The following treatment of integral transforms in applied mathematics is directed primarily toward senior and graduate students in engineering and applied science. It assumes a basic knowledge of complex variables and contour integration, gamma and Bessel functions, partial differential equations, and continuum mechanics. Examples and exercises are drawn from the fields of electric circuits, mechanical vibration and wave motion, heat conduction, and fluid mechanics. It is not essential that the student have a detailed familiarity with all of these fields, but knowledge of at least some of them is important for motivation (terms that may be unfamiliar to the student are listed in the Glossary, p. 89). The unstarred exercises, including those posed parenthetically in the text, form an integral part of the treatment; the starred exercises and sections are rather more difficult than those that are unstarred.

I have found that all of the material, plus supplementary material on asymptotic methods, can be covered in a single quarter by first-year graduate students (the minimum preparation of these students includes the equivalent of one-quarter courses on each of complex variables and partial differential equations); a semester allows either a separate treatment of contour integration or a more thorough treatment of asymptotic methods. The material in Chapter 4 and Sections 5.5 through 5.7 could be omitted in an undergraduate course for students with an inadequate knowledge of Bessel functions.

The exercises and, with a few exceptions, the examples require only those transform pairs listed in the Tables in Appendix 2.

Scope and purpose of the book

This book grew out of an undergraduate course in the University of Manchester, in which the author attempted to expound the most useful facts about Fourier series and integrals. It seemed to him on planning the course that a satisfactory account must make use of functions like the delta function of Dirac which are outside the usual scope of function theory. Now, Laurent Schwartz in his Théorie des distributions* has evolved a rigorous theory of these, while Professor Temple has given a version of the theory (Generalised functions) which appears to be more readily intelligible to students. With some slight further simplifications the author found that the theory of generalised functions was accessible to undergraduates in their final year, and that it greatly curtails the labour of understanding Fourier transforms, as well as making available a technique for their asymptotic estimation which seems superior to previous techniques. This is an approach in which the theory of Fourier series appears as a special case, the Fourier transform of a periodic function being a ‘row of delta functions’.

The book which grew out of the course therefore covers not only the principal results concerning Fourier transforms and Fourier series, but also serves as an introduction to the theory of generalised functions, whose general properties as well as those useful in Fourier analysis are derived, simply but without any departure from rigorous standards of mathematical proof.