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

psr:-TBXLXKB4-5
  skos:prefLabel "elliptic gamma function"@en, "fonction gamma elliptique"@fr ;
  a skos:Concept ;
  skos:broader psr:-KRLWRR7V-H .

psr:-M73B26XF-8
  skos:prefLabel "Hadjicostas's formula"@en, "formule de Hadjicostas"@fr ;
  a skos:Concept ;
  skos:related psr:-KRLWRR7V-H .

psr:-LCW4TM3S-M
  skos:prefLabel "Hölder's theorem"@en, "théorème de Hölder"@fr ;
  a skos:Concept ;
  skos:broader psr:-KRLWRR7V-H .

psr:-KRLWRR7V-H
  skos:narrower psr:-N6NT23ZF-K, psr:-FB95XBPX-P, psr:-ZKSBD6VF-K, psr:-LCW4TM3S-M, psr:-TBXLXKB4-5, psr:-F203S6GV-C, psr:-HNLXNRMF-6 ;
  skos:definition """On définit la <b>fonction gamma</b>, et notée par la lettre grecque <span class="texhtml">Γ</span> (gamma majuscule) de la façon suivante. Pour tout <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}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>z</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle z}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="z"></span> de partie réelle strictement positive, on pose
<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 \\\\Gamma (z)=\\\\int _{0}^{+\\\\infty }t^{z-1}\\\\,\\\\mathrm {e} ^{-t}\\\\,\\\\mathrm {d} t}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi mathvariant="normal">Γ<!-- Γ --></mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>z</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/>            <mo>+</mo>
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <msup>
<br/>          <mi>t</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>z</mi>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mspace width="thinmathspace"></mspace>
<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>t</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="normal">d</mi>
<br/>        </mrow>
<br/>        <mi>t</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\Gamma (z)=\\\\int _{0}^{+\\\\infty }t^{z-1}\\\\,\\\\mathrm {e} ^{-t}\\\\,\\\\mathrm {d} t}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dda418eae53e5f5b015d875f91e4c74c69db8df" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:23.439ex; height:6.009ex;" alt="{\\\\displaystyle \\\\Gamma (z)=\\\\int _{0}^{+\\\\infty }t^{z-1}\\\\,\\\\mathrm {e} ^{-t}\\\\,\\\\mathrm {d} t}"></span>.</dd></dl>
<br/>C'est une intégrale paramétrée par <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}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>z</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle z}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="z"></span>, l'intégration se faisant sur <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 t}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>t</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle t}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="t"></span>. Cette intégrale impropre converge absolument sur le demi-plan complexe où la partie réelle est strictement positive, et une intégration par parties montre que
<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 \\\\Gamma (z+1)=z\\\\;\\\\Gamma (z)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi mathvariant="normal">Γ<!-- Γ --></mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>z</mi>
<br/>        <mo>+</mo>
<br/>        <mn>1</mn>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mi>z</mi>
<br/>        <mspace width="thickmathspace"></mspace>
<br/>        <mi mathvariant="normal">Γ<!-- Γ --></mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>z</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\Gamma (z+1)=z\\\\;\\\\Gamma (z)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e30dfc17c5872fc48af21d874981072d769c7a7" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:17.535ex; height:2.843ex;" alt="{\\\\displaystyle \\\\Gamma (z+1)=z\\\\;\\\\Gamma (z)}"></span>.</dd></dl>
<br/>Cette fonction peut être prolongée analytiquement en une fonction méromorphe sur l'ensemble des nombres complexes, excepté pour <i>z</i>&nbsp;=&nbsp;0,&nbsp; −1, −2, −3… qui sont des pôles. C'est ce prolongement qu'on appelle généralement «&nbsp;fonction gamma&nbsp;». L'unicité du prolongement analytique permet de montrer que la fonction prolongée vérifie encore l'équation fonctionnelle précédente. Cela permet une définition plus simple, à partir de l'intégrale, et un calcul de proche en proche de Γ pour <i>z</i> – 1, <i>z</i> – 2,&nbsp;<abbr class="abbr" title="et cetera">etc.</abbr> 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Fonction_gamma">https://fr.wikipedia.org/wiki/Fonction_gamma</a>)"""@fr, """In mathematics, the <b>gamma function</b> (represented by <span class="texhtml">Γ</span>, the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers.  For every positive integer <span class="texhtml mvar" style="font-style:italic;">n</span>, <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 \\\\Gamma (n)=(n-1)!\\\\,.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi mathvariant="normal">Γ<!-- Γ --></mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>n</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>n</mi>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mn>1</mn>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>!</mo>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\Gamma (n)=(n-1)!\\\\,.}</annotation>
<br/>  </semantics>
<br/></math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/223eac31f4207bed0200d4b88e9a95f6f3e5c6a7" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -0.838ex; width:16.643ex; height:2.843ex;" alt="{\\\\displaystyle \\\\Gamma (n)=(n-1)!\\\\,.}"></div>
<br/>Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral:
<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 \\\\Gamma (z)=\\\\int _{0}^{\\\\infty }t^{z-1}e^{-t}{\\	ext{ d}}t,\\\\ \\\\qquad \\\\Re (z)>0\\\\,.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi mathvariant="normal">Γ<!-- Γ --></mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>z</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/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <msup>
<br/>          <mi>t</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>z</mi>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <msup>
<br/>          <mi>e</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>−<!-- − --></mo>
<br/>            <mi>t</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mtext>&nbsp;d</mtext>
<br/>        </mrow>
<br/>        <mi>t</mi>
<br/>        <mo>,</mo>
<br/>        <mtext>&nbsp;</mtext>
<br/>        <mspace width="2em"></mspace>
<br/>        <mi mathvariant="normal">ℜ<!-- ℜ --></mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>z</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>&gt;</mo>
<br/>        <mn>0</mn>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\Gamma (z)=\\\\int _{0}^{\\\\infty }t^{z-1}e^{-t}{\\	ext{ d}}t,\\\\ \\\\qquad \\\\Re (z)&gt;0\\\\,.}</annotation>
<br/>  </semantics>
<br/></math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47fa0f0aaaabee2c7e6d27fce3781a7106fd8fa1" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -2.338ex; width:38.394ex; height:5.843ex;" alt="{\\\\displaystyle \\\\Gamma (z)=\\\\int _{0}^{\\\\infty }t^{z-1}e^{-t}{\\	ext{ d}}t,\\\\ \\\\qquad \\\\Re (z)>0\\\\,.}"></div>
<br/>The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles.
<br/>The gamma function has no zeros, so the reciprocal gamma function <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1050945101">.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,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span role="math" class="sfrac tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">Γ(<i>z</i>)</span></span></span> is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential function:
<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 \\\\Gamma (z)={\\\\mathcal {M}}\\\\{e^{-x}\\\\}(z)\\\\,.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi mathvariant="normal">Γ<!-- Γ --></mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>z</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mo fence="false" stretchy="false">{</mo>
<br/>        <msup>
<br/>          <mi>e</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>−<!-- − --></mo>
<br/>            <mi>x</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo fence="false" stretchy="false">}</mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>z</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\Gamma (z)={\\\\mathcal {M}}\\\\{e^{-x}\\\\}(z)\\\\,.}</annotation>
<br/>  </semantics>
<br/></math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ff87d616e9b3b6214865daeacf081701dc9cbda" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -0.838ex; width:20.03ex; height:3.009ex;" alt="{\\\\displaystyle \\\\Gamma (z)={\\\\mathcal {M}}\\\\{e^{-x}\\\\}(z)\\\\,.}"></div>
<br/>Other extensions of the factorial function do exist, but the gamma function is the most popular and useful. It is a component in various probability-distribution functions, and as such it is applicable in the fields of probability and statistics, as well as combinatorics.
<br/> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Gamma_function">https://en.wikipedia.org/wiki/Gamma_function</a>)"""@en ;
  skos:related psr:-W127WDLN-J, psr:-J8PV8QXJ-5, psr:-MPKZZL79-2, psr:-M73B26XF-8 ;
  skos:altLabel "intégrale d'Euler de seconde espèce"@fr, "Euler integral of the second kind"@en ;
  skos:broader psr:-FH1H1FB9-1, psr:-L5F0NSBD-W ;
  skos:prefLabel "gamma function"@en, "fonction gamma"@fr ;
  skos:inScheme psr: ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Fonction_gamma>, <https://en.wikipedia.org/wiki/Gamma_function> ;
  a skos:Concept .

psr:-L5F0NSBD-W
  skos:prefLabel "fonction méromorphe"@fr, "meromorphic function"@en ;
  a skos:Concept ;
  skos:narrower psr:-KRLWRR7V-H .

psr:-ZKSBD6VF-K
  skos:prefLabel "multiplication theorem"@en, "théorème de la multiplication"@fr ;
  a skos:Concept ;
  skos:broader psr:-KRLWRR7V-H .

psr:-FB95XBPX-P
  skos:prefLabel "Fransén-Robinson constant"@en, "constante de Fransén-Robinson"@fr ;
  a skos:Concept ;
  skos:broader psr:-KRLWRR7V-H .

psr: a skos:ConceptScheme .
psr:-HNLXNRMF-6
  skos:prefLabel "fonction gamma p-adique"@fr, "p-adic gamma function"@en ;
  a skos:Concept ;
  skos:broader psr:-KRLWRR7V-H .

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

psr:-F203S6GV-C
  skos:prefLabel "Euler integral"@en, "intégrale d'Euler"@fr ;
  a skos:Concept ;
  skos:broader psr:-KRLWRR7V-H .

psr:-MPKZZL79-2
  skos:prefLabel "Barnes G-function"@en, "fonction G de Barnes"@fr ;
  a skos:Concept ;
  skos:related psr:-KRLWRR7V-H .

psr:-FH1H1FB9-1
  skos:prefLabel "special function"@en, "fonction spéciale"@fr ;
  a skos:Concept ;
  skos:narrower psr:-KRLWRR7V-H .

psr:-N6NT23ZF-K
  skos:prefLabel "Chowla-Selberg formula"@en, "formule de Chowla-Selberg"@fr ;
  a skos:Concept ;
  skos:broader psr:-KRLWRR7V-H .

psr:-W127WDLN-J
  skos:prefLabel "factorielle"@fr, "factorial"@en ;
  a skos:Concept ;
  skos:related psr:-KRLWRR7V-H .