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

psr:-S8K0HGZN-F
  skos:prefLabel "fonction transcendante"@fr, "transcendental function"@en ;
  a skos:Concept ;
  skos:broader psr:-WSV4W5WP-1 .

psr:-HD65CQDF-2
  skos:prefLabel "trigonometric function"@en, "fonction trigonométrique"@fr ;
  a skos:Concept ;
  skos:broader psr:-WSV4W5WP-1 .

psr:-NG5DMZ5W-1
  skos:prefLabel "fonction hyperbolique réciproque"@fr, "inverse hyperbolic function"@en ;
  a skos:Concept ;
  skos:broader psr:-WSV4W5WP-1 .

psr:-WX4R0T02-R
  skos:prefLabel "Jensen's formula"@en, "formule de Jensen"@fr ;
  a skos:Concept ;
  skos:related psr:-WSV4W5WP-1 .

psr:-R92BT00M-4
  skos:prefLabel "hyperbolic function"@en, "fonction hyperbolique"@fr ;
  a skos:Concept ;
  skos:broader psr:-WSV4W5WP-1 .

psr:-ST0RJ5D8-4
  skos:prefLabel "fonction holomorphe"@fr, "holomorphic function"@en ;
  a skos:Concept ;
  skos:broader psr:-WSV4W5WP-1 .

psr:-CT7WBSPV-M
  skos:prefLabel "approximant de Padé"@fr, "Padé approximant"@en ;
  a skos:Concept ;
  skos:related psr:-WSV4W5WP-1 .

psr: a skos:ConceptScheme .
psr:-WSV4W5WP-1
  skos:inScheme psr: ;
  skos:related psr:-SL16JB2H-1, psr:-FH1H1FB9-1, psr:-WX4R0T02-R, psr:-CT7WBSPV-M ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Analytic_function>, <https://fr.wikipedia.org/wiki/Fonction_analytique> ;
  skos:broader psr:-MDFZ99KQ-Q, psr:-RN57KZJ9-9 ;
  skos:narrower psr:-ST0RJ5D8-4, psr:-P36V4MHV-V, psr:-HD65CQDF-2, psr:-R92BT00M-4, psr:-DSXFBSBG-2, psr:-NG5DMZ5W-1, psr:-S8K0HGZN-F ;
  skos:definition """En mathématiques, et plus précisément en analyse, une <b>fonction analytique</b> est une fonction d'une variable réelle ou complexe qui est développable en série entière au voisinage de chacun des points de son domaine de définition, c'est-à-dire que pour tout <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 x_{0}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>x</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle x_{0}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="x_{0}"></span> de ce domaine, il existe une suite <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 (a_{n})}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo stretchy="false">(</mo>
<br/>        <msub>
<br/>          <mi>a</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle (a_{n})}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18bc33c7c35d82b00f88d3a9103ed4738cde41f9" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:4.258ex; height:2.843ex;" alt="(a_n)"></span>  donnant une expression de la fonction, valable pour tout <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 x}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>x</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle x}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="x"></span> assez proche de <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 x_{0}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>x</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle x_{0}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="x_{0}"></span>, sous la forme d'une série convergente&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 f(x)=\\\\sum _{n=0}^{+\\\\infty }a_{n}(x-x_{0})^{n}.}">
<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>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>            <mo>=</mo>
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>+</mo>
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </munderover>
<br/>        <msub>
<br/>          <mi>a</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo>−<!-- − --></mo>
<br/>        <msub>
<br/>          <mi>x</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <msup>
<br/>          <mo stretchy="false">)</mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f(x)=\\\\sum _{n=0}^{+\\\\infty }a_{n}(x-x_{0})^{n}.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07041ca9e7515c44e31c77b7cc792d8fd9aabe86" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:23.935ex; height:7.176ex;" alt="f(x)=\\\\sum _{{n=0}}^{{+\\\\infty }}a_{n}(x-x_{0})^{n}."></span></dd></dl>
<br/>Toute fonction analytique est dérivable de dérivée analytique, ce qui implique que toute fonction analytique est indéfiniment dérivable, mais la réciproque est fausse en analyse réelle. En revanche, en analyse complexe, toute fonction simplement dérivable sur un ouvert est analytique et vérifie de nombreuses autres propriétés. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Fonction_analytique">https://fr.wikipedia.org/wiki/Fonction_analytique</a>)"""@fr, """In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if its Taylor series about <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 x_{0}}">
         <br/>  <semantics>
         <br/>    <mrow class="MJX-TeXAtom-ORD">
         <br/>      <mstyle displaystyle="true" scriptlevel="0">
         <br/>        <msub>
         <br/>          <mi>x</mi>
         <br/>          <mrow class="MJX-TeXAtom-ORD">
         <br/>            <mn>0</mn>
         <br/>          </mrow>
         <br/>        </msub>
         <br/>      </mstyle>
         <br/>    </mrow>
         <br/>    <annotation encoding="application/x-tex">{\\\\displaystyle x_{0}}</annotation>
         <br/>  </semantics>
         <br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="x_{0}"></span> converges to the function in some neighborhood for every <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 x_{0}}">
         <br/>  <semantics>
         <br/>    <mrow class="MJX-TeXAtom-ORD">
         <br/>      <mstyle displaystyle="true" scriptlevel="0">
         <br/>        <msub>
         <br/>          <mi>x</mi>
         <br/>          <mrow class="MJX-TeXAtom-ORD">
         <br/>            <mn>0</mn>
         <br/>          </mrow>
         <br/>        </msub>
         <br/>      </mstyle>
         <br/>    </mrow>
         <br/>    <annotation encoding="application/x-tex">{\\\\displaystyle x_{0}}</annotation>
         <br/>  </semantics>
         <br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="x_{0}"></span> in its domain. It is important to note that it's a neighborhood and not just at some point <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 x_{0}}">
         <br/>  <semantics>
         <br/>    <mrow class="MJX-TeXAtom-ORD">
         <br/>      <mstyle displaystyle="true" scriptlevel="0">
         <br/>        <msub>
         <br/>          <mi>x</mi>
         <br/>          <mrow class="MJX-TeXAtom-ORD">
         <br/>            <mn>0</mn>
         <br/>          </mrow>
         <br/>        </msub>
         <br/>      </mstyle>
         <br/>    </mrow>
         <br/>    <annotation encoding="application/x-tex">{\\\\displaystyle x_{0}}</annotation>
         <br/>  </semantics>
         <br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="x_{0}"></span>, since every differentiable function has at least a tangent line at every point, which is its Taylor series of order 1. So just having a polynomial expansion at singular points is not enough, and the Taylor series must also converge to the function on points adjacent to <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 x_{0}}">
         <br/>  <semantics>
         <br/>    <mrow class="MJX-TeXAtom-ORD">
         <br/>      <mstyle displaystyle="true" scriptlevel="0">
         <br/>        <msub>
         <br/>          <mi>x</mi>
         <br/>          <mrow class="MJX-TeXAtom-ORD">
         <br/>            <mn>0</mn>
         <br/>          </mrow>
         <br/>        </msub>
         <br/>      </mstyle>
         <br/>    </mrow>
         <br/>    <annotation encoding="application/x-tex">{\\\\displaystyle x_{0}}</annotation>
         <br/>  </semantics>
         <br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="x_{0}"></span> to be considered an analytic function. As a counterexample see the Fabius function. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Analytic_function">https://en.wikipedia.org/wiki/Analytic_function</a>)"""@en ;
  skos:prefLabel "analytic function"@en, "fonction analytique"@fr ;
  a skos:Concept .

psr:-P36V4MHV-V
  skos:prefLabel "fonction zêta de Riemann"@fr, "Riemann zeta function"@en ;
  a skos:Concept ;
  skos:broader psr:-WSV4W5WP-1 .

psr:-DSXFBSBG-2
  skos:prefLabel "exponential function"@en, "fonction exponentielle"@fr ;
  a skos:Concept ;
  skos:broader psr:-WSV4W5WP-1 .

psr:-SL16JB2H-1
  skos:prefLabel "Carlson's theorem"@en, "théorème de Carlson"@fr ;
  a skos:Concept ;
  skos:related psr:-WSV4W5WP-1 .

psr:-MDFZ99KQ-Q
  skos:prefLabel "fonction numérique"@fr, "real-valued function"@en ;
  a skos:Concept ;
  skos:narrower psr:-WSV4W5WP-1 .

psr:-FH1H1FB9-1
  skos:prefLabel "special function"@en, "fonction spéciale"@fr ;
  a skos:Concept ;
  skos:related psr:-WSV4W5WP-1 .

psr:-RN57KZJ9-9
  skos:prefLabel "analyse complexe"@fr, "complex analysis"@en ;
  a skos:Concept ;
  skos:narrower psr:-WSV4W5WP-1 .