Archiv der Mathematik

, Volume 93, Issue 1, pp 37–45 | Cite as

An ultimate extremely accurate formula for approximation of the factorial function

Article
First Online:
Received:

Abstract

We prove in this paper that for every x ≥ 0,
$$\sqrt{2\pi e}\cdot e^{-\omega}\left( \frac{x+\omega}{e}\right) ^{x+\frac {1}{2}} < \Gamma(x+1)\leq\alpha\cdot\sqrt{2\pi e}\cdot e^{-\omega}\left( \frac{x+\omega}{e}\right)^{x+\frac{1}{2}}$$
where \({\omega=(3-\sqrt{3})/6}\) and α = 1.072042464..., then
$$\beta\cdot\sqrt{2\pi e}\cdot e^{-\zeta}\left(\frac{x+\zeta}{e}\right)^{x+\frac{1}{2}}\leq\Gamma(x+1) < \sqrt{2\pi e}\cdot e^{-\zeta}\left( \frac{x+\zeta}{e}\right)^{x+\frac{1}{2}},$$
where \({\zeta=(3+\sqrt{3})/6}\) and β = 0.988503589... Besides the simplicity, our new formulas are very accurate, if we take into account that they are much stronger than Burnside’s formula, which is considered one of the best approximation formulas ever known having a simple form.

Mathematics Subject Classification (2000)

Primary: 40A25 Secondary 26D07 

Keywords

Factorial function Gamma function Digamma function Numeric series Stirling’s formula Burnside’s formula and inequalities 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Andrews G., Askey R., Roy R.: Special functions, Encyclopedia of Mathematics and Its Applications 71. Cambridge University Press, London (1999)Google Scholar
  2. 2.
    Batir N.: Sharp inequalities for factorial n. Proyecciones 27, 97–102 (2008)MathSciNetGoogle Scholar
  3. 3.
    Batir N.: Inequalities for the gamma function. Arch. Math. 91, 554–563 (2008)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Burnside W.: A rapidly convergent series for log N!. Messenger Math. 46, 157–159 (1917)Google Scholar
  5. 5.
    Hsu L.C.: A new constructive proof of the Stirling formula. J. Math. Res. Exposition 17, 5–7 (1997)MATHMathSciNetGoogle Scholar
  6. 6.
    J. O’Connor and E. F. Robertson, James Stirling, MacTutor History of Mathematics Archive.Google Scholar
  7. 7.
    Sandor J., Debnath L.: On certain inequalities involving the constant e and their applications, J. Math. Anal. Appl. 249, 569–582 (2000)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Schuster W.: Improving Stirling’s formula. Arch. Math. 77, 170–176 (2001)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Srivastava H.M., Choi J.: Series Associated with the Zeta and Related Functions. Kluwer, Boston (2001)MATHGoogle Scholar
  10. 10.
    Y. Weissman, An improved analytical approximation to n!, Amer. J. Phys. 51 (1983).Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Sciences and ArtsValahia University of TârgovişteTârgovişteRomania

Personalised recommendations