@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:-XQZPCT5B-R
  skos:prefLabel "série L de Dirichlet"@fr, "Dirichlet L-series"@en ;
  a skos:Concept ;
  skos:narrower psr:-BH3DV7MT-2 .

psr: a skos:ConceptScheme .
psr:-BH3DV7MT-2
  skos:prefLabel "Dirichlet beta function"@en, "fonction bêta de Dirichlet"@fr ;
  skos:inScheme psr: ;
  a skos:Concept ;
  skos:broader psr:-XQZPCT5B-R, psr:-FH1H1FB9-1 ;
  skos:definition """En mathématiques, la <b>fonction β de Dirichlet</b>, aussi appelée fonction ζ de Catalan, est un des exemples les plus simples de fonction L, après la fonction zêta de Riemann. C'est la fonction L de Dirichlet associée au caractère de Dirichlet alterné de période 4.
<br/>Elle est définie, pour tout complexe <span class="texhtml mvar" style="font-style:italic;">s</span> de partie réelle strictement positive, par la série&nbsp;:
<br/>
<br/><dl><dd><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 \\eta (s)=\\\\sum _{n=0}^{\\\\infty }{\\rac {(-1)^{n}}{(2n+1)^{s}}}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>β<!-- β --></mi>
<br/>        <mo stretchy="false">(</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/>            <mrow>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mn>1</mn>
<br/>              <msup>
<br/>                <mo stretchy="false">)</mo>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi>n</mi>
<br/>                </mrow>
<br/>              </msup>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mn>2</mn>
<br/>              <mi>n</mi>
<br/>              <mo>+</mo>
<br/>              <mn>1</mn>
<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 \\eta (s)=\\\\sum _{n=0}^{\\\\infty }{\\rac {(-1)^{n}}{(2n+1)^{s}}}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32628be466e83329848e892f940cbbb3479e62fb" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.281ex; height:6.843ex;" alt="{\\\\displaystyle \\eta (s)=\\\\sum _{n=0}^{\\\\infty }{\\rac {(-1)^{n}}{(2n+1)^{s}}}}"></span>,</dd></dl></dd></dl>
<br/>ou par l'intégrale
<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 \\eta (s)={\\rac {1}{\\\\Gamma (s)}}\\\\int _{0}^{\\\\infty }{\\rac {x^{s-1}\\\\mathrm {e} ^{x}}{\\\\mathrm {e} ^{2x}+1}}\\\\,dx}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>β<!-- β --></mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>s</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mn>1</mn>
<br/>            <mrow>
<br/>              <mi mathvariant="normal">Γ<!-- Γ --></mi>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>s</mi>
<br/>              <mo stretchy="false">)</mo>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<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/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <msup>
<br/>                <mi>x</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi>s</mi>
<br/>                  <mo>−<!-- − --></mo>
<br/>                  <mn>1</mn>
<br/>                </mrow>
<br/>              </msup>
<br/>              <msup>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi mathvariant="normal">e</mi>
<br/>                </mrow>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi>x</mi>
<br/>                </mrow>
<br/>              </msup>
<br/>            </mrow>
<br/>            <mrow>
<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>x</mi>
<br/>                </mrow>
<br/>              </msup>
<br/>              <mo>+</mo>
<br/>              <mn>1</mn>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mi>d</mi>
<br/>        <mi>x</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\eta (s)={\\rac {1}{\\\\Gamma (s)}}\\\\int _{0}^{\\\\infty }{\\rac {x^{s-1}\\\\mathrm {e} ^{x}}{\\\\mathrm {e} ^{2x}+1}}\\\\,dx}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32715560169dff6b9d63f7154de1ec6812c26b6f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.431ex; height:6.509ex;" alt="{\\\\displaystyle \\eta (s)={\\rac {1}{\\\\Gamma (s)}}\\\\int _{0}^{\\\\infty }{\\rac {x^{s-1}\\\\mathrm {e} ^{x}}{\\\\mathrm {e} ^{2x}+1}}\\\\,dx}"></span>.</dd> 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Fonction_b%C3%AAta_de_Dirichlet">https://fr.wikipedia.org/wiki/Fonction_b%C3%AAta_de_Dirichlet</a>)"""@fr, """In mathematics, the Dirichlet beta function (also known as the Catalan beta function) is a special function, closely related to the Riemann zeta function. It is a particular Dirichlet L-function, the L-function for the alternating character of period four. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Dirichlet_beta_function">https://en.wikipedia.org/wiki/Dirichlet_beta_function</a>)"""@en ;
  dc:modified "2023-08-21"^^xsd:date ;
  skos:altLabel "fonction ζ de Catalan"@fr, "Catalan beta function"@en, "fonction β de Dirichlet"@fr ;
  dc:created "2023-08-04"^^xsd:date ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Fonction_b%C3%AAta_de_Dirichlet>, <https://en.wikipedia.org/wiki/Dirichlet_beta_function> .

psr:-FH1H1FB9-1
  skos:prefLabel "special function"@en, "fonction spéciale"@fr ;
  a skos:Concept ;
  skos:narrower psr:-BH3DV7MT-2 .