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CF. for. In probability theory and statistics, the hyperbolic secant distribution is a continuous probability distribution whose probability density function and characteristic function are proportional to the hyperbolic secant function. The hyperbolic secant function is equivalent to the reciprocal hyperbolic cosine, and thus this distribution ...
The hyperbolic functions represent an expansion of trigonometry beyond the circular functions. Both types depend on an argument, either circular angle or hyperbolic angle . Since the area of a circular sector with radius r and angle u (in radians) is r2u/2, it will be equal to u when r = √2. In the diagram, such a circle is tangent to the ...
Twice the area of the purple triangle is the stereographic projection s = tan 1 / 2 ϕ = tanh 1 / 2 ψ. The blue point has coordinates (cosh ψ, sinh ψ). The red point has coordinates (cos ϕ, sin ϕ). The purple point has coordinates (0, s). The integral of the hyperbolic secant function defines the Gudermannian function:
Alternately hyperbolic angle is the area of a sector of the hyperbola Some authors call the inverse hyperbolic functions hyperbolic area functions. [ 1] Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. It also occurs in the solutions of many linear differential equations (such as the equation ...
More detail may be found on the following pages for the lists of integrals : Gradshteyn, Ryzhik, Geronimus, Tseytlin, Jeffrey, Zwillinger, and Moll 's (GR) Table of Integrals, Series, and Products contains a large collection of results. An even larger, multivolume table is the Integrals and Series by Prudnikov, Brychkov, and Marichev (with ...
In probability theory and statistics, the logistic distribution is a continuous probability distribution. Its cumulative distribution function is the logistic function, which appears in logistic regression and feedforward neural networks. It resembles the normal distribution in shape but has heavier tails (higher kurtosis ).
A sigmoid function is any mathematical function whose graph has a characteristic S-shaped or sigmoid curve . A common example of a sigmoid function is the logistic function shown in the first figure and defined by the formula: [ 1] Other standard sigmoid functions are given in the Examples section. In some fields, most notably in the context of ...
In mathematics, the Korteweg–De Vries (KdV) equation is a partial differential equation (PDE) which serves as a mathematical model of waves on shallow water surfaces. It is particularly notable as the prototypical example of an integrable PDE and exhibits many of the expected behaviors for an integrable PDE, such as a large number of explicit ...