Defintion

Definition of Hyperbolic Functions

Hyperbolic functions are analogue to the circular trigonometric functions, but related to the hyperbola rather than the circle.
The hyperbolic sine function, or sinh, is the function defined as sinh(x) = 0.5(e^x - e^-x).
The hyperbolic cosine function, or cosh, is the function defined as cosh(x) = 0.5(e^x + e^-x).
The hyperbolic tangent function, or tanh, is defined as the ratio of the hyperbolic sine to the hyperbolic cosine: tanh(x) = sinh(x) / cosh(x).
The hyperbolic functions are all functions of real variables.
The hyperbolic functions often appear in solutions to differential equations and geometry problems involving hyperbolic shapes.
The hyperbolic functions, much like the circular trigonometric functions, have similar identities that relate them to each other.

The function sinh(x) is odd; that is, sinh(-x) = -sinh(x).
The function cosh(x) is even; that is, cosh(-x) = cosh(x).
The function tanh(x) is odd.
For all real values of x, cosh^2(x) - sinh^2(x) = 1. This is similar to the Pythagorean identity for the circular functions, cos^2(t) + sin^2(t) = 1.
The graphs of hyperbolic functions are hyperbolae.
The functions sinh(x) and cosh(x) are inverses of each other.

Hyperbolic functions can be combined and manipulated just like any other functions to simplify or rearrange expressions.
The derivatives and integrals of the hyperbolic functions can be found using the standard rules of calculus.
Hyperbolic functions are particularly useful in integration due to the identities that enable integration by substitution and by parts.