|
%I #25 Dec 01 2020 11:35:36
%S 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,
%T 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,
%U 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
%N Decimal expansion of gamma_1(3/4), the first generalized Stieltjes constant at 3/4 (negated).
%H G. C. Greubel, <a href="/A254348/b254348.txt">Table of n, a(n) for n = 0..5000</a>
%H Iaroslav V. Blagouchine, <a href="http://arxiv.org/abs/1401.3724">A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments</a>, arXiv:1401.3724 [math.NT], 2015.
%H Iaroslav V. Blagouchine, <a href="http://www.researchgate.net/publication/259743013_A_theorem_for_the_closed-form_evaluation_of_the_first_generalized_Stieltjes_constant_at_rational_arguments_and_some_related_summations">A theorem ... (same title)</a>, Journal of Number Theory Volume 148, March 2015, Pages 537-592.
%H Iaroslav V. Blagouchine, <a href="https://doi.org/10.1007/s11139-013-9528-5">Rediscovery of Malmsten’s integrals, their evaluation by contour integration methods and some related results</a>, The Ramanujan Journal October 2014, Volume 35, Issue 1, pp 21-110.
%H Iaroslav V. Blagouchine, <a href="http://www.researchgate.net/publication/257381156_Rediscovery_of_Malmsten%27s_integrals_their_evaluation_by_contour_integration_methods_and_some_related_results">Rediscovery of Malmsten’s integrals: Full PDF text</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HurwitzZetaFunction.html">Hurwitz Zeta Function</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StieltjesConstants.html">Stieltjes Constants</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Stieltjes_constants">Stieltjes constants</a>
%F 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).
%e -0.39129890240454977423987419218929637145038973196714...
%p 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
%t 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
%t (* Or, from Mma version 7 up: *) RealDigits[StieltjesGamma[1, 3/4], 10, 103] // First
%Y 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)).
%K nonn,cons,easy
%O 0,1
%A _Jean-François Alcover_, Jan 29 2015
|
