# 19.4: Stirling's Approximation

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The polynomial coefficient, $$C$$, is a function of the factorials of large numbers. Since $$N!$$ quickly becomes very large as $$N$$ increases, it is often impractical to evaluate $$N!$$ from the definition,

$N!=\left(N\right)\left(N-1\right)\left(N-2\right)\dots \left(3\right)\left(2\right)\left(1\right) \nonumber$

Fortunately, an approximation, known as Stirling’s formula or Stirling’s approximation is available. Stirling’s approximation is a product of factors. Depending on the application and the required accuracy, one or two of these factors can often be taken as unity. Stirling’s approximation is

$N!\approx N^N \left(2\pi N\right)^{1/2}\mathrm{exp}\left(-N\right)\mathrm{exp}\left(\frac{1}{12N}\right)\approx N^N\left(2\pi N\right)^{1/2}\mathrm{exp}\left(-N\right)\approx N^N\mathrm{exp}\left(-N\right) \nonumber$

In many statistical thermodynamic arguments, the important quantity is the natural logarithm of $$N!$$ or its derivative, $${d ~ { \ln N!\ }}/{dN}$$. In such cases, the last version of Stirling’s approximation is usually adequate, even though it affords a rather poor approximation for $$N!$$ itself.

This page titled 19.4: Stirling's Approximation is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.