Mikusiński Operators

Abstract

The development of the Laplace transformation in the preceding chapter does not directly allow us to include such useful ideas as impulses, not even the Dirac delta function. In order to overcome some of these difficulties, we introduce the operational calculus of Mikusiński. This operational calculus is closely connected with the Laplace transformation, since it is based upon the convolution. It gives us a model for the development of any other, similar, operational calculi. In a sense it generalizes the Laplace transformation and it provides an introduction to generalized functions. We develop some of the theory in this chapter and then provide examples of applications to differential and integral equations. The development of the theory parallels completely the development of the rational numbers by construction from the integers. It is this parallel which places the work on a firm theoretical basis.