The version of the formula typically used in applications is
Specifying the constant in the O(ln n) error term gives 1/2ln(2πn), yielding the more precise formula:
where the sign ~ means that the two quantities are asymptotic: their ratio tends to 1 as n tends to infinity.
One may also give simple bounds valid for all positive integers n, rather than only for large n:
Roughly speaking, the simplest version of Stirling's formula can be quickly obtained by approximating the sum
with an integral:
The full formula, together with precise estimates of its error, can be derived as follows.
The right-hand side of this equation minus
is the approximation by the trapezoid rule of the integral
and the error in this approximation is given by the Euler–Maclaurin formula:
where Bk is a Bernoulli number, and Rm,n is the remainder term in the Euler–Maclaurin formula.
Take limits to find that
Denote this limit as y.
Because the remainder Rm,n in the Euler–Maclaurin formula satisfies
where big-O notation is used, combining the equations above yields the approximation formula in its logarithmic form:
Taking the exponential of both sides and choosing any positive integer m, one obtains a formula involving an unknown quantity e. For m = 1, the formula is
The quantity e can be found by taking the limit on both sides as n tends to infinity and using Wallis' product, which shows that e = √2π.
Therefore, one obtains Stirling's formula:
An alternative derivation
An alternative formula for n!
using the gamma function is
(as can be seen by repeated integration by parts).
Rewriting and changing variables x = ny, one obtains
Applying Laplace's method one has
which recovers Stirling's formula:
In fact, further corrections can also be obtained using Laplace's method.
For example, computing two-order expansion using Laplace's method yields
and gives Stirling's formula to two orders:
Speed of convergence and error estimates
Stirling's formula is in fact the first approximation to the following series (now called the Stirling series):
An explicit formula for the coefficients in this series was given by G. Nemes.
The first graph in this section shows the relative error vs. n, for 1 through all 5 terms listed above.
As n → ∞, the error in the truncated series is asymptotically equal to the first omitted term.
This is an example of an asymptotic expansion.
It is not a convergent series; for any particular value of n there are only so many terms of the series that improve accuracy, after which accuracy worsens.
This is shown in the next graph, which shows the relative error versus the number of terms in the series, for larger numbers of terms.
More precisely, let S(n, t) be the Stirling series to t terms evaluated at n. The graphs show
which, when small, is essentially the relative error.
Writing Stirling's series in the form
it is known that the error in truncating the series is always of the opposite sign and at most the same magnitude as the first omitted term.
More precise bounds, due to Robbins, valid for all positive integers n are
Stirling's formula for the gamma function
For all positive integers,
where Γ denotes the gamma function.
However, the gamma function, unlike the factorial, is more broadly defined for all complex numbers other than non-positive integers; nevertheless, Stirling's formula may still be applied.
If Re(z) > 0, then
Repeated integration by parts gives
where the expansion is identical to that of Stirling' series above for n!, except that n is replaced with z-1.
A further application of this asymptotic expansion is for complex argument z with constant Re(z).
See for example the Stirling formula applied in Im(z) = t of the Riemann–Siegel theta function on the straight line 1/4 + it.
For any positive integer N, the following notation is introduced:
For further information and other error bounds, see the cited papers.
A convergent version of Stirling's formula
Obtaining a convergent version of Stirling's formula entails evaluating Raabe's formula:
One way to do this is by means of a convergent series of inverted rising exponentials.
where s(n, k) denotes the Stirling numbers of the first kind.
From this one obtains a version of Stirling's series
which converges when Re(x) > 0.
Versions suitable for calculators
and its equivalent form
This approximation is good to more than 8 decimal digits for z with a real part greater than 8.
Robert H. Windschitl suggested it in 2002 for computing the gamma function with fair accuracy on calculators with limited program or register memory.
Gergő Nemes proposed in 2007 an approximation which gives the same number of exact digits as the Windschitl approximation but is much simpler:
for x ≥ 0.
The equivalent approximation for ln n!
has an asymptotic error of 1/1400n and is given by
The approximation may be made precise by giving paired upper and lower bounds; one such inequality is
Estimating central effect in the binomial distribution
The formula was first discovered by Abraham de Moivre in the form
Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Stirling's approximation.