*Fourier Series and Boundary Value Problems* is a classic textbook that was first published in 1941. This edition, the seventh, is a revision of the 2001 edition. Since the third edition James Ward Brown has been co-author, and he has taken over responsibility for revisions since Churchill’s death. The book provides an introductory treatment of Fourier series and their application to boundary value problems in partial differential equations. In its present form, the book is designed to achieve two objectives. The first is to introduce the concept of orthogonal sets of functions and their use in representing arbitrary functions. The second objective is to present the method of separation of variables in the solution of boundary value problems. A basic example is the use of Fourier series to solve the heat equation.

This book is aimed toward students who have had basic courses in ordinary differential equations and advanced calculus. Although it’s completely appropriate for mathematics students, I expect that the primary audience now consists of physics and engineering students. The first summer I was a graduate student, I was assigned as a grader for a course based on — well, an earlier edition of — this book. The course instructor took me aside and warned me that this would actually require me to compute things. (I gather this had been an issue with previous graduate student graders, what with calculation being then somewhat out of fashion.) At that time I very much enjoyed the assignment and the satisfaction of computing Fourier series, expanding functions in series of Bessel functions and Legendre polynomials and solving Sturm-Liouville systems. I think — based on the papers I graded — that most of the students had a similar experience.

This edition shows evidence of considerable revision. The scope of the text has changed very little, although some of the chapters have been rearranged. Most changes seem to be the result of feedback based on classroom experience. The first two chapters now concentrate on Fourier series and their convergence. The chapter on orthogonal functions was rewritten and repositioned so that it is more self-contained and appears in the text right before it is needed. The large collections of exercises have been broken up and moved around so that exercises are matched more clearly with relevant sections of the text.

This continues to be an attractive textbook that is probably still the best choice for learning this subject. The revisions in this most recent edition only enhance its value.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.