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

Author

Stanislav Sykora, Oct 03 2014

Keywords

• nonn
• cons
• easy

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

Examples

			0.09465032062247697727187848272191072247626297176354162323298972411890...
		

Links

• Stanislav Sykora, Table of n, a(n) for n = -1..2000
• Wikipedia, Digamma function.

Crossrefs

Cf. A113319, A001620.

Programs

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 *)

PARI

real(psi(I))

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)