Complex-variable distribution theory for Laplace and z transforms

The author proposes a generalisation of the theory of generalised functions, also known as the theory of distributions, by extending the theory to include generalised functions of a complex variable, both in the complex plane associated with continuous-time functions and that with discrete-time functions. The generalisation provides, among others, mathematical justifications of the properties of recently introduced generalised Dirac-delta impulses, using the principles of distribution theory. Properties of generalised functions of a complex variable are explored both in the Laplace domain associated with continuous-time functions and the z domain associated with discrete-time functions. Shifting of distributions, scaling, derivation, convolution with distributions and convolution with ordinary functions are evaluated in Laplace and z domains. Three-dimensional generalisations of sequences leading to generalised impulses, and of test functions in Laplace and z domains are presented. New expanded Laplace and z transforms are obtained using the proposed generalisation.