = 2*log(2)/3 - Integral_{t=0..infinity} cosh(t)/t - sinh(t)/t - (cos(sqrt(3)*t)*cosh(t/2))/(1 - cosh(t) + sinh(t)) + (cos(sqrt(3)*t)*sinh(t/2))/(1 - cosh(t) + sinh(t)).

%F J = gamma + (1/6)*Sum_{t>=1} (6*t^3-4*t^2-4*t-1)/(t*(t+1)*(2t+1)*(t^2+t+1)).

%F Equals 1 + 2*log(2)/3 - Psi(0, 5/2 - i*sqrt(3)/2)/2 - Psi(0, 5/2 + i*sqrt(3)/2)/2. - _Vaclav Kotesovec_, Nov 26 2020

%e J = 0.677024679102993347...

%p evalf(1 + 2*log(2)/3 - Psi(0, 5/2 - sqrt(3)*I/2)/2 - Psi(0, 5/2 + sqrt(3)*I/2)/2, 100); # _Vaclav Kotesovec_, Nov 26 2020

%t RealDigits[N[Re[2 Log[2]/3 - PolyGamma[0, 1/2 + I Sqrt[3]/2]], 110]][[1]]

%t Chop[N[1 + 2*Log[2]/3 - PolyGamma[0, 5/2 - I*Sqrt[3]/2]/2 - PolyGamma[0, 5/2 + I*Sqrt[3]/2]/2, 120]] (* _Vaclav Kotesovec_, Nov 26 2020 *)

%o (PARI) 2*log(2)/3 - real(psi(1/2 + I*sqrt(3)/2)) \\ _Michel Marcus_, Nov 25 2020

%Y Cf. A097663, A097663-A097676, A268046, A268086, A257958, A257959, A338815, A339237.

%K nonn,cons

%O 0,1

%A _Artur Jasinski_, Nov 25 2020