@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:-MPTBN71B-H
  dc:modified "2023-08-16"^^xsd:date ;
  skos:definition """In mathematics, the <b>associated Legendre polynomials</b> are the canonical solutions of the <b>general Legendre equation</b>
<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 \\\\left(1-x^{2}\\
ight){\\rac {d^{2}}{dx^{2}}}P_{\\\\ell }^{m}(x)-2x{\\rac {d}{dx}}P_{\\\\ell }^{m}(x)+\\\\left[\\\\ell (\\\\ell +1)-{\\rac {m^{2}}{1-x^{2}}}\\
ight]P_{\\\\ell }^{m}(x)=0,}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow>
<br/>          <mo>(</mo>
<br/>          <mrow>
<br/>            <mn>1</mn>
<br/>            <mo>−<!-- − --></mo>
<br/>            <msup>
<br/>              <mi>x</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mn>2</mn>
<br/>              </mrow>
<br/>            </msup>
<br/>          </mrow>
<br/>          <mo>)</mo>
<br/>        </mrow>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <msup>
<br/>              <mi>d</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mn>2</mn>
<br/>              </mrow>
<br/>            </msup>
<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/>        <msubsup>
<br/>          <mi>P</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>ℓ<!-- ℓ --></mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>m</mi>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mn>2</mn>
<br/>        <mi>x</mi>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mi>d</mi>
<br/>            <mrow>
<br/>              <mi>d</mi>
<br/>              <mi>x</mi>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <msubsup>
<br/>          <mi>P</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>ℓ<!-- ℓ --></mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>m</mi>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>+</mo>
<br/>        <mrow>
<br/>          <mo>[</mo>
<br/>          <mrow>
<br/>            <mi>ℓ<!-- ℓ --></mi>
<br/>            <mo stretchy="false">(</mo>
<br/>            <mi>ℓ<!-- ℓ --></mi>
<br/>            <mo>+</mo>
<br/>            <mn>1</mn>
<br/>            <mo stretchy="false">)</mo>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mfrac>
<br/>                <msup>
<br/>                  <mi>m</mi>
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mn>2</mn>
<br/>                  </mrow>
<br/>                </msup>
<br/>                <mrow>
<br/>                  <mn>1</mn>
<br/>                  <mo>−<!-- − --></mo>
<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/>          </mrow>
<br/>          <mo>]</mo>
<br/>        </mrow>
<br/>        <msubsup>
<br/>          <mi>P</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>ℓ<!-- ℓ --></mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>m</mi>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mn>0</mn>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\left(1-x^{2}\\
ight){\\rac {d^{2}}{dx^{2}}}P_{\\\\ell }^{m}(x)-2x{\\rac {d}{dx}}P_{\\\\ell }^{m}(x)+\\\\left[\\\\ell (\\\\ell +1)-{\\rac {m^{2}}{1-x^{2}}}\\
ight]P_{\\\\ell }^{m}(x)=0,}</annotation>
<br/>  </semantics>
<br/></math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/914d8bc4e6fd78db7db32609c345663927ad11ab" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -2.505ex; width:70.365ex; height:6.343ex;" alt="{\\\\displaystyle \\\\left(1-x^{2}\\
ight){\\rac {d^{2}}{dx^{2}}}P_{\\\\ell }^{m}(x)-2x{\\rac {d}{dx}}P_{\\\\ell }^{m}(x)+\\\\left[\\\\ell (\\\\ell +1)-{\\rac {m^{2}}{1-x^{2}}}\\
ight]P_{\\\\ell }^{m}(x)=0,}"></div>
<br/>or equivalently
<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 {\\rac {d}{dx}}\\\\left[\\\\left(1-x^{2}\\
ight){\\rac {d}{dx}}P_{\\\\ell }^{m}(x)\\
ight]+\\\\left[\\\\ell (\\\\ell +1)-{\\rac {m^{2}}{1-x^{2}}}\\
ight]P_{\\\\ell }^{m}(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/>            <mi>d</mi>
<br/>            <mrow>
<br/>              <mi>d</mi>
<br/>              <mi>x</mi>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mrow>
<br/>          <mo>[</mo>
<br/>          <mrow>
<br/>            <mrow>
<br/>              <mo>(</mo>
<br/>              <mrow>
<br/>                <mn>1</mn>
<br/>                <mo>−<!-- − --></mo>
<br/>                <msup>
<br/>                  <mi>x</mi>
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mn>2</mn>
<br/>                  </mrow>
<br/>                </msup>
<br/>              </mrow>
<br/>              <mo>)</mo>
<br/>            </mrow>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mfrac>
<br/>                <mi>d</mi>
<br/>                <mrow>
<br/>                  <mi>d</mi>
<br/>                  <mi>x</mi>
<br/>                </mrow>
<br/>              </mfrac>
<br/>            </mrow>
<br/>            <msubsup>
<br/>              <mi>P</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi>ℓ<!-- ℓ --></mi>
<br/>              </mrow>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi>m</mi>
<br/>              </mrow>
<br/>            </msubsup>
<br/>            <mo stretchy="false">(</mo>
<br/>            <mi>x</mi>
<br/>            <mo stretchy="false">)</mo>
<br/>          </mrow>
<br/>          <mo>]</mo>
<br/>        </mrow>
<br/>        <mo>+</mo>
<br/>        <mrow>
<br/>          <mo>[</mo>
<br/>          <mrow>
<br/>            <mi>ℓ<!-- ℓ --></mi>
<br/>            <mo stretchy="false">(</mo>
<br/>            <mi>ℓ<!-- ℓ --></mi>
<br/>            <mo>+</mo>
<br/>            <mn>1</mn>
<br/>            <mo stretchy="false">)</mo>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mfrac>
<br/>                <msup>
<br/>                  <mi>m</mi>
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mn>2</mn>
<br/>                  </mrow>
<br/>                </msup>
<br/>                <mrow>
<br/>                  <mn>1</mn>
<br/>                  <mo>−<!-- − --></mo>
<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/>          </mrow>
<br/>          <mo>]</mo>
<br/>        </mrow>
<br/>        <msubsup>
<br/>          <mi>P</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>ℓ<!-- ℓ --></mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>m</mi>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mn>0</mn>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\rac {d}{dx}}\\\\left[\\\\left(1-x^{2}\\
ight){\\rac {d}{dx}}P_{\\\\ell }^{m}(x)\\
ight]+\\\\left[\\\\ell (\\\\ell +1)-{\\rac {m^{2}}{1-x^{2}}}\\
ight]P_{\\\\ell }^{m}(x)=0,}</annotation>
<br/>  </semantics>
<br/></math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e8c86f72c4096479b00d25aaa318916c46e36c9" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -2.505ex; width:60.185ex; height:6.343ex;" alt="{\\\\displaystyle {\\rac {d}{dx}}\\\\left[\\\\left(1-x^{2}\\
ight){\\rac {d}{dx}}P_{\\\\ell }^{m}(x)\\
ight]+\\\\left[\\\\ell (\\\\ell +1)-{\\rac {m^{2}}{1-x^{2}}}\\
ight]P_{\\\\ell }^{m}(x)=0,}"></div>
<br/>where the indices <i>ℓ</i> and <i>m</i> (which are integers) are referred to as the degree and order of the associated Legendre polynomial respectively. This equation has nonzero solutions that are nonsingular on <span class="texhtml">[−1, 1]</span> only if <i>ℓ</i> and <i>m</i> are integers with 0 ≤ <i>m</i> ≤ <i>ℓ</i>, or with trivially equivalent negative values. When in addition <i>m</i> is even, the function is a polynomial. When <i>m</i> is zero and <i>ℓ</i> integer, these functions are identical to the Legendre polynomials.  In general, when <i>ℓ</i> and <i>m</i> are integers, the regular solutions are sometimes called "associated Legendre polynomials", even though they are not polynomials when <i>m</i> is odd. The fully general class of functions with arbitrary real or complex values of <i>ℓ</i> and <i>m</i> are Legendre functions.  In that case the parameters are usually labelled with Greek letters. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Associated_Legendre_polynomials">https://en.wikipedia.org/wiki/Associated_Legendre_polynomials</a>)"""@en, """En mathématiques, un <b>polynôme associé de Legendre</b>, noté <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 P_{\\\\ell }^{m}(x)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msubsup>
<br/>          <mi>P</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>ℓ<!-- ℓ --></mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>m</mi>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle P_{\\\\ell }^{m}(x)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a9b97f26a184f3993b6af4811375ec8b60e757b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.636ex; height:3.009ex;" alt="{\\\\displaystyle P_{\\\\ell }^{m}(x)}"></span>, est une solution particulière de l'équation générale de Legendre[</span>ref 1]</span>&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 (1-x^{2})\\\\,y''-2xy'+\\\\left(\\\\ell (\\\\ell +1)-{\\rac {m^{2}}{1-x^{2}}}\\
ight)\\\\,y=0,}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo stretchy="false">(</mo>
<br/>        <mn>1</mn>
<br/>        <mo>−<!-- − --></mo>
<br/>        <msup>
<br/>          <mi>x</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <msup>
<br/>          <mi>y</mi>
<br/>          <mo>″</mo>
<br/>        </msup>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mn>2</mn>
<br/>        <mi>x</mi>
<br/>        <msup>
<br/>          <mi>y</mi>
<br/>          <mo>′</mo>
<br/>        </msup>
<br/>        <mo>+</mo>
<br/>        <mrow>
<br/>          <mo>(</mo>
<br/>          <mrow>
<br/>            <mi>ℓ<!-- ℓ --></mi>
<br/>            <mo stretchy="false">(</mo>
<br/>            <mi>ℓ<!-- ℓ --></mi>
<br/>            <mo>+</mo>
<br/>            <mn>1</mn>
<br/>            <mo stretchy="false">)</mo>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mfrac>
<br/>                <msup>
<br/>                  <mi>m</mi>
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mn>2</mn>
<br/>                  </mrow>
<br/>                </msup>
<br/>                <mrow>
<br/>                  <mn>1</mn>
<br/>                  <mo>−<!-- − --></mo>
<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/>          </mrow>
<br/>          <mo>)</mo>
<br/>        </mrow>
<br/>        <mspace width="thinmathspace"></mspace>
<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 (1-x^{2})\\\\,y''-2xy'+\\\\left(\\\\ell (\\\\ell +1)-{\\rac {m^{2}}{1-x^{2}}}\\
ight)\\\\,y=0,}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74c19c23578b85526956205c66f269b168902769" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:48.972ex; height:6.343ex;" alt="{\\\\displaystyle (1-x^{2})\\\\,y''-2xy'+\\\\left(\\\\ell (\\\\ell +1)-{\\rac {m^{2}}{1-x^{2}}}\\
ight)\\\\,y=0,}"></span></dd></dl>
<br/>laquelle n'a de solution régulière que sur l'intervalle [–1, 1] et si –<i>m</i> ≤ ℓ ≤ <i>m</i> avec ℓ et <i>m</i> entiers. Elle se réduit à l'équation différentielle de Legendre si <i>m</i> = 0. 
<br/>Cette fonction est un polynôme si <i>m</i> est un entier <i>pair</i>. Toutefois, l’appellation de «&nbsp;polynôme&nbsp;», bien qu'incorrecte, est quand même conservée dans le cas où <i>m</i> est un entier <i>impair</i>.  
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Polyn%C3%B4me_associ%C3%A9_de_Legendre">https://fr.wikipedia.org/wiki/Polyn%C3%B4me_associ%C3%A9_de_Legendre</a>)"""@fr ;
  skos:broader psr:-FH1H1FB9-1, psr:-N2QX9K1Z-L ;
  dc:created "2023-08-16"^^xsd:date ;
  a skos:Concept ;
  skos:inScheme psr: ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Polyn%C3%B4me_associ%C3%A9_de_Legendre>, <https://en.wikipedia.org/wiki/Associated_Legendre_polynomials> ;
  skos:prefLabel "associated Legendre polynomial"@en, "polynôme associé de Legendre"@fr .

psr:-N2QX9K1Z-L
  skos:prefLabel "orthogonal polynomials"@en, "polynômes orthogonaux"@fr ;
  a skos:Concept ;
  skos:narrower psr:-MPTBN71B-H .

psr:-FH1H1FB9-1
  skos:prefLabel "special function"@en, "fonction spéciale"@fr ;
  a skos:Concept ;
  skos:narrower psr:-MPTBN71B-H .