- Stirling's approximation
In
mathematics , Stirling's approximation (or Stirling's formula) is an approximation for largefactorial s. It is named in honour of James Stirling.The formula is written as
:
Roughly, this means that these quantities approximate each other for all sufficiently large integers "n". More precisely, Stirling's formula says that
:
or
:
or
:
or
:
Derivation
The formula, together with precise estimates of its error, can be derived as follows. Instead of approximating "n"!, one considers its
natural logarithm ::
The right-hand side of this equation is (almost) the approximation of the integral using the
trapezoid rule ,and the error in this approximation is given by theEuler–Maclaurin formula ::
where "B""k" is
Bernoulli number and "R"m,n is the remainder term in the Euler–Maclaurin formula.Take limits to find that:
Denote this limit by "y". Because the remainder "R""m","n" in the
Euler–Maclaurin formula satisfies:
where we use
Big-O notation , combining the equations above yields the approximation formula in its logarithmic form::
Taking the exponential of both sides, and choosing any positive integer "m", we get a formula involving an unknown quantity "e""y".For "m"="1", the formula is
:
The quantity "e""y" can be found by taking the limit on both sides as "n" tends to infinity and using Wallis' product,which shows that . Therefore, we get Stirling's formula:
:
The formula may also be obtained by repeated
integration by parts , and the leading term can be found through themethod of steepest descent .Stirling's formula, without the factor that is often irrelevant in applications, can be quickly obtained by approximating the sum:
with an integral:
:
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 . It is not aconvergent series ; for any "particular" value of "n" there are only so many terms of the series that improve accuracy, after which point accuracy actually gets worse. Let be the Stirling series to terms evaluated at . The graph shows , which, when small, is essentially the relative error.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.
tirling's formula for the Gamma function
For all positive integers,
:
However, the
Gamma function , unlike the factorial, is more broadly defined for all complex numbers other than non-positive integers; nevertheless, Stirling's formula may still be applied. If then:
Repeated integration by parts gives the asymptotic expansion
:
where "B""n" 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 when the first "m" terms are used. The corresponding approximation may now be written::
A convergent version of Stirling's formula
Thomas Bayes showed, in a letter toJohn Canton published by theRoyal Society in1763 , that Stirling's formula did not give aconvergent series . [http://www.york.ac.uk/depts/maths/histstat/letter.pdf]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 .
A version suitable for calculators
The approximation
:
or equivalently,
:
can be obtained by rearranging Stirling's extended formula and observing a coincidence between the resultant power series and the
Taylor series expansion of thehyperbolic 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:
:
or equivalently,
:
History
The formula was first discovered by
Abraham de Moivre in the form:Stirling's contribution consisted of showing that the constantis . The more precise versions are due toJacques Binet .ee 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
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