Formelsammlung Mathematik: Arkusfunktionen
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Formelsammlung Mathematik
Definition der Arkusfunktionen durch den Logarithmus
arcsin
z
=
−
i
log
(
1
−
z
2
+
i
z
)
{\displaystyle \arcsin z=-\mathrm {i} \,\log \left({\sqrt {1-z^{2}}}+\mathrm {i} z\right)}
arccos
z
=
−
i
log
(
z
+
i
1
−
z
2
)
{\displaystyle \arccos z=-\mathrm {i} \,\log \left(z+\mathrm {i} \,{\sqrt {1-z^{2}}}\right)}
arctan
z
=
i
2
(
log
(
1
−
i
z
)
−
log
(
1
+
i
z
)
)
{\displaystyle \arctan z={\frac {\mathrm {i} }{2}}{\Big (}\log(1-\mathrm {i} z)-\log(1+\mathrm {i} z){\Big )}}
für
z
≠
±
i
{\displaystyle z\neq \pm \mathrm {i} \,}
arccot
z
=
{
arctan
(
1
z
)
z
≠
0
π
2
z
=
0
{\displaystyle \operatorname {arccot} z=\left\{{\begin{matrix}\arctan \left({\frac {1}{z}}\right)&&z\neq 0\\{\frac {\pi }{2}}&&z=0\end{matrix}}\right.}
arcsec
z
=
arccos
(
1
z
)
{\displaystyle \operatorname {arcsec} z=\arccos \left({\frac {1}{z}}\right)}
für
z
≠
0
{\displaystyle z\neq 0\,}
arccsc
z
=
arcsin
(
1
z
)
{\displaystyle \operatorname {arccsc} z=\arcsin \left({\frac {1}{z}}\right)}
für
z
≠
0
{\displaystyle z\neq 0\,}
Argument iz
arcsin
(
i
z
)
=
i
arsinh
z
arsinh
(
i
z
)
=
i
arcsin
z
arctan
(
i
z
)
=
i
artanh
z
artanh
(
i
z
)
=
i
arctan
z
arccot
(
i
z
)
=
−
i
arcoth
z
arcoth
(
i
z
)
=
−
i
arccot
z
arccsc
(
i
z
)
=
−
i
arcsch
z
arcsch
(
i
z
)
=
−
i
arccsc
z
{\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 Winkelfunktion mit einer Arkusfunktion
f
∘
g
{\displaystyle f\circ g}
arcsin
{\displaystyle \arcsin \!}
arccos
{\displaystyle \arccos \!}
arctan
{\displaystyle \arctan \!}
arccot
{\displaystyle \operatorname {arccot} \!}
arcsec
{\displaystyle \operatorname {arcsec} \!}
arccsc
{\displaystyle \operatorname {arccsc} \!}
sin
{\displaystyle \sin \!}
z
{\displaystyle z\!}
1
−
z
2
{\displaystyle {\sqrt {1-z^{2}}}}
z
1
+
z
2
{\displaystyle {\frac {z}{\sqrt {1+z^{2}}}}}
1
z
1
+
1
z
2
{\displaystyle {\frac {1}{z\,{\sqrt {1+{\frac {1}{z^{2}}}}}}}}
1
−
1
z
2
{\displaystyle {\sqrt {1-{\frac {1}{z^{2}}}}}}
1
z
{\displaystyle {\frac {1}{z}}}
cos
{\displaystyle \cos \!}
1
−
z
2
{\displaystyle {\sqrt {1-z^{2}}}}
z
{\displaystyle z\!}
1
1
+
z
2
{\displaystyle {\frac {1}{\sqrt {1+z^{2}}}}}
1
1
+
1
z
2
{\displaystyle {\frac {1}{\sqrt {1+{\frac {1}{z^{2}}}}}}}
1
z
{\displaystyle {\frac {1}{z}}}
1
−
1
z
2
{\displaystyle {\sqrt {1-{\frac {1}{z^{2}}}}}}
tan
{\displaystyle \tan \!}
z
1
−
z
2
{\displaystyle {\frac {z}{\sqrt {1-z^{2}}}}}
1
−
z
2
z
{\displaystyle {\frac {\sqrt {1-z^{2}}}{z}}}
z
{\displaystyle z\!}
1
z
{\displaystyle {\frac {1}{z}}}
z
1
−
1
z
2
{\displaystyle z\,{\sqrt {1-{\frac {1}{z^{2}}}}}}
1
z
1
−
1
z
2
{\displaystyle {\frac {1}{z\,{\sqrt {1-{\frac {1}{z^{2}}}}}}}}
cot
{\displaystyle \cot \!}
1
−
z
2
z
{\displaystyle {\frac {\sqrt {1-z^{2}}}{z}}}
z
1
−
z
2
{\displaystyle {\frac {z}{\sqrt {1-z^{2}}}}}
1
z
{\displaystyle {\frac {1}{z}}}
z
{\displaystyle z\!}
1
z
1
−
1
z
2
{\displaystyle {\frac {1}{z\,{\sqrt {1-{\frac {1}{z^{2}}}}}}}}
z
1
−
1
z
2
{\displaystyle z\,{\sqrt {1-{\frac {1}{z^{2}}}}}}
sec
{\displaystyle \sec \!}
1
1
−
z
2
{\displaystyle {\frac {1}{\sqrt {1-z^{2}}}}}
1
z
{\displaystyle {\frac {1}{z}}}
1
+
z
2
{\displaystyle {\sqrt {1+z^{2}}}}
1
+
1
z
2
{\displaystyle {\sqrt {1+{\frac {1}{z^{2}}}}}}
z
{\displaystyle z\!}
1
1
−
1
z
2
{\displaystyle {\frac {1}{\sqrt {1-{\frac {1}{z^{2}}}}}}}
csc
{\displaystyle \csc \!}
1
z
{\displaystyle {\frac {1}{z}}}
1
1
−
z
2
{\displaystyle {\frac {1}{\sqrt {1-z^{2}}}}}
1
+
z
2
z
{\displaystyle {\frac {\sqrt {1+z^{2}}}{z}}}
z
1
+
1
z
2
{\displaystyle z\,{\sqrt {1+{\frac {1}{z^{2}}}}}}
1
1
−
1
z
2
{\displaystyle {\frac {1}{\sqrt {1-{\frac {1}{z^{2}}}}}}}
z
{\displaystyle z\!}
Komplementärbeziehungen
arcsin
(
z
)
+
arccos
(
z
)
=
π
2
{\displaystyle \arcsin(z)+\arccos(z)={\frac {\pi }{2}}}
arctan
(
z
)
+
arccot
(
z
)
=
{
π
2
Re
(
z
)
>
0
z
∈
i
]
−
1
,
0
]
z
∈
i
]
1
,
∞
[
−
π
2
Re
(
z
)
<
0
z
∈
i
]
−
∞
,
−
1
[
z
∈
i
]
0
,
1
[
{\displaystyle \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.}
arcsec
(
z
)
+
arccsc
(
z
)
=
π
2
{\displaystyle \operatorname {arcsec}(z)+\operatorname {arccsc}(z)={\frac {\pi }{2}}}
für
z
≠
0
{\displaystyle z\neq 0\,}
für 
für 
für 
![{\displaystyle \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.}](./_assets_/eb734a37dd21ce173a46342d1cc64c92/4999e0495d3495e649c033b5e4f6d5b8978e767c.svg)
für