Re-writing Laplace and z transforms

A generalization of the Dirac-delta impulse is shown to extend the domain of convergence of Laplace and z transforms to functions that had hitherto had no transform. An historic anomaly is in fact revealed and a a solution thereof proposed. Novel generalized spectral analysis transforms and power spectra are shown to call for such a generalization of the Dirac-delta impulse. The suggested generalization, extending considerably the domains of convergence of Laplace and z transform domains calls for the re-writing of Laplace and z transforms. Transforms of basic Jitnctions such as a two-sided infinite duration exponential now exist. Transforms of such basic functions such as the Heaviside unit step function is now re-written to be a true generalization of the Fourier transform. Bilateral Laplace and z transform are now given their true power and importance, leading no doubt to a new interest and intensive activity in their theory and to their applications in most scientific and engineering domains. A complex-variable Distribution Theory for Laplace and z transforms recently proposed is briefly introduced.