@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:-JL9Z4260-Z
  skos:prefLabel "trigonométrie"@fr, "trigonometry"@en ;
  a skos:Concept ;
  skos:narrower psr:-HXW20HTQ-1 .

psr:-HXW20HTQ-1
  skos:broader psr:-RN57KZJ9-9, psr:-JL9Z4260-Z ;
  skos:definition """<b>Euler's formula</b>, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number&nbsp;<span class="texhtml mvar" style="font-style:italic;">x</span>:
<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 e^{ix}=\\\\cos x+i\\\\sin x,}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msup>
<br/>          <mi>e</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>            <mi>x</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>=</mo>
<br/>        <mi>cos</mi>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mi>x</mi>
<br/>        <mo>+</mo>
<br/>        <mi>i</mi>
<br/>        <mi>sin</mi>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mi>x</mi>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle e^{ix}=\\\\cos x+i\\\\sin x,}</annotation>
<br/>  </semantics>
<br/></math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aab1fcd1a6db5cc6678bb9cbd871580eeeb86eda" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -0.671ex; width:19.999ex; height:3.009ex;" alt="{\\\\displaystyle e^{ix}=\\\\cos x+i\\\\sin x,}"></div>
<br/>where <span class="texhtml mvar" style="font-style:italic;">e</span> is the base of the natural logarithm, <span class="texhtml mvar" style="font-style:italic;">i</span> is the imaginary unit, and <span class="texhtml">cos</span> and <span class="texhtml">sin</span> are the trigonometric functions cosine and sine respectively. This complex exponential function is sometimes denoted <span class="texhtml">cis <i>x</i></span> ("cosine plus <i>i</i> sine"). The formula is still valid if <span class="texhtml mvar" style="font-style:italic;">x</span> is a complex number, and so some authors refer to the more general complex version as Euler's formula. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Euler%27s_formula">https://en.wikipedia.org/wiki/Euler%27s_formula</a>)"""@en, """La <b>formule d'Euler</b> est une égalité mathématique, attribuée au mathématicien suisse Leonhard Euler. Elle s'écrit, pour tout nombre réel <span class="texhtml"><i>x</i></span>,
<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 \\\\mathrm {e} ^{\\\\mathrm {i} \\\\,x}=\\\\cos x+\\\\mathrm {i} \\\\,\\\\sin x}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msup>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">e</mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi mathvariant="normal">i</mi>
<br/>            </mrow>
<br/>            <mspace width="thinmathspace"></mspace>
<br/>            <mi>x</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>=</mo>
<br/>        <mi>cos</mi>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mi>x</mi>
<br/>        <mo>+</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="normal">i</mi>
<br/>        </mrow>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mi>sin</mi>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mi>x</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathrm {e} ^{\\\\mathrm {i} \\\\,x}=\\\\cos x+\\\\mathrm {i} \\\\,\\\\sin x}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1329706416e696b8f2438128c5cdf8b74e0f57f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.505ex; width:19.81ex; height:2.843ex;" alt="{\\\\displaystyle \\\\mathrm {e} ^{\\\\mathrm {i} \\\\,x}=\\\\cos x+\\\\mathrm {i} \\\\,\\\\sin x}"></span></center>
<br/>et se généralise aux <span class="texhtml mvar" style="font-style:italic;">x</span> complexes.
<br/>Ici, le nombre <span class="texhtml">e</span> est la base des logarithmes naturels, <span class="texhtml">i</span> est l'unité imaginaire, <span class="texhtml">sin</span> et <span class="texhtml">cos</span> sont des fonctions trigonométriques. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Formule_d%27Euler">https://fr.wikipedia.org/wiki/Formule_d%27Euler</a>)"""@fr ;
  skos:prefLabel "Euler's formula"@en, "formule d'Euler"@fr ;
  skos:inScheme psr: ;
  dc:modified "2023-08-02"^^xsd:date ;
  a skos:Concept ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Formule_d%27Euler>, <https://en.wikipedia.org/wiki/Euler%27s_formula> .

psr:-RN57KZJ9-9
  skos:prefLabel "analyse complexe"@fr, "complex analysis"@en ;
  a skos:Concept ;
  skos:narrower psr:-HXW20HTQ-1 .