A257958 Decimal expansion of the Digamma function at 1/Pi, negated.

3, 2, 9, 0, 2, 1, 3, 9, 6, 0, 1, 7, 3, 2, 2, 4, 0, 9, 0, 8, 4, 3, 0, 9, 0, 8, 4, 5, 5, 4, 0, 0, 1, 9, 0, 3, 7, 4, 0, 2, 1, 9, 3, 2, 8, 2, 0, 0, 7, 0, 1, 6, 1, 2, 9, 3, 8, 8, 9, 5, 3, 1, 8, 3, 7, 5, 5, 3, 7, 5, 6, 6, 5, 3, 3, 7, 1, 7, 9, 1, 2, 9, 1, 5, 3, 2, 8, 7, 7, 1, 1, 1, 6, 9, 3, 5, 6, 7, 3, 1, 6, 6, 9

Offset: 1

Author

Iaroslav V. Blagouchine, May 14 2015

Keywords

• nonn
• cons

Comments

The reference gives an interesting series representation with rational coefficients for Psi(1/Pi) = -log(Pi) - Pi/2 - 1/2 - 1/8 - 1/72 + 1/64 +7/400 + 7/576 + 643/94080 + 103/30720 + ...

Examples

			-3.2902139601732240908430908455400190374021932820070161...
		

Links

• Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/Pi, Mathematics of Computation (AMS), 2015.

Crossrefs

Cf. A257955, A257957, A257959, A155968, A256165, A256166, A256167, A255888, A256609, A255306, A256610, A256612, A256611, A256066, A256614, A256615, A256616.

Programs

Maple

evalf(Psi(1/Pi), 120);

Mathematica

RealDigits[PolyGamma[1/Pi], 10, 120][[1]]

PARI

default(realprecision, 120); psi(1/Pi)

Formula

Int_0^infinity x*dx/[(x^2+1)(exp(2x)-1)] = -Pi/2-Psi(1/Pi) = -1.5707...+ 3.2902.. = 1.71941... - R. J. Mathar, Aug 14 2023