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

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

Stanislav Sykora, Table of n, a(n) for n = -1..2000

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

Cf. A113319, A001620.

Sequence in context: A322088 A343199 A139720 * A194826 A309610 A198989

Adjacent sequences: A248174 A248175 A248176 * A248178 A248179 A248180

Keyword

nonn,cons,easy

Author

Stanislav Sykora, Oct 03 2014

Status

approved