On the \(n\)-dimensional inversion Laplace transform of retarded, Lorentz invariant functions (Q1260872)

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 \(r\)-th-order derivative of the one- dimensional Laplace transform. This formula can be understood as a generalization of the one-dimensional formula due to \textit{D. V. Widder} [Trans. Am. Math. Soc. 36, 107-200 (1934; Zbl 0008.30603)]. 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. Our main theorem can be related to a result due to \textit{E. L. Post} [ibid. 32, 723-781 (1930; JFM 56.0349.01)] and we also obtain an equivalent Leray formula which expresses the Laplace transform of retarded, Lorentz invariant functions by means of the \(m\)-th-order derivative of a \(K_ 0\)-transform.