Formelsammlung Mathematik: Areafunktionen
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Formelsammlung Mathematik
Definition der Areafunktionen durch den Logarithmus
arsinh
(
z
)
=
ln
(
z
+
z
2
+
1
)
{\displaystyle \operatorname {arsinh} (z)=\ln \left(z+{\sqrt {z^{2}+1}}\right)}
arcosh
(
z
)
=
ln
(
z
+
z
−
1
z
+
1
)
{\displaystyle \operatorname {arcosh} (z)=\ln \left(z+{\sqrt {z-1}}\,{\sqrt {z+1}}\right)}
artanh
(
z
)
=
1
2
(
ln
(
1
+
z
)
−
ln
(
1
−
z
)
)
{\displaystyle \operatorname {artanh} (z)={\frac {1}{2}}{\Big (}\ln(1+z)-\ln(1-z){\Big )}}
für
z
≠
±
1
{\displaystyle z\neq \pm 1\,}
arcoth
(
z
)
=
{
artanh
(
1
z
)
z
≠
0
i
π
2
z
=
0
{\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.}
arsech
(
z
)
=
ln
(
1
+
1
−
z
2
z
)
{\displaystyle \operatorname {arsech} (z)=\ln \left({\frac {1+{\sqrt {1-z^{2}}}}{z}}\right)}
arcsch
(
z
)
=
ln
(
1
+
1
+
z
2
z
)
{\displaystyle \operatorname {arcsch} (z)=\ln \left({\frac {1+{\sqrt {1+z^{2}}}}{z}}\right)}
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 Hyperbelfunktion mit einer Areafunktion
arsinh
{\displaystyle \operatorname {arsinh} \!}
arcosh
{\displaystyle \operatorname {arcosh} \!}
artanh
{\displaystyle \operatorname {artanh} \!}
arcoth
{\displaystyle \operatorname {arcoth} \!}
arsech
{\displaystyle \operatorname {arsech} \!}
arcsch
{\displaystyle \operatorname {arcsch} \!}
sinh
{\displaystyle \sinh \!}
z
{\displaystyle z\!}
z
−
1
z
+
1
{\displaystyle {\sqrt {z-1}}\;{\sqrt {z+1}}}
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
z
−
1
1
z
+
1
{\displaystyle {\sqrt {{\frac {1}{z}}-1}}\;{\sqrt {{\frac {1}{z}}+1}}}
1
z
{\displaystyle {\frac {1}{z}}}
cosh
{\displaystyle \cosh \!}
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}}}}}}
tanh
{\displaystyle \tanh \!}
z
1
+
z
2
{\displaystyle {\frac {z}{\sqrt {1+z^{2}}}}}
z
−
1
z
+
1
z
{\displaystyle {\frac {{\sqrt {z-1}}\;{\sqrt {z+1}}}{z}}}
z
{\displaystyle z\!}
1
z
{\displaystyle {\frac {1}{z}}}
z
1
z
−
1
1
z
+
1
{\displaystyle z\,{\sqrt {{\frac {1}{z}}-1}}\;{\sqrt {{\frac {1}{z}}+1}}}
1
z
1
+
1
z
2
{\displaystyle {\frac {1}{z\,{\sqrt {1+{\frac {1}{z^{2}}}}}}}}
coth
{\displaystyle \coth \!}
1
+
z
2
z
{\displaystyle {\frac {\sqrt {1+z^{2}}}{z}}}
z
z
−
1
z
+
1
{\displaystyle {\frac {z}{{\sqrt {z-1}}\;{\sqrt {z+1}}}}}
1
z
{\displaystyle {\frac {1}{z}}}
z
{\displaystyle z\!}
1
z
1
1
z
−
1
1
z
+
1
{\displaystyle {\frac {1}{z}}{\frac {1}{{\sqrt {{\frac {1}{z}}-1}}\;{\sqrt {{\frac {1}{z}}+1}}}}}
z
1
+
1
z
2
{\displaystyle z\,{\sqrt {1+{\frac {1}{z^{2}}}}}}
sech
{\displaystyle \operatorname {sech} \!}
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}}}}}}}
csch
{\displaystyle \operatorname {csch} \!}
1
z
{\displaystyle {\frac {1}{z}}}
1
z
−
1
z
+
1
{\displaystyle {\frac {1}{{\sqrt {z-1}}\;{\sqrt {z+1}}}}}
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
z
−
1
1
z
+
1
{\displaystyle {\frac {1}{{\sqrt {{\frac {1}{z}}-1}}\;{\sqrt {{\frac {1}{z}}+1}}}}}
z
{\displaystyle z\!}
für 
