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:
Derivation
Roughly speaking, the simplest version of Stirling's formula can be quickly obtained by approximating the sum
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
and the error in this approximation is given by the Euler–Maclaurin formula:
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:
Alternative derivations
An alternative formula for using the gamma function is
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 two changes 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 obtain
Thus we get Stirling's formula to two orders:
Complex-analytic version
A complex-analysis version of this method[4] 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):[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
Writing Stirling's series in the form
Other bounds, due to Robbins,[8] valid for all positive integers are
Stirling's formula for the gamma function
For all positive integers,
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:
For further information and other error bounds, see the cited papers.
A convergent version of Stirling's formula
Thomas Bayes showed, in a letter to John Canton published by the Royal Society in 1763, that Stirling's formula did not give a convergent series.[12] Obtaining a convergent version of Stirling's formula entails evaluating Binet's formula:
One way to do this is by means of a convergent series of inverted rising factorials. If
Versions suitable for calculators
The approximation
Gergő Nemes proposed in 2007 an approximation which gives the same number of exact digits as the Windschitl approximation but is much simpler:[15]
An alternative approximation for the gamma function stated by Srinivasa Ramanujan in Ramanujan's lost notebook[16] is
The approximation may be made precise by giving paired upper and lower bounds; one such inequality is[17][18][19][20]
History
The formula was first discovered by Abraham de Moivre[2] in the form
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]
See also
References
- ^ Dutka, Jacques (1991), "The early history of the factorial function", Archive for History of Exact Sciences, 43 (3): 225–249, doi:10.1007/BF00389433, S2CID 122237769
- ^ a b Le Cam, L. (1986), "The central limit theorem around 1935", Statistical Science, 1 (1): 78–96, doi:10.1214/ss/1177013818, JSTOR 2245503, MR 0833276; 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."
- ^ a b 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, JSTOR 2331714,
I consider that the fact that Stirling showed that De Moivre's arithmetical constant was does not entitle him to claim the theorem, [...]
- ^ Flajolet, Philippe; Sedgewick, Robert (2009), Analytic Combinatorics, Cambridge, UK: Cambridge University Press, p. 555, doi:10.1017/CBO9780511801655, ISBN 978-0-521-89806-5, MR 2483235, S2CID 27509971
- ^ 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. ISBN 978-0-387-98931-0.
- ^ Robbins, Herbert (1955), "A Remark on Stirling's Formula", The American Mathematical Monthly, 62 (1): 26–29, doi:10.2307/2308012, JSTOR 2308012
- ^ 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, MR 1221957
- ^ G. Nemes, Error bounds and exponential improvements for the asymptotic expansions of the gamma function and its reciprocal, Proc. Roy. Soc. Edinburgh Sect. A 145 (2015), 571–596.
- ^ Bayes, Thomas (24 November 1763), "A letter from the late Reverend Mr. Thomas Bayes, F. R. S. to John Canton, M. A. and F. R. S." (PDF), Philosophical Transactions of the Royal Society of London, Series I, 53: 269, Bibcode:1763RSPT...53..269B, archived (PDF) from the original on 2012-01-28, retrieved 2012-03-01
- ^ Artin, Emil (2015). The Gamma Function. Dover. p. 24.
- ^ Toth, V. T. Programmable Calculators: Calculators and the Gamma Function (2006) Archived 2005-12-31 at the Wayback Machine.
- ^ Nemes, Gergő (2010), "New asymptotic expansion for the Gamma function", Archiv der Mathematik, 95 (2): 161–169, doi:10.1007/s00013-010-0146-9, S2CID 121820640
- ^ Ramanujan, Srinivasa, Lost Notebook and Other Unpublished Papers, p. 339 – via Internet Archive
- ^ Karatsuba, Ekatherina A. (2001), "On the asymptotic representation of the Euler gamma function by Ramanujan", Journal of Computational and Applied Mathematics, 135 (2): 225–240, Bibcode:2001JCoAM.135..225K, doi:10.1016/S0377-0427(00)00586-0, MR 1850542
- ^ 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, S2CID 119530041
- ^ 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.
- ^ Mortici, Cristinel (2011), "On Ramanujan's large argument formula for the gamma function", Ramanujan J., 26 (2): 185–192, doi:10.1007/s11139-010-9281-y, S2CID 120371952.
Further reading
- Abramowitz, M. & Stegun, I. (2002), Handbook of Mathematical Functions
- Paris, R. B. & Kaminski, D. (2001), Asymptotics and Mellin–Barnes Integrals, New York: Cambridge University Press, ISBN 978-0-521-79001-7
- Whittaker, E. T. & Watson, G. N. (1996), A Course in Modern Analysis (4th ed.), New York: Cambridge University Press, ISBN 978-0-521-58807-2
- Romik, Dan (2000), "Stirling's approximation for : the ultimate short proof?", The American Mathematical Monthly, 107 (6): 556–557, doi:10.2307/2589351, JSTOR 2589351, MR 1767064
- Li, Yuan-Chuan (July 2006), "A note on an identity of the gamma function and Stirling's formula", Real Analysis Exchange, 32 (1): 267–271, MR 2329236
- ^ For example, a program in Mathematica:
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]]