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The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic functions that does not involve complex numbers. The graph of the function a cosh( x / a ) is the catenary , the curve formed by a uniform flexible chain, hanging freely between two fixed points under uniform gravity.
The argument to the hyperbolic functions is a hyperbolic angle measure. In mathematics, the inverse hyperbolic functions are inverses of the hyperbolic functions, analogous to the inverse circular functions. There are six in common use: inverse hyperbolic sine, inverse hyperbolic cosine, inverse hyperbolic tangent, inverse hyperbolic cosecant ...
Many other mathematical objects have their origin in the hyperbola, such as hyperbolic paraboloids (saddle surfaces), hyperboloids ("wastebaskets"), hyperbolic geometry (Lobachevsky's celebrated non-Euclidean geometry), hyperbolic functions (sinh, cosh, tanh, etc.), and gyrovector spaces (a geometry proposed for use in both relativity and ...
The following is a list of integrals (anti-derivative functions) of hyperbolic functions. For a complete list of integral functions, see list of integrals. In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration.
The Gudermannian function is a sigmoid function, and as such is sometimes used as an activation function in machine learning. The (scaled and shifted) Gudermannian function is the cumulative distribution function of the hyperbolic secant distribution. A function based on the Gudermannian provides a good model for the shape of spiral galaxy arms ...
The hyperbolastic rate equation of type II, denoted by H2, is defined as = (() ()),where is the hyperbolic tangent function, is the carrying capacity, and both and > jointly determine the growth rate.
For each inverse hyperbolic integration formula below there is a corresponding formula in the list of integrals of inverse trigonometric functions. The ISO 80000-2 standard uses the prefix "ar-" rather than "arc-" for the inverse hyperbolic functions; we do that here.
Sigmoid functions most often show a return value (y axis) in the range 0 to 1. Another commonly used range is from −1 to 1. A wide variety of sigmoid functions including the logistic and hyperbolic tangent functions have been used as the activation function of artificial neurons.