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# Difference between revisions of "Stirling's approximation"

Carl McBride (talk | contribs) m (Added applications section.) |
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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 *B*_{1} = −1/2, *B*_{2} = 1/6, *B*_{3} = 0, *B*_{4} = −1/30, *B*_{5} = 0, *B*_{6} = 1/42, *B*_{7} = 0, *B*_{8} = −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*,