Stirling's approximation#Stirling's formula for the gamma function

Roughly speaking, the simplest version of Stirling's formula can be quickly obtained by approximating the sum

$$\backslash ln(n!)\; =\; \backslash sum\_\{j=1\}^n\; \backslash ln\; j$$

with an integral:

$$\backslash sum\_\{j=1\}^n\; \backslash ln\; j\; \backslash approx\; \backslash int\_1^n\; \backslash ln\; x\; \backslash ,\{\backslash rm\; d\}x\; =\; n\backslash ln\; n\; -\; n\; +\; 1.$$

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

$$\backslash tfrac\{1\}\{2\}(\backslash ln\; 1\; +\; \backslash ln\; n)\; =\; \backslash tfrac\{1\}\{2\}\backslash ln\; n$$

is the approximation by the trapezoid rule of the integral

$$\backslash ln(n!)\; -\; \backslash tfrac\{1\}\{2\}\backslash ln\; n\; \backslash approx\; \backslash int\_1^n\; \backslash ln\; x\backslash ,\{\backslash rm\; d\}x\; =\; n\; \backslash ln\; n\; -\; n\; +\; 1,$$

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

$$\backslash begin\{align\}$$

\ln(n!) - \tfrac{1}{2}\ln n & = \tfrac{1}{2}\ln 1 + \ln 2 + \ln 3 + \cdots + \ln(n-1) + \tfrac{1}{2}\ln n\\

& = n \ln n - n + 1 + \sum_{k=2}^{m} \frac{(-1)^k B_k}{k(k-1)} \left( \frac{1}{n^{k-1}} - 1 \right) + R_{m,n},

\end{align}

where $B\_k$ is a Bernoulli number, and {{math|R_m,n}} is the remainder term in the Euler–Maclaurin formula. Take limits to find that

$$\backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash left(\; \backslash ln(n!)\; -\; n\; \backslash ln\; n\; +\; n\; -\; \backslash tfrac\{1\}\{2\}\backslash ln\; n\; \backslash right)\; =\; 1\; -\; \backslash sum\_\{k=2\}^\{m\}\; \backslash frac\{(-1)^k\; B\_k\}\{k(k-1)\}\; +\; \backslash lim\_\{n\; \backslash to\; \backslash infty\}\; R\_\{m,n\}.$$

Denote this limit as $y$. Because the remainder {{math|R_m,n}} in the Euler–Maclaurin formula satisfies

$$R\_\{m,n\}\; =\; \backslash lim\_\{n\; \backslash to\; \backslash infty\}\; R\_\{m,n\}\; +\; O\; \backslash left(\; \backslash frac\{1\}\{n^m\}\; \backslash right),$$

where big-O notation is used, combining the equations above yields the approximation formula in its logarithmic form:

$$\backslash ln(n!)\; =\; n\; \backslash ln\; \backslash left(\; \backslash frac\{n\}\{e\}\; \backslash right)\; +\; \backslash tfrac\{1\}\{2\}\backslash ln\; n\; +\; y\; +\; \backslash sum\_\{k=2\}^\{m\}\; \backslash frac\{(-1)^k\; B\_k\}\{k(k-1)n^\{k-1\}\}\; +\; O\; \backslash left(\; \backslash frac\{1\}\{n^m\}\; \backslash right).$$

Taking the exponential of both sides and choosing any positive integer $m$, one obtains a formula involving an unknown quantity $e^y$. For {{math|m {{=}} 1}}, the formula is

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

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 $e^y=\backslash sqrt\{2\backslash pi\}$. Therefore, one obtains Stirling's formula:

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

An alternative formula for $n!$ using the gamma function is

$$n!\; =\; \backslash int\_0^\backslash infty\; x^n\; e^\{-x\}\backslash ,\{\backslash rm\; d\}x.$$

(as can be seen by repeated integration by parts). Rewriting and changing variables {{math|x {{=}} ny}}, one obtains

$$n!\; =\; \backslash int\_0^\backslash infty\; e^\{n\backslash ln\; x-x\}\backslash ,\{\backslash rm\; d\}x\; =\; e^\{n\; \backslash ln\; n\}\; n\; \backslash int\_0^\backslash infty\; e^\{n(\backslash ln\; y\; -y)\}\backslash ,\{\backslash rm\; d\}y.$$

Applying Laplace's method one has

$$\backslash int\_0^\backslash infty\; e^\{n(\backslash ln\; y\; -y)\}\backslash ,\{\backslash rm\; d\}y\; \backslash sim\; \backslash sqrt\{\backslash frac\{2\backslash pi\}\{n\}\}\; e^\{-n\},$$

which recovers Stirling's formula:

$$n!\; \backslash sim\; e^\{n\; \backslash ln\; n\}\; n\; \backslash sqrt\{\backslash frac\{2\backslash pi\}\{n\}\}\; e^\{-n\}$$

= \sqrt{2\pi n}\left(\frac{n}{e}\right)^n.

In fact, further corrections can also be obtained using Laplace's method. From previous result, we know that $\backslash Gamma(x)\; \backslash sim\; x^x\; e^\{-x\}$, so we "peel off" this dominant term, then perform two changes of variables, to obtain:$$x^\{-x\}e^x\backslash Gamma(x)\; =\; \backslash int\_\backslash R\; e^\{x(1+t-e^t)\}dt$$To verify this: $\backslash int\_\backslash R\; e^\{x(1+t-e^t)\}dt\; \backslash overset\{t\; \backslash mapsto\; \backslash ln\; t\}\{=\}\; e^x\; \backslash int\_0^\backslash infty\; t^\{x-1\}\; e^\{-xt\}\; dt\; \backslash overset\{t\; \backslash mapsto\; t/x\}\{=\}\; x^\{-x\}\; e^x\; \backslash int\_0^\backslash infty\; e^\{-t\}\; t^\{x-1\}\; dt\; =\; x^\{-x\}\; e^x\; \backslash Gamma(x)$.

Now the function $t\; \backslash mapsto\; 1+t\; -\; e^t$ 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 $1+t-e^t\; =\; -\backslash tau^2/2$. This equation cannot be solved in closed form, but it can be solved by serial expansion, which gives us $t\; =\; \backslash tau\; -\; \backslash tau^2/6\; +\; \backslash tau^3/36\; +\; a\_4\; \backslash tau^4\; +\; O(\backslash tau^5)$. Now plug back to the equation to obtain$$x^\{-x\}e^x\backslash Gamma(x)\; =\; \backslash int\_\backslash R\; e^\{-x\backslash tau^2/2\}(1-\backslash tau/3\; +\; \backslash tau^2/12\; +\; 4a\_4\; \backslash tau^3\; +\; O(\backslash tau^4))\; d\backslash tau\; =\; \backslash sqrt\{2\backslash pi\}(x^\{-1/2\}\; +\; x^\{-3/2\}/12)\; +\; O(x^\{-5/2\})$$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 $t\; =\; \backslash tau\; +\; \backslash cdots$, which can be obtained programmatically.{{NoteTag|note=For example, a program in Mathematica:

series = tau - tau^2/6 + tau^3/36 + tau^4*a + tau^5*b;

(*pick the right a,b to make the series equal 0 at higher orders*)

Series[tau^2/2 + 1 + t - Exp[t] /. t -> series, {tau, 0, 8}]

(*now do the integral*)

integral = Integrate[Exp[-x*tau^2/2] * D[series /. a -> 0 /. b -> 0, tau], {tau, -Infinity, Infinity}];

Simplify[integral/Sqrt[2*Pi]*Sqrt[x]]

|name=mathematica-program|content=content|text=text}}

Thus we get Stirling's formula to two orders:$$n!\; =\; \backslash sqrt\{2\backslash pi\; n\}\backslash left(\backslash frac\{n\}\{e\}\backslash right)^n\; \backslash left(1\; +\; \backslash frac\{1\}\{12\; n\}+O\backslash left(\backslash frac\{1\}\{n^2\}\backslash right)\; \backslash right).$$

A complex-analysis version of this method{{r|flajolet-sedgewick}} is to consider $\backslash frac\{1\}\{n!\}$ as a Taylor coefficient of the exponential function $e^z\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{z^n\}\{n!\}$, computed by Cauchy's integral formula as

$$\backslash frac\{1\}\{n!\}\; =\; \backslash frac\{1\}\{2\backslash pi\; i\}\; \backslash oint\backslash limits\_\{|z|=r\}\; \backslash frac\{e^z\}\{z^\{n+1\}\}\; \backslash ,\; \backslash mathrm\; dz.$$

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.

An alternative version uses the fact that the Poisson distribution converges to a normal distribution by the Central Limit Theorem.{{Cite book |last=MacKay |first=David J. C. |title=Information theory, inference, and learning algorithms |date=2019 |publisher=Cambridge University Press |isbn=978-0-521-64298-9 |edition=22nd printing |location=Cambridge}}

Since the Poisson distribution with parameter $\backslash lambda$ converges to a normal distribution with mean $\backslash lambda$ and variance $\backslash lambda$, their density functions will be approximately the same:

$\backslash frac\{\backslash exp(-\backslash mu)\backslash mu^x\}\{x!\}\backslash approx\; \backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\backslash mu\}\}\backslash exp\backslash left(-\backslash frac\{1\}\{2\}\backslash left(\backslash frac\{x-\backslash mu\}\{\backslash sqrt\{\backslash mu\}\}\backslash right)^\{2\}\backslash right)$

Evaluating this expression at the mean, at which the approximation is particularly accurate, simplifies this expression to:

$\backslash frac\{\backslash exp(-\backslash mu)\backslash mu^\backslash mu\}\{\backslash mu!\}\backslash approx\; \backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\backslash mu\}\}$

Taking logs then results in:

$-\backslash mu+\backslash mu\backslash ln\backslash mu-\backslash ln\backslash mu!\backslash approx\; -\backslash frac\{1\}\{2\}\backslash ln\; 2\backslash pi\backslash mu$

which can easily be rearranged to give:

$\backslash ln\backslash mu!\backslash approx\; \backslash mu\backslash ln\backslash mu\; -\; \backslash mu\; +\; \backslash frac\{1\}\{2\}\backslash ln\; 2\backslash pi\backslash mu$

Evaluating at $\backslash mu=n$ gives the usual, more precise form of Stirling's approximation.

File:Stirling series relative error.svg

Stirling's formula is in fact the first approximation to the following series (now called the Stirling series):{{r|nist}}

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

An explicit formula for the coefficients in this series was given by G. Nemes.{{r|Nemes2010-2}} Further terms are listed in the On-Line Encyclopedia of Integer Sequences as {{OEIS link|A001163}} and {{OEIS link|A001164}}. The first graph in this section shows the relative error vs. $n$, for 1 through all 5 terms listed above. (Bender and Orszag{{Cite book |last1=Bender |first1=Carl M. |title=Advanced mathematical methods for scientists and engineers. 1: Asymptotic methods and perturbation theory |last2=Orszag |first2=Steven A. |date=2009 |publisher=Springer |isbn=978-0-387-98931-0 |edition=Nachdr. |location=New York, NY}} p. 218) gives the asymptotic formula for the coefficients:$$A\_\{2\; j+1\}\; \backslash sim(-1)^j\; 2(2\; j)\; !\; /(2\; \backslash pi)^\{2(j+1)\}$$which shows that it grows superexponentially, and that by the ratio test the radius of convergence is zero.

File:Stirling error vs number of terms.svg

As {{math|n → ∞}}, 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 {{math|S(n, t)}} be the Stirling series to $t$ terms evaluated at $n$. The graphs show

$$\backslash left\; |\; \backslash ln\; \backslash left\; (\backslash frac\{S(n,\; t)\}\{n!\}\; \backslash right)\; \backslash right\; |,$$

which, when small, is essentially the relative error.

Writing Stirling's series in the form

$$\backslash ln(n!)\; \backslash sim\; n\backslash ln\; n\; -\; n\; +\; \backslash tfrac12\backslash ln(2\backslash pi\; n)\; +\backslash frac\{1\}\{12n\}\; -\; \backslash frac\{1\}\{360n^3\}\; +\; \backslash frac\{1\}\{1260n^5\}\; -\; \backslash frac\{1\}\{1680n^7\}\; +\; \backslash cdots,$$

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.{{Citation needed|date=December 2024}}

Other bounds, due to Robbins,{{r|Robbins1955}} valid for all positive integers $n$ are

This upper bound corresponds to stopping the above series for $\backslash ln(n!)$ after the $\backslash frac\{1\}\{n\}$ term. The lower bound is weaker than that obtained by stopping the series after the $\backslash frac\{1\}\{n^3\}$ term. A looser version of this bound is that $\backslash frac\{n!\; e^n\}\{n^\{n+\backslash frac12\}\}\; \backslash in\; (\backslash sqrt\{2\; \backslash pi\},\; e]$ for all $n\; \backslash ge\; 1$.

For all positive integers,

$$n!\; =\; \backslash Gamma(n\; +\; 1),$$

where {{math|Γ}} 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 {{math|Re(z) > 0}}, then

$$\backslash ln\backslash Gamma\; (z)\; =\; z\backslash ln\; z\; -\; z\; +\; \backslash tfrac12\backslash ln\backslash frac\{2\backslash pi\}\{z\}\; +\; \backslash int\_0^\backslash infty\backslash frac\{2\backslash arctan\backslash left(\backslash frac\{t\}\{z\}\backslash right)\}\{e^\{2\backslash pi\; t\}-1\}\backslash ,\{\backslash rm\; d\}t.$$

Repeated integration by parts gives

$$\backslash begin\{align\}$$

\ln\Gamma(z) \sim z\ln z - z + \tfrac12\ln\frac{2\pi}{z}

+ \sum_{n=1}^{N-1} \frac{B_{2n}}{2n(2n-1)z^{2n-1}} \\

= z\ln z - z + \tfrac12\ln\frac{2\pi}{z}

+\frac{1}{12z}

-\frac{1}{360z^3}

+\frac{1}{1260z^5}+\dots

,\end{align}

where $B\_n$ is the $n$th Bernoulli number (note that the limit of the sum as $N\; \backslash to\; \backslash infty$ is not convergent, so this formula is just an asymptotic expansion). The formula is valid for $z$ large enough in absolute value, when {{math|{{abs|arg(z)}} < π − ε}}, where {{mvar|ε}} is positive, with an error term of {{math|O(z^{−2N+ 1})}}. The corresponding approximation may now be written:

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

where the expansion is identical to that of Stirling's series above for $n!$, except that $n$ is replaced with {{math|z − 1}}.{{r|spiegel}}

A further application of this asymptotic expansion is for complex argument {{mvar|z}} with constant {{math|Re(z)}}. See for example the Stirling formula applied in {{math|Im(z) {{=}} t}} of the Riemann–Siegel theta function on the straight line {{math|{{sfrac|1|4}} + it}}.

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.{{r|bayes-canton}} Obtaining a convergent version of Stirling's formula entails evaluating Binet's formula:

$$\backslash int\_0^\backslash infty\; \backslash frac\{2\backslash arctan\backslash left(\backslash frac\{t\}\{x\}\backslash right)\}\{e^\{2\backslash pi\; t\}-1\}\backslash ,\{\backslash rm\; d\}t\; =\; \backslash ln\backslash Gamma(x)\; -\; x\backslash ln\; x\; +\; x\; -\; \backslash tfrac12\backslash ln\backslash frac\{2\backslash pi\}\{x\}.$$

One way to do this is by means of a convergent series of inverted rising factorials. If

$$z^\{\backslash bar\; n\}\; =\; z(z\; +\; 1)\; \backslash cdots\; (z\; +\; n\; -\; 1),$$

then

$$\backslash int\_0^\backslash infty\; \backslash frac\{2\backslash arctan\backslash left(\backslash frac\{t\}\{x\}\backslash right)\}\{e^\{2\backslash pi\; t\}\; -\; 1\}\backslash ,\{\backslash rm\; d\}t\; =\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{c\_n\}\{(x\; +\; 1)^\{\backslash bar\; n\}\},$$

where

$$c\_n\; =\; \backslash frac\{1\}\{n\}\; \backslash int\_0^1\; x^\{\backslash bar\; n\}\; \backslash left(x\; -\; \backslash tfrac\{1\}\{2\}\backslash right)\backslash ,\{\backslash rm\; d\}x\; =\; \backslash frac\{1\}\{2n\}\backslash sum\_\{k=1\}^n\; \backslash frac\{k|s(n,\; k)|\}\{(k\; +\; 1)(k\; +\; 2)\},$$

where {{math|s(n, k)}} denotes the Stirling numbers of the first kind. From this one obtains a version of Stirling's series

$$\backslash begin\{align\}$$

\ln\Gamma(x) &= x\ln x - x + \tfrac12\ln\frac{2\pi}{x} + \frac{1}{12(x+1)} + \frac{1}{12(x+1)(x+2)} + \\

&\quad + \frac{59}{360(x+1)(x+2)(x+3)} + \frac{29}{60(x+1)(x+2)(x+3)(x+4)} + \cdots,

\end{align}

which converges when {{math|Re(x) > 0}}.

Stirling's formula may also be given in convergent form as{{cite book |last1=Artin |first1=Emil |title=The Gamma Function |date=2015 |page = 24|publisher=Dover }}

\Gamma(x)=\sqrt{2\pi}x^{x-\frac{1}{2}}e^{-x+\mu(x)}

where

\mu\left(x\right)=\sum_{n=0}^{\infty}\left(\left(x+n+\frac{1}{2}\right)\ln\left(1+\frac{1}{x+n}\right)-1\right).

The approximation

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

and its equivalent form

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

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 {{mvar|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.{{r|toth}}

Gergő Nemes proposed in 2007 an approximation which gives the same number of exact digits as the Windschitl approximation but is much simpler:{{r|Nemes2010}}

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

or equivalently,

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

An alternative approximation for the gamma function stated by Srinivasa Ramanujan in Ramanujan's lost notebook{{citation|url=https://archive.org/details/lost-notebook/page/n337/|title=Lost Notebook and Other Unpublished Papers|first=Srinivasa|last=Ramanujan|date=14 August 1920 |author-link=Srinivasa Ramanujan|via=Internet Archive|page=339}} is

$$\backslash Gamma(1+x)\; \backslash approx\; \backslash sqrt\{\backslash pi\}\; \backslash left(\backslash frac\{x\}\{e\}\backslash right)^x\; \backslash left(\; 8x^3\; +\; 4x^2\; +\; x\; +\; \backslash frac\{1\}\{30\}\; \backslash right)^\{\backslash frac\{1\}\{6\}\}$$

for {{math|x ≥ 0}}. The equivalent approximation for {{math|ln n!}} has an asymptotic error of {{math|{{sfrac|1|1400n³}}}} and is given by

$$\backslash ln\; n!\; \backslash approx\; n\backslash ln\; n\; -\; n\; +\; \backslash tfrac\{1\}\{6\}\backslash ln(8n^3\; +\; 4n^2\; +\; n\; +\; \backslash tfrac\{1\}\{30\})\; +\; \backslash tfrac\{1\}\{2\}\backslash ln\backslash pi\; .$$

The approximation may be made precise by giving paired upper and lower bounds; one such inequality is{{r|E.A.Karatsuba|Mortici2011-1|Mortici2011-2|Mortici2011-3}}

The formula was first discovered by Abraham de Moivre{{r|LeCam1986}} in the form

$$n!\; \backslash sim\; [\{\backslash rm\; constant\}]\; \backslash cdot\; n^\{n+\backslash frac12\}\; e^\{-n\}.$$

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 $\backslash sqrt\{2\backslash pi\}$.{{r|Pearson1924}}

• Lanczos approximation
• Spouge's approximation

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• {{citation |last1=Paris |first1=R. B. |last2=Kaminski |first2=D. |name-list-style=amp |title=Asymptotics and Mellin–Barnes Integrals |year=2001 |publisher=Cambridge University Press |location=New York |isbn=978-0-521-79001-7 |url-access=registration |url=https://archive.org/details/asymptoticsmelli0000pari }}
• {{citation |last1=Whittaker |first1=E. T. |last2=Watson |first2=G. N. |name-list-style=amp |title=A Course in Modern Analysis |year=1996 |edition=4th |publisher=Cambridge University Press |location=New York |isbn=978-0-521-58807-2 }}
• {{citation

| last = Romik | first = Dan

| doi = 10.2307/2589351

| issue = 6

| journal = The American Mathematical Monthly

| mr = 1767064

| pages = 556–557

| title = Stirling's approximation for $n!$: the ultimate short proof?

| volume = 107

| year = 2000| jstor = 2589351

}}

• {{citation

| last = Li | first = Yuan-Chuan

| date = July 2006

| issue = 1

| journal = Real Analysis Exchange

| mr = 2329236

| pages = 267–271

| title = A note on an identity of the gamma function and Stirling's formula

| url = https://projecteuclid.org/euclid.rae/1184700051

| volume = 32}}

{{Notelist}}

{{Commons category}}

• {{SpringerEOM | title=Stirling_formula | id=Stirling_formula&oldid=44695}}
• [http://www.luschny.de/math/factorial/approx/SimpleCases.html Peter Luschny, Approximation formulas for the factorial function n!]
• {{MathWorld | urlname=StirlingsApproximation | title=Stirling's Approximation | mode=cs2}}
• {{PlanetMath | urlname=StirlingsApproximation | title=Stirling's approximation}}

{{Calculus topics}}

Category:Approximations

Category:Asymptotic analysis

Category:Analytic number theory

Category:Gamma and related functions

Category:Theorems in mathematical analysis