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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.
Pages in category "Hyperbolic functions" The following 25 pages are in this category, out of 25 total. This list may not reflect recent changes. ...
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.
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.
Hyperbolic angle is used as the independent variable for the hyperbolic functions sinh, cosh, and tanh, because these functions may be premised on hyperbolic analogies to the corresponding circular (trigonometric) functions by regarding a hyperbolic angle as defining a hyperbolic triangle.
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 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 ...
Hyperbolic may refer to: of or pertaining to a hyperbola, a type of smooth curve lying in a plane in mathematics Hyperbolic geometry, a non-Euclidean geometry; Hyperbolic functions, analogues of ordinary trigonometric functions, defined using the hyperbola; of or pertaining to hyperbole, the use of exaggeration as a rhetorical device or figure ...