In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of . 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:
where the big O notation means that, for all sufficiently large values of , the difference between and will be at most proportional to the logarithm. 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 The error term in either base can be expressed more precisely as , corresponding to an approximate formula for the factorial itself,
Here the sign means that the two quantities are asymptotic, that is, that their ratio tends to 1 as tends to infinity. The following version of the bound holds for all , rather than only asymptotically:
Derivation
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. Instead of approximating , one considers its natural logarithm, as this is a slowly varying function:
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 is a Bernoulli number, and is the remainder term in the Euler–Maclaurin formula. Take limits to find that
Denote this limit as . Because the remainder 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 , one obtains a formula involving an unknown quantity . For , the formula is
The quantity can be found by taking the limit on both sides as tends to infinity and using Wallis' product, which shows that . Therefore, one obtains Stirling's formula:
Alternative derivations
An alternative formula for using the gamma function is
(as can be seen by repeated integration by parts). Rewriting and changing variables , one obtains
Applying Laplace's method one has
which recovers Stirling's formula:
Higher orders
In fact, further corrections can also be obtained using Laplace's method. From previous result, we know that , so we "peel off" this dominant term, then perform a change of variables, to obtain:Now the function is unimodal, with maximum value zero. Locally around zero, it looks like , which is why we are able to perform Laplace's method. In order to extend Laplace's method to higher orders, we perform another change of variables by . This equation cannot be solved in closed form, but it can be solved by serial expansion, which gives us . Now plug back to the equation to obtainnotice how we don't need to actually find , since it is cancelled out by the integral. Higher orders can be achieved by computing more terms in .
Thus we get Stirling's formula to two orders:
Complex-analytic version
A complex-analysis version of this method is to consider as a Taylor coefficient of the exponential function , computed by Cauchy's integral formula as
This line integral can then be approximated using the saddle-point method with an appropriate choice of contour radius . The dominant portion of the integral near the saddle point is then approximated by a real integral and Laplace's method, while the remaining portion of the integral can be bounded above to give an error term.
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. Further terms are listed in the On-Line Encyclopedia of Integer Sequences as and . The first graph in this section shows the relative error vs. , for 1 through all 5 terms listed above. (Bender and Orszag p. 218) gives the asymptotic formula for the coefficients:which shows that it grows superexponentially, and that by ratio test the radius of convergence is zero.