This approximation is useful in the aspect of calculating factorials for large numbers. In fields like thermodynamics and statistical mechanics, it’s essential to assume a large number of molecules,
n!log2n!lnn!≈2πn(en)n≈nlog2n−nlog2e≈nlnn−n
and a more simplified expression like the one above is also used. The proof below is somewhat less rigorous analytically but is sufficient if one only emphasizes the facts for application.
When n is sufficiently large, the terms following 2n(x−n)2 can be ignored as the denominator increases too rapidly. Thus, the result below is obtained.
f(x)≈nlnn−n−2n(x−n)2
Furthermore, if n is significantly large, the integration of the Gaussian curve from 0 to ∞, and one from −∞ to ∞, have no significant difference. Consequently, for a sufficiently large n,
n!=enlnn−n∫0∞e−(x−n)2/2n+⋯dx≈enlnn−n∫−∞∞e−(x−n)2/2ndx
by Gaussian integration, ∫−∞∞e−(x−n)2/2ndx=2πn holds, leading to the following formula.
n!≈enlnn−n2πn
Taking the logarithm of both sides yields the formula below.
lnn!≈nlnn−n+21ln2πn
If n is very large, then the following approximation holds.
lnn!≈nlnn−n
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Stephen J. Blundell and Katherine M. Blundell, 열 물리학(Concepts in Thermal Physics, 이재우 역) (2nd Edition, 2014), p12 ↩︎
Stephen J. Blundell and Katherine M. Blundell, 열 물리학(Concepts in Thermal Physics, 이재우 역) (2nd Edition, 2014), p591-593 ↩︎