A survey on the Mellin transform and its applicationsA survey on the Mellin transform and its Applications

Abstract

The Mellin transform is one of the most popular transforms which was initiated by Hjal- mar Mellin in the end of the nineteenth century. D. E. Knuth, together with De Bruijn, introduced it in the orbit of discrete mathematics in the mid 60's, however, Flajolet's school systematized and applied the Mellin transform to a myriad problems of analytic combinatorics and analysis of algorithms. Recently, the Mellin transform has found its way into information theory. The popularity of this transform stems from two properties. It allows the reduction of certain functional equations to algebraic ones, and it provides a direct mapping between asymptotic expansions of a function near zero or infinity and the set singularities of the transform in the complex analysis. The latter asymptotic property is crucial for applications in discrete mathematics, analysis of algorithms, analytic combinatorics and analytic information theory (see [5, 6, 7, 16]). The main idea in this project is to give a little survey on the Mellin transform and its applications