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Stirling's Approximation

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Stirling's approximation gives an approximate value for the factorial function n! or the gamma function Gamma(n) for n>>1. The approximation can most simply be derived for n an integer by approximating the sum over the terms of the factorial with an integral, so that

lnn!=ln1+ln2+...+lnn
(1)
=sum_(k=1)^(n)lnk
(2)
approx int_1^nlnxdx
(3)
=[xlnx-x]_1^n
(4)
=nlnn-n+1
(5)
approx nlnn-n.
(6)

The equation can also be derived using the integral definition of the factorial,

n!=int_0^inftye^(-x)x^ndx.
(7)

Note that the derivative of the logarithm of the integrand can be written

d/(dx)ln(e^(-x)x^n)=d/(dx)(nlnx-x)=n/x-1.
(8)

The integrand is sharply peaked with the contribution important only near x=n. Therefore, let x=n+xi where xi<<n, and write

ln(x^ne^(-x))=nlnx-x
(9)
=nln(n+xi)-(n+xi).
(10)

Now,

ln(n+xi)=ln[n(1+xi/n)]
(11)
=lnn+ln(1+xi/n)
(12)
=lnn+xi/n-1/2(xi^2)/(n^2)+...,
(13)

so

ln(x^ne^(-x))=nln(n+xi)-(n+xi)
(14)
=nlnn+xi-1/2(xi^2)/n-n-xi+...
(15)
=nlnn-n-(xi^2)/(2n)+....
(16)

Taking the exponential of each side then gives

x^ne^(-x) approx e^(nlnn)e^(-n)e^(-xi^2/2n)
(17)
=n^ne^(-n)e^(-xi^2/2n).
(18)

Plugging into the integral expression for n! then gives

n! approx int_(-n)^inftyn^ne^(-n)e^(-xi^2/2n)dxi
(19)
approx n^ne^(-n)int_(-infty)^inftye^(-xi^2/2n)dxi.
(20)

Evaluating the integral gives

n! approx n^ne^(-n)sqrt(2pin)
(21)
=sqrt(2pi)n^(n+1/2)e^(-n)
(22)

(Wells 1986, p. 45). Taking the logarithm of both sides then gives

lnn! approx nlnn-n+1/2ln(2pin)
(23)
=(n+1/2)lnn-n+1/2ln(2pi).
(24)

This is Stirling's series with only the first term retained and, for large n, it reduces to Stirling's approximation

lnn! approx nlnn-n.
(25)

Taking successive terms of |_n^n/n!_|, where |_x_| is the floor function, gives the sequence 1, 2, 4, 10, 26, 64, 163, 416, 1067, 2755, ... (OEIS A055775).

Stirling's approximation can be extended to the double inequality

sqrt(2pi)n^(n+1/2)e^(-n+1/(12n+1))<n!<sqrt(2pi)n^(n+1/2)e^(-n+1/(12n))
(26)

(Robbins 1955, Feller 1968).

Gosper has noted that a better approximation to n! (i.e., one which approximates the terms in Stirling's series instead of truncating them) is given by

n! approx sqrt((2n+1/3)pi)n^ne^(-n).
(27)

Considering n a real number so that lim_(n->0)n^n=1, the equation (27) also gives a much closer approximation to the factorial of 0, 0!=1, yielding sqrt(pi/3) approx 1.02333 instead of 0 obtained with the conventional Stirling approximation.

See also

Binet's Log Gamma Formulas, Factorial, Gamma Function, Log Gamma Function, Stirling's Series

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References

Feller, W. "Stirling's Formula." §2.9 in An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, pp. 50-53, 1968.Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 86-88, 2003.Robbins, H. "A Remark of Stirling's Formula." Amer. Math. Monthly 62, 26-29, 1955.Sloane, N. J. A. Sequence A055775 in "The On-Line Encyclopedia of Integer Sequences."Stirling, J. Methodus differentialis, sive tractatus de summation et interpolation serierum infinitarium. London, 1730. English translation by Holliday, J. The Differential Method: A Treatise of the Summation and Interpolation of Infinite Series. 1749.Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 45, 1986.Whittaker, E. T. and Robinson, G. "Stirling's Approximation to the Factorial." §70 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 138-140, 1967.

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Stirling's Approximation

Cite this as:

Weisstein, Eric W. "Stirling's Approximation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/StirlingsApproximation.html

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