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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Stirling's approximation (or Stirling's formula) is an
asymptotic In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates Limit of a function#Limits at infinity, tends to infinity. In pro ...
approximation for
factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times ...
s. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related but less precise result was first stated by Abraham de Moivre. One way of stating the approximation involves the
logarithm In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...
of the factorial: \ln(n!) = n\ln n - n +O(\ln n), where the
big O notation Big ''O'' notation is a mathematical notation that describes the asymptotic analysis, limiting behavior of a function (mathematics), function when the Argument of a function, argument tends towards a particular value or infinity. Big O is a memb ...
means that, for all sufficiently large values of n, the difference between \ln(n!) and n\ln n-n will be at most proportional to the logarithm of n. In computer science applications such as the worst-case lower bound for comparison sorting, it is convenient to instead use the
binary logarithm In mathematics, the binary logarithm () is the exponentiation, power to which the number must be exponentiation, raised to obtain the value . That is, for any real number , :x=\log_2 n \quad\Longleftrightarrow\quad 2^x=n. For example, th ...
, giving the equivalent form \log_2 (n!) = n\log_2 n - n\log_2 e +O(\log_2 n). The error term in either base can be expressed more precisely as \tfrac12\log(2\pi n)+O(\tfrac1n), corresponding to an approximate formula for the factorial itself, n! \sim \sqrt\left(\frac\right)^n. Here the sign \sim means that the two quantities are asymptotic, that is, their ratio tends to 1 as n tends to infinity.


Derivation

Roughly speaking, the simplest version of Stirling's formula can be quickly obtained by approximating the sum \ln(n!) = \sum_^n \ln j with an
integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
: \sum_^n \ln j \approx \int_1^n \ln x \,x = n\ln n - n + 1. The full formula, together with precise estimates of its error, can be derived as follows. Instead of approximating n!, one considers its
natural logarithm The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...
, as this is a slowly varying function: \ln(n!) = \ln 1 + \ln 2 + \cdots + \ln n. The right-hand side of this equation minus \tfrac(\ln 1 + \ln n) = \tfrac\ln n is the approximation by the trapezoid rule of the integral \ln(n!) - \tfrac\ln n \approx \int_1^n \ln x\,x = n \ln n - n + 1, and the error in this approximation is given by the Euler–Maclaurin formula: \begin \ln(n!) - \tfrac\ln n & = \tfrac\ln 1 + \ln 2 + \ln 3 + \cdots + \ln(n-1) + \tfrac\ln n\\ & = n \ln n - n + 1 + \sum_^ \frac \left( \frac - 1 \right) + R_, \end where B_k is a Bernoulli number, and is the remainder term in the Euler–Maclaurin formula. Take limits to find that \lim_ \left( \ln(n!) - n \ln n + n - \tfrac\ln n \right) = 1 - \sum_^ \frac + \lim_ R_. Denote this limit as y. Because the remainder in the Euler–Maclaurin formula satisfies R_ = \lim_ R_ + O \left( \frac \right), where big-O notation is used, combining the equations above yields the approximation formula in its logarithmic form: \ln(n!) = n \ln \left( \frac \right) + \tfrac\ln n + y + \sum_^ \frac + O \left( \frac \right). Taking the exponential of both sides and choosing any positive integer m, one obtains a formula involving an unknown quantity e^y. For , the formula is n! = e^y \sqrt \left( \frac \right)^n \left( 1 + O \left( \frac \right) \right). 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 e^y=\sqrt. Therefore, one obtains Stirling's formula: n! = \sqrt \left( \frac \right)^n \left( 1 + O \left( \frac \right) \right).


Alternative derivations

An alternative formula for n! using the
gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
is n! = \int_0^\infty x^n e^\,x. (as can be seen by repeated integration by parts). Rewriting and changing variables , one obtains n! = \int_0^\infty e^\,x = e^ n \int_0^\infty e^\,y. Applying Laplace's method one has \int_0^\infty e^\,y \sim \sqrt e^, which recovers Stirling's formula: n! \sim e^ n \sqrt e^ = \sqrt\left(\frac\right)^n.


Higher orders

In fact, further corrections can also be obtained using Laplace's method. From previous result, we know that \Gamma(x) \sim x^x e^, so we "peel off" this dominant term, then perform two changes of variables, to obtain:x^e^x\Gamma(x) = \int_\R e^dtTo verify this: \int_\R e^dt \overset e^x \int_0^\infty t^ e^ dt \overset x^ e^x \int_0^\infty e^ t^ dt = x^ e^x \Gamma(x). Now the function t \mapsto 1+t - e^t is unimodal, with maximum value zero. Locally around zero, it looks like -t^2/2, which is why we are able to perform Laplace's method. In order to extend Laplace's method to higher orders, we perform another change of variables by 1+t-e^t = -\tau^2/2. This equation cannot be solved in closed form, but it can be solved by serial expansion, which gives us t = \tau - \tau^2/6 + \tau^3/36 + a_4 \tau^4 + O(\tau^5) . Now plug back to the equation to obtainx^e^x\Gamma(x) = \int_\R e^(1-\tau/3 + \tau^2/12 + 4a_4 \tau^3 + O(\tau^4)) d\tau = \sqrt(x^ + x^/12) + O(x^)notice how we don't need to actually find a_4, since it is cancelled out by the integral. Higher orders can be achieved by computing more terms in t = \tau + \cdots, which can be obtained programmatically. Thus we get Stirling's formula to two orders: n! = \sqrt\left(\frac\right)^n \left(1 + \frac+O\left(\frac\right) \right).


Complex-analytic version

A complex-analysis version of this method is to consider \frac as a Taylor coefficient of the exponential function e^z = \sum_^\infty \frac, computed by
Cauchy's integral formula In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary o ...
as \frac = \frac \oint\limits_ \frac \, \mathrm dz. This line integral can then be approximated using the saddle-point method with an appropriate choice of contour radius r = r_n. The dominant portion of the integral near the saddle point is then approximated by a real integral and Laplace's method, while the remaining portion of the integral can be bounded above to give an error term.


Using the Central Limit Theorem and the Poisson distribution

An alternative version uses the fact that the
Poisson distribution In probability theory and statistics, the Poisson distribution () is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known const ...
converges to a
normal distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac ...
by the Central Limit Theorem. Since the Poisson distribution with parameter \lambda converges to a normal distribution with mean \lambda and variance \lambda, their density functions will be approximately the same: \frac\approx \frac\exp\left(-\frac\left(\frac\right)^\right) Evaluating this expression at the mean, at which the approximation is particularly accurate, simplifies this expression to: \frac\approx \frac Taking logs then results in: -\mu+\mu\ln\mu-\ln\mu!\approx -\frac\ln 2\pi\mu which can easily be rearranged to give: \ln\mu!\approx \mu\ln\mu - \mu + \frac\ln 2\pi\mu Evaluating at \mu=n gives the usual, more precise form of Stirling's approximation.


Speed of convergence and error estimates

Stirling's formula is in fact the first approximation to the following series (now called the Stirling series): n! \sim \sqrt\left(\frac\right)^n \left(1 +\frac+\frac - \frac -\frac+ \cdots \right). An explicit formula for the coefficients in this series was given by G. Nemes. Further terms are listed in the
On-Line Encyclopedia of Integer Sequences The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to th ...
as and . The first graph in this section shows the relative error vs. n, for 1 through all 5 terms listed above. (Bender and Orszag p. 218) gives the asymptotic formula for the coefficients:A_ \sim(-1)^j 2(2 j) ! /(2 \pi)^which shows that it grows superexponentially, and that by the
ratio test In mathematics, the ratio test is a convergence tests, test (or "criterion") for the convergent series, convergence of a series (mathematics), series :\sum_^\infty a_n, where each term is a real number, real or complex number and is nonzero wh ...
the radius of convergence is zero. As , the error in the truncated series is asymptotically equal to the first omitted term. This is an example of an
asymptotic expansion In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation ...
. It is not a convergent series; for any ''particular'' value of n there are only so many terms of the series that improve accuracy, after which accuracy worsens. This is shown in the next graph, which shows the relative error versus the number of terms in the series, for larger numbers of terms. More precisely, let be the Stirling series to t terms evaluated at n. The graphs show \left , \ln \left (\frac \right) \right , , which, when small, is essentially the relative error. Writing Stirling's series in the form \ln(n!) \sim n\ln n - n + \tfrac12\ln(2\pi n) +\frac - \frac + \frac - \frac + \cdots, it is known that the error in truncating the series is always of the opposite sign and at most the same magnitude as the first omitted term. Other bounds, due to Robbins, valid for all positive integers n are \sqrt\left(\frac\right)^n e^ < n! < \sqrt\left(\frac\right)^n e^. This upper bound corresponds to stopping the above series for \ln(n!) after the \frac term. The lower bound is weaker than that obtained by stopping the series after the \frac term. A looser version of this bound is that \frac \in (\sqrt, e] for all n \ge 1.


Stirling's formula for the gamma function

For all positive integers, n! = \Gamma(n + 1), where denotes the
gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
. 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 \ln\Gamma (z) = z\ln z - z + \tfrac12\ln\frac + \int_0^\infty\frac\,t. Repeated integration by parts gives \begin \ln\Gamma(z) \sim z\ln z - z + \tfrac12\ln\frac + \sum_^ \frac \\ = z\ln z - z + \tfrac12\ln\frac +\frac -\frac +\frac+\dots ,\end where B_n is the nth Bernoulli number (note that the limit of the sum as N \to \infty is not convergent, so this formula is just an
asymptotic expansion In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation ...
). The formula is valid for z large enough in absolute value, when , where is positive, with an error term of . The corresponding approximation may now be written: \Gamma(z) = \sqrt\,^z \left(1 + O\left(\frac\right)\right). where the expansion is identical to that of Stirling's series above for n!, except that n is replaced with . A further application of this asymptotic expansion is for complex argument with constant . See for example the Stirling formula applied in of the Riemann–Siegel theta function on the straight line .


A convergent version of Stirling's formula

Thomas Bayes Thomas Bayes ( , ; 7 April 1761) was an English statistician, philosopher and Presbyterian minister who is known for formulating a specific case of the theorem that bears his name: Bayes' theorem. Bayes never published what would become his m ...
showed, in a letter to John Canton published by the
Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...
in 1763, that Stirling's formula did not give a convergent series. Obtaining a convergent version of Stirling's formula entails evaluating Binet's formula: \int_0^\infty \frac\,t = \ln\Gamma(x) - x\ln x + x - \tfrac12\ln\frac. One way to do this is by means of a convergent series of inverted rising factorials. If z^ = z(z + 1) \cdots (z + n - 1), then \int_0^\infty \frac\,t = \sum_^\infty \frac, where c_n = \frac \int_0^1 x^ \left(x - \tfrac\right)\,x = \frac\sum_^n \frac, where denotes the Stirling numbers of the first kind. From this one obtains a version of Stirling's series \begin \ln\Gamma(x) &= x\ln x - x + \tfrac12\ln\frac + \frac + \frac + \\ &\quad + \frac + \frac + \cdots, \end which converges when . Stirling's formula may also be given in convergent form as \Gamma(x)=\sqrtx^e^ where \mu\left(x\right)=\sum_^\left(\left(x+n+\frac\right)\ln\left(1+\frac\right)-1\right).


Versions suitable for calculators

The approximation \Gamma(z) \approx \sqrt \left(\frac \sqrt \right)^z and its equivalent form 2\ln\Gamma(z) \approx \ln(2\pi) - \ln z + z \left(2\ln z + \ln\left(z\sinh\frac + \frac\right) - 2\right) can be obtained by rearranging Stirling's extended formula and observing a coincidence between the resultant
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
and the
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
expansion of the
hyperbolic sine In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a unit circle, circle with a unit radius, the points form the right ha ...
function. This approximation is good to more than 8 decimal digits for 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. 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 \left(\frac \left(z + \frac\right)\right)^z, or equivalently, \ln\Gamma(z) \approx \tfrac \left(\ln(2\pi) - \ln z\right) + z\left(\ln\left(z + \frac\right) - 1\right). An alternative approximation for the gamma function stated by Srinivasa Ramanujan in Ramanujan's lost notebook is \Gamma(1+x) \approx \sqrt \left(\frac\right)^x \left( 8x^3 + 4x^2 + x + \frac \right)^ for . The equivalent approximation for has an asymptotic error of and is given by \ln n! \approx n\ln n - n + \tfrac\ln(8n^3 + 4n^2 + n + \tfrac) + \tfrac\ln\pi . The approximation may be made precise by giving paired upper and lower bounds; one such inequality is \sqrt \left(\frac\right)^x \left( 8x^3 + 4x^2 + x + \frac \right)^ < \Gamma(1+x) < \sqrt \left(\frac\right)^x \left( 8x^3 + 4x^2 + x + \frac \right)^.


History

The formula was first discovered by Abraham de Moivre in the form n! \sim [] \cdot n^ e^. De Moivre gave an approximate rational-number expression for the natural logarithm of the constant. Stirling's contribution consisted of showing that the constant is precisely \sqrt .


See also

* Lanczos approximation * Spouge's approximation


References


Further reading

* * * * *


External links

*
Peter Luschny, ''Approximation formulas for the factorial function n!''
* * {{Calculus topics Approximations Asymptotic analysis Analytic number theory Gamma and related functions Theorems in mathematical analysis