Lectures on Functional Equations and Their Applications (Dover Books on Mathematics) by J. Aczel (PDF)

Description

This systematic treatment of one of the oldest topics of mathematical analysis — the solution of functional equations — features numerous detailed proofs. Thorough and elementary in its approach, it will benefit upper-level undergraduates as well as graduate students.The text is divided into two parts: equations for functions of a single variable and equations for functions of several variables. Starting with equations that can be solved by simple substitutions, the first part examines the solution of equations by determining the values of the unknown function on a dense set. It also surveys equations with several unknown functions and methods of reduction to differential and integral equations. The second part begins with simple equations and advances to composite equations, equations with several unknown functions of several variables, and reduction to partial differential equations. The text concludes with explorations of vector and matrix equations.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Great Dover value of Aczel’s 1960 Debrecen Lectures, with a priceless chronological bibliography stretching from 1747 to 1965. The field is, of course, chaotic and pell-mell, but the author succeeds in burnishing it with insight of an almost grand design. A must-have for serious users of functional equations, albeit not the obvious go-to book.

⭐It has been a classic since its first publication in 1966. The book is essential for anyone working with or using functional equations in mathematics. It is the foundation of the field.

⭐This book on functional equations is written by one of the experts on the topic and it is intended for upper level undergraduate and graduate students. As such, it does a superb job in the presentation of the material. It is well written, clear, and pedagogical. It covers almost all topics that you would like to know about although, since this was originally published in 1966, it does not cover more recent material (such as theory of iterations and chaos). However, this is not really a problem since there are many other books which cover these recent developments and the book already contains a large amount of information. It also contains a huge bibliography starting at page 383 and ending at page 496 (covering the years 1747 to 1965, the year before its publication) listing works per year of publication.The only minor disadvantage of the book is the lack of unsolved problems at the end of each chapter. Therefore, if you want to solve problems related to the theory of the book, you will not find any and you must use another source. At the end of the book (and before the bibliography) there is an epilogue entitled `Concluding Remarks. Some Unsolved Problems.’ This epilogue does not contain practice problems. Instead it discusses some open research problems at the time of the original publication. Despite this minor disadvantage of not including practice problems, this is a great book that I strongly recommend to anyone interested in this area of analysis.Incidentally, I should add the following comment: if you are not an upper level undergraduate student yet, you can still gain a lot from this book. However, you may want to read another book first, such as

⭐, to learn the most essential theory and the most important of the functional equations. This latter book also contains some of the recent developments and a really large number of solved and unsolved problems (taken from mathematical competitions).

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