@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .

psr: a skos:ConceptScheme .
psr:-BDBLSF4B-Q
  skos:definition """En physique mathématique, on appelle <b>équation de Mathieu</b> une équation mise en évidence par Émile Mathieu au <abbr class="abbr" title="19ᵉ siècle"><span class="romain">XIX</span><sup style="font-size:72%">e</sup></abbr>&nbsp;siècle.
<br/>C'est un cas particulier de l'équation de Hill&nbsp;: <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 {\\rac {d^{2}x}{dt^{2}}}+G(t)x=0}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <msup>
<br/>                <mi>d</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>2</mn>
<br/>                </mrow>
<br/>              </msup>
<br/>              <mi>x</mi>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mi>d</mi>
<br/>              <msup>
<br/>                <mi>t</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>2</mn>
<br/>                </mrow>
<br/>              </msup>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>+</mo>
<br/>        <mi>G</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>t</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mi>x</mi>
<br/>        <mo>=</mo>
<br/>        <mn>0</mn>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\rac {d^{2}x}{dt^{2}}}+G(t)x=0}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5498ed7d715aeb45ad4a61d6219d37265764cbbd" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.171ex; width:17.345ex; height:6.009ex;" alt="{\\\\displaystyle {\\rac {d^{2}x}{dt^{2}}}+G(t)x=0}"></span> où <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 G(t)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>G</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>t</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle G(t)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3d6c09ba5569413364689bf4837c7b71ef0892f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:4.476ex; height:2.843ex;" alt="G(t)"></span> est une fonction périodique, avec&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 G(t)=a-2q\\\\cos(2t)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>G</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>t</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mi>a</mi>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mn>2</mn>
<br/>        <mi>q</mi>
<br/>        <mi>cos</mi>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mn>2</mn>
<br/>        <mi>t</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle G(t)=a-2q\\\\cos(2t)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56150266f137ae2370a67c0547912516dd8c3c26" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:21.186ex; height:2.843ex;" alt="{\\\\displaystyle G(t)=a-2q\\\\cos(2t)}"></span>, périodique de période <span class="texhtml"><i>T</i>=π</span>.</dd></dl>
<br/>Son comportement est assez particulier (résonance paramétrique, existence de sous-harmoniques, etc.). Émile Mathieu l'a rencontrée (1865) en étudiant les vibrations d'une membrane elliptique.
<br/>Ses solutions seront appelées les <b>fonctions de Mathieu</b>. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/%C3%89quation_de_Mathieu">https://fr.wikipedia.org/wiki/%C3%89quation_de_Mathieu</a>)"""@fr, """In mathematics, <b>Mathieu functions</b>, sometimes called <b>angular Mathieu functions</b>, are solutions of Mathieu's differential equation
<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 {\\rac {d^{2}y}{dx^{2}}}+(a-2q\\\\cos(2x))y=0,}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <msup>
<br/>                <mi>d</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>2</mn>
<br/>                </mrow>
<br/>              </msup>
<br/>              <mi>y</mi>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mi>d</mi>
<br/>              <msup>
<br/>                <mi>x</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>2</mn>
<br/>                </mrow>
<br/>              </msup>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>+</mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>a</mi>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mn>2</mn>
<br/>        <mi>q</mi>
<br/>        <mi>cos</mi>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mn>2</mn>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mi>y</mi>
<br/>        <mo>=</mo>
<br/>        <mn>0</mn>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\rac {d^{2}y}{dx^{2}}}+(a-2q\\\\cos(2x))y=0,}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8e4d30875926f1274fbee986fa475625fde88eb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.171ex; width:29.251ex; height:6.009ex;" alt="{\\\\displaystyle {\\rac {d^{2}y}{dx^{2}}}+(a-2q\\\\cos(2x))y=0,}"></span></dd></dl>
<br/>where <span class="texhtml mvar" style="font-style:italic;">a, q</span> are real-valued parameters. Since we may add <span class="texhtml">π/2</span> to <span class="texhtml mvar" style="font-style:italic;">x</span> to change the sign of <span class="texhtml mvar" style="font-style:italic;">q</span>, it is a usual convention to set <span class="texhtml"><i>q</i> ≥ 0</span>.
<br/> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Mathieu_function">https://en.wikipedia.org/wiki/Mathieu_function</a>)"""@en ;
  a skos:Concept ;
  skos:inScheme psr: ;
  skos:broader psr:-FH1H1FB9-1 ;
  skos:prefLabel "fonction de Mathieu"@fr, "Mathieu function"@en ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Mathieu_function>, <https://fr.wikipedia.org/wiki/%C3%89quation_de_Mathieu> .

psr:-FH1H1FB9-1
  skos:prefLabel "special function"@en, "fonction spéciale"@fr ;
  a skos:Concept ;
  skos:narrower psr:-BDBLSF4B-Q .