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A248177 Decimal expansion of the real part of psi(i), i being the imaginary unit.

%I A248177 #26 Dec 07 2023 05:24:57
%S A248177 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,
%T A248177 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,
%U A248177 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
%N A248177 Decimal expansion of the real part of psi(i), i being the imaginary unit.
%C A248177 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
%H A248177 Stanislav Sykora, <a href="/A248177/b248177.txt">Table of n, a(n) for n = -1..2000</a>
%H A248177 Wikipedia, <a href="http://en.wikipedia.org/wiki/Digamma_function">Digamma function</a>.
%F A248177 psi(i) = -EulerGamma - Sum_{k>=0} ((k-1)/(k+1)/(k^2+1)) + A113319*i, where EulerGamma is the Euler-Mascheroni constant (A001620).
%F A248177 Equals 3/4 - EulerGamma - 2*Sum_{k>=2} 1/(k*(k^4 - 1)). - _Vaclav Kotesovec_, Dec 24 2020
%F A248177 From _Amiram Eldar_, May 20 2022: (Start)
%F A248177 Equals Sum_{n>=1} 1/(n^3+n) - EulerGamma.
%F A248177 Equals 1/2 - EulerGamma + Sum_{n>=1} (-1)^(n+1) * (zeta(2*n+1) - 1). (End)
%e A248177 0.09465032062247697727187848272191072247626297176354162323298972411890...
%p A248177 Re(Psi(I)) ; evalf(%) ; # _R. J. Mathar_, Oct 18 2019
%t A248177 RealDigits[N[Re[PolyGamma[0, I]], 105]][[1]]  (* _Vaclav Kotesovec_, Oct 04 2014 *)
%o A248177 (PARI) real(psi(I))
%Y A248177 Cf. A113319, A001620.
%K A248177 nonn,cons,easy
%O A248177 0,2
%A A248177 _Stanislav Sykora_, Oct 03 2014