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

psr:-BLP2HLSP-6
  skos:prefLabel "calcul intégral"@fr, "integral calculus"@en ;
  a skos:Concept ;
  skos:narrower psr:-HMVJL63M-G .

psr: a skos:ConceptScheme .
psr:-HMVJL63M-G
  skos:altLabel "Euler-Poisson integral"@en ;
  skos:inScheme psr: ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Gaussian_integral>, <https://fr.wikipedia.org/wiki/Int%C3%A9grale_de_Gauss> ;
  skos:prefLabel "intégrale de Gauss"@fr, "Gaussian integral"@en ;
  skos:definition """The <b>Gaussian integral</b>, also known as the <b>Euler–Poisson integral</b>, is the integral of the Gaussian function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f(x)=e^{-x^{2}}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>f</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <msup>
<br/>          <mi>e</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>−<!-- − --></mo>
<br/>            <msup>
<br/>              <mi>x</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mn>2</mn>
<br/>              </mrow>
<br/>            </msup>
<br/>          </mrow>
<br/>        </msup>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f(x)=e^{-x^{2}}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bed0b77b34cab03996deb42d464becab2f05636" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:11.882ex; height:3.509ex;" alt="{\\\\displaystyle f(x)=e^{-x^{2}}}"></span> over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is
<br/><div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\int _{-\\\\infty }^{\\\\infty }e^{-x^{2}}\\\\,dx={\\\\sqrt {\\\\pi }}.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msubsup>
<br/>          <mo>∫<!-- ∫ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>−<!-- − --></mo>
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <msup>
<br/>          <mi>e</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>−<!-- − --></mo>
<br/>            <msup>
<br/>              <mi>x</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mn>2</mn>
<br/>              </mrow>
<br/>            </msup>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mi>d</mi>
<br/>        <mi>x</mi>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <msqrt>
<br/>            <mi>π<!-- π --></mi>
<br/>          </msqrt>
<br/>        </mrow>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\int _{-\\\\infty }^{\\\\infty }e^{-x^{2}}\\\\,dx={\\\\sqrt {\\\\pi }}.}</annotation>
<br/>  </semantics>
<br/></math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b06d446e3c625f48f318811eabdfe5902b11508a" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -2.505ex; width:19.145ex; height:6.009ex;" alt="{\\\\displaystyle \\\\int _{-\\\\infty }^{\\\\infty }e^{-x^{2}}\\\\,dx={\\\\sqrt {\\\\pi }}.}"></div>
<br/> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Gaussian_integral">https://en.wikipedia.org/wiki/Gaussian_integral</a>)"""@en, """En mathématiques, une <b>intégrale de Gauss</b> est l'intégrale d'une fonction gaussienne sur l'ensemble des réels. Sa valeur est reliée à la constante π par la formule
<br/>
<br/><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\int _{-\\\\infty }^{+\\\\infty }\\\\mathrm {e} ^{-\\\\alpha \\\\,x^{2}}\\\\mathrm {d} x={\\\\sqrt {\\rac {\\\\pi }{\\\\alpha }}},}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msubsup>
<br/>          <mo>∫<!-- ∫ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>−<!-- − --></mo>
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>+</mo>
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <msup>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">e</mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>−<!-- − --></mo>
<br/>            <mi>α<!-- α --></mi>
<br/>            <mspace width="thinmathspace"></mspace>
<br/>            <msup>
<br/>              <mi>x</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mn>2</mn>
<br/>              </mrow>
<br/>            </msup>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="normal">d</mi>
<br/>        </mrow>
<br/>        <mi>x</mi>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <msqrt>
<br/>            <mfrac>
<br/>              <mi>π<!-- π --></mi>
<br/>              <mi>α<!-- α --></mi>
<br/>            </mfrac>
<br/>          </msqrt>
<br/>        </mrow>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\int _{-\\\\infty }^{+\\\\infty }\\\\mathrm {e} ^{-\\\\alpha \\\\,x^{2}}\\\\mathrm {d} x={\\\\sqrt {\\rac {\\\\pi }{\\\\alpha }}},}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04d3380599b5e66ea05121f938fdef688cc45b89" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.671ex; width:22.774ex; height:6.343ex;" alt="\\\\int _{{-\\\\infty }}^{{+\\\\infty }}{\\\\mathrm  e}^{{-\\\\alpha \\\\,x^{2}}}{\\\\mathrm  d}x={\\\\sqrt  {{\\rac  {\\\\pi }{\\\\alpha }}}},"></span></dd></dl>
<br/>où <i>α</i> est un paramètre réel strictement positif. Elle intervient dans la définition de la loi de probabilité appelée loi gaussienne, ou loi normale. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Int%C3%A9grale_de_Gauss">https://fr.wikipedia.org/wiki/Int%C3%A9grale_de_Gauss</a>)"""@fr ;
  dc:created "2023-08-16"^^xsd:date ;
  a skos:Concept ;
  skos:broader psr:-BLP2HLSP-6 ;
  dc:modified "2023-08-24"^^xsd:date ;
  skos:related psr:-XQF2FLQF-B .

psr:-XQF2FLQF-B
  skos:prefLabel "normal distribution"@en, "loi normale"@fr ;
  a skos:Concept ;
  skos:related psr:-HMVJL63M-G .