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A082633 Decimal expansion of the 1st Stieltjes constant gamma_1 (negated). 103
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
The Stieltjes constants are named after the Dutch mathematician Thomas Joannes Stieltjes (1856-1894). - Amiram Eldar, Jun 16 2021
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 166.
LINKS
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.
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.
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
EXAMPLE
-0.0728158454836767248605863758749...
MAPLE
evalf(gamma(1)) ; # R. J. Mathar, Sep 15 2013
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 *)
PROG
(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
(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
CROSSREFS
Sequence in context: A253383 A010506 A197845 * A121239 A348362 A201322
KEYWORD
cons,nonn
AUTHOR
Benoit Cloitre, May 24 2003
EXTENSIONS
More terms from Eric W. Weisstein, Jul 14 2003
STATUS
approved

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Last modified June 17 15:57 EDT 2024. Contains 373463 sequences. (Running on oeis4.)