@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:-NHFK3Q1R-H
  skos:prefLabel "fonction L"@fr, "L-function"@en ;
  a skos:Concept ;
  skos:narrower psr:-XNLCMG3Q-2 .

psr:-XNLCMG3Q-2
  dc:modified "2023-08-24"^^xsd:date ;
  skos:broader psr:-NHFK3Q1R-H, psr:-FH1H1FB9-1 ;
  skos:definition """In mathematics, the Lerch zeta function, sometimes called the Hurwitz–Lerch zeta function, is a special function that generalizes the Hurwitz zeta function and the polylogarithm. It is named after Czech mathematician Mathias Lerch, who published a paper about the function in 1887. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Lerch_zeta_function">https://en.wikipedia.org/wiki/Lerch_zeta_function</a>)"""@en, """En mathématiques, la <b>fonction zêta de Lerch</b>, ou <b>fonction zêta de Hurwitz-Lerch</b> est une fonction spéciale qui généralise la fonction zêta de Hurwitz et le polylogarithme, nommée d'après le mathématicien Mathias Lerch. Elle est définie comme somme d'une série comme suit&nbsp;:
<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 L(\\\\lambda ,\\\\alpha ,s)=\\\\sum _{n=0}^{\\\\infty }{\\rac {\\\\mathrm {e} ^{2\\\\pi \\\\mathrm {i} \\\\lambda n}}{(n+\\\\alpha )^{s}}}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>L</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>λ<!-- λ --></mi>
<br/>        <mo>,</mo>
<br/>        <mi>α<!-- α --></mi>
<br/>        <mo>,</mo>
<br/>        <mi>s</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>            <mo>=</mo>
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </munderover>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <msup>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi mathvariant="normal">e</mi>
<br/>              </mrow>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mn>2</mn>
<br/>                <mi>π<!-- π --></mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi mathvariant="normal">i</mi>
<br/>                </mrow>
<br/>                <mi>λ<!-- λ --></mi>
<br/>                <mi>n</mi>
<br/>              </mrow>
<br/>            </msup>
<br/>            <mrow>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>n</mi>
<br/>              <mo>+</mo>
<br/>              <mi>α<!-- α --></mi>
<br/>              <msup>
<br/>                <mo stretchy="false">)</mo>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi>s</mi>
<br/>                </mrow>
<br/>              </msup>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle L(\\\\lambda ,\\\\alpha ,s)=\\\\sum _{n=0}^{\\\\infty }{\\rac {\\\\mathrm {e} ^{2\\\\pi \\\\mathrm {i} \\\\lambda n}}{(n+\\\\alpha )^{s}}}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c66ebf9c83809737043eb74cd034020add3a7b58" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:25.605ex; height:6.843ex;" alt="{\\\\displaystyle L(\\\\lambda ,\\\\alpha ,s)=\\\\sum _{n=0}^{\\\\infty }{\\rac {\\\\mathrm {e} ^{2\\\\pi \\\\mathrm {i} \\\\lambda n}}{(n+\\\\alpha )^{s}}}}"></span>.</dd></dl>
<br/>La fonction zêta de Lerch est reliée à la fonction transcendante de Lerch, définie par la formule&nbsp;:
<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 \\\\Phi (z,s,\\\\alpha )=\\\\sum _{n=0}^{\\\\infty }{\\rac {z^{n}}{(n+\\\\alpha )^{s}}}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi mathvariant="normal">Φ<!-- Φ --></mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>z</mi>
<br/>        <mo>,</mo>
<br/>        <mi>s</mi>
<br/>        <mo>,</mo>
<br/>        <mi>α<!-- α --></mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>            <mo>=</mo>
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </munderover>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <msup>
<br/>              <mi>z</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi>n</mi>
<br/>              </mrow>
<br/>            </msup>
<br/>            <mrow>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>n</mi>
<br/>              <mo>+</mo>
<br/>              <mi>α<!-- α --></mi>
<br/>              <msup>
<br/>                <mo stretchy="false">)</mo>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi>s</mi>
<br/>                </mrow>
<br/>              </msup>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\Phi (z,s,\\\\alpha )=\\\\sum _{n=0}^{\\\\infty }{\\rac {z^{n}}{(n+\\\\alpha )^{s}}}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ce334a553e1d950a0235b2aa2290236bbdf3ed4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:25.433ex; height:6.843ex;" alt="{\\\\displaystyle \\\\Phi (z,s,\\\\alpha )=\\\\sum _{n=0}^{\\\\infty }{\\rac {z^{n}}{(n+\\\\alpha )^{s}}}}"></span></dd></dl>
<br/>par l'identité&nbsp;:
<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 \\\\Phi (\\\\mathrm {e} ^{2\\\\pi \\\\mathrm {i} \\\\lambda },s,\\\\alpha )=L(\\\\lambda ,\\\\alpha ,s)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi mathvariant="normal">Φ<!-- Φ --></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/>            <mn>2</mn>
<br/>            <mi>π<!-- π --></mi>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi mathvariant="normal">i</mi>
<br/>            </mrow>
<br/>            <mi>λ<!-- λ --></mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>,</mo>
<br/>        <mi>s</mi>
<br/>        <mo>,</mo>
<br/>        <mi>α<!-- α --></mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mi>L</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>λ<!-- λ --></mi>
<br/>        <mo>,</mo>
<br/>        <mi>α<!-- α --></mi>
<br/>        <mo>,</mo>
<br/>        <mi>s</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\Phi (\\\\mathrm {e} ^{2\\\\pi \\\\mathrm {i} \\\\lambda },s,\\\\alpha )=L(\\\\lambda ,\\\\alpha ,s)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a02e9db19cc081f428cf48e85d4ff8b12131562" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:25.069ex; height:3.176ex;" alt="{\\\\displaystyle \\\\Phi (\\\\mathrm {e} ^{2\\\\pi \\\\mathrm {i} \\\\lambda },s,\\\\alpha )=L(\\\\lambda ,\\\\alpha ,s)}"></span>.</dd> 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Fonction_z%C3%AAta_de_Lerch">https://fr.wikipedia.org/wiki/Fonction_z%C3%AAta_de_Lerch</a>)"""@fr ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Lerch_zeta_function>, <https://fr.wikipedia.org/wiki/Fonction_z%C3%AAta_de_Lerch> ;
  skos:altLabel "fonction zêta de Hurwitz-Lerch"@fr, "Hurwitz-Lerch zeta function"@en ;
  skos:prefLabel "fonction zêta de Lerch"@fr, "Lerch zeta function"@en ;
  skos:inScheme psr: ;
  a skos:Concept ;
  dc:created "2023-08-04"^^xsd:date .

psr:-FH1H1FB9-1
  skos:prefLabel "special function"@en, "fonction spéciale"@fr ;
  a skos:Concept ;
  skos:narrower psr:-XNLCMG3Q-2 .