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[[File:Mplwp factorial gamma stirling.svg|thumb|right|upright=1.35|Comparison of Stirling's approximation with the factorial]]
In [[mathematics]], '''Stirling's approximation''' (or '''Stirling's formula''') is an approximation for [[factorial]]s. It is a good approximation, leading to accurate results even for small values of {{mvar|n}}. It is named after [[James Stirling (mathematician)|James Stirling]], though it was first stated by [[Abraham de Moivre]].<ref>
{{citation 
 | last = Dutka
 | first = Jacques
 | year = 1991
 | title = The early history of the factorial function
 | journal = [[Archive for History of Exact Sciences]] 
 | volume = 43
 | issue = 3
 | pages = 225&ndash;249
 | doi = 10.1007/BF00389433}}</ref><ref name="LeCam1986">{{citation |doi=10.1214/ss/1177013818 |doi-access=free |last=Le Cam |first=L. |title=The central limit theorem around 1935 |journal=Statistical Science |volume=1 |issue=1 |pages=78–96 [p. 81] |year=1986 |mr=833276 |quote=The result, obtained using a formula originally proved by de Moivre but now called Stirling's formula, occurs in his `Doctrine of Chances' of 1733. }}.{{Unreliable source?|date=May 2009}}</ref><ref name="Pearson1924">{{citation |last=Pearson |first=Karl |year=1924 |title=Historical note on the origin of the normal curve of errors |journal=Biometrika |volume=16 |issue=3/4 |pages=402–404 [p. 403] |quote=I consider that the fact that Stirling showed that De Moivre's arithmetical constant was {{sqrt|2''{{pi}}''}} does not entitle him to claim the theorem, [...] |doi=10.2307/2331714|jstor=2331714 }}</ref>

The version of the formula typically used in applications is
:<math>\ln n! = n\ln n - n +O(\ln n)</math>
(in [[big O notation]], as <math>n\to\infty</math>), or, by changing the base of the logarithm (for instance in the [[Comparison sort#Number of comparisons required to sort a list|worst-case lower bound for comparison sorting]]),

:<math>\log_2 n! = n\log_2 n - n\log_2 e +O(\log_2 n).</math>

Specifying the constant in the {{math|''O''(ln ''n'')}} error term gives {{math|{{sfrac|1|2}}ln(2''πn'')}}, yielding the more precise formula:

:<math>n! \sim \sqrt{2 \pi n}\left(\frac{n}{e}\right)^n,</math>

where the sign ~ means that the two quantities are [[Asymptotic analysis|asymptotic]]: their ratio tends to 1 as {{mvar|n}} tends to infinity.

One may also give simple bounds valid for all positive integers {{mvar|n}}, rather than only for large {{mvar|n}}:

:<math>\sqrt{2\pi}\ n^{n+\frac12}e^{-n} \le n! \le e\ n^{n+\frac12}e^{-n}</math>

for <math>n=1,2,3,\ldots</math>. These follow from the more [[Stirling's approximation#Speed of convergence and error estimates|precise error bounds]] discussed below.

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

:<math>\ln n! = \sum_{j=1}^n \ln j</math>

with an [[integral]]:

:<math>\sum_{j=1}^n \ln j \approx \int_1^n \ln x \,{\rm d}x = n\ln n - n + 1.</math>

The full formula, together with precise estimates of its error, can be derived as follows. Instead of approximating {{math|''n''!}}, one considers its [[natural logarithm]] as this is a [[slowly varying function]]:

:<math>\ln n! = \ln 1 + \ln 2 + \cdots + \ln n.</math>

The right-hand side of this equation minus

:<math>\tfrac{1}{2}(\ln 1 + \ln n) = \tfrac{1}{2}\ln n,</math>

is the approximation by the [[trapezoid rule]] of the integral

:<math>\ln n! - \tfrac{1}{2}\ln n \approx \int_1^n \ln x\,{\rm d}x = n \ln n - n + 1,</math>

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

:<math>\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}</math>

where {{mvar|B<sub>k</sub>}} is a [[Bernoulli number]] and {{math|''R''<sub>''m'',''n''</sub>}} is the remainder term in the Euler–Maclaurin formula. Take limits to find that

:<math>\lim_{n \to \infty} \left( \ln n! - n \ln n + n - \tfrac{1}{2}\ln n \right) = 1 - \sum_{k=2}^{m} \frac{(-1)^k B_k}{k(k-1)} + \lim_{n \to \infty} R_{m,n}.</math>

Denote this limit as {{mvar|y}}.  Because the remainder {{math|''R''<sub>''m'',''n''</sub>}} in the [[Euler–Maclaurin formula]] satisfies

:<math>R_{m,n} = \lim_{n \to \infty} R_{m,n} + O \left( \frac{1}{n^m} \right),</math>

where we use [[big-O notation]], combining the equations above yields the approximation formula in its logarithmic form:

:<math>\ln n! = n \ln \left( \frac{n}{e} \right) + \tfrac{1}{2}\ln n + y + \sum_{k=2}^{m} \frac{(-1)^k B_k}{k(k-1)n^{k-1}} + O \left( \frac{1}{n^m} \right).</math>

Taking the exponential of both sides, and choosing any positive integer {{mvar|m}}, we get a formula involving an unknown quantity {{mvar|e<sup>y</sup>}}. For {{math|''m'' {{=}} 1}}, the formula is

:<math>n! = e^{y} \sqrt{n} \left( \frac{n}{e} \right)^n \left( 1 + O \left( \frac{1}{n} \right) \right).</math>

The quantity {{mvar|e<sup>y</sup>}} can be found by taking the limit on both sides as {{mvar|n}} tends to infinity and using [[Wallis product|Wallis' product]], which shows that {{math|''e<sup>y</sup>'' {{=}} {{sqrt|2''π''}}}}. Therefore, we get Stirling's formula:

:<math>n! = \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n \left( 1 + O \left( \frac{1}{n} \right) \right).</math>

== An alternative derivation ==
An alternative formula for {{math|''n''!}} using the [[gamma function]] is
:<math> n! = \int_0^\infty x^n e^{-x}\,{\rm d}x.</math>
(as can be seen by repeated integration by parts). Rewriting and changing variables {{math|''x'' {{=}} ''ny''}} one gets
:<math> n! = \int_0^\infty e^{n\ln x-x}\,{\rm d}x = e^{n \ln n} n \int_0^\infty e^{n(\ln y -y)}\,{\rm d}y.</math>
Applying [[Laplace's method]] we have:
:<math>\int_0^\infty e^{n(\ln y -y)}\,{\rm d}y \sim  \sqrt{\frac{2\pi}{n}} e^{-n}</math>
which recovers Stirling's formula,
:<math> n! \sim   e^{n \ln n} n \sqrt{\frac{2\pi}{n}} e^{-n}
= \sqrt{2\pi n}\left(\frac{n}{e}\right)^n.
</math>
In fact further corrections can also be obtained using Laplace's method. For example, computing two-order expansion using Laplace's method yields
:<math>\int_0^\infty e^{n(\ln y-y)}\,{\rm d}y \sim  \sqrt{\frac{2\pi}{n}} e^{-n}
\left(1+\frac{1}{12 n}\right)</math>
and gives Stirling's formula to two orders,
:<math> n! \sim   e^{n \ln n} n \sqrt{\frac{2\pi}{n}} e^{-n}\left(1+\frac{1}{12 n}\right)
= \sqrt{2\pi n}\left(\frac{n}{e}\right)^n \left(1+ \frac{1}{12 n}\right).
</math>

A complex-analysis version of this method<ref>Phillipe Flajolet and Robert Sedgewick, ''Analytic Combinatorics'', p. 555</ref> is to consider <math>\frac{1}{n!}</math> as a [[Taylor series|Taylor coefficient]] of the exponential function <math>e^z = \sum_{n=0}^\infty \frac{z^n}{n!}</math>, computed by [[Cauchy's integral formula]] as:

:<math>\frac{1}{n!} \ = \ \frac{1}{2\pi i} \oint_{\!\!\!\!\!|z|=r}\, \frac{e^z}{z^{n+1}} \, \mathrm dz. </math>

This line integral can then be approximated using the [[Method of steepest descent|saddle point method]] with an appropriate choice of countour radius <math>r=r_n</math>. 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 ==
[[File:Stirling series relative error.svg|thumb|upright=1.8|The relative error in a truncated Stirling series vs. {{mvar|n}}, for 0 to 5 terms. The kinks in the curves represent points where the truncated series coincides with {{math|&Gamma;(''n'' + 1)}}.]]

Stirling's formula is in fact the first approximation to the following series (now called the '''Stirling series'''<ref>{{cite web|url=https://dlmf.nist.gov/5.11.i|title=NIST Digital Library of Mathematical Functions|authors= F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, and B. V. Saunders, eds.}}</ref>):

:<math>
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).</math>

An explicit formula for the coefficients in this series was given by G. Nemes.<ref name="Nemes2010-2">{{Citation|last=Nemes|first=Gergő|year=2010|title=On the Coefficients of the Asymptotic Expansion of n!|journal=Journal of Integer Sequences|volume=13|issue=6|pages=5 pp|issn=|doi=|postscript=}}</ref>{{efn|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 [[Approximation error|relative error]] vs. {{mvar|n}}, for 1 through all 5 terms listed above.

[[File:Stirling error vs number of terms.svg|thumb|upright=1.8|The relative error in a truncated Stirling series vs. the number of terms used]]

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 {{mvar|n}} there are only so many terms of the series that improve accuracy, after which point accuracy actually gets worse.  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 {{mvar|t}} terms evaluated at&nbsp;{{mvar|n}}.  The graphs show 
:<math>\left | \ln \left (\frac{S(n, t)}{n!} \right) \right |, </math>
which, when small, is essentially the relative error.

Writing Stirling's series in the form

:<math>\ln n! \sim n\ln n - n + \tfrac12\ln(2\pi n) +\frac{1}{12n} - \frac{1}{360n^3} + \frac{1}{1260n^5} - \frac{1}{1680n^7} + \cdots,</math>

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.

More precise bounds, due to Robbins,<ref name="Robbins1955">{{Citation|last=Robbins|first=Herbert|year=1955|title=A Remark on Stirling's Formula|journal=The American Mathematical Monthly|volume=62|issue=1|pages=26–29 pp|issn=|doi=10.2307/2308012|postscript=|jstor=2308012}}</ref> valid for all positive integers {{mvar|n}} are

:<math>\sqrt{2\pi}\ n^{n+\frac12}e^{-n} e^{\frac{1}{12n + 1}} < n! < \sqrt{2\pi}\ n^{n+\frac12}e^{-n} e^{\frac{1}{12n}}. </math>

==Stirling's formula for the gamma function==
For all positive integers,

:<math>n! = \Gamma(n+1),</math>

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

:<math>\ln\Gamma (z) = z\ln z - z + \tfrac12\ln\frac{2\pi}{z} + \int_0^\infty\frac{2\arctan\left(\frac{t}{z}\right)}{e^{2\pi t}-1}\,{\rm d}t.</math>

Repeated integration by parts gives

:<math>\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}}</math>

where {{mvar|B<sub>n</sub>}} is the {{mvar|n}}th [[Bernoulli number]] (note that the limit of the sum as <math>\scriptstyle N \to \infty</math> is not convergent, so this formula is just an [[asymptotic expansion]]). The formula is valid for {{mvar|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''<sup>−2''N''+ 1</sup>)}}. The corresponding approximation may now be written:

:<math>\Gamma(z) = \sqrt{\frac{2\pi}{z}}\,{\left(\frac{z}{e}\right)}^z \left(1 + O\left(\frac{1}{z}\right)\right).</math>

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''}}.

== Error bounds ==
For any positive integer {{mvar|N}}, we introduce the following notation:

:<math>\ln\Gamma(z) = z\ln z - z + \tfrac12\ln\frac{2\pi}{z} + \sum\limits_{n=1}^{N-1}{\frac{{B_{2n}}}{{2n\left({2n - 1}\right)z^{2n - 1} }}} + R_N(z)</math>

and

:<math>\Gamma(z) = \sqrt{\frac{{2\pi}}{z}} \left({\frac{z}{e}}\right)^z  \left({\sum\limits_{n=0}^{N-1}{\frac{{a_n}}{{z^n}}} + \widetilde{R}_N(z)}\right).</math>

Then <ref>F. W. Schäfke, A. Sattler, Restgliedabschätzungen für die Stirlingsche Reihe, ''Note. Mat.'' '''10''' (1990), 453–470.</ref><ref>G. Nemes, Error bounds and exponential improvements for the asymptotic expansions of the gamma function and its reciprocal, ''Proc. Roy. Soc. Edinburgh Sect. A'' '''145''' (2015), 571–596.</ref>

:<math>\begin{align}
|R_N(z)| &\le
  \frac{|B_{2N}|}{2N(2N-1)|z|^{2N-1}}
  \begin{cases}
    1                                      & \text{ if } |\arg z| \leq \frac{\pi}{4}, \\
    |\csc(\arg z)|                         & \text{ if } \frac{\pi}{4}<|\arg z| < \frac{\pi}{2}, \\
    \sec^{2N}\left(\tfrac{\arg z}{2}\right) & \text{ if } |\arg z| < \pi,
  \end{cases} \\[6pt]
  \left |\widetilde{R}_N(z) \right | &\le
    \left(\frac{\left |a_N \right |}{|z|^N} + \frac{\left |a_{N+1} \right |}{|z|^{N+1}}\right)
    \begin{cases}
      1              & \text{ if } |\arg z| \leq \frac{\pi}{4}, \\
      |\csc(\arg z)| & \text{ if } \frac{\pi}{4} < |\arg z| < \frac{\pi}{2}.
    \end{cases}
\end{align}
</math>

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]].<ref>{{cite web |url=http://www.york.ac.uk/depts/maths/histstat/letter.pdf |title=Archived copy |access-date=2012-03-01 |url-status=live |archive-url=https://web.archive.org/web/20120128050439/http://www.york.ac.uk/depts/maths/histstat/letter.pdf |archive-date=2012-01-28 |df= }}</ref>  Obtaining a convergent version of Stirling's formula entails evaluating [[Gamma function#Raabe's formula|Raabe's formula]]:

:<math>\int_0^\infty \frac{2\arctan\left(\frac{t}{x}\right)}{e^{2\pi t}-1}\,{\rm d}t = \ln\Gamma(x) - x\ln x + x - \tfrac12\ln\frac{2\pi}{x}.</math>

One way to do this is by means of a convergent series of inverted [[Pochhammer symbol#Alternate notations|rising exponentials]]. If

:<math>z^{\bar n} = z(z+1) \cdots (z+n-1)</math>

then

:<math>\int_0^\infty \frac{2\arctan\left(\frac{t}{x}\right)}{e^{2\pi t}-1}\,{\rm d}t = \sum_{n=1}^\infty \frac{c_n}{(x+1)^{\bar n}}</math>

where

:<math> c_n = \frac{1}{n} \int_0^1 x^{\bar n} \left(x - \tfrac{1}{2}\right)\,{\rm d}x = \frac{1}{2n}\sum_{k=1}^n \frac{k|s(n,k)|}{(k + 1)(k + 2)}</math>

where {{math|''s''(''n'',&nbsp;''k'')}} denotes the [[Stirling numbers of the first kind]]. From this we obtain a version of Stirling's series

:<math>\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}</math>

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

==Versions suitable for calculators==
The approximation

:<math>\Gamma(z) \approx \sqrt{\frac{2 \pi}{z}} \left(\frac{z}{e} \sqrt{z \sinh\frac{1}{z} + \frac{1}{810z^6} } \right)^z</math>

and its equivalent form

:<math>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)</math>

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.<ref>[http://www.rskey.org/gamma.htm Toth, V. T. ''Programmable Calculators: Calculators and the Gamma Function'' (2006)] {{webarchive|url=https://web.archive.org/web/20051231063913/http://www.rskey.org/gamma.htm |date=2005-12-31 }}</ref>

Gergő Nemes proposed in 2007 an approximation which gives the same number of exact digits as the Windschitl approximation but is much simpler:<ref name="Nemes2010">{{Citation|last=Nemes|first=Gergő|year=2010|title=New asymptotic expansion for the Gamma function|journal=Archiv der Mathematik|volume=95|issue=2|pages=161–169|issn=0003-889X|doi=10.1007/s00013-010-0146-9|postscript=.}}</ref>

:<math>\Gamma(z) \approx \sqrt{\frac{2\pi}{z} } \left(\frac{1}{e} \left(z + \frac{1}{12z - \frac{1}{10z}}\right)\right)^z,</math>

or equivalently,

:<math> \ln\Gamma(z) \approx \tfrac{1}{2} \left(\ln(2\pi) - \ln z\right) + z\left(\ln\left(z + \frac{1}{12z - \frac{1}{10z}}\right) - 1\right). </math>

An alternative approximation for the Gamma function stated by [[Srinivasa Ramanujan]] ([[Ramanujan's lost notebook#References|Ramanujan 1988]]{{clarify|This is obviously supposed to be a citation but its target is not clear|date=May 2018}}) is

:<math>\Gamma(1+x) \approx \sqrt{\pi} \left(\frac{x}{e}\right)^x  \left( 8x^3 + 4x^2 + x + \frac{1}{30} \right)^{\frac{1}{6}}</math>

for {{math|''x'' ≥ 0}}. The equivalent approximation for {{math|ln ''n''!}} has an asymptotic error of {{math|{{sfrac|1|1400''n''<sup>3</sup>}}}} and is given by

:<math>\ln n! \approx n\ln n - n + \tfrac{1}{6}\ln(8n^3 + 4n^2 + n + \tfrac{1}{30}) + \tfrac{1}{2}\ln\pi .</math>

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

:<math> \sqrt{\pi} \left(\frac{x}{e}\right)^x \left( 8x^3 + 4x^2 + x + \frac{1}{100} \right)^{1/6} < \Gamma(1+x) <  \sqrt{\pi} \left(\frac{x}{e}\right)^x \left( 8x^3 + 4x^2 + x + \frac{1}{30} \right)^{1/6}.</math><ref name="E.A.Karatsuba">{{Citation|last=Karatsuba|first=Ekatherina|year=2001|title=On the asymptotic representation of the Euler gamma function by Ramanujan|journal=Journal of Computational and Applied Mathematics|volume=135|issue=2|pages=225–240|doi=10.1016/S0377-0427(00)00586-0}}</ref><ref name="Mortici2011-1">{{Citation|last=Mortici|first=Cristinel|year=2011|title=Ramanujan's estimate for the gamma function via monotonicity arguments|journal=Ramanujan J.|volume=25|pages=149–154}}</ref><ref name="Mortici2011-2">{{Citation|last=Mortici|first=Cristinel|year=2011|title=Improved asymptotic formulas for the gamma function|journal=Comput. Math. Appl.|volume=61|pages=3364–3369}}</ref><ref name="Mortici2011-3">{{Citation|last=Mortici|first=Cristinel|year=2011|title=On Ramanujan's large argument formula for the gamma function|journal=Ramanujan J.|volume=26|pages=185–192}}</ref>

== History ==
The formula was first discovered by [[Abraham de Moivre]]<ref name="LeCam1986" /> in the form

:<math>n! \sim [{\rm constant}] \cdot n^{n+\frac12} e^{-n}.</math>

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 <math>\sqrt{2\pi} </math>.<ref name="Pearson1924" />

==See also==
* [[Lanczos approximation]]
* [[Spouge's approximation]]

== Notes ==
{{notelist}}

==References==
{{Reflist|30em}}

*{{citation |last=Olver |first=F. W. J. |last2= Olde Daalhuis |first2=A. B. |last3=Lozier |first3=D. W. |last4=Schneider |first4=B. I. |last5=Boisvert |first5=R. F. |last6=Clark |first6=C. W. |last7=Miller |first7=B. R. |last8=Saunders |first8=B. V. |lastauthoramp=yes |title=NIST Digital Library of Mathematical Functions |version=Release 1.0.13 of 2016-09-16 |url=http://dlmf.nist.gov/5.11}}
*{{citation |last=Abramowitz |first=M. |last2=Stegun |first2=I. |lastauthoramp=yes |title=Handbook of Mathematical Functions |year=2002 |url=http://www.math.hkbu.edu.hk/support/aands/toc.htm }}
*{{citation
 | last = Nemes
 | first = G.
 | title = New asymptotic expansion for the Gamma function
 | journal = Archiv der Mathematik
 | volume = 95
 | year = 2010
 | issue = 2
 | pages = 161–169
 | doi = 10.1007/s00013-010-0146-9}}
*{{citation |last=Paris |first=R. B. |last2=Kaminski |first2=D. |lastauthoramp=yes |chapter= |title=Asymptotics and the Mellin–Barnes Integrals |year=2001 |publisher=Cambridge University Press |location=New York |isbn=978-0-521-79001-7 }}
*{{citation |last=Whittaker |first=E. T. |last2=Watson |first2=G. N. |lastauthoramp=yes |title=A Course in Modern Analysis |year=1996 |edition=4th |publisher=Cambridge University Press |location=New York |isbn=978-0-521-58807-2 }}
* Dan Romik, ''Stirling’s Approximation for n!: The Ultimate Short Proof?'', The American Mathematical Monthly, Vol. 107, No. 6 (Jun. – Jul., 2000), 556–557.
* Y.-C. Li, ''A Note on an Identity of The Gamma Function and Stirling’s Formula'', Real Analysis Exchang, Vol. 32(1), 2006/2007, pp.&nbsp;267–272.

==External links==
{{Commons category}}

* {{springer|title=Stirling formula|id=p/s087830}}
* [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}}
* {{PlanetMath | urlname=StirlingsApproximation | title=Stirling's approximation}}

[[Category:Approximations]]
[[Category:Asymptotic analysis]]
[[Category:Analytic number theory]]
[[Category:Gamma and related functions]]
[[Category:Theorems in analysis]]