@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:-VTR5XXB2-M
  skos:prefLabel "identité"@fr, "identity"@en ;
  a skos:Concept ;
  skos:narrower psr:-LW5DJF41-Z .

psr:-LW5DJF41-Z
  skos:exactMatch <https://en.wikipedia.org/wiki/Heine%27s_identity> ;
  skos:inScheme psr: ;
  dc:created "2023-07-13"^^xsd:date ;
  skos:prefLabel "Heine's identity"@en, "identité de Heine"@fr ;
  dc:modified "2023-07-13"^^xsd:date ;
  skos:definition """In mathematical analysis, <b>Heine's identity</b>, named after Heinrich Eduard Heine is a Fourier expansion of a reciprocal square root which Heine presented as
<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 {1}{\\\\sqrt {z-\\\\cos \\\\psi }}}={\\rac {\\\\sqrt {2}}{\\\\pi }}\\\\sum _{m=-\\\\infty }^{\\\\infty }Q_{m-{\\rac {1}{2}}}(z)e^{im\\\\psi }}">
<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/>            <msqrt>
<br/>              <mi>z</mi>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mi>cos</mi>
<br/>              <mo>⁡<!-- ⁡ --></mo>
<br/>              <mi>ψ<!-- ψ --></mi>
<br/>            </msqrt>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <msqrt>
<br/>              <mn>2</mn>
<br/>            </msqrt>
<br/>            <mi>π<!-- π --></mi>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>m</mi>
<br/>            <mo>=</mo>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </munderover>
<br/>        <msub>
<br/>          <mi>Q</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>m</mi>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mfrac>
<br/>                <mn>1</mn>
<br/>                <mn>2</mn>
<br/>              </mfrac>
<br/>            </mrow>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>z</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <msup>
<br/>          <mi>e</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>            <mi>m</mi>
<br/>            <mi>ψ<!-- ψ --></mi>
<br/>          </mrow>
<br/>        </msup>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\rac {1}{\\\\sqrt {z-\\\\cos \\\\psi }}}={\\rac {\\\\sqrt {2}}{\\\\pi }}\\\\sum _{m=-\\\\infty }^{\\\\infty }Q_{m-{\\rac {1}{2}}}(z)e^{im\\\\psi }}</annotation>
<br/>  </semantics>
<br/></math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c104654c883fab9174c6992e2f13eb80950421a" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -3.171ex; width:39.139ex; height:7.176ex;" alt="{\\\\displaystyle {\\rac {1}{\\\\sqrt {z-\\\\cos \\\\psi }}}={\\rac {\\\\sqrt {2}}{\\\\pi }}\\\\sum _{m=-\\\\infty }^{\\\\infty }Q_{m-{\\rac {1}{2}}}(z)e^{im\\\\psi }}"></div>
<br/>where <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 Q_{m-{\\rac {1}{2}}}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>Q</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>m</mi>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mfrac>
<br/>                <mn>1</mn>
<br/>                <mn>2</mn>
<br/>              </mfrac>
<br/>            </mrow>
<br/>          </mrow>
<br/>        </msub>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle Q_{m-{\\rac {1}{2}}}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7aad128ea5db4f1df33e45f0877a46cf85bfaa03" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.171ex; width:6.295ex; height:4.009ex;" alt=" Q_{m-\\rac12}"></span> is a Legendre function of the second kind, which has degree, <i>m</i>&nbsp;−&nbsp;<style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="frac" role="math"><span class="num">1</span>⁄<span class="den">2</span></span>, a half-integer, and argument, <i>z</i>,  real and greater than one.  This expression can be generalized for arbitrary half-integer powers as follows
<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 (z-\\\\cos \\\\psi )^{n-{\\rac {1}{2}}}={\\\\sqrt {\\rac {2}{\\\\pi }}}{\\rac {(z^{2}-1)^{\\rac {n}{2}}}{\\\\Gamma ({\\rac {1}{2}}-n)}}\\\\sum _{m=-\\\\infty }^{\\\\infty }{\\rac {\\\\Gamma (m-n+{\\rac {1}{2}})}{\\\\Gamma (m+n+{\\rac {1}{2}})}}Q_{m-{\\rac {1}{2}}}^{n}(z)e^{im\\\\psi },}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>z</mi>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mi>cos</mi>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mi>ψ<!-- ψ --></mi>
<br/>        <msup>
<br/>          <mo stretchy="false">)</mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mfrac>
<br/>                <mn>1</mn>
<br/>                <mn>2</mn>
<br/>              </mfrac>
<br/>            </mrow>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <msqrt>
<br/>            <mfrac>
<br/>              <mn>2</mn>
<br/>              <mi>π<!-- π --></mi>
<br/>            </mfrac>
<br/>          </msqrt>
<br/>        </mrow>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mo stretchy="false">(</mo>
<br/>              <msup>
<br/>                <mi>z</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>2</mn>
<br/>                </mrow>
<br/>              </msup>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mn>1</mn>
<br/>              <msup>
<br/>                <mo stretchy="false">)</mo>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mfrac>
<br/>                    <mi>n</mi>
<br/>                    <mn>2</mn>
<br/>                  </mfrac>
<br/>                </mrow>
<br/>              </msup>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mi mathvariant="normal">Γ<!-- Γ --></mi>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mfrac>
<br/>                  <mn>1</mn>
<br/>                  <mn>2</mn>
<br/>                </mfrac>
<br/>              </mrow>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mi>n</mi>
<br/>              <mo stretchy="false">)</mo>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>m</mi>
<br/>            <mo>=</mo>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<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/>              <mi mathvariant="normal">Γ<!-- Γ --></mi>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>m</mi>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mi>n</mi>
<br/>              <mo>+</mo>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mfrac>
<br/>                  <mn>1</mn>
<br/>                  <mn>2</mn>
<br/>                </mfrac>
<br/>              </mrow>
<br/>              <mo stretchy="false">)</mo>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mi mathvariant="normal">Γ<!-- Γ --></mi>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>m</mi>
<br/>              <mo>+</mo>
<br/>              <mi>n</mi>
<br/>              <mo>+</mo>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mfrac>
<br/>                  <mn>1</mn>
<br/>                  <mn>2</mn>
<br/>                </mfrac>
<br/>              </mrow>
<br/>              <mo stretchy="false">)</mo>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <msubsup>
<br/>          <mi>Q</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>m</mi>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mfrac>
<br/>                <mn>1</mn>
<br/>                <mn>2</mn>
<br/>              </mfrac>
<br/>            </mrow>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>z</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <msup>
<br/>          <mi>e</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>            <mi>m</mi>
<br/>            <mi>ψ<!-- ψ --></mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle (z-\\\\cos \\\\psi )^{n-{\\rac {1}{2}}}={\\\\sqrt {\\rac {2}{\\\\pi }}}{\\rac {(z^{2}-1)^{\\rac {n}{2}}}{\\\\Gamma ({\\rac {1}{2}}-n)}}\\\\sum _{m=-\\\\infty }^{\\\\infty }{\\rac {\\\\Gamma (m-n+{\\rac {1}{2}})}{\\\\Gamma (m+n+{\\rac {1}{2}})}}Q_{m-{\\rac {1}{2}}}^{n}(z)e^{im\\\\psi },}</annotation>
<br/>  </semantics>
<br/></math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7cc1fb994acf465755f68ae6fc4aba60cdfc217" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -3.338ex; width:68.527ex; height:8.009ex;" alt="{\\\\displaystyle (z-\\\\cos \\\\psi )^{n-{\\rac {1}{2}}}={\\\\sqrt {\\rac {2}{\\\\pi }}}{\\rac {(z^{2}-1)^{\\rac {n}{2}}}{\\\\Gamma ({\\rac {1}{2}}-n)}}\\\\sum _{m=-\\\\infty }^{\\\\infty }{\\rac {\\\\Gamma (m-n+{\\rac {1}{2}})}{\\\\Gamma (m+n+{\\rac {1}{2}})}}Q_{m-{\\rac {1}{2}}}^{n}(z)e^{im\\\\psi },}"></div>
<br/>where <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 \\\\scriptstyle \\\\,\\\\Gamma }">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mstyle displaystyle="false" scriptlevel="1">
<br/>          <mspace width="thinmathspace"></mspace>
<br/>          <mi mathvariant="normal">Γ<!-- Γ --></mi>
<br/>        </mstyle>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\scriptstyle \\\\,\\\\Gamma }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/379030399b3f6f4134bf28e6d4ef12c0d21ffca2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.414ex; height:1.676ex;" alt="\\\\scriptstyle\\\\,\\\\Gamma"></span> is the Gamma function. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Heine%27s_identity">https://en.wikipedia.org/wiki/Heine%27s_identity</a>)"""@en ;
  a skos:Concept ;
  skos:broader psr:-FH1H1FB9-1, psr:-VTR5XXB2-M .

psr:-FH1H1FB9-1
  skos:prefLabel "special function"@en, "fonction spéciale"@fr ;
  a skos:Concept ;
  skos:narrower psr:-LW5DJF41-Z .