Stirling's approximation
approximation for factorials
In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of
n
{\displaystyle n}
. It is named after James Stirling, though a related but less precise result was first stated by Abraham de Moivre.
One way of stating the approximation involves the logarithm of the factorial:
ln
n
!
=
n
ln
n
−
n
+
O
(
ln
n
)
,
{\displaystyle \ln n!=n\ln n-n+O(\ln n),}
where the big O notation means that, for all sufficiently large values of
n
{\displaystyle n}
, the difference between
ln
n
!
{\displaystyle \ln n!}
and
n
ln
n
−
n
{\displaystyle n\ln n-n}
will be at most proportional to the logarithm of
n
{\displaystyle n}
. In computer science applications such as the worst-case lower bound for comparison sorting, it is convenient to instead use the binary logarithm, giving the equivalent form
log
2
n
!
=
n
log
2
n
−
n
log
2
e
+
O
(
log
2
n
)
.
{\displaystyle \log _{2}n!=n\log _{2}n-n\log _{2}e+O(\log _{2}n).}
The error term in either base can be expressed more precisely as
1
2
log
(
2
π
n
)
+
O
(
1
n
)
{\displaystyle {\tfrac {1}{2}}\log(2\pi n)+O({\tfrac {1}{n}})}
, corresponding to an approximate formula for the factorial itself,
n
!
∼
2
π
n
(
n
e
)
n
.
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