Stirling approximation

A useful approximation for $N!$ is provided by the Stirling formula:

$$\ln N! = N \ln N - N + \ln \sqrt{2 \pi N} + o\left (\frac{1}{N} \right)$$ (15.1)

For large values of $N$, a sufficiently accurate formula is also obtained by neglecting the term $\ln \sqrt{2 \pi N}$ in , which leads to the familiar expression used in statistical physics:

$$\ln N \simeq N \ln N - N.$$ (15.2)

The origin of this expression can be understood by expressing:

$$\ln N! = \sum_{n=1}^N \ln n.$$ (15.3)

The value of this sum is the shaded area in Fig. , which can be approximated by the integral of $\ln x$:

$$\int_1^N dx \ln x = N \ln N - N + 1.$$ (15.4)

Figure: Approximation of $\ln N!$ (shaded area) by the integral of $\ln x$ between 1 and $N$.

The Stirling formula  is quite accurate also for relatively small values of N. The error in Eq.  is of order $\ln N$, which becomes completely negligible when compared to $N \ln N - N$ for $N \simeq 10^{23}$.

	(r)1-2 $N$	$\ln N!$	$N \ln N - N + \ln \sqrt{2 \pi N}$	$N \ln N - N$
	2	0.6931	0.6518
	3	1.7918	1.7641
	4	3.1781	3.1573
	$\dots$	$\dots$	$\dots$
	10	15.1044	15.0961	13.0258
	$\dots$	$\dots$	$\dots$
	100	363.7394	363.7385	360.5170