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.
Comparison of Stirling's approximation with the factorial
One way of stating the approximation involves the logarithm of the factorial:
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 estimatesEdit
The relative error in a truncated Stirling series vs. , for 0 to 5 terms. The kinks in the curves represent points where the truncated series coincides with Γ(n + 1).
Stirling's formula is in fact the first approximation to the following series (now called the Stirling series):
The relative error in a truncated Stirling series vs. the number of terms used
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 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 terms evaluated at . 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 are
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 is the th Bernoulli number (note that the limit of the sum as is not convergent, so this formula is just an asymptotic expansion). The formula is valid for large enough in absolute value, when |arg(z)| < π − ε, where ε is positive, with an error term of O(z−2N+ 1). The corresponding approximation may now be written:
where the expansion is identical to that of Stirling's series above for , except that 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 , the following notation is introduced:
can be obtained by rearranging Stirling's extended formula and observing a coincidence between the resultant power series and the Taylor series expansion of the hyperbolic sine function. 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:
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