C.2 Improved Stirling Approximation to n!
627
n! !
√
n
n
e
n ∞
−
√
n
e
−y 2 /2 dy .
The same arguments that allow us to make a Gaussian approximation to I (y) allow
us to extend the lower limit of our integral from −
√
n to −∞, so that
n! !
√
n
n
e
n ∞
−∞
e
−y 2 /2 dy ,
or
n! !
√
2πn
n
e
n
,
(C.2.5)
as the definite integral has the value
√
2π . Further improvements to the approximation for n!, and comments on them have been given by Mermin. 1
1 N. D. Mermin, Amer. J. Phys. 52, 362 (1984).
627
n! !
√
n
n
e
n ∞
−
√
n
e
−y 2 /2 dy .
The same arguments that allow us to make a Gaussian approximation to I (y) allow
us to extend the lower limit of our integral from −
√
n to −∞, so that
n! !
√
n
n
e
n ∞
−∞
e
−y 2 /2 dy ,
or
n! !
√
2πn
n
e
n
,
(C.2.5)
as the definite integral has the value
√
2π . Further improvements to the approximation for n!, and comments on them have been given by Mermin. 1
1 N. D. Mermin, Amer. J. Phys. 52, 362 (1984).