What is "hyperbolic function"

n. (context mathematics English) A function that is derived from some arithmetic operations on the exponential function with base e and the inverse function, and was named after the corresponding similar trigonometric function.

In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular functions. The basic hyperbolic functions are the hyperbolic sine "sinh" ( or ), and the hyperbolic cosine "cosh" , from which are derived the hyperbolic tangent "tanh" ( or ), hyperbolic cosecant "csch" or "cosech" ( or ), hyperbolic secant "sech" ( or ), and hyperbolic cotangent "coth" ( or ), 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. 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.

Hyperbolic functions occur in 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.

In complex analysis, the hyperbolic functions arise as the imaginary parts of sine and cosine. When considered defined by a complex variable, the hyperbolic functions are rational functions of exponentials, and are hence holomorphic.

Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert. Riccati used Sc. and Cc. ([co]sinus circulare) to refer to circular functions and Sh. and Ch. ([co]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 French and Russian.