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An asymptotic representation which provides approximate values of the factorials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087830/s0878301.png" /> and of the [[Gamma-function|gamma-function]] for large values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087830/s0878302.png" />. This representation has the form
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{{MSC|33B15}}
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087830/s0878303.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$
 
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\newcommand{\abs}[1]{\left|#1\right|}
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087830/s0878304.png" />. The asymptotic equalities
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\newcommand{\Re}{\mathrm{Re}}
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087830/s0878305.png" /></td> </tr></table>
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An asymptotic representation which provides approximate values of the
 
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factorials $n! = 1 \ldots n$ and of the
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087830/s0878306.png" /></td> </tr></table>
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[[Gamma-function|gamma-function]] for large values of $n$. This
 
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representation has the form
hold, and mean that when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087830/s0878307.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087830/s0878308.png" />, the ratio of the left- and right-hand sides tends to one.
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$$
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n! = \sqrt{2\pi n}\; n^n e^{-n} e^{\theta(n)}, \tag{$^*$}
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$$
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where $\abs{\theta(n)} < 1/12n$. The asymptotic equalities
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$$
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n! \approx  \sqrt{2\pi n}\; n^n e^{-n},  \quad n \rightarrow \infty,
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$$
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$$
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\Gamma(z+1) \approx  \sqrt{2\pi z}\; z^z e^{-z},  \quad
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z \rightarrow \infty,\; |{\arg z}|<\pi,
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$$
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hold, and mean that when $n\rightarrow\infty$ or $z \rightarrow \infty$, $|{\arg z}|<\pi$, the ratio of the left- and right-hand sides tends to one.
  
 
The representation (*) was established by J. Stirling (1730).
 
The representation (*) was established by J. Stirling (1730).
 
 
  
 
====Comments====
 
====Comments====
See [[Gamma-function|Gamma-function]] for the corresponding asymptotic series (Stirling series) and additional references.
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See
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[[Gamma-function|Gamma-function]] for the corresponding asymptotic
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series (Stirling series) and additional references.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"N.G. de Bruijn,   "Asymptotic methods in analysis" , Dover, reprint (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"G. Marsaglia,   J.C.W. Marsaglia,   "A new derivation of Stirling's approximation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087830/s0878309.png" />" ''Amer. Math. Monthly'' , '''97''' (1990) pp. 826–829</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"V. Namias,   "A simple derivation of Stirling's asymptotic series" ''Amer. Math. Monthly'' , '''93''' (1986) pp. 25–29</TD></TR></table>
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{|
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|-
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|valign="top"|{{Ref|Br}}||valign="top"| N.G. de Bruijn, "Asymptotic methods in analysis", Dover, reprint (1981)
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|-
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|valign="top"|{{Ref|MaMa}}||valign="top"| G. Marsaglia, J.C.W. Marsaglia, "A new derivation of Stirling's approximation of $n!$" ''Amer. Math. Monthly'', '''97''' (1990) pp. 826–829
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|-
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|valign="top"|{{Ref|Na}}||valign="top"| V. Namias, "A simple derivation of Stirling's asymptotic series" ''Amer. Math. Monthly'', '''93''' (1986) pp. 25–29
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|-
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|}

Latest revision as of 15:21, 14 February 2020

2020 Mathematics Subject Classification: Primary: 33B15 [MSN][ZBL]

$$ \newcommand{\abs}[1]{\left|#1\right|} \newcommand{\Re}{\mathrm{Re}} $$ An asymptotic representation which provides approximate values of the factorials $n! = 1 \ldots n$ and of the gamma-function for large values of $n$. This representation has the form $$ n! = \sqrt{2\pi n}\; n^n e^{-n} e^{\theta(n)}, \tag{$^*$} $$ where $\abs{\theta(n)} < 1/12n$. The asymptotic equalities $$ n! \approx \sqrt{2\pi n}\; n^n e^{-n}, \quad n \rightarrow \infty, $$ $$ \Gamma(z+1) \approx \sqrt{2\pi z}\; z^z e^{-z}, \quad z \rightarrow \infty,\; |{\arg z}|<\pi, $$ hold, and mean that when $n\rightarrow\infty$ or $z \rightarrow \infty$, $|{\arg z}|<\pi$, the ratio of the left- and right-hand sides tends to one.

The representation (*) was established by J. Stirling (1730).

Comments

See Gamma-function for the corresponding asymptotic series (Stirling series) and additional references.

References

[Br] N.G. de Bruijn, "Asymptotic methods in analysis", Dover, reprint (1981)
[MaMa] G. Marsaglia, J.C.W. Marsaglia, "A new derivation of Stirling's approximation of $n!$" Amer. Math. Monthly, 97 (1990) pp. 826–829
[Na] V. Namias, "A simple derivation of Stirling's asymptotic series" Amer. Math. Monthly, 93 (1986) pp. 25–29
How to Cite This Entry:
Stirling formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stirling_formula&oldid=13618
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article