Laplace's Integral, the Gamma Function, and Beyond

Abstract

It is impossible to overplay this function's pervasive role in mathematics. From its humble origin as a complex-valued generalization of the shifted factorial function to its more sophisticated guise as the Mellin transform of e-x, the gamma function surfaces in the study of special functions everywhere. On the other hand it can be argued, following Weierstrass, that the reciprocal of the gamma function is a more natural player, one equally deserving of a central part. In this article we focus on Laplace's basic result and its numerous ramifications. Our first goal is to provide two simple proofs of his identity. We avoid popular techniques: messy contour integration (the kind that deals with branch points) and the theory of Fourier transforms, which invariably invokes the Fourier inversion formula. The only facts about the gamma function that we use follow from Euler's definition. These are its recurrence relation