Graf funkce argsinh x Hyperbolometrické funkce jsou funkce inverzní k funkcím hyperbolickým . Jedná se o funkce argument hyperbolického sinu (argsinh x ), argument hyperbolického kosinu (argcosh x ), argument hyperbolického tangens (argtanh x ) a argument hyperbolického kotangens (argcoth x ).
Funkce y = arg sinh x {\displaystyle y=\arg \sinh x}
x ∈ R {\displaystyle x\in \mathbb {R} }
Obor hodnot y ∈ R {\displaystyle y\in \mathbb {R} }
Parita Lichá (inverzní funkce k liché funkci je lichá funkce)
Identita arg sinh x = ln ( x + x 2 + 1 ) {\displaystyle \arg \sinh x=\ln(x+{\sqrt {x^{2}+1}})}
Funkce y = arg cosh x {\displaystyle y=\arg \cosh x}
1 ≤ x < ∞ {\displaystyle 1\leq x<\infty }
Obor hodnot 0 ≤ y < ∞ {\displaystyle 0\leq y<\infty }
Parita Ani lichá ani sudá
Identita arg cosh x = ln ( x + x 2 − 1 ) {\displaystyle \arg \cosh x=\ln(x+{\sqrt {x^{2}-1}})}
Funkce y = arg tanh x {\displaystyle y=\arg \tanh x}
− 1 < x < 1 {\displaystyle -1<x<1} | x | < 1 {\displaystyle |x|<1}
Obor hodnot y ∈ R {\displaystyle y\in \mathbb {R} }
Parita Lichá (inverzní funkce k liché funkci je lichá funkce)
Identita arg tanh x = 1 2 ln 1 + x 1 − x {\displaystyle \arg \tanh x={\frac {1}{2}}\ln {\frac {1+x}{1-x}}}
Funkce y = arg coth x {\displaystyle y=\arg \coth x}
| x | > 1 {\displaystyle |x|>1}
Obor hodnot y = R − { 0 } {\displaystyle y=\mathbb {R} -\{0\}}
Parita Lichá (inverzní funkce k liché funkci je lichá funkce)
Identita arg coth x = 1 2 ln x + 1 x − 1 {\displaystyle \arg \coth x={\frac {1}{2}}\ln {\frac {x+1}{x-1}}}
Identity arg sinh x {\displaystyle \arg \sinh x} = arg cosh x 2 + 1 ( x ≥ 0 ) {\displaystyle =\arg \cosh {\sqrt {x^{2}+1}}\ \ \ \ \ \ \ (x\geq 0)} = − arg cosh x 2 + 1 ( x < 0 ) {\displaystyle =-\arg \cosh {\sqrt {x^{2}+1}}\ \ \ \ \ (x<0)} = arg tanh x x 2 + 1 {\displaystyle =\arg \tanh {\frac {x}{\sqrt {x^{2}+1}}}}
arg cosh x = arg sinh x 2 − 1 = arg tanh x 2 − 1 x ( x ≥ 0 ) {\displaystyle \arg \cosh x=\arg \sinh {\sqrt {x^{2}-1}}=\arg \tanh {\frac {\sqrt {x^{2}-1}}{x}}\ \ \ \ \ (x\geq 0)}
arg tanh x = sinh x 1 − x 2 ( x ≥ 0 ) {\displaystyle \arg \tanh x=\sinh {\frac {x}{\sqrt {1-x^{2}}}}\ \ \ \ \ (x\geq 0)}
arg tanh x {\displaystyle \arg \tanh x} = arg sinh x 1 − x 2 ( | x | < 1 ) {\displaystyle =\arg \sinh {\frac {x}{\sqrt {1-x^{2}}}}\ \ \ \ \ \ \ (|x|<1)} = arg cosh 1 1 − x 2 ( 0 ≤ x < 1 ) {\displaystyle =\arg \cosh {\frac {1}{\sqrt {1-x^{2}}}}\ \ \ \ \ (0\leq x<1)} = − arg cosh 1 1 − x 2 ( − 1 < x ≤ 0 ) {\displaystyle =-\arg \cosh {\frac {1}{\sqrt {1-x^{2}}}}\ \ \ \ \ (-1<x\leq 0)} = arg coth 1 x ( − 1 < x < 1 , x ≠ 0 ) {\displaystyle =\arg \coth {\frac {1}{x}}\ \ \ \ \ (-1<x<1,x\not =0)}
arg coth x {\displaystyle \arg \coth x} = arg sinh 1 x 2 − 1 ( x > 1 ) {\displaystyle =\arg \sinh {\frac {1}{\sqrt {x^{2}-1}}}\ \ \ \ \ (x>1)} = − arg sinh 1 x 2 − 1 ( x < − 1 ) {\displaystyle =-\arg \sinh {\frac {1}{\sqrt {x^{2}-1}}}\ \ \ \ \ (x<-1)} = arg cosh x x 2 − 1 ( x > 1 ) {\displaystyle =\arg \cosh {\frac {x}{\sqrt {x^{2}-1}}}\ \ \ \ \ (x>1)} = arg tanh 1 x ( | x | > 1 ) {\displaystyle =\arg \tanh {\frac {1}{x}}\ \ \ \ \ (|x|>1)}
arg sinh x ± arg sinh y = arg sinh ( x 1 + y 2 ± y 1 + x 2 ) {\displaystyle \arg \sinh x\pm \arg \sinh y=\arg \sinh(x{\sqrt {1+y^{2}}}\pm y{\sqrt {1+x^{2}}})}
arg cosh x ± arg cosh y = arg cosh ( x y ± ( 1 + x 2 ) ( y 2 − 1 ) ) ( x ≥ 1 , y ≥ 1 ) {\displaystyle \arg \cosh x\pm \arg \cosh y=\arg \cosh(xy\pm {\sqrt {(1+x^{2})(y^{2}-1)}})\ \ \ \ \ (x\geq 1,y\geq 1)}
arg tanh x ± arg tanh y = arg tanh x ± y 1 ± x y ( | x | < 1 , | y | < 1 ) {\displaystyle \arg \tanh x\pm \arg \tanh y=\arg \tanh {\frac {x\pm y}{1\pm xy}}\ \ \ \ \ (|x|<1,|y|<1)}
Derivace ( arg sinh x ) ′ = 1 1 + x 2 {\displaystyle (\arg \sinh x)'={\frac {1}{\sqrt {1+x^{2}}}}}
( arg cosh x ) ′ = 1 x 2 − 1 ( x > 1 ) {\displaystyle (\arg \cosh x)'={\frac {1}{\sqrt {x^{2}-1}}}\ \ \ \ \ (x>1)}
( arg tanh x ) ′ = 1 1 − x 2 ( | x | < 1 ) {\displaystyle (\arg \tanh x)'={\frac {1}{1-x^{2}}}\ \ \ \ \ (|x|<1)}
( arg coth x ) ′ = 1 1 − x 2 ( | x | > 1 ) {\displaystyle (\arg \coth x)'={\frac {1}{1-x^{2}}}\ \ \ \ \ (|x|>1)}
∫ 1 1 + x 2 d x = arg sinh x + C {\displaystyle \int {\frac {1}{\sqrt {1+x^{2}}}}{\rm {d}}x=\arg \sinh x+C}
∫ 1 x 2 − 1 d x = arg cosh x + C ( x > 1 ) {\displaystyle \int {\frac {1}{\sqrt {x^{2}-1}}}{\rm {d}}x=\arg \cosh x+C\ \ \ \ \ (x>1)}
∫ 1 1 − x 2 d x {\displaystyle \int {\frac {1}{1-x^{2}}}{\rm {d}}x} = arg tanh x + C ( | x | < 1 ) {\displaystyle =\arg \tanh x+C\ \ \ \ \ (|x|<1)} = arg coth x + C ( | x | > 1 ) {\displaystyle =\arg \coth x+C\ \ \ \ \ (|x|>1)}
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