Inverse Hyperbolic Function - Composition of Hyperbolic and Inverse Hyperbolic Functions

Composition of Hyperbolic and Inverse Hyperbolic Functions

\begin{align} &\sinh(\operatorname{arcosh}\,x) = \sqrt{x^{2} - 1} \quad \text{for} \quad |x| > 1 \\ &\sinh(\operatorname{artanh}\,x) = \frac{x}{\sqrt{1-x^{2}}} \quad \text{for} \quad -1 < x < 1 \\ &\cosh(\operatorname{arsinh}\,x) = \sqrt{1+x^{2}} \\ &\cosh(\operatorname{artanh}\,x) = \frac{1}{\sqrt{1-x^{2}}} \quad \text{for} \quad -1 < x < 1 \\ &\tanh(\operatorname{arsinh}\,x) = \frac{x}{\sqrt{1+x^{2}}} \\ &\tanh(\operatorname{arcosh}\,x) = \frac{\sqrt{x^{2} - 1}}{x} \quad \text{for} \quad |x| > 1 \end{align}

Read more about this topic:  Inverse Hyperbolic Function

Famous quotes containing the words composition of, composition, inverse and/or functions:

    “Modern Western thought will pass into history and be incorporated in it, will have its influence and its place, just as our body will pass into the composition of grass, of sheep, of cutlets, and of men. We do not like that kind of immortality, but what is to be done about it?”
    —Alexander Herzen (1812–1870)

    “Give a scientist a problem and he will probably provide a solution; historians and sociologists, by contrast, can offer only opinions. Ask a dozen chemists the composition of an organic compound such as methane, and within a short time all twelve will have come up with the same solution of CH4. Ask, however, a dozen economists or sociologists to provide policies to reduce unemployment or the level of crime and twelve widely differing opinions are likely to be offered.”
    —Derek Gjertsen, British scientist, author. Science and Philosophy: Past and Present, ch. 3, Penguin (1989)

    “The quality of moral behaviour varies in inverse ratio to the number of human beings involved.”
    —Aldous Huxley (1894–1963)

    “Let us stop being afraid. Of our own thoughts, our own minds. Of madness, our own or others’. Stop being afraid of the mind itself, its astonishing functions and fandangos, its complications and simplifications, the wonderful operation of its machinery—more wonderful because it is not machinery at all or predictable.”
    —Kate Millett (b. 1934)