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