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Mathematical Communications, Vol. 16 No. 1, 2011.

Original scientific paper

Improving Stirling's formula

Necdet Batir orcid id orcid.org/0000-0003-0125-497X ; Department of Mathematics, Faculty of Arts and Sciences, Nevşehir University, Nevşehir, Turkey

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page 105-114

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Abstract

We calculated the optimal values of the real parameters $a$ and $b$ in such a way that the asymptotic formula
\[
n!\sim e^{-a}\left(\frac{n+a}{e}\right)^n \sqrt{2\pi(n+b)}\,\,\,(as\,\,\, n\to \infty)
\]
gives the best accurate values for $n!$. Our estimations improve the classical Stirling and Burnside's formulas and their several recent improvements due to the author and C. Mortici. Apart from their simplicities and beauties our formulas give very accurate values for factorial $n$. Also, our results lead to new upper and lower bounds for the gamma function and recover some published inequalities for the gamma function.

Keywords

Stirling formula; Burnside's formula; gamma function; digamma function; inequalities

Hrčak ID:

68627

URI

https://hrcak.srce.hr/68627

Publication date:

10.6.2011.

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