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

1,2

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.

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.

Example

-1.3534596808049415177086871691780644035912862890363466...

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 *)

Crossrefs

Cf. A001620 (gamma), A082633 (gamma_1), A254331 (gamma_1(1/3)), A254345 (gamma_1(2/3)), A254347 (gamma_1(1/4)), A254348 (gamma_1(3/4)), A254349 (gamma_1(1/6)), A254350 (gamma_1(5/6)), A251866 (gamma_1(1/5)), A255188 (gamma_1(1/8)), A255189 (gamma_1(1/12)).

Sequence in context: A077973 A374624 A210606 * A175999 A236965 A259684

Adjacent sequences: A254324 A254325 A254326 * A254328 A254329 A254330

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Jan 28 2015

Status

approved