On the monotonicity and convexity of the remainder of the Stirling formula

The Stirling’s estimation to [Formula: see text](N!) is typically introduced to students as a step in the derivation of the statistical expression for the heat capacity. However, naïve application of this estimation leads to wrong conclusions. In this paper, firstly, the heat capacity of some semiconductor compounds was calculated using exponential Boltzmann distribution and compared with experimental data. It has shown a disagreement between experimental results and those calculated. Secondly, by applying the more exact Stirling formula, an analytical formulation of Boltzmann statistics using Lambert W function is shown to be a very good model and proves an excellent agreement between calculated and experimental data for heat capacity over the entire temperature range.

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Interpolation by means of Given Differences (Central-Difference Notation)

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500013717 ◽

1932 ◽

Vol 3 (1) ◽

pp. 1-5

Author(s):

W. F. Sheppard

Keyword(s):

Mean Difference ◽

Stirling Formula ◽

Central Difference ◽

Difference Formula ◽

Group A ◽

The Mean ◽

The Difference ◽

Mean Differences ◽

Group B

Formulae of interpolation in terms of given central differences might be regarded as falling into two groups, A and B. In group A, the simplest cases are those in which each given difference is one of the two which in the difference table lie nearest to the preceding given difference; the differences are all natural differences (i.e., are not mean differences), and are all expressed in the centraldifference notation. Any such formula can be a central-difference formula for a certain range of the variable: but that is a matter with which we are only incidentally concerned. What I have to do is to examine the formula as determined by the series of differences given. I have then to see how the formula is affected when an ordinary difference is replaced by a mean difference. This brings us to group B, which comprises two formulae only: the Newton-Stirling formula, which expresses the required quantity in terms of a tabulated value and its central differences; and the Newton-Bessel formula, which expresses it in terms of the mean of two tabulated values and the central differences of this mean.

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Best estimates of the generalized Stirling formula

Applied Mathematics and Computation ◽

10.1016/j.amc.2009.12.013 ◽

2010 ◽

Vol 215 (11) ◽

pp. 4044-4048 ◽

Cited By ~ 22

Author(s):

Cristinel Mortici

Keyword(s):

Stirling Formula ◽

Best Estimates

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New improvements of the Stirling formula

Applied Mathematics and Computation ◽

10.1016/j.amc.2010.06.007 ◽

2010 ◽

Vol 217 (2) ◽

pp. 699-704 ◽

Cited By ~ 20

Author(s):

Cristinel Mortici

Keyword(s):

Stirling Formula

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A generalization of Mortici lemma

Creative Mathematics and Informatics ◽

10.37193/cmi.2012.02.02 ◽

2012 ◽

Vol 21 (2) ◽

pp. 129-134

Author(s):

VASILE BERINDE ◽

Keyword(s):

Gamma Function ◽

Asymptotic Expansions ◽

Asymptotic Integration ◽

Stirling Formula ◽

Monotonic Functions ◽

Factorial Function ◽

Approximation Formulas ◽

Appl Math ◽

Best Estimates

The aim of this note is to obtain a generalization of a very simple, elegant but powerful convergence lemma introduced by Mortici [Mortici, C., Best estimates of the generalized Stirling formula, Appl. Math. Comp., 215 (2010), No. 11, 4044–4048; Mortici, C., Product approximations via asymptotic integration, Amer. Math. Monthly, 117 (2010), No. 5, 434–441; Mortici, C., An ultimate extremely accurate formula for approximation of the factorial function, Arch. Math. (Basel), 93 (2009), No. 1, 37–45; Mortici, C., Complete monotonic functions associated with gamma function and applications, Carpathian J. Math., 25 (2009), No. 2, 186–191] and exploited by him and other authors in an impressive number of recent and very recent papers devoted to constructing asymptotic expansions, accelerating famous sequences in mathematics, developing approximation formulas for factorials that improve various classical results etc. We illustrate the new result by some important particular cases and also indicate a way for using it in similar contexts.

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