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## Open Mathematics

2010 | 8 | 3 | 488-499
Tytuł artykułu

### An accurate approximation of zeta-generalized-Euler-constant functions

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EN
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EN
Zeta-generalized-Euler-constant functions, $$\gamma \left( s \right): = \sum\limits_{k = 1}^\infty {\left( {\frac{1} {{k^s }} - \int_k^{k + 1} {\frac{{dx}} {{x^s }}} } \right)}$$ and $$\tilde \gamma \left( s \right): = \sum\limits_{k = 1}^\infty {\left( { - 1} \right)^{k + 1} \left( {\frac{1} {{k^s }} - \int_k^{k + 1} {\frac{{dx}} {{x^s }}} } \right)}$$ defined on the closed interval [0, ∞), where γ(1) is the Euler-Mascheroni constant and $$\tilde \gamma$$(1) = ln $$\frac{4} {\pi }$$, are studied and estimated with high accuracy.
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EN
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Tom
Numer
Strony
488-499
Opis fizyczny
Daty
wydano
2010-06-01
online
2010-05-30
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autor
Bibliografia
• [1] Abramowitz M., Stegun I. A., Handbook of Mathematical Functions, 9th ed., Dover Publications, N.Y., 1972
• [2] Lampret V., The Euler-Maclaurin and Taylor Formulas: Twin, Elementary Derivations, Math. Mag., 2001, 74, 109–122
• [3] Lampret V., Constructing the Euler-Maclaurin formula - celebrating Euler’s 300th birthday, Int. J. Math. Stat., 2007, 1, 60–85
• [4] Lampret V., Approximating real Pochhammer products: A comparison with powers, Cent. Eur. J. Math., 2009, 7, 493–505 http://dx.doi.org/10.2478/s11533-009-0036-1
• [5] Sîntămărian A., A generalization of Euler’s constant, Numer. Algorithms, 2007, 46, 141–151 http://dx.doi.org/10.1007/s11075-007-9132-0
• [6] Sîntămărian A., Some inequalities regarding a generalization of Euler’s constant, J. Inequal. Pure Appl. Math., 2008, 9(2), 46
• [7] Sondow J., Double integrals for Euler’s constant and ln $$\frac{4} {\pi }$$ and an analog of Hadjicosta’s formula, Amer. Math. Monthly, 2005, 112, 61–65 http://dx.doi.org/10.2307/30037385
• [8] Sondow J., Hadjicostas P., The generalized-Euler-constant function γ(z) and a generalization of Somos’s quadratic recurrence constant, J. Math. Anal. Appl., 2007, 332, 292–314 http://dx.doi.org/10.1016/j.jmaa.2006.09.081
• [9] Wolfram S., Mathematica, Version 6.0, Wolfram Research, Inc., 1988–2008
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