@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:-SDBHRQ52-J
  skos:definition """En mathématiques, la  <b>fonction d'erreur</b>  (aussi appelée <b>fonction d'erreur de Gauss</b>) est une fonction entière utilisée en analyse. Cette fonction se note <b>erf</b> et fait partie des fonctions spéciales. Elle est définie par&nbsp;:
<br/><br>
<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 \\\\operatorname {erf} (x)={\\rac {2}{\\\\sqrt {\\\\pi }}}\\\\int _{0}^{x}\\\\mathrm {e} ^{-t^{2}}\\\\,\\\\mathrm {d} t.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>erf</mi>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mn>2</mn>
<br/>            <msqrt>
<br/>              <mi>π<!-- π --></mi>
<br/>            </msqrt>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <msubsup>
<br/>          <mo>∫<!-- ∫ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>x</mi>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <msup>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">e</mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>−<!-- − --></mo>
<br/>            <msup>
<br/>              <mi>t</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mn>2</mn>
<br/>              </mrow>
<br/>            </msup>
<br/>          </mrow>
<br/>        </msup>
<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 \\\\operatorname {erf} (x)={\\rac {2}{\\\\sqrt {\\\\pi }}}\\\\int _{0}^{x}\\\\mathrm {e} ^{-t^{2}}\\\\,\\\\mathrm {d} t.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/774b4326ed046c630a1b4eb7e84d25747901cb45" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.838ex; width:24.543ex; height:6.343ex;" alt="{\\\\displaystyle \\\\operatorname {erf} (x)={\\rac {2}{\\\\sqrt {\\\\pi }}}\\\\int _{0}^{x}\\\\mathrm {e} ^{-t^{2}}\\\\,\\\\mathrm {d} t.}"></span></center>
<br/><br>
<br/>La fonction erf intervient régulièrement dans le domaine des probabilités et statistiques, ainsi que dans les problèmes de diffusion (de la chaleur ou de la matière). 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Fonction_d%27erreur">https://fr.wikipedia.org/wiki/Fonction_d%27erreur</a>)"""@fr, """In mathematics, the <b>error function</b> (also called the <b>Gauss error function</b>), often denoted by <b><span class="texhtml">erf</span></b>, is a complex function of a complex variable defined as:
<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 \\\\operatorname {erf} z={\\rac {2}{\\\\sqrt {\\\\pi }}}\\\\int _{0}^{z}e^{-t^{2}}\\\\,\\\\mathrm {d} t.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>erf</mi>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mi>z</mi>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mn>2</mn>
<br/>            <msqrt>
<br/>              <mi>π<!-- π --></mi>
<br/>            </msqrt>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <msubsup>
<br/>          <mo>∫<!-- ∫ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>z</mi>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <msup>
<br/>          <mi>e</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>−<!-- − --></mo>
<br/>            <msup>
<br/>              <mi>t</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mn>2</mn>
<br/>              </mrow>
<br/>            </msup>
<br/>          </mrow>
<br/>        </msup>
<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 \\\\operatorname {erf} z={\\rac {2}{\\\\sqrt {\\\\pi }}}\\\\int _{0}^{z}e^{-t^{2}}\\\\,\\\\mathrm {d} t.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4750918c7292089fdeabff08c35aabfa349a2fc5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:22.759ex; height:6.343ex;" alt="{\\\\displaystyle \\\\operatorname {erf} z={\\rac {2}{\\\\sqrt {\\\\pi }}}\\\\int _{0}^{z}e^{-t^{2}}\\\\,\\\\mathrm {d} t.}"></span></dd></dl>
<br/>Some authors define <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 \\\\operatorname {erf} }">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>erf</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\operatorname {erf} }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a8ee8591a5cc9057c9c646d1d07b848c1d68e81" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.153ex; width:2.809ex; height:2.176ex;" alt="\\\\operatorname {erf} "></span> without the factor of <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 2/{\\\\sqrt {\\\\pi }}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mn>2</mn>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mo>/</mo>
<br/>        </mrow>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <msqrt>
<br/>            <mi>π<!-- π --></mi>
<br/>          </msqrt>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle 2/{\\\\sqrt {\\\\pi }}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba9f93f46aee9590ff4d81c71a537c20eb238a06" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.593ex; height:3.009ex;" alt="{\\\\displaystyle 2/{\\\\sqrt {\\\\pi }}}"></span>.
<br/>This nonelementary integral is a sigmoid function that occurs often in probability, statistics, and partial differential equations. In many of these applications, the function argument is a real number. If the function argument is real, then the function value is also real.
<br/>In statistics, for non-negative values of <span class="texhtml mvar" style="font-style:italic;">x</span>, the error function has the following interpretation: for a random variable <span class="texhtml mvar" style="font-style:italic;">Y</span> that is normally distributed with mean 0 and standard deviation <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1050945101">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span role="math" class="sfrac tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span></span></span></span>, <span class="texhtml">erf <i>x</i></span> is the probability that <span class="texhtml mvar" style="font-style:italic;">Y</span> falls in the range <span class="texhtml">[−<i>x</i>, <i>x</i>]</span>.
<br/>Two closely related functions are the <b>complementary error function</b> (<b><span class="texhtml">erfc</span></b>) defined as
<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 \\\\operatorname {erfc} z=1-\\\\operatorname {erf} z,}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>erfc</mi>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mi>z</mi>
<br/>        <mo>=</mo>
<br/>        <mn>1</mn>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mi>erf</mi>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mi>z</mi>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\operatorname {erfc} z=1-\\\\operatorname {erf} z,}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ff4545622108c21fb7cedc0fe91c39903f2101" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.043ex; height:2.509ex;" alt="{\\\\displaystyle \\\\operatorname {erfc} z=1-\\\\operatorname {erf} z,}"></span></dd></dl>
<br/>and the <b>imaginary error function</b> (<b><span class="texhtml">erfi</span></b>) defined as
<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 \\\\operatorname {erfi} z=-i\\\\operatorname {erf} iz,}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>erfi</mi>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mi>z</mi>
<br/>        <mo>=</mo>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mi>i</mi>
<br/>        <mi>erf</mi>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mi>i</mi>
<br/>        <mi>z</mi>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\operatorname {erfi} z=-i\\\\operatorname {erf} iz,}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bc6c7a6c3d5727f1859f80db7a219bfa743b11a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.455ex; height:2.509ex;" alt="{\\\\displaystyle \\\\operatorname {erfi} z=-i\\\\operatorname {erf} iz,}"></span></dd></dl>
<br/>where <span class="texhtml mvar" style="font-style:italic;">i</span> is the imaginary unit. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Error_function">https://en.wikipedia.org/wiki/Error_function</a>)"""@en ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Fonction_d%27erreur>, <https://en.wikipedia.org/wiki/Error_function> ;
  skos:altLabel "fonction d'erreur de Gauss"@fr, "Gauss error function"@en ;
  skos:prefLabel "error function"@en, "fonction d'erreur"@fr ;
  skos:broader psr:-GLLWFCMV-S, psr:-VZ83B143-L ;
  dc:modified "2023-08-17"^^xsd:date ;
  skos:inScheme psr: ;
  a skos:Concept .

psr:-VZ83B143-L
  skos:prefLabel "fonction hypergéométrique"@fr, "hypergeometric function"@en ;
  a skos:Concept ;
  skos:narrower psr:-SDBHRQ52-J .

psr:-GLLWFCMV-S
  skos:prefLabel "entire function"@en, "fonction entière"@fr ;
  a skos:Concept ;
  skos:narrower psr:-SDBHRQ52-J .