A HISTORICAL DEVELOPMENT OF THE LAPLACE TRANSFORM IN MODERN OPERATIONAL CALCULUS WITH APPLICATIONS TO MATHEMATICS, PHYSICS, AND TECHNOLOGY

This thesis is a source book for the teacher of undergraduate mathematics and engineering, and contains: (1) A short biography of Oliver Heaviside and a natural development of the Operational Calculus with examples. This can serve as supplementary material to any course at the calculus or elementary differential equation level, and to any basic engineering course concerned with transients in mechanical and electrical systems. (2) An algebraic structure of the Laplace transform in relation to Heaviside's Operational Calculus based on Arthur Erdelyi's Operational Calculus and Generalized Functions and Jan Mikusinski's Operational Calculus. Erdelyi has given us an algebraic structure of the Operational Calculus in general, including some background material for the Laplace Transform. His approach is that of Mikusinski's, and is a fine example of connections between analysis and algebra. This work of Arthur Erdelyi, presented by him as Mikusinski's theory of convolution quotients, is interpreted as applied solely to the Laplace transform. This then is expanded to an extension of the Laplace transform to a class of mathematical functions called "locally integrable" which, in general, are not Laplace transformable via complex variable theory. Included in this algebra is a treatment of the Delta function and other impulse functions.;Included in this discussion of Heaviside's contributions is an account of the mathematical justification of Heaviside's methods by Bromwich who used contour integration techniques to justify Heaviside's results, and of Carson's introduction of the Laplace transform as a mathematical basis for most of Heaviside's methods in the area of the operational calculus. Reference material is presented which contains Laplace transform examples and comparisons with other integral transforms.;Although there are a number of mathematical justifications of the operational calculus available, it is this writer's view that none are as practical and as easy to use as Mikusinski's theory of convolution quotients as presented by Erdelyi and incorporated into the thesis of the writer. Furthermore, as presented in this thesis, Laplace theory and Mikusinski's theory are shown to be completely compatible with each other, with the latter viewed as an extension of the former, resulting in an approach to the operational calculus that increases the utilitarian value of both theories.;At the end of this thesis are suggestions as to where and how various sections of it can be used with no additional mathematical preparation required on the part of the undergraduate student in typical existing undergraduate courses in mathematics and engineering.;Most undergraduate teachers of elementary differential equations are aware of the Laplace transform based operational calculus as used by scientists and engineers to solve linear differential equations. However, they may not be familiar with Heaviside's contributions in this area, nor familiar with Mikusinski's theory of convolution quotients. This thesis is written with this teacher in mind, and suggestions offered in the use of this thesis as a source book.