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), the derivatives of sinh(t) and cosh(t) are cosh(t) and sinh(t). Hyperbolic functions are used to express the angle of parallelism in hyperbolic geometry. They are used to express Lorentz boosts as hyperbolic rotations in special relativity. 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, and fluid dynamics. The basic hyperbolic functions are:
hyperbolic sine "sinh" (), hyperbolic cosine "cosh" (), from which are derived:
hyperbolic tangent "tanh" (), hyperbolic cotangent "coth" (), hyperbolic secant "sech" (), hyperbolic cosecant "csch" or "cosech" () corresponding to the derived trigonometric functions. The inverse hyperbolic functions are:
inverse hyperbolic sine "arsinh" (also denoted "sinh−1", "asinh" or sometimes "arcsinh") inverse hyperbolic cosine "arcosh" (also denoted "cosh−1", "acosh" or sometimes "arccosh") inverse hyperbolic tangent "artanh" (also denoted "tanh−1", "atanh" or sometimes "arctanh") inverse hyperbolic cotangent "arcoth" (also denoted "coth−1", "acoth" or sometimes "arccoth") inverse hyperbolic secant "arsech" (also denoted "sech−1", "asech" or sometimes "arcsech") inverse hyperbolic cosecant "arcsch" (also denoted "arcosech", "csch−1", "cosech−1","acsch", "acosech", or sometimes "arccsch" or "arccosech")
The hyperbolic functions take an argument called a hyperbolic angle. The magnitude of a hyperbolic angle is the area of its hyperbolic sector to xy = 1. 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.
History The first known calculation of a hyperbolic trigonometry problem is attributed to Gerardus Mercator when issuing the Mercator map projection circa 1566. It requires tabulating solutions to a transcendental equation involving hyperbolic functions. The first to suggest a similarity between the sector of the circle and that of the hyperbola was Isaac Newton in his 1687 Principia Mathematica. Roger Cotes suggested to modify the trigonometric functions using the imaginary unit i = − 1 {\displaystyle i={\sqrt {-1}}} to obtain an oblate spheroid from a prolate one. Hyperbolic functions were formally introduced in 1757 by Vincenzo Riccati. 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. As early as 1759, Daviet de Foncenex showed the interchangeability of the trigonometric and hyperbolic functions using the imaginary unit and extended de Moivre's formula to hyperbolic functions. During the 1760s, Johann Heinrich Lambert systematized the use functions and provided exponential expressions in various publications. Lambert credited Riccati for the terminology and names of the functions, but altered the abbreviations to those used today.
Notation
Definitions
With hyperbolic angle u, the hyperbolic functions sinh and cosh can be defined with the exponential function eu. In the figure
A = ( e − u , e u ) , B = ( e u , e − u ) , O A + O B = O C {\displaystyle A=(e^{-u},e^{u}),\ B=(e^{u},\ e^{-u}),\ OA+OB=OC} .
Exponential definitions
Hyperbolic sine: the odd part of the exponential function, that is, sinh x = e x − e − x 2 = e 2 x − 1 2 e x . {\displaystyle \sinh x={\frac {e^{x}-e^{-x}}{2}}={\frac {e^{2x}-1}{2e^{x}}}.}
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