@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:-CXXJ6GSD-R
  skos:inScheme psr: ;
  skos:prefLabel "fonction de Bessel"@fr, "Bessel function"@en ;
  skos:altLabel "fonction cylindrique"@fr, "cylinder function"@en ;
  dc:modified "2023-08-17"^^xsd:date ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Fonction_de_Bessel>, <https://en.wikipedia.org/wiki/Bessel_function> ;
  a skos:Concept ;
  skos:definition """En mathématiques, et plus précisément en analyse, les <b>fonctions de Bessel</b>, appelées aussi quelquefois fonctions cylindriques, découvertes par le mathématicien suisse Daniel Bernoulli, portent le nom du mathématicien allemand Friedrich Wilhelm Bessel. Bessel développa l'analyse de ces fonctions en 1816 dans le cadre de ses études du mouvement des planètes induit par l'interaction gravitationnelle, généralisant les découvertes antérieures de Bernoulli. Ces fonctions sont des solutions canoniques <i>y</i>(<i>x</i>) de l'<b>équation différentielle de Bessel</b>&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 x^{2}{\\rac {\\\\mathrm {d} ^{2}y}{\\\\mathrm {d} x^{2}}}+x{\\rac {\\\\mathrm {d} y}{\\\\mathrm {d} x}}+(x^{2}-\\\\alpha ^{2})y=0}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msup>
<br/>          <mi>x</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <msup>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi mathvariant="normal">d</mi>
<br/>                </mrow>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>2</mn>
<br/>                </mrow>
<br/>              </msup>
<br/>              <mi>y</mi>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi mathvariant="normal">d</mi>
<br/>              </mrow>
<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/>        <mi>x</mi>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi mathvariant="normal">d</mi>
<br/>              </mrow>
<br/>              <mi>y</mi>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi mathvariant="normal">d</mi>
<br/>              </mrow>
<br/>              <mi>x</mi>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>+</mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <msup>
<br/>          <mi>x</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>−<!-- − --></mo>
<br/>        <msup>
<br/>          <mi>α<!-- α --></mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mi>y</mi>
<br/>        <mo>=</mo>
<br/>        <mn>0</mn>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle x^{2}{\\rac {\\\\mathrm {d} ^{2}y}{\\\\mathrm {d} x^{2}}}+x{\\rac {\\\\mathrm {d} y}{\\\\mathrm {d} x}}+(x^{2}-\\\\alpha ^{2})y=0}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c8620838bd3b0eaec71b14493876763d904bf3b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:32.357ex; height:6.176ex;" alt="x^{2}{\\rac  {{\\\\mathrm  d}^{2}y}{{\\\\mathrm  d}x^{2}}}+x{\\rac  {{\\\\mathrm  d}y}{{\\\\mathrm  d}x}}+(x^{2}-\\\\alpha ^{2})y=0"></span></dd></dl>
<br/>pour tout nombre réel ou complexe α. Le plus souvent, α est un entier naturel (alors appelé l'<i>ordre</i> de la fonction), ou un demi-entier. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Fonction_de_Bessel">https://fr.wikipedia.org/wiki/Fonction_de_Bessel</a>)"""@fr, """essel functions</b>, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions <span class="texhtml"><i>y</i>(<i>x</i>)</span> of Bessel's differential equation
<br/><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 x^{2}{\\rac {d^{2}y}{dx^{2}}}+x{\\rac {dy}{dx}}+\\\\left(x^{2}-\\\\alpha ^{2}\\
ight)y=0}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msup>
<br/>          <mi>x</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msup>
<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/>        <mi>x</mi>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mi>d</mi>
<br/>              <mi>y</mi>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mi>d</mi>
<br/>              <mi>x</mi>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>+</mo>
<br/>        <mrow>
<br/>          <mo>(</mo>
<br/>          <mrow>
<br/>            <msup>
<br/>              <mi>x</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mn>2</mn>
<br/>              </mrow>
<br/>            </msup>
<br/>            <mo>−<!-- − --></mo>
<br/>            <msup>
<br/>              <mi>α<!-- α --></mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mn>2</mn>
<br/>              </mrow>
<br/>            </msup>
<br/>          </mrow>
<br/>          <mo>)</mo>
<br/>        </mrow>
<br/>        <mi>y</mi>
<br/>        <mo>=</mo>
<br/>        <mn>0</mn>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle x^{2}{\\rac {d^{2}y}{dx^{2}}}+x{\\rac {dy}{dx}}+\\\\left(x^{2}-\\\\alpha ^{2}\\
ight)y=0}</annotation>
<br/>  </semantics>
<br/></math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f37ea24b1f82f74dbc1f8cf1348020726772f891" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:32.912ex; height:6.009ex;" alt="{\\\\displaystyle x^{2}{\\rac {d^{2}y}{dx^{2}}}+x{\\rac {dy}{dx}}+\\\\left(x^{2}-\\\\alpha ^{2}\\
ight)y=0}"></div>
<br/>for an arbitrary complex number <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 \\\\alpha }">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>α<!-- α --></mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\alpha }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="\\\\alpha "></span>, which represents the <i>order</i> of the Bessel function. Although <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 \\\\alpha }">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>α<!-- α --></mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\alpha }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="\\\\alpha "></span> and <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 -\\\\alpha }">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo>−<!-- − --></mo>
<br/>        <mi>α<!-- α --></mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle -\\\\alpha }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6807df17ce845a127ca43612e2ffd5cf62b7adc6" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.296ex; height:2.176ex;" alt="-\\\\alpha "></span> produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of <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 \\\\alpha }">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>α<!-- α --></mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\alpha }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="\\\\alpha "></span>. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Bessel_function">https://en.wikipedia.org/wiki/Bessel_function</a>)"""@en ;
  skos:broader psr:-VZ83B143-L .

psr:-VZ83B143-L
  skos:prefLabel "fonction hypergéométrique"@fr, "hypergeometric function"@en ;
  a skos:Concept ;
  skos:narrower psr:-CXXJ6GSD-R .