A254327 Decimal expansion of gamma_1(1/2), the first generalized Stieltjes constant at 1/2 (negated).
1, 3, 5, 3, 4, 5, 9, 6, 8, 0, 8, 0, 4, 9, 4, 1, 5, 1, 7, 7, 0, 8, 6, 8, 7, 1, 6, 9, 1, 7, 8, 0, 6, 4, 4, 0, 3, 5, 9, 1, 2, 8, 6, 2, 8, 9, 0, 3, 6, 3, 4, 6, 6, 1, 1, 6, 7, 4, 3, 8, 3, 8, 8, 6, 2, 6, 8, 0, 4, 6, 2, 0, 2, 4, 5, 9, 2, 3, 8, 4, 3, 8, 5, 9, 7, 0, 9, 3, 5, 2, 3, 1, 9, 6, 7, 9, 0, 3, 7, 3, 0, 5, 8, 7, 7
Offset: 1
Examples
-1.3534596808049415177086871691780644035912862890363466...
Links
- G. C. Greubel, Table of n, a(n) for n = 1..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 MathWorld, Hurwitz Zeta Function.
- Eric Weisstein's MathWorld, Stieltjes Constants.
- Wikipedia, Stieltjes constants.
Crossrefs
Programs
-
Maple
evalf(int((coth(Pi*x)-1)*(-2*arctan(2*x)+2*x*log(1/4+x^2))/(1+4*x^2), x = 0..infinity) - log(2) - (1/2)*log(2)^2, 120); # Vaclav Kotesovec, Jan 28 2015 evalf(gamma(1) - log(2)^2 - 2*gamma*log(2), 120); # Vaclav Kotesovec, Jan 29 2015 (faster)
-
Mathematica
gamma1[1/2] = StieltjesGamma[1] - Log[2]^2 - 2*EulerGamma*Log[2]; RealDigits[ gamma1[1/2], 10, 105] // First (* = StieltjesGamma[1, 1/2] expanded *)
-
PARI
Stieltjes(n)=my(a=log(2)); a^n/(n+1)*sumalt(k=1, (-1)^k/k*bernpol(n+1, log(k)/a)) Stieltjes(1)-log(2)^2-2*Euler*log(2) \\ Charles R Greathouse IV, Feb 04 2025
Formula
Equals Gamma(1) - log(2)^2 - 2*gamma*log(2).
Equals Integral_{0..oo} (coth(Pi*x)-1) * (-2*arctan(2*x) + 2*x*log(1/4+x^2)) / (1+4*x^2) dx - log(2) - log(2)^2/2.