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

• Autorzy: Vito Lampret
• Języki publikacji: EN

Abstrakty

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.

Słowa kluczowe

EN

Alternating; Convergence acceleration; Estimate; Generalized-Euler-constant-function; Inequality; Series; Zeta

Kategorie tematyczne

• 65B15: Euler-Maclaurin formula
• 40A05: Convergence and divergence of series and sequences
• 33F05: Numerical approximation and evaluation
• 11Y60: Evaluation of constants
• 33E20: Other functions defined by series and integrals
• 40A25: Approximation to limiting values (summation of series, etc.)

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2010
• Tom: 8
• Numer: 3
• Strony: 488-499

Daty

wydano

2010-06-01

online

2010-05-30

Twórcy

autor

Vito Lampret

• University of Ljubljana

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

Identyfikatory

DOI

10.2478/s11533-010-0030-7