Even though the numerical computation of integrals is a topic that has been investigated for many centuries, current textbooks have failed to describe certain facets in detail. One of these missing areas is the special case of when the integrand oscillates rapidly. Deaño et al.’s book is a first significant step toward filling this gap.
The first two chapters provide an introduction to the problem at hand; explain the fundamental differences from the classical setting where the integrand does not oscillate strongly; and establish the analytical background on which the associated algorithms are based. In particular, this part contains explanations for why the classical approaches are usually unsuitable for this class of problems, and why one commonly considers the asymptotic behavior of the problem as the oscillation frequency increases and not as the number of quadrature nodes grows.
The next four chapters are then devoted to the motivation, description, and investigation of the four most practically important classes of methods, namely the Filon and Levin methods (chapter 3), the extended Filon method with its special cases Filon–Jacobi and Filon–Clenshaw–Curtis (chapter 4), the numerical steepest descent method (chapter 5), and the complex-valued Gaussian approach (chapter 6). In none of these cases are the algorithms listed explicitly in some pseudocode form or a similar way, though hints to helpful literature are included. The authors also provide useful remarks concerning the effort required to apply the algorithms in common use cases.
Finally, chapters 7 and 8 contain some further remarks on special variants of the problem and a comparison of their respective performances for typical application scenarios. This gives the reader looking at a specific problem a good deal of help when making a decision in favor of a certain algorithm.
As far as content, it is a mathematical textbook. However, for the most part, the writing does not follow the conventional definition-theorem-proof style usually found in such books; rather, many parts of the book are written in prose. The authors have managed to do this without sacrificing too much mathematical rigor, and their style certainly makes the reading easier for novices in the field (to whom the book is mainly addressed), but advanced readers may find a traditionally written text more efficient.
In any case, the book provides an excellent introduction to this still not fully matured area. Any scientist who needs to deal with oscillatory integrals and who has at least some basic knowledge of numerical mathematics can draw plenty of useful information from it.