On the N-Dimensional Inversion Laplace Transform of Retarded, Lorentz Invariant Functions

Abstract

The purpose of this paper is to obtain n-dimensional inversion Laplace transform of retarded, Lorentz invariant functions by means of the passage to the limit of the rth-order derivative of the one-dimensional Laplace transform. This formula (IV.2) can be understood as a generalization of the one-dimensional formula due to Widder [Trans. Amer. Math. Soc.32 (1930)]. This topic is intimately related to the generalized differentiation, the symbolic treatment of the differential equations with constant coefficients and its application to important physical problems (cf. Leibnitz, Pincherle, Liouville, Riemann, Boole, 1-leaviside, and others). Our main theorem (Theorem 15, formula (IV.2)) can be related to a result due to E. Post [6] and we also obtain an equivalent Leray′s formula (cf. (VI.I)) and (VI.2)) which expresses the Laplace transform of retarded, Lorentz invariant functions by means of the mth-order derivative of a K0-transform. Our method consists, essentially, in the following two steps. First: the obtainment of an analog of Bochner′s formula for a Laplace transform of the form (II.1), where φ is a function of the Lorentz distance, whose support is contained in the closure of the domain t0 > 0, t20 − t21 − ··· −t2n − 1 > 0. Formula (II.2) permits us to evaluate n-dimensional integrals by means of a one-dimensional K-transform. This last result was already employed to solve partial differential equations of the hyperbolic type (cf. [A. Gonzalez Dominguez and E. E. Trione, Adv. Math.31 (1979), 51-62]). Second: The passage to the limit of the rth-order derivative of the one-dimensional Laplace transform (via the K-transform). The previous conclusions are related to the classical Functional Analysis and Probability (i.e., the theory of moments, the classical Weierstrass theorem approximation, on compact sets, of continuous functions by polynomials and the inversion of Laplace-Stieltjes integrals). Finally, by appealing to the analytical continuation, we can extend our results to the distributional n-dimensional Laplace integrals.