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

By P.P.G. Dyke

This complicated undergraduate/graduate textbook offers an easy-to-read account of Fourier sequence, wavelets and Laplace transforms. It gains many labored examples with all options supplied.

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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.

It is linearity that enables us to add results together to deduce other more complicated ones and is so basic that we state it as a theorem and prove it first. 1 (Linearity) If and are two functions whose Laplace transform exists, then where and are arbitrary constants. Proof where we have assumed that so that where This proves the theorem. Here we shall concentrate on those properties of the Laplace transform that do not involve the calculus. The first of these takes the form of another theorem because of its generality.

4 where the length of the arrow is unity. 4) with we see that which is consistent with the area under being unity. Fig. 3The “top hat” function Fig. 4The Dirac- function We now ask ourselves what is the Laplace transform of ? Does it exist? 4) with , a perfectly valid choice of gives However, we progress with care. This is good advice when dealing with generalised functions. Let us take the Laplace transform of the top hat function defined mathematically by The calculation proceeds as follows:- As hence which as .