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}
Funkce y = arg cosh x {\displaystyle y=\arg \cosh x}
Funkce y = arg tanh x {\displaystyle y=\arg \tanh x}
Funkce y = arg coth x {\displaystyle y=\arg \coth x}
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 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)}
( 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)}