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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
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 ...
The Euler integral of the second kind is the gamma function [2] = For positive integers m and n , the two integrals can be expressed in terms of factorials and binomial coefficients : B ( n , m ) = ( n − 1 ) !
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.
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 ...
In mathematical logic, Gödel's β function is a function used to permit quantification over finite sequences of natural numbers in formal theories of arithmetic. The β function is used, in particular, in showing that the class of arithmetically definable functions is closed under primitive recursion, and therefore includes all primitive recursive functions.
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The Beta function, also called the Euler beta function or the Euler integral of the first kind, is a special function in mathematics. Beta function may also refer to: Beta function (physics), details the running of the coupling strengths; Dirichlet beta function, closely related to the Riemann zeta function