The Fourier Transform and an Inversion Formula for Laplace Transforms

摘要

In the present paper, we prove that the imaginary part of the analytic continuation of a double Laplace transform to the negative axis coincides, up to a constant, with the function to which this double Laplace transformation was applied (Corollary 1 of Theorem 1) for a very general class of functions Z(x): if r(s) = LL(Z( ? ))(s), L(Z( ?))(s) = {_0~∞ e~(sx) Z(x) dx, s ∈ (0. ∞), then πZ(s) = - Im r_(An)(-s), r_(An)(s) = r(s), s ∈ (0, +∞), where r_(An)(p),p ∈ C, denotes the analytic continuation of the function r(p), p ∈ D = {p : Rep > 0}, to the left half-plane with violation of analyticity at, perhaps, a finite number of points. Such an analytic continuation (analytic continuation to the upper half-plane) is assumed to exist and the function r_(an)(p) is analytic on the whole real axis with the possible exception of zero.