Formelsammlung Mathematik: Arkusfunktionen

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Definition der Arkusfunktionen durch den Logarithmus

$\arcsin z=-\mathrm {i} \,\log \left({\sqrt {1-z^{2}}}+\mathrm {i} z\right)$

$\arccos z=-\mathrm {i} \,\log \left(z+\mathrm {i} \,{\sqrt {1-z^{2}}}\right)$

$\arctan z={\frac {\mathrm {i} }{2}}{\Big (}\log(1-\mathrm {i} z)-\log(1+\mathrm {i} z){\Big )}$ für $z\neq \pm \mathrm {i} \,$

$\operatorname {arccot} z=\left\{{\begin{matrix}\arctan \left({\frac {1}{z}}\right)&&z\neq 0\\{\frac {\pi }{2}}&&z=0\end{matrix}}\right.$

$\operatorname {arcsec} z=\arccos \left({\frac {1}{z}}\right)$ für $z\neq 0\,$

$\operatorname {arccsc} z=\arcsin \left({\frac {1}{z}}\right)$ für $z\neq 0\,$

Argument iz

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

Verkettung einer Winkelfunktion mit einer Arkusfunktion

$f\circ g$	$\arcsin \!$	$\arccos \!$	$\arctan \!$	$\operatorname {arccot} \!$	$\operatorname {arcsec} \!$	$\operatorname {arccsc} \!$
$\sin \!$	$z\!$	${\sqrt {1-z^{2}}}$	${\frac {z}{\sqrt {1+z^{2}}}}$	${\frac {1}{z\,{\sqrt {1+{\frac {1}{z^{2}}}}}}}$	${\sqrt {1-{\frac {1}{z^{2}}}}}$	${\frac {1}{z}}$
$\cos \!$	${\sqrt {1-z^{2}}}$	$z\!$	${\frac {1}{\sqrt {1+z^{2}}}}$	${\frac {1}{\sqrt {1+{\frac {1}{z^{2}}}}}}$	${\frac {1}{z}}$	${\sqrt {1-{\frac {1}{z^{2}}}}}$
$\tan \!$	${\frac {z}{\sqrt {1-z^{2}}}}$	${\frac {\sqrt {1-z^{2}}}{z}}$	$z\!$	${\frac {1}{z}}$	$z\,{\sqrt {1-{\frac {1}{z^{2}}}}}$	${\frac {1}{z\,{\sqrt {1-{\frac {1}{z^{2}}}}}}}$
$\cot \!$	${\frac {\sqrt {1-z^{2}}}{z}}$	${\frac {z}{\sqrt {1-z^{2}}}}$	${\frac {1}{z}}$	$z\!$	${\frac {1}{z\,{\sqrt {1-{\frac {1}{z^{2}}}}}}}$	$z\,{\sqrt {1-{\frac {1}{z^{2}}}}}$
$\sec \!$	${\frac {1}{\sqrt {1-z^{2}}}}$	${\frac {1}{z}}$	${\sqrt {1+z^{2}}}$	${\sqrt {1+{\frac {1}{z^{2}}}}}$	$z\!$	${\frac {1}{\sqrt {1-{\frac {1}{z^{2}}}}}}$
$\csc \!$	${\frac {1}{z}}$	${\frac {1}{\sqrt {1-z^{2}}}}$	${\frac {\sqrt {1+z^{2}}}{z}}$	$z\,{\sqrt {1+{\frac {1}{z^{2}}}}}$	${\frac {1}{\sqrt {1-{\frac {1}{z^{2}}}}}}$	$z\!$

Komplementärbeziehungen

$\arcsin(z)+\arccos(z)={\frac {\pi }{2}}$

$\arctan(z)+\operatorname {arccot}(z)=\left\{{\begin{matrix}{\frac {\pi }{2}}&&\operatorname {Re} (z)>0&&z\in \mathrm {i} \,]-1,0]&&z\in \mathrm {i} \,]1,\infty [\\\\-{\frac {\pi }{2}}&&\operatorname {Re} (z)<0&&z\in \mathrm {i} \,]-\infty ,-1[&&z\in \mathrm {i} \,]0,1[\end{matrix}}\right.$

$\operatorname {arcsec}(z)+\operatorname {arccsc}(z)={\frac {\pi }{2}}$ für $z\neq 0\,$