Hyperbolic Functions

The hyperbolic functions , , , , , (hyperbolic sine, hyperbolic cosine, hyperbolic tangent, hyperbolic cosecant, hyperbolic secant, and hyperbolic cotangent) are analogs of the trigonometric functions, obtained by removing factors of from their formulas in terms of . For example,

(1)

so

(2)

Note that alternate notations are sometimes used, as summarized in the following table.

	alternate notations
	(Gradshteyn and Ryzhik 2000, p. xxvii)
	(Gradshteyn and Ryzhik 2000, p. xxvii)
	(Gradshteyn and Ryzhik 2000, p. xxvii)
	(Gradshteyn and Ryzhik 2000, p. xxvii)

The hyperbolic functions share many properties with the corresponding trigonometric functions. In fact, just as the circle can be represented parametrically by

			(3)
			(4)

a rectangular hyperbola (or, more specifically, its right branch) can be analogously represented by

			(5)
			(6)

where is the hyperbolic cosine and is the hyperbolic sine.

The parameter also has an area interpretation analogous to the angle parameter for the unit circle. On the unit rectangular hyperbola , the signed area bounded by the positive -axis, the ray from the origin to , and the hyperbolic arc between them is . Thus is twice this signed area.

The hyperbolic functions arise in many problems of mathematics and mathematical physics in which integrals involving arise (whereas the trigonometric functions involve ). For instance, the hyperbolic sine arises in the gravitational potential of a cylinder and the calculation of the Roche limit. The hyperbolic cosine function is the shape of a hanging cable (the so-called catenary). The hyperbolic tangent arises in the calculation of and rapidity of special relativity. All three appear in the Schwarzschild metric using external isotropic Kruskal coordinates in general relativity. The hyperbolic secant arises in the profile of a laminar jet. The hyperbolic cotangent arises in the Langevin function for magnetic polarization.

The hyperbolic functions are defined by

			(7)
			(8)
			(9)
			(10)
			(11)
			(12)
			(13)
			(14)
			(15)
			(16)

For arguments multiplied by ,

(17)

(18)

The hyperbolic functions satisfy many identities analogous to the trigonometric identities (which can be inferred using Osborn's rule) such as

			(19)
			(20)
			(21)

See also Beyer (1987, p. 168).

Some half-angle formulas are

			(22)
			(23)

where .

Some double-angle formulas are

			(24)
			(25)
			(26)

Identities for complex arguments include

			(27)
			(28)

The absolute squares for complex arguments are

			(29)
			(30)