A254348 Decimal expansion of gamma_1(3/4), the first generalized Stieltjes constant at 3/4 (negated).

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

Offset

0,1

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

Iaroslav V. Blagouchine, A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments, arXiv:1401.3724 [math.NT], 2015.

Iaroslav V. Blagouchine, A theorem ... (same title), Journal of Number Theory Volume 148, March 2015, Pages 537-592.

Iaroslav V. Blagouchine, Rediscovery of Malmsten’s integrals, their evaluation by contour integration methods and some related results, The Ramanujan Journal October 2014, Volume 35, Issue 1, pp 21-110.

Iaroslav V. Blagouchine, Rediscovery of Malmsten’s integrals: Full PDF text.

Eric Weisstein's World of Mathematics, Hurwitz Zeta Function.

Eric Weisstein's World of Mathematics, Stieltjes Constants.

Wikipedia, Stieltjes constants

Formula

Equals integral_[0..infinity] (4*(-6*arctan(4*x/3) + 4*x*log(9/16 + x^2)))/((-1 + exp(2*Pi*x))*(9 + 16*x^2)) dx -(2/3 + (1/2)*log(4/3))*log(4/3).

Example

-0.39129890240454977423987419218929637145038973196714...

Maple

evalf(int((4*(-6*arctan(4*x*(1/3))+4*x*log(9/16+x^2)))/((-1+exp(2*Pi*x))*(16*x^2+9)), x = 0..infinity) - (2/3+(1/2)*log(4/3))*log(4/3), 120); # Vaclav Kotesovec, Jan 29 2015

Mathematica

gamma1[3/4] = (1/2)*(-Log[4]^2 + EulerGamma*(Pi - 2*Log[8]) - 2*Log[4]*Log[2*Pi] + Pi*Log[(8*Pi*Gamma[3/4]^2)/Gamma[1/4]^2] - 2*(Log[2*Pi]^2 - Log[Pi]*Log[8*Pi] - StieltjesGamma[1] + Derivative[2, 0][Zeta][0, 1/2])) // Re; RealDigits[gamma1[3/4], 10, 103] // First
(* Or, from Mma version 7 up: *) RealDigits[StieltjesGamma[1, 3/4], 10, 103] // First

Crossrefs

Cf. A001620 (gamma), A082633 (gamma_1), A254327 (gamma_1(1/2)), A254331 (gamma_1(1/3)), A254345 (gamma_1(2/3)), A254347 (gamma_1(1/4)), A254349 (gamma_1(1/6)), A254350 (gamma_1(5/6)), A251866 (gamma_1(1/5)), A255188 (gamma_1(1/8)), A255189 (gamma_1(1/12)).

Sequence in context: A016673 A304022 A103556 * A168399 A290375 A245081

Adjacent sequences: A254345 A254346 A254347 * A254349 A254350 A254351

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Jan 29 2015

Status

approved