The Gamma and Beta Functions

Abstract

Euler discovered the gamma function, Γ( x ), when he extended the domain of the factorial function. Thus Γ( x ) is a meromorphic function equal to ( x − 1)! when x is a positive integer. The gamma function has several representations, but the two most important, found by Euler, represent it as an infinite integral and as a limit of a finite product. We take the second as the definition. Instead of viewing the beta function as a function, it is more illuminating to think of it as a class of integrals – integrals that can be evaluated in terms of gamma functions. We therefore often refer to beta functions as beta integrals. In this chapter, we develop some elementary properties of the beta and gamma functions. We give more than one proof for some results. Often, one proof generalizes and others do not. We briefly discuss the finite field analogs of the gamma and beta functions. These are called Gauss and Jacobi sums and are important in number theory. We show how they can be used to prove Fermat's theorem that a prime of the form 4 n + 1 is expressible as a sum of two squares. We also treat a simple multidimensional extension of a beta integral, due to Dirichlet, from which the volume of an n -dimensional ellipsoid can be deduced.