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Definition der Areafunktionen durch den Logarithmus

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

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

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

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

${\displaystyle \operatorname {arsech} (z)=\ln \left({\frac {1+{\sqrt {1-z^{2}}}}{z}}\right)}$

${\displaystyle \operatorname {arcsch} (z)=\ln \left({\frac {1+{\sqrt {1+z^{2}}}}{z}}\right)}$

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

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