Funciones hiperbolicas

The hyperbolic functions are:
* Hyperbolic sine:

* Hyperbolic cosine:

* Hyperbolic tangent:

* Hyperbolic cotangent:

* Hyperbolic secant:

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

* Hyperbolic cosine:

* Hyperbolic tangent:

* Hyperbolic cotangent:

* Hyperbolicsecant:

* Hyperbolic cosecant:

where i is the imaginary unit defined as i2 = −1.
The complex forms in the definitions above derive from Euler's formula.
Note that, 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 morecommon.
[edit] Useful relations

Hence:

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 issimilar to the Pythagorean trigonometric identity. One also has

for the other functions.
The hyperbolic tangent is the solution to the nonlinear boundary value problem[5]:

It can be shown that thearea under the curve of cosh x is always equal to the arc length:[6]

[edit] Inverse functions as logarithms

[edit] Derivatives

[edit] Standard Integrals
For a full list of integrals ofhyperbolic functions, see list of integrals of hyperbolic functions

Where C is the constant of integration.
[edit] Taylor series expressions
It is possible to express the above functions as Taylorseries:

The function sinh x has a Taylor series expression with only odd exponents for x. Thus it is an odd function, that is, −sinh x = sinh(−x), and sinh 0 = 0.

The function cosh x has a Taylorseries expression with only even exponents for x. Thus it is an even function, that is, symmetric with respect to the y-axis. The sum of the sinh and cosh series is the infinite series expression of...