“inverse hyperbolic function”的概念、定义、翻译、参考文献-科学参考

inverse hyperbolic function

Mathematics

Each of the hyperbolic functions sinh and tanh is strictly increasing throughout the whole of its domain ℝ, so in each case an inverse function exists. In the case of cosh, the function has to be restricted to a suitable domain (see inverse function), taken to be [0, ∞). The domain of the inverse function is, in each case, the image of the original function (after the restriction of the domain, in the case of cosh). The inverse functions obtained are: cosh⁻¹:[1, ∞)→ [0, ∞); sinh⁻¹: ℝ → ℝ; tanh⁻¹: (−1, 1) → ℝ. These functions are given by the formulae:

$\begin{array}{l} {cosh}^{- 1} x = ln (x + \sqrt{x^{2} - 1}) for x \geq 1, \\ {sinh}^{- 1} x = ln (x + \sqrt{x^{2} + 1}) for all x, \\ {tanh}^{- 1} x = ln \sqrt{\frac{1 + x}{1 - x}} for - 1 < x < 1. \end{array}$

The following derivatives can be obtained:

$\begin{array}{l} \frac{d}{d x} ({cosh}^{- 1} x) = \frac{1}{\sqrt{x^{2} - 1}} (x \neq 1), \\ \frac{d}{d x} ({sinh}^{- 1} x) = \frac{1}{\sqrt{x^{2} + 1}}, \\ \frac{d}{d x} ({tanh}^{- 1} x) = \frac{1}{1 - x^{2}} . \end{array}$

单词	inverse hyperbolic function
释义	inverse hyperbolic function Mathematics Each of the hyperbolic functions sinh and tanh is strictly increasing throughout the whole of its domain ℝ, so in each case an inverse function exists. In the case of cosh, the function has to be restricted to a suitable domain (see inverse function), taken to be [0, ∞). The domain of the inverse function is, in each case, the image of the original function (after the restriction of the domain, in the case of cosh). The inverse functions obtained are: cosh⁻¹:[1, ∞)→ [0, ∞); sinh⁻¹: ℝ → ℝ; tanh⁻¹: (−1, 1) → ℝ. These functions are given by the formulae: $\begin{array}{l} {cosh}^{- 1} x = ln (x + \sqrt{x^{2} - 1}) for x \geq 1, \\ {sinh}^{- 1} x = ln (x + \sqrt{x^{2} + 1}) for all x, \\ {tanh}^{- 1} x = ln \sqrt{\frac{1 + x}{1 - x}} for - 1 < x < 1. \end{array}$ The following derivatives can be obtained: $\begin{array}{l} \frac{d}{d x} ({cosh}^{- 1} x) = \frac{1}{\sqrt{x^{2} - 1}} (x \neq 1), \\ \frac{d}{d x} ({sinh}^{- 1} x) = \frac{1}{\sqrt{x^{2} + 1}}, \\ \frac{d}{d x} ({tanh}^{- 1} x) = \frac{1}{1 - x^{2}} . \end{array}$
随便看	semiochemical semiology semiotics semi-parametric model semi-permeable membrane semipermeable membrane semipolar bond semiprime semi-regular polyhedron semi-regular tessellation semiregular variable semiring Semite semi-Thue system semi-variogram semi-vertical angle Semnan Spaceport sempiternity Semtex Sen, Amartya (1933–) Senanayake, Don Stephen (1884–1952) Senate, Roman Senate, US Seneca, Lucius Annaeus (4 bc–ad 65) Seneca, Marcus Annaeus (55)