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A248177
Decimal expansion of the real part of psi(i), i being the imaginary unit.
7
0, 9, 4, 6, 5, 0, 3, 2, 0, 6, 2, 2, 4, 7, 6, 9, 7, 7, 2, 7, 1, 8, 7, 8, 4, 8, 2, 7, 2, 1, 9, 1, 0, 7, 2, 2, 4, 7, 6, 2, 6, 2, 9, 7, 1, 7, 6, 3, 5, 4, 1, 6, 2, 3, 2, 3, 2, 9, 8, 9, 7, 2, 4, 1, 1, 8, 9, 0, 5, 1, 1, 4, 7, 5, 9, 2, 8, 0, 6, 5, 3, 3, 8, 3, 4, 7, 0, 7, 0, 9, 4, 9, 5, 4, 5, 3, 6, 7, 1, 8, 1, 3, 7, 6, 4
OFFSET
0,2
COMMENTS
For real b, Im(psi(b*i)) = 1/(2*b) + Pi*coth(Pi*b)/2, but no such closed formula is known for the real part (see Wikipedia link). - Vaclav Kotesovec, Dec 24 2020
LINKS
Wikipedia, Digamma function.
FORMULA
psi(i) = -EulerGamma - Sum_{k>=0} ((k-1)/(k+1)/(k^2+1)) + A113319*i, where EulerGamma is the Euler-Mascheroni constant (A001620).
Equals 3/4 - EulerGamma - 2*Sum_{k>=2} 1/(k*(k^4 - 1)). - Vaclav Kotesovec, Dec 24 2020
From Amiram Eldar, May 20 2022: (Start)
Equals Sum_{n>=1} 1/(n^3+n) - EulerGamma.
Equals 1/2 - EulerGamma + Sum_{n>=1} (-1)^(n+1) * (zeta(2*n+1) - 1). (End)
EXAMPLE
0.09465032062247697727187848272191072247626297176354162323298972411890...
MAPLE
Re(Psi(I)) ; evalf(%) ; # R. J. Mathar, Oct 18 2019
MATHEMATICA
RealDigits[N[Re[PolyGamma[0, I]], 105]][[1]] (* Vaclav Kotesovec, Oct 04 2014 *)
PROG
(PARI) real(psi(I))
CROSSREFS
Sequence in context: A322088 A343199 A139720 * A194826 A309610 A198989
KEYWORD
nonn,cons,easy
AUTHOR
Stanislav Sykora, Oct 03 2014
STATUS
approved