@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr:-VHDD6KJX-8
  skos:prefLabel "analytic number theory"@en, "théorie analytique des nombres"@fr ;
  a skos:Concept ;
  skos:narrower psr:-B0SJH805-9 .

psr: a skos:ConceptScheme .
psr:-W127WDLN-J
  skos:prefLabel "factorielle"@fr, "factorial"@en ;
  a skos:Concept ;
  skos:narrower psr:-B0SJH805-9 .

psr:-B0SJH805-9
  a skos:Concept ;
  skos:broader psr:-VHDD6KJX-8, psr:-W127WDLN-J ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Stirling%27s_approximation>, <https://fr.wikipedia.org/wiki/Formule_de_Stirling> ;
  skos:altLabel "Stirling's formula"@en ;
  dc:modified "2024-10-18"^^xsd:date ;
  skos:prefLabel "Stirling's approximation"@en, "formule de Stirling"@fr ;
  skos:definition """In mathematics, <b>Stirling's approximation</b> (or <b>Stirling's formula</b>) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle n}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>n</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle n}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="n"></span>. 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: <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\ln(n!)=n\\\\ln n-n+O(\\\\ln n),}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>ln</mi>         <mo>⁡<!-- ⁡ --></mo>         <mo stretchy="false">(</mo>         <mi>n</mi>         <mo>!</mo>         <mo stretchy="false">)</mo>         <mo>=</mo>         <mi>n</mi>         <mi>ln</mi>         <mo>⁡<!-- ⁡ --></mo>         <mi>n</mi>         <mo>−<!-- − --></mo>         <mi>n</mi>         <mo>+</mo>         <mi>O</mi>         <mo stretchy="false">(</mo>         <mi>ln</mi>         <mo>⁡<!-- ⁡ --></mo>         <mi>n</mi>         <mo stretchy="false">)</mo>         <mo>,</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\ln(n!)=n\\\\ln n-n+O(\\\\ln n),}</annotation>   </semantics> </math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e36df40589da1d3ac3da57cd2e786e97cb4cac31" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.418ex; height:2.843ex;" alt="{\\\\displaystyle \\\\ln(n!)=n\\\\ln n-n+O(\\\\ln n),}"></div> where the big O notation means that, for all sufficiently large values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle n}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>n</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle n}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="n"></span>, the difference between <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\ln(n!)}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>ln</mi>         <mo>⁡<!-- ⁡ --></mo>         <mo stretchy="false">(</mo>         <mi>n</mi>         <mo>!</mo>         <mo stretchy="false">)</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\ln(n!)}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d264f09c80b779e9028658543d5e31ae647cacf1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.79ex; height:2.843ex;" alt="{\\\\displaystyle \\\\ln(n!)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle n\\\\ln n-n}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>n</mi>         <mi>ln</mi>         <mo>⁡<!-- ⁡ --></mo>         <mi>n</mi>         <mo>−<!-- − --></mo>         <mi>n</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle n\\\\ln n-n}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e38807f2f9c7f462113eee75d0be2e1b5166313a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.738ex; height:2.343ex;" alt="{\\\\displaystyle n\\\\ln n-n}"></span> 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 <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\log _{2}(n!)=n\\\\log _{2}n-n\\\\log _{2}e+O(\\\\log _{2}n).}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>log</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msub>         <mo>⁡<!-- ⁡ --></mo>         <mo stretchy="false">(</mo>         <mi>n</mi>         <mo>!</mo>         <mo stretchy="false">)</mo>         <mo>=</mo>         <mi>n</mi>         <msub>           <mi>log</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msub>         <mo>⁡<!-- ⁡ --></mo>         <mi>n</mi>         <mo>−<!-- − --></mo>         <mi>n</mi>         <msub>           <mi>log</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msub>         <mo>⁡<!-- ⁡ --></mo>         <mi>e</mi>         <mo>+</mo>         <mi>O</mi>         <mo stretchy="false">(</mo>         <msub>           <mi>log</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msub>         <mo>⁡<!-- ⁡ --></mo>         <mi>n</mi>         <mo stretchy="false">)</mo>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\log _{2}(n!)=n\\\\log _{2}n-n\\\\log _{2}e+O(\\\\log _{2}n).}</annotation>   </semantics> </math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fa4c11db5e9758b508117d276bf7fb509fa4dc4" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.561ex; height:2.843ex;" alt="{\\\\displaystyle \\\\log _{2}(n!)=n\\\\log _{2}n-n\\\\log _{2}e+O(\\\\log _{2}n).}"></div> The error term in either base can be expressed more precisely as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle {\\	frac {1}{2}}\\\\log(2\\\\pi n)+O({\\	frac {1}{n}})}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <mstyle displaystyle="false" scriptlevel="0">             <mfrac>               <mn>1</mn>               <mn>2</mn>             </mfrac>           </mstyle>         </mrow>         <mi>log</mi>         <mo>⁡<!-- ⁡ --></mo>         <mo stretchy="false">(</mo>         <mn>2</mn>         <mi>π<!-- π --></mi>         <mi>n</mi>         <mo stretchy="false">)</mo>         <mo>+</mo>         <mi>O</mi>         <mo stretchy="false">(</mo>         <mrow class="MJX-TeXAtom-ORD">           <mstyle displaystyle="false" scriptlevel="0">             <mfrac>               <mn>1</mn>               <mi>n</mi>             </mfrac>           </mstyle>         </mrow>         <mo stretchy="false">)</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle {\\	frac {1}{2}}\\\\log(2\\\\pi n)+O({\\	frac {1}{n}})}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b125b6276f948662e1497c2f6231d66dfb8ecee" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.961ex; height:3.509ex;" alt="{\\\\displaystyle {\\	frac {1}{2}}\\\\log(2\\\\pi n)+O({\\	frac {1}{n}})}"></span>, corresponding to an approximate formula for the factorial itself, <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle n!\\\\sim {\\\\sqrt {2\\\\pi n}}\\\\left({\\rac {n}{e}}\\
ight)^{n}.}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>n</mi>         <mo>!</mo>         <mo>∼<!-- ∼ --></mo>         <mrow class="MJX-TeXAtom-ORD">           <msqrt>             <mn>2</mn>             <mi>π<!-- π --></mi>             <mi>n</mi>           </msqrt>         </mrow>         <msup>           <mrow>             <mo>(</mo>             <mrow class="MJX-TeXAtom-ORD">               <mfrac>                 <mi>n</mi>                 <mi>e</mi>               </mfrac>             </mrow>             <mo>)</mo>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msup>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle n!\\\\sim {\\\\sqrt {2\\\\pi n}}\\\\left({\\rac {n}{e}}\\
ight)^{n}.}</annotation>   </semantics> </math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b3c28f23e205ed542a2b9bbeff5c56db3881877" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.837ex; height:4.843ex;" alt="{\\\\displaystyle n!\\\\sim {\\\\sqrt {2\\\\pi n}}\\\\left({\\rac {n}{e}}\\
ight)^{n}.}"></div> Here the sign <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\sim }">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mo>∼<!-- ∼ --></mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\sim }</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afcc42adfcfdc24d5c4c474869e5d8eaa78d1173" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="\\\\sim "></span> means that the two quantities are asymptotic, that is, that their ratio tends to 1 as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle n}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>n</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle n}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="n"></span> tends to infinity. The following version of the bound holds for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle n\\\\geq 1}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>n</mi>         <mo>≥<!-- ≥ --></mo>         <mn>1</mn>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle n\\\\geq 1}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="n\\\\geq 1"></span>, rather than only asymptotically: <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle {\\\\sqrt {2\\\\pi n}}\\\\ \\\\left({\\rac {n}{e}}\\
ight)^{n}e^{\\rac {1}{12n+1}}<n!<{\\\\sqrt {2\\\\pi n}}\\\\ \\\\left({\\rac {n}{e}}\\
ight)^{n}e^{\\rac {1}{12n}}.}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <msqrt>             <mn>2</mn>             <mi>π<!-- π --></mi>             <mi>n</mi>           </msqrt>         </mrow>         <mtext> </mtext>         <msup>           <mrow>             <mo>(</mo>             <mrow class="MJX-TeXAtom-ORD">               <mfrac>                 <mi>n</mi>                 <mi>e</mi>               </mfrac>             </mrow>             <mo>)</mo>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msup>         <msup>           <mi>e</mi>           <mrow class="MJX-TeXAtom-ORD">             <mfrac>               <mn>1</mn>               <mrow>                 <mn>12</mn>                 <mi>n</mi>                 <mo>+</mo>                 <mn>1</mn>               </mrow>             </mfrac>           </mrow>         </msup>         <mo>&lt;</mo>         <mi>n</mi>         <mo>!</mo>         <mo>&lt;</mo>         <mrow class="MJX-TeXAtom-ORD">           <msqrt>             <mn>2</mn>             <mi>π<!-- π --></mi>             <mi>n</mi>           </msqrt>         </mrow>         <mtext> </mtext>         <msup>           <mrow>             <mo>(</mo>             <mrow class="MJX-TeXAtom-ORD">               <mfrac>                 <mi>n</mi>                 <mi>e</mi>               </mfrac>             </mrow>             <mo>)</mo>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msup>         <msup>           <mi>e</mi>           <mrow class="MJX-TeXAtom-ORD">             <mfrac>               <mn>1</mn>               <mrow>                 <mn>12</mn>                 <mi>n</mi>               </mrow>             </mfrac>           </mrow>         </msup>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle {\\\\sqrt {2\\\\pi n}}\\\\ \\\\left({\\rac {n}{e}}\\
ight)^{n}e^{\\rac {1}{12n+1}}&lt;n!&lt;{\\\\sqrt {2\\\\pi n}}\\\\ \\\\left({\\rac {n}{e}}\\
ight)^{n}e^{\\rac {1}{12n}}.}</annotation>   </semantics> </math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b75b90f83727d024724ab6aea4627bb785855c87" class="mwe-math-
<br/>fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:44.426ex; height:5.176ex;" alt="{\\\\displaystyle {\\\\sqrt {2\\\\pi n}}\\\\ \\\\left({\\rac {n}{e}}\\
ight)^{n}e^{\\rac {1}{12n+1}}<n!<{\\\\sqrt {2\\\\pi n}}\\\\ \\\\left({\\rac {n}{e}}\\
ight)^{n}e^{\\rac {1}{12n}}.}"></div>
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Stirling%27s_approximation">https://en.wikipedia.org/wiki/Stirling%27s_approximation</a>)"""@en, """La <b>formule de Stirling</b>, du nom du mathématicien écossais James Stirling, donne un équivalent de la factorielle d'un entier naturel <i>n </i>quand <i>n </i>tend vers l'infini :  <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\lim _{n\\	o +\\\\infty }{n\\\\,! \\\\over {\\\\sqrt {2\\\\pi n}}\\\\;\\\\left({n}/{\\
m {e}}\\
ight)^{n}}=1}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <munder>           <mo movablelimits="true" form="prefix">lim</mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo stretchy="false">→<!-- → --></mo>             <mo>+</mo>             <mi mathvariant="normal">∞<!-- ∞ --></mi>           </mrow>         </munder>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mrow>               <mi>n</mi>               <mspace width="thinmathspace"></mspace>               <mo>!</mo>             </mrow>             <mrow>               <mrow class="MJX-TeXAtom-ORD">                 <msqrt>                   <mn>2</mn>                   <mi>π<!-- π --></mi>                   <mi>n</mi>                 </msqrt>               </mrow>               <mspace width="thickmathspace"></mspace>               <msup>                 <mrow>                   <mo>(</mo>                   <mrow>                     <mrow class="MJX-TeXAtom-ORD">                       <mi>n</mi>                     </mrow>                     <mrow class="MJX-TeXAtom-ORD">                       <mo>/</mo>                     </mrow>                     <mrow class="MJX-TeXAtom-ORD">                       <mrow class="MJX-TeXAtom-ORD">                         <mi mathvariant="normal">e</mi>                       </mrow>                     </mrow>                   </mrow>                   <mo>)</mo>                 </mrow>                 <mrow class="MJX-TeXAtom-ORD">                   <mi>n</mi>                 </mrow>               </msup>             </mrow>           </mfrac>         </mrow>         <mo>=</mo>         <mn>1</mn>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\lim _{n\\	o +\\\\infty }{n\\\\,! \\\\over {\\\\sqrt {2\\\\pi n}}\\\\;\\\\left({n}/{\\
m {e}}\\
ight)^{n}}=1}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/813d65092ba303cb3e15f5a4ab5c6e992c29eeb8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.123ex; height:6.509ex;" alt="\\\\lim _{{n\\	o +\\\\infty }}{n\\\\,! \\\\over {\\\\sqrt  {2\\\\pi n}}\\\\;\\\\left({n}/{{\\
m {e}}}\\
ight)^{{n}}}=1"></span></center> que l'on trouve souvent écrite ainsi</span> :  <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle n\\\\,!\\\\sim {\\\\sqrt {2\\\\pi n}}\\\\,\\\\left({n \\\\over {\\
m {e}}}\\
ight)^{n}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>n</mi>         <mspace width="thinmathspace"></mspace>         <mo>!</mo>         <mo>∼<!-- ∼ --></mo>         <mrow class="MJX-TeXAtom-ORD">           <msqrt>             <mn>2</mn>             <mi>π<!-- π --></mi>             <mi>n</mi>           </msqrt>         </mrow>         <mspace width="thinmathspace"></mspace>         <msup>           <mrow>             <mo>(</mo>             <mrow class="MJX-TeXAtom-ORD">               <mfrac>                 <mi>n</mi>                 <mrow class="MJX-TeXAtom-ORD">                   <mrow class="MJX-TeXAtom-ORD">                     <mi mathvariant="normal">e</mi>                   </mrow>                 </mrow>               </mfrac>             </mrow>             <mo>)</mo>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msup>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle n\\\\,!\\\\sim {\\\\sqrt {2\\\\pi n}}\\\\,\\\\left({n \\\\over {\\
m {e}}}\\
ight)^{n}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a33528d86b7d1ee471838c3bdd7dc8951cc30c6" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.964ex; height:4.843ex;" alt="n\\\\,!\\\\sim {\\\\sqrt  {2\\\\pi n}}\\\\,\\\\left({n \\\\over {{\\
m {e}}}}\\
ight)^{n}"></span></center> où le nombre <span class="texhtml">e</span> désigne la base de l'exponentielle. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Formule_de_Stirling">https://fr.wikipedia.org/wiki/Formule_de_Stirling</a>)"""@fr ;
  skos:inScheme psr: ;
  dc:created "2023-08-03"^^xsd:date .