@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: a skos:ConceptScheme .
psr:-X5J7XXLL-2
  skos:prefLabel "multivalued function"@en, "fonction multivaluée"@fr ;
  a skos:Concept ;
  skos:narrower psr:-HZ1BNPV5-S .

psr:-HZ1BNPV5-S
  skos:definition """In mathematics, the <b>Lambert <span class="texhtml mvar" style="font-style:italic;">W</span> function</b>, also called the <b>omega function</b> or <b>product logarithm</b>, is a multivalued function, namely the branches of the converse relation of the function <span class="texhtml"><i>f</i>(<i>w</i>) = <i>we</i><sup><i>w</i></sup></span>, where <span class="texhtml mvar" style="font-style:italic;">w</span> is any complex number and <span class="texhtml"><i>e</i><sup><i>w</i></sup></span> is the exponential function. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Lambert_W_function">https://en.wikipedia.org/wiki/Lambert_W_function</a>)"""@en, """En mathématiques, et plus précisément en analyse, la <b>fonction W de Lambert</b>, nommée ainsi d'après Jean-Henri Lambert, et parfois aussi appelée la <b>fonction Oméga</b>, est la réciproque de la fonction de variable complexe <span class="texhtml mvar" style="font-style:italic;">f</span> définie par <span class="texhtml"><i>f</i>(<i>w</i>) = <i>w</i> e<sup><i>w</i></sup></span>, c'est-à-dire
<br/>que pour tous nombres complexes <i>z</i> et <i>w</i>, nous avons&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 z=w\\\\mathrm {e} ^{w}\\\\;\\\\Longleftrightarrow \\\\;w=W(z).}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>z</mi>
<br/>        <mo>=</mo>
<br/>        <mi>w</mi>
<br/>        <msup>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">e</mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>w</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mspace width="thickmathspace"></mspace>
<br/>        <mo stretchy="false">⟺<!-- ⟺ --></mo>
<br/>        <mspace width="thickmathspace"></mspace>
<br/>        <mi>w</mi>
<br/>        <mo>=</mo>
<br/>        <mi>W</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>z</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle z=w\\\\mathrm {e} ^{w}\\\\;\\\\Longleftrightarrow \\\\;w=W(z).}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f90dc1c50546b37ae383eefbde6a9c0d4a9b6783" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:25.931ex; height:2.843ex;" alt="{\\\\displaystyle z=w\\\\mathrm {e} ^{w}\\\\;\\\\Longleftrightarrow \\\\;w=W(z).}"></span></center>
<br/>Puisque la fonction <span class="texhtml mvar" style="font-style:italic;">f</span> n'est pas injective, <span class="texhtml mvar" style="font-style:italic;">W</span> est une fonction multivaluée ou «&nbsp;multiforme&nbsp;» qui comprend deux branches pour les valeurs réelles <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 x\\\\geqslant -{\\rac {1}{\\\\mathrm {e} }}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>x</mi>
<br/>        <mo>⩾<!-- ⩾ --></mo>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mn>1</mn>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi mathvariant="normal">e</mi>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle x\\\\geqslant -{\\rac {1}{\\\\mathrm {e} }}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6f8617601f9b2ebe09c1ab5cc24d0bd408883ed" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:8.235ex; height:5.176ex;" alt="{\\\\displaystyle x\\\\geqslant -{\\rac {1}{\\\\mathrm {e} }}}"></span>. Une des branches, la branche principale, <span class="texhtml"><i>W</i><sub>0</sub></span> peut être prolongée analytiquement en dehors de <span class="texhtml">]−∞, –<span class="texhtml"><span style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;">1</span><span style="position:absolute;left:-10000px;top:auto;width:1px;height:1px;overflow:hidden">/</span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">e</span></span></span>]</span>. Pour tout nombre complexe <span class="texhtml"><i>z </i>∉ ]−∞, –<span class="texhtml"><span style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;">1</span><span style="position:absolute;left:-10000px;top:auto;width:1px;height:1px;overflow:hidden">/</span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">e</span></span></span>]</span>, on a&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 W_{0}(z)\\\\mathrm {e} ^{W_{0}(z)}=z\\\\,.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>W</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>z</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <msup>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">e</mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <msub>
<br/>              <mi>W</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mn>0</mn>
<br/>              </mrow>
<br/>            </msub>
<br/>            <mo stretchy="false">(</mo>
<br/>            <mi>z</mi>
<br/>            <mo stretchy="false">)</mo>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>=</mo>
<br/>        <mi>z</mi>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle W_{0}(z)\\\\mathrm {e} ^{W_{0}(z)}=z\\\\,.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2528793efe9cfc264a76e0e1bb66e4caadee065f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:17.062ex; height:3.343ex;" alt="{\\\\displaystyle W_{0}(z)\\\\mathrm {e} ^{W_{0}(z)}=z\\\\,.}"></span></center>
<br/>La fonction <span class="texhtml mvar" style="font-style:italic;">W</span> de Lambert ne peut pas être exprimée à l'aide de fonctions élémentaires.
<br/> 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Fonction_W_de_Lambert">https://fr.wikipedia.org/wiki/Fonction_W_de_Lambert</a>)"""@fr ;
  skos:broader psr:-X5J7XXLL-2, psr:-FH1H1FB9-1 ;
  dc:modified "2023-08-16"^^xsd:date ;
  skos:prefLabel "Lambert W function"@en, "fonction W de Lambert"@fr ;
  skos:altLabel "omega function"@en, "product logarithm"@en, "fonction Oméga"@fr ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Fonction_W_de_Lambert>, <https://en.wikipedia.org/wiki/Lambert_W_function> ;
  skos:inScheme psr: ;
  dc:created "2023-08-16"^^xsd:date ;
  a skos:Concept .

psr:-FH1H1FB9-1
  skos:prefLabel "special function"@en, "fonction spéciale"@fr ;
  a skos:Concept ;
  skos:narrower psr:-HZ1BNPV5-S .