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A254327 Decimal expansion of gamma_1(1/2), the first generalized Stieltjes constant at 1/2 (negated). 10
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
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.
Eric Weisstein's MathWorld, Hurwitz Zeta Function.
Eric Weisstein's MathWorld, 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: A077950 A077973 A210606 * A175999 A236965 A259684
KEYWORD
nonn,cons,easy
AUTHOR
STATUS
approved

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Last modified July 20 15:46 EDT 2024. Contains 374459 sequences. (Running on oeis4.)