Mellin transform

Abstract

In contrast to Fourier and Laplace transformations that were introduced to solve physical problems, Mellin’s transformation arose in a mathematical context. In fact, the first occurrence of the transformation is found in a memoir by Riemann in which he used it to study the famous Zeta function. References concerning this work and its further extension by M. Cahen are given in Ref. [1]. However, it is the Finnish mathematician, R. H. Mellin (1854-1933), who was the first to give a systematic formulation of the transformation and its inverse. Working in the theory of special functions, he developed applications to the solution of hypergeometric differential equations and to the derivation of asymptotic expansions. The Mellin contribution gives a prominent place to the theory of analytic functions and relies essentially on Cauchy’s theorem and the method of residues. A biography of R. H. Mellin including a sketch of his works can be found in Ref. [2]. Actually, the Mellin transformation can also be placed in another framework, which in some respects conforms more closely to the original ideas of Riemann.