@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .

psr: a skos:ConceptScheme .
psr:-TQC1MD4T-P
  skos:definition """In mathematics, the <b>Abel transform</b>, named for Niels Henrik Abel, is an integral transform often used in the analysis of spherically symmetric or axially symmetric functions. The Abel transform of a function <i>f</i>(<i>r</i>) is given by
<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 F(y)=2\\\\int _{y}^{\\\\infty }{\\rac {f(r)r}{\\\\sqrt {r^{2}-y^{2}}}}\\\\,dr.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>F</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>y</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mn>2</mn>
<br/>        <msubsup>
<br/>          <mo>∫<!-- ∫ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>y</mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mi>f</mi>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>r</mi>
<br/>              <mo stretchy="false">)</mo>
<br/>              <mi>r</mi>
<br/>            </mrow>
<br/>            <msqrt>
<br/>              <msup>
<br/>                <mi>r</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>2</mn>
<br/>                </mrow>
<br/>              </msup>
<br/>              <mo>−<!-- − --></mo>
<br/>              <msup>
<br/>                <mi>y</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>2</mn>
<br/>                </mrow>
<br/>              </msup>
<br/>            </msqrt>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mi>d</mi>
<br/>        <mi>r</mi>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle F(y)=2\\\\int _{y}^{\\\\infty }{\\rac {f(r)r}{\\\\sqrt {r^{2}-y^{2}}}}\\\\,dr.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fd9a4facb7296b05265f4f4036c650ac2b662ee" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.697ex; height:7.009ex;" alt="{\\\\displaystyle F(y)=2\\\\int _{y}^{\\\\infty }{\\rac {f(r)r}{\\\\sqrt {r^{2}-y^{2}}}}\\\\,dr.}"></span></dd></dl>
<br/>Assuming that <i>f</i>(<i>r</i>) drops to zero more quickly than 1/<i>r</i>, the inverse Abel transform is given by
<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 f(r)=-{\\rac {1}{\\\\pi }}\\\\int _{r}^{\\\\infty }{\\rac {dF}{dy}}\\\\,{\\rac {dy}{\\\\sqrt {y^{2}-r^{2}}}}.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>f</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>r</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mn>1</mn>
<br/>            <mi>π<!-- π --></mi>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <msubsup>
<br/>          <mo>∫<!-- ∫ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>r</mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mi>d</mi>
<br/>              <mi>F</mi>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mi>d</mi>
<br/>              <mi>y</mi>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mi>d</mi>
<br/>              <mi>y</mi>
<br/>            </mrow>
<br/>            <msqrt>
<br/>              <msup>
<br/>                <mi>y</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>2</mn>
<br/>                </mrow>
<br/>              </msup>
<br/>              <mo>−<!-- − --></mo>
<br/>              <msup>
<br/>                <mi>r</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>2</mn>
<br/>                </mrow>
<br/>              </msup>
<br/>            </msqrt>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f(r)=-{\\rac {1}{\\\\pi }}\\\\int _{r}^{\\\\infty }{\\rac {dF}{dy}}\\\\,{\\rac {dy}{\\\\sqrt {y^{2}-r^{2}}}}.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25e534cb87d2ce0e47650b412e17203a8c9865dd" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.47ex; height:6.676ex;" alt="{\\\\displaystyle f(r)=-{\\rac {1}{\\\\pi }}\\\\int _{r}^{\\\\infty }{\\rac {dF}{dy}}\\\\,{\\rac {dy}{\\\\sqrt {y^{2}-r^{2}}}}.}"> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Abel_transform">https://en.wikipedia.org/wiki/Abel_transform</a>)"""@en ;
  a skos:Concept ;
  skos:inScheme psr: ;
  skos:broader psr:-ZSN127JX-M ;
  skos:prefLabel "Abel transform"@en, "transformée d'Abel"@fr ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Abel_transform> .

psr:-ZSN127JX-M
  skos:prefLabel "opérateur intégral"@fr, "integral transform"@en ;
  a skos:Concept ;
  skos:narrower psr:-TQC1MD4T-P .