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In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral
In mathematics, there are two types of Euler integral: [1]. The Euler integral of the first kind is the beta function (,) = = () (+); The Euler integral of the second kind is the gamma function [2] =
For every odd positive integer +, the following equation holds: [3] (+) = ()!() +where is the n-th Euler Number.This yields: =,() =,() =,() =For the values of the Dirichlet beta function at even positive integers no elementary closed form is known, and no method has yet been found for determining the arithmetic nature of even beta values (similarly to the Riemann zeta function at odd integers ...
Beta functions are usually computed in some kind of approximation scheme. An example is perturbation theory , where one assumes that the coupling parameters are small. One can then make an expansion in powers of the coupling parameters and truncate the higher-order terms (also known as higher loop contributions, due to the number of loops in ...
the golden ratio 1.618... in mathematics, art, and architecture; Euler's totient function in number theory; the argument of a complex number in mathematics; the value of a plane angle in physics and mathematics; the angle to the z axis in spherical coordinates (mathematics) epoch or phase difference between two waves or vectors
Selberg's formula implies Dixon's identity for well poised hypergeometric series, and some special cases of Dyson's conjecture. This is a corollary of Aomoto. This is a corollary of Aomoto. Aomoto's integral formula
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In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] or (0, 1) in terms of two positive parameters, denoted by alpha (α) and beta (β), that appear as exponents of the variable and its complement to 1, respectively, and control the shape of the distribution.