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_{x>=0} (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).
Equals (-EulerGamma*Pi + 6*EulerGamma*log(2) + 7*log(2)^2 + 5*Pi*log(Pi) - 4*Pi*log(Gamma(1/4)) - 8*Pi*log(Gamma(3/4)) - 2*StieltjesGamma(1))/2. - Artur Jasinski, Mar 07 2026
EXAMPLE
-0.39129890240454977423987419218929637145038973196714...
MAPLE
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
MATHEMATICA
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
(* Or, from Mma version 7 up: *) RealDigits[StieltjesGamma[1, 3/4], 10, 103] // First
(* Or *) RealDigits[(-EulerGamma Pi + 6 EulerGamma Log[2] + 7 Log[2]^2 + 5 Pi Log[Pi] - 4 Pi Log[Gamma[1/4]] - 8 Pi Log[Gamma[3/4]] - 2 StieltjesGamma[1])/2, 10, 105] (* Artur Jasinski, Mar 07 2026 *)
CROSSREFS
KEYWORD
AUTHOR
Jean-François Alcover, Jan 29 2015
STATUS
approved
