The Beta function was Örst studied by Euler and Legendre and was given its name by Jacques Binet. Just as the gamma function for integers describes factorials, the beta function can deÖne a binomial coe¢ - cient after adjusting indices.The beta function was the Örst known scattering amplitude in string theory,Örst conjectured by Gabriele Veneziano. It also occurs in the theory of the preferential attachment process, a type of stochastic urn process.The incomplete beta function is a generalization of the beta function that replaces the deÖnite integral of the beta function with an indeÖnite integral.The situation is analogous to the incomplete gamma function being a generalization of the gamma function.

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