“Hyperbolic Functions”的概念、定义、习题解题方法及翻译-数学辞典

单词

Hyperbolic Functions

释义

Hyperbolic Functions

The hyperbolic functions sinh, cosh, tanh, csch, sech, coth (Hyperbolic Sine, Hyperbolic Cosine, etc.) share manyproperties with the corresponding Circular Functions. The hyperbolic functions arise in many problems of mathematicsand mathematical physics in which integrals involving arise (whereas the Circular Functions involve).

For instance, the Hyperbolic Sine arises in the gravitational potential of a cylinder and the calculation of the Roche limit. TheHyperbolic Cosine function is the shape of a hanging cable (the so-called Catenary). The HyperbolicTangent arises in the calculation of magnetic moment and rapidity ofspecial relativity. All three appear in the Schwarzschild metric using external isotropic Kruskal coordinates ingeneral relativity. The Hyperbolic Secant arises in the profile of a laminarjet. The Langevin function for magnetic polarization.

The hyperbolic functions are defined by

			(1)
			(2)
			(3)
			(4)
			(5)
			(6)

For purely Imaginary arguments,

(7)

(8)

The hyperbolic functions satisfy many identities anomalous to the trigonometric identities (which can be inferred using Osborne's Rule) such as

			(9)
			(10)
			(11)

See also Beyer (1987, p. 168). Some half-angle Formulas are

			(12)
			(13)

Some double-angle Formulas are

			(14)
			(15)

Identities for Complex arguments include

			(16)
			(17)

The Absolute Squares for Complex arguments are

			(18)
			(19)

Integrals involving hyperbolic functions include

			(20)


			(21)

, then

			(22)

			(23)

Let

, and

and


			(24)
			(25)
			(26)
			(27)

See also Hyperbolic Cosecant, Hyperbolic Cosine, Hyperbolic Cotangent, Generalized HyperbolicFunctions, Hyperbolic Inverse Functions, Hyperbolic Secant, Hyperbolic Sine, Hyperbolic Tangent,Hyperbolic Inverse Functions, Osborne's Rule

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Hyperbolic Functions.'' §4.5 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 83-86, 1972.

Beyer, W. H. ``Hyperbolic Function.'' CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 168-186, 1987.

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 126-131, 1967.

Yates, R. C. ``Hyperbolic Functions.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 113-118, 1952.

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