Hyperbolometrická funkce

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 ${\displaystyle y=\arg \sinh x}$

${\displaystyle x\in \mathbb {R} }$

${\displaystyle y\in \mathbb {R} }$

Lichá (inverzní funkce k liché funkci je lichá funkce)

${\displaystyle \arg \sinh x=\ln(x+{\sqrt {x^{2}+1}})}$

Funkce ${\displaystyle y=\arg \cosh x}$

${\displaystyle 1\leq x<\infty }$

${\displaystyle 0\leq y<\infty }$

Ani lichá ani sudá

${\displaystyle \arg \cosh x=\ln(x+{\sqrt {x^{2}-1}})}$

Funkce ${\displaystyle y=\arg \tanh x}$

${\displaystyle -1<x<1}$ resp. ${\displaystyle |x|<1}$

${\displaystyle y\in \mathbb {R} }$

Lichá (inverzní funkce k liché funkci je lichá funkce)

${\displaystyle \arg \tanh x={\frac {1}{2}}\ln {\frac {1+x}{1-x}}}$

Funkce ${\displaystyle y=\arg \coth x}$

${\displaystyle |x|>1}$

${\displaystyle y=\mathbb {R} -\{0\}}$

Lichá (inverzní funkce k liché funkci je lichá funkce)

${\displaystyle \arg \coth x={\frac {1}{2}}\ln {\frac {x+1}{x-1}}}$

${\displaystyle \arg \sinh x}$	${\displaystyle =\arg \cosh {\sqrt {x^{2}+1}}\ \ \ \ \ \ \ (x\geq 0)}$
	${\displaystyle =-\arg \cosh {\sqrt {x^{2}+1}}\ \ \ \ \ (x<0)}$
	${\displaystyle =\arg \tanh {\frac {x}{\sqrt {x^{2}+1}}}}$

${\displaystyle \arg \cosh x=\arg \sinh {\sqrt {x^{2}-1}}=\arg \tanh {\frac {\sqrt {x^{2}-1}}{x}}\ \ \ \ \ (x\geq 0)}$

${\displaystyle \arg \tanh x=\sinh {\frac {x}{\sqrt {1-x^{2}}}}\ \ \ \ \ (x\geq 0)}$

${\displaystyle \arg \tanh x}$	${\displaystyle =\arg \sinh {\frac {x}{\sqrt {1-x^{2}}}}\ \ \ \ \ \ \ (|x|<1)}$
	${\displaystyle =\arg \cosh {\frac {1}{\sqrt {1-x^{2}}}}\ \ \ \ \ (0\leq x<1)}$
	${\displaystyle =-\arg \cosh {\frac {1}{\sqrt {1-x^{2}}}}\ \ \ \ \ (-1<x\leq 0)}$
	${\displaystyle =\arg \coth {\frac {1}{x}}\ \ \ \ \ (-1<x<1,x\not =0)}$

${\displaystyle \arg \coth x}$	${\displaystyle =\arg \sinh {\frac {1}{\sqrt {x^{2}-1}}}\ \ \ \ \ (x>1)}$
	${\displaystyle =-\arg \sinh {\frac {1}{\sqrt {x^{2}-1}}}\ \ \ \ \ (x<-1)}$
	${\displaystyle =\arg \cosh {\frac {x}{\sqrt {x^{2}-1}}}\ \ \ \ \ (x>1)}$
	${\displaystyle =\arg \tanh {\frac {1}{x}}\ \ \ \ \ (|x|>1)}$

${\displaystyle \arg \sinh x\pm \arg \sinh y=\arg \sinh(x{\sqrt {1+y^{2}}}\pm y{\sqrt {1+x^{2}}})}$

${\displaystyle \arg \cosh x\pm \arg \cosh y=\arg \cosh(xy\pm {\sqrt {(1+x^{2})(y^{2}-1)}})\ \ \ \ \ (x\geq 1,y\geq 1)}$

${\displaystyle \arg \tanh x\pm \arg \tanh y=\arg \tanh {\frac {x\pm y}{1\pm xy}}\ \ \ \ \ (|x|<1,|y|<1)}$

${\displaystyle (\arg \sinh x)'={\frac {1}{\sqrt {1+x^{2}}}}}$

${\displaystyle (\arg \cosh x)'={\frac {1}{\sqrt {x^{2}-1}}}\ \ \ \ \ (x>1)}$

${\displaystyle (\arg \tanh x)'={\frac {1}{1-x^{2}}}\ \ \ \ \ (|x|<1)}$

${\displaystyle (\arg \coth x)'={\frac {1}{1-x^{2}}}\ \ \ \ \ (|x|>1)}$

${\displaystyle \int {\frac {1}{\sqrt {1+x^{2}}}}{\rm {d}}x=\arg \sinh x+C}$

${\displaystyle \int {\frac {1}{\sqrt {x^{2}-1}}}{\rm {d}}x=\arg \cosh x+C\ \ \ \ \ (x>1)}$

${\displaystyle \int {\frac {1}{1-x^{2}}}{\rm {d}}x}$	${\displaystyle =\arg \tanh x+C\ \ \ \ \ (|x|<1)}$
	${\displaystyle =\arg \coth x+C\ \ \ \ \ (|x|>1)}$

• Obrázky, zvuky či videa k tématu hyperbolometrická funkce na Wikimedia Commons