A339135 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).

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Offset: 0

Author

Artur Jasinski, Nov 25 2020

Keywords

• nonn
• cons

Comments

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

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

Examples

			J = 0.677024679102993347...
		

Crossrefs

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

Programs

Maple

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

Mathematica

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

PARI

2*log(2)/3 - real(psi(1/2 + I*sqrt(3)/2)) \\ Michel Marcus, Nov 25 2020

Formula

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

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

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.

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

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).

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

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

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

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).

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)).

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)).

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