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

psr: a skos:ConceptScheme .
psr:-CL3SVR3H-P
  skos:altLabel "transformation de Stieltjes"@fr, "Stieltjes transformation"@en ;
  a skos:Concept ;
  skos:inScheme psr: ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Transform%C3%A9e_de_Stieltjes>, <https://en.wikipedia.org/wiki/Stieltjes_transformation> ;
  skos:broader psr:-ZSN127JX-M ;
  skos:prefLabel "transformée de Stieltjes"@fr, "Stieltjes transform"@en ;
  skos:definition """In mathematics, the <b>Stieltjes transformation</b> <span class="texhtml"><i>S</i><sub><i>ρ</i></sub>(<i>z</i>)</span> of a measure of density <span class="texhtml"><i>ρ</i></span> on a real interval <span class="texhtml mvar" style="font-style:italic;">I</span> is the function of the complex variable <span class="texhtml mvar" style="font-style:italic;">z</span> defined outside <span class="texhtml mvar" style="font-style:italic;">I</span> by the formula
<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 S_{\\
ho }(z)=\\\\int _{I}{\\rac {\\
ho (t)\\\\,dt}{z-t}},\\\\qquad z\\\\in \\\\mathbb {C} \\\\setminus I.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>S</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>ρ<!-- ρ --></mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>z</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <msub>
<br/>          <mo>∫<!-- ∫ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>I</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mi>ρ<!-- ρ --></mi>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>t</mi>
<br/>              <mo stretchy="false">)</mo>
<br/>              <mspace width="thinmathspace"></mspace>
<br/>              <mi>d</mi>
<br/>              <mi>t</mi>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mi>z</mi>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mi>t</mi>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>,</mo>
<br/>        <mspace width="2em"></mspace>
<br/>        <mi>z</mi>
<br/>        <mo>∈<!-- ∈ --></mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="double-struck">C</mi>
<br/>        </mrow>
<br/>        <mo class="MJX-variant">∖<!-- ∖ --></mo>
<br/>        <mi>I</mi>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle S_{\\
ho }(z)=\\\\int _{I}{\\rac {\\
ho (t)\\\\,dt}{z-t}},\\\\qquad z\\\\in \\\\mathbb {C} \\\\setminus I.}</annotation>
<br/>  </semantics>
<br/></math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/009d656128fd181fff566bc403273f69d3ce3261" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -2.338ex; width:33.672ex; height:6.176ex;" alt="{\\\\displaystyle S_{\\
ho }(z)=\\\\int _{I}{\\rac {\\
ho (t)\\\\,dt}{z-t}},\\\\qquad z\\\\in \\\\mathbb {C} \\\\setminus I.}"></div>
<br/>Under certain conditions we can reconstitute the density function <span class="texhtml"><i>ρ</i></span> starting from its Stieltjes transformation thanks to the inverse formula of Stieltjes-Perron. For example, if the density <span class="texhtml"><i>ρ</i></span> is continuous throughout <span class="texhtml mvar" style="font-style:italic;">I</span>, one will have inside this interval
<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 \\
ho (x)=\\\\lim _{\\\\varepsilon \\	o 0^{+}}{\\rac {S_{\\
ho }(x-i\\\\varepsilon )-S_{\\
ho }(x+i\\\\varepsilon )}{2i\\\\pi }}.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>ρ<!-- ρ --></mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <munder>
<br/>          <mo movablelimits="true" form="prefix">lim</mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>ε<!-- ε --></mi>
<br/>            <mo stretchy="false">→<!-- → --></mo>
<br/>            <msup>
<br/>              <mn>0</mn>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mo>+</mo>
<br/>              </mrow>
<br/>            </msup>
<br/>          </mrow>
<br/>        </munder>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <msub>
<br/>                <mi>S</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi>ρ<!-- ρ --></mi>
<br/>                </mrow>
<br/>              </msub>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>x</mi>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mi>i</mi>
<br/>              <mi>ε<!-- ε --></mi>
<br/>              <mo stretchy="false">)</mo>
<br/>              <mo>−<!-- − --></mo>
<br/>              <msub>
<br/>                <mi>S</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi>ρ<!-- ρ --></mi>
<br/>                </mrow>
<br/>              </msub>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>x</mi>
<br/>              <mo>+</mo>
<br/>              <mi>i</mi>
<br/>              <mi>ε<!-- ε --></mi>
<br/>              <mo stretchy="false">)</mo>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mn>2</mn>
<br/>              <mi>i</mi>
<br/>              <mi>π<!-- π --></mi>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\
ho (x)=\\\\lim _{\\\\varepsilon \\	o 0^{+}}{\\rac {S_{\\
ho }(x-i\\\\varepsilon )-S_{\\
ho }(x+i\\\\varepsilon )}{2i\\\\pi }}.}</annotation>
<br/>  </semantics>
<br/></math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daf9f840f537ae695ba01b1940a250835794656" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -2.338ex; width:37.328ex; height:6.176ex;" alt="{\\\\displaystyle \\
ho (x)=\\\\lim _{\\\\varepsilon \\	o 0^{+}}{\\rac {S_{\\
ho }(x-i\\\\varepsilon )-S_{\\
ho }(x+i\\\\varepsilon )}{2i\\\\pi }}.}"></div>
<br/>
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Stieltjes_transformation">https://en.wikipedia.org/wiki/Stieltjes_transformation</a>)"""@en, """En mathématiques, la <b>transformée de Stieltjes</b> d'une mesure à densité <span class="texhtml">ρ</span> sur un intervalle <span class="texhtml"><i>I</i></span> est une fonction de la variable complexe <i>z</i>, définie à l'extérieur de cet intervalle par la formule&nbsp;: 
<br/>
<br/><center><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 S_{\\
ho }\\\\left(z\\
ight)=\\\\int _{I}{\\rac {\\
ho \\\\left(t\\
ight)}{z-t}}\\\\,\\\\mathrm {d} t.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>S</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>ρ<!-- ρ --></mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mrow>
<br/>          <mo>(</mo>
<br/>          <mi>z</mi>
<br/>          <mo>)</mo>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <msub>
<br/>          <mo>∫<!-- ∫ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>I</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mi>ρ<!-- ρ --></mi>
<br/>              <mrow>
<br/>                <mo>(</mo>
<br/>                <mi>t</mi>
<br/>                <mo>)</mo>
<br/>              </mrow>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mi>z</mi>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mi>t</mi>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="normal">d</mi>
<br/>        </mrow>
<br/>        <mi>t</mi>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle S_{\\
ho }\\\\left(z\\
ight)=\\\\int _{I}{\\rac {\\
ho \\\\left(t\\
ight)}{z-t}}\\\\,\\\\mathrm {d} t.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f471eeab6ae1b6b2d3e69be58fcb71733b96e2c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.401ex; height:6.176ex;" alt="{\\\\displaystyle S_{\\
ho }\\\\left(z\\
ight)=\\\\int _{I}{\\rac {\\
ho \\\\left(t\\
ight)}{z-t}}\\\\,\\\\mathrm {d} t.}"></span></center>
<br/>Sous certaines conditions on peut reconstituer la densité d'origine à partir de sa transformée grâce à la formule d'inversion de Stieltjes-Perron. Par exemple, si la densité <span class="texhtml">ρ</span> est continue sur <span class="texhtml"><i>I</i></span>, on aura à l'intérieur de cet intervalle&nbsp;: 
<br/><span style="display: block; margin-left:1.6em;"><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 \\
ho \\\\left(x\\
ight)=\\\\lim _{\\\\varepsilon \\	o 0^{+}}\\\\,{\\rac {S_{\\
ho }\\\\left(x-{\\
m {i}}\\\\varepsilon \\
ight)-S_{\\
ho }\\\\left(x+{\\
m {i}}\\\\varepsilon \\
ight)}{2{\\
m {i}}\\\\pi }}.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>ρ<!-- ρ --></mi>
<br/>        <mrow>
<br/>          <mo>(</mo>
<br/>          <mi>x</mi>
<br/>          <mo>)</mo>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <munder>
<br/>          <mo movablelimits="true" form="prefix">lim</mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>ε<!-- ε --></mi>
<br/>            <mo stretchy="false">→<!-- → --></mo>
<br/>            <msup>
<br/>              <mn>0</mn>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mo>+</mo>
<br/>              </mrow>
<br/>            </msup>
<br/>          </mrow>
<br/>        </munder>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <msub>
<br/>                <mi>S</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi>ρ<!-- ρ --></mi>
<br/>                </mrow>
<br/>              </msub>
<br/>              <mrow>
<br/>                <mo>(</mo>
<br/>                <mrow>
<br/>                  <mi>x</mi>
<br/>                  <mo>−<!-- − --></mo>
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mrow class="MJX-TeXAtom-ORD">
<br/>                      <mi mathvariant="normal">i</mi>
<br/>                    </mrow>
<br/>                  </mrow>
<br/>                  <mi>ε<!-- ε --></mi>
<br/>                </mrow>
<br/>                <mo>)</mo>
<br/>              </mrow>
<br/>              <mo>−<!-- − --></mo>
<br/>              <msub>
<br/>                <mi>S</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi>ρ<!-- ρ --></mi>
<br/>                </mrow>
<br/>              </msub>
<br/>              <mrow>
<br/>                <mo>(</mo>
<br/>                <mrow>
<br/>                  <mi>x</mi>
<br/>                  <mo>+</mo>
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mrow class="MJX-TeXAtom-ORD">
<br/>                      <mi mathvariant="normal">i</mi>
<br/>                    </mrow>
<br/>                  </mrow>
<br/>                  <mi>ε<!-- ε --></mi>
<br/>                </mrow>
<br/>                <mo>)</mo>
<br/>              </mrow>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mn>2</mn>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi mathvariant="normal">i</mi>
<br/>                </mrow>
<br/>              </mrow>
<br/>              <mi>π<!-- π --></mi>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\
ho \\\\left(x\\
ight)=\\\\lim _{\\\\varepsilon \\	o 0^{+}}\\\\,{\\rac {S_{\\
ho }\\\\left(x-{\\
m {i}}\\\\varepsilon \\
ight)-S_{\\
ho }\\\\left(x+{\\
m {i}}\\\\varepsilon \\
ight)}{2{\\
m {i}}\\\\pi }}.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cb6cdf2843c099ac9ef6fc4db05496855904f40" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.565ex; height:6.176ex;" alt="{\\\\displaystyle \\
ho \\\\left(x\\
ight)=\\\\lim _{\\\\varepsilon \\	o 0^{+}}\\\\,{\\rac {S_{\\
ho }\\\\left(x-{\\
m {i}}\\\\varepsilon \\
ight)-S_{\\
ho }\\\\left(x+{\\
m {i}}\\\\varepsilon \\
ight)}{2{\\
m {i}}\\\\pi }}.}"> </center>
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Transform%C3%A9e_de_Stieltjes">https://fr.wikipedia.org/wiki/Transform%C3%A9e_de_Stieltjes</a>)"""@fr .

psr:-ZSN127JX-M
  skos:prefLabel "opérateur intégral"@fr, "integral transform"@en ;
  a skos:Concept ;
  skos:narrower psr:-CL3SVR3H-P .