@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:-T36RJ8T9-6
  skos:prefLabel "fonction zêta de Hurwitz"@fr, "Hurwitz zeta function"@en ;
  skos:related psr:-NHDPQMVR-B, psr:-WRXGXT2X-P ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Hurwitz_zeta_function>, <https://fr.wikipedia.org/wiki/Fonction_z%C3%AAta_de_Hurwitz> ;
  dc:created "2023-08-04"^^xsd:date ;
  a skos:Concept ;
  skos:broader psr:-FH1H1FB9-1, psr:-NHFK3Q1R-H ;
  skos:definition """In mathematics, the <b>Hurwitz zeta function</b> is one of the many zeta functions. It is formally defined for complex variables <span class="texhtml mvar" style="font-style:italic;">s</span> with <span class="texhtml">Re(<i>s</i>) &gt; 1</span> and <span class="texhtml"><i>a</i> ≠ 0, −1, −2, …</span> by
<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 \\\\zeta (s,a)=\\\\sum _{n=0}^{\\\\infty }{\\rac {1}{(n+a)^{s}}}.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>ζ<!-- ζ --></mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>s</mi>
<br/>        <mo>,</mo>
<br/>        <mi>a</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/>            <mn>1</mn>
<br/>            <mrow>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>n</mi>
<br/>              <mo>+</mo>
<br/>              <mi>a</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/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\zeta (s,a)=\\\\sum _{n=0}^{\\\\infty }{\\rac {1}{(n+a)^{s}}}.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb4eac48bd910e74269c705e7ff14be757772e0c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:22.86ex; height:6.843ex;" alt="{\\\\displaystyle \\\\zeta (s,a)=\\\\sum _{n=0}^{\\\\infty }{\\rac {1}{(n+a)^{s}}}.}"></span></dd></dl>
<br/>This series is absolutely convergent for the given values of <span class="texhtml mvar" style="font-style:italic;">s</span> and <span class="texhtml mvar" style="font-style:italic;">a</span> and can be extended to a meromorphic function defined for all <span class="texhtml"><i>s</i> ≠ 1</span>. The Riemann zeta function is <span class="texhtml">ζ(<i>s</i>,1)</span>. The Hurwitz zeta function is named after Adolf Hurwitz, who introduced it in 1882.
<br/> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Hurwitz_zeta_function">https://en.wikipedia.org/wiki/Hurwitz_zeta_function</a>)"""@en, """En mathématiques, la <b>fonction zêta de Hurwitz</b> est une des nombreuses fonctions zêta.
<br/>Elle est définie, pour toute valeur <span class="texhtml mvar" style="font-style:italic;">q</span> du paramètre, nombre complexe de partie réelle strictement positive, par la série suivante, convergeant vers une fonction holomorphe sur le demi-plan des complexes <span class="texhtml mvar" style="font-style:italic;">s</span> tels que <span class="texhtml">Re(<i>s</i>) &gt; 1</span>&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 \\\\zeta (s,q)=\\\\sum _{k=0}^{\\\\infty }(k+q)^{-s}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>ζ<!-- ζ --></mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>s</mi>
<br/>        <mo>,</mo>
<br/>        <mi>q</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>k</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/>        <mo stretchy="false">(</mo>
<br/>        <mi>k</mi>
<br/>        <mo>+</mo>
<br/>        <mi>q</mi>
<br/>        <msup>
<br/>          <mo stretchy="false">)</mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>−<!-- − --></mo>
<br/>            <mi>s</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\zeta (s,q)=\\\\sum _{k=0}^{\\\\infty }(k+q)^{-s}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e1116ced298a5fa5b576d9ae8564cffbf73ff90" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.171ex; width:21.764ex; height:7.009ex;" alt="{\\\\displaystyle \\\\zeta (s,q)=\\\\sum _{k=0}^{\\\\infty }(k+q)^{-s}}"></span>.</dd></dl>
<br/>Par prolongement analytique, <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 \\\\zeta (\\\\cdot ,q)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>ζ<!-- ζ --></mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mo>⋅<!-- ⋅ --></mo>
<br/>        <mo>,</mo>
<br/>        <mi>q</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\zeta (\\\\cdot ,q)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/669023ba851d02318411f631b7785ed3196a35a0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:5.655ex; height:2.843ex;" alt="{\\\\displaystyle \\\\zeta (\\\\cdot ,q)}"></span> s'étend en une fonction méromorphe sur le plan complexe, d'unique pôle <span class="texhtml"><i>s</i> = 1</span>.
<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 \\\\zeta (\\\\cdot ,1)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>ζ<!-- ζ --></mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mo>⋅<!-- ⋅ --></mo>
<br/>        <mo>,</mo>
<br/>        <mn>1</mn>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\zeta (\\\\cdot ,1)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8766b32573a4ce38d5233e1d12ac9d028439909d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:5.748ex; height:2.843ex;" alt="{\\\\displaystyle \\\\zeta (\\\\cdot ,1)}"></span> est la fonction zêta de Riemann.
<br/> 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Fonction_z%C3%AAta_de_Hurwitz">https://fr.wikipedia.org/wiki/Fonction_z%C3%AAta_de_Hurwitz</a>)"""@fr ;
  dc:modified "2023-08-04"^^xsd:date ;
  skos:inScheme psr: .

psr:-NHFK3Q1R-H
  skos:prefLabel "fonction L"@fr, "L-function"@en ;
  a skos:Concept ;
  skos:narrower psr:-T36RJ8T9-6 .

psr:-WRXGXT2X-P
  skos:prefLabel "fonction polylogarithme"@fr, "polylogarithm"@en ;
  a skos:Concept ;
  skos:related psr:-T36RJ8T9-6 .

psr:-NHDPQMVR-B
  skos:prefLabel "rational zeta series"@en, "série zêta rationnelle"@fr ;
  a skos:Concept ;
  skos:related psr:-T36RJ8T9-6 .

psr: a skos:ConceptScheme .
psr:-FH1H1FB9-1
  skos:prefLabel "special function"@en, "fonction spéciale"@fr ;
  a skos:Concept ;
  skos:narrower psr:-T36RJ8T9-6 .