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:

    “Viewed freely, the English language is the accretion and growth of every dialect, race, and range of time, and is both the free and compacted composition of all.”
    —Walt Whitman (1819–1892)

    “The naive notion that a mother naturally acquires the complex skills of childrearing simply because she has given birth now seems as absurd to me as enrolling in a nine-month class in composition and imagining that at the end of the course you are now prepared to begin writing War and Peace.”
    —Mary Kay Blakely (20th century)

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

    “Nobody is so constituted as to be able to live everywhere and anywhere; and he who has great duties to perform, which lay claim to all his strength, has, in this respect, a very limited choice. The influence of climate upon the bodily functions ... extends so far, that a blunder in the choice of locality and climate is able not only to alienate a man from his actual duty, but also to withhold it from him altogether, so that he never even comes face to face with it.”
    —Friedrich Nietzsche (1844–1900)