# Straight talking on linearity

Fourier and Laplace Transforms. First edition
November 26, 2004

The importance of mathematical transforms lies in their power to describe, in easy-to-understand symbols, the precise response of certain classes of systems to arbitrary inputs. These systems may arise in applications ranging from engineering to social science. The sole constraint is that they must be linear, that is, the response to the sum of two inputs must be the sum of the responses to each input, and it helps if the system is homogeneous; transforms can easily describe the acoustics in a concert hall but may be messy for sound waves in an atmosphere where temperature varies appreciably.

Despite the linearity restriction, transform representations have long been fundamental for the basic scientific insights they bring. But nowadays they are especially valuable for the quality control they exert over the output of computational codes that may in principle work for more general nonlinear systems.

This text describes the basic transforms that are most useful today, whether they be continuous (suitable for differential equations) or discrete (suitable for signal processing). Each chapter is liberally punctuated with exercises and examples, and is prefaced with learning objectives and ends with a self-test. Lucidity for the novice scientist is the keynote, the necessary rigour being prescribed quite digestibly. Moreover, the ordering is right: first Fourier series, accessible to A-level students, then Fourier transforms, then Laplace transforms.

Alas, the beautiful mathematical unification of these methods is missing because the authors eschew the theory of integration of functions of a complex variable. Only with this increasingly unfashionable tool can a unified theory of transforms be presented. Here, its adoption would have obviated reliance on the unintuitive theory of generalised functions or distributions.

Had the authors been able to include complex integration as well, they could have concluded with a call to arms to mathematicians to better unify the “complex variable” and “generalised function” approaches, something that could have a far greater practical payoff than Fermat’s theorem.

John Ockendon is lecturer in mathematics, Oxford University.

## Fourier and Laplace Transforms. First edition

Author - R. J. Beerends, H. G. ter Morsche, J. C. van den Berg and E. M. van de Vrie
Publisher - Cambridge University Press
Pages - 447
Price - £32.50
ISBN - 0 521 53441 0

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