@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:-RBFVN7DN-2
  skos:prefLabel "mathematical constant"@en, "constante mathématique"@fr ;
  a skos:Concept ;
  skos:narrower psr:-SH08CC7M-L .

psr:-SH08CC7M-L
  dc:created "2023-08-03"^^xsd:date ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Constante_de_Gelfond>, <https://en.wikipedia.org/wiki/Gelfond%27s_constant> ;
  skos:broader psr:-L3LNPG9M-Q, psr:-RBFVN7DN-2 ;
  skos:definition """En mathématiques, la <b>constante de Gelfond</b> est le nombre réel transcendant <span class="texhtml">e<sup>π</sup></span>, c'est-à-dire <span class="texhtml">e</span> à la puissance <span class="texhtml">π</span>. Sa transcendance fut démontrée en 1929 par Alexandre Gelfond. C'est un cas particulier de son théorème de 1934. En effet, les nombres <span class="texhtml">–1</span> (différent de 0 et 1) et <span class="texhtml">–i</span> (non rationnel) sont algébriques, or  <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 {\\
m {e}}^{\\\\pi }=({\\
m {e}}^{{\\
m {i}}\\\\pi })^{-{\\
m {i}}}=(-1)^{-{\\
m {i}}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msup>           <mrow class="MJX-TeXAtom-ORD">             <mrow class="MJX-TeXAtom-ORD">               <mi mathvariant="normal">e</mi>             </mrow>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mi>π<!-- π --></mi>           </mrow>         </msup>         <mo>=</mo>         <mo stretchy="false">(</mo>         <msup>           <mrow class="MJX-TeXAtom-ORD">             <mrow class="MJX-TeXAtom-ORD">               <mi mathvariant="normal">e</mi>             </mrow>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mrow class="MJX-TeXAtom-ORD">               <mrow class="MJX-TeXAtom-ORD">                 <mi mathvariant="normal">i</mi>               </mrow>             </mrow>             <mi>π<!-- π --></mi>           </mrow>         </msup>         <msup>           <mo stretchy="false">)</mo>           <mrow class="MJX-TeXAtom-ORD">             <mo>−<!-- − --></mo>             <mrow class="MJX-TeXAtom-ORD">               <mrow class="MJX-TeXAtom-ORD">                 <mi mathvariant="normal">i</mi>               </mrow>             </mrow>           </mrow>         </msup>         <mo>=</mo>         <mo stretchy="false">(</mo>         <mo>−<!-- − --></mo>         <mn>1</mn>         <msup>           <mo stretchy="false">)</mo>           <mrow class="MJX-TeXAtom-ORD">             <mo>−<!-- − --></mo>             <mrow class="MJX-TeXAtom-ORD">               <mrow class="MJX-TeXAtom-ORD">                 <mi mathvariant="normal">i</mi>               </mrow>             </mrow>           </mrow>         </msup>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle {\\
m {e}}^{\\\\pi }=({\\
m {e}}^{{\\
m {i}}\\\\pi })^{-{\\
m {i}}}=(-1)^{-{\\
m {i}}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0f2d0aa72ee2226f23bc901d92aaaf8f78a4b27" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.593ex; height:3.176ex;" alt="{\\\\displaystyle {\\
m {e}}^{\\\\pi }=({\\
m {e}}^{{\\
m {i}}\\\\pi })^{-{\\
m {i}}}=(-1)^{-{\\
m {i}}}}"></span></dd></dl> (En considérant, la détermination principale de l'argument). Cette constante fut mentionnée dans le septième problème de Hilbert. Une constante reliée est la constante de Gelfond-Schneider, 2<sup><span class="racine">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span></sup>. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Constante_de_Gelfond">https://fr.wikipedia.org/wiki/Constante_de_Gelfond</a>)"""@fr, """In mathematics, <b>Gelfond's constant</b>, named after Aleksandr Gelfond, is <span class="texhtml"><i>e</i><sup><i>π</i></sup></span>, that is, <span class="texhtml mvar" style="font-style:italic;">e</span> raised to the power  <span class="texhtml mvar" style="font-style:italic;">π</span>. Like both <span class="texhtml mvar" style="font-style:italic;">e</span> and <span class="texhtml mvar" style="font-style:italic;">π</span>, this constant is a transcendental number. This was first established by Gelfond and may now be considered as an application of the Gelfond–Schneider theorem, noting that <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle e^{\\\\pi }=(e^{i\\\\pi })^{-i}=(-1)^{-i},}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msup>           <mi>e</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>π<!-- π --></mi>           </mrow>         </msup>         <mo>=</mo>         <mo stretchy="false">(</mo>         <msup>           <mi>e</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>i</mi>             <mi>π<!-- π --></mi>           </mrow>         </msup>         <msup>           <mo stretchy="false">)</mo>           <mrow class="MJX-TeXAtom-ORD">             <mo>−<!-- − --></mo>             <mi>i</mi>           </mrow>         </msup>         <mo>=</mo>         <mo stretchy="false">(</mo>         <mo>−<!-- − --></mo>         <mn>1</mn>         <msup>           <mo stretchy="false">)</mo>           <mrow class="MJX-TeXAtom-ORD">             <mo>−<!-- − --></mo>             <mi>i</mi>           </mrow>         </msup>         <mo>,</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle e^{\\\\pi }=(e^{i\\\\pi })^{-i}=(-1)^{-i},}</annotation>   </semantics> </math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaacf09b00028e08f0d4faf73886b130f4959119" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.672ex; height:3.176ex;" alt="{\\\\displaystyle e^{\\\\pi }=(e^{i\\\\pi })^{-i}=(-1)^{-i},}"></div> where <span class="texhtml mvar" style="font-style:italic;">i</span> is the imaginary unit. Since <span class="texhtml">−<i>i</i></span> is algebraic but not rational, <span class="texhtml"><i>e</i><sup><i>π</i></sup></span> is transcendental. The constant was mentioned in Hilbert's seventh problem. A related constant is <span class="texhtml">2<sup><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</sup></span></span></span>, known as the Gelfond–Schneider constant. The related value <span class="texhtml mvar" style="font-style:italic;">π</span> + <span class="texhtml"><i>e</i><sup><i>π</i></sup></span> is also irrational.  
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Gelfond%27s_constant">https://en.wikipedia.org/wiki/Gelfond%27s_constant</a>)"""@en ;
  skos:prefLabel "Gelfond's constant"@en, "constante de Gelfond"@fr ;
  skos:inScheme psr: ;
  a skos:Concept ;
  dc:modified "2024-10-18"^^xsd:date .

psr:-L3LNPG9M-Q
  skos:prefLabel "nombre transcendant"@fr, "transcendental number"@en ;
  a skos:Concept ;
  skos:narrower psr:-SH08CC7M-L .