Approximation of Sondow’s generalized-Euler-constant function on the interval [−1, 1]
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Abstract
Using the Euler-Maclaurin (Boole/Hermite) summation formula, the generalized-Euler-Sondow-constant function γ(z),
where \({\gamma(-1)=\ln\frac{4}{\pi}}\) and γ(1) is the Euler-Mascheroni constant, is estimated accurately.
$$ \gamma(z):=\sum_{k=1}^{\infty}z^{k-1}\left(\frac{1}{k}-\ln\frac{k+1}{k}\right) \qquad (-1\le z\le 1),$$
Keywords
Alternating Euler constant Approximation Estimate Euler constant Generalized-Euler-Sondow-constant function Inequality SeriesMathematics Subject Classification (2000)
Primary 33E20 33F05 65D20 Secondary 11Y60 40A05 40A25 40A30 65B15Preview
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Rferences
- 1.Abramowitz M., Stegun I.A.: Handbook of Mathematical Functions, 9th edn. Dover, New York (1972)MATHGoogle Scholar
- 2.Brass H.: Quadraturverfahren. Vandenhoeck & Ruprecht, Göttingen (1977)MATHGoogle Scholar
- 3.Lampret V.: An invitation to Hermite’s integration and summation: a comparison between Hermite’s and Simpson’s rules. SIAM Rev. 46, 311–328 (2004)MATHCrossRefMathSciNetGoogle Scholar
- 4.Lampret V.: Constructing the Euler-Maclaurin formula—celebrating Euler’s 300th birthday. Int. J. Math. Stat. 1, 60–85 (2007)MATHMathSciNetGoogle Scholar
- 5.Pilehrood H.K., Pilehrood H.T.: Arithmetical properties of some series with logarithmic coefficients. Math. Z. 255(1), 117–131 (2007)MATHCrossRefMathSciNetGoogle Scholar
- 6.Sondow J., Hadjicostas P.: The generalized-Euler-constant function γ(z) and a generalization of Somos’s quadratic recurrence constant. J. Math. Anal. Appl. 332(1), 292–314 (2007)MATHCrossRefMathSciNetGoogle Scholar
- 7.Wolfram, S.: Mathematica, Version 6.0. Wolfram Research, Inc. (1988–2008)Google Scholar
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