Beta function

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

The beta function was studied by Euler and Legendre and was given its name by Jacques Binet; its symbol Β is a Greek capital beta.

A key property of the beta function is its close relationship to the gamma function: one has that[1]

The beta function is also closely related to binomial coefficients. When x (or y, by symmetry) is a positive integer, it follows from the definition of the gamma function Γ that[2]

The stated identity may be seen as a particular case of the identity for the integral of a convolution. Taking

for large x and large y. If on the other hand x is large and y is fixed, then

The integral defining the beta function may be rewritten in a variety of ways, including the following:

The beta function satisfies several identities analogous to corresponding identities for binomial coefficients, including a version of Pascal's identity

Evaluations at particular points may simplify significantly; for example,

Euler's integral for the beta function may be converted into an integral over the Pochhammer contour C as

This Pochhammer contour integral converges for all values of α and β and so gives the analytic continuation of the beta function.

Just as the gamma function for integers describes factorials, the beta function can define a binomial coefficient after adjusting indices:

Moreover, for integer n, Β can be factored to give a closed form interpolation function for continuous values of k:

Interestingly, their integral representations closely relate as the definite integral of trigonometric functions with product of its power and multiple-angle:[6]

The incomplete beta function, a generalization of the beta function, is defined as

For x = 1, the incomplete beta function coincides with the complete beta function. The relationship between the two functions is like that between the gamma function and its generalization the incomplete gamma function.

The regularized incomplete beta function (or regularized beta function for short) is defined in terms of the incomplete beta function and the complete beta function:

The beta function can be extended to a function with more than two arguments:

This multivariate beta function is used in the definition of the Dirichlet distribution. Its relationship to the beta function is analogous to the relationship between multinomial coefficients and binomial coefficients.

The beta function is useful in computing and representing the scattering amplitude for Regge trajectories. Furthermore, it was the first known scattering amplitude in string theory, first conjectured by Gabriele Veneziano. It also occurs in the theory of the preferential attachment process, a type of stochastic urn process. The beta function is also important in statistics, e.g. for the Beta distribution and Beta prime distribution. As briefly alluded to previously, the beta function is closely tied with the gamma function and plays an important role in calculus.

Even if unavailable directly, the complete and incomplete beta function values can be calculated using functions commonly included in spreadsheet or computer algebra systems. In Excel, for example, the complete beta value can be calculated from the GammaLn function:

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Value = BetaDist(x, a, b) * Exp(GammaLn(a) + GammaLn(b) − GammaLn(a + b))