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Comparison of Stirling's approximation with the factorial
This line integral can then be approximated using the saddle point method with an appropriate choice of countour 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
The relative error in a truncated Stirling series vs. n, 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):
An explicit formula for the coefficients in this series was given by G. Nemes.[a] The first graph in this section shows the relative error vs. n, for 1 through all 5 terms listed above.
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 n there are only so many terms of the series that improve accuracy, after which point accuracy actually gets worse. 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
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 Bn is the nth 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 z large enough in absolute value when || < ? - ?, where ? is positive, with an error term of O(z-2N+ 1). The corresponding approximation may now be written:
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 + it.
For any positive integer N, we introduce the following notation:
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.
Gerg? Nemes proposed in 2007 an approximation which gives the same number of exact digits as the Windschitl approximation but is much simpler:
^ abLe Cam, L. (1986), "The central limit theorem around 1935", Statistical Science, 1 (1): 78-96 [p. 81], doi:10.1214/ss/1177013818, MR0833276, The result, obtained using a formula originally proved by de Moivre but now called Stirling's formula, occurs in his `Doctrine of Chances' of 1733..[unreliable source?]
^ abPearson, Karl (1924), "Historical note on the origin of the normal curve of errors", Biometrika, 16 (3/4): 402-404 [p. 403], doi:10.2307/2331714, JSTOR2331714, I consider that the fact that Stirling showed that De Moivre's arithmetical constant was does not entitle him to claim the theorem, [...]
^Phillipe Flajolet and Robert Sedgewick, Analytic Combinatorics, p. 555
^Karatsuba, Ekatherina (2001), "On the asymptotic representation of the Euler gamma function by Ramanujan", Journal of Computational and Applied Mathematics, 135 (2): 225-240, doi:10.1016/S0377-0427(00)00586-0
^Mortici, Cristinel (2011), "Ramanujan's estimate for the gamma function via monotonicity arguments", Ramanujan J., 25: 149-154
^Mortici, Cristinel (2011), "Improved asymptotic formulas for the gamma function", Comput. Math. Appl., 61: 3364-3369
^Mortici, Cristinel (2011), "On Ramanujan's large argument formula for the gamma function", Ramanujan J., 26: 185-192