Jump to navigation Jump to search

General

Display information for equation id:math.224631.1 on revision:224631

* Page found: Stirling's approximation (eq math.224631.1)

(force rerendering)

Occurrences on the following pages:

Hash: 4406031ad950dcd31c4193db40c50582

TeX (original user input): 

n! \sim \sqrt{2 \pi n}\left(\frac{n}{e}\right)^n

TeX (checked): 

n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}

LaTeXML (experimental; uses MathML) rendering

MathML (3.693 KB / 846 B) :

n ! ∼ 2 ⁢ π ⁢ n ⁢ ( n e ) n similar-to 𝑛 2 𝜋 𝑛 superscript 𝑛 𝑒 𝑛 {\displaystyle n!\sim{\sqrt{2\pi n}}\left({\frac{n}{e}}\right)^{n}} 
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p1.1.m1.1" class="ltx_Math" alttext="{\displaystyle n!\sim{\sqrt{2\pi n}}\left({\frac{n}{e}}\right)^{n}}" display="inline">
  <semantics id="p1.1.m1.1a">
    <mrow id="p1.1.m1.1.9" xref="p1.1.m1.1.9.cmml">
      <mrow id="p1.1.m1.1.9.1" xref="p1.1.m1.1.9.1.cmml">
        <mi id="p1.1.m1.1.1" xref="p1.1.m1.1.1.cmml">n</mi>
        <mo lspace="0pt" rspace="3.5pt" id="p1.1.m1.1.2" xref="p1.1.m1.1.2.cmml">!</mo>
      </mrow>
      <mo id="p1.1.m1.1.3" xref="p1.1.m1.1.3.cmml">∼</mo>
      <mrow id="p1.1.m1.1.9.2" xref="p1.1.m1.1.9.2.cmml">
        <msqrt id="p1.1.m1.1.4" xref="p1.1.m1.1.4.cmml">
          <mrow id="p1.1.m1.1.4.2" xref="p1.1.m1.1.4.2.cmml">
            <mn id="p1.1.m1.1.4.2.1" xref="p1.1.m1.1.4.2.1.cmml">2</mn>
            <mo id="p1.1.m1.1.4.2.4" xref="p1.1.m1.1.4.2.4.cmml">⁢</mo>
            <mi id="p1.1.m1.1.4.2.2" xref="p1.1.m1.1.4.2.2.cmml">π</mi>
            <mo id="p1.1.m1.1.4.2.4a" xref="p1.1.m1.1.4.2.4.cmml">⁢</mo>
            <mi id="p1.1.m1.1.4.2.3" xref="p1.1.m1.1.4.2.3.cmml">n</mi>
          </mrow>
        </msqrt>
        <mo id="p1.1.m1.1.9.2.1" xref="p1.1.m1.1.9.2.1.cmml">⁢</mo>
        <msup id="p1.1.m1.1.9.2.2" xref="p1.1.m1.1.9.2.2.cmml">
          <mrow id="p1.1.m1.1.9.2.2.2" xref="p1.1.m1.1.6.cmml">
            <mo id="p1.1.m1.1.5" xref="p1.1.m1.1.6.cmml">(</mo>
            <mstyle displaystyle="true" id="p1.1.m1.1.6" xref="p1.1.m1.1.6.cmml">
              <mfrac id="p1.1.m1.1.6a" xref="p1.1.m1.1.6.cmml">
                <mi id="p1.1.m1.1.6.2" xref="p1.1.m1.1.6.2.cmml">n</mi>
                <mi id="p1.1.m1.1.6.3" xref="p1.1.m1.1.6.3.cmml">e</mi>
              </mfrac>
            </mstyle>
            <mo id="p1.1.m1.1.7" xref="p1.1.m1.1.6.cmml">)</mo>
          </mrow>
          <mi id="p1.1.m1.1.8.1" xref="p1.1.m1.1.8.1.cmml">n</mi>
        </msup>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content" id="p1.1.m1.1b">
      <apply id="p1.1.m1.1.9.cmml" xref="p1.1.m1.1.9">
        <csymbol cd="latexml" id="p1.1.m1.1.3.cmml" xref="p1.1.m1.1.3">similar-to</csymbol>
        <apply id="p1.1.m1.1.9.1.cmml" xref="p1.1.m1.1.9.1">
          <factorial id="p1.1.m1.1.2.cmml" xref="p1.1.m1.1.2"/>
          <ci id="p1.1.m1.1.1.cmml" xref="p1.1.m1.1.1">𝑛</ci>
        </apply>
        <apply id="p1.1.m1.1.9.2.cmml" xref="p1.1.m1.1.9.2">
          <times id="p1.1.m1.1.9.2.1.cmml" xref="p1.1.m1.1.9.2.1"/>
          <apply id="p1.1.m1.1.4.cmml" xref="p1.1.m1.1.4">
            <root id="p1.1.m1.1.4a.cmml" xref="p1.1.m1.1.4"/>
            <apply id="p1.1.m1.1.4.2.cmml" xref="p1.1.m1.1.4.2">
              <times id="p1.1.m1.1.4.2.4.cmml" xref="p1.1.m1.1.4.2.4"/>
              <cn type="integer" id="p1.1.m1.1.4.2.1.cmml" xref="p1.1.m1.1.4.2.1">2</cn>
              <ci id="p1.1.m1.1.4.2.2.cmml" xref="p1.1.m1.1.4.2.2">𝜋</ci>
              <ci id="p1.1.m1.1.4.2.3.cmml" xref="p1.1.m1.1.4.2.3">𝑛</ci>
            </apply>
          </apply>
          <apply id="p1.1.m1.1.9.2.2.cmml" xref="p1.1.m1.1.9.2.2">
            <csymbol cd="ambiguous" id="p1.1.m1.1.9.2.2.1.cmml" xref="p1.1.m1.1.9.2.2">superscript</csymbol>
            <apply id="p1.1.m1.1.6.cmml" xref="p1.1.m1.1.9.2.2.2">
              <divide id="p1.1.m1.1.6.1.cmml" xref="p1.1.m1.1.9.2.2.2"/>
              <ci id="p1.1.m1.1.6.2.cmml" xref="p1.1.m1.1.6.2">𝑛</ci>
              <ci id="p1.1.m1.1.6.3.cmml" xref="p1.1.m1.1.6.3">𝑒</ci>
            </apply>
            <ci id="p1.1.m1.1.8.1.cmml" xref="p1.1.m1.1.8.1">𝑛</ci>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation encoding="application/x-tex" id="p1.1.m1.1c">{\displaystyle n!\sim{\sqrt{2\pi n}}\left({\frac{n}{e}}\right)^{n}}</annotation>
  </semantics>
</math>

SVG (6.969 KB / 2.736 KB) :

n factorial tilde StartRoot 2 times pi times n EndRoot times left-parenthesis StartFraction n Over e EndFraction right-parenthesis Superscript n

SVG (MathML can be enabled via browser plugin) rendering

MathML (1.013 KB / 398 B) :

n ! ∼ 2 π n ( n e ) n {\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}} 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" 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>&#x223C;<!-- ∼ --></mo>
        <mrow class="MJX-TeXAtom-ORD">
          <msqrt>
            <mn>2</mn>
            <mi>&#x03C0;<!-- 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>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}}</annotation>
  </semantics>
</math>

SVG (5.673 KB / 2.522 KB) :

{\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}}

Translations to Computer Algebra Systems

Translation to Maple

In Maple:

Translation to Mathematica

In Mathematica:

Similar pages

Calculated based on the variables occurring on the entire Stirling's approximation page

Identifiers

MathML observations

0results

0results

no statistics present please run the maintenance script ExtractFeatures.php

0 results

0 results

Retrieved from "https://en.formulasearchengine.com/wiki/Special:FormulaInfo"