MellinTransform 2008-05-14 18:11:37 2008-10-30 08:10:23 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here The \emph{Mellin transform} is an integral transform defined as follows: \[ F(s) = \int_0^\infty f(t) t^{s-1} \, dt \] Intuitively, it may be viewed as a continuous analogue of a power series --- instead of synthetizing a function by summing multiples of integer powers, we integrate over all real powers. This transform is closely related to the Laplace transform --- if we make a change of variables $t = e^{-r}$ and define $g$ by $f(e^{-r}) = g(r)$, then the above integral becomes \[ F(s) = -\int_{-\infty}^{+\infty} g(r) e^{-rs} \, dr , \] which is a bilateral Laplace transform. (more to come)