OFFSET
0,1
COMMENTS
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).
LINKS
Iaroslav V. Blagouchine, A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations, Journal of Number Theory, vol. 148, pp. 537-592 and vol. 151, pp. 276-277, 2015. arXiv version, arXiv:1401.3724 [math.NT], 2014.
Peter Luschny, Illustration of the real Stieltjes gamma function.
FORMULA
c = -Re((4/3)*Pi*Integral_{-oo..oo} log(1/2+i*z)^(3/2)/(exp(-Pi*z)+exp(Pi*z))^2 dz).
EXAMPLE
0.2754347245639200799552878777978068357987023238863074873733211475133063441...
MAPLE
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.
KEYWORD
nonn,cons
AUTHOR
Peter Luschny, Apr 09 2018
STATUS
approved