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

hyperbolic function

Mathematics

Any of the functions hyperbolic sine (sinh), hyperbolic cosine (cosh), and hyperbolic tangent (tanh), and their reciprocals cosech, sech, and coth, defined as follows:

$\begin{array}{l} cosh x = \frac{1}{2} (e^{x} + e^{- x}), sinh x = \frac{1}{2} (e^{x} - e^{- x}), \\ tanh x = \frac{sinh x}{cosh x}, coth x = \frac{cosh x}{sinh x} (x \neq 0), \\ sech x = \frac{1}{cosh x}, cosech x = \frac{1}{sinh x} (x \neq 0) . \end{array}$

The functions derive their name from the possibility of using x = acosht, y = bsinht (tϵ‎ℝ) as parametric equations for (one branch of) a hyperbola. (The pronunciation of these functions causes difficulty. For instance, tanh may be pronounced as ‘tansh’ or ‘than’ (with the ‘th’ as in ‘thing’); and sinh may be pronounced as ‘shine’ or ‘sinch’. Many of the formulae satisfied by the hyperbolic functions are similar to corresponding formulae for the trigonometric functions, but some changes of sign must be noted (see Osborne’s rule). For example:

$\begin{array}{l} {cosh}^{2} x = 1 + {sinh}^{2} x, \\ {sech}^{2} x = 1 - {tanh}^{2} x, \\ sinh (x + y) = sinh x cosh y + cosh x sinh y, \\ cosh (x + y) = cosh x cosh y + sinh x sinh y, \\ sinh 2 x = 2 sinh x cosh x, \\ cosh 2 x = {cosh}^{2} x + {sinh}^{2} x . \end{array}$

Since cosh(−x)=coshx and sinh(−x)= −sinhx, cosh is an even function and sinh is an odd function. The graphs of cosh x and sinh x are shown below.

Graph of hyperbolic cosine

Graph of hyperbolic sine

The graphs of the other hyperbolic functions are:

Graph of hyperbolic tangent

Graph of hyperbolic cotangent

Graph of hyperbolic secant

Graph of hyperbolic cosecant

The following derivatives are readily established:

$\frac{d}{d x} (cosh x) = sinh x, \frac{d}{d x} (sinh x) = cosh x, \frac{d}{d x} (tanh x) = {sech}^{2} x .$

See also inverse hyperbolic function.

单词	hyperbolic function
释义	hyperbolic function Mathematics Any of the functions hyperbolic sine (sinh), hyperbolic cosine (cosh), and hyperbolic tangent (tanh), and their reciprocals cosech, sech, and coth, defined as follows: $\begin{array}{l} cosh x = \frac{1}{2} (e^{x} + e^{- x}), sinh x = \frac{1}{2} (e^{x} - e^{- x}), \\ tanh x = \frac{sinh x}{cosh x}, coth x = \frac{cosh x}{sinh x} (x \neq 0), \\ sech x = \frac{1}{cosh x}, cosech x = \frac{1}{sinh x} (x \neq 0) . \end{array}$ The functions derive their name from the possibility of using x = acosht, y = bsinht (tϵ‎ℝ) as parametric equations for (one branch of) a hyperbola. (The pronunciation of these functions causes difficulty. For instance, tanh may be pronounced as ‘tansh’ or ‘than’ (with the ‘th’ as in ‘thing’); and sinh may be pronounced as ‘shine’ or ‘sinch’. Many of the formulae satisfied by the hyperbolic functions are similar to corresponding formulae for the trigonometric functions, but some changes of sign must be noted (see Osborne’s rule). For example: $\begin{array}{l} {cosh}^{2} x = 1 + {sinh}^{2} x, \\ {sech}^{2} x = 1 - {tanh}^{2} x, \\ sinh (x + y) = sinh x cosh y + cosh x sinh y, \\ cosh (x + y) = cosh x cosh y + sinh x sinh y, \\ sinh 2 x = 2 sinh x cosh x, \\ cosh 2 x = {cosh}^{2} x + {sinh}^{2} x . \end{array}$ Since cosh(−x)=coshx and sinh(−x)= −sinhx, cosh is an even function and sinh is an odd function. The graphs of cosh x and sinh x are shown below. Graph of hyperbolic cosine Graph of hyperbolic sine The graphs of the other hyperbolic functions are: Graph of hyperbolic tangent Graph of hyperbolic cotangent Graph of hyperbolic secant Graph of hyperbolic cosecant The following derivatives are readily established: $\frac{d}{d x} (cosh x) = sinh x, \frac{d}{d x} (sinh x) = cosh x, \frac{d}{d x} (tanh x) = {sech}^{2} x .$ See also inverse hyperbolic function.
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