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A339135
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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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10
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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, 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, 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
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OFFSET
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0,1
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COMMENTS
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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.
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LINKS
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FORMULA
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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
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EXAMPLE
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J = 0.677024679102993347...
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(PARI) 2*log(2)/3 - real(psi(1/2 + I*sqrt(3)/2)) \\ Michel Marcus, Nov 25 2020
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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