%I #46 Jun 11 2024 04:57:59
%S 1,3,5,3,4,5,9,6,8,0,8,0,4,9,4,1,5,1,7,7,0,8,6,8,7,1,6,9,1,7,8,0,6,4,
%T 4,0,3,5,9,1,2,8,6,2,8,9,0,3,6,3,4,6,6,1,1,6,7,4,3,8,3,8,8,6,2,6,8,0,
%U 4,6,2,0,2,4,5,9,2,3,8,4,3,8,5,9,7,0,9,3,5,2,3,1,9,6,7,9,0,3,7,3,0,5,8,7,7
%N Decimal expansion of gamma_1(1/2), the first generalized Stieltjes constant at 1/2 (negated).
%H G. C. Greubel, <a href="/A254327/b254327.txt">Table of n, a(n) for n = 1..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="http://dx.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 MathWorld, <a href="http://mathworld.wolfram.com/HurwitzZetaFunction.html">Hurwitz Zeta Function</a>.
%H Eric Weisstein's MathWorld, <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 Gamma(1) - log(2)^2 - 2*gamma*log(2).
%F Equals Integral_{0..oo} (coth(Pi*x)-1) * (-2*arctan(2*x) + 2*x*log(1/4+x^2)) / (1+4*x^2) dx - log(2) - log(2)^2/2.
%e -1.3534596808049415177086871691780644035912862890363466...
%p evalf(int((coth(Pi*x)-1)*(-2*arctan(2*x)+2*x*log(1/4+x^2))/(1+4*x^2), x = 0..infinity) - log(2) - (1/2)*log(2)^2, 120); # _Vaclav Kotesovec_, Jan 28 2015
%p evalf(gamma(1) - log(2)^2 - 2*gamma*log(2), 120); # _Vaclav Kotesovec_, Jan 29 2015 (faster)
%t gamma1[1/2] = StieltjesGamma[1] - Log[2]^2 - 2*EulerGamma*Log[2]; RealDigits[ gamma1[1/2], 10, 105] // First (* = StieltjesGamma[1, 1/2] expanded *)
%Y Cf. A001620 (gamma), A082633 (gamma_1), A254331 (gamma_1(1/3)), A254345 (gamma_1(2/3)), A254347 (gamma_1(1/4)), A254348 (gamma_1(3/4)), 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 1,2
%A _Jean-François Alcover_, Jan 28 2015