Hyperbolic Function

Hyperbolic Function

In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The basic hyperbolic functions are the hyperbolic sine "sinh" ( /ˈsɪntʃ/ or /ˈʃaɪn/), and the hyperbolic cosine "cosh" ( /ˈkɒʃ/), from which are derived the hyperbolic tangent "tanh" ( /ˈtæntʃ/), and so on, corresponding to the derived trigonometric functions. The inverse hyperbolic functions are the area hyperbolic sine "arsinh" (also called "asinh" or sometimes "arcsinh") and so on.

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 some important linear differential equations, for example the equation defining a catenary, of some cubic equations, and of Laplace's equation in Cartesian coordinates. The latter is important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity.

The hyperbolic functions take real values for a real argument called a hyperbolic angle. In complex analysis, they are simply rational functions of exponentials, and so are meromorphic.

Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert. Riccati used Sc. and Cc. (sinus circulare) to refer to circular functions and Sh. and Ch. (sinus hyperbolico) to refer to hyperbolic functions. Lambert adopted the names but altered the abbreviations to what they are today. The abbreviations sh and ch are still used in some other languages, like European French and Russian.