C.2 Improved Stirling Approximation to n!
625
d
dn
ln n! =
1
m
ln[(n + m)(n + m − 1) · · · (n + 1)]
1
m
ln
n
m
= ln n
C.2 Improved Stirling Approximation to n!
Euler’s gamma function, denoted (x), is a continuous function that gives the values
for the discrete function n! for positive integer arguments. This function is defined
as
≡
∞
0
t
x−1 e
−t dt .
(C.2.1)
If we begin with + 1) =
∞
0
t
x e
−t dt and integrate by parts, we obtain +
1) = xx(x) for arbitrary values of x. If we now let x be n, we see that we can iterate
the integration-by-parts process to obtain
+ 1) = nn(n) = · · · = n(n − 1)(n − 2) · · · 2 · (1) ,
with
(1) =
∞
0
t
1−1 e
−t dt = 1 ,
so that
+ 1) = n! .
(C.2.2)
If n is very large, then the major contribution to the integral defining n! comes
from those values of t lying near t = n, and extending to either side of t = n by
±
√
n. By substituting t = n + y
√
n into the defining integral for n! we obtain
n! =
∞
0
t
n e
−t dt =
√
n
∞
−
√
n
e
−(n+y
√
n) (n + y
√
n)
n dy ,
which we can rewrite as
n! =
√
n
n
e
n ∞
−
√
n
e
−y
√
n
1 +
y
√
n
n
dy .
(C.2.3)