CHAPTER XIII - THE LAPLACE TRANSFORMATION

Abstract

This chapter discusses the Laplace transformation. In practice, the solution of linear differential equations describing transients in linear networks often presents a long and difficult problem, which becomes more difficult as the order of the equations becomes higher. The Laplace transformation or operational calculus is a combination of methods of mathematical analysis that considerably simplify and speed up the process of calculating, by algebraicizing differential equations and subsequently interpreting the results obtained. The idea of the Laplace transformation consists in transferring a solution from the region of the functions of a real variable, the time domain, to the region of the functions of a complex variable, the complex frequency domain, where the operations are simplified. After the operations have been carried out on the functions of the complex variable, the solution is converted back to the time domain. The close relationship between the Laplace and Fourier transformations, thus, introduces a physical significance into the concept of the transform as a generalized form of spectrum. This affinity extends to the basic properties of the Laplace and Fourier transformations.