Appendix C
Stirling Approximations
C.1 Lowest-Level Approximation to n!
We begin with the expression for n!, which is
n! = n(n − 1)(n − 2) · · · (2)(1) .
(C.1.1)
The natural logarithm of n! can be written (see Fig. C.1)
ln n! = ln n + ln(n − 1) + · · · + ln 2 + ln 1
=
n
m=1
ln m .
(C.1.2)
The spirit of the Stirling approximation is to replace the summation in the last
line of Eq. (C.1.2) by an integral, so that
ln n! !
n
1
dx ln x.
(C.1.3)
This integral can be integrated by parts, as in your first year calculus course, to give
ln n! =
x ln x − x
n
1
,
or
ln n! ! n ln n − n + 1 ,
(C.1.4)
© Springer Nature Switzerland AG 2021
F. R. W. McCourt, Statistical Thermodynamics for Pure and Applied Sciences,
https://doi.org/10.1007/978-3-030-52006-9
623
Stirling Approximations
C.1 Lowest-Level Approximation to n!
We begin with the expression for n!, which is
n! = n(n − 1)(n − 2) · · · (2)(1) .
(C.1.1)
The natural logarithm of n! can be written (see Fig. C.1)
ln n! = ln n + ln(n − 1) + · · · + ln 2 + ln 1
=
n
m=1
ln m .
(C.1.2)
The spirit of the Stirling approximation is to replace the summation in the last
line of Eq. (C.1.2) by an integral, so that
ln n! !
n
1
dx ln x.
(C.1.3)
This integral can be integrated by parts, as in your first year calculus course, to give
ln n! =
x ln x − x
n
1
,
or
ln n! ! n ln n − n + 1 ,
(C.1.4)
© Springer Nature Switzerland AG 2021
F. R. W. McCourt, Statistical Thermodynamics for Pure and Applied Sciences,
https://doi.org/10.1007/978-3-030-52006-9
623