A082633 Decimal expansion of the 1st Stieltjes constant gamma_1 (negated).
0, 7, 2, 8, 1, 5, 8, 4, 5, 4, 8, 3, 6, 7, 6, 7, 2, 4, 8, 6, 0, 5, 8, 6, 3, 7, 5, 8, 7, 4, 9, 0, 1, 3, 1, 9, 1, 3, 7, 7, 3, 6, 3, 3, 8, 3, 3, 4, 3, 3, 7, 9, 5, 2, 5, 9, 9, 0, 0, 6, 5, 5, 9, 7, 4, 1, 4, 0, 1, 4, 3, 3, 5, 7, 1, 5, 1, 1, 4, 8, 4, 8, 7, 8, 0, 8, 6, 9, 2, 8, 2, 4, 4, 8, 4, 4, 0, 1, 4, 6, 0, 4
Offset: 0
Examples
-0.0728158454836767248605863758749...
References
- Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 166.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..2500
- Dorin Andrica and Eugen J. Ionascu, On the number of polynomials with coefficients in [n], An. St. Univ. Ovidius Constanta, Volume XXII (2014), fascicola 1.
- G. H. Hardy, Note on Dr. Vacca's series for gamma, Quart. J. Pure Appl. Math., Vol. 43 (1912), pp. 215-216. [Available only in the USA]
- Krzysztof Maślanka and Andrzej Koleżyński, The High Precision Numerical Calculation of Stieltjes Constants. Simple and Fast Algorithm, arXiv preprint, arXiv:2210.04609 [math.NT], 2022.
- Marc Prévost, Expansion of generalized Stieltjes constants in terms of derivatives of Hurwitz zeta-functions, arXiv:2305.15806 [math.NA], 2023.
- Sandeep Tyagi, High precision computation and a new asymptotic formula for the generalized Stieltjes constants, arXiv preprint, arXiv:2212.07956 [math.NA], 2022.
- Eric Weisstein's World of Mathematics, Stieltjes Constants.
- Wikipedia, Stieltjes constants.
Crossrefs
Programs
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Maple
evalf(gamma(1)) ; # R. J. Mathar, Sep 15 2013
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Mathematica
Prepend[RealDigits[c=N[StieltjesGamma[1], 120], 10][[1]], 0] N[EulerGamma^2 - Residue[Zeta[s]^3, {s, 1}]/3, 100] (* Vaclav Kotesovec, Jan 07 2017 *) -
PARI
intnum(x=0,oo,(1/tanh(Pi*x)-1)*(x*log(1+x^2)-2*atan(x))/(2*(1+x^2))) \\ Charles R Greathouse IV, Mar 10 2016
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PARI
Stieltjes(n)=my(a=log(2)); a^n/(n+1)*sumalt(k=1,(-1)^k/k*subst(bernpol(n+1),'x,log(k)/a)) Stieltjes(1) \\ Charles R Greathouse IV, Feb 23 2022
Formula
Equals lim_{y->infinity} y*(Im(zeta(1+i/y))+y).
Equals lim_{n->infinity} (((log(n))^2)/2 - Sum_{k=2..n} (log(k))/k). - Warut Roonguthai, Aug 04 2005
Equals Integral_{0..infinity} (coth(Pi*x)-1)*(x*log(1+x^2)-2*arctan(x))/(2*(1+x^2)) dx. - Jean-François Alcover, Jan 28 2015
Using the abbreviations a = log(z^2 + 1/4)/2, b = arctan(2*z) and c = cosh(Pi*z) then gamma_1 = -(Pi/2)*Integral_{0..infinity} (a^2 - b^2)/c^2. The general case is for n >= 0 (which includes Euler's gamma as gamma_0) gamma_n = -(Pi/(n+1))* Integral_{0..infinity} sigma(n+1)/c^2, where sigma(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n,2*k)*b^(2*k)*a^(n-2*k). - Peter Luschny, Apr 19 2018
Equals log(2)^2/6 - log(2)*gamma/2 + (1/(2*log(2))) * Sum_{k>=1} (-1)^k * log(k)^2/k, where gamma is Euler's constant (A001620) (Hardy, 1912). - Amiram Eldar, Jun 09 2023
Equals Sum_{j>=1} Zeta'(2*j + 1) / (2*j + 1). - Peter Luschny, Jun 16 2023
Extensions
More terms from Eric W. Weisstein, Jul 14 2003
Comments