# Formelsammlung Mathematik: Hyperbelfunktionen

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### Definition von Hyperbel- und Winkelfunktionen durch die e-Funktion

${\displaystyle \sin(z)={\frac {e^{\mathrm {i} z}-e^{-\mathrm {i} z}}{2\mathrm {i} }}}$
${\displaystyle \sinh(z)={\frac {e^{z}-e^{-z}}{2}}}$
${\displaystyle \sin(\mathrm {i} z)=\mathrm {i} \sinh(z)\,}$
${\displaystyle \sinh(\mathrm {i} z)=\mathrm {i} \sin(z)\,}$
${\displaystyle \cos(z)={\frac {e^{\mathrm {i} z}+e^{-\mathrm {i} z}}{2}}}$
${\displaystyle \cosh(z)={\frac {e^{z}+e^{-z}}{2}}}$
${\displaystyle \cos(\mathrm {i} z)=\cosh(z)\,}$
${\displaystyle \cosh(\mathrm {i} z)=\cos(z)\,}$
${\displaystyle \tan(z)={\frac {1}{\mathrm {i} }}\,{\frac {e^{\mathrm {i} z}-e^{-\mathrm {i} z}}{e^{\mathrm {i} z}+e^{-\mathrm {i} z}}}}$
${\displaystyle \tanh(z)={\frac {e^{z}-e^{-z}}{e^{z}+e^{-z}}}}$
${\displaystyle \tan(\mathrm {i} z)=\mathrm {i} \tanh(z)\,}$
${\displaystyle \tanh(\mathrm {i} z)=\mathrm {i} \tan(z)\,}$
${\displaystyle \cot(z)=\mathrm {i} \,{\frac {e^{iz}+e^{-iz}}{e^{iz}-e^{-iz}}}}$
${\displaystyle \coth(z)={\frac {e^{z}+e^{-z}}{e^{z}-e^{-z}}}}$
${\displaystyle \cot(\mathrm {i} z)={\frac {1}{\mathrm {i} }}\coth(z)\,}$
${\displaystyle \coth(\mathrm {i} z)={\frac {1}{\mathrm {i} }}\cot(z)\,}$
${\displaystyle \sec(z)={\frac {2}{e^{\mathrm {i} z}+e^{-\mathrm {i} z}}}}$
${\displaystyle \operatorname {sech} (z)={\frac {2}{e^{z}+e^{-z}}}}$
${\displaystyle \sec(\mathrm {i} z)=\operatorname {sech} (z)}$
${\displaystyle \operatorname {sech} (\mathrm {i} z)=\sec(z)}$
${\displaystyle \csc(z)={\frac {2\mathrm {i} }{e^{\mathrm {i} z}-e^{-\mathrm {i} z}}}}$
${\displaystyle \operatorname {csch} (z)={\frac {2}{e^{z}-e^{-z}}}}$
${\displaystyle \csc(\mathrm {i} z)={\frac {1}{\mathrm {i} }}\,\operatorname {csch} (z)}$
${\displaystyle \operatorname {csch} (\mathrm {i} z)={\frac {1}{\mathrm {i} }}\,\csc(z)}$

### Weitere Identitäten mit Hyperbelfunktionen

${\displaystyle \operatorname {csch} (2z)=\coth(z)-\coth(2z)}$