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:
which is often written as
(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
Then, we can apply Euler-Maclaurin formula by putting f(x) = ln(x) to find a approximation of the value of ln(n!).
where Bk is Bernoulli number and R is the remainder of Euler-Maclaurin formula.
Then, we can then take limit on both sides,
Let the above limit be y and compound the above two formula, we get the approximation formula in its logarithmic form:
where O(f(n)) is Big-O notation.
Just take exponential on both sides, and take an positive integer m, say 1. We got the formula with an unknown term ey.
The unknown term ey can be found by taking limit on both side as n tends to infinity and using Wallis' product. One can estimate the value of ey is . Therefore, we got Stirling's formula:
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,
with
Stirling's formula is in fact the first approximation to the following series (now called the Stirling series):
As , 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:
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
- Γ(z + 1) = Π(z) = z!
defined for all complex numbers other than non-positive integers. If then
Repeated integration by parts gives the asymptotic expansion
where Bn is the nth Bernoulli number. The formula is valid for z large enough in absolute value when , 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
One way to do this is by means of a convergent series of inverted rising exponentials. If , then
where
From this we obtain a version of Stirling's series
-
which converges when .
History
The formula was first discovered by Abraham de Moivre in the form
Stirling's contribution consisted of showing that the "constant" is . 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
Last updated: 06-02-2005 02:15:34