@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .

psr:-G83Z11RF-C
  skos:prefLabel "intégrale de Selberg"@fr, "Selberg integral"@en ;
  a skos:Concept ;
  skos:related psr:-J8PV8QXJ-5 .

psr:-KRLWRR7V-H
  skos:prefLabel "fonction gamma"@fr, "gamma function"@en ;
  a skos:Concept ;
  skos:related psr:-J8PV8QXJ-5 .

psr: a skos:ConceptScheme .
psr:-J8PV8QXJ-5
  skos:altLabel "Euler integral of the first kind"@en, "intégrale d'Euler de première espèce"@fr ;
  skos:related psr:-KRLWRR7V-H, psr:-G83Z11RF-C ;
  skos:definition """In mathematics, the <b>beta function</b>, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral
<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 \\\\mathrm {B} (z_{1},z_{2})=\\\\int _{0}^{1}t^{z_{1}-1}(1-t)^{z_{2}-1}\\\\,dt}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
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<br/>          <mi mathvariant="normal">B</mi>
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<br/>            <mn>1</mn>
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<br/>        <mo>,</mo>
<br/>        <msub>
<br/>          <mi>z</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <msubsup>
<br/>          <mo>∫<!-- ∫ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>1</mn>
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<br/>        </msubsup>
<br/>        <msup>
<br/>          <mi>t</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <msub>
<br/>              <mi>z</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mn>1</mn>
<br/>              </mrow>
<br/>            </msub>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mn>1</mn>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mi>t</mi>
<br/>        <msup>
<br/>          <mo stretchy="false">)</mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <msub>
<br/>              <mi>z</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mn>2</mn>
<br/>              </mrow>
<br/>            </msub>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mi>d</mi>
<br/>        <mi>t</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathrm {B} (z_{1},z_{2})=\\\\int _{0}^{1}t^{z_{1}-1}(1-t)^{z_{2}-1}\\\\,dt}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af2e0e1ed581918ee9225e0f011d791df29591d0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:33.555ex; height:6.176ex;" alt="{\\\\displaystyle \\\\mathrm {B} (z_{1},z_{2})=\\\\int _{0}^{1}t^{z_{1}-1}(1-t)^{z_{2}-1}\\\\,dt}"></span></dd></dl>
<br/>for complex number inputs 
<br/><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 z_{1},z_{2}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>z</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>,</mo>
<br/>        <msub>
<br/>          <mi>z</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle z_{1},z_{2}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd9e6b30dcd6b3d0ba19760e47ed97204c0a6cbb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:5.305ex; height:2.009ex;" alt="{\\\\displaystyle z_{1},z_{2}}"></span> such that <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 \\\\Re (z_{1}),\\\\Re (z_{2})>0}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi mathvariant="normal">ℜ<!-- ℜ --></mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <msub>
<br/>          <mi>z</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>,</mo>
<br/>        <mi mathvariant="normal">ℜ<!-- ℜ --></mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <msub>
<br/>          <mi>z</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>&gt;</mo>
<br/>        <mn>0</mn>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\Re (z_{1}),\\\\Re (z_{2})&gt;0}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e586772938f1fe6118c8c9f32ef6e485c4957189" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:17.033ex; height:2.843ex;" alt="{\\\\displaystyle \\\\Re (z_{1}),\\\\Re (z_{2})>0}"></span>. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Beta_function">https://en.wikipedia.org/wiki/Beta_function</a>)"""@en, """En mathématiques, la <b>fonction bêta</b> est une des deux intégrales d'Euler, définie pour tous nombres complexes <span class="texhtml mvar" style="font-style:italic;">x</span> et <span class="texhtml mvar" style="font-style:italic;">y</span> de parties réelles strictement positives par&nbsp;:
<br/>
<br/><center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\mathrm {B} (x,y)=\\\\int _{0}^{1}t^{x-1}(1-t)^{y-1}\\\\mathrm {d} t,}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="normal">B</mi>
<br/>        </mrow>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo>,</mo>
<br/>        <mi>y</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <msubsup>
<br/>          <mo>∫<!-- ∫ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <msup>
<br/>          <mi>t</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>x</mi>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mn>1</mn>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mi>t</mi>
<br/>        <msup>
<br/>          <mo stretchy="false">)</mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>y</mi>
<br/>            <mo>−<!-- − --></mo>
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<br/>        </msup>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="normal">d</mi>
<br/>        </mrow>
<br/>        <mi>t</mi>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathrm {B} (x,y)=\\\\int _{0}^{1}t^{x-1}(1-t)^{y-1}\\\\mathrm {d} t,}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/424c235329fa714e2405331608f1916e6a91ceec" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.671ex; height:6.176ex;" alt="{\\\\displaystyle \\\\mathrm {B} (x,y)=\\\\int _{0}^{1}t^{x-1}(1-t)^{y-1}\\\\mathrm {d} t,}"></span></center>
<br/>et éventuellement prolongée analytiquement à tout le plan complexe à l'exception des entiers négatifs. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Fonction_b%C3%AAta">https://fr.wikipedia.org/wiki/Fonction_b%C3%AAta</a>)"""@fr ;
  skos:broader psr:-FH1H1FB9-1 ;
  skos:inScheme psr: ;
  skos:prefLabel "fonction bêta"@fr, "beta function"@en ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Fonction_b%C3%AAta>, <https://en.wikipedia.org/wiki/Beta_function> ;
  a skos:Concept .

psr:-FH1H1FB9-1
  skos:prefLabel "special function"@en, "fonction spéciale"@fr ;
  a skos:Concept ;
  skos:narrower psr:-J8PV8QXJ-5 .