The Inverse Laplace Transform and Analytic Pseudo-Differential Operators

By comparing the Laplace transform L with the differential operator D, we obtain a formula for the inverse Laplace transform Ly1 s 1rp.Vy1 cos pD.VL, where V is a unitary transformation operator. This helps us obtain an explicit spectral representation of L. Some applications of the above relation are dis - cussed.