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A301816 Decimal expansion of the real Stieltjes gamma function at x = 1/2.

%I A301816 #28 Apr 12 2018 04:57:00
%S A301816 2,7,5,4,3,4,7,2,4,5,6,3,9,2,0,0,7,9,9,5,5,2,8,7,8,7,7,7,9,7,8,0,6,8,
%T A301816 3,5,7,9,8,7,0,2,3,2,3,8,8,6,3,0,7,4,8,7,3,7,3,3,2,1,1,4,7,5,1,3,3,0,
%U A301816 6,3,4,4,1,7,3,0,6,4,6,8,8,2,2,3,5,9,2
%N A301816 Decimal expansion of the real Stieltjes gamma function at x = 1/2.
%C A301816 Define the real Stieltjes gamma function (this is not a standard notion) as Sti(x) = -2*Pi*I(x+1)/(x+1) where I(x) = Integral_{-infinity..+infinity} log(1/2+i*z)^x/(exp(-Pi*z) + exp(Pi*z))^2 dz and i is the imaginary unit. We look here at the real part of Sti(x).
%H A301816 Iaroslav V. Blagouchine, <a href="http://dx.doi.org/10.1016/j.jnt.2014.08.009">A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations</a>, Journal of Number Theory, vol. 148, pp. 537-592 and vol. 151, pp. 276-277, 2015. <a href="http://arxiv.org/abs/1401.3724">arXiv version</a>, arXiv:1401.3724 [math.NT], 2014.
%H A301816 Peter Luschny, <a href="/A301816/a301816_1.pdf">Illustration of the real Stieltjes gamma function.</a>
%F A301816 c = -Re((4/3)*Pi*Integral_{-oo..oo} log(1/2+i*z)^(3/2)/(exp(-Pi*z)+exp(Pi*z))^2 dz).
%e A301816 0.2754347245639200799552878777978068357987023238863074873733211475133063441...
%p A301816 Sti := x -> (-4*Pi/(x + 1))*int(log(1/2 + I*z)^(x + 1)/(exp(-Pi*z) + exp(Pi*z))^2, z=0..64): Sti(1/2): Re(evalf(%, 100)); # Note that this is an approximation which needs a larger domain of integration and higher precision if used for more values than are in the Data section.
%Y A301816 Sti(0) = A001620 (Euler's constant gamma) (cf. A262235/A075266),
%Y A301816 Sti(1/2) = A301816,
%Y A301816 Sti(1) = A082633 (Stieltjes constant gamma_1) (cf. A262382/A262383),
%Y A301816 Sti(3/2) = A301817,
%Y A301816 Sti(2) = A086279 (Stieltjes constant gamma_2) (cf. A262384/A262385),
%Y A301816 Sti(3) = A086280 (Stieltjes constant gamma_3) (cf. A262386/A262387),
%Y A301816 Sti(4) = A086281, Sti(5) = A086282, Sti(6) = A183141, Sti(7) = A183167,
%Y A301816 Sti(8) = A183206, Sti(9) = A184853, Sti(10) = A184854.
%K A301816 nonn,cons
%O A301816 0,1
%A A301816 _Peter Luschny_, Apr 09 2018