The hyperbolic functions are similar to the trigonometric functions, often called circular functions, in that they play an important role in problems in mathematics and the mathematics of physics. The hyperbolic functions are involved in mathematical problems in which integrals involving (1 + x²) arise whereas trigonometric functions involve functions with integrals of the type (1 - x²). Although the hyperbolic functions are similar to trigonometric functions, their definitions are much more straightforward. As the trigonometric functions yield parameters describing the unit circle, the hyperbolic functions yield parameters describing the standard hyperbola: x² - y² = 1 for x > 1. For every trigonometric function there is a corresponding hyperbolic function, though they are not necessarily identical.

Like the trigonometric functions the hyperbolic functions are periodic because they are all elliptic functions. Elliptic functions can be doubly periodic, singly periodic, or non-periodic, which is often referred to as trivially periodic. The exponential function...