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A Course in Analysis, Vol. IV: Fourier Analysis, Ordinary Differential Equations, Calculus of Variations

Niels Jacob and Krstian P. Evans
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
John Ross
, on
See our reviews of the previous volumes I, II, and III
A Course in Analysis is a series authored by Niels Jacob and Kristian Evans. There are seven volumes planned in the series: this is the fourth volume, subtitled Fourier Analysis, Ordinary Differential Equations, Calculus of Variations. While the first three volumes covered more standard material for undergraduate mathematical study, Volume IV begins to cover more advanced material. Nevertheless, the authors are quick to mention that this material is still vital “for every serious student of analysis,” and they pitch the book to this advanced undergraduate analysis specialist. I find the authors’ pitch to hit the mark: the expository writing is very easy to follow, and a tremendous level of detail is provided, tied together in cohesive and understandable narratives.
The three big topics in the subtitle comprise the three major sections of this text (which, to put in the context of the entire series, are labeled as Part 8, Part 9, and Part 10). Each section includes appropriate orientation to the topic, as well as a good assortment of problems for the reader to attempt. Full solutions for many of the problems are provided – the problem solutions, along with the readability of the text, make this an excellent book for an advanced student looking for a high-level class (or independent study) in these select topics, for either an advanced undergraduate or (first- or second-year) graduate student.
Part 8: Fourier Analysis begins with a chapter placing Fourier analysis in its historical context, both in terms of Fourier’s contemporaries and in terms of the mathematical theory that developed from Fourier’s work. The book then offers a robust treatment of the Fourier series and the Fourier transform. The text does a good job of seeding some select advanced topics that will be returned to later, such as Legendre or Hermite polynomials. Knowledge of the Lebesgue integral and some complex analysis are assumed.
Part 9: Ordinary Differential Equations covers lots of standard ODE material for both single equations and systems of equations, including the Picard-Lindelöf theorem and the Peano theorem, and Frobenius theory. Advanced topics include Sturm-Liouville problems and the hypergeometric differential equations. Phase diagrams and flows are discussed, but that discussion is saved for the very end of this section. It is worth mentioning (as the authors are quick to admit) that modern ODE theory has been indelibly marked by the advent of computing power and mathematical software. In an effort to keep the entire Course in Analysis consistent, the authors deemphasize software approaches to DEs, instead focusing on deriving the analytic theory. As such, the contents of this part are appropriate for dedicated analysis students --- but a text in DEs that utilizes technology would offer a more balanced approach for general students seeking to learn the theory of differential equations.
Part 10: Introduction to the Calculus of Variations aims to introduce the readers to the basic ideas of the theory: meatier results will be left to Volume V or Volume VI. As such, this is the shortest section in this volume and focuses primarily on Euler-Lagrange equations, the second variation, and solving partial differential equations of the first order.
All told, A Course in Analysis IV is a very accessible text that serves as an introduction and orientation to some advanced topics in analysis. It nicely continues the first three offerings in this series.


John Ross is an assistant professor of mathematics at Southwestern University.