Science Fair Project Encyclopedia

Science Fair Project Encyclopedia Contents Page

Categories: Asymptotic analysis | Number theory | Special functions

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.

Contents

1 Derivation

2 Speed of convergence and error estimates

3 Stirling's formula for the gamma function

4 A convergent version of Stirling's formula

5 History

6 References

Derivation

The formula, together with precise estimates of its error, can be derived as follows. Instead of approximating n!, one considers the natural logarithm

$\ln n! = \ln 1 + \ln 2 + \ldots + \ln n$

Then, we can apply Euler-Maclaurin formula by putting f(x) = ln(x) to find a approximation of the value of ln(n!).

$\ln (n-1)! = n \ln n - n + 1 - \frac{\ln n}{2} + \sum_{k=2}^{m} \frac{B_k {(-1)}^k}{k(k-1)} ( \frac{1}{n^{k-1}} - 1 ) + R$

where B_k is Bernoulli number and R is the remainder of Euler-Maclaurin formula.

Then, we can then take limit on both sides,

$\lim_{n \to \infty} ( \ln n! - n \ln n + n - \frac{\ln n}{2} ) = 1 + \sum_{k=2}^{m} \frac{B_k {(-1)}^k}{k(k-1)} + \lim_{n \to \infty} R$

Let the above limit be y and compound the above two formula, we get the approximation formula in its logarithmic form:

$\ln n! = (n+\frac{1}{2}) \ln n - n + y + \sum_{k=2}^{m} \frac{B_k {(-1)}^k}{k(k-1)n^{k-1}} + O(\frac{1}{n^m})$

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 e^y.

$n! = e^y \sqrt{n}~{( \frac{n}{e} )}^n ( 1 + O(\frac{1}{n}) )$

The unknown term e^y can be found by taking limit on both side as n tends to infinity and using Wallis' product. One can estimate the value of e^y is $\sqrt{2 \pi}$ . Therefore, we got Stirling's formula:

$n! = \sqrt{2 \pi n}~{( \frac{n}{e} )}^n ( 1 + O(\frac{1}{n}) )$

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

Γ(z + 1) = Π(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 B_n 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

$n!\sim [{\rm constant}]\cdot n^{n+1/2} e^{-n}$

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

Categories: Asymptotic analysis | Number theory | Special functions

Last updated: 06-02-2005 02:15:34

Science Fair Project Encyclopedia Contents Page

10-26-2009 08:16:03
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details

All Science Fair Projects