GermanWikipedia - /page/entry.mu

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<title>Hyperbelfunktion</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Hyperbelfunktion</h1>

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Die Hyperbelfunktionen sind die korrespondierenden Funktionen der <a href="Kreisfunktion" class="mw-redirect" title="Kreisfunktion">Kreisfunktionen</a> (die auch als Winkel- oder trigonometrische Funktionen bezeichnet werden),

allerdings nicht am Einheitskreis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}=1}">

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<annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec84b90236512e8d27ff1a8f7707b60b63327de1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.7ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}=1}" loading="lazy">, sondern an der <a href="Hyperbel_(Mathematik)" title="Hyperbel (Mathematik)">Einheitshyperbel</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}-y^{2}=1}">

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<annotation encoding="application/x-tex">{\displaystyle x^{2}-y^{2}=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66b6dc015284f725f6537eb50fe4b20348cf0a5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.7ex; height:3.009ex;" alt="{\displaystyle x^{2}-y^{2}=1}" loading="lazy">.

Wie eng diese Funktionen miteinander verwandt sind, erschließt sich noch deutlicher in der komplexen Zahlenebene. Sie wird durch die Relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathrm {i} y)^{2}=-y^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle (\mathrm {i} y)^{2}=-y^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e91c0a25421b0656b1bb4de90f73df8d2ce83208.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.787ex; height:3.176ex;" alt="{\displaystyle (\mathrm {i} y)^{2}=-y^{2}}" loading="lazy"> vermittelt. So gilt z. B. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\mathrm {i} x)=\cosh x}">

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<annotation encoding="application/x-tex">{\displaystyle \cos(\mathrm {i} x)=\cosh x}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0dc09b497381f9218e032c739b9f41139999446a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.116ex; height:2.843ex;" alt="{\displaystyle \cos(\mathrm {i} x)=\cosh x}" loading="lazy">.

Folgende Funktionen gehören zu den Hyperbelfunktionen:

<ul><li><a href="Sinus_hyperbolicus_und_Kosinus_hyperbolicus" title="Sinus hyperbolicus und Kosinus hyperbolicus">Hyperbelsinus</a> oder lat. Sinus hyperbolicus (Formelzeichen: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sinh }">

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<annotation encoding="application/x-tex">{\displaystyle \sinh }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb6649e190c30e91b280cc02b27acdfe00055e58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.148ex; height:2.176ex;" alt="{\displaystyle \sinh }" loading="lazy">)</li>

<li><a href="Sinus_hyperbolicus_und_Kosinus_hyperbolicus" title="Sinus hyperbolicus und Kosinus hyperbolicus">Hyperbelkosinus</a> oder lat. Cosinus hyperbolicus (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cosh }">

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<annotation encoding="application/x-tex">{\displaystyle \cosh }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/011ccafd311d38f9bbc0fcf3f7b13f7d81469d84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.404ex; height:2.176ex;" alt="{\displaystyle \cosh }" loading="lazy">)</li>

<li><a href="Tangens_hyperbolicus_und_Kotangens_hyperbolicus" title="Tangens hyperbolicus und Kotangens hyperbolicus">Hyperbeltangens</a> oder lat. Tangens hyperbolicus (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tanh }">

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<annotation encoding="application/x-tex">{\displaystyle \tanh }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1494d5d222bc80846ce45e804a38b0ea37e846b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.652ex; height:2.176ex;" alt="{\displaystyle \tanh }" loading="lazy">)</li>

<li><a href="Tangens_hyperbolicus_und_Kotangens_hyperbolicus" title="Tangens hyperbolicus und Kotangens hyperbolicus">Hyperbelkotangens</a> oder lat. Cotangens hyperbolicus (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \coth }">

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<annotation encoding="application/x-tex">{\displaystyle \coth }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b3111993c8de3e6524f16291f1777a34652e15b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.392ex; height:2.176ex;" alt="{\displaystyle \coth }" loading="lazy">)</li>

<li><a href="Sekans_hyperbolicus_und_Kosekans_hyperbolicus" title="Sekans hyperbolicus und Kosekans hyperbolicus">Hyperbelsekans</a> oder lat. Sekans hyperbolicus (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sech} }">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {sech} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb8545d4ea8fb301a8ea17b3a6096d6c0bbc9380.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.274ex; height:2.176ex;" alt="{\displaystyle \operatorname {sech} }" loading="lazy">)</li>

<li><a href="Sekans_hyperbolicus_und_Kosekans_hyperbolicus" title="Sekans hyperbolicus und Kosekans hyperbolicus">Hyperbelkosekans</a> oder lat. Cosekans hyperbolicus (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {csch} }">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {csch} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/819e1186e01d44792c05ad673466f556ffbd97fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.274ex; height:2.176ex;" alt="{\displaystyle \operatorname {csch} }" loading="lazy">).</li></ul>

In der deutschen Sprache werden noch sehr häufig die lateinischen Namen verwendet, mit teils eingedeutschter Schreibweise.

Sinus hyperbolicus und Kosinus hyperbolicus sind für alle <a href="Komplexe_Zahlen" class="mw-redirect" title="Komplexe Zahlen">komplexen Zahlen</a> definiert und auf dem gesamten <a href="Gebiet_(Mathematik)" title="Gebiet (Mathematik)">Gebiet</a> der komplexen Zahlen <a href="Holomorphe_Funktion" title="Holomorphe Funktion">holomorph</a>. Die übrigen Hyperbelfunktionen haben Pole auf der imaginären Achse.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Definition_über_die_Exponentialfunktion">Definition über die Exponentialfunktion</h3></div>

Mittels der <a href="Exponentialfunktion" title="Exponentialfunktion">Exponentialfunktion</a> können <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sinh }">

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<annotation encoding="application/x-tex">{\displaystyle \sinh }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb6649e190c30e91b280cc02b27acdfe00055e58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.148ex; height:2.176ex;" alt="{\displaystyle \sinh }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cosh }">

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<annotation encoding="application/x-tex">{\displaystyle \cosh }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/011ccafd311d38f9bbc0fcf3f7b13f7d81469d84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.404ex; height:2.176ex;" alt="{\displaystyle \cosh }" loading="lazy"> wie folgt definiert werden:

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<annotation encoding="application/x-tex">{\displaystyle \sinh(z):={\frac {\mathrm {e} ^{z}-\mathrm {e} ^{-z}}{2}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7a197966da8a1255750d76a386246fc8250dfe0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.814ex; height:5.509ex;" alt="{\displaystyle \sinh(z):={\frac {\mathrm {e} ^{z}-\mathrm {e} ^{-z}}{2}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \cosh(z):={\frac {\mathrm {e} ^{z}+\mathrm {e} ^{-z}}{2}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e0a2f0087d7c1b0bd76792dd9842fde1c7e6bf2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.07ex; height:5.509ex;" alt="{\displaystyle \cosh(z):={\frac {\mathrm {e} ^{z}+\mathrm {e} ^{-z}}{2}}}" loading="lazy"></dd></dl>

Daher sind die hyperbolischen Funktionen periodisch (mit rein imaginärer Periode).

Die <a href="Potenzreihe" title="Potenzreihe">Potenzreihen</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cosh }">

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<annotation encoding="application/x-tex">{\displaystyle \cosh }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/011ccafd311d38f9bbc0fcf3f7b13f7d81469d84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.404ex; height:2.176ex;" alt="{\displaystyle \cosh }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sinh }">

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<annotation encoding="application/x-tex">{\displaystyle \sinh }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb6649e190c30e91b280cc02b27acdfe00055e58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.148ex; height:2.176ex;" alt="{\displaystyle \sinh }" loading="lazy"> lauten

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sinh(z)&=z+{\frac {z^{3}}{3!}}+{\frac {z^{5}}{5!}}+{\frac {z^{7}}{7!}}+\dots =\sum _{n=0}^{\infty }{\frac {z^{2n+1}}{(2n+1)!}}\\\cosh(z)&=1+{\frac {z^{2}}{2!}}+{\frac {z^{4}}{4!}}+{\frac {z^{6}}{6!}}+\dots =\sum _{n=0}^{\infty }{\frac {z^{2n}}{(2n)!}}\,,\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/074828a70be576876144f22a2ce63d9e7b66c8ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:51.959ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}\sinh(z)&=z+{\frac {z^{3}}{3!}}+{\frac {z^{5}}{5!}}+{\frac {z^{7}}{7!}}+\dots =\sum _{n=0}^{\infty }{\frac {z^{2n+1}}{(2n+1)!}}\\\cosh(z)&=1+{\frac {z^{2}}{2!}}+{\frac {z^{4}}{4!}}+{\frac {z^{6}}{6!}}+\dots =\sum _{n=0}^{\infty }{\frac {z^{2n}}{(2n)!}}\,,\end{aligned}}}" loading="lazy"></dd></dl>

wobei der Ausdruck <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!}">

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<annotation encoding="application/x-tex">{\displaystyle n!}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bae971720be3cc9b8d82f4cdac89cb89877514a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:2.176ex;" alt="{\displaystyle n!}" loading="lazy"> für die <a href="Fakult%C3%A4t_(Mathematik)" title="Fakultät (Mathematik)">Fakultät</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">, das Produkt der ersten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> <a href="Nat%C3%BCrliche_Zahlen" class="mw-redirect" title="Natürliche Zahlen">natürlichen Zahlen</a> steht. Im Gegensatz zu den Potenzreihenentwicklungen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos }">

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<annotation encoding="application/x-tex">{\displaystyle \cos }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e473a3de151d75296f141f9f482fe59d582a7509.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.111ex; height:1.676ex;" alt="{\displaystyle \cos }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin }">

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<annotation encoding="application/x-tex">{\displaystyle \sin }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee55beec18afd710e7ab767964b915b020c65093.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.856ex; height:2.176ex;" alt="{\displaystyle \sin }" loading="lazy"> haben alle Terme ein positives Vorzeichen.

<div class="mw-heading mw-heading3"><h3 id="Geometrische_Definition_mit_Hilfe_der_Hyperbel">Geometrische Definition mit Hilfe der Hyperbel</h3></div>

Wegen ihrer Verwendung zur Parametrisierung der <a href="Hyperbel_(Mathematik)" title="Hyperbel (Mathematik)">Einheitshyperbel</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}-y^{2}=1}">

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<annotation encoding="application/x-tex">{\displaystyle x^{2}-y^{2}=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66b6dc015284f725f6537eb50fe4b20348cf0a5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.7ex; height:3.009ex;" alt="{\displaystyle x^{2}-y^{2}=1}" loading="lazy">:

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<annotation encoding="application/x-tex">{\displaystyle x=\cosh(t),y=\sinh(t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c85154c2bb7148e4d96ca7472b5be9a079821f0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.566ex; height:2.843ex;" alt="{\displaystyle x=\cosh(t),y=\sinh(t)}" loading="lazy"></dd></dl>

werden sie Hyperbelfunktionen genannt, in Analogie zu den Kreisfunktionen <a href="Sinus_und_Kosinus" title="Sinus und Kosinus">Sinus und Kosinus</a>, die den <a href="Kreis" title="Kreis">Einheitskreis</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}=1}">

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<annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec84b90236512e8d27ff1a8f7707b60b63327de1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.7ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}=1}" loading="lazy"> parametrisieren:

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<annotation encoding="application/x-tex">{\displaystyle x=\cos(t),y=\sin(t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db0038dcb95f40333a7d85104dc005fc97996c8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.981ex; height:2.843ex;" alt="{\displaystyle x=\cos(t),y=\sin(t)}" loading="lazy"></dd></dl>

Die Funktionen stellen eine Verbindung her zwischen der Fläche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">, die von einer vom Nullpunkt ausgehenden Geraden und ihrem Spiegelbild an der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">-Achse sowie der Hyperbel eingeschlossen wird, und der Länge verschiedener Strecken.

Dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sinh(A)}">

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<annotation encoding="application/x-tex">{\displaystyle \sinh(A)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a5fe51c75731ba83553dfe5899ddd68c8a275e30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.701ex; height:2.843ex;" alt="{\displaystyle \sinh(A)}" loading="lazy"> die (positive) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">-Koordinate des Schnittpunkts der Geraden mit der Hyperbel und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cosh(A)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cosh(A)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7df1eb662664ba7915c764d9d3dcc13560dafcd5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.956ex; height:2.843ex;" alt="{\displaystyle \cosh(A)}" loading="lazy"> die dazugehörige <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">-Koordinate; <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tanh(A)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \tanh(A)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba61084a3c9efe714515fd685836a6bca230f7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.205ex; height:2.843ex;" alt="{\displaystyle \tanh(A)}" loading="lazy"> ist die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">-Koordinate der Geraden bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=1}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee42176e76ae6b56d68c42ced807e08b962a2b54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=1}" loading="lazy">, d. h. die Steigung der Geraden.

Berechnet man die Fläche durch <a href="Integralrechnung" title="Integralrechnung">Integration</a>, erhält man die Darstellung mit Hilfe der Exponentialfunktion.

<div class="mw-heading mw-heading2"><h2 id="Eigenschaften_der_reellen_Hyperbelfunktionen">Eigenschaften der reellen Hyperbelfunktionen</h2></div>

<ul><li>Für alle <a href="Reelle_Zahlen" class="mw-redirect" title="Reelle Zahlen">reellen Zahlen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> sind auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sinh(x)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \sinh(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83fc0bbb4285d27d164b0b82ac864a957f24e875.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.287ex; height:2.843ex;" alt="{\displaystyle \sinh(x)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cosh(x)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \cosh(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/17450bb663b0b6cf7eb46bcbef41236a21dffe6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.543ex; height:2.843ex;" alt="{\displaystyle \cosh(x)}" loading="lazy"> reell.</li>

<li>Die <a href="Reelle_Funktion" class="mw-redirect" title="Reelle Funktion">reelle Funktion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sinh }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sinh }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb6649e190c30e91b280cc02b27acdfe00055e58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.148ex; height:2.176ex;" alt="{\displaystyle \sinh }" loading="lazy"> ist streng monoton steigend und besitzt in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}" loading="lazy"> ihren einzigen Wendepunkt.</li>

<li>Die reelle Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cosh }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cosh }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/011ccafd311d38f9bbc0fcf3f7b13f7d81469d84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.404ex; height:2.176ex;" alt="{\displaystyle \cosh }" loading="lazy"> ist auf dem Intervall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\infty ,0]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (-\infty ,0]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0241c015ef4c611c9c9aeafb395e7c4a16178405.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.88ex; height:2.843ex;" alt="{\displaystyle (-\infty ,0]}" loading="lazy"> streng monoton fallend, auf dem Intervall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,\infty )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,\infty )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8dc2d914c2df66bc0f7893bfb8da36766650fe47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.072ex; height:2.843ex;" alt="{\displaystyle [0,\infty )}" loading="lazy"> <a href="Streng_monoton_wachsende_Funktion" class="mw-redirect" title="Streng monoton wachsende Funktion">streng monoton steigend</a> und besitzt bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}" loading="lazy"> ein globales Minimum.</li></ul>

Wegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sinh ,\cosh \colon \mathbb {R} \mapsto \mathbb {R} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sinh ,\cosh \colon \mathbb {R} \mapsto \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9f16538c91d98579f7f99aa7698a2076f2707e79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.59ex; height:2.509ex;" alt="{\displaystyle \sinh ,\cosh \colon \mathbb {R} \mapsto \mathbb {R} }" loading="lazy"> gelten alle Eigenschaften der komplexen Hyperbelfunktionen, die im nachfolgenden Absatz aufgeführt sind, auch für die Funktionen, die auf die reellen Zahlen eingeschränkt sind.

<div class="mw-heading mw-heading2"><h2 id="Eigenschaften_der_komplexen_Hyperbelfunktionen">Eigenschaften der komplexen Hyperbelfunktionen</h2></div>

Für alle komplexen Zahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z,z_{1},z_{2}}">

<msub>

</mrow>

</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle z,z_{1},z_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12be55db7c573f60fa97d429cf9f244e98272f77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.427ex; height:2.009ex;" alt="{\displaystyle z,z_{1},z_{2}}" loading="lazy"> gilt:

<div class="mw-heading mw-heading3"><h3 id="Symmetrie_und_Periodizität">Symmetrie und Periodizität</h3></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sinh(-z)=-\sinh(z)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8bdd1b597428ebb083c00bc644268e8917aa0c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.193ex; height:2.843ex;" alt="{\displaystyle \sinh(-z)=-\sinh(z)}" loading="lazy">, d. h., sinh ist eine <a href="Gerade_und_ungerade_Funktionen" title="Gerade und ungerade Funktionen">ungerade Funktion</a>.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cosh(z)=\cosh(-z)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b734f76192618084f5282dd4a214ee2dff312e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.509ex; height:2.843ex;" alt="{\displaystyle \cosh(z)=\cosh(-z)}" loading="lazy">, d. h., cosh ist eine gerade Funktion.</li></ul>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sinh(z)=\sinh(z+2\pi \mathrm {i} )\quad {\text{ und }}\quad \cosh(z)=\cosh(z+2\pi \mathrm {i} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/022c9e11231bc9fedc1874755eea5b8e9a65cb21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:56.925ex; height:2.843ex;" alt="{\displaystyle \sinh(z)=\sinh(z+2\pi \mathrm {i} )\quad {\text{ und }}\quad \cosh(z)=\cosh(z+2\pi \mathrm {i} )}" loading="lazy">,</li></ul>

d. h., es liegt rein „imaginäre Periodizität“ vor mit minimaler Periodenlänge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/73efd1f6493490b058097060a572606d2c550a06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Additionstheoreme">Additionstheoreme</h3></div>

<msub>

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</msub>

<msub>

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</msub>

<msub>

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<msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle \sinh(z_{1}\pm z_{2})=\sinh(z_{1})\cdot \cosh(z_{2})\pm \sinh(z_{2})\cdot \cosh(z_{1})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5060776b78c18e11f34525c6bd143fc02fbdbbfb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.248ex; height:2.843ex;" alt="{\displaystyle \sinh(z_{1}\pm z_{2})=\sinh(z_{1})\cdot \cosh(z_{2})\pm \sinh(z_{2})\cdot \cosh(z_{1})}" loading="lazy"></li>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

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</msub>

<msub>

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<msub>

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<msub>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \cosh(z_{1}\pm z_{2})=\cosh(z_{1})\cdot \cosh(z_{2})\pm \sinh(z_{1})\cdot \sinh(z_{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ae5e3e4fe79bfacdd608ce401cb6627351b94ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.504ex; height:2.843ex;" alt="{\displaystyle \cosh(z_{1}\pm z_{2})=\cosh(z_{1})\cdot \cosh(z_{2})\pm \sinh(z_{1})\cdot \sinh(z_{2})}" loading="lazy"></li>

<msub>

</mrow>

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<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

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</mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tanh(z_{1}\pm z_{2})={\frac {\tanh(z_{1})\pm \tanh(z_{2})}{1\pm \tanh(z_{1})\tanh(z_{2})}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8654de9ae606a3039a25778e79c9514efbac3de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.091ex; height:6.509ex;" alt="{\displaystyle \tanh(z_{1}\pm z_{2})={\frac {\tanh(z_{1})\pm \tanh(z_{2})}{1\pm \tanh(z_{1})\tanh(z_{2})}}}" loading="lazy"></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Zusammenhänge">Zusammenhänge</h3></div>

<msup>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\cosh }^{2}(z)-{\sinh }^{2}(z)=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24bc4f6738b3ddae7862a50f2082d3ab63a326cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.556ex; height:3.176ex;" alt="{\displaystyle {\cosh }^{2}(z)-{\sinh }^{2}(z)=1}" loading="lazy"></dd></dl>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \cosh z+\sinh z\ =\mathrm {e} ^{z}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/42189b1c20526ada49811df5ccb3e1216810d903.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:20.056ex; height:2.509ex;" alt="{\displaystyle \cosh z+\sinh z\ =\mathrm {e} ^{z}}" loading="lazy"></dd></dl>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \cosh z-\sinh z\ =\mathrm {e} ^{-z}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ddd5f1d1ae18aae508f383411ae39ae8b2855cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.334ex; height:2.676ex;" alt="{\displaystyle \cosh z-\sinh z\ =\mathrm {e} ^{-z}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Ableitung">Ableitung</h3></div>

Die <a href="Differentialrechnung" title="Differentialrechnung">Ableitung</a> des Sinus hyperbolicus lautet:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}{\sinh }(z)=\cosh(z)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8f8a3cf3508d47205ef08f1dc7dd87fb8136667a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.662ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}{\sinh }(z)=\cosh(z)}" loading="lazy">.</dd></dl>

Die Ableitung des Kosinus hyperbolicus lautet:

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</mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}{\cosh }(z)=\sinh(z)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04b243a37ffec471bdf41a15784927d86a67d3b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.662ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}{\cosh }(z)=\sinh(z)}" loading="lazy">.</dd></dl>

Die Ableitung des Tangens hyperbolicus lautet:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}{\tanh }(z)=1-{\tanh }^{2}(z)={\frac {1}{\cosh ^{2}(z)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9edf8939be8fc7f2c45d2e5a971c8ee6b92709fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.761ex; height:6.509ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}{\tanh }(z)=1-{\tanh }^{2}(z)={\frac {1}{\cosh ^{2}(z)}}}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Differentialgleichung">Differentialgleichung</h3></div>

Die Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sinh(z)}">

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<annotation encoding="application/x-tex">{\displaystyle \sinh(z)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3bbee4f9f46ba1c22c8a14aa4bce49f4e7af4bec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.046ex; height:2.843ex;" alt="{\displaystyle \sinh(z)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cosh(z)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cosh(z)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2075ae5e2034bd0954ac61f6cc5bf2304b067dfc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.301ex; height:2.843ex;" alt="{\displaystyle \cosh(z)}" loading="lazy"> bilden wie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{z}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{z}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4772def31b56e642df3e4d1160cadff3d80ba45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.085ex; height:2.343ex;" alt="{\displaystyle e^{z}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-z}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{-z}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bc74f35dad4a02e1253aa1baa2320294896b07f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.364ex; height:2.509ex;" alt="{\displaystyle e^{-z}}" loading="lazy"> eine Lösungsbasis (<a href="Fundamentalsystem_(Mathematik)" title="Fundamentalsystem (Mathematik)">Fundamentalsystem</a>) der linearen <a href="Differentialgleichung" title="Differentialgleichung">Differentialgleichung</a>

<mfrac>

<msup>

</mrow>

</mrow>

</msup>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} ^{2}}{\mathrm {d} z^{2}}}f(z)=f(z)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65e8990b40a5d4e8dc467e9271a775078d45f1b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.724ex; height:6.009ex;" alt="{\displaystyle {\frac {\mathrm {d} ^{2}}{\mathrm {d} z^{2}}}f(z)=f(z)}" loading="lazy">.</dd></dl>

Fordert man allgemein für die beiden Basislösungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}(z)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{i}(z)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8a2cb96cc7e0ebf8e346c7766dd29fd986ed82a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.836ex; height:2.843ex;" alt="{\displaystyle f_{i}(z)}" loading="lazy"> dieser Differentialgleichung zweiter Ordnung noch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}(0)=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{1}(0)=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/242b32e4414ccf5b1a4338a5f39f7b461bf9f2df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.426ex; height:2.843ex;" alt="{\displaystyle f_{1}(0)=0}" loading="lazy">,<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}'(0)=1}">

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{1}'(0)=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e06361019a3ab2bb9b1f865e18dd16ffaeb54758.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.426ex; height:3.009ex;" alt="{\displaystyle f_{1}'(0)=1}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{2}(0)=1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{2}(0)=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ecaad96926c7ec3d4200ce18bcfd002a9db5eb18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.426ex; height:2.843ex;" alt="{\displaystyle f_{2}(0)=1}" loading="lazy">,<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{2}'(0)=0}">

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{2}'(0)=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6f2dfd39ef7a905cc24a5779c77813bd6f53460c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.426ex; height:3.009ex;" alt="{\displaystyle f_{2}'(0)=0}" loading="lazy">, so sind sie bereits eindeutig durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sinh }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sinh }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb6649e190c30e91b280cc02b27acdfe00055e58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.148ex; height:2.176ex;" alt="{\displaystyle \sinh }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cosh }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cosh }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/011ccafd311d38f9bbc0fcf3f7b13f7d81469d84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.404ex; height:2.176ex;" alt="{\displaystyle \cosh }" loading="lazy"> festgelegt. Sprich, diese Eigenschaft kann ebenfalls als Definition dieser beiden Hyperbelfunktionen herangezogen werden.

<div class="mw-heading mw-heading2"><h2 id="Bijektivität_der_komplexen_Hyperbelfunktionen">Bijektivität der komplexen Hyperbelfunktionen</h2></div>

Es seien folgende Teilmengen der komplexen Zahlen definiert:

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A:=\{z\in \mathbb {C} \mid -\pi /2<\operatorname {Im} \,z<\pi /2\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e21a7e10867760c03e4b899deda4cea67b1f436f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.314ex; height:2.843ex;" alt="{\displaystyle A:=\{z\in \mathbb {C} \mid -\pi /2<\operatorname {Im} \,z<\pi /2\}}" loading="lazy"></dd>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B:=\{z\in \mathbb {C} \mid \operatorname {Re} \,z\neq 0\vee |\operatorname {Im} \,z|<1\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4a65186c7d0c90f6288f188f5d74d0c30a79cda.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.407ex; height:2.843ex;" alt="{\displaystyle B:=\{z\in \mathbb {C} \mid \operatorname {Re} \,z\neq 0\vee |\operatorname {Im} \,z|<1\}}" loading="lazy"></dd></dl>

Dann bildet die komplexe Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sinh }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sinh }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb6649e190c30e91b280cc02b27acdfe00055e58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.148ex; height:2.176ex;" alt="{\displaystyle \sinh }" loading="lazy"> den „Streifen“ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> <a href="Bijektiv" class="mw-redirect" title="Bijektiv">bijektiv</a> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> ab.

Es seien folgende Teilmengen der komplexen Zahlen definiert:

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A:=\{z\in \mathbb {C} \mid 0<\operatorname {Im} \,z<\pi \}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b3bf7e06241b99ad552828f0e3b99d8c7df00973.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.687ex; height:2.843ex;" alt="{\displaystyle A:=\{z\in \mathbb {C} \mid 0<\operatorname {Im} \,z<\pi \}}" loading="lazy"></dd>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B:=\{z\in \mathbb {C} \mid \operatorname {Im} \,z\neq 0\vee |\operatorname {Re} \,z|<1\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c227a5b36322c5884a650caf6a8934b2ff91382f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.407ex; height:2.843ex;" alt="{\displaystyle B:=\{z\in \mathbb {C} \mid \operatorname {Im} \,z\neq 0\vee |\operatorname {Re} \,z|<1\}}" loading="lazy"></dd></dl>

Dann bildet die komplexe Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cosh }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cosh }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/011ccafd311d38f9bbc0fcf3f7b13f7d81469d84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.404ex; height:2.176ex;" alt="{\displaystyle \cosh }" loading="lazy"> den „Streifen“ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> <a href="Bijektiv" class="mw-redirect" title="Bijektiv">bijektiv</a> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> ab.

<div class="mw-heading mw-heading2"><h2 id="Historische_Notation">Historische Notation</h2></div>

In deutschsprachiger Literatur wurden zur Unterscheidung von den trigonometrischen Funktionen die Hyperbelfunktionen lange Zeit in <a href="Fraktur_(Schrift)" title="Fraktur (Schrift)">Frakturschrift</a> dargestellt – mit initialer Großschreibung und ohne abschließendes h:<a href="#cite_note-1">[1]</a>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {Sin}}\,x\,{\widehat {=}}\,\sinh x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ae685c69e74b5f34f60b26053cf5a9be035179d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.352ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {Sin}}\,x\,{\widehat {=}}\,\sinh x}" loading="lazy"></dd>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {Cos}}\,x\,{\widehat {=}}\,\cosh x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c6c5863b8e4ada83c94d5a9c60ad9d03eddd731e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.399ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {Cos}}\,x\,{\widehat {=}}\,\cosh x}" loading="lazy"></dd>

</mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {Tan}}\,x\ /\ {\mathfrak {Tg}}\,x\,{\widehat {=}}\,\tanh x\ /\ \operatorname {tgh} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1a9a672d5f56642388880488f9806679354a05e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.585ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {Tan}}\,x\ /\ {\mathfrak {Tg}}\,x\,{\widehat {=}}\,\tanh x\ /\ \operatorname {tgh} x}" loading="lazy"></dd>

</mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {Cot}}\,x\ /\ {\mathfrak {Ctg}}\,x\,{\widehat {=}}\,\coth x\ /\ \operatorname {ctgh} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cdca7c430d91c6d098afafdbff46f271e05d4100.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.432ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {Cot}}\,x\ /\ {\mathfrak {Ctg}}\,x\,{\widehat {=}}\,\coth x\ /\ \operatorname {ctgh} x}" loading="lazy"></dd>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {Sec}}\,x\,{\widehat {=}}\,\operatorname {sech} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/31aa4a771591a09e450081fd74f976cef33c825f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.44ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {Sec}}\,x\,{\widehat {=}}\,\operatorname {sech} x}" loading="lazy"></dd>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {Csc}}\,x\,{\widehat {=}}\,\operatorname {csch} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/514164006204ade1b81d46bad6e3f964d322914e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.036ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {Csc}}\,x\,{\widehat {=}}\,\operatorname {csch} x}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Alternative_Namen">Alternative Namen</h2></div>

<ul><li>Für die Hyperbelfunktionen ist auch der Name hyperbolische Funktionen gebräuchlich.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sinh }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb6649e190c30e91b280cc02b27acdfe00055e58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.148ex; height:2.176ex;" alt="{\displaystyle \sinh }" loading="lazy"> sind auch die Namen hsin, Hyperbelsinus und Sinus hyperbolicus gebräuchlich.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cosh }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/011ccafd311d38f9bbc0fcf3f7b13f7d81469d84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.404ex; height:2.176ex;" alt="{\displaystyle \cosh }" loading="lazy"> sind auch die Namen hcos, Hyperbelcosinus und Cosinus hyperbolicus gebräuchlich. Der Graph entspricht der <a href="Katenoide" class="mw-redirect" title="Katenoide">Kettenlinie</a> (Katenoide).</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Abgeleitete_Funktionen">Abgeleitete Funktionen</h2></div>

<ul><li><a href="Tangens_hyperbolicus" class="mw-redirect" title="Tangens hyperbolicus">Tangens hyperbolicus</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tanh(x):={\frac {\sinh(x)}{\cosh(x)}}}">

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tanh(x):={\frac {\sinh(x)}{\cosh(x)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/42b16165cbc32c535f72d4fd99fbb32afc4e57b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.915ex; height:6.509ex;" alt="{\displaystyle \tanh(x):={\frac {\sinh(x)}{\cosh(x)}}}" loading="lazy"></li>

<li><a href="Cotangens_hyperbolicus" class="mw-redirect" title="Cotangens hyperbolicus">Cotangens hyperbolicus</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \coth(x):={\frac {1}{\tanh(x)}}={\frac {\cosh(x)}{\sinh(x)}}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \coth(x):={\frac {1}{\tanh(x)}}={\frac {\cosh(x)}{\sinh(x)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/454af068495dfa98d95211837f9d1339e19a8222.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.381ex; height:6.509ex;" alt="{\displaystyle \coth(x):={\frac {1}{\tanh(x)}}={\frac {\cosh(x)}{\sinh(x)}}}" loading="lazy"></li>

<li><a href="Secans_hyperbolicus" class="mw-redirect" title="Secans hyperbolicus">Secans hyperbolicus</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sech} (x):={\frac {1}{\cosh(x)}}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {sech} (x):={\frac {1}{\cosh(x)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4473cddb998bc65ba8270d90137f68f4ca187b98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.537ex; height:6.009ex;" alt="{\displaystyle \operatorname {sech} (x):={\frac {1}{\cosh(x)}}}" loading="lazy"></li>

<li><a href="Kosecans_hyperbolicus" class="mw-redirect" title="Kosecans hyperbolicus">Kosecans hyperbolicus</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {csch} (x):={\frac {1}{\sinh(x)}}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {csch} (x):={\frac {1}{\sinh(x)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68e4e0e7fc4cc956f4e813ad0406c96a43e05e22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.281ex; height:6.009ex;" alt="{\displaystyle \operatorname {csch} (x):={\frac {1}{\sinh(x)}}}" loading="lazy"></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Umrechnungstabelle">Umrechnungstabelle</h2></div>

<th>Funktion

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sinh }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb6649e190c30e91b280cc02b27acdfe00055e58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.148ex; height:2.176ex;" alt="{\displaystyle \sinh }" loading="lazy">

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cosh }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/011ccafd311d38f9bbc0fcf3f7b13f7d81469d84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.404ex; height:2.176ex;" alt="{\displaystyle \cosh }" loading="lazy">

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tanh }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1494d5d222bc80846ce45e804a38b0ea37e846b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.652ex; height:2.176ex;" alt="{\displaystyle \tanh }" loading="lazy">

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \coth }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b3111993c8de3e6524f16291f1777a34652e15b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.392ex; height:2.176ex;" alt="{\displaystyle \coth }" loading="lazy">

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {sech} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb8545d4ea8fb301a8ea17b3a6096d6c0bbc9380.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.274ex; height:2.176ex;" alt="{\displaystyle \operatorname {sech} }" loading="lazy">

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {csch} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/819e1186e01d44792c05ad673466f556ffbd97fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.274ex; height:2.176ex;" alt="{\displaystyle \operatorname {csch} }" loading="lazy">

</th></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sinh(x)=}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e84eddc489cd940c909a7325e3c2c1179c5e0034.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.74ex; height:2.843ex;" alt="{\displaystyle \sinh(x)=}" loading="lazy">

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sinh(x)\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/169e1dc0b9ebb0dabeabc40aae3c6fbf42018ab0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.674ex; height:2.843ex;" alt="{\displaystyle \sinh(x)\,}" loading="lazy">

</td>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn}(x){\sqrt {\cosh ^{2}(x)-1}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d26cd92ba8935f41d38596b4cbbb6152327e3eaf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:21.434ex; height:4.843ex;" alt="{\displaystyle \operatorname {sgn} (x){\sqrt {\cosh ^{2}(x)-1}}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\tanh(x)}{\sqrt {1-\tanh ^{2}(x)}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/348c563cece776d66f9ceaf149ebb2c117b04979.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:16.008ex; height:8.509ex;" alt="{\displaystyle {\frac {\tanh(x)}{\sqrt {1-\tanh ^{2}(x)}}}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {sgn}(x)}{\sqrt {\coth ^{2}(x)-1}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/652e194675ddf96b86f829629ecd614fbd9fa3bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:15.748ex; height:8.509ex;" alt="{\displaystyle {\frac {\operatorname {sgn} (x)}{\sqrt {\coth ^{2}(x)-1}}}}" loading="lazy">

</td>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn}(x){\frac {\sqrt {1-\operatorname {sech} ^{2}(x)}}{\operatorname {sech} (x)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e75b1098baa3414340d9479096f7c28b2c9a05ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.14ex; height:8.509ex;" alt="{\displaystyle \operatorname {sgn} (x){\frac {\sqrt {1-\operatorname {sech} ^{2}(x)}}{\operatorname {sech} (x)}}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\operatorname {csch} (x)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4af1a21cadbee41984c47962fc2ef52317bcc699.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:8.249ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{\operatorname {csch} (x)}}}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cosh(x)=}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2009de9ded22e667ba55ae6c44b5971aa7c33d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.996ex; height:2.843ex;" alt="{\displaystyle \cosh(x)=}" loading="lazy">

</th>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,{\sqrt {1+\sinh ^{2}(x)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7a530c29f4431871c81cda9a515d169ffc8a99d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:15.055ex; height:4.843ex;" alt="{\displaystyle \,{\sqrt {1+\sinh ^{2}(x)}}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,\cosh(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24022eb08fed697984f3e96e2c0d9eb5791e0c16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.93ex; height:2.843ex;" alt="{\displaystyle \,\cosh(x)}" loading="lazy">

</td>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,{\frac {1}{\sqrt {1-\tanh ^{2}(x)}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/82f4b8425ca12bf608f36fe1001dfe741aae39b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:16.395ex; height:8.009ex;" alt="{\displaystyle \,{\frac {1}{\sqrt {1-\tanh ^{2}(x)}}}}" loading="lazy">

</td>

<mfrac>

<mrow>

<mrow>

</mrow>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,{\frac {\left|\coth(x)\right|}{\sqrt {\coth ^{2}(x)-1}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a85cccc2d898a9ded72f6bdd5b9c4efaa0c1d4f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:16.135ex; height:8.509ex;" alt="{\displaystyle \,{\frac {\left|\coth(x)\right|}{\sqrt {\coth ^{2}(x)-1}}}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,{\frac {1}{\operatorname {sech} (x)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/595e1191769087ecab185bb2b14de78ff6938106.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:8.636ex; height:6.009ex;" alt="{\displaystyle \,{\frac {1}{\operatorname {sech} (x)}}}" loading="lazy">

</td>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

<mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,{\frac {\sqrt {1+\operatorname {csch} ^{2}(x)}}{\left|\operatorname {csch} (x)\right|}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6f805942b83dbbc33285f518da00f2563253e1f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.017ex; height:8.509ex;" alt="{\displaystyle \,{\frac {\sqrt {1+\operatorname {csch} ^{2}(x)}}{\left|\operatorname {csch} (x)\right|}}}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tanh(x)=}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6fac719b1c7d5a8bad19b8f6534c932a838bc4a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.244ex; height:2.843ex;" alt="{\displaystyle \tanh(x)=}" loading="lazy">

</th>

<mfrac>

<mrow>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,{\frac {\sinh(x)}{\sqrt {1+\sinh ^{2}(x)}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4a8423ca42d9b94b2cb14c76bb63dc74b0da7aab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:15.891ex; height:8.509ex;" alt="{\displaystyle \,{\frac {\sinh(x)}{\sqrt {1+\sinh ^{2}(x)}}}}" loading="lazy">

</td>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,\operatorname {sgn}(x){\frac {\sqrt {\cosh ^{2}(x)-1}}{\cosh(x)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dd81564b2bdc2c4dff609a546c2025d0eb8b0ec0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.657ex; height:8.509ex;" alt="{\displaystyle \,\operatorname {sgn} (x){\frac {\sqrt {\cosh ^{2}(x)-1}}{\cosh(x)}}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,\tanh(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f7c777a29695c4b5595c3525d70a5112fcb45a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.178ex; height:2.843ex;" alt="{\displaystyle \,\tanh(x)}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,{\frac {1}{\coth(x)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/60a4c0c3469d643c75fabc4225e1f14c368e2613.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:8.754ex; height:6.009ex;" alt="{\displaystyle \,{\frac {1}{\coth(x)}}}" loading="lazy">

</td>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,\operatorname {sgn}(x){\sqrt {1-\operatorname {sech} ^{2}(x)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db1eab6d645ab9828e937bb18a340456c66f178a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:21.691ex; height:4.843ex;" alt="{\displaystyle \,\operatorname {sgn} (x){\sqrt {1-\operatorname {sech} ^{2}(x)}}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,{\frac {\operatorname {sgn}(x)}{\sqrt {1+\operatorname {csch} ^{2}(x)}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33cc67af0ea5260fa2a0ef650066b2dd98a7256f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:16.017ex; height:8.509ex;" alt="{\displaystyle \,{\frac {\operatorname {sgn} (x)}{\sqrt {1+\operatorname {csch} ^{2}(x)}}}}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \coth(x)=}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a920946ffc290431a16e4f4ed281cb1ad3fdcef4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.984ex; height:2.843ex;" alt="{\displaystyle \coth(x)=}" loading="lazy">

</th>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,{\frac {\sqrt {1+\sinh ^{2}(x)}}{\sinh(x)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e00f828451d1bcf8019a248b5bc94cd768d06c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.891ex; height:8.509ex;" alt="{\displaystyle \,{\frac {\sqrt {1+\sinh ^{2}(x)}}{\sinh(x)}}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,\operatorname {sgn}(x){\frac {\cosh(x)}{\sqrt {\cosh ^{2}(x)-1}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/37bfb3ff9cafbb9ed830cc0a11b71dd1068569a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:22.657ex; height:8.509ex;" alt="{\displaystyle \,\operatorname {sgn} (x){\frac {\cosh(x)}{\sqrt {\cosh ^{2}(x)-1}}}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,{\frac {1}{\tanh(x)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/150d63ad5c38a00b5224db2924f38714901f2707.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:9.014ex; height:6.009ex;" alt="{\displaystyle \,{\frac {1}{\tanh(x)}}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,\coth(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/966987983127feb705664139f6573e1626a2afe8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.918ex; height:2.843ex;" alt="{\displaystyle \,\coth(x)}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,{\frac {\operatorname {sgn}(x)}{\sqrt {1-\operatorname {sech} ^{2}(x)}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9c1b2d712a2548d36a418aa55089fb6d96f870d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:16.017ex; height:8.509ex;" alt="{\displaystyle \,{\frac {\operatorname {sgn} (x)}{\sqrt {1-\operatorname {sech} ^{2}(x)}}}}" loading="lazy">

</td>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,\operatorname {sgn}(x){\sqrt {1+\operatorname {csch} ^{2}(x)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe984a42e2f9c314278a1d83aa39bd0b84a66851.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:21.691ex; height:4.843ex;" alt="{\displaystyle \,\operatorname {sgn} (x){\sqrt {1+\operatorname {csch} ^{2}(x)}}}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {sech} (x)=}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6eab8e9c3646a1ef51e64106d25ae406793a5935.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.866ex; height:2.843ex;" alt="{\displaystyle \operatorname {sech} (x)=}" loading="lazy">

</th>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,{\frac {1}{\sqrt {1+\sinh ^{2}(x)}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/871fa6b1d2996533277e08021a513a30a86c3c21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:15.891ex; height:8.009ex;" alt="{\displaystyle \,{\frac {1}{\sqrt {1+\sinh ^{2}(x)}}}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,{\frac {1}{\cosh(x)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/634336c5caab268611ef3bee1dba7d6bb157ec50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:8.766ex; height:6.009ex;" alt="{\displaystyle \,{\frac {1}{\cosh(x)}}}" loading="lazy">

</td>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,{\sqrt {1-\tanh ^{2}(x)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7e553e1d4f94197719f76166d1f09adf454446f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:15.559ex; height:4.843ex;" alt="{\displaystyle \,{\sqrt {1-\tanh ^{2}(x)}}}" loading="lazy">

</td>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

<mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,{\frac {\sqrt {\coth ^{2}(x)-1}}{\left|\coth(x)\right|}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f1feebf17083bf8a4cdd53b81ae14ab2adcf76d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.135ex; height:8.509ex;" alt="{\displaystyle \,{\frac {\sqrt {\coth ^{2}(x)-1}}{\left|\coth(x)\right|}}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,\operatorname {sech} (x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f934c26ce369ef29e47942ddc206a9c9a6f1d782.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.8ex; height:2.843ex;" alt="{\displaystyle \,\operatorname {sech} (x)}" loading="lazy">

</td>

<mfrac>

<mrow>

<mrow>

</mrow>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,{\frac {\left|\operatorname {csch} (x)\right|}{\sqrt {1+\operatorname {csch} ^{2}(x)}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61f6ad45a722d6ae774f21b76ed3f4f533828181.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:16.017ex; height:8.509ex;" alt="{\displaystyle \,{\frac {\left|\operatorname {csch} (x)\right|}{\sqrt {1+\operatorname {csch} ^{2}(x)}}}}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {csch} (x)=}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/349bb755fd211ff7b114a5d010b943513c6f81b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.866ex; height:2.843ex;" alt="{\displaystyle \operatorname {csch} (x)=}" loading="lazy">

</th>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,{\frac {1}{\sinh(x)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5583e1c13a1ceb9a954ab41049f757a27d0197a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:8.51ex; height:6.009ex;" alt="{\displaystyle \,{\frac {1}{\sinh(x)}}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,{\frac {\operatorname {sgn}(x)}{\sqrt {\cosh ^{2}(x)-1}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0bf190de2d56a30567bd4052306a251768140cfd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:16.147ex; height:8.509ex;" alt="{\displaystyle \,{\frac {\operatorname {sgn} (x)}{\sqrt {\cosh ^{2}(x)-1}}}}" loading="lazy">

</td>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,{\frac {\sqrt {1-\tanh ^{2}(x)}}{\tanh(x)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b85f46717dc478534e9c7dd3c0391a9ffba297f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.395ex; height:8.509ex;" alt="{\displaystyle \,{\frac {\sqrt {1-\tanh ^{2}(x)}}{\tanh(x)}}}" loading="lazy">

</td>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,\operatorname {sgn}(x){\sqrt {\coth ^{2}(x)-1}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/67e7831ef45c2eeca09a4da5450801e85ff9f5e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:21.809ex; height:4.843ex;" alt="{\displaystyle \,\operatorname {sgn} (x){\sqrt {\coth ^{2}(x)-1}}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,\operatorname {sgn}(x){\frac {\operatorname {sech} (x)}{\sqrt {1-\operatorname {sech} ^{2}(x)}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0aeb0abf97b9e647ad3c5c93ef1493573d3a27f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:22.527ex; height:8.509ex;" alt="{\displaystyle \,\operatorname {sgn} (x){\frac {\operatorname {sech} (x)}{\sqrt {1-\operatorname {sech} ^{2}(x)}}}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,\operatorname {csch} (x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec2d9ff0f5c252e39cc00649195feff78c15587b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.8ex; height:2.843ex;" alt="{\displaystyle \,\operatorname {csch} (x)}" loading="lazy">

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Cauchysche_Reihen">Cauchysche Reihen</h2></div>

Analog zum <a href="Basler_Problem#Eulers_erste_Lösung" title="Basler Problem">Eulerschen Beweis des Basler Problems</a> können unendliche Produktreihen für den Sinus Hyperbolicus und den Cosinus Hyperbolicus aufgestellt werden:

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{x}}\sinh(x)=\prod _{n=1}^{\infty }{\biggl (}1+{\frac {x^{2}}{n^{2}\pi ^{2}}}{\biggr )}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4e511277bc5c04a0a87937834dd608c0cb265c4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.51ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{x}}\sinh(x)=\prod _{n=1}^{\infty }{\biggl (}1+{\frac {x^{2}}{n^{2}\pi ^{2}}}{\biggr )}}" loading="lazy"></dd>

</mrow>

</mrow>

</munderover>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cosh(x)=\prod _{n=1}^{\infty }{\biggl [}1+{\frac {4x^{2}}{(2n-1)^{2}\pi ^{2}}}{\biggr ]}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10f96cfff1913f2188d5a035e4dc353bf1876308.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.221ex; height:6.843ex;" alt="{\displaystyle \cosh(x)=\prod _{n=1}^{\infty }{\biggl [}1+{\frac {4x^{2}}{(2n-1)^{2}\pi ^{2}}}{\biggr ]}}" loading="lazy"></dd></dl>

Die erste gezeigte Funktion stellt die nicht normierte Variante des Hyperbolischen <a href="Sinc-Funktion" title="Sinc-Funktion">Kardinalsinus</a> dar.

Die Summen der diskreten <a href="Cauchy-Verteilung" title="Cauchy-Verteilung">Cauchy-Verteilung</a> ergeben die Hyperbelfunktionen:

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tanh(x)=\sum _{n=1}^{\infty }{\frac {8x}{(2n-1)^{2}\pi ^{2}+4x^{2}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/658dbeb529a436cd5f0a6811244d2856e7623066.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.667ex; height:6.843ex;" alt="{\displaystyle \tanh(x)=\sum _{n=1}^{\infty }{\frac {8x}{(2n-1)^{2}\pi ^{2}+4x^{2}}}}" loading="lazy"></dd>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(x)=\coth(x)-{\frac {1}{x}}=\sum _{n=1}^{\infty }{\frac {2x}{n^{2}\pi ^{2}+x^{2}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0bdf085c367d52df3439ab2fa5d72c012cb5f94f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.096ex; height:6.843ex;" alt="{\displaystyle L(x)=\coth(x)-{\frac {1}{x}}=\sum _{n=1}^{\infty }{\frac {2x}{n^{2}\pi ^{2}+x^{2}}}}" loading="lazy"></dd>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {sech} (x)=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}(8n-4)\pi }{(2n-1)^{2}\pi ^{2}+4x^{2}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c0baca2d06c310a972c81b7fa3dad89b5ff7606d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.288ex; height:7.009ex;" alt="{\displaystyle \operatorname {sech} (x)=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}(8n-4)\pi }{(2n-1)^{2}\pi ^{2}+4x^{2}}}}" loading="lazy"></dd>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{x}}-\operatorname {csch} (x)=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}2x}{n^{2}\pi ^{2}+x^{2}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd0d159754712d9d92c3c1129c42628873fa831e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.686ex; height:7.009ex;" alt="{\displaystyle {\frac {1}{x}}-\operatorname {csch} (x)=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}2x}{n^{2}\pi ^{2}+x^{2}}}}" loading="lazy"></dd></dl>

Alle sechs nun gezeigten Reihen sind für alle reellen Werte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> konvergent!

Der Buchstabe L steht für die <a href="Langevin-Funktion" title="Langevin-Funktion">Langevin-Funktion</a>, welche in der Elektrodynamik bei der Beschreibung des <a href="Paramagnetismus" title="Paramagnetismus">Paramagnetismus</a> und in der statistischen Thermodynamik bei der Beschreibung der <a href="Thermische_Energie" title="Thermische Energie">Wärmeenergie</a> eine essentielle Rolle spielt und einen Spezialfall der <a href="Brillouin-Funktion" title="Brillouin-Funktion">Brillouin-Funktionen</a> bildet. Und generell gilt für alle reellen Zahlen a, b und c mit dem Kriterium <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4ac-b^{2}>0}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 4ac-b^{2}>0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/518c207fb90ddb262637565bdc76184f6dfbfd45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.552ex; height:2.843ex;" alt="{\displaystyle 4ac-b^{2}>0}" loading="lazy"> folgende Formel:

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=-\infty }^{\infty }{\frac {1}{a\,n^{2}+b\,n+c}}={\frac {2\,\pi \sinh {\bigl (}{\tfrac {1}{a}}{\sqrt {4ac-b^{2}}}\,\pi {\bigr )}}{{\sqrt {4ac-b^{2}}}\,{\bigl [}\cosh {\bigl (}{\tfrac {1}{a}}{\sqrt {4ac-b^{2}}}\,\pi {\bigr )}-\cos {\bigl (}{\tfrac {b}{a}}\pi {\bigr )}{\bigr ]}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7c64d25b5cedbf1f69d0c7ccafedaf5fd88ce67a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:69.672ex; height:8.176ex;" alt="{\displaystyle \sum _{n=-\infty }^{\infty }{\frac {1}{a\,n^{2}+b\,n+c}}={\frac {2\,\pi \sinh {\bigl (}{\tfrac {1}{a}}{\sqrt {4ac-b^{2}}}\,\pi {\bigr )}}{{\sqrt {4ac-b^{2}}}\,{\bigl [}\cosh {\bigl (}{\tfrac {1}{a}}{\sqrt {4ac-b^{2}}}\,\pi {\bigr )}-\cos {\bigl (}{\tfrac {b}{a}}\pi {\bigr )}{\bigr ]}}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Umkehrfunktionen">Umkehrfunktionen</h2></div>

Die <a href="Umkehrfunktion" title="Umkehrfunktion">Umkehrfunktionen</a> der Hyperbelfunktionen heißen <a href="Area-Funktion" class="mw-redirect" title="Area-Funktion">Area-Funktionen</a>.

Siehe auch:

<a href="Kreis-_und_Hyperbelfunktionen" title="Kreis- und Hyperbelfunktionen">Zusammenhang mit den Kreisfunktionen</a>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li><a href="Ilja_Nikolajewitsch_Bronschtein" title="Ilja Nikolajewitsch Bronschtein">Ilja N. Bronstein</a>: <a href="Taschenbuch_der_Mathematik" title="Taschenbuch der Mathematik">Taschenbuch der Mathematik</a>. Deutsch (Harri).</li>

<li>Nickos Papadatos: The characteristic function of the discrete Cauchy distribution. Department of Mathematics, National and Kapodistrian University of Athens, Panepistemiopolis, 157 84 Athens, Greece, 2022</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>

<div class="sisterproject" style="margin:0.1em 0 0 0;"><div class="noresize noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"></div><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Hyperbolic_functions?uselang=de">Commons: Hyperbolic functions</a> – Sammlung von Bildern, Videos und Audiodateien</div>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> <a href="Stefan_Hildebrandt" title="Stefan Hildebrandt">Stefan Hildebrandt</a>: <cite style="font-style:italic">Analysis</cite>. <a href="Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>, 2002, ISBN 978-3-540-42838-1, S. 243, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-662-05694-3">10.1007/978-3-662-05694-3</a> (<a rel="nofollow" class="external text" href="https://books.google.de/books?id=XXEdBgAAQBAJ&pg=PA243#v=onepage">eingeschränkte Vorschau</a> in der Google-Buchsuche).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Hyperbelfunktion&rft.au=Stefan+Hildebrandt&rft.btitle=Analysis&rft.date=2002&rft.doi=10.1007%2F978-3-662-05694-3&rft.genre=book&rft.isbn=9783540428381&rft.pages=243&rft.pub=Springer" style="display:none">

</li>

</ol>

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<div class="klappleiste-kopf"><a href="Trigonometrische_Funktion" title="Trigonometrische Funktion">Trigonometrische Funktion</a></div>

<a href="Trigonometrische_Funktion" title="Trigonometrische Funktion">Primäre trigonometrische Funktionen</a>

<a href="Sinus_und_Kosinus" title="Sinus und Kosinus">Sinus und Kosinus</a> |

<a href="Tangens_und_Kotangens" title="Tangens und Kotangens">Tangens und Kotangens</a> |

<a href="Sekans_und_Kosekans" title="Sekans und Kosekans">Sekans und Kosekans</a>

<a href="Arkusfunktion" title="Arkusfunktion">Umkehrfunktionen (Arkusfunktionen)</a>

<a href="Arkussinus_und_Arkuskosinus" title="Arkussinus und Arkuskosinus">Arkussinus und Arkuskosinus</a> |

<a href="Arkustangens_und_Arkuskotangens" title="Arkustangens und Arkuskotangens">Arkustangens und Arkuskotangens</a> |

<a href="Arkussekans_und_Arkuskosekans" title="Arkussekans und Arkuskosekans">Arkussekans und Arkuskosekans</a>

<a class="mw-selflink selflink">Hyperbelfunktionen</a>

<a href="Sinus_hyperbolicus_und_Kosinus_hyperbolicus" title="Sinus hyperbolicus und Kosinus hyperbolicus">Sinus hyperbolicus und Kosinus hyperbolicus</a> |

<a href="Tangens_hyperbolicus_und_Kotangens_hyperbolicus" title="Tangens hyperbolicus und Kotangens hyperbolicus">Tangens hyperbolicus und Kotangens hyperbolicus</a> |

<a href="Sekans_hyperbolicus_und_Kosekans_hyperbolicus" title="Sekans hyperbolicus und Kosekans hyperbolicus">Sekans hyperbolicus und Kosekans hyperbolicus</a>

<a href="Areafunktion" title="Areafunktion">Areafunktionen</a>

<a href="Areasinus_hyperbolicus_und_Areakosinus_hyperbolicus" title="Areasinus hyperbolicus und Areakosinus hyperbolicus">Areasinus hyperbolicus und Areakosinus hyperbolicus</a> |

<a href="Areatangens_hyperbolicus_und_Areakotangens_hyperbolicus" title="Areatangens hyperbolicus und Areakotangens hyperbolicus">Areatangens hyperbolicus und Areakotangens hyperbolicus</a> |

<a href="Areasekans_hyperbolicus_und_Areakosekans_hyperbolicus" title="Areasekans hyperbolicus und Areakosekans hyperbolicus">Areasekans hyperbolicus und Areakosekans hyperbolicus</a>

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