P.P.G. Dyke - An Introduction to Laplace Transforms and Fourier Series

School of Computing and Mathematics, Plymouth University, Plymouth, UK

Phil Dyke

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As a discipline, mathematics encompasses a vast range of subjects. In pure mathematics an important concept is the idea of an axiomatic system whereby axioms are proposed and theorems are proved by invoking these axioms logically. These activities are often of little interest to the applied mathematician to whom the pure mathematics of algebraic structures will seem like tinkering with axioms for hours in order to prove the obvious. To the engineer, this kind of pure mathematics is even more of an anathema. The value of knowing about such structures lies in the ability to generalise the obvious to other areas. These generalisations are notoriously unpredictable and are often very surprising. Indeed, many say that there is no such thing as non-applicable mathematics, just mathematics whose application has yet to be found.

The Laplace transform expresses the conflict between pure and applied mathematics splendidly. There is a temptation to begin a book such as this on linear algebra outlining the theorems and properties of normed spaces. This would indeed provide a sound basis for future results. However most applied mathematicians and all engineers would probably turn off. On the other hand, engineering texts present the Laplace transform as a toolkit of results with little attention being paid to the underlying mathematical structure, regions of validity or restrictions. What has been decided here is to give a brief introduction to the underlying pure mathematical structures, enough it is hoped for the pure mathematician to appreciate what kind of creature the Laplace transform is, whilst emphasising applications and giving plenty of examples. The point of view from which this book is written is therefore definitely that of the applied mathematician. However, pure mathematical asides, some of which can be quite extensive, will occur. It remains the view of this author that Laplace transforms only come alive when they are used to solve real problems. Those who strongly disagree with this will find pure mathematics textbooks on integral transforms much more to their liking.

The main area of pure mathematics needed to understand the fundamental properties of Laplace transforms is analysis and, to a lesser extent the normed vector space. Analysis, in particular integration, is needed from the start as it governs the existence conditions for the Laplace transform itself; however as is soon apparent, calculations involving Laplace transforms can take place without explicit knowledge of analysis. Normed vector spaces and associated linear algebra put the Laplace transform on a firm theoretical footing, but can be left until a little later in a book aimed at second year undergraduate mathematics students.

The definition of the Laplace transform could hardly be more straightforward. Given a suitable function the Laplace transform, written is defined by This bald statement may satisfy most engineers, but not mathematicians. The question of what constitutes a suitable function will now be addressed. The integral on the right has infinite range and hence is what is called an improper integral. This too needs careful handling. The notation is used to denote the Laplace transform of the function .

Another way of looking at the Laplace transform is as a mapping from points in the domain to points in the domain. Pictorially, Fig. indicates this mapping process.

The time domain will contain all those functions whose Laplace transform exists, whereas the frequency domain contains all the images . Another aspect of Laplace transforms that needs mentioning at this stage is that the variable often has to take complex values. This means that is a function of a complex variable, which in turn places restrictions on the (real) function given that the improper integral must converge. Much of the analysis involved in dealing with the image of the function in the plane is therefore complex analysis which may be quite new to some readers.

As has been said earlier, engineers are quite happy to use Laplace transforms to help solve a variety of problems without questioning the convergence of the improper integrals. This goes for some applied mathematicians too. The argument seems to be on the lines that if it gives what looks a reasonable answer, then fine. In our view, this takes the engineers maxim if it aint broke, dont fix it too far. This is primarily a mathematics textbook, therefore in this opening chapter we shall be more mathematically explicit than is customary in books on Laplace transforms. In Chap. there is some more pure mathematics when Fourier series are introduced. That is there for similar reasons. One mathematical question that ought to be asked concerns uniqueness. Given a function , its Laplace transform is surely unique from the well defined nature of the improper integral. However, is it possible for two different functions to have the same Laplace transform? To put the question a different but equivalent way, is there a function