Funciones hiperbolicas

Hyperbolic function - Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Hyperbolic_function

Hyperbolic function
From Wikipedia, the free encyclopedia

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/ or /ˈθæn/[citation needed] ), 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")[1] and so on. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) formthe 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, and 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 takereal 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.[2] Riccati used Sc. and Cc. ([co]sinus circulare) to refer to circular functions and Sh. and Ch. ([co]sinus hyperbolico) torefer to hyperbolic functions. Lambert adopted the names but altered the abbreviations to what they are today.[3] The abbreviations sh and ch are still used in some other languages, like French and Russian.

A ray through the origin intercepts the hyperbola in the point , where is twice the area between the ray and the -axis. For points on the hyperbola below the -axis, the area is considerednegative (see animated version with comparison with the trigonometric (circular) functions).

Contents
1 Standard algebraic expressions 2 Useful relations 3 Inverse functions as logarithms 4 Derivatives 5 Standard Integrals 6 Taylor series expressions 7 Comparison with circular trigonometric functions 8 Relationship to the exponential function 9 Hyperbolic functions for complex numbers 10 Seealso 11 References 12 External links

Standard algebraic expressions
The hyperbolic functions are: Hyperbolic sine:

Hyperbolic cosine:

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Hyperbolic function - Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Hyperbolic_function

Hyperbolic tangent:

sinh, cosh and tanh

Hyperbolic cotangent:

Hyperbolic secant:

csch, sech and cothHyperbolic cosecant:

Hyperbolic functions can be introduced via imaginary circular angles: Hyperbolic sine:

Hyperbolic cosine:

Hyperbolic tangent:

Hyperbolic cotangent:

(a) cosh(x) is the average of ex and e−x

Hyperbolic secant:

Hyperbolic cosecant:

where i is the imaginary unit defined as i2 = −1. The complex forms in the definitions above derive from Euler's formula. Notethat, by convention, sinh2 x means (sinh x)2, not sinh(sinh x); similarly for the other hyperbolic functions when used with positive exponents. Another notation for the hyperbolic cotangent function is ctnh x, though coth x is far more common.

Useful relations

(b) sinh(x) is half the difference ofex and −e−x Hyperbolic functions (a) cosh and (b) sinh obtained using exponential x −xfunctions e and e

Hence:

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Hyperbolic function - Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Hyperbolic_function

It can be seen that cosh x and sech x are even functions; the others are odd functions.

Hyperbolic sine and cosine satisfy the identity

which is similar to the Pythagorean trigonometric identity. One also has

for the other...