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

psr:-K1N5P6P9-9
  skos:definition """Définie par&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 \\\\coth(x)={\\rac {\\\\operatorname {cosh} (x)}{\\\\operatorname {sinh} (x)}}={\\rac {{\\
m {e}}^{x}+{\\
m {e}}^{-x}}{{\\
m {e}}^{x}-{\\
m {e}}^{-x}}}={\\rac {{\\
m {e}}^{2x}+1}{{\\
m {e}}^{2x}-1}}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>coth</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/>            <mrow>
<br/>              <mi>cosh</mi>
<br/>              <mo>⁡<!-- ⁡ --></mo>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>x</mi>
<br/>              <mo stretchy="false">)</mo>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mi>sinh</mi>
<br/>              <mo>⁡<!-- ⁡ --></mo>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>x</mi>
<br/>              <mo stretchy="false">)</mo>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <msup>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mi mathvariant="normal">e</mi>
<br/>                  </mrow>
<br/>                </mrow>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi>x</mi>
<br/>                </mrow>
<br/>              </msup>
<br/>              <mo>+</mo>
<br/>              <msup>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mi mathvariant="normal">e</mi>
<br/>                  </mrow>
<br/>                </mrow>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mo>−<!-- − --></mo>
<br/>                  <mi>x</mi>
<br/>                </mrow>
<br/>              </msup>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <msup>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mi mathvariant="normal">e</mi>
<br/>                  </mrow>
<br/>                </mrow>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi>x</mi>
<br/>                </mrow>
<br/>              </msup>
<br/>              <mo>−<!-- − --></mo>
<br/>              <msup>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mi mathvariant="normal">e</mi>
<br/>                  </mrow>
<br/>                </mrow>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mo>−<!-- − --></mo>
<br/>                  <mi>x</mi>
<br/>                </mrow>
<br/>              </msup>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <msup>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mi mathvariant="normal">e</mi>
<br/>                  </mrow>
<br/>                </mrow>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>2</mn>
<br/>                  <mi>x</mi>
<br/>                </mrow>
<br/>              </msup>
<br/>              <mo>+</mo>
<br/>              <mn>1</mn>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <msup>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mi mathvariant="normal">e</mi>
<br/>                  </mrow>
<br/>                </mrow>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>2</mn>
<br/>                  <mi>x</mi>
<br/>                </mrow>
<br/>              </msup>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mn>1</mn>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\coth(x)={\\rac {\\\\operatorname {cosh} (x)}{\\\\operatorname {sinh} (x)}}={\\rac {{\\
m {e}}^{x}+{\\
m {e}}^{-x}}{{\\
m {e}}^{x}-{\\
m {e}}^{-x}}}={\\rac {{\\
m {e}}^{2x}+1}{{\\
m {e}}^{2x}-1}}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/927a142fa5e0baac33cf2586db8a2f0414c3cc9a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.671ex; width:42.436ex; height:6.509ex;" alt="\\\\coth(x)={\\rac  {\\\\operatorname {cosh}(x)}{\\\\operatorname {sinh}(x)}}={\\rac  {{{\\
m {e}}}^{x}+{{\\
m {e}}}^{{-x}}}{{{\\
m {e}}}^{x}-{{\\
m {e}}}^{{-x}}}}={\\rac  {{{\\
m {e}}}^{{2x}}+1}{{{\\
m {e}}}^{{2x}}-1}}"></span></dd></dl>
<br/><span class="texhtml">coth</span> est une bijection de classe <span class="texhtml"><i>C</i><sup>∞</sup></span> de ℝ* dans <span class="texhtml">]–∞, –1[∪]1, +∞[</span>. Sa dérivée est <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 {\\rac {-1}{\\\\operatorname {sinh} ^{2}}}=1-\\\\coth ^{2}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mn>1</mn>
<br/>            </mrow>
<br/>            <msup>
<br/>              <mi>sinh</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mn>2</mn>
<br/>              </mrow>
<br/>            </msup>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <mn>1</mn>
<br/>        <mo>−<!-- − --></mo>
<br/>        <msup>
<br/>          <mi>coth</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\rac {-1}{\\\\operatorname {sinh} ^{2}}}=1-\\\\coth ^{2}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45465467d23ef7d82501e7fdd47482a88f70474c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:18.586ex; height:5.676ex;" alt="{\\rac  {-1}{\\\\operatorname {sinh}^{2}}}=1-\\\\coth ^{2}"></span>. Son application réciproque est l'argument cotangente hyperbolique. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Fonction_hyperbolique#Cotangente_hyperbolique">https://fr.wikipedia.org/wiki/Fonction_hyperbolique#Cotangente_hyperbolique</a>)"""@fr ;
  a skos:Concept ;
  skos:inScheme psr: ;
  skos:broader psr:-R92BT00M-4 ;
  skos:prefLabel "cotangente hyperbolique"@fr, "hyperbolic cotangent"@en ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Fonction_hyperbolique#Cotangente_hyperbolique> .

psr: a skos:ConceptScheme .
psr:-R92BT00M-4
  skos:prefLabel "hyperbolic function"@en, "fonction hyperbolique"@fr ;
  a skos:Concept ;
  skos:narrower psr:-K1N5P6P9-9 .