Mathematical Tools for Physics

2—Infinite Series 41

2.6 Stirling’s Approximation
The Gamma function for positive integers is a factorial. A clever use of infinite series and Gaussian integrals
provides a useful approximate value for the factorial of largen.

n!∼

√

2 πnnne−n for largen (15)

Start from the Gamma function ofn+ 1.

n! = Γ(n+ 1) =

∫∞

0

dttne−t=

∫∞

0

dte−t+nlnt

The integrand starts at zero, increases, and drops back down to zero ast→∞. The graph roughly resembles a
Gaussian, and I can make this more precise by expanding the exponent around the point where it is a maximum.
The largest contribution to the whole integral comes from the region near this point. Differentiate the exponent
to find the maximum:

t

t=n= 5

t^5 e−t

21. 06

d

dt

(

−t+nlnt

)

=−1 +

n

t

= 0 gives t=n

Expand about this point

f(t) =−t+nlnt=f(n) + (t−n)f′(n) + (t−n)^2 f′′(n)/2 +···

=−n+nlnn+ 0 + (t−n)^2 (−n/n^2 )/2 +···

Keep terms only to the second order and the integral is approximately

n!∼

∫∞

0

dte−n+nlnn−(t−n)

(^2) / 2 n
=nne−n

∫∞

0

dte−(t−n)

(^2) / 2 n
(16)