In mathematics , hyperbolic functions are analogs of the ordinary trigonometric functions defined for the hyperbola rather than on the circle : just as the points cos t , sin t form a circle with a unit radius , the points cosh t , sinh t form the right half of the equilateral hyperbola. Hyperbolic functions occur in the solutions of many linear differential equations for example, the equation defining a catenary , of some cubic equations , in calculations of angles and distances in hyperbolic geometry , and of Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics , including electromagnetic theory , heat transfer , fluid dynamics , and special relativity. The inverse hyperbolic functions are:. The hyperbolic functions take a real argument called a hyperbolic angle.

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The hyperbolic functions , , , , , hyperbolic sine , hyperbolic cosine , hyperbolic tangent , hyperbolic cosecant , hyperbolic secant , and hyperbolic cotangent are analogs of the circular functions , defined by removing s appearing in the complex exponentials.

For example,. The hyperbolic functions share many properties with the corresponding circular functions. In fact, just as the circle can be represented parametrically by. The hyperbolic functions arise in many problems of mathematics and 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. 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. For arguments multiplied by ,. The hyperbolic functions satisfy many identities analogous to the trigonometric identities which can be inferred using Osborn's rule such as.

The absolute squares for complex arguments are. Abramowitz, M. New York: Dover, pp. Anderson, J. New York: Springer-Verlag, pp. Beyer, W. Coxeter, H. Geometry Revisited. Washington, DC: Math. Harris, J. Jeffrey, A. Orlando, FL: Academic Press, pp. Yates, R. Ann Arbor, MI: J. Edwards, pp. Zwillinger, D. Weisstein, Eric W.

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