Hyperbolic functions

In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than 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 unit hyperbola. Also, similarly to how the derivatives of sin(t) and cos(t) are cos(t) and –sin(t) respectively, the derivatives of sinh(t) and cosh(t) are cosh(t) and +sinh(t) respectively.

Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. They also occur in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and 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 basic hyperbolic functions are:^[1]

• hyperbolic sine "sinh" (/ˈsɪŋ, ˈsɪntʃ, ˈʃaɪn/),^[2]
• hyperbolic cosine "cosh" (/ˈkɒʃ, ˈkoʊʃ/),^[3]

from which are derived:^[4]

• hyperbolic tangent "tanh" (/ˈtæŋ, ˈtæntʃ, ˈθæn/),^[5]
• hyperbolic cotangent "coth" (/ˈkɒθ, ˈkoʊθ/),^[6][7]
• hyperbolic secant "sech" (/ˈsɛtʃ, ˈʃɛk/),^[8]
• hyperbolic cosecant "csch" or "cosech" (/ˈkoʊsɛtʃ, ˈkoʊʃɛk/^[3])

corresponding to the derived trigonometric functions.

The inverse hyperbolic functions are:

• area hyperbolic sine "arsinh" (also denoted "sinh⁻¹", "asinh" or sometimes "arcsinh")^[9][10][11]
• area hyperbolic cosine "arcosh" (also denoted "cosh⁻¹", "acosh" or sometimes "arccosh")
• area hyperbolic tangent "artanh" (also denoted "tanh⁻¹", "atanh" or sometimes "arctanh")
• area hyperbolic cotangent "arcoth" (also denoted "coth⁻¹", "acoth" or sometimes "arccoth")
• area hyperbolic secant "arsech" (also denoted "sech⁻¹", "asech" or sometimes "arcsech")
• area hyperbolic cosecant "arcsch" (also denoted "arcosech", "csch⁻¹", "cosech⁻¹","acsch", "acosech", or sometimes "arccsch" or "arccosech")

The hyperbolic functions take a real argument called a hyperbolic angle. The size of a hyperbolic angle is twice the area of its hyperbolic sector. The hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector.

In complex analysis, the hyperbolic functions arise when applying the ordinary sine and cosine functions to an imaginary angle. The hyperbolic sine and the hyperbolic cosine are entire functions. As a result, the other hyperbolic functions are meromorphic in the whole complex plane.

By Lindemann–Weierstrass theorem, the hyperbolic functions have a transcendental value for every non-zero algebraic value of the argument.^[12]

Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert.^[13] Riccati used Sc. and Cc. (sinus/cosinus circulare) to refer to circular functions and Sh. and Ch. (sinus/cosinus hyperbolico) to refer to hyperbolic functions. Lambert adopted the names, but altered the abbreviations to those used today.^[14] The abbreviations sh, ch, th, cth are also currently used, depending on personal preference.