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The hyperbolic functions sinhz, coshz, tanhz, cschz, sechz, cothz (hyperbolic sine, hyperbolic cosine, hyperbolic tangent, hyperbolic cosecant, hyperbolic secant, and hyperbolic cotangent) are analogs of the circular functions, defined by removing is appearing in the complex exponentials. For example,

 cosz=1/2(e^(iz)+e^(-iz)),

(1)

so

 coshz=1/2(e^z+e^(-z)).

(2)

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

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

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

where cosht is the hyperbolic cosine and sinht is the hyperbolic sine.

The hyperbolic functions arise in many problems of mathematics and mathematical physics in which integrals involving sqrt(1+x^2) arise (whereas the circular functions involve sqrt(1-x^2)). 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

For arguments multiplied by i,

 sinh(iz)=isinz

(17)

 cosh(iz)=cosz.

(18)

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

See also Beyer (1987, p. 168).

Some half-angle formulas are

where z=x+iy.

Some Double-Angle Formulas are

Identities for complex arguments include

The absolute squares for complex arguments are

SEE ALSO: Double-Angle Formulas, Fibonacci Hyperbolic Functions, Half-Angle Formulas, Hyperbolic Cosecant, Hyperbolic Cosine, Hyperbolic Cotangent, Generalized Hyperbolic Functions, Hyperbolic Secant, Hyperbolic Sine, Hyperbolic Tangent, Inverse Hyperbolic Functions, Osborn's Rule REFERENCES:

Abramowitz, M. and Stegun, I. 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.

Anderson, J. W. "Trigonometry in the Hyperbolic Plane." §5.7 in Hyperbolic Geometry. New York: Springer-Verlag, pp. 146-151, 1999.

Beyer, W. H. "Hyperbolic Function." CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 168-186 and 219, 1987.

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

Harris, J. W. and Stocker, H. "Hyperbolic Functions." Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 245-262, 1998.

Jeffrey, A. "Hyperbolic Identities." §2.5 in Handbook of Mathematical Formulas and Integrals, 2nd ed. Orlando, FL: Academic Press, pp. 117-122, 2000.

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

Zwillinger, D. (Ed.). "Hyperbolic Functions." §6.7 in CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 476-481 1995.

Referenced on Wolfram|Alpha: Hyperbolic Functions CITE THIS AS:

Weisstein, Eric W. "Hyperbolic Functions." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HyperbolicFunctions.html

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