Mathematische Ergänzungen: Hyperbelfunktionen [ Bearbeiten ] Es werden oft unterschiedliche Grenzfälle betrachtet, bei denen Hyperbelfunktionen
genähert werden müssen. Daher wiederholen wir hier kurz deren Grenzwertverhalten
und verwenden dabei insbesondere
e
±
x
∼
x
≪
1
1
±
x
+
1
2
x
2
±
1
3
!
x
3
{\displaystyle e^{\pm x}{\underset {x\ll 1}{\sim }}1\pm x+{\frac {1}{2}}x^{2}\pm {\frac {1}{3!}}x^{3}}
sinh
x
=
1
2
(
e
x
−
e
−
x
)
∼
x
≪
1
x
−
1
3
!
x
3
{\displaystyle \sinh x={\frac {1}{2}}\left(e^{x}-e^{-x}\right){\underset {x\ll 1}{\sim }}x-{\frac {1}{3!}}x^{3}}
sinh
x
=
1
2
(
e
x
−
e
−
x
)
∼
x
≫
1
1
2
e
x
{\displaystyle \sinh x={\frac {1}{2}}\left(e^{x}-e^{-x}\right){\underset {x\gg 1}{\sim }}{\frac {1}{2}}e^{x}}
cosh
x
=
1
2
(
e
x
+
e
−
x
)
∼
x
≪
1
1
+
1
2
x
2
{\displaystyle \cosh x={\frac {1}{2}}\left(e^{x}+e^{-x}\right){\underset {x\ll 1}{\sim }}1+{\frac {1}{2}}x^{2}}
cosh
x
=
1
2
(
e
x
+
e
−
x
)
∼
x
≫
1
1
2
e
x
{\displaystyle \cosh x={\frac {1}{2}}\left(e^{x}+e^{-x}\right){\underset {x\gg 1}{\sim }}{\frac {1}{2}}e^{x}}
tanh
x
=
sinh
x
cosh
x
∼
x
≪
1
x
−
1
3
!
x
3
1
+
1
2
x
2
≈
(
x
−
1
3
!
x
3
)
(
1
−
1
2
x
2
)
≈
x
−
1
3
x
3
{\displaystyle \tanh x={\frac {\sinh x}{\cosh x}}{\underset {x\ll 1}{\sim }}{\frac {x-{\frac {1}{3!}}x^{3}}{1+{\frac {1}{2}}x^{2}}}\approx \left(x-{\frac {1}{3!}}x^{3}\right)\left(1-{\frac {1}{2}}x^{2}\right)\approx x-{\frac {1}{3}}x^{3}}
tanh
x
=
sinh
x
cosh
x
∼
x
≫
1
1
{\displaystyle \tanh x={\frac {\sinh x}{\cosh x}}{\underset {x\gg 1}{\sim }}1}
coth
x
=
cosh
x
sinh
x
∼
x
≪
1
1
x
(
1
−
1
3
x
2
)
≈
1
x
(
1
+
1
3
x
2
)
=
1
x
+
x
3
{\displaystyle \coth x={\frac {\cosh x}{\sinh x}}{\underset {x\ll 1}{\sim }}{\frac {1}{x\left(1-{\frac {1}{3}}x^{2}\right)}}\approx {\frac {1}{x}}\left(1+{\frac {1}{3}}x^{2}\right)={\frac {1}{x}}+{\frac {x}{3}}}
cosh
2
x
−
sinh
2
x
=
1
{\displaystyle \cosh ^{2}x-\sinh ^{2}x=1}
d
d
x
tanh
x
=
1
cosh
2
x
{\displaystyle {\frac {d}{dx}}\tanh x={\frac {1}{\cosh ^{2}x}}}
