Stirling's approximation explained

In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. 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 of the factorial:$$\backslash ln(n!)\; =\; n\backslash ln\; n\; -\; n\; +O(\backslash ln\; n),$$where the big O notation 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. In computer science applications such as the worst-case lower bound for comparison sorting, it is convenient to instead use the binary logarithm, giving the equivalent form$$\backslash log\_2\; (n!)\; =\; n\backslash log\_2\; n\; -\; n\backslash log\_2\; e\; +O(\backslash log\_2\; n).$$ The error term in either base can be expressed more precisely as \tfrac 12log ( 2 \pi n) + O(\tfrac 1 n) , corresponding to an approximate formula for the factorial itself,$$n!\; \backslash sim\; \backslash sqrt\backslash left(\backslash frac\backslash right)^n.$$Here the sign \sim means that the two quantities are asymptotic, that is, that their ratio tends to 1 as n tends to infinity. The following version of the bound holds for all n \ge 1 , rather than only asymptotically:

Derivation

Roughly speaking, the simplest version of Stirling's formula can be quickly obtained by approximating the sum$$\backslash ln(n!)\; =\; \backslash sum\_^n\; \backslash ln\; j$$with an integral:$$\backslash sum\_^n\; \backslash ln\; j\; \backslash approx\; \backslash int\_1^n\; \backslash ln\; x\; \backslash ,x\; =\; n\backslash ln\; n\; -\; n\; +\; 1.$$

The full formula, together with precise estimates of its error, can be derived as follows. Instead of approximating

n! , one considers its natural logarithm, as this is a slowly varying function:$$\backslash ln(n!)\; =\; \backslash ln\; 1\; +\; \backslash ln\; 2\; +\; \backslash cdots\; +\; \backslash ln\; n.$$

The right-hand side of this equation minus$$\backslash tfrac(\backslash ln\; 1\; +\; \backslash ln\; n)\; =\; \backslash tfrac\backslash ln\; n$$is the approximation by the trapezoid rule of the integral$$\backslash ln(n!)\; -\; \backslash tfrac\backslash ln\; n\; \backslash approx\; \backslash int\_1^n\; \backslash ln\; x\backslash ,x\; =\; n\; \backslash ln\; n\; -\; n\; +\; 1,$$

and the error in this approximation is given by the Euler–Maclaurin formula:

where

B _{k} is a Bernoulli number, and is the remainder term in the Euler–Maclaurin formula. Take limits to find that$$\backslash lim\_\; \backslash left(\backslash ln(n!)\; -\; n\; \backslash ln\; n\; +\; n\; -\; \backslash tfrac\backslash ln\; n\; \backslash right)\; =\; 1\; -\; \backslash sum\_^\; \backslash frac\; +\; \backslash lim\_\; R\_.$$

Denote this limit as

y . Because the remainder in the Euler–Maclaurin formula satisfies$$R\_\; =\; \backslash lim\_\; R\_\; +\; O\; \backslash left(\backslash frac\; \backslash right),$$

where big-O notation is used, combining the equations above yields the approximation formula in its logarithmic form:$$\backslash ln(n!)\; =\; n\; \backslash ln\; \backslash left(\backslash frac\; \backslash right)\; +\; \backslash tfrac\backslash ln\; n\; +\; y\; +\; \backslash sum\_^\; \backslash frac\; +\; O\; \backslash left(\backslash frac\; \backslash right).$$

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$$n!\; =\; e^y\; \backslash sqrt\; \backslash left(\backslash frac\; \backslash right)^n\; \backslash left(1\; +\; O\; \backslash left(\backslash frac\; \backslash right)\; \backslash right).$$

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{2\pi}} . Therefore, one obtains Stirling's formula:$$n!\; =\; \backslash sqrt\; \backslash left(\backslash frac\; \backslash right)^n\; \backslash left(1\; +\; O\; \backslash left(\backslash frac\; \backslash right)\; \backslash right).$$

Alternative derivations

An alternative formula for

n! using the gamma function is$$n!\; =\; \backslash int\_0^\backslash infty\; x^n\; e^\backslash ,x.$$(as can be seen by repeated integration by parts). Rewriting and changing variables, one obtains$$n!\; =\; \backslash int\_0^\backslash infty\; e^\backslash ,x\; =\; e^\; n\; \backslash int\_0^\backslash infty\; e^\backslash ,y.$$Applying Laplace's method one has$$\backslash int\_0^\backslash infty\; e^\backslash ,y\; \backslash sim\; \backslash sqrt\; e^,$$which recovers Stirling's formula:$$n!\; \backslash sim\; e^\; n\; \backslash sqrt\; e^=\; \backslash sqrt\backslash left(\backslash frac\backslash right)^n.$$

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 ^{-x} , so we "peel off" this dominant term, then perform two changes of variables, to obtain:$$x^e^x\backslash Gamma(x)\; =\; \backslash int\_\backslash R\; e^dt$$To verify this: \int _{\R} e ^{x( 1+ t - e ^{t)}} dt \overset{t \mapsto ln t}{ = } e ^{x} \int ^{inf ty} _{0} t ^{x-1} e ^{-xt} dt \overset{t \mapsto t/x}{ = } x ^{-x} e ^{x} \int ^{inf ty} _{0} e ^{-t} t ^{x-1} dt = x ^{-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 obtain$$x^e^x\backslash Gamma(x)\; =\; \backslash int\_\backslash R\; e^(1-\backslash tau/3\; +\; \backslash tau^2/12\; +\; 4a\_4\; \backslash tau^3\; +\; O(\backslash tau^4))\; d\backslash tau\; =\; \backslash sqrt(x^\; +\; x^/12)\; +\; O(x^)$$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 + … , which can be obtained programmatically.

Thus we get Stirling's formula to two orders:$$n!\; =\; \backslash sqrt\backslash left(\backslash frac\backslash right)^n\; \backslash left(1\; +\; \backslash frac+O\backslash left(\backslash frac\backslash right)\; \backslash right).$$

Complex-analytic version

A complex-analysis version of this method is to consider

(1)/(n!) as a Taylor coefficient of the exponential function e ^{z} = \sum ^{inf ty} _{n =0} (z ^{n})/(n!) , computed by Cauchy's integral formula as$$\backslash frac\; =\; \backslash frac\; \backslash oint\backslash limits\_$$

=r

\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.

Speed of convergence and error estimates

Stirling's formula is in fact the first approximation to the following series (now called the Stirling series):$$n!\; \backslash sim\; \backslash sqrt\backslash left(\backslash frac\backslash right)^n\; \backslash left(1\; +\backslash frac+\backslash frac\; -\; \backslash frac\; -\backslash frac+\; \backslash cdots\; \backslash right).$$

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 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^[1] p. 218) gives the asymptotic formula for the coefficients:$$A\_\; \backslash sim(-1)^j\; 2(2\; j)\; !\; /(2\; \backslash pi)^$$which shows that it grows superexponentially, and that by ratio test 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. 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 $$\backslash left\; |\; \backslash ln\; \backslash left\; (\backslash frac\; \backslash right)\; \backslash right\; |,$$which, when small, is essentially the relative error.

Writing Stirling's series in the form$$\backslash ln(n!)\; \backslash sim\; n\backslash ln\; n\; -\; n\; +\; \backslash tfrac12\backslash ln(2\backslash pi\; n)\; +\backslash frac\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash frac\; +\; \backslash cdots,$$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 areThis upper bound corresponds to stopping the above series for ln (n!) after the (1)/(n) term. The lower bound is weaker than that obtained by stopping the series after the (1)/(n ^{3}) term. A looser version of this bound is that (n! e ^{n})/(n ^{n + (12)/()}) \in (\sqrt{ 2 \pi}, e] for all n \ge 1 .

Stirling's formula for the gamma function

For all positive integers,$$n!\; =\; \backslash Gamma(n\; +\; 1),$$where denotes the gamma function.

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$$\backslash ln\backslash Gamma\; (z)\; =\; z\backslash ln\; z\; -\; z\; +\; \backslash tfrac12\backslash ln\backslash frac\; +\; \backslash int\_0^\backslash infty\backslash frac\backslash ,t.$$

Repeated integration by parts gives$$\backslash ln\backslash Gamma(z)\; \backslash sim\; z\backslash ln\; z\; -\; z\; +\; \backslash tfrac12\backslash ln\backslash frac\; +\; \backslash sum\_^\; \backslash frac,$$

where

B _{n} is the n th Bernoulli number (note that the limit of the sum as N \to inf ty is not convergent, so this formula is just an asymptotic expansion). 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:$$\backslash Gamma(z)\; =\; \backslash sqrt\backslash ,^z\; \backslash left(1\; +\; O\backslash left(\backslash frac\backslash right)\backslash right).$$

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 .

Error bounds

For any positive integer

N , the following notation is introduced:$$\backslash ln\backslash Gamma(z)\; =\; z\backslash ln\; z\; -\; z\; +\; \backslash tfrac12\backslash ln\backslash frac\; +\; \backslash sum\backslash limits\_^\; +\; R\_N(z)$$and$$\backslash Gamma(z)\; =\; \backslash sqrt\; \backslash left(\backslash right)^z\; \backslash left(\backslash right).$$

Then

\times \begin 1 & \text \left|\arg z\right| \leq \frac, \\ \left|\csc(\arg z)\right| & \text \frac < \left|\arg z\right| < \frac, \\ \sec^\left(\tfrac\right) & \text \left|\arg z\right| < \pi, \end \\[6pt]

\left |\widetilde_N(z) \right | &\le \left(\frac

^N

+ \frac

^

\right)\times \begin 1 & \text \left|\arg z\right| \leq \frac, \\ \left|\csc(2\arg z)\right| & \text \frac < \left|\arg z\right| < \frac. \end\end

For further information and other error bounds, see the cited papers.

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. Obtaining a convergent version of Stirling's formula entails evaluating Binet's formula:$$\backslash int\_0^\backslash infty\; \backslash frac\backslash ,t\; =\; \backslash ln\backslash Gamma(x)\; -\; x\backslash ln\; x\; +\; x\; -\; \backslash tfrac12\backslash ln\backslash frac.$$

One way to do this is by means of a convergent series of inverted rising factorials. If$$z^\; =\; z(z\; +\; 1)\; \backslash cdots\; (z\; +\; n\; -\; 1),$$then$$\backslash int\_0^\backslash infty\; \backslash frac\backslash ,t\; =\; \backslash sum\_^\backslash infty\; \backslash frac,$$where$$c\_n\; =\; \backslash frac\; \backslash int\_0^1\; x^\; \backslash left(x\; -\; \backslash tfrac\backslash right)\backslash ,x\; =\; \backslash frac\backslash sum\_^n\; \backslash frac,$$where denotes the Stirling numbers of the first kind. From this one obtains a version of Stirling's serieswhich converges when . Stirling's formula may also be given in convergent form as^[2] $$\backslash Gamma(x)=\backslash sqrtx^e^$$where $$\backslash mu\backslash left(x\backslash right)=\backslash sum\_^\backslash left(\backslash left(x+n+\backslash frac\backslash right)\backslash ln\backslash left(1+\backslash frac\backslash right)-1\backslash right).$$

Versions suitable for calculators

The approximation$$\backslash Gamma(z)\; \backslash approx\; \backslash sqrt\; \backslash left(\backslash frac\; \backslash sqrt\; \backslash right)^z$$and its equivalent form$$2\backslash ln\backslash Gamma(z)\; \backslash approx\; \backslash ln(2\backslash pi)\; -\; \backslash ln\; z\; +\; z\; \backslash left(2\backslash ln\; z\; +\; \backslash ln\backslash left(z\backslash sinh\backslash frac\; +\; \backslash frac\backslash right)\; -\; 2\backslash right)$$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 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:$$\backslash Gamma(z)\; \backslash approx\; \backslash sqrt\; \backslash left(\backslash frac\; \backslash left(z\; +\; \backslash frac\backslash right)\backslash right)^z,$$or equivalently,$$\backslash ln\backslash Gamma(z)\; \backslash approx\; \backslash tfrac\; \backslash left(\backslash ln(2\backslash pi)\; -\; \backslash ln\; z\backslash right)\; +\; z\backslash left(\backslash ln\backslash left(z\; +\; \backslash frac\backslash right)\; -\; 1\backslash right).$$

An alternative approximation for the gamma function stated by Srinivasa Ramanujan in Ramanujan's lost notebook is$$\backslash Gamma(1+x)\; \backslash approx\; \backslash sqrt\; \backslash left(\backslash frac\backslash right)^x\; \backslash left(8x^3\; +\; 4x^2\; +\; x\; +\; \backslash frac\; \backslash right)^$$for . The equivalent approximation for has an asymptotic error of and is given by$$\backslash ln\; n!\; \backslash approx\; n\backslash ln\; n\; -\; n\; +\; \backslash tfrac\backslash ln(8n^3\; +\; 4n^2\; +\; n\; +\; \backslash tfrac)\; +\; \backslash tfrac\backslash ln\backslash pi\; .$$

The approximation may be made precise by giving paired upper and lower bounds; one such inequality is

History

The formula was first discovered by Abraham de Moivre in the form$$n!\; \backslash sim\; [\{\backslash rm\; constant\}]\; \backslash cdot\; n^\; e^.$$

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{ 2 \pi} .

See also

• Lanczos approximation
• Spouge's approximation

Further reading

1. Book: Bender . Carl M. . Advanced mathematical methods for scientists and engineers. 1: Asymptotic methods and perturbation theory . Orszag . Steven A. . 2009 . Springer . 978-0-387-98931-0 . Nachdr. . New York, NY.
2. Book: Artin . Emil . The Gamma Function . 2015 . 24. Dover .

External links

• • Peter Luschny, Approximation formulas for the factorial function n!