Stirling's approximation

The ratio of (ln n!) to (n ln n − n) approaches unity as n increases.

In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for large factorials. It is named in honour of James Stirling.

The formula is written as

$n!\approx {\sqrt {2\pi n}}\,{\frac {n^{n}}{e^{n}}}.$

Roughly, this means that these quantities approximate each other for all sufficiently large integers n. More precisely, Stirling's formula says that

$\lim _{n\rightarrow \infty }{n! \over {\sqrt {2\pi n}}\;{\frac {n^{n}}{e^{n}}}}=1.$

or

$\lim _{n\rightarrow \infty }{\frac {e^{n}n!}{n^{n}{\sqrt {n}}}}={\sqrt {2\pi }}$

(See limit, square root, π, e.)

Derivation

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

$\ln(n!)=\ln 1+\ln 2+\cdots +\ln n.$

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

$\ln(n-1)!=n\ln n-n+1+{\frac {\ln n}{2}}+\sum _{k=2}^{m}{\frac {B_{k}{(-1)}^{k}}{k(k-1)}}\left({\frac {1}{n^{k-1}}}-1\right)+R$

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

We can then take limits on both sides,

$\lim _{n\to \infty }\left(\ln n!-n\ln n+n-{\frac {\ln n}{2}}\right)=1+\sum _{k=2}^{m}{\frac {B_{k}{(-1)}^{k}}{k(k-1)}}+\lim _{n\to \infty }R.$

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

$\ln n!=\left(n+{\frac {1}{2}}\right)\ln n-n+y+\sum _{k=2}^{m}{\frac {B_{k}{(-1)}^{k}}{k(k-1)n^{k-1}}}+O\left({\frac {1}{n^{m}}}\right)$

where O(f(n)) is Big-O notation.

Just take the exponential on both sides, and choose any positive integer m, say 1. We get the formula with an unknown term e^y:

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

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

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

The formula may also be obtained by repeated integration by parts, and the leading term can be found through the method of steepest descent. The general formula (without the n^1/2 term) may be quickly obtained by approximating the sum

$\ln N!=\sum _{1}^{N}\ln n$

with an integral:

$\sum _{n=1}^{N}\ln n\approx \int _{1}^{N}\ln n\,dn=N\ln N-N+1.$

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):

$n!={\sqrt {2\pi n}}\left({n \over e}\right)^{n}\left(1+{1 \over 12n}+{1 \over 288n^{2}}-{139 \over 51840n^{3}}-{571 \over 2488320n^{4}}+\cdots \right).$

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:

$\ln n!=n\ln n-n+{1 \over 2}\ln(2\pi n)+{1 \over 12n}-{1 \over 360n^{3}}+{1 \over 1260n^{5}}-{1 \over 1680n^{7}}+\cdots .$

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

$\ln \Gamma (z)=\left(z-{\frac {1}{2}}\right)\ln z-z+{\frac {\ln {2\pi }}{2}}+2\int _{0}^{\infty }{\frac {\arctan {\frac {t}{z}}}{\exp(2\pi t)-1}}\,dt.$

Repeated integration by parts gives the asymptotic expansion

$\ln \Gamma (z)=\left(z-{\frac {1}{2}}\right)\ln z-z+{\frac {\ln {2\pi }}{2}}+\sum _{n=1}^{\infty }{\frac {B_{2n}}{2n(2n-1)z^{2n-1}}}$

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. The corresponding approximation may now be written:

$\Gamma (z)={\sqrt {\frac {2\pi }{z}}}~{\left({\frac {z}{e}}\right)}^{z}\left(1+O\left({\frac {1}{z}}\right)\right).$

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.[1]

Obtaining a convergent version of Stirling's formula entails evaluating

$\int _{0}^{\infty }{\frac {2\arctan {\frac {t}{z}}}{\exp(2\pi t)-1}}\,dt=\ln \Gamma (z)-\left(z-{\frac {1}{2}}\right)\ln z+z-{\frac {1}{2}}\ln(2\pi ).$

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

$c_{n}={\frac {1}{n}}\int _{0}^{1}x^{\overline {n}}\left(x-{\frac {1}{2}}\right)\,dx.$

From this we obtain a version of Stirling's series

$\ln \Gamma (z)=\left(z-{\frac {1}{2}}\right)\ln z-z+{\frac {\ln {2\pi }}{2}}$

${}+{\frac {1}{12(z+1)}}+{\frac {1}{12(z+1)(z+2)}}+{\frac {59}{360(z+1)(z+2)(z+3)}}+{\frac {29}{60(z+1)(z+2)(z+3)(z+4)}}+\cdots$

which converges when $\Re (z)>0$.

A version suitable for calculators

The approximation

$\Gamma (z)\approx {\sqrt {\frac {2\pi }{z}}}\left({\frac {z}{e}}{\sqrt {z\sinh {\frac {1}{z}}+{\frac {1}{810z^{6}}}}}\right)^{z},$

or equivalently,

$2\ln \Gamma (z)\approx \ln(2\pi )-\ln z+z\left(2\ln z+\ln \left(z\sinh {\frac {1}{z}}+{\frac {1}{810z^{6}}}\right)-2\right),$

can be obtained by rearranging Stirling's extended formula and observing a coincidence between the resultant power series and the Taylor series expansion of the hyperbolic sine function. This approximation is good to more than 8 decimal digits for z with a real part greater than 8. Robert H. Windschitl suggested it in 2002 for computing the Gamma function with fair accuracy on calculators with limited program or register memory (see ref. 'Toth').

Gergő Nemes proposed in 2007 an approximation which gives the same number of exact digits as the Windschitl approximation but is much simpler:

$\Gamma (z)\approx {\sqrt {\frac {2\pi }{z}}}\left({\frac {1}{e}}\left(z+{\frac {1}{12z-{\frac {1}{10z}}}}\right)\right)^{z},$

or equivalently,

$\ln \Gamma (z)\approx {\frac {1}{2}}\left(\ln(2\pi )-\ln z\right)+z\left(\ln \left(z+{\frac {1}{12z-{\frac {1}{10z}}}}\right)-1\right).$

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.

The "first-order" version of Stirling's approximation, $n!=n^{n}$ , was used by Max Planck in his 1901 article on the black body radiation formula. It linked Planck's concept of energy elements to the black body radiation formula for very large numbers of energy elements and oscillators. The approximation was often used in quantum theory, for example by Debye and de Broglie. Einstein and Bose took a different approach. For very large n, the graph of the probability expression Planck obtained using the "first order" Stirling's formula, plotted in a logarithmic coordinate system, is almost parallel to the line obtained direct from the idea of separated light quanta.

However, the system entropy calculated by using the "first order" approximation is different and the ratio gets strongly nonlinear for small n. One can only speculate that a similar total effect on entropy could be obtained by introducing the uncertainty principle and the photon spin as well as other quantities which were unknown at the time when the old quantum theory was created. Unfortunately, the experimental verification of the link between the "first order" Stirling's approximation and modern physical theories is still missing.

See also

• Lanczos approximation
• Spouge's approximation

References

• Abramowitz, 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 0-521-58807-3
• Toth, V. T. Programmable Calculators: Calculators and the Gamma Function. http://www.rskey.org/gamma.htm, modified 2006
• Weisstein, Eric W. "Stirling's Approximation". MathWorld.
• Stirling's approximation at PlanetMath.