Properties
The beta function is symmetric, meaning that
When x and y are positive integers, it follows trivially from the definition of the gamma function that:
It has many other forms, including:
where is a truncated power function and the star denotes convolution. The second identity shows in particular . Some of these identities, e.g. the trigonometric formula, can be applied to deriving the volume of an n-ball in Cartesian coordinates.
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, can be integrated to give a closed form, an interpolation function for continuous values of k:
The beta function 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.
Read more about this topic: Beta Function
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