HYPERBOLIC FUNCTIONS

HYPERBOLIC FUNCTIONS

A ray through the unit hyperbola in the point , where is twice the area between the ray, the hyperbola, and the -axis. For points on the hyperbola below the -axis, the area is considered negative (see

animated version with comparison with the trigonometric (circular) functions).

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/ ), [1] and the hyperbolic cosine "cosh" (/ ˈkɒʃ/ ),[2] from which are derived the hyperbolic tangent "tanh" ( / ˈtæntʃ/ or /ˈθæn/ ), [3] hyperbolic cosecant "csch" or "cosech" (/ ˈkoʊʃɛk/ [2] or /

ˈkoʊsɛtʃ/ ), hyperbolic secant "sech" ( /ˈʃɛk/ or

/ ˈsɛtʃ/ ),[4] and hyperbolic cotangent "coth" (/

ˈkoʊθ/ or / ˈkɒθ/ ), [5][6] corresponding to the derived trigonometric functions.

The inverse hyperbolic functions are the area hyperbolic sine "arsinh" (also called "asinh" or sometimes "arcsinh") [7][8] 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 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 .

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. [9] 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. [10] The abbreviations sh and ch are still used in some other languages, like French and Russian.