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] (-12*arctan(3*x/2) + 9*x*log(4/9 + x^2))/((-1 + exp(2*Pi*x))*(4 + 9*x^2)) dx - (3/4 + (1/2)*log(3/2))*log(3/2).
EXAMPLE
-0.5989062842859892925678760212692502566639134078175714915865...
MAPLE
evalf(int((-12*arctan(3*x*(1/2))+9*x*log(4/9+x^2))/((-1+exp(2*Pi*x))*(9*x^2+4)), x = 0..infinity) - (3/4+(1/2)*log(3/2))*log(3/2), 120); # Vaclav Kotesovec, Jan 29 2015
MATHEMATICA
gamma1[2/3] = (1/12)*(Sqrt[3]*Pi*(2*EulerGamma + Log[(576*Pi^5)/Gamma[1/6]^6]) - 6*(EulerGamma*Log[27] + Log[3]*Log[18*Pi] - 2*StieltjesGamma[1] + Derivative[2, 0][Zeta][0, 1/3] + Derivative[2, 0][Zeta][0, 2/3])) // Re; RealDigits[gamma1[2/3], 10, 105] // First
(* Or, from Mma version 7 up: *) RealDigits[StieltjesGamma[1, 2/3], 10, 105] // First
CROSSREFS
KEYWORD
AUTHOR
Jean-François Alcover, Jan 29 2015
STATUS
approved