Formelsammlung Mathematik: Areafunktionen

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Definition der Areafunktionen durch den Logarithmus

${\displaystyle \operatorname {arsinh} \,z=\log \left(z+{\sqrt {z^{2}+1}}\right)}$

${\displaystyle \operatorname {arcosh} \,z=\log \left(z+{\sqrt {z-1}}\,{\sqrt {z+1}}\right)}$

${\displaystyle \operatorname {artanh} \,z={\frac {1}{2}}{\Big (}\log(1+z)-\log(1-z){\Big )}}$ für ${\displaystyle z\neq \pm 1\,}$

${\displaystyle \operatorname {arcoth} \,z=\left\{{\begin{matrix}\operatorname {artanh} \left({\frac {1}{z}}\right)&&z\neq 0\\{\frac {\mathrm {i} \pi }{2}}&&z=0\end{matrix}}\right.}$

${\displaystyle \operatorname {arsech} \,z=\operatorname {arcosh} \left({\frac {1}{z}}\right)}$ für ${\displaystyle z\neq 0\,}$

${\displaystyle \operatorname {arcsch} \,z=\operatorname {arsinh} \left({\frac {1}{z}}\right)}$ für ${\displaystyle z\neq 0\,}$

Argument iz

${\displaystyle {\begin{matrix}\arcsin(\mathrm {i} z)&=&\;\;\,\mathrm {i} \;\operatorname {arsinh} \,z&\qquad &\operatorname {arsinh} (\mathrm {i} z)&=&\;\;\mathrm {i} \;\arcsin z\\\arctan(\mathrm {i} z)&=&\;\;\,\,\mathrm {i} \;\operatorname {artanh} \,z&\qquad &\operatorname {artanh} (\mathrm {i} z)&=&\;\;\;\mathrm {i} \;\arctan z\\\operatorname {arccot} (\mathrm {i} z)&=&-\mathrm {i} \;\operatorname {arcoth} \,z&\qquad &\operatorname {arcoth} (\mathrm {i} z)&=&-\mathrm {i} \;\operatorname {arccot} z\\\operatorname {arccsc} (\mathrm {i} z)&=&-\mathrm {i} \;\operatorname {arcsch} \,z&\qquad &\operatorname {arcsch} (\mathrm {i} z)&=&-\mathrm {i} \;\operatorname {arccsc} z\\\end{matrix}}}$

Verkettung einer Hyperbelfunktion mit einer Areafunktion

${\displaystyle \operatorname {arsinh} \!}$ ${\displaystyle \operatorname {arcosh} \!}$ ${\displaystyle \operatorname {artanh} \!}$ ${\displaystyle \operatorname {arcoth} \!}$ ${\displaystyle \operatorname {arsech} \!}$ ${\displaystyle \operatorname {arcsch} \!}$
${\displaystyle \sinh \!}$ ${\displaystyle z\!}$ ${\displaystyle {\sqrt {z-1}}\;{\sqrt {z+1}}}$ ${\displaystyle {\frac {z}{\sqrt {1-z^{2}}}}}$ ${\displaystyle {\frac {1}{z\,{\sqrt {1-{\frac {1}{z^{2}}}}}}}}$ ${\displaystyle {\sqrt {{\frac {1}{z}}-1}}\;{\sqrt {{\frac {1}{z}}+1}}}$ ${\displaystyle {\frac {1}{z}}}$
${\displaystyle \cosh \!}$ ${\displaystyle {\sqrt {1+z^{2}}}}$ ${\displaystyle z\!}$ ${\displaystyle {\frac {1}{\sqrt {1-z^{2}}}}}$ ${\displaystyle {\frac {1}{\sqrt {1-{\frac {1}{z^{2}}}}}}}$ ${\displaystyle {\frac {1}{z}}}$ ${\displaystyle {\sqrt {1+{\frac {1}{z^{2}}}}}}$
${\displaystyle \tanh \!}$ ${\displaystyle {\frac {z}{\sqrt {1+z^{2}}}}}$ ${\displaystyle {\frac {{\sqrt {z-1}}\;{\sqrt {z+1}}}{z}}}$ ${\displaystyle z\!}$ ${\displaystyle {\frac {1}{z}}}$ ${\displaystyle z\,{\sqrt {{\frac {1}{z}}-1}}\;{\sqrt {{\frac {1}{z}}+1}}}$ ${\displaystyle {\frac {1}{z\,{\sqrt {1+{\frac {1}{z^{2}}}}}}}}$
${\displaystyle \coth \!}$ ${\displaystyle {\frac {\sqrt {1+z^{2}}}{z}}}$ ${\displaystyle {\frac {z}{{\sqrt {z-1}}\;{\sqrt {z+1}}}}}$ ${\displaystyle {\frac {1}{z}}}$ ${\displaystyle z\!}$ ${\displaystyle {\frac {1}{z}}{\frac {1}{{\sqrt {{\frac {1}{z}}-1}}\;{\sqrt {{\frac {1}{z}}+1}}}}}$ ${\displaystyle z\,{\sqrt {1+{\frac {1}{z^{2}}}}}}$
${\displaystyle \operatorname {sech} \!}$ ${\displaystyle {\frac {1}{\sqrt {1+z^{2}}}}}$ ${\displaystyle {\frac {1}{z}}}$ ${\displaystyle {\sqrt {1-z^{2}}}}$ ${\displaystyle {\sqrt {1-{\frac {1}{z^{2}}}}}}$ ${\displaystyle z\!}$ ${\displaystyle {\frac {1}{\sqrt {1+{\frac {1}{z^{2}}}}}}}$
${\displaystyle \operatorname {csch} \!}$ ${\displaystyle {\frac {1}{z}}}$ ${\displaystyle {\frac {1}{{\sqrt {z-1}}\;{\sqrt {z+1}}}}}$ ${\displaystyle {\frac {\sqrt {1-z^{2}}}{z}}}$ ${\displaystyle z\,{\sqrt {1-{\frac {1}{z^{2}}}}}}$ ${\displaystyle {\frac {1}{{\sqrt {{\frac {1}{z}}-1}}\;{\sqrt {{\frac {1}{z}}+1}}}}}$ ${\displaystyle z\!}$