Special Transformations

Abstract

In functional analysis, the Laplace transform is a powerful technique for analyzing linear time-invariant systems. In actual, physical systems, the Laplace transform is often interpreted as a transformation from the time-domain point of view, in which inputs and outputs are understood as functions of time, to the frequency-domain point of view, where the same inputs and outputs are seen as functions of complex angular frequency, or radians per unit time. This transformation not only provides a fundamentally different way to understand the behavior of the system, but it also drastically reduces the complexity of the mathematical calculations required to analyze the system. The Laplace transform has many important Operations Research applications as well as applications in control engineering, physics, optics, signal processing and probability theory. The Laplace transform is used to analyze continuous-time systems whereas its discrete-time counterpart is the Z transform. The Z transform among other applications is used frequently in discrete probability theory and stochastic processes, combinatorics and optimization. In this chapter, we will present an overview of these transformations from differential/difference equation systems’ viewpoint.