Stirling's approximation

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approximation (or Stirling's formula) is an <a href="https://en-m-wikipedia-org.translate.goog/wiki/Asymptotic_analysis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Asymptotic analysis">asymptotic</a> approximation for <a href="https://en-m-wikipedia-org.translate.goog/wiki/Factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Factorial">factorials</a>. It is a good approximation, leading to accurate results even for small values of <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">. It is named after <a href="https://en-m-wikipedia-org.translate.goog/wiki/James_Stirling_(mathematician)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="James Stirling (mathematician)">James Stirling</a>, though a related but less precise result was first stated by <a href="https://en-m-wikipedia-org.translate.goog/wiki/Abraham_de_Moivre?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abraham de Moivre">Abraham de Moivre</a>.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-dutka-1">[1]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-LeCam1986-2">[2]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Pearson1924-3">[3]</a> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Mplwp_factorial_gamma_stirling.svg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Mplwp_factorial_gamma_stirling.svg/300px-Mplwp_factorial_gamma_stirling.svg.png" decoding="async" width="300" height="200" class="mw-file-element" srcset="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Mplwp_factorial_gamma_stirling.svg/450px-Mplwp_factorial_gamma_stirling.svg.png 1.5x,https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Mplwp_factorial_gamma_stirling.svg/600px-Mplwp_factorial_gamma_stirling.svg.png 2x" data-file-width="600" data-file-height="400"></a> <figcaption> Comparison of Stirling's approximation with the factorial </figcaption> </figure> One way of stating the approximation involves the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Logarithm?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Logarithm">logarithm</a> of the factorial: <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e36df40589da1d3ac3da57cd2e786e97cb4cac31" class="mwe-math-fallback-image-display mw-invert skin-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),}"> where the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Big_O_notation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Big O notation">big O notation</a> means that, for all sufficiently large values of <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, the difference between <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d264f09c80b779e9028658543d5e31ae647cacf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.79ex; height:2.843ex;" alt="{\displaystyle \ln(n!)}"> and <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e38807f2f9c7f462113eee75d0be2e1b5166313a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.738ex; height:2.343ex;" alt="{\displaystyle n\ln n-n}"> will be at most proportional to the logarithm of <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">. In computer science applications such as the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Comparison_sort?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Number_of_comparisons_required_to_sort_a_list" title="Comparison sort">worst-case lower bound for comparison sorting</a>, it is convenient to instead use the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Binary_logarithm?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Binary logarithm">binary logarithm</a>, giving the equivalent form <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fa4c11db5e9758b508117d276bf7fb509fa4dc4" class="mwe-math-fallback-image-display mw-invert skin-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).}"> The error term in either base can be expressed more precisely as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}\log(2\pi n)+O({\tfrac {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 </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 {\tfrac {1}{2}}\log(2\pi n)+O({\tfrac {1}{n}})} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b125b6276f948662e1497c2f6231d66dfb8ecee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.961ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}\log(2\pi n)+O({\tfrac {1}{n}})}">, corresponding to an approximate formula for the factorial itself, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{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 </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({\frac {n}{e}}\right)^{n}.} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b3c28f23e205ed542a2b9bbeff5c56db3881877" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.837ex; height:4.843ex;" alt="{\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}.}"> Here the sign <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afcc42adfcfdc24d5c4c474869e5d8eaa78d1173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle \sim }"> means that the two quantities are <a href="https://en-m-wikipedia-org.translate.goog/wiki/Asymptotic_analysis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Asymptotic analysis">asymptotic</a>, that is, that their ratio tends to 1 as <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> tends to infinity. The following version of the bound <a class="mw-selflink-fragment" href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Speed_of_convergence_and_error_estimates">holds for all <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}">, rather than only asymptotically</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2\pi n}}\ \left({\frac {n}{e}}\right)^{n}e^{\left({\frac {1}{12n}}-{\frac {1}{360n^{3}}}\right)}<n!<{\sqrt {2\pi n}}\ \left({\frac {n}{e}}\right)^{n}e^{\frac {1}{12n}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 2 </mn> <mi> 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"> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mrow> <mn> 12 </mn> <mi> n </mi> </mrow> </mfrac> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mrow> <mn> 360 </mn> <msup> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> < </mo> <mi> n </mi> <mo> ! </mo> <mo> < </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 2 </mn> <mi> 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({\frac {n}{e}}\right)^{n}e^{\left({\frac {1}{12n}}-{\frac {1}{360n^{3}}}\right)}<n!<{\sqrt {2\pi n}}\ \left({\frac {n}{e}}\right)^{n}e^{\frac {1}{12n}}.} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a37064928396b06c50c33f83ca9089f88abe3627" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:50.858ex; height:6.343ex;" alt="{\displaystyle {\sqrt {2\pi n}}\ \left({\frac {n}{e}}\right)^{n}e^{\left({\frac {1}{12n}}-{\frac {1}{360n^{3}}}\right)}<n!<{\sqrt {2\pi n}}\ \left({\frac {n}{e}}\right)^{n}e^{\frac {1}{12n}}.}"> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"> <input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"> <div class="toctitle" lang="en" dir="ltr"> <h2 id="mw-toc-heading">Contents</h2><label class="toctogglelabel" for="toctogglecheckbox"></label> </div> <ul> <li class="toclevel-1 tocsection-1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Derivation">1 Derivation</a></li> <li class="toclevel-1 tocsection-2"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Alternative_derivations">2 Alternative derivations</a> <ul> <li class="toclevel-2 tocsection-3"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Higher_orders">2.1 Higher orders</a></li> <li class="toclevel-2 tocsection-4"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Complex-analytic_version">2.2 Complex-analytic version</a></li> </ul></li> <li class="toclevel-1 tocsection-5"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Speed_of_convergence_and_error_estimates">3 Speed of convergence and error estimates</a></li> <li class="toclevel-1 tocsection-6"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Stirling's_formula_for_the_gamma_function">4 Stirling's formula for the gamma function</a></li> <li class="toclevel-1 tocsection-7"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Error_bounds">5 Error bounds</a></li> <li class="toclevel-1 tocsection-8"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#A_convergent_version_of_Stirling's_formula">6 A convergent version of Stirling's formula</a></li> <li class="toclevel-1 tocsection-9"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Versions_suitable_for_calculators">7 Versions suitable for calculators</a></li> <li class="toclevel-1 tocsection-10"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#History">8 History</a></li> <li class="toclevel-1 tocsection-11"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#See_also">9 See also</a></li> <li class="toclevel-1 tocsection-12"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#References">10 References</a></li> <li class="toclevel-1 tocsection-13"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Further_reading">11 Further reading</a></li> <li class="toclevel-1 tocsection-14"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#External_links">12 External links</a></li> </ul> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"> <h2 id="Derivation">Derivation</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Stirling's_approximation&action=edit&section=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Derivation" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-1 collapsible-block" id="mf-section-1"> Roughly speaking, the simplest version of Stirling's formula can be quickly obtained by approximating the sum <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(n!)=\sum _{j=1}^{n}\ln j}"><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> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <mi> ln </mi> <mo> ⁡ </mo> <mi> j </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \ln(n!)=\sum _{j=1}^{n}\ln j} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b37812cd938e78035eb60bbaf3ce1ccfa624615" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:15.915ex; height:7.176ex;" alt="{\displaystyle \ln(n!)=\sum _{j=1}^{n}\ln j}"> </noscript><span class="lazy-image-placeholder" style="width: 15.915ex;height: 7.176ex;vertical-align: -3.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b37812cd938e78035eb60bbaf3ce1ccfa624615" data-alt="{\displaystyle \ln(n!)=\sum _{j=1}^{n}\ln j}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  with an <a href="https://en-m-wikipedia-org.translate.goog/wiki/Integral?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Integral">integral</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{j=1}^{n}\ln j\approx \int _{1}^{n}\ln x\,{\rm {d}}x=n\ln n-n+1.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <mi> ln </mi> <mo> ⁡ </mo> <mi> j </mi> <mo> ≈ </mo> <msubsup> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msubsup> <mi> ln </mi> <mo> ⁡ </mo> <mi> x </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> </mrow> <mi> x </mi> <mo> = </mo> <mi> n </mi> <mi> ln </mi> <mo> ⁡ </mo> <mi> n </mi> <mo> − </mo> <mi> n </mi> <mo> + </mo> <mn> 1. </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sum _{j=1}^{n}\ln j\approx \int _{1}^{n}\ln x\,{\rm {d}}x=n\ln n-n+1.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a35e8e0205988cf8552eec558d121f764d7e4021" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:38.346ex; height:7.176ex;" alt="{\displaystyle \sum _{j=1}^{n}\ln j\approx \int _{1}^{n}\ln x\,{\rm {d}}x=n\ln n-n+1.}"> </noscript><span class="lazy-image-placeholder" style="width: 38.346ex;height: 7.176ex;vertical-align: -3.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a35e8e0205988cf8552eec558d121f764d7e4021" data-alt="{\displaystyle \sum _{j=1}^{n}\ln j\approx \int _{1}^{n}\ln x\,{\rm {d}}x=n\ln n-n+1.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  The full formula, together with precise estimates of its error, can be derived as follows. Instead of approximating <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> <mo> ! </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n!} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae971720be3cc9b8d82f4cdac89cb89877514a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:2.176ex;" alt="{\displaystyle n!}"> </noscript><span class="lazy-image-placeholder" style="width: 2.042ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae971720be3cc9b8d82f4cdac89cb89877514a6" data-alt="{\displaystyle n!}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , one considers its <a href="https://en-m-wikipedia-org.translate.goog/wiki/Natural_logarithm?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Natural logarithm">natural logarithm</a>, as this is a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Slowly_varying_function?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Slowly varying function">slowly varying function</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(n!)=\ln 1+\ln 2+\cdots +\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> ln </mi> <mo> ⁡ </mo> <mn> 1 </mn> <mo> + </mo> <mi> ln </mi> <mo> ⁡ </mo> <mn> 2 </mn> <mo> + </mo> <mo> ⋯ </mo> <mo> + </mo> <mi> ln </mi> <mo> ⁡ </mo> <mi> n </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \ln(n!)=\ln 1+\ln 2+\cdots +\ln n.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef550ba2dbfb9b226fa3fb08d7c8eee738f3ced5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.479ex; height:2.843ex;" alt="{\displaystyle \ln(n!)=\ln 1+\ln 2+\cdots +\ln n.}"> </noscript><span class="lazy-image-placeholder" style="width: 31.479ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef550ba2dbfb9b226fa3fb08d7c8eee738f3ced5" data-alt="{\displaystyle \ln(n!)=\ln 1+\ln 2+\cdots +\ln n.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  The right-hand side of this equation minus <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}(\ln 1+\ln n)={\tfrac {1}{2}}\ln 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> <mo stretchy="false"> ( </mo> <mi> ln </mi> <mo> ⁡ </mo> <mn> 1 </mn> <mo> + </mo> <mi> ln </mi> <mo> ⁡ </mo> <mi> n </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mstyle> </mrow> <mi> ln </mi> <mo> ⁡ </mo> <mi> n </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\tfrac {1}{2}}(\ln 1+\ln n)={\tfrac {1}{2}}\ln n} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ff466fe2a6a445eb057ba2a96e93848c2b4116c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:22.383ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}(\ln 1+\ln n)={\tfrac {1}{2}}\ln n}"> </noscript><span class="lazy-image-placeholder" style="width: 22.383ex;height: 3.509ex;vertical-align: -1.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ff466fe2a6a445eb057ba2a96e93848c2b4116c" data-alt="{\displaystyle {\tfrac {1}{2}}(\ln 1+\ln n)={\tfrac {1}{2}}\ln n}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  is the approximation by the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Trapezoid_rule?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Trapezoid rule">trapezoid rule</a> of the integral <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(n!)-{\tfrac {1}{2}}\ln n\approx \int _{1}^{n}\ln x\,{\rm {d}}x=n\ln n-n+1,}"><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> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mstyle> </mrow> <mi> ln </mi> <mo> ⁡ </mo> <mi> n </mi> <mo> ≈ </mo> <msubsup> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msubsup> <mi> ln </mi> <mo> ⁡ </mo> <mi> x </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> </mrow> <mi> x </mi> <mo> = </mo> <mi> n </mi> <mi> ln </mi> <mo> ⁡ </mo> <mi> n </mi> <mo> − </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \ln(n!)-{\tfrac {1}{2}}\ln n\approx \int _{1}^{n}\ln x\,{\rm {d}}x=n\ln n-n+1,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6a6b30d17f28e2c139ac56d455fcbc7d4eb29d6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:45.717ex; height:5.843ex;" alt="{\displaystyle \ln(n!)-{\tfrac {1}{2}}\ln n\approx \int _{1}^{n}\ln x\,{\rm {d}}x=n\ln n-n+1,}"> </noscript><span class="lazy-image-placeholder" style="width: 45.717ex;height: 5.843ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6a6b30d17f28e2c139ac56d455fcbc7d4eb29d6" data-alt="{\displaystyle \ln(n!)-{\tfrac {1}{2}}\ln n\approx \int _{1}^{n}\ln x\,{\rm {d}}x=n\ln n-n+1,}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  and the error in this approximation is given by the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Euler%E2%80%93Maclaurin_formula?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Euler–Maclaurin formula">Euler–Maclaurin formula</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\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{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi> ln </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> ! </mo> <mo stretchy="false"> ) </mo> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mstyle> </mrow> <mi> ln </mi> <mo> ⁡ </mo> <mi> n </mi> </mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mstyle> </mrow> <mi> ln </mi> <mo> ⁡ </mo> <mn> 1 </mn> <mo> + </mo> <mi> ln </mi> <mo> ⁡ </mo> <mn> 2 </mn> <mo> + </mo> <mi> ln </mi> <mo> ⁡ </mo> <mn> 3 </mn> <mo> + </mo> <mo> ⋯ </mo> <mo> + </mo> <mi> ln </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mstyle> </mrow> <mi> ln </mi> <mo> ⁡ </mo> <mi> n </mi> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mi> n </mi> <mi> ln </mi> <mo> ⁡ </mo> <mi> n </mi> <mo> − </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> <mo> + </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false"> ( </mo> <mo> − </mo> <mn> 1 </mn> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <msub> <mi> B </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mrow> <mrow> <mi> k </mi> <mo stretchy="false"> ( </mo> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mrow> </mfrac> </mrow> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <msup> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> </mfrac> </mrow> <mo> − </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <msub> <mi> R </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> </msub> <mo> , </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}\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{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/559acb779d1d9e2ca828e3f8ebec37ebee2a8b6e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:69.729ex; height:10.843ex;" alt="{\displaystyle {\begin{aligned}\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{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 69.729ex;height: 10.843ex;vertical-align: -4.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/559acb779d1d9e2ca828e3f8ebec37ebee2a8b6e" data-alt="{\displaystyle {\begin{aligned}\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{aligned}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{k}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> B </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle B_{k}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a6457760e36cf45e1471e33bcc1536cb4802fb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.853ex; height:2.509ex;" alt="{\displaystyle B_{k}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.853ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a6457760e36cf45e1471e33bcc1536cb4802fb9" data-alt="{\displaystyle B_{k}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Bernoulli_number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Bernoulli number">Bernoulli number</a>, and Rm,n is the remainder term in the Euler–Maclaurin formula. Take limits to find that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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}.}"><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> <mi mathvariant="normal"> ∞ </mi> </mrow> </munder> <mrow> <mo> ( </mo> <mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mstyle> </mrow> <mi> ln </mi> <mo> ⁡ </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mn> 1 </mn> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false"> ( </mo> <mo> − </mo> <mn> 1 </mn> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <msub> <mi> B </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mrow> <mrow> <mi> k </mi> <mo stretchy="false"> ( </mo> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mrow> </mfrac> </mrow> <mo> + </mo> <munder> <mo movablelimits="true" form="prefix"> lim </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo stretchy="false"> → </mo> <mi mathvariant="normal"> ∞ </mi> </mrow> </munder> <msub> <mi> R </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> </msub> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \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}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1de677b895e298f6e743494ca20df4f6c33349f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:67.195ex; height:7.009ex;" alt="{\displaystyle \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}.}"> </noscript><span class="lazy-image-placeholder" style="width: 67.195ex;height: 7.009ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1de677b895e298f6e743494ca20df4f6c33349f" data-alt="{\displaystyle \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}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  Denote this limit as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> y </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle y} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> </noscript><span class="lazy-image-placeholder" style="width: 1.155ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" data-alt="{\displaystyle y}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Because the remainder Rm,n in the Euler–Maclaurin formula satisfies <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{m,n}=\lim _{n\to \infty }R_{m,n}+O\left({\frac {1}{n^{m}}}\right),}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> R </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> </msub> <mo> = </mo> <munder> <mo movablelimits="true" form="prefix"> lim </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo stretchy="false"> → </mo> <mi mathvariant="normal"> ∞ </mi> </mrow> </munder> <msub> <mi> R </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> </msub> <mo> + </mo> <mi> O </mi> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <msup> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> </mrow> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle R_{m,n}=\lim _{n\to \infty }R_{m,n}+O\left({\frac {1}{n^{m}}}\right),} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/385ea6e95b78cc25641016606c4d729fb1d9dfc3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.885ex; height:6.176ex;" alt="{\displaystyle R_{m,n}=\lim _{n\to \infty }R_{m,n}+O\left({\frac {1}{n^{m}}}\right),}"> </noscript><span class="lazy-image-placeholder" style="width: 30.885ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/385ea6e95b78cc25641016606c4d729fb1d9dfc3" data-alt="{\displaystyle R_{m,n}=\lim _{n\to \infty }R_{m,n}+O\left({\frac {1}{n^{m}}}\right),}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  where <a href="https://en-m-wikipedia-org.translate.goog/wiki/Big-O_notation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Big-O notation">big-O notation</a> is used, combining the equations above yields the approximation formula in its logarithmic form: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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).}"><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> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> n </mi> <mi> e </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mstyle> </mrow> <mi> ln </mi> <mo> ⁡ </mo> <mi> n </mi> <mo> + </mo> <mi> y </mi> <mo> + </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false"> ( </mo> <mo> − </mo> <mn> 1 </mn> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <msub> <mi> B </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mrow> <mrow> <mi> k </mi> <mo stretchy="false"> ( </mo> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <msup> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> + </mo> <mi> O </mi> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <msup> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> </mrow> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \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).} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28b026d2e5f73bf6ce88c47c248844d4cf622a73" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:63.818ex; height:7.009ex;" alt="{\displaystyle \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).}"> </noscript><span class="lazy-image-placeholder" style="width: 63.818ex;height: 7.009ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28b026d2e5f73bf6ce88c47c248844d4cf622a73" data-alt="{\displaystyle \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).}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  Taking the exponential of both sides and choosing any positive integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> m </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> </noscript><span class="lazy-image-placeholder" style="width: 2.04ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" data-alt="{\displaystyle m}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , one obtains a formula involving an unknown quantity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{y}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle e^{y}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ff26085237c8dc6802eba0882a8aea22e890183" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.133ex; height:2.343ex;" alt="{\displaystyle e^{y}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.133ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ff26085237c8dc6802eba0882a8aea22e890183" data-alt="{\displaystyle e^{y}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . For m = 1, the formula is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!=e^{y}{\sqrt {n}}\left({\frac {n}{e}}\right)^{n}\left(1+O\left({\frac {1}{n}}\right)\right).}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> ! </mo> <mo> = </mo> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <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> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> O </mi> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mi> n </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n!=e^{y}{\sqrt {n}}\left({\frac {n}{e}}\right)^{n}\left(1+O\left({\frac {1}{n}}\right)\right).} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3280cd1e77e7ec163297311e38d28a2bcbc33c1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.486ex; height:6.176ex;" alt="{\displaystyle n!=e^{y}{\sqrt {n}}\left({\frac {n}{e}}\right)^{n}\left(1+O\left({\frac {1}{n}}\right)\right).}"> </noscript><span class="lazy-image-placeholder" style="width: 33.486ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3280cd1e77e7ec163297311e38d28a2bcbc33c1" data-alt="{\displaystyle n!=e^{y}{\sqrt {n}}\left({\frac {n}{e}}\right)^{n}\left(1+O\left({\frac {1}{n}}\right)\right).}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  The quantity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{y}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle e^{y}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ff26085237c8dc6802eba0882a8aea22e890183" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.133ex; height:2.343ex;" alt="{\displaystyle e^{y}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.133ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ff26085237c8dc6802eba0882a8aea22e890183" data-alt="{\displaystyle e^{y}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  can be found by taking the limit on both sides as <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> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  tends to infinity and using <a href="https://en-m-wikipedia-org.translate.goog/wiki/Wallis_product?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Wallis product">Wallis' product</a>, which shows that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{y}={\sqrt {2\pi }}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> </mrow> </msup> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 2 </mn> <mi> π </mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle e^{y}={\sqrt {2\pi }}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa0fcfb78cc1c5ec93e19890d73eb3631fbfb6aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.662ex; height:3.009ex;" alt="{\displaystyle e^{y}={\sqrt {2\pi }}}"> </noscript><span class="lazy-image-placeholder" style="width: 9.662ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa0fcfb78cc1c5ec93e19890d73eb3631fbfb6aa" data-alt="{\displaystyle e^{y}={\sqrt {2\pi }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Therefore, one obtains Stirling's formula: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!={\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\left(1+O\left({\frac {1}{n}}\right)\right).}"><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 </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> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> O </mi> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mi> n </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n!={\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\left(1+O\left({\frac {1}{n}}\right)\right).} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ac088bf420d3ad4257f7593a0c1af48b0cf22c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.847ex; height:6.176ex;" alt="{\displaystyle n!={\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\left(1+O\left({\frac {1}{n}}\right)\right).}"> </noscript><span class="lazy-image-placeholder" style="width: 33.847ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ac088bf420d3ad4257f7593a0c1af48b0cf22c" data-alt="{\displaystyle n!={\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\left(1+O\left({\frac {1}{n}}\right)\right).}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"> <h2 id="Alternative_derivations">Alternative derivations</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Stirling's_approximation&action=edit&section=2&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Alternative derivations" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-2 collapsible-block" id="mf-section-2"> An alternative formula for <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> <mo> ! </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n!} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae971720be3cc9b8d82f4cdac89cb89877514a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:2.176ex;" alt="{\displaystyle n!}"> </noscript><span class="lazy-image-placeholder" style="width: 2.042ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae971720be3cc9b8d82f4cdac89cb89877514a6" data-alt="{\displaystyle n!}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  using the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Gamma_function?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gamma function">gamma function</a> is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!=\int _{0}^{\infty }x^{n}e^{-x}\,{\rm {d}}x.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> ! </mo> <mo> = </mo> <msubsup> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∞ </mi> </mrow> </msubsup> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mi> x </mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> </mrow> <mi> x </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n!=\int _{0}^{\infty }x^{n}e^{-x}\,{\rm {d}}x.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec3d16f1466e7bef37e4674bce9f3ca8baa5a43b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.605ex; height:5.843ex;" alt="{\displaystyle n!=\int _{0}^{\infty }x^{n}e^{-x}\,{\rm {d}}x.}"> </noscript><span class="lazy-image-placeholder" style="width: 19.605ex;height: 5.843ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec3d16f1466e7bef37e4674bce9f3ca8baa5a43b" data-alt="{\displaystyle n!=\int _{0}^{\infty }x^{n}e^{-x}\,{\rm {d}}x.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  (as can be seen by repeated integration by parts). Rewriting and changing variables x = ny, one obtains <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> ! </mo> <mo> = </mo> <msubsup> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∞ </mi> </mrow> </msubsup> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mi> ln </mi> <mo> ⁡ </mo> <mi> x </mi> <mo> − </mo> <mi> x </mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> </mrow> <mi> x </mi> <mo> = </mo> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mi> ln </mi> <mo> ⁡ </mo> <mi> n </mi> </mrow> </msup> <mi> n </mi> <msubsup> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∞ </mi> </mrow> </msubsup> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo stretchy="false"> ( </mo> <mi> ln </mi> <mo> ⁡ </mo> <mi> y </mi> <mo> − </mo> <mi> y </mi> <mo stretchy="false"> ) </mo> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> </mrow> <mi> y </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 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.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5c1d168cbbe92d515939df7b8703b02bac5aa0f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:47.257ex; height:5.843ex;" alt="{\displaystyle 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.}"> </noscript><span class="lazy-image-placeholder" style="width: 47.257ex;height: 5.843ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5c1d168cbbe92d515939df7b8703b02bac5aa0f" data-alt="{\displaystyle 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.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  Applying <a href="https://en-m-wikipedia-org.translate.goog/wiki/Laplace%27s_method?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Laplace's method">Laplace's method</a> one has <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{\infty }e^{n(\ln y-y)}\,{\rm {d}}y\sim {\sqrt {\frac {2\pi }{n}}}e^{-n},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∞ </mi> </mrow> </msubsup> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo stretchy="false"> ( </mo> <mi> ln </mi> <mo> ⁡ </mo> <mi> y </mi> <mo> − </mo> <mi> y </mi> <mo stretchy="false"> ) </mo> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> </mrow> <mi> y </mi> <mo> ∼ </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn> 2 </mn> <mi> π </mi> </mrow> <mi> n </mi> </mfrac> </msqrt> </mrow> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mi> n </mi> </mrow> </msup> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \int _{0}^{\infty }e^{n(\ln y-y)}\,{\rm {d}}y\sim {\sqrt {\frac {2\pi }{n}}}e^{-n},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cb2e06231b3f5f3502bb001fface31c74836498" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.794ex; height:6.176ex;" alt="{\displaystyle \int _{0}^{\infty }e^{n(\ln y-y)}\,{\rm {d}}y\sim {\sqrt {\frac {2\pi }{n}}}e^{-n},}"> </noscript><span class="lazy-image-placeholder" style="width: 28.794ex;height: 6.176ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cb2e06231b3f5f3502bb001fface31c74836498" data-alt="{\displaystyle \int _{0}^{\infty }e^{n(\ln y-y)}\,{\rm {d}}y\sim {\sqrt {\frac {2\pi }{n}}}e^{-n},}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  which recovers Stirling's formula: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!\sim e^{n\ln n}n{\sqrt {\frac {2\pi }{n}}}e^{-n}={\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> ! </mo> <mo> ∼ </mo> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mi> ln </mi> <mo> ⁡ </mo> <mi> n </mi> </mrow> </msup> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn> 2 </mn> <mi> π </mi> </mrow> <mi> n </mi> </mfrac> </msqrt> </mrow> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mi> n </mi> </mrow> </msup> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 2 </mn> <mi> 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 e^{n\ln n}n{\sqrt {\frac {2\pi }{n}}}e^{-n}={\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b91b7ad167af44b3b4bb07390fce77ba1ba23879" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.998ex; height:6.176ex;" alt="{\displaystyle n!\sim e^{n\ln n}n{\sqrt {\frac {2\pi }{n}}}e^{-n}={\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}.}"> </noscript><span class="lazy-image-placeholder" style="width: 36.998ex;height: 6.176ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b91b7ad167af44b3b4bb07390fce77ba1ba23879" data-alt="{\displaystyle n!\sim e^{n\ln n}n{\sqrt {\frac {2\pi }{n}}}e^{-n}={\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  <div class="mw-heading mw-heading3"> <h3 id="Higher_orders">Higher orders</h3> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Stirling's_approximation&action=edit&section=3&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Higher orders" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> In fact, further corrections can also be obtained using Laplace's method. From previous result, we know that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (x)\sim x^{x}e^{-x}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> Γ </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> ∼ </mo> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msup> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mi> x </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \Gamma (x)\sim x^{x}e^{-x}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65cbf7864fae628b3dbee15f718cedf1167b6b03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.727ex; height:3.009ex;" alt="{\displaystyle \Gamma (x)\sim x^{x}e^{-x}}"> </noscript><span class="lazy-image-placeholder" style="width: 13.727ex;height: 3.009ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65cbf7864fae628b3dbee15f718cedf1167b6b03" data-alt="{\displaystyle \Gamma (x)\sim x^{x}e^{-x}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , so we "peel off" this dominant term, then perform two changes of variables, to obtain: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{-x}e^{x}\Gamma (x)=\int _{\mathbb {R} }e^{x(1+t-e^{t})}dt}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mi> x </mi> </mrow> </msup> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msup> <mi mathvariant="normal"> Γ </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> </mrow> </msub> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo> + </mo> <mi> t </mi> <mo> − </mo> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> t </mi> </mrow> </msup> <mo stretchy="false"> ) </mo> </mrow> </msup> <mi> d </mi> <mi> t </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x^{-x}e^{x}\Gamma (x)=\int _{\mathbb {R} }e^{x(1+t-e^{t})}dt} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16c287e185c84a185918e750559381cf0a02e94f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.801ex; height:5.676ex;" alt="{\displaystyle x^{-x}e^{x}\Gamma (x)=\int _{\mathbb {R} }e^{x(1+t-e^{t})}dt}"> </noscript><span class="lazy-image-placeholder" style="width: 27.801ex;height: 5.676ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16c287e185c84a185918e750559381cf0a02e94f" data-alt="{\displaystyle x^{-x}e^{x}\Gamma (x)=\int _{\mathbb {R} }e^{x(1+t-e^{t})}dt}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> To verify this: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\mathbb {R} }e^{x(1+t-e^{t})}dt{\overset {t\mapsto \ln t}{=}}e^{x}\int _{0}^{\infty }t^{x-1}e^{-xt}dt{\overset {t\mapsto t/x}{=}}x^{-x}e^{x}\int _{0}^{\infty }e^{-t}t^{x-1}dt=x^{-x}e^{x}\Gamma (x)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> </mrow> </msub> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo> + </mo> <mi> t </mi> <mo> − </mo> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> t </mi> </mrow> </msup> <mo stretchy="false"> ) </mo> </mrow> </msup> <mi> d </mi> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo> = </mo> <mrow> <mi> t </mi> <mo stretchy="false"> ↦ </mo> <mi> ln </mi> <mo> ⁡ </mo> <mi> t </mi> </mrow> </mover> </mrow> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msup> <msubsup> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∞ </mi> </mrow> </msubsup> <msup> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mi> x </mi> <mi> t </mi> </mrow> </msup> <mi> d </mi> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo> = </mo> <mrow> <mi> t </mi> <mo stretchy="false"> ↦ </mo> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mi> x </mi> </mrow> </mover> </mrow> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mi> x </mi> </mrow> </msup> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msup> <msubsup> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∞ </mi> </mrow> </msubsup> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mi> t </mi> </mrow> </msup> <msup> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> <mi> d </mi> <mi> t </mi> <mo> = </mo> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mi> x </mi> </mrow> </msup> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msup> <mi mathvariant="normal"> Γ </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \int _{\mathbb {R} }e^{x(1+t-e^{t})}dt{\overset {t\mapsto \ln t}{=}}e^{x}\int _{0}^{\infty }t^{x-1}e^{-xt}dt{\overset {t\mapsto t/x}{=}}x^{-x}e^{x}\int _{0}^{\infty }e^{-t}t^{x-1}dt=x^{-x}e^{x}\Gamma (x)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/232a803997500953b97e1a5a9f6836039210d192" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:75.156ex; height:5.843ex;" alt="{\displaystyle \int _{\mathbb {R} }e^{x(1+t-e^{t})}dt{\overset {t\mapsto \ln t}{=}}e^{x}\int _{0}^{\infty }t^{x-1}e^{-xt}dt{\overset {t\mapsto t/x}{=}}x^{-x}e^{x}\int _{0}^{\infty }e^{-t}t^{x-1}dt=x^{-x}e^{x}\Gamma (x)}"> </noscript><span class="lazy-image-placeholder" style="width: 75.156ex;height: 5.843ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/232a803997500953b97e1a5a9f6836039210d192" data-alt="{\displaystyle \int _{\mathbb {R} }e^{x(1+t-e^{t})}dt{\overset {t\mapsto \ln t}{=}}e^{x}\int _{0}^{\infty }t^{x-1}e^{-xt}dt{\overset {t\mapsto t/x}{=}}x^{-x}e^{x}\int _{0}^{\infty }e^{-t}t^{x-1}dt=x^{-x}e^{x}\Gamma (x)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Now the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\mapsto 1+t-e^{t}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> t </mi> <mo stretchy="false"> ↦ </mo> <mn> 1 </mn> <mo> + </mo> <mi> t </mi> <mo> − </mo> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> t </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle t\mapsto 1+t-e^{t}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df85098ac94220d1b2f6f7f33d40c86ca1b969bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.046ex; height:2.676ex;" alt="{\displaystyle t\mapsto 1+t-e^{t}}"> </noscript><span class="lazy-image-placeholder" style="width: 14.046ex;height: 2.676ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df85098ac94220d1b2f6f7f33d40c86ca1b969bb" data-alt="{\displaystyle t\mapsto 1+t-e^{t}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is unimodal, with maximum value zero. Locally around zero, it looks like <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -t^{2}/2}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> − </mo> <msup> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle -t^{2}/2} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87a40dd6615f474e0416abe2054ab61156437375" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.027ex; height:3.176ex;" alt="{\displaystyle -t^{2}/2}"> </noscript><span class="lazy-image-placeholder" style="width: 6.027ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87a40dd6615f474e0416abe2054ab61156437375" data-alt="{\displaystyle -t^{2}/2}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+t-e^{t}=-\tau ^{2}/2}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn> 1 </mn> <mo> + </mo> <mi> t </mi> <mo> − </mo> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> t </mi> </mrow> </msup> <mo> = </mo> <mo> − </mo> <msup> <mi> τ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 1+t-e^{t}=-\tau ^{2}/2} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdc08f8f909f9fa9fb1c11a37fe2532327ffedf9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.136ex; height:3.176ex;" alt="{\displaystyle 1+t-e^{t}=-\tau ^{2}/2}"> </noscript><span class="lazy-image-placeholder" style="width: 19.136ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdc08f8f909f9fa9fb1c11a37fe2532327ffedf9" data-alt="{\displaystyle 1+t-e^{t}=-\tau ^{2}/2}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . This equation cannot be solved in closed form, but it can be solved by serial expansion, which gives us <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=\tau -\tau ^{2}/6+\tau ^{3}/36+a_{4}\tau ^{4}+O(\tau ^{5})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> t </mi> <mo> = </mo> <mi> τ </mi> <mo> − </mo> <msup> <mi> τ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 6 </mn> <mo> + </mo> <msup> <mi> τ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 36 </mn> <mo> + </mo> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msub> <msup> <mi> τ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msup> <mo> + </mo> <mi> O </mi> <mo stretchy="false"> ( </mo> <msup> <mi> τ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 5 </mn> </mrow> </msup> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle t=\tau -\tau ^{2}/6+\tau ^{3}/36+a_{4}\tau ^{4}+O(\tau ^{5})} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0309cb95f6a07fb980fc574633583ad027ea72e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.429ex; height:3.176ex;" alt="{\displaystyle t=\tau -\tau ^{2}/6+\tau ^{3}/36+a_{4}\tau ^{4}+O(\tau ^{5})}"> </noscript><span class="lazy-image-placeholder" style="width: 37.429ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0309cb95f6a07fb980fc574633583ad027ea72e" data-alt="{\displaystyle t=\tau -\tau ^{2}/6+\tau ^{3}/36+a_{4}\tau ^{4}+O(\tau ^{5})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Now plug back to the equation to obtain <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{-x}e^{x}\Gamma (x)=\int _{\mathbb {R} }e^{-x\tau ^{2}/2}(1-\tau /3+\tau ^{2}/12+4a_{4}\tau ^{3}+O(\tau ^{4}))d\tau ={\sqrt {2\pi }}(x^{-1/2}+x^{-3/2}/12)+O(x^{-5/2})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mi> x </mi> </mrow> </msup> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msup> <mi mathvariant="normal"> Γ </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> </mrow> </msub> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mi> x </mi> <msup> <mi> τ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> </mrow> </msup> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo> − </mo> <mi> τ </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 3 </mn> <mo> + </mo> <msup> <mi> τ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 12 </mn> <mo> + </mo> <mn> 4 </mn> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msub> <msup> <mi> τ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mo> + </mo> <mi> O </mi> <mo stretchy="false"> ( </mo> <msup> <mi> τ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msup> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mi> d </mi> <mi> τ </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 2 </mn> <mi> π </mi> </msqrt> </mrow> <mo stretchy="false"> ( </mo> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 3 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 12 </mn> <mo stretchy="false"> ) </mo> <mo> + </mo> <mi> O </mi> <mo stretchy="false"> ( </mo> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 5 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> </mrow> </msup> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x^{-x}e^{x}\Gamma (x)=\int _{\mathbb {R} }e^{-x\tau ^{2}/2}(1-\tau /3+\tau ^{2}/12+4a_{4}\tau ^{3}+O(\tau ^{4}))d\tau ={\sqrt {2\pi }}(x^{-1/2}+x^{-3/2}/12)+O(x^{-5/2})} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53413a29c58763f2cc53f7f32fce6513f57ee2b1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:99.464ex; height:5.676ex;" alt="{\displaystyle x^{-x}e^{x}\Gamma (x)=\int _{\mathbb {R} }e^{-x\tau ^{2}/2}(1-\tau /3+\tau ^{2}/12+4a_{4}\tau ^{3}+O(\tau ^{4}))d\tau ={\sqrt {2\pi }}(x^{-1/2}+x^{-3/2}/12)+O(x^{-5/2})}"> </noscript><span class="lazy-image-placeholder" style="width: 99.464ex;height: 5.676ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53413a29c58763f2cc53f7f32fce6513f57ee2b1" data-alt="{\displaystyle x^{-x}e^{x}\Gamma (x)=\int _{\mathbb {R} }e^{-x\tau ^{2}/2}(1-\tau /3+\tau ^{2}/12+4a_{4}\tau ^{3}+O(\tau ^{4}))d\tau ={\sqrt {2\pi }}(x^{-1/2}+x^{-3/2}/12)+O(x^{-5/2})}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> notice how we don't need to actually find <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{4}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle a_{4}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9fde542c1a0ee6390f05d9c0a58e9de213e4415" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{4}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.284ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9fde542c1a0ee6390f05d9c0a58e9de213e4415" data-alt="{\displaystyle a_{4}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , since it is cancelled out by the integral. Higher orders can be achieved by computing more terms in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=\tau +\cdots }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> t </mi> <mo> = </mo> <mi> τ </mi> <mo> + </mo> <mo> ⋯ </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle t=\tau +\cdots } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/914f0925da4067a78b1976f1bd9be2966c62d1ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.704ex; height:2.176ex;" alt="{\displaystyle t=\tau +\cdots }"> </noscript><span class="lazy-image-placeholder" style="width: 10.704ex;height: 2.176ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/914f0925da4067a78b1976f1bd9be2966c62d1ba" data-alt="{\displaystyle t=\tau +\cdots }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , which can be obtained programmatically.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-mathematica-program-4">[note 1]</a> Thus we get Stirling's formula to two orders: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!={\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\left(1+{\frac {1}{12n}}+O\left({\frac {1}{n^{2}}}\right)\right).}"><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 </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> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mrow> <mn> 12 </mn> <mi> n </mi> </mrow> </mfrac> </mrow> <mo> + </mo> <mi> O </mi> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <msup> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n!={\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\left(1+{\frac {1}{12n}}+O\left({\frac {1}{n^{2}}}\right)\right).} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cfc1fbd35d1f53867d7cce1047d3ceaf6741f4e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.298ex; height:6.176ex;" alt="{\displaystyle n!={\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\left(1+{\frac {1}{12n}}+O\left({\frac {1}{n^{2}}}\right)\right).}"> </noscript><span class="lazy-image-placeholder" style="width: 42.298ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cfc1fbd35d1f53867d7cce1047d3ceaf6741f4e" data-alt="{\displaystyle n!={\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\left(1+{\frac {1}{12n}}+O\left({\frac {1}{n^{2}}}\right)\right).}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  <div class="mw-heading mw-heading3"> <h3 id="Complex-analytic_version">Complex-analytic version</h3> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Stirling's_approximation&action=edit&section=4&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Complex-analytic version" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> A complex-analysis version of this method<a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-flajolet-sedgewick-5">[4]</a> is to consider <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{n!}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {1}{n!}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13afd849707d08294eb548c3947c0f29801a4ad2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.878ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{n!}}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.878ex;height: 5.343ex;vertical-align: -2.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13afd849707d08294eb548c3947c0f29801a4ad2" data-alt="{\displaystyle {\frac {1}{n!}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  as a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Taylor_series?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Taylor series">Taylor coefficient</a> of the exponential function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{z}=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> z </mi> </mrow> </msup> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∞ </mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle e^{z}=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62afa4bdef0bbdf72a026e48d8089103e84fa37d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.071ex; height:6.843ex;" alt="{\displaystyle e^{z}=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}}"> </noscript><span class="lazy-image-placeholder" style="width: 12.071ex;height: 6.843ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62afa4bdef0bbdf72a026e48d8089103e84fa37d" data-alt="{\displaystyle e^{z}=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , computed by <a href="https://en-m-wikipedia-org.translate.goog/wiki/Cauchy%27s_integral_formula?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Cauchy's integral formula">Cauchy's integral formula</a> as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{n!}}={\frac {1}{2\pi i}}\oint \limits _{|z|=r}{\frac {e^{z}}{z^{n+1}}}\,\mathrm {d} z.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mi> π </mi> <mi> i </mi> </mrow> </mfrac> </mrow> <munder> <mo> ∮ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> = </mo> <mi> r </mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> z </mi> </mrow> </msup> <msup> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> z </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {1}{n!}}={\frac {1}{2\pi i}}\oint \limits _{|z|=r}{\frac {e^{z}}{z^{n+1}}}\,\mathrm {d} z.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c8e7fbd26bd3cb1707d6e67604a7f64734ba243" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:22.662ex; height:7.843ex;" alt="{\displaystyle {\frac {1}{n!}}={\frac {1}{2\pi i}}\oint \limits _{|z|=r}{\frac {e^{z}}{z^{n+1}}}\,\mathrm {d} z.}"> </noscript><span class="lazy-image-placeholder" style="width: 22.662ex;height: 7.843ex;vertical-align: -4.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c8e7fbd26bd3cb1707d6e67604a7f64734ba243" data-alt="{\displaystyle {\frac {1}{n!}}={\frac {1}{2\pi i}}\oint \limits _{|z|=r}{\frac {e^{z}}{z^{n+1}}}\,\mathrm {d} z.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  This line integral can then be approximated using the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Method_of_steepest_descent?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Method of steepest descent">saddle-point method</a> with an appropriate choice of contour radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=r_{n}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> r </mi> <mo> = </mo> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r=r_{n}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/402caef90569e54e6e8864e299f869dd849df91e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.414ex; height:2.009ex;" alt="{\displaystyle r=r_{n}}"> </noscript><span class="lazy-image-placeholder" style="width: 6.414ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/402caef90569e54e6e8864e299f869dd849df91e" data-alt="{\displaystyle r=r_{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . 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. </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"> <h2 id="Speed_of_convergence_and_error_estimates">Speed of convergence and error estimates</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Stirling's_approximation&action=edit&section=5&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Speed of convergence and error estimates" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-3 collapsible-block" id="mf-section-3"> <figure class="mw-default-size" typeof="mw:File/Thumb"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Stirling_series_relative_error.svg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Stirling_series_relative_error.svg/400px-Stirling_series_relative_error.svg.png" decoding="async" width="400" height="286" class="mw-file-element" data-file-width="740" data-file-height="530"> </noscript><span class="lazy-image-placeholder" style="width: 400px;height: 286px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Stirling_series_relative_error.svg/400px-Stirling_series_relative_error.svg.png" data-width="400" data-height="286" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Stirling_series_relative_error.svg/600px-Stirling_series_relative_error.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/Stirling_series_relative_error.svg/800px-Stirling_series_relative_error.svg.png 2x" data-class="mw-file-element"> </a> <figcaption> The relative error in a truncated Stirling series vs. <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> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , for 0 to 5 terms. The kinks in the curves represent points where the truncated series coincides with Γ(n + 1). </figcaption> </figure> Stirling's formula is in fact the first approximation to the following series (now called the Stirling series):<a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-nist-6">[5]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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).}"><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 </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> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mrow> <mn> 12 </mn> <mi> n </mi> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mrow> <mn> 288 </mn> <msup> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 139 </mn> <mrow> <mn> 51840 </mn> <msup> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 571 </mn> <mrow> <mn> 2488320 </mn> <msup> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> + </mo> <mo> ⋯ </mo> </mrow> <mo> ) </mo> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 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).} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b2cf3c9b6fe771f6b3c1179a846ed77f19f4f89" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:71.968ex; height:6.176ex;" alt="{\displaystyle 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).}"> </noscript><span class="lazy-image-placeholder" style="width: 71.968ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b2cf3c9b6fe771f6b3c1179a846ed77f19f4f89" data-alt="{\displaystyle 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).}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  An explicit formula for the coefficients in this series was given by G. Nemes.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Nemes2010-2-7">[6]</a> Further terms are listed in the <a href="https://en-m-wikipedia-org.translate.goog/wiki/On-Line_Encyclopedia_of_Integer_Sequences?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a> as <a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://oeis.org/A001163" class="extiw" title="oeis:A001163">A001163</a> and <a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://oeis.org/A001164" class="extiw" title="oeis:A001164">A001164</a>. The first graph in this section shows the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Approximation_error?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Approximation error">relative error</a> vs. <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> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , for 1 through all 5 terms listed above. (Bender and Orszag<a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-8">[7]</a> p. 218) gives the asymptotic formula for the coefficients: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{2j+1}\sim (-1)^{j}2(2j)!/(2\pi )^{2(j+1)}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ∼ </mo> <mo stretchy="false"> ( </mo> <mo> − </mo> <mn> 1 </mn> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msup> <mn> 2 </mn> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> j </mi> <mo stretchy="false"> ) </mo> <mo> ! </mo> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> π </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo stretchy="false"> ( </mo> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle A_{2j+1}\sim (-1)^{j}2(2j)!/(2\pi )^{2(j+1)}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e99c9ce19477b939bdfffc501b95a1ed9df9b3a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.68ex; height:3.509ex;" alt="{\displaystyle A_{2j+1}\sim (-1)^{j}2(2j)!/(2\pi )^{2(j+1)}}"> </noscript><span class="lazy-image-placeholder" style="width: 30.68ex;height: 3.509ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e99c9ce19477b939bdfffc501b95a1ed9df9b3a" data-alt="{\displaystyle A_{2j+1}\sim (-1)^{j}2(2j)!/(2\pi )^{2(j+1)}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> which shows that it grows superexponentially, and that by <a href="https://en-m-wikipedia-org.translate.goog/wiki/Ratio_test?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ratio test">ratio test</a> the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Radius_of_convergence?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Radius of convergence">radius of convergence</a> is zero. <figure class="mw-default-size" typeof="mw:File/Thumb"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Stirling_error_vs_number_of_terms.svg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Stirling_error_vs_number_of_terms.svg/400px-Stirling_error_vs_number_of_terms.svg.png" decoding="async" width="400" height="271" class="mw-file-element" data-file-width="736" data-file-height="499"> </noscript><span class="lazy-image-placeholder" style="width: 400px;height: 271px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Stirling_error_vs_number_of_terms.svg/400px-Stirling_error_vs_number_of_terms.svg.png" data-width="400" data-height="271" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Stirling_error_vs_number_of_terms.svg/600px-Stirling_error_vs_number_of_terms.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Stirling_error_vs_number_of_terms.svg/800px-Stirling_error_vs_number_of_terms.svg.png 2x" data-class="mw-file-element"> </a> <figcaption> The relative error in a truncated Stirling series vs. the number of terms used </figcaption> </figure> As n → ∞, the error in the truncated series is asymptotically equal to the first omitted term. This is an example of an <a href="https://en-m-wikipedia-org.translate.goog/wiki/Asymptotic_expansion?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Asymptotic expansion">asymptotic expansion</a>. It is not a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Convergent_series?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Convergent series">convergent series</a>; for any particular value of <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> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  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 S(n, t) be the Stirling series to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> t </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle t} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> </noscript><span class="lazy-image-placeholder" style="width: 0.84ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" data-alt="{\displaystyle t}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  terms evaluated at  <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> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . The graphs show <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\ln \left({\frac {S(n,t)}{n!}}\right)\right|,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo> | </mo> <mrow> <mi> ln </mi> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> S </mi> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> | </mo> </mrow> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left|\ln \left({\frac {S(n,t)}{n!}}\right)\right|,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/451995218e426dea9963cd62fd1caaa712d0ae8c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.101ex; height:6.509ex;" alt="{\displaystyle \left|\ln \left({\frac {S(n,t)}{n!}}\right)\right|,}"> </noscript><span class="lazy-image-placeholder" style="width: 15.101ex;height: 6.509ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/451995218e426dea9963cd62fd1caaa712d0ae8c" data-alt="{\displaystyle \left|\ln \left({\frac {S(n,t)}{n!}}\right)\right|,}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  which, when small, is essentially the relative error. Writing Stirling's series in the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(n!)\sim n\ln n-n+{\tfrac {1}{2}}\ln(2\pi n)+{\frac {1}{12n}}-{\frac {1}{360n^{3}}}+{\frac {1}{1260n^{5}}}-{\frac {1}{1680n^{7}}}+\cdots ,}"><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> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mstyle> </mrow> <mi> ln </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> π </mi> <mi> n </mi> <mo stretchy="false"> ) </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mrow> <mn> 12 </mn> <mi> n </mi> </mrow> </mfrac> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mrow> <mn> 360 </mn> <msup> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mrow> <mn> 1260 </mn> <msup> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 5 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mrow> <mn> 1680 </mn> <msup> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 7 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> + </mo> <mo> ⋯ </mo> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \ln(n!)\sim n\ln n-n+{\tfrac {1}{2}}\ln(2\pi n)+{\frac {1}{12n}}-{\frac {1}{360n^{3}}}+{\frac {1}{1260n^{5}}}-{\frac {1}{1680n^{7}}}+\cdots ,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ea7279df14c37194e55c825ac63e749ba4c2a98" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:76.307ex; height:5.509ex;" alt="{\displaystyle \ln(n!)\sim n\ln n-n+{\tfrac {1}{2}}\ln(2\pi n)+{\frac {1}{12n}}-{\frac {1}{360n^{3}}}+{\frac {1}{1260n^{5}}}-{\frac {1}{1680n^{7}}}+\cdots ,}"> </noscript><span class="lazy-image-placeholder" style="width: 76.307ex;height: 5.509ex;vertical-align: -2.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ea7279df14c37194e55c825ac63e749ba4c2a98" data-alt="{\displaystyle \ln(n!)\sim n\ln n-n+{\tfrac {1}{2}}\ln(2\pi n)+{\frac {1}{12n}}-{\frac {1}{360n^{3}}}+{\frac {1}{1260n^{5}}}-{\frac {1}{1680n^{7}}}+\cdots ,}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  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,<a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Robbins1955-9">[8]</a> valid for all positive integers <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> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  are <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}e^{\frac {1}{12n+1}}<n!<{\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}e^{\frac {1}{12n}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 2 </mn> <mi> 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> <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> < </mo> <mi> n </mi> <mo> ! </mo> <mo> < </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 2 </mn> <mi> 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> <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({\frac {n}{e}}\right)^{n}e^{\frac {1}{12n+1}}<n!<{\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}e^{\frac {1}{12n}}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7326478542ba974e9a5b120756b3ca0cd834a404" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:43.265ex; height:5.176ex;" alt="{\displaystyle {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}e^{\frac {1}{12n+1}}<n!<{\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}e^{\frac {1}{12n}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 43.265ex;height: 5.176ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7326478542ba974e9a5b120756b3ca0cd834a404" data-alt="{\displaystyle {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}e^{\frac {1}{12n+1}}<n!<{\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}e^{\frac {1}{12n}}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  This upper bound corresponds to stopping the above series for <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> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d264f09c80b779e9028658543d5e31ae647cacf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.79ex; height:2.843ex;" alt="{\displaystyle \ln(n!)}"> </noscript><span class="lazy-image-placeholder" style="width: 5.79ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d264f09c80b779e9028658543d5e31ae647cacf1" data-alt="{\displaystyle \ln(n!)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  after the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{n}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mi> n </mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {1}{n}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0aefecf48d43fdedd71e318ae6129bd67be252" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.231ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{n}}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.231ex;height: 5.176ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0aefecf48d43fdedd71e318ae6129bd67be252" data-alt="{\displaystyle {\frac {1}{n}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  term. The lower bound is weaker than that obtained by stopping the series after the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{n^{3}}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <msup> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {1}{n^{3}}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b069e94735197bb88b1e47acf1165c341c799a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:3.285ex; height:5.509ex;" alt="{\displaystyle {\frac {1}{n^{3}}}}"> </noscript><span class="lazy-image-placeholder" style="width: 3.285ex;height: 5.509ex;vertical-align: -2.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b069e94735197bb88b1e47acf1165c341c799a0" data-alt="{\displaystyle {\frac {1}{n^{3}}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  term. A looser version of this bound is that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {n!e^{n}}{n^{n+{\frac {1}{2}}}}}\in ({\sqrt {2\pi }},e]}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> n </mi> <mo> ! </mo> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> </mrow> <msup> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> </msup> </mfrac> </mrow> <mo> ∈ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 2 </mn> <mi> π </mi> </msqrt> </mrow> <mo> , </mo> <mi> e </mi> <mo stretchy="false"> ] </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {n!e^{n}}{n^{n+{\frac {1}{2}}}}}\in ({\sqrt {2\pi }},e]} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/809b4def5f2cd3a4f66ee3651b61f8c5d1f7e5b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:17.171ex; height:6.843ex;" alt="{\displaystyle {\frac {n!e^{n}}{n^{n+{\frac {1}{2}}}}}\in ({\sqrt {2\pi }},e]}"> </noscript><span class="lazy-image-placeholder" style="width: 17.171ex;height: 6.843ex;vertical-align: -3.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/809b4def5f2cd3a4f66ee3651b61f8c5d1f7e5b8" data-alt="{\displaystyle {\frac {n!e^{n}}{n^{n+{\frac {1}{2}}}}}\in ({\sqrt {2\pi }},e]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  for all <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> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}"> </noscript><span class="lazy-image-placeholder" style="width: 5.656ex;height: 2.343ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe" data-alt="{\displaystyle n\geq 1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"> <h2 id="Stirling's_formula_for_the_gamma_function">Stirling's formula for the gamma function</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Stirling's_approximation&action=edit&section=6&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Stirling's formula for the gamma function" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-4 collapsible-block" id="mf-section-4"> For all positive integers, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!=\Gamma (n+1),}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> ! </mo> <mo> = </mo> <mi mathvariant="normal"> Γ </mi> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n!=\Gamma (n+1),} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07589a814a57c666e6403264017e580514e75bc3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.447ex; height:2.843ex;" alt="{\displaystyle n!=\Gamma (n+1),}"> </noscript><span class="lazy-image-placeholder" style="width: 14.447ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07589a814a57c666e6403264017e580514e75bc3" data-alt="{\displaystyle n!=\Gamma (n+1),}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  where Γ denotes the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Gamma_function?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gamma function">gamma function</a>. 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 Re(z) > 0, then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln \Gamma (z)=z\ln z-z+{\tfrac {1}{2}}\ln {\frac {2\pi }{z}}+\int _{0}^{\infty }{\frac {2\arctan \left({\frac {t}{z}}\right)}{e^{2\pi t}-1}}\,{\rm {d}}t.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> ln </mi> <mo> ⁡ </mo> <mi mathvariant="normal"> Γ </mi> <mo stretchy="false"> ( </mo> <mi> z </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> z </mi> <mi> ln </mi> <mo> ⁡ </mo> <mi> z </mi> <mo> − </mo> <mi> z </mi> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mstyle> </mrow> <mi> ln </mi> <mo> ⁡ </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 2 </mn> <mi> π </mi> </mrow> <mi> z </mi> </mfrac> </mrow> <mo> + </mo> <msubsup> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∞ </mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 2 </mn> <mi> arctan </mi> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> t </mi> <mi> z </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mi> π </mi> <mi> t </mi> </mrow> </msup> <mo> − </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> </mrow> <mi> t </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \ln \Gamma (z)=z\ln z-z+{\tfrac {1}{2}}\ln {\frac {2\pi }{z}}+\int _{0}^{\infty }{\frac {2\arctan \left({\frac {t}{z}}\right)}{e^{2\pi t}-1}}\,{\rm {d}}t.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c398a644d2ccc7ffaf82be5478403b9a5baea09b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:52.456ex; height:6.676ex;" alt="{\displaystyle \ln \Gamma (z)=z\ln z-z+{\tfrac {1}{2}}\ln {\frac {2\pi }{z}}+\int _{0}^{\infty }{\frac {2\arctan \left({\frac {t}{z}}\right)}{e^{2\pi t}-1}}\,{\rm {d}}t.}"> </noscript><span class="lazy-image-placeholder" style="width: 52.456ex;height: 6.676ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c398a644d2ccc7ffaf82be5478403b9a5baea09b" data-alt="{\displaystyle \ln \Gamma (z)=z\ln z-z+{\tfrac {1}{2}}\ln {\frac {2\pi }{z}}+\int _{0}^{\infty }{\frac {2\arctan \left({\frac {t}{z}}\right)}{e^{2\pi t}-1}}\,{\rm {d}}t.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  Repeated integration by parts gives <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln \Gamma (z)\sim z\ln z-z+{\tfrac {1}{2}}\ln {\frac {2\pi }{z}}+\sum _{n=1}^{N-1}{\frac {B_{2n}}{2n(2n-1)z^{2n-1}}},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> ln </mi> <mo> ⁡ </mo> <mi mathvariant="normal"> Γ </mi> <mo stretchy="false"> ( </mo> <mi> z </mi> <mo stretchy="false"> ) </mo> <mo> ∼ </mo> <mi> z </mi> <mi> ln </mi> <mo> ⁡ </mo> <mi> z </mi> <mo> − </mo> <mi> z </mi> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mstyle> </mrow> <mi> ln </mi> <mo> ⁡ </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 2 </mn> <mi> π </mi> </mrow> <mi> z </mi> </mfrac> </mrow> <mo> + </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> N </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi> B </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mi> n </mi> </mrow> </msub> <mrow> <mn> 2 </mn> <mi> n </mi> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <msup> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \ln \Gamma (z)\sim z\ln z-z+{\tfrac {1}{2}}\ln {\frac {2\pi }{z}}+\sum _{n=1}^{N-1}{\frac {B_{2n}}{2n(2n-1)z^{2n-1}}},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d257681480aa1bed0237cb9239f1f722a1786e7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:53.564ex; height:7.343ex;" alt="{\displaystyle \ln \Gamma (z)\sim z\ln z-z+{\tfrac {1}{2}}\ln {\frac {2\pi }{z}}+\sum _{n=1}^{N-1}{\frac {B_{2n}}{2n(2n-1)z^{2n-1}}},}"> </noscript><span class="lazy-image-placeholder" style="width: 53.564ex;height: 7.343ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d257681480aa1bed0237cb9239f1f722a1786e7" data-alt="{\displaystyle \ln \Gamma (z)\sim z\ln z-z+{\tfrac {1}{2}}\ln {\frac {2\pi }{z}}+\sum _{n=1}^{N-1}{\frac {B_{2n}}{2n(2n-1)z^{2n-1}}},}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> B </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle B_{n}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f568bf6d34e97b9fdda0dc7e276d6c4501d2045" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.982ex; height:2.509ex;" alt="{\displaystyle B_{n}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.982ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f568bf6d34e97b9fdda0dc7e276d6c4501d2045" data-alt="{\displaystyle B_{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is the <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> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> th <a href="https://en-m-wikipedia-org.translate.goog/wiki/Bernoulli_number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Bernoulli number">Bernoulli number</a> (note that the limit of the sum as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\to \infty }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> N </mi> <mo stretchy="false"> → </mo> <mi mathvariant="normal"> ∞ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle N\to \infty } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e23159ea0d291e21c5709a6dd7486bed7f18febe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.001ex; height:2.176ex;" alt="{\displaystyle N\to \infty }"> </noscript><span class="lazy-image-placeholder" style="width: 8.001ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e23159ea0d291e21c5709a6dd7486bed7f18febe" data-alt="{\displaystyle N\to \infty }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is not convergent, so this formula is just an <a href="https://en-m-wikipedia-org.translate.goog/wiki/Asymptotic_expansion?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Asymptotic expansion">asymptotic expansion</a>). The formula is valid for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> z </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> </noscript><span class="lazy-image-placeholder" style="width: 1.088ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" data-alt="{\displaystyle z}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  large enough in absolute value, when |arg(z)| < π − ε, where ε is positive, with an error term of O(z−2N+ 1). The corresponding approximation may now be written: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (z)={\sqrt {\frac {2\pi }{z}}}\,{\left({\frac {z}{e}}\right)}^{z}\left(1+O\left({\frac {1}{z}}\right)\right).}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> Γ </mi> <mo stretchy="false"> ( </mo> <mi> z </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn> 2 </mn> <mi> π </mi> </mrow> <mi> z </mi> </mfrac> </msqrt> </mrow> <mspace width="thinmathspace"></mspace> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> z </mi> <mi> e </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> z </mi> </mrow> </msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> O </mi> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \Gamma (z)={\sqrt {\frac {2\pi }{z}}}\,{\left({\frac {z}{e}}\right)}^{z}\left(1+O\left({\frac {1}{z}}\right)\right).} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ed4a7a21192f91f42e90e0f2dbc5cbc851a089a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.617ex; height:6.343ex;" alt="{\displaystyle \Gamma (z)={\sqrt {\frac {2\pi }{z}}}\,{\left({\frac {z}{e}}\right)}^{z}\left(1+O\left({\frac {1}{z}}\right)\right).}"> </noscript><span class="lazy-image-placeholder" style="width: 35.617ex;height: 6.343ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ed4a7a21192f91f42e90e0f2dbc5cbc851a089a" data-alt="{\displaystyle \Gamma (z)={\sqrt {\frac {2\pi }{z}}}\,{\left({\frac {z}{e}}\right)}^{z}\left(1+O\left({\frac {1}{z}}\right)\right).}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  where the expansion is identical to that of Stirling's series above for <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> <mo> ! </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n!} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae971720be3cc9b8d82f4cdac89cb89877514a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:2.176ex;" alt="{\displaystyle n!}"> </noscript><span class="lazy-image-placeholder" style="width: 2.042ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae971720be3cc9b8d82f4cdac89cb89877514a6" data-alt="{\displaystyle n!}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , except that <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> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is replaced with z − 1.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-spiegel-10">[9]</a> A further application of this asymptotic expansion is for complex argument z with constant Re(z). See for example the Stirling formula applied in Im(z) = t of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Riemann%E2%80%93Siegel_theta_function?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Riemann–Siegel theta function">Riemann–Siegel theta function</a> on the straight line <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠1/4⁠ + it. </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"> <h2 id="Error_bounds">Error bounds</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Stirling's_approximation&action=edit&section=7&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Error bounds" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-5 collapsible-block" id="mf-section-5"> For any positive integer <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> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"> </noscript><span class="lazy-image-placeholder" style="width: 2.064ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" data-alt="{\displaystyle N}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , the following notation is introduced: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln \Gamma (z)=z\ln z-z+{\tfrac {1}{2}}\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)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> ln </mi> <mo> ⁡ </mo> <mi mathvariant="normal"> Γ </mi> <mo stretchy="false"> ( </mo> <mi> z </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> z </mi> <mi> ln </mi> <mo> ⁡ </mo> <mi> z </mi> <mo> − </mo> <mi> z </mi> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mstyle> </mrow> <mi> ln </mi> <mo> ⁡ </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 2 </mn> <mi> π </mi> </mrow> <mi> z </mi> </mfrac> </mrow> <mo> + </mo> <munderover> <mo movablelimits="false"> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> N </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi> B </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mi> n </mi> </mrow> </msub> <mrow> <mn> 2 </mn> <mi> n </mi> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <msup> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> + </mo> <msub> <mi> R </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> N </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> z </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \ln \Gamma (z)=z\ln z-z+{\tfrac {1}{2}}\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)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19a5d992f27054d73eb99a5ab9d9f15acfbec0a3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:62.885ex; height:7.343ex;" alt="{\displaystyle \ln \Gamma (z)=z\ln z-z+{\tfrac {1}{2}}\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)}"> </noscript><span class="lazy-image-placeholder" style="width: 62.885ex;height: 7.343ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19a5d992f27054d73eb99a5ab9d9f15acfbec0a3" data-alt="{\displaystyle \ln \Gamma (z)=z\ln z-z+{\tfrac {1}{2}}\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)}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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).}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> Γ </mi> <mo stretchy="false"> ( </mo> <mi> z </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn> 2 </mn> <mi> π </mi> </mrow> <mi> z </mi> </mfrac> </msqrt> </mrow> <msup> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> z </mi> <mi> e </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> z </mi> </mrow> </msup> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <munderover> <mo movablelimits="false"> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> N </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <msup> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> </mfrac> </mrow> <mo> + </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> R </mi> <mo> ~ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> N </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> z </mi> <mo stretchy="false"> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \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).} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56d62236ddd9cbfabb820fba18895c77655c433b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:40.331ex; height:7.509ex;" alt="{\displaystyle \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).}"> </noscript><span class="lazy-image-placeholder" style="width: 40.331ex;height: 7.509ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56d62236ddd9cbfabb820fba18895c77655c433b" data-alt="{\displaystyle \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).}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  Then<a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-schafke-sattler-11">[10]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-nemes15-12">[11]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}|R_{N}(z)|&\leq {\frac {|B_{2N}|}{2N(2N-1)|z|^{2N-1}}}\times {\begin{cases}1&{\text{ if }}\left|\arg z\right|\leq {\frac {\pi }{4}},\\\left|\csc(\arg z)\right|&{\text{ if }}{\frac {\pi }{4}}<\left|\arg z\right|<{\frac {\pi }{2}},\\\sec ^{2N}\left({\tfrac {\arg z}{2}}\right)&{\text{ if }}\left|\arg z\right|<\pi ,\end{cases}}\\[6pt]\left|{\widetilde {R}}_{N}(z)\right|&\leq \left({\frac {\left|a_{N}\right|}{|z|^{N}}}+{\frac {\left|a_{N+1}\right|}{|z|^{N+1}}}\right)\times {\begin{cases}1&{\text{ if }}\left|\arg z\right|\leq {\frac {\pi }{4}},\\\left|\csc(2\arg z)\right|&{\text{ if }}{\frac {\pi }{4}}<\left|\arg z\right|<{\frac {\pi }{2}}.\end{cases}}\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <msub> <mi> R </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> N </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> z </mi> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> </mtd> <mtd> <mi></mi> <mo> ≤ </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <msub> <mi> B </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mi> N </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mi> N </mi> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> N </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> z </mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mi> N </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> { </mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn> 1 </mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>  if  </mtext> </mrow> <mrow> <mo> | </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mi> z </mi> </mrow> <mo> | </mo> </mrow> <mo> ≤ </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> π </mi> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> </mtd> </mtr> <mtr> <mtd> <mrow> <mo> | </mo> <mrow> <mi> csc </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mi> arg </mi> <mo> ⁡ </mo> <mi> z </mi> <mo stretchy="false"> ) </mo> </mrow> <mo> | </mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>  if  </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> π </mi> <mn> 4 </mn> </mfrac> </mrow> <mo> < </mo> <mrow> <mo> | </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mi> z </mi> </mrow> <mo> | </mo> </mrow> <mo> < </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi> sec </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mi> N </mi> </mrow> </msup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> </mstyle> </mrow> <mo> ) </mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>  if  </mtext> </mrow> <mrow> <mo> | </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mi> z </mi> </mrow> <mo> | </mo> </mrow> <mo> < </mo> <mi> π </mi> <mo> , </mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo> | </mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> R </mi> <mo> ~ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> N </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> z </mi> <mo stretchy="false"> ) </mo> </mrow> <mo> | </mo> </mrow> </mtd> <mtd> <mi></mi> <mo> ≤ </mo> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo> | </mo> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> N </mi> </mrow> </msub> <mo> | </mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> z </mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> N </mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo> | </mo> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> N </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> | </mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> z </mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> N </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> { </mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn> 1 </mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>  if  </mtext> </mrow> <mrow> <mo> | </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mi> z </mi> </mrow> <mo> | </mo> </mrow> <mo> ≤ </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> π </mi> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> </mtd> </mtr> <mtr> <mtd> <mrow> <mo> | </mo> <mrow> <mi> csc </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> arg </mi> <mo> ⁡ </mo> <mi> z </mi> <mo stretchy="false"> ) </mo> </mrow> <mo> | </mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>  if  </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> π </mi> <mn> 4 </mn> </mfrac> </mrow> <mo> < </mo> <mrow> <mo> | </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mi> z </mi> </mrow> <mo> | </mo> </mrow> <mo> < </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> . </mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}|R_{N}(z)|&\leq {\frac {|B_{2N}|}{2N(2N-1)|z|^{2N-1}}}\times {\begin{cases}1&{\text{ if }}\left|\arg z\right|\leq {\frac {\pi }{4}},\\\left|\csc(\arg z)\right|&{\text{ if }}{\frac {\pi }{4}}<\left|\arg z\right|<{\frac {\pi }{2}},\\\sec ^{2N}\left({\tfrac {\arg z}{2}}\right)&{\text{ if }}\left|\arg z\right|<\pi ,\end{cases}}\\[6pt]\left|{\widetilde {R}}_{N}(z)\right|&\leq \left({\frac {\left|a_{N}\right|}{|z|^{N}}}+{\frac {\left|a_{N+1}\right|}{|z|^{N+1}}}\right)\times {\begin{cases}1&{\text{ if }}\left|\arg z\right|\leq {\frac {\pi }{4}},\\\left|\csc(2\arg z)\right|&{\text{ if }}{\frac {\pi }{4}}<\left|\arg z\right|<{\frac {\pi }{2}}.\end{cases}}\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49a2ef6c072d31d6b521b5e9094485cb362a6e3a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:70.404ex; height:19.176ex;" alt="{\displaystyle {\begin{aligned}|R_{N}(z)|&\leq {\frac {|B_{2N}|}{2N(2N-1)|z|^{2N-1}}}\times {\begin{cases}1&{\text{ if }}\left|\arg z\right|\leq {\frac {\pi }{4}},\\\left|\csc(\arg z)\right|&{\text{ if }}{\frac {\pi }{4}}<\left|\arg z\right|<{\frac {\pi }{2}},\\\sec ^{2N}\left({\tfrac {\arg z}{2}}\right)&{\text{ if }}\left|\arg z\right|<\pi ,\end{cases}}\\[6pt]\left|{\widetilde {R}}_{N}(z)\right|&\leq \left({\frac {\left|a_{N}\right|}{|z|^{N}}}+{\frac {\left|a_{N+1}\right|}{|z|^{N+1}}}\right)\times {\begin{cases}1&{\text{ if }}\left|\arg z\right|\leq {\frac {\pi }{4}},\\\left|\csc(2\arg z)\right|&{\text{ if }}{\frac {\pi }{4}}<\left|\arg z\right|<{\frac {\pi }{2}}.\end{cases}}\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 70.404ex;height: 19.176ex;vertical-align: -9.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49a2ef6c072d31d6b521b5e9094485cb362a6e3a" data-alt="{\displaystyle {\begin{aligned}|R_{N}(z)|&\leq {\frac {|B_{2N}|}{2N(2N-1)|z|^{2N-1}}}\times {\begin{cases}1&{\text{ if }}\left|\arg z\right|\leq {\frac {\pi }{4}},\\\left|\csc(\arg z)\right|&{\text{ if }}{\frac {\pi }{4}}<\left|\arg z\right|<{\frac {\pi }{2}},\\\sec ^{2N}\left({\tfrac {\arg z}{2}}\right)&{\text{ if }}\left|\arg z\right|<\pi ,\end{cases}}\\[6pt]\left|{\widetilde {R}}_{N}(z)\right|&\leq \left({\frac {\left|a_{N}\right|}{|z|^{N}}}+{\frac {\left|a_{N+1}\right|}{|z|^{N+1}}}\right)\times {\begin{cases}1&{\text{ if }}\left|\arg z\right|\leq {\frac {\pi }{4}},\\\left|\csc(2\arg z)\right|&{\text{ if }}{\frac {\pi }{4}}<\left|\arg z\right|<{\frac {\pi }{2}}.\end{cases}}\end{aligned}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  For further information and other error bounds, see the cited papers. </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"> <h2 id="A_convergent_version_of_Stirling's_formula">A convergent version of Stirling's formula</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Stirling's_approximation&action=edit&section=8&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: A convergent version of Stirling's formula" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-6 collapsible-block" id="mf-section-6"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Thomas_Bayes?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Thomas Bayes">Thomas Bayes</a> showed, in a letter to <a href="https://en-m-wikipedia-org.translate.goog/wiki/John_Canton?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="John Canton">John Canton</a> published by the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Royal_Society?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Royal Society">Royal Society</a> in 1763, that Stirling's formula did not give a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Convergent_series?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Convergent series">convergent series</a>.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-bayes-canton-13">[12]</a> Obtaining a convergent version of Stirling's formula entails evaluating <a href="https://en-m-wikipedia-org.translate.goog/wiki/Gamma_function?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Raabe's_formula" title="Gamma function">Binet's formula</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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-{\tfrac {1}{2}}\ln {\frac {2\pi }{x}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∞ </mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 2 </mn> <mi> arctan </mi> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> t </mi> <mi> x </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mi> π </mi> <mi> t </mi> </mrow> </msup> <mo> − </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> </mrow> <mi> t </mi> <mo> = </mo> <mi> ln </mi> <mo> ⁡ </mo> <mi mathvariant="normal"> Γ </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> − </mo> <mi> x </mi> <mi> ln </mi> <mo> ⁡ </mo> <mi> x </mi> <mo> + </mo> <mi> x </mi> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mstyle> </mrow> <mi> ln </mi> <mo> ⁡ </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 2 </mn> <mi> π </mi> </mrow> <mi> x </mi> </mfrac> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \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-{\tfrac {1}{2}}\ln {\frac {2\pi }{x}}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d912992e78260a9dbbd0ba446b96be2d2dfca212" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:53.593ex; height:6.676ex;" alt="{\displaystyle \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-{\tfrac {1}{2}}\ln {\frac {2\pi }{x}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 53.593ex;height: 6.676ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d912992e78260a9dbbd0ba446b96be2d2dfca212" data-alt="{\displaystyle \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-{\tfrac {1}{2}}\ln {\frac {2\pi }{x}}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  One way to do this is by means of a convergent series of inverted <a href="https://en-m-wikipedia-org.translate.goog/wiki/Rising_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Rising factorial">rising factorials</a>. If <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{\bar {n}}=z(z+1)\cdots (z+n-1),}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> ¯ </mo> </mover> </mrow> </mrow> </msup> <mo> = </mo> <mi> z </mi> <mo stretchy="false"> ( </mo> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> ⋯ </mo> <mo stretchy="false"> ( </mo> <mi> z </mi> <mo> + </mo> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z^{\bar {n}}=z(z+1)\cdots (z+n-1),} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b8f38ab7b86f762cedbb9395458a1afb36b0578" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.675ex; height:3.343ex;" alt="{\displaystyle z^{\bar {n}}=z(z+1)\cdots (z+n-1),}"> </noscript><span class="lazy-image-placeholder" style="width: 28.675ex;height: 3.343ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b8f38ab7b86f762cedbb9395458a1afb36b0578" data-alt="{\displaystyle z^{\bar {n}}=z(z+1)\cdots (z+n-1),}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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}}}},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∞ </mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 2 </mn> <mi> arctan </mi> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> t </mi> <mi> x </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mi> π </mi> <mi> t </mi> </mrow> </msup> <mo> − </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> </mrow> <mi> t </mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∞ </mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mrow> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> + </mo> <mn> 1 </mn> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> ¯ </mo> </mover> </mrow> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \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}}}},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad3f584996c74270de2e6681eee0f6bf2f07ef91" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:36.688ex; height:7.343ex;" alt="{\displaystyle \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}}}},}"> </noscript><span class="lazy-image-placeholder" style="width: 36.688ex;height: 7.343ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad3f584996c74270de2e6681eee0f6bf2f07ef91" data-alt="{\displaystyle \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}}}},}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  where <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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)}},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mi> n </mi> </mfrac> </mrow> <msubsup> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msubsup> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> ¯ </mo> </mover> </mrow> </mrow> </msup> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mstyle> </mrow> </mrow> <mo> ) </mo> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> </mrow> <mi> x </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mi> n </mi> </mrow> </mfrac> </mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> s </mi> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> , </mo> <mi> k </mi> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> </mrow> <mrow> <mo stretchy="false"> ( </mo> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mi> k </mi> <mo> + </mo> <mn> 2 </mn> <mo stretchy="false"> ) </mo> </mrow> </mfrac> </mrow> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 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)}},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bcdb1c8323f6e91e60b7038b1fb3e6be337fd11" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:52.288ex; height:6.843ex;" alt="{\displaystyle 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)}},}"> </noscript><span class="lazy-image-placeholder" style="width: 52.288ex;height: 6.843ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bcdb1c8323f6e91e60b7038b1fb3e6be337fd11" data-alt="{\displaystyle 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)}},}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  where s(n, k) denotes the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling_numbers_of_the_first_kind?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Stirling numbers of the first kind">Stirling numbers of the first kind</a>. From this one obtains a version of Stirling's series <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\ln \Gamma (x)&=x\ln x-x+{\tfrac {1}{2}}\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{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi> ln </mi> <mo> ⁡ </mo> <mi mathvariant="normal"> Γ </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> </mtd> <mtd> <mi></mi> <mo> = </mo> <mi> x </mi> <mi> ln </mi> <mo> ⁡ </mo> <mi> x </mi> <mo> − </mo> <mi> x </mi> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mstyle> </mrow> <mi> ln </mi> <mo> ⁡ </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 2 </mn> <mi> π </mi> </mrow> <mi> x </mi> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mrow> <mn> 12 </mn> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> + </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mrow> <mn> 12 </mn> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> + </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> + </mo> <mn> 2 </mn> <mo stretchy="false"> ) </mo> </mrow> </mfrac> </mrow> <mo> + </mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mspace width="1em"></mspace> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 59 </mn> <mrow> <mn> 360 </mn> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> + </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> + </mo> <mn> 2 </mn> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> + </mo> <mn> 3 </mn> <mo stretchy="false"> ) </mo> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 29 </mn> <mrow> <mn> 60 </mn> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> + </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> + </mo> <mn> 2 </mn> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> + </mo> <mn> 3 </mn> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> + </mo> <mn> 4 </mn> <mo stretchy="false"> ) </mo> </mrow> </mfrac> </mrow> <mo> + </mo> <mo> ⋯ </mo> <mo> , </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}\ln \Gamma (x)&=x\ln x-x+{\tfrac {1}{2}}\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{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e92480dc9ee30c76420a560f402a4e660d5c3c93" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:79.748ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}\ln \Gamma (x)&=x\ln x-x+{\tfrac {1}{2}}\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{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 79.748ex;height: 12.176ex;vertical-align: -5.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e92480dc9ee30c76420a560f402a4e660d5c3c93" data-alt="{\displaystyle {\begin{aligned}\ln \Gamma (x)&=x\ln x-x+{\tfrac {1}{2}}\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{aligned}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  which converges when Re(x) > 0. Stirling's formula may also be given in convergent form as<a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-14">[13]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (x)={\sqrt {2\pi }}x^{x-{\frac {1}{2}}}e^{-x+\mu (x)}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> Γ </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 2 </mn> <mi> π </mi> </msqrt> </mrow> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> </msup> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mi> x </mi> <mo> + </mo> <mi> μ </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \Gamma (x)={\sqrt {2\pi }}x^{x-{\frac {1}{2}}}e^{-x+\mu (x)}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22fb124c11382df1c29111d531098cdfd56654fe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.429ex; height:4.009ex;" alt="{\displaystyle \Gamma (x)={\sqrt {2\pi }}x^{x-{\frac {1}{2}}}e^{-x+\mu (x)}}"> </noscript><span class="lazy-image-placeholder" style="width: 25.429ex;height: 4.009ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22fb124c11382df1c29111d531098cdfd56654fe" data-alt="{\displaystyle \Gamma (x)={\sqrt {2\pi }}x^{x-{\frac {1}{2}}}e^{-x+\mu (x)}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  where <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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).}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> μ </mi> <mrow> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∞ </mi> </mrow> </munderover> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> ln </mi> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mrow> <mi> x </mi> <mo> + </mo> <mi> n </mi> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> − </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \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).} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0972561e44aa7b84fd9cb35f77abbecd8049b731" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:50.203ex; height:6.843ex;" alt="{\displaystyle \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).}"> </noscript><span class="lazy-image-placeholder" style="width: 50.203ex;height: 6.843ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0972561e44aa7b84fd9cb35f77abbecd8049b731" data-alt="{\displaystyle \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).}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"> <h2 id="Versions_suitable_for_calculators">Versions suitable for calculators</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Stirling's_approximation&action=edit&section=9&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Versions suitable for calculators" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-7 collapsible-block" id="mf-section-7"> The approximation <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (z)\approx {\sqrt {\frac {2\pi }{z}}}\left({\frac {z}{e}}{\sqrt {z\sinh {\frac {1}{z}}+{\frac {1}{810z^{6}}}}}\right)^{z}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> Γ </mi> <mo stretchy="false"> ( </mo> <mi> z </mi> <mo stretchy="false"> ) </mo> <mo> ≈ </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn> 2 </mn> <mi> π </mi> </mrow> <mi> z </mi> </mfrac> </msqrt> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> z </mi> <mi> e </mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi> z </mi> <mi> sinh </mi> <mo> ⁡ </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mrow> <mn> 810 </mn> <msup> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 6 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> z </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \Gamma (z)\approx {\sqrt {\frac {2\pi }{z}}}\left({\frac {z}{e}}{\sqrt {z\sinh {\frac {1}{z}}+{\frac {1}{810z^{6}}}}}\right)^{z}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8117fdf62ad1e1991e255eefde12ec06ab4ce6eb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.091ex; height:6.343ex;" alt="{\displaystyle \Gamma (z)\approx {\sqrt {\frac {2\pi }{z}}}\left({\frac {z}{e}}{\sqrt {z\sinh {\frac {1}{z}}+{\frac {1}{810z^{6}}}}}\right)^{z}}"> </noscript><span class="lazy-image-placeholder" style="width: 39.091ex;height: 6.343ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8117fdf62ad1e1991e255eefde12ec06ab4ce6eb" data-alt="{\displaystyle \Gamma (z)\approx {\sqrt {\frac {2\pi }{z}}}\left({\frac {z}{e}}{\sqrt {z\sinh {\frac {1}{z}}+{\frac {1}{810z^{6}}}}}\right)^{z}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  and its equivalent form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn> 2 </mn> <mi> ln </mi> <mo> ⁡ </mo> <mi mathvariant="normal"> Γ </mi> <mo stretchy="false"> ( </mo> <mi> z </mi> <mo stretchy="false"> ) </mo> <mo> ≈ </mo> <mi> ln </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> π </mi> <mo stretchy="false"> ) </mo> <mo> − </mo> <mi> ln </mi> <mo> ⁡ </mo> <mi> z </mi> <mo> + </mo> <mi> z </mi> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mi> ln </mi> <mo> ⁡ </mo> <mi> z </mi> <mo> + </mo> <mi> ln </mi> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mi> sinh </mi> <mo> ⁡ </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mrow> <mn> 810 </mn> <msup> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 6 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> − </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 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)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b70f63ef14c40546863b26a0635f6a8cd199992a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:66.045ex; height:6.176ex;" alt="{\displaystyle 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)}"> </noscript><span class="lazy-image-placeholder" style="width: 66.045ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b70f63ef14c40546863b26a0635f6a8cd199992a" data-alt="{\displaystyle 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)}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  can be obtained by rearranging Stirling's extended formula and observing a coincidence between the resultant <a href="https://en-m-wikipedia-org.translate.goog/wiki/Power_series?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Power series">power series</a> and the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Taylor_series?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Taylor series">Taylor series</a> expansion of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Hyperbolic_sine?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Hyperbolic sine">hyperbolic sine</a> function. This approximation is good to more than 8 decimal digits for 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-toth-15">[14]</a> Gergő Nemes proposed in 2007 an approximation which gives the same number of exact digits as the Windschitl approximation but is much simpler:<a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Nemes2010-16">[15]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (z)\approx {\sqrt {\frac {2\pi }{z}}}\left({\frac {1}{e}}\left(z+{\frac {1}{12z-{\frac {1}{10z}}}}\right)\right)^{z},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> Γ </mi> <mo stretchy="false"> ( </mo> <mi> z </mi> <mo stretchy="false"> ) </mo> <mo> ≈ </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn> 2 </mn> <mi> π </mi> </mrow> <mi> z </mi> </mfrac> </msqrt> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mi> e </mi> </mfrac> </mrow> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mrow> <mn> 12 </mn> <mi> z </mi> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mrow> <mn> 10 </mn> <mi> z </mi> </mrow> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> z </mi> </mrow> </msup> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \Gamma (z)\approx {\sqrt {\frac {2\pi }{z}}}\left({\frac {1}{e}}\left(z+{\frac {1}{12z-{\frac {1}{10z}}}}\right)\right)^{z},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bf6a375a8346a2ff56722096601b85a4347863c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:38.767ex; height:7.843ex;" alt="{\displaystyle \Gamma (z)\approx {\sqrt {\frac {2\pi }{z}}}\left({\frac {1}{e}}\left(z+{\frac {1}{12z-{\frac {1}{10z}}}}\right)\right)^{z},}"> </noscript><span class="lazy-image-placeholder" style="width: 38.767ex;height: 7.843ex;vertical-align: -3.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bf6a375a8346a2ff56722096601b85a4347863c" data-alt="{\displaystyle \Gamma (z)\approx {\sqrt {\frac {2\pi }{z}}}\left({\frac {1}{e}}\left(z+{\frac {1}{12z-{\frac {1}{10z}}}}\right)\right)^{z},}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  or equivalently, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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).}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> ln </mi> <mo> ⁡ </mo> <mi mathvariant="normal"> Γ </mi> <mo stretchy="false"> ( </mo> <mi> z </mi> <mo stretchy="false"> ) </mo> <mo> ≈ </mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mstyle> </mrow> <mrow> <mo> ( </mo> <mrow> <mi> ln </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> π </mi> <mo stretchy="false"> ) </mo> <mo> − </mo> <mi> ln </mi> <mo> ⁡ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mi> z </mi> <mrow> <mo> ( </mo> <mrow> <mi> ln </mi> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mrow> <mn> 12 </mn> <mi> z </mi> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mrow> <mn> 10 </mn> <mi> z </mi> </mrow> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> − </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \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).} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f05deb7cc6fb2336e1a3b2e62b68e7334251b0a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:59.05ex; height:7.843ex;" alt="{\displaystyle \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).}"> </noscript><span class="lazy-image-placeholder" style="width: 59.05ex;height: 7.843ex;vertical-align: -3.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f05deb7cc6fb2336e1a3b2e62b68e7334251b0a" data-alt="{\displaystyle \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).}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  An alternative approximation for the gamma function stated by <a href="https://en-m-wikipedia-org.translate.goog/wiki/Srinivasa_Ramanujan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Srinivasa Ramanujan">Srinivasa Ramanujan</a> in <a href="https://en-m-wikipedia-org.translate.goog/wiki/Ramanujan%27s_lost_notebook?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ramanujan's lost notebook">Ramanujan's lost notebook</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-17">[16]</a> is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> Γ </mi> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo> + </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> ≈ </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi> π </mi> </msqrt> </mrow> <msup> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> x </mi> <mi> e </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 8 </mn> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mo> + </mo> <mn> 4 </mn> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mi> x </mi> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 30 </mn> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \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}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5a63e515690cf06b6bd412329fe7c1a625b74ea" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:46.336ex; height:7.176ex;" alt="{\displaystyle \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}}}"> </noscript><span class="lazy-image-placeholder" style="width: 46.336ex;height: 7.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5a63e515690cf06b6bd412329fe7c1a625b74ea" data-alt="{\displaystyle \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}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  for x ≥ 0. The equivalent approximation for ln n! has an asymptotic error of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/1400n3⁠ and is given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln n!\approx n\ln n-n+{\tfrac {1}{6}}\ln(8n^{3}+4n^{2}+n+{\tfrac {1}{30}})+{\tfrac {1}{2}}\ln \pi .}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> ln </mi> <mo> ⁡ </mo> <mi> n </mi> <mo> ! </mo> <mo> ≈ </mo> <mi> n </mi> <mi> ln </mi> <mo> ⁡ </mo> <mi> n </mi> <mo> − </mo> <mi> n </mi> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> </mstyle> </mrow> <mi> ln </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mn> 8 </mn> <msup> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mo> + </mo> <mn> 4 </mn> <msup> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn> 1 </mn> <mn> 30 </mn> </mfrac> </mstyle> </mrow> <mo stretchy="false"> ) </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mstyle> </mrow> <mi> ln </mi> <mo> ⁡ </mo> <mi> π </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \ln n!\approx n\ln n-n+{\tfrac {1}{6}}\ln(8n^{3}+4n^{2}+n+{\tfrac {1}{30}})+{\tfrac {1}{2}}\ln \pi .} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c561b7b2741bd7253fc53b352eb1c1b07967fd5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:54.649ex; height:3.676ex;" alt="{\displaystyle \ln n!\approx n\ln n-n+{\tfrac {1}{6}}\ln(8n^{3}+4n^{2}+n+{\tfrac {1}{30}})+{\tfrac {1}{2}}\ln \pi .}"> </noscript><span class="lazy-image-placeholder" style="width: 54.649ex;height: 3.676ex;vertical-align: -1.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c561b7b2741bd7253fc53b352eb1c1b07967fd5" data-alt="{\displaystyle \ln n!\approx n\ln n-n+{\tfrac {1}{6}}\ln(8n^{3}+4n^{2}+n+{\tfrac {1}{30}})+{\tfrac {1}{2}}\ln \pi .}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  The approximation may be made precise by giving paired upper and lower bounds; one such inequality is<a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-E.A.Karatsuba-18">[17]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Mortici2011-1-19">[18]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Mortici2011-2-20">[19]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Mortici2011-3-21">[20]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\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}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi> π </mi> </msqrt> </mrow> <msup> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> x </mi> <mi> e </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 8 </mn> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mo> + </mo> <mn> 4 </mn> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mi> x </mi> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 100 </mn> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 6 </mn> </mrow> </msup> <mo> < </mo> <mi mathvariant="normal"> Γ </mi> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo> + </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> < </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi> π </mi> </msqrt> </mrow> <msup> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> x </mi> <mi> e </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 8 </mn> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mo> + </mo> <mn> 4 </mn> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mi> x </mi> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 30 </mn> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 6 </mn> </mrow> </msup> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\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}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70abbc9761518df503553faca29aac924e88cdb4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:87.812ex; height:6.676ex;" alt="{\displaystyle {\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}.}"> </noscript><span class="lazy-image-placeholder" style="width: 87.812ex;height: 6.676ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70abbc9761518df503553faca29aac924e88cdb4" data-alt="{\displaystyle {\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}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(8)"> <h2 id="History">History</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Stirling's_approximation&action=edit&section=10&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: History" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-8 collapsible-block" id="mf-section-8"> The formula was first discovered by <a href="https://en-m-wikipedia-org.translate.goog/wiki/Abraham_de_Moivre?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abraham de Moivre">Abraham de Moivre</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-LeCam1986-2">[2]</a> in the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!\sim [{\rm {constant}}]\cdot n^{n+{\frac {1}{2}}}e^{-n}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> ! </mo> <mo> ∼ </mo> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> c </mi> <mi mathvariant="normal"> o </mi> <mi mathvariant="normal"> n </mi> <mi mathvariant="normal"> s </mi> <mi mathvariant="normal"> t </mi> <mi mathvariant="normal"> a </mi> <mi mathvariant="normal"> n </mi> <mi mathvariant="normal"> t </mi> </mrow> </mrow> <mo stretchy="false"> ] </mo> <mo> ⋅ </mo> <msup> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> </msup> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mi> n </mi> </mrow> </msup> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n!\sim [{\rm {constant}}]\cdot n^{n+{\frac {1}{2}}}e^{-n}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a51109a2de3b04535d0b8065ec095be411bab47" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.403ex; height:4.009ex;" alt="{\displaystyle n!\sim [{\rm {constant}}]\cdot n^{n+{\frac {1}{2}}}e^{-n}.}"> </noscript><span class="lazy-image-placeholder" style="width: 26.403ex;height: 4.009ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a51109a2de3b04535d0b8065ec095be411bab47" data-alt="{\displaystyle n!\sim [{\rm {constant}}]\cdot n^{n+{\frac {1}{2}}}e^{-n}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2\pi }}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 2 </mn> <mi> π </mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\sqrt {2\pi }}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a9b009153bbbb3273a7e7279cb6b084fd650a80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.43ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2\pi }}}"> </noscript><span class="lazy-image-placeholder" style="width: 4.43ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a9b009153bbbb3273a7e7279cb6b084fd650a80" data-alt="{\displaystyle {\sqrt {2\pi }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> .<a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Pearson1924-3">[3]</a> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(9)"> <h2 id="See_also">See also</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Stirling's_approximation&action=edit&section=11&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: See also" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-9 collapsible-block" id="mf-section-9"> <ul> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Lanczos_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Lanczos approximation">Lanczos approximation</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Spouge%27s_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Spouge's approximation">Spouge's approximation</a></li> </ul> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(10)"> <h2 id="References">References</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Stirling's_approximation&action=edit&section=12&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: References" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-10 collapsible-block" id="mf-section-10"> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style> <div class="reflist"> <div class="mw-references-wrap mw-references-columns"> <ol class="references"> <li id="cite_note-dutka-1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-dutka_1-0">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFDutka1991" class="citation cs2">Dutka, Jacques (1991), "The early history of the factorial function", <a href="https://en-m-wikipedia-org.translate.goog/wiki/Archive_for_History_of_Exact_Sciences?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Archive for History of Exact Sciences">Archive for History of Exact Sciences</a>, 43 (3): 225–249, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1007%252FBF00389433">10.1007/BF00389433</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/S2CID_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://api.semanticscholar.org/CorpusID:122237769">122237769</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archive+for+History+of+Exact+Sciences&rft.atitle=The+early+history+of+the+factorial+function&rft.volume=43&rft.issue=3&rft.pages=225-249&rft.date=1991&rft_id=info%3Adoi%2F10.1007%2FBF00389433&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122237769%23id-name%3DS2CID&rft.aulast=Dutka&rft.aufirst=Jacques&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling%27s+approximation" class="Z3988"></li> <li id="cite_note-LeCam1986-2">^ <a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-LeCam1986_2-0">a</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-LeCam1986_2-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLe_Cam1986" class="citation cs2"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Lucien_Le_Cam?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Lucien Le Cam">Le Cam, L.</a> (1986), "The central limit theorem around 1935", Statistical Science, 1 (1): 78–96, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1214%252Fss%252F1177013818">10.1214/ss/1177013818</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/JSTOR_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.jstor.org/stable/2245503">2245503</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/MR_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mathscinet.ams.org/mathscinet-getitem?mr%3D0833276">0833276</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Statistical+Science&rft.atitle=The+central+limit+theorem+around+1935&rft.volume=1&rft.issue=1&rft.pages=78-96&rft.date=1986&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D833276%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2245503%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1214%2Fss%2F1177013818&rft.aulast=Le+Cam&rft.aufirst=L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling%27s+approximation" class="Z3988">; see p. 81, "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."</li> <li id="cite_note-Pearson1924-3">^ <a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Pearson1924_3-0">a</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Pearson1924_3-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPearson1924" class="citation cs2">Pearson, Karl (1924), "Historical note on the origin of the normal curve of errors", Biometrika, 16 (3/4): 402–404 [p. 403], <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.2307%252F2331714">10.2307/2331714</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/JSTOR_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.jstor.org/stable/2331714">2331714</a>, <q>I consider that the fact that Stirling showed that De Moivre's arithmetical constant was <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2\pi }}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 2 </mn> <mi> π </mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\sqrt {2\pi }}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a9b009153bbbb3273a7e7279cb6b084fd650a80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.43ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2\pi }}}"> </noscript><span class="lazy-image-placeholder" style="width: 4.43ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a9b009153bbbb3273a7e7279cb6b084fd650a80" data-alt="{\displaystyle {\sqrt {2\pi }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  does not entitle him to claim the theorem, [...]</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Biometrika&rft.atitle=Historical+note+on+the+origin+of+the+normal+curve+of+errors&rft.volume=16&rft.issue=3%2F4&rft.pages=402-404+p.+403&rft.date=1924&rft_id=info%3Adoi%2F10.2307%2F2331714&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2331714%23id-name%3DJSTOR&rft.aulast=Pearson&rft.aufirst=Karl&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling%27s+approximation" class="Z3988"></li> <li id="cite_note-flajolet-sedgewick-5"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-flajolet-sedgewick_5-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFlajoletSedgewick2009" class="citation cs2">Flajolet, Philippe; Sedgewick, Robert (2009), <a href="https://en-m-wikipedia-org.translate.goog/wiki/Analytic_Combinatorics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Analytic Combinatorics">Analytic Combinatorics</a>, Cambridge, UK: Cambridge University Press, p. 555, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1017%252FCBO9780511801655">10.1017/CBO9780511801655</a>, <a 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(2001), "On the asymptotic representation of the Euler gamma function by Ramanujan", Journal of Computational and Applied Mathematics, 135 (2): 225–240, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Bibcode_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ui.adsabs.harvard.edu/abs/2001JCoAM.135..225K">2001JCoAM.135..225K</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1016%252FS0377-0427%252800%252900586-0">10.1016/S0377-0427(00)00586-0</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/MR_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mathscinet.ams.org/mathscinet-getitem?mr%3D1850542">1850542</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Computational+and+Applied+Mathematics&rft.atitle=On+the+asymptotic+representation+of+the+Euler+gamma+function+by+Ramanujan&rft.volume=135&rft.issue=2&rft.pages=225-240&rft.date=2001&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1850542%23id-name%3DMR&rft_id=info%3Adoi%2F10.1016%2FS0377-0427%2800%2900586-0&rft_id=info%3Abibcode%2F2001JCoAM.135..225K&rft.aulast=Karatsuba&rft.aufirst=Ekatherina+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling%27s+approximation" class="Z3988"></li> <li id="cite_note-Mortici2011-1-19"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Mortici2011-1_19-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMortici2011" class="citation cs2">Mortici, Cristinel (2011), "Ramanujan's estimate for the gamma function via monotonicity arguments", Ramanujan J., 25 (2): 149–154, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1007%252Fs11139-010-9265-y">10.1007/s11139-010-9265-y</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/S2CID_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://api.semanticscholar.org/CorpusID:119530041">119530041</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Ramanujan+J.&rft.atitle=Ramanujan%27s+estimate+for+the+gamma+function+via+monotonicity+arguments&rft.volume=25&rft.issue=2&rft.pages=149-154&rft.date=2011&rft_id=info%3Adoi%2F10.1007%2Fs11139-010-9265-y&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119530041%23id-name%3DS2CID&rft.aulast=Mortici&rft.aufirst=Cristinel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling%27s+approximation" class="Z3988"></li> <li id="cite_note-Mortici2011-2-20"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Mortici2011-2_20-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMortici2011" class="citation cs2">Mortici, Cristinel (2011), "Improved asymptotic formulas for the gamma function", Comput. Math. Appl., 61 (11): 3364–3369, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1016%252Fj.camwa.2011.04.036">10.1016/j.camwa.2011.04.036</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Comput.+Math.+Appl.&rft.atitle=Improved+asymptotic+formulas+for+the+gamma+function&rft.volume=61&rft.issue=11&rft.pages=3364-3369&rft.date=2011&rft_id=info%3Adoi%2F10.1016%2Fj.camwa.2011.04.036&rft.aulast=Mortici&rft.aufirst=Cristinel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling%27s+approximation" class="Z3988">.</li> <li id="cite_note-Mortici2011-3-21"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Mortici2011-3_21-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMortici2011" class="citation cs2">Mortici, Cristinel (2011), "On Ramanujan's large argument formula for the gamma function", Ramanujan J., 26 (2): 185–192, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1007%252Fs11139-010-9281-y">10.1007/s11139-010-9281-y</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/S2CID_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://api.semanticscholar.org/CorpusID:120371952">120371952</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Ramanujan+J.&rft.atitle=On+Ramanujan%27s+large+argument+formula+for+the+gamma+function&rft.volume=26&rft.issue=2&rft.pages=185-192&rft.date=2011&rft_id=info%3Adoi%2F10.1007%2Fs11139-010-9281-y&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120371952%23id-name%3DS2CID&rft.aulast=Mortici&rft.aufirst=Cristinel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling%27s+approximation" class="Z3988">.</li> </ol> </div> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(11)"> <h2 id="Further_reading">Further reading</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Stirling's_approximation&action=edit&section=13&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Further reading" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-11 collapsible-block" id="mf-section-11"> <ul> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAbramowitzStegun2002" class="citation cs2">Abramowitz, M. & Stegun, I. (2002), <a href="https://en-m-wikipedia-org.translate.goog/wiki/Abramowitz_and_Stegun?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abramowitz and Stegun">Handbook of Mathematical Functions</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Mathematical+Functions&rft.date=2002&rft.aulast=Abramowitz&rft.aufirst=M.&rft.au=Stegun%2C+I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling%27s+approximation" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFParisKaminski2001" class="citation cs2">Paris, R. B. & Kaminski, D. (2001), <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/asymptoticsmelli0000pari">Asymptotics and Mellin–Barnes Integrals</a>, New York: Cambridge University Press, <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-521-79001-7?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-521-79001-7"><bdi>978-0-521-79001-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Asymptotics+and+Mellin%E2%80%93Barnes+Integrals&rft.place=New+York&rft.pub=Cambridge+University+Press&rft.date=2001&rft.isbn=978-0-521-79001-7&rft.aulast=Paris&rft.aufirst=R.+B.&rft.au=Kaminski%2C+D.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fasymptoticsmelli0000pari&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling%27s+approximation" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWhittakerWatson1996" class="citation cs2">Whittaker, E. T. & Watson, G. N. (1996), A Course in Modern Analysis (4th ed.), New York: Cambridge University Press, <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-521-58807-2?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-521-58807-2"><bdi>978-0-521-58807-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Course+in+Modern+Analysis&rft.place=New+York&rft.edition=4th&rft.pub=Cambridge+University+Press&rft.date=1996&rft.isbn=978-0-521-58807-2&rft.aulast=Whittaker&rft.aufirst=E.+T.&rft.au=Watson%2C+G.+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling%27s+approximation" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRomik2000" class="citation cs2">Romik, Dan (2000), "Stirling's approximation for <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> <mo> ! </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n!} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae971720be3cc9b8d82f4cdac89cb89877514a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:2.176ex;" alt="{\displaystyle n!}"> </noscript><span class="lazy-image-placeholder" style="width: 2.042ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae971720be3cc9b8d82f4cdac89cb89877514a6" data-alt="{\displaystyle n!}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> : the ultimate short proof?", <a href="https://en-m-wikipedia-org.translate.goog/wiki/The_American_Mathematical_Monthly?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="The American Mathematical Monthly">The American Mathematical Monthly</a>, 107 (6): 556–557, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.2307%252F2589351">10.2307/2589351</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/JSTOR_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.jstor.org/stable/2589351">2589351</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/MR_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mathscinet.ams.org/mathscinet-getitem?mr%3D1767064">1767064</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=Stirling%27s+approximation+for+MATH+RENDER+ERROR%3A+the+ultimate+short+proof%3F&rft.volume=107&rft.issue=6&rft.pages=556-557&rft.date=2000&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1767064%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2589351%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2589351&rft.aulast=Romik&rft.aufirst=Dan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling%27s+approximation" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLi2006" class="citation cs2">Li, Yuan-Chuan (July 2006), <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://projecteuclid.org/euclid.rae/1184700051">"A note on an identity of the gamma function and Stirling's formula"</a>, Real Analysis Exchange, 32 (1): 267–271, <a href="https://en-m-wikipedia-org.translate.goog/wiki/MR_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mathscinet.ams.org/mathscinet-getitem?mr%3D2329236">2329236</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Real+Analysis+Exchange&rft.atitle=A+note+on+an+identity+of+the+gamma+function+and+Stirling%27s+formula&rft.volume=32&rft.issue=1&rft.pages=267-271&rft.date=2006-07&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2329236%23id-name%3DMR&rft.aulast=Li&rft.aufirst=Yuan-Chuan&rft_id=https%3A%2F%2Fprojecteuclid.org%2Feuclid.rae%2F1184700051&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling%27s+approximation" class="Z3988"></li> </ul> <div class="mw-references-wrap"> <ol class="references"> <li id="cite_note-mathematica-program-4"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling's_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-mathematica-program_4-0">^</a> For example, a program in Mathematica: <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"> <pre>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]] </pre> </div></li> </ol> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"> <div class="reflist reflist-lower-alpha"> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(12)"> <h2 id="External_links">External links</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Stirling's_approximation&action=edit&section=14&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: External links" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-12 collapsible-block" id="mf-section-12"> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style> <style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style> <div class="side-box side-box-right plainlinks sistersitebox"> <style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"> <noscript> <img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" data-file-width="1024" data-file-height="1376"> </noscript><span class="lazy-image-placeholder" style="width: 30px;height: 40px;" data-src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" data-alt="" data-width="30" data-height="40" data-srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-class="mw-file-element">  </div> <div class="side-box-text plainlist"> Wikimedia Commons has media related to <a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://commons.wikimedia.org/wiki/Category:Stirling%2527s_approximation" class="extiw" title="commons:Category:Stirling's approximation">Stirling's approximation</a>. </div> </div> </div> <ul> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.encyclopediaofmath.org/index.php?title%3DStirling_formula">"Stirling_formula"</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Encyclopedia_of_Mathematics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/European_Mathematical_Society?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Stirling_formula&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DStirling_formula&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling%27s+approximation" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://www.luschny.de/math/factorial/approx/SimpleCases.html">Peter Luschny, Approximation formulas for the factorial function n!</a></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs2"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Eric_W._Weisstein?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Eric W. Weisstein">Weisstein, Eric W.</a>, <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mathworld.wolfram.com/StirlingsApproximation.html">"Stirling's Approximation"</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/MathWorld?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="MathWorld">MathWorld</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Stirling%27s+Approximation&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FStirlingsApproximation.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling%27s+approximation" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://planetmath.org/StirlingsApproximation">Stirling's approximation</a> at <a 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data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li> <li class="interlanguage-link interwiki-be mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://be.wikipedia.org/wiki/%25D0%25A4%25D0%25BE%25D1%2580%25D0%25BC%25D1%2583%25D0%25BB%25D0%25B0_%25D0%25A1%25D1%2582%25D1%258B%25D1%2580%25D0%25BB%25D1%2596%25D0%25BD%25D0%25B3%25D0%25B0" title="Формула Стырлінга – Belarusian" lang="be" hreflang="be" data-title="Формула Стырлінга" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li> <li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://bg.wikipedia.org/wiki/%25D0%25A4%25D0%25BE%25D1%2580%25D0%25BC%25D1%2583%25D0%25BB%25D0%25B0_%25D0%25BD%25D0%25B0_%25D0%25A1%25D1%2582%25D1%258A%25D1%2580%25D0%25BB%25D0%25B8%25D0%25BD%25D0%25B3" title="Формула на Стърлинг – Bulgarian" lang="bg" hreflang="bg" data-title="Формула на Стърлинг" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ca.wikipedia.org/wiki/F%25C3%25B3rmula_de_Stirling" title="Fórmula de Stirling – Catalan" lang="ca" hreflang="ca" data-title="Fórmula de Stirling" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li> <li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cs.wikipedia.org/wiki/Stirling%25C5%25AFv_vzorec" title="Stirlingův vzorec – Czech" lang="cs" hreflang="cs" data-title="Stirlingův vzorec" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li> <li class="interlanguage-link interwiki-de mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://de.wikipedia.org/wiki/Stirlingformel" title="Stirlingformel – German" lang="de" hreflang="de" data-title="Stirlingformel" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li> <li class="interlanguage-link interwiki-el mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://el.wikipedia.org/wiki/%25CE%25A4%25CF%258D%25CF%2580%25CE%25BF%25CF%2582_%25CE%25A3%25CF%2584%25CE%25AF%25CF%2581%25CE%25BB%25CE%25B9%25CE%25BD%25CE%25B3%25CE%25BA" title="Τύπος Στίρλινγκ – Greek" lang="el" hreflang="el" data-title="Τύπος Στίρλινγκ" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li> <li class="interlanguage-link interwiki-es mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://es.wikipedia.org/wiki/F%25C3%25B3rmula_de_Stirling" title="Fórmula de Stirling – Spanish" lang="es" hreflang="es" data-title="Fórmula de Stirling" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li> <li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://eu.wikipedia.org/wiki/Stirlingen_hurbilketa" title="Stirlingen hurbilketa – Basque" lang="eu" hreflang="eu" data-title="Stirlingen hurbilketa" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li> <li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fa.wikipedia.org/wiki/%25D8%25AA%25D9%2582%25D8%25B1%25DB%258C%25D8%25A8_%25D8%25A7%25D8%25B3%25D8%25AA%25D8%25B1%25D9%2584%25DB%258C%25D9%2586%25DA%25AF" title="تقریب استرلینگ – Persian" lang="fa" hreflang="fa" data-title="تقریب استرلینگ" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li> <li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fr.wikipedia.org/wiki/Formule_de_Stirling" title="Formule de Stirling – French" lang="fr" hreflang="fr" data-title="Formule de Stirling" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li> <li class="interlanguage-link interwiki-gl mw-list-item"><a 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title="Стирлинг формуласы – Kazakh" lang="kk" hreflang="kk" data-title="Стирлинг формуласы" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li> <li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://lt.wikipedia.org/wiki/Stirlingo_formul%25C4%2597" title="Stirlingo formulė – Lithuanian" lang="lt" hreflang="lt" data-title="Stirlingo formulė" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li> <li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hu.wikipedia.org/wiki/Stirling-formula" title="Stirling-formula – Hungarian" lang="hu" hreflang="hu" data-title="Stirling-formula" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li> <li 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Stirling's approximation - Wikipedia