Simultaneous double transformations of functions depending on space and time

Abstract

It is shown that performing simultaneously two transformations on functions of space and time (for instance a Fourier transform on the space variable and a Laplace transform on the time variable) can be easier than performing them one after the other when the variables are combined in invariant quantities. This is naturally also true when performing two inverse transforms simultaneously, when the conjugated variables are combined into a propagator. An immediate application is found in the computation of the solutions of partial differential equations. This article contains several general examples of such "simultaneous double transforms" for arbitrary analytic functions of space and time.