Stirling's approximation: Difference between revisions

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:<math>\ln N! \sim \int_1^N \ln x\,dx \sim N \ln N -N .</math>
:<math>\ln N! \sim \int_1^N \ln x\,dx \sim N \ln N -N .</math>
==Applications in statistical mechanics==
*[[Ideal gas Helmholtz energy function]]
[[Category: Mathematics]]
[[Category: Mathematics]]

Revision as of 11:03, 7 July 2008

James Stirling (1692-1770, Scotland)

Because of Euler-MacLaurin formula

where B1 = −1/2, B2 = 1/6, B3 = 0, B4 = −1/30, B5 = 0, B6 = 1/42, B7 = 0, B8 = −1/30, ... are the Bernoulli numbers, and R is an error term which is normally small for suitable values of p.

Then, for large N,

Applications in statistical mechanics