A254348 Decimal expansion of gamma_1(3/4), the first generalized Stieltjes constant at 3/4 (negated).
3, 9, 1, 2, 9, 8, 9, 0, 2, 4, 0, 4, 5, 4, 9, 7, 7, 4, 2, 3, 9, 8, 7, 4, 1, 9, 2, 1, 8, 9, 2, 9, 6, 3, 7, 1, 4, 5, 0, 3, 8, 9, 7, 3, 1, 9, 6, 7, 1, 4, 0, 7, 6, 6, 2, 7, 7, 3, 0, 7, 1, 0, 8, 6, 9, 7, 1, 7, 9, 3, 9, 5, 0, 6, 0, 4, 7, 1, 3, 3, 2, 6, 4, 3, 2, 7, 8, 2, 7, 5, 6, 2, 2, 1, 9, 7, 5, 8, 8, 1, 4, 7, 8
Offset: 0
Examples
-0.39129890240454977423987419218929637145038973196714...
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
Crossrefs
Programs
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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
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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
Formula
Equals integral_[0..infinity] (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).