@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:-VZ83B143-L
  skos:prefLabel "fonction hypergéométrique"@fr, "hypergeometric function"@en ;
  a skos:Concept ;
  skos:narrower psr:-DD268JMV-3 .

psr:-DD268JMV-3
  dc:created "2023-08-17"^^xsd:date ;
  skos:prefLabel "Kelvin function"@en, "fonction de Kelvin-Bessel"@fr ;
  skos:broader psr:-VZ83B143-L ;
  dc:modified "2023-08-17"^^xsd:date ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Kelvin_functions>, <https://fr.wikipedia.org/wiki/Fonction_de_Kelvin-Bessel> ;
  a skos:Concept ;
  skos:definition """Les fonctions de Kelvin-Bessel sont des fonctions mathématiques obtenues à partir des fonctions de Bessel, en prenant comme argument pour ces dernières les racines carrées d'un nombre imaginaire pur. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Fonction_de_Kelvin-Bessel">https://fr.wikipedia.org/wiki/Fonction_de_Kelvin-Bessel</a>)"""@fr, """In applied mathematics, the <b>Kelvin functions</b> ber<sub><i>ν</i></sub>(<i>x</i>) and bei<sub><i>ν</i></sub>(<i>x</i>) are the real and imaginary parts, respectively, of
<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 J_{\\
u }\\\\left(xe^{\\rac {3\\\\pi i}{4}}\\
ight),\\\\,}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>J</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>ν<!-- ν --></mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mrow>
<br/>          <mo>(</mo>
<br/>          <mrow>
<br/>            <mi>x</mi>
<br/>            <msup>
<br/>              <mi>e</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mfrac>
<br/>                  <mrow>
<br/>                    <mn>3</mn>
<br/>                    <mi>π<!-- π --></mi>
<br/>                    <mi>i</mi>
<br/>                  </mrow>
<br/>                  <mn>4</mn>
<br/>                </mfrac>
<br/>              </mrow>
<br/>            </msup>
<br/>          </mrow>
<br/>          <mo>)</mo>
<br/>        </mrow>
<br/>        <mo>,</mo>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle J_{\\
u }\\\\left(xe^{\\rac {3\\\\pi i}{4}}\\
ight),\\\\,}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/835bc567e7fb12f2203c9b813243da9babba0874" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:12.997ex; height:6.176ex;" alt="J_\\
u \\\\left (x e^{\\rac{3 \\\\pi i}{4}} \\
ight ),\\\\,"></span></dd></dl>
<br/>where <i>x</i> is real, and <span class="texhtml"><i>J<sub>ν</sub></i>(<i>z</i>)</span>, is the <i>ν</i><sup>th</sup> order Bessel function of the first kind. Similarly, the functions ker<sub>ν</sub>(<i>x</i>) and kei<sub>ν</sub>(<i>x</i>) are the real and imaginary parts, respectively, of
<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 K_{\\
u }\\\\left(xe^{\\rac {\\\\pi i}{4}}\\
ight),\\\\,}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>K</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>ν<!-- ν --></mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mrow>
<br/>          <mo>(</mo>
<br/>          <mrow>
<br/>            <mi>x</mi>
<br/>            <msup>
<br/>              <mi>e</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mfrac>
<br/>                  <mrow>
<br/>                    <mi>π<!-- π --></mi>
<br/>                    <mi>i</mi>
<br/>                  </mrow>
<br/>                  <mn>4</mn>
<br/>                </mfrac>
<br/>              </mrow>
<br/>            </msup>
<br/>          </mrow>
<br/>          <mo>)</mo>
<br/>        </mrow>
<br/>        <mo>,</mo>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle K_{\\
u }\\\\left(xe^{\\rac {\\\\pi i}{4}}\\
ight),\\\\,}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b169c9549a952624e55eb527c751d7cc6f6d705" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:13.013ex; height:6.176ex;" alt="K_\\
u \\\\left (x e^{\\rac{\\\\pi i}{4}} \\
ight ),\\\\,"></span></dd></dl>
<br/>where <span class="texhtml"><i>K<sub>ν</sub></i>(<i>z</i>)</span> is the <i>ν</i><sup>th</sup> order modified Bessel function of the second kind.
<br/>These functions are named after William Thomson, 1st Baron Kelvin.
<br/>While the Kelvin functions are defined as the real and imaginary parts of Bessel functions with <i>x</i> taken to be real, the functions can be analytically continued for complex arguments <span class="texhtml"><i>xe</i><sup><i>iφ</i></sup>, 0 ≤ <i>φ</i> &lt; 2<i>π</i>.</span> With the exception of ber<sub><i>n</i></sub>(<i>x</i>) and bei<sub><i>n</i></sub>(<i>x</i>) for integral <i>n</i>, the Kelvin functions have a branch point at <i>x</i>&nbsp;=&nbsp;0.
<br/>Below, <span class="texhtml">Γ(<i>z</i>)</span> is the gamma function and <span class="texhtml"><i>ψ</i>(<i>z</i>)</span> is the digamma function.
<br/> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Kelvin_functions">https://en.wikipedia.org/wiki/Kelvin_functions</a>)"""@en ;
  skos:inScheme psr: .