Stirling's Approximation

In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for large factorials. It is named in honour of James Stirling. Formally, it states:
\lim_{n \rightarrow \infty} {n!\over \sqrt{2 \pi n} \; \left(\frac{n}{e}\right)^{n} } = 1
which is often written as
n! \sim \sqrt{2 \pi n} \; \left(\frac{n}{e}\right)^{n}
(See limit, square root, π, e.) For large n, the right hand side is a good approximation for n!, and much faster and easier to calculate. For example, the formula gives for 30! the approximation 2.6452 × 1032 while the correct value is about 2.6525 × 1032. The error is less than 0.3% in this case.

Derivation

The formula, together with precise estimates of its error, can be derived as follows. Instead of approximating n!, one considers the natural logarithm
ln(n!) = ln(1) + ln(2) + ... + ln(n);
the Euler-Maclaurin formula gives estimates for sums like these. The goal, then, is to show the approximation formula in its logarithmic form:
\ln n! \approx \left(n+\frac{1}{2}\right)\ln n - n +\ln\left(\sqrt{2\pi}\right)
The formula may also be obtained by repeated integration by parts. The leading term can be found through the method of steepest descent.

Speed of convergence and error estimates

More precisely,
n! = \sqrt{2 \pi n} \; \left(\frac{n}{e}\right)^{n}e^{\lambda_n}
with
\frac{1}{12n+1} < \lambda_n < \frac{1}{12n}.
Stirling's formula is in fact the first approximation to the following series (now called the Stirling series): 
   n!=\sqrt{2\pi n}\left({n\over e}\right)^n   \left(    1    +{1\over12n}    +{1\over288n^2}    -{139\over51840n^3}    -{571\over2488320n^4}    + \cdots   \right)
As n \to \infty, the error in the truncated series is asymptotically equal to the first omitted term. This is an example of an asymptotic expansion. The asymptotic expansion of the logarithm is also called Stirling's series: 
   \ln n!=n\ln n - n + {1\over 2}\ln(2\pi n)    +{1\over12n}    -{1\over360n^3}    +{1\over1260n^5}    -{1\over 1680n^7}    +\cdots
In this case, it is known that the error in truncating the series is always of the same sign and at most the same magnitude as the first omitted term.

Stirling's formula for the gamma function

Stirling's formula may also be applied to the gamma function
\Gamma(z+1) = \Pi(z) = z!
defined for all complex numbers other than non-positive integers. If \Re(z) > 0 then
\ln \Gamma (z) = (z-\frac12)\ln z -z + \frac{\ln {2 \pi}}{2} + 2 \int_0^\infty \frac{\arctan \frac{t}{z}}{\exp(2 \pi t)-1} dt
Repeated integration by parts gives the asymptotic expansion
\ln \Gamma (z) = (z-\frac12)\ln z -z + \frac{\ln {2 \pi}}{2} + \sum_{n=1}^\infty \frac{B_{2n}}{2n(2n-1)z^{2n-1}}
where Bn is the nth Bernoulli number. The formula is valid for z large enough in absolute value when |\arg z| < \pi - \epsilon, where ε is positive, with an error term of O(z^{-m - 1/2}) when the first m terms are used.

A convergent version of Stirling's formula

Obtaining a convergent version of Stirling's formula entails evaluating
\int_0^\infty \frac{2\arctan \frac{t}{z}}{\exp(2 \pi t)-1}\, dt
= \ln\Gamma (z) - (z-\frac12)\ln z +z - \frac12\ln(2\pi). One way to do this is by means of a convergent series of inverted rising exponentials. If z^{\overline n} = z(z+1) \cdots (z+n-1), then
\int_0^\infty \frac{2\arctan \frac{t}{z}}{\exp(2 \pi t)-1} \, dt
= \sum_{n=1}^\infty \frac{c_n}{(z+1)^{\overline n}} where
n c_n = \int_0^1 x^{\overline n}(x-\frac12)\, dx.
From this we obtain a version of Stirling's series
\ln \Gamma (z) = (z-\frac12)\ln z -z + \frac{\ln {2 \pi}}{2} +
\frac{1}{12(z+1)} + \frac{1}{12(z+1)(z+2)} + \frac{29}{60(z+1)(z+2)(z+3)} + \cdots
which converges when \Re(z)>0.

History

The formula was first discovered by Abraham de Moivre in the form
n!\sim constant}\cdot n^{n+1/2} e^{-n}
Stirling's contribution consisted of showing that the "constant" is \sqrt{2\pi}. The more precise versions are due to Jacques Binet.

References

  • Abromowitz, M. and Stegun, I., Handbook of Mathematical Functions, http://www.math.hkbu.edu.hk/support/aands/toc.htm
  • Paris, R. B., and Kaminsky, D., Asymptotics and the Mellin-Barnes Integrals, Cambridge University Press, 2001
  • Whittaker, E. T., and Watson, G. N., A Course in Modern Analysis, fourth edition, Cambridge University Press, 1963. ISBN 0521588073

 

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