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An Introduction to Laplace Transforms and Fourier Series

Phil Dyke
Publication Date: 
Number of Pages: 
Springer Undergraduate Mathematics Series
[Reviewed by
Allen Stenger
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This is an awkwardly-positioned introductory text on Laplace transforms, that also includes some Fourier analysis, differential equations, and complex analysis material. It is aimed at second-year undergraduates, and assumes little beyond the techniques of calculus.

The Preface (p. vii) says that the book’s goal is to use the Laplace transform “as a particularly good vehicle for exhibiting fundamental ideas such as a mapping, linearity, an operator, a kernel and an image.” This is probably overly-ambitious for second-year students, and in any case the book doesn’t follow through: most of these topics are not mentioned explicitly, and are not explained when they are. The treatment is very much old-school cookbookery, where we go through the formalism to get useful results but don’t investigate conditions on their validity.

The narrative is chatty and easy to follow. There are a reasonable number of exercises, of reasonable difficulty. A feature that many students will appreciate is that the back of the book contains complete worked solutions to all exercises.

The first three chapters cover ordinary differential equations and Laplace transforms, and the next three chapters cover partial differential equations and Fourier series and transforms. This material is very classical, and appears in many books of mathematical physics and engineering mathematics in nearly the same form as it is here (including the same applications, taken from mechanics and electricity). One weakness of the present book’s approach is that it does not develop the mathematical models or differential equations, but takes them as given and solves them.

There are two additional chapters that seem to be tacked on. The last chapter covers complex analysis and its use to invert Laplace transforms. This is also very classical, although it seems out of place here because in applications we just look up the inverse transform in a good table. The next-to-last chapter (and new to this second edition) is on wavelets. This is not classical, although wavelets have now moved into the mainstream and there is good coverage in newer Fourier analysis books. The coverage in the present book is good as far as it goes, but that’s not very far: there are no applications, and only Haar wavelets are covered. For an introductory book I like Pereyra & Ward’s 2012 Harmonic Analysis: From Fourier to Wavelets, although its prerequisites are more substantial than the present book’s.

Bottom line: easy to follow, but limited and not very original. The useful material here would usually not be a whole course in itself, but would be covered as chapters in a larger applied-math book, such as Kreyszig’s Advanced Engineering Mathematics. Those books usually also develop the models rather than taking them for granted.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.