@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:-RN7T2RV9-J
  skos:exactMatch <https://en.wikipedia.org/wiki/Confluent_hypergeometric_function>, <https://fr.wikipedia.org/wiki/Fonction_hyperg%C3%A9om%C3%A9trique_confluente> ;
  a skos:Concept ;
  skos:prefLabel "confluent hypergeometric function"@en, "fonction hypergéométrique confluente"@fr ;
  dc:modified "2023-07-27"^^xsd:date ;
  skos:inScheme psr: ;
  skos:definition """In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity. The term confluent refers to the merging of singular points of families of differential equations; confluere is Latin for "to flow together". There are several common standard forms of confluent hypergeometric functions :
         <br/>- Kummer's (confluent hypergeometric) function <i>M</i>(<i>a</i>, <i>b</i>, <i>z</i>), introduced by Kummer (1837), is a solution to Kummer's differential equation. This is also known as the confluent hypergeometric function of the first kind. There is a different and unrelated Kummer's function bearing the same name.
<br/>- Whittaker functions (for Edmund Taylor Whittaker) are solutions to Whittaker's equation.
<br/>- Coulomb wave functions are solutions to the Coulomb wave equation.
<br/>The Kummer functions, Whittaker functions, and Coulomb wave functions are essentially the same, and differ from each other only by elementary functions and change of variables.
<br/>- Tricomi's (confluent hypergeometric) function <i>U</i>(<i>a</i>, <i>b</i>, <i>z</i>) introduced by Francesco Tricomi (1947), sometimes denoted by Ψ(<i>a</i>; <i>b</i>; <i>z</i>), is another solution to Kummer's equation. This is also known as the confluent hypergeometric function of the second kind. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Confluent_hypergeometric_function">https://en.wikipedia.org/wiki/Confluent_hypergeometric_function</a>)"""@en, """La <b>fonction hypergéométrique confluente</b> (ou fonction de Kummer) est&nbsp;:
<br/><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 _{1}F_{1}(a;c;z)=\\\\sum _{n=0}^{\\\\infty }{\\rac {(a)_{n}}{(c)_{n}}}{\\rac {z^{n}}{n!}}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi></mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <msub>
<br/>          <mi>F</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>a</mi>
<br/>        <mo>;</mo>
<br/>        <mi>c</mi>
<br/>        <mo>;</mo>
<br/>        <mi>z</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>            <mo>=</mo>
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </munderover>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>a</mi>
<br/>              <msub>
<br/>                <mo stretchy="false">)</mo>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi>n</mi>
<br/>                </mrow>
<br/>              </msub>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>c</mi>
<br/>              <msub>
<br/>                <mo stretchy="false">)</mo>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi>n</mi>
<br/>                </mrow>
<br/>              </msub>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <msup>
<br/>              <mi>z</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi>n</mi>
<br/>              </mrow>
<br/>            </msup>
<br/>            <mrow>
<br/>              <mi>n</mi>
<br/>              <mo>!</mo>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle _{1}F_{1}(a;c;z)=\\\\sum _{n=0}^{\\\\infty }{\\rac {(a)_{n}}{(c)_{n}}}{\\rac {z^{n}}{n!}}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3660d8e154c922c7f28f74009e30a07a814043e2" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.884ex; height:6.843ex;" alt="{\\\\displaystyle _{1}F_{1}(a;c;z)=\\\\sum _{n=0}^{\\\\infty }{\\rac {(a)_{n}}{(c)_{n}}}{\\rac {z^{n}}{n!}}}"></span>
<br/>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 (a)_{n}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>a</mi>
<br/>        <msub>
<br/>          <mo stretchy="false">)</mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle (a)_{n}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e0810e7e636eef31bcf5a3df5a2b77b217a641b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.258ex; height:2.843ex;" alt="{\\\\displaystyle (a)_{n}}"></span> désigne le symbole de Pochhammer.
<br/>Elle est solution de l'équation différentielle d'ordre deux, appelée équation de Kummer&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 z{\\rac {\\\\mathrm {d} ^{2}u(z)}{\\\\mathrm {d} z^{2}}}+(c-z){\\rac {\\\\mathrm {d} u(z)}{\\\\mathrm {d} z}}-au(z)=0}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>z</mi>
<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>u</mi>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>z</mi>
<br/>              <mo stretchy="false">)</mo>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi mathvariant="normal">d</mi>
<br/>              </mrow>
<br/>              <msup>
<br/>                <mi>z</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>c</mi>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mi>z</mi>
<br/>        <mo stretchy="false">)</mo>
<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>u</mi>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>z</mi>
<br/>              <mo stretchy="false">)</mo>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi mathvariant="normal">d</mi>
<br/>              </mrow>
<br/>              <mi>z</mi>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mi>a</mi>
<br/>        <mi>u</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>z</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mn>0</mn>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle z{\\rac {\\\\mathrm {d} ^{2}u(z)}{\\\\mathrm {d} z^{2}}}+(c-z){\\rac {\\\\mathrm {d} u(z)}{\\\\mathrm {d} z}}-au(z)=0}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae48a66c195a6d8b539c2d83962d4a1e3588964a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:36.997ex; height:6.343ex;" alt="{\\\\displaystyle z{\\rac {\\\\mathrm {d} ^{2}u(z)}{\\\\mathrm {d} z^{2}}}+(c-z){\\rac {\\\\mathrm {d} u(z)}{\\\\mathrm {d} z}}-au(z)=0}"></span></dd></dl>
<br/>Elle est aussi définie par&nbsp;:
<br/><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 _{1}F_{1}(a;c;z)=M(a;c;z)=}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi></mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <msub>
<br/>          <mi>F</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>a</mi>
<br/>        <mo>;</mo>
<br/>        <mi>c</mi>
<br/>        <mo>;</mo>
<br/>        <mi>z</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mi>M</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>a</mi>
<br/>        <mo>;</mo>
<br/>        <mi>c</mi>
<br/>        <mo>;</mo>
<br/>        <mi>z</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle _{1}F_{1}(a;c;z)=M(a;c;z)=}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c537d808270f5fe529fee9bee863da1986f260e8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.001ex; height:2.843ex;" alt="{\\\\displaystyle _{1}F_{1}(a;c;z)=M(a;c;z)=}"></span>
<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 {1}{\\\\mathrm {B} (a,c-a)}}{\\\\int _{0}^{1}u^{a-1}(1-u)^{c-a-1}\\\\mathrm {e} ^{zu}\\\\,du}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mn>1</mn>
<br/>            <mrow>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi mathvariant="normal">B</mi>
<br/>              </mrow>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>a</mi>
<br/>              <mo>,</mo>
<br/>              <mi>c</mi>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mi>a</mi>
<br/>              <mo stretchy="false">)</mo>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <msubsup>
<br/>            <mo>∫<!-- ∫ --></mo>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mn>0</mn>
<br/>            </mrow>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mn>1</mn>
<br/>            </mrow>
<br/>          </msubsup>
<br/>          <msup>
<br/>            <mi>u</mi>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi>a</mi>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mn>1</mn>
<br/>            </mrow>
<br/>          </msup>
<br/>          <mo stretchy="false">(</mo>
<br/>          <mn>1</mn>
<br/>          <mo>−<!-- − --></mo>
<br/>          <mi>u</mi>
<br/>          <msup>
<br/>            <mo stretchy="false">)</mo>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi>c</mi>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mi>a</mi>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mn>1</mn>
<br/>            </mrow>
<br/>          </msup>
<br/>          <msup>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi mathvariant="normal">e</mi>
<br/>            </mrow>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi>z</mi>
<br/>              <mi>u</mi>
<br/>            </mrow>
<br/>          </msup>
<br/>          <mspace width="thinmathspace"></mspace>
<br/>          <mi>d</mi>
<br/>          <mi>u</mi>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\rac {1}{\\\\mathrm {B} (a,c-a)}}{\\\\int _{0}^{1}u^{a-1}(1-u)^{c-a-1}\\\\mathrm {e} ^{zu}\\\\,du}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48d2a4e1582c7b701bef01c0982b3be0cda5ddf5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:38.311ex; height:6.509ex;" alt="{\\\\displaystyle {\\rac {1}{\\\\mathrm {B} (a,c-a)}}{\\\\int _{0}^{1}u^{a-1}(1-u)^{c-a-1}\\\\mathrm {e} ^{zu}\\\\,du}}"></span></dd></dl>
<br/><br>
<br/>Les fonctions de Bessel, la fonction gamma incomplète, les fonctions génératrices des moments des distributions bêta et bêta prime, les fonctions cylindre parabolique ou encore les polynômes d'Hermite et les polynômes de Laguerre peuvent être représentés à l'aide de fonctions hypergéométriques confluentes (cf. Slater). Whittaker a introduit des fonctions <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_{\\\\mu ,\\
u }(z)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>M</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>μ<!-- μ --></mi>
<br/>            <mo>,</mo>
<br/>            <mi>ν<!-- ν --></mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>z</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle M_{\\\\mu ,\\
u }(z)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c1799d6cd19baca8aa8bf0c964bc2070dfbae1c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.704ex; height:3.009ex;" alt="{\\\\displaystyle M_{\\\\mu ,\\
u }(z)}"></span> et <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 W_{\\\\mu ,\\
u }(z)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>W</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>μ<!-- μ --></mi>
<br/>            <mo>,</mo>
<br/>            <mi>ν<!-- ν --></mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>z</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle W_{\\\\mu ,\\
u }(z)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/667883f673492d04aa2e21db272ab1dc8e92df1b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.643ex; height:3.009ex;" alt="{\\\\displaystyle W_{\\\\mu ,\\
u }(z)}"></span> qui sont également liées aux fonctions hypergéométriques confluentes. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Fonction_hyperg%C3%A9om%C3%A9trique_confluente">https://fr.wikipedia.org/wiki/Fonction_hyperg%C3%A9om%C3%A9trique_confluente</a>)"""@fr ;
  dc:created "2023-07-27"^^xsd:date ;
  skos:broader psr:-VZ83B143-L .

psr:-VZ83B143-L
  skos:prefLabel "fonction hypergéométrique"@fr, "hypergeometric function"@en ;
  a skos:Concept ;
  skos:narrower psr:-RN7T2RV9-J .