@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .

psr:-XTHQQMRV-S
  dc:modified "2024-10-18"^^xsd:date ;
  skos:broader psr:-RBFVN7DN-2, psr:-P36V4MHV-V ;
  skos:prefLabel "Stieltjes constants"@en, "constantes de Stieltjes"@fr ;
  skos:definition """In mathematics, the <b>Stieltjes constants</b> are the numbers <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 _{k}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>γ<!-- γ --></mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>k</mi>           </mrow>         </msub>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\gamma _{k}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fe92cc350733480812673fa7deeaf7ad0bf70f1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.293ex; height:2.176ex;" alt="\\\\gamma _{k}"></span> that occur in the Laurent series expansion of the Riemann zeta function:  <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 (1+s)={\\rac {1}{s}}+\\\\sum _{n=0}^{\\\\infty }{\\rac {(-1)^{n}}{n!}}\\\\gamma _{n}s^{n}.}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>ζ<!-- ζ --></mi>         <mo stretchy="false">(</mo>         <mn>1</mn>         <mo>+</mo>         <mi>s</mi>         <mo stretchy="false">)</mo>         <mo>=</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <mi>s</mi>           </mfrac>         </mrow>         <mo>+</mo>         <munderover>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>=</mo>             <mn>0</mn>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mi mathvariant="normal">∞<!-- ∞ --></mi>           </mrow>         </munderover>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mrow>               <mo stretchy="false">(</mo>               <mo>−<!-- − --></mo>               <mn>1</mn>               <msup>                 <mo stretchy="false">)</mo>                 <mrow class="MJX-TeXAtom-ORD">                   <mi>n</mi>                 </mrow>               </msup>             </mrow>             <mrow>               <mi>n</mi>               <mo>!</mo>             </mrow>           </mfrac>         </mrow>         <msub>           <mi>γ<!-- γ --></mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msub>         <msup>           <mi>s</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msup>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\zeta (1+s)={\\rac {1}{s}}+\\\\sum _{n=0}^{\\\\infty }{\\rac {(-1)^{n}}{n!}}\\\\gamma _{n}s^{n}.}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a280f91c8fbdf2eb9c0a27b3fbb346f1d973acfb" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.89ex; height:6.843ex;" alt="{\\\\displaystyle \\\\zeta (1+s)={\\rac {1}{s}}+\\\\sum _{n=0}^{\\\\infty }{\\rac {(-1)^{n}}{n!}}\\\\gamma _{n}s^{n}.}"></span></dd></dl> The constant <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 _{0}=\\\\gamma =0.577\\\\dots }">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>γ<!-- γ --></mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>0</mn>           </mrow>         </msub>         <mo>=</mo>         <mi>γ<!-- γ --></mi>         <mo>=</mo>         <mn>0.577</mn>         <mo>…<!-- … --></mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\gamma _{0}=\\\\gamma =0.577\\\\dots }</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c516df52b1c79d218fc85f6bdf1f0d974192719" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.125ex; height:2.676ex;" alt="\\\\gamma _{0}=\\\\gamma =0.577\\\\dots "></span> is known as the Euler–Mascheroni constant. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Stieltjes_constants">https://en.wikipedia.org/wiki/Stieltjes_constants</a>)"""@en, """En mathématiques, les <b>constantes de Stieltjes</b> (nommées d'après le mathématicien néerlandais Thomas Joannes Stieltjes) sont les nombres qui interviennent dans le développement en série de Laurent de la fonction zêta de Riemann :  <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 \\\\zeta (s)={\\rac {1}{s-1}}+\\\\sum _{n=0}^{+\\\\infty }{\\rac {(-1)^{n}}{n!}}\\\\gamma _{n}\\\\;(s-1)^{n}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>ζ<!-- ζ --></mi>         <mo stretchy="false">(</mo>         <mi>s</mi>         <mo stretchy="false">)</mo>         <mo>=</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <mrow>               <mi>s</mi>               <mo>−<!-- − --></mo>               <mn>1</mn>             </mrow>           </mfrac>         </mrow>         <mo>+</mo>         <munderover>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>=</mo>             <mn>0</mn>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mo>+</mo>             <mi mathvariant="normal">∞<!-- ∞ --></mi>           </mrow>         </munderover>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mrow>               <mo stretchy="false">(</mo>               <mo>−<!-- − --></mo>               <mn>1</mn>               <msup>                 <mo stretchy="false">)</mo>                 <mrow class="MJX-TeXAtom-ORD">                   <mi>n</mi>                 </mrow>               </msup>             </mrow>             <mrow>               <mi>n</mi>               <mo>!</mo>             </mrow>           </mfrac>         </mrow>         <msub>           <mi>γ<!-- γ --></mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msub>         <mspace width="thickmathspace"></mspace>         <mo stretchy="false">(</mo>         <mi>s</mi>         <mo>−<!-- − --></mo>         <mn>1</mn>         <msup>           <mo stretchy="false">)</mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msup>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\zeta (s)={\\rac {1}{s-1}}+\\\\sum _{n=0}^{+\\\\infty }{\\rac {(-1)^{n}}{n!}}\\\\gamma _{n}\\\\;(s-1)^{n}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6d41773430c4044acd4af85a700d2fd9b76608d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.629ex; height:7.176ex;" alt="{\\\\displaystyle \\\\zeta (s)={\\rac {1}{s-1}}+\\\\sum _{n=0}^{+\\\\infty }{\\rac {(-1)^{n}}{n!}}\\\\gamma _{n}\\\\;(s-1)^{n}}"></span>.</center> On démontre que chaque <span class="texhtml">γ<sub><i>n</i></sub></span> est donné par une limite :  <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 \\\\gamma _{n}=\\\\lim _{m\\
ightarrow \\\\infty }{\\\\left(\\\\left(\\\\sum _{k=1}^{m}{\\rac {(\\\\ln k)^{n}}{k}}\\
ight)-{\\rac {(\\\\ln m)^{n+1}}{n+1}}\\
ight)}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>γ<!-- γ --></mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msub>         <mo>=</mo>         <munder>           <mo movablelimits="true" form="prefix">lim</mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>m</mi>             <mo stretchy="false">→<!-- → --></mo>             <mi mathvariant="normal">∞<!-- ∞ --></mi>           </mrow>         </munder>         <mrow class="MJX-TeXAtom-ORD">           <mrow>             <mo>(</mo>             <mrow>               <mrow>                 <mo>(</mo>                 <mrow>                   <munderover>                     <mo>∑<!-- ∑ --></mo>                     <mrow class="MJX-TeXAtom-ORD">                       <mi>k</mi>                       <mo>=</mo>                       <mn>1</mn>                     </mrow>                     <mrow class="MJX-TeXAtom-ORD">                       <mi>m</mi>                     </mrow>                   </munderover>                   <mrow class="MJX-TeXAtom-ORD">                     <mfrac>                       <mrow>                         <mo stretchy="false">(</mo>                         <mi>ln</mi>                         <mo>⁡<!-- ⁡ --></mo>                         <mi>k</mi>                         <msup>                           <mo stretchy="false">)</mo>                           <mrow class="MJX-TeXAtom-ORD">                             <mi>n</mi>                           </mrow>                         </msup>                       </mrow>                       <mi>k</mi>                     </mfrac>                   </mrow>                 </mrow>                 <mo>)</mo>               </mrow>               <mo>−<!-- − --></mo>               <mrow class="MJX-TeXAtom-ORD">                 <mfrac>                   <mrow>                     <mo stretchy="false">(</mo>                     <mi>ln</mi>                     <mo>⁡<!-- ⁡ --></mo>                     <mi>m</mi>                     <msup>                       <mo stretchy="false">)</mo>                       <mrow class="MJX-TeXAtom-ORD">                         <mi>n</mi>                         <mo>+</mo>                         <mn>1</mn>                       </mrow>                     </msup>                   </mrow>                   <mrow>                     <mi>n</mi>                     <mo>+</mo>                     <mn>1</mn>                   </mrow>                 </mfrac>               </mrow>             </mrow>             <mo>)</mo>           </mrow>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\gamma _{n}=\\\\lim _{m\\
ightarrow \\\\infty }{\\\\left(\\\\left(\\\\sum _{k=1}^{m}{\\rac {(\\\\ln k)^{n}}{k}}\\
ight)-{\\rac {(\\\\ln m)^{n+1}}{n+1}}\\
ight)}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56b494419523fce49caede1e2090cfb297c456da" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.315ex; height:7.509ex;" alt="{\\\\displaystyle \\\\gamma _{n}=\\\\lim _{m\\
ightarrow \\\\infty }{\\\\left(\\\\left(\\\\sum _{k=1}^{m}{\\rac {(\\\\ln k)^{n}}{k}}\\
ight)-{\\rac {(\\\\ln m)^{n+1}}{n+1}}\\
ight)}}"></span></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 \\\\gamma _{0}=\\\\gamma \\\\ =0{,}577...}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>γ<!-- γ --></mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>0</mn>           </mrow>         </msub>         <mo>=</mo>         <mi>γ<!-- γ --></mi>         <mtext> </mtext>         <mo>=</mo>         <mn>0,577...</mn>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\gamma _{0}=\\\\gamma \\\\ =0{,}577...}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3187f784b077b1e943c89497398faf14bdfbe05d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.536ex; height:2.676ex;" alt="{\\\\displaystyle \\\\gamma _{0}=\\\\gamma \\\\ =0{,}577...}"></span> est la constante d'Euler-Mascheroni. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Constantes_de_Stieltjes">https://fr.wikipedia.org/wiki/Constantes_de_Stieltjes</a>)"""@fr ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Constantes_de_Stieltjes>, <https://en.wikipedia.org/wiki/Stieltjes_constants> ;
  skos:inScheme psr: ;
  dc:created "2023-08-04"^^xsd:date ;
  a skos:Concept .

psr:-P36V4MHV-V
  skos:prefLabel "fonction zêta de Riemann"@fr, "Riemann zeta function"@en ;
  a skos:Concept ;
  skos:narrower psr:-XTHQQMRV-S .

psr:-RBFVN7DN-2
  skos:prefLabel "mathematical constant"@en, "constante mathématique"@fr ;
  a skos:Concept ;
  skos:narrower psr:-XTHQQMRV-S .

psr: a skos:ConceptScheme .