Properties and Application of Beta Function

The Beta function was first studied by Euler and Legendre and was given its name by Jacques Bi-net just as the gamma function for integers describes Factorials, the gamma function can define a binomial coefficient after adjusting indices. 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 preferred attachment process, a type of stochastic urn process. The incomplete beta function is a generalization of the beta function that replaces the definite integral of the beta function with an indefinite integral. The situation is analogous to the incomplete gamma function being a generalization of the gamma function.