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 . It is named after James Stirling, though a related but less precise result was first stated by Abraham de Moivre.[1][2][3]
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,
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
where 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 . 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 , one obtains a formula involving an unknown quantity . For m = 1, 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:
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 two changes of variables, to obtain:
To verify this: . 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 obtain
notice 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:
series=tau-tau^2/6+tau^3/36+tau^4*a+tau^5*b;(*pick the right a,b to make the series equal 0 at higher orders*)Series[tau^2/2+1+t-Exp[t]/.t->series,{tau,0,8}](*now do the integral*)integral=Integrate[Exp[-x*tau^2/2]*D[series/.a->0/.b->0,tau],{tau,-Infinity,Infinity}];Simplify[integral/Sqrt[2*Pi]*Sqrt[x]]
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):[5]
An explicit formula for the coefficients in this series was given by G. Nemes.[6] Further terms are listed in the On-Line Encyclopedia of Integer Sequences as A001163 and A001164. The first graph in this section shows the relative error vs. , for 1 through all 5 terms listed above. (Bender and Orszag[7] p.218) gives the asymptotic formula for the coefficients:
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,[8] 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.[9]
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.
Error bounds
For any positive integer , the following notation is introduced:
which converges when Re(x) > 0. Stirling's formula may also be given in convergent form as[13]
where
Versions suitable for calculators
The approximation
and its equivalent form
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.[14]
Gergő Nemes proposed in 2007 an approximation which gives the same number of exact digits as the Windschitl approximation but is much simpler:[15]
De Moivre gave an approximate rational-number expression for the natural logarithm of the constant. Stirling's contribution consisted of showing that the constant is precisely .[3]
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1 2 Le Cam, L. (1986), "The central limit theorem around 1935", Statistical Science, 1 (1): 78–96, doi:10.1214/ss/1177013818, JSTOR2245503, MR0833276 ; see p. 81, "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."
1 2 Pearson, 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, [...]
↑ Olver, F. W. J.; Olde Daalhuis, A. B.; Lozier, D. W.; Schneider, B. I.; Boisvert, R. F.; Clark, C. W.; Miller, B. R. & Saunders, B. V., "5.11 Gamma function properties: Asymptotic Expansions", NIST Digital Library of Mathematical Functions, Release 1.0.13 of 2016-09-16
↑ Nemes, Gergő (2010), "On the coefficients of the asymptotic expansion of ", Journal of Integer Sequences, 13 (6): 5
↑ Bender, Carl M.; Orszag, Steven A. (2009). Advanced mathematical methods for scientists and engineers. 1: Asymptotic methods and perturbation theory (Nachdr.ed.). New York, NY: Springer. ISBN978-0-387-98931-0.
↑ Robbins, Herbert (1955), "A Remark on Stirling's Formula", The American Mathematical Monthly, 62 (1): 26–29, doi:10.2307/2308012, JSTOR2308012
↑ Spiegel, M. R. (1999), Mathematical handbook of formulas and tables, McGraw-Hill, p.148
↑ Schäfke, F. W.; Sattler, A. (1990), "Restgliedabschätzungen für die Stirlingsche Reihe", Note di Matematica, 10 (suppl. 2): 453–470, MR1221957
↑ G. Nemes, Error bounds and exponential improvements for the asymptotic expansions of the gamma function and its reciprocal, Proc. Roy. Soc. Edinburgh Sect. A145 (2015), 571–596.
↑ Mortici, Cristinel (2011), "Ramanujan's estimate for the gamma function via monotonicity arguments", Ramanujan J., 25 (2): 149–154, doi:10.1007/s11139-010-9265-y, S2CID119530041
↑ Mortici, Cristinel (2011), "Improved asymptotic formulas for the gamma function", Comput. Math. Appl., 61 (11): 3364–3369, doi:10.1016/j.camwa.2011.04.036 .
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