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Fundamental Mathematical Analysis

Robert Magnus
Publication Date: 
Number of Pages: 
Springer Undergraduate Mathematics Series
[Reviewed by
John Ross
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Fundamental Mathematical Analysis offers a carefully measured, comprehensive look at the analysis of functions of one variable. The author, Robert Magnus, offers careful justification of what constitutes (and does not constitute) a fundamental topic.  Each topic is explored in detail, and the text features appropriate historical context, reasoned proofs, and additional exploratory topics in the form of “nuggets.” The result is a rich but demanding text that is appropriate for advanced undergraduate students who have had a moderate amount of previous experience writing and reading proofs. 
A title such as Fundamental Mathematical Analysis carries weight, and Magnus takes care to justify the “fundamental” nature of the topics in the text. Of course, the text contains coverage of each of the major topics in any Real Analysis text: a crash course in real numbers, sequences, series, functions, continuity, differentiation, and integration. However, this only reaches up through the first half of the text: my copy of this textbook has 426 pages of mathematical content, with the Fundamental Theorem of Calculus appearing squarely on page 216. The second half of the text contains chapters on elementary transcendentals (namely, the trigonometric, logarithmic, and exponential functions); techniques of integration; an introduction to the algebra of complex numbers; function sequences and series; and improper integration. Of these later chapters, Chapter 11 (Function Sequences and Function Series) is the most robust, spanning 75 pages and covering uniform convergence, various power series, and Taylor’s theorem. 
It is also important to consider what is omitted from this text. Magnus omits many ideas that appear just beyond the upper boundary of classic real analysis: subjects like the Lebesgue integral, multivariable functions, and complex analysis are nowhere to be found. Other omissions include topics that are adjacent to real analysis, though they might appear in other peer texts. For example, gone are those topics that arise from topology, including any conversation of open and closed sets, compactness, or the Heine-Borel theorem (the closest brush in this direction involves a section on the “limit points of sets” found within Chapter 3). Gone, too, are conversations about countable or uncountable sets. Finally, gone is any explicit construction of the real numbers. Magnus takes great care to justify these omissions, making well-reasoned arguments in each case that they are tangential, not fundamental, to the study of analysis.  
In terms of style and format, this text is excellent. There is an incredible level of detail and granularity to the chapter/section/subsection divisions in the table of contents. Exercises occur at the end of each subsection, and they are by and large stated at the appropriate level of difficulty. Historical context or quotes are occasionally given to help guide a conversation or elucidate where an idea came from. One of my favorite elements of this text are the extra, exploratory sections known as “nuggets” (so-named to represent a “nugget of wisdom”). These are entire sections, scattered throughout the text, that do a momentary deep dive into an advanced topic. Nuggets include topics such as continued fractions, approximation by step functions, and Riemann’s rearrangement theorem, to name three. These serve a clear role as supplemental material for class projects, or as a tempting morsel for an interested student.
There is a question of who this text will best serve. I believe that this text will best serve undergraduate students who can accept a high degree of rigor, and have solid previous experience (one semester minimum, likely a full year) reading and writing proofs. The exercises offer few “truly elementary” proofs for the readers to work on. As an example of this: after the formal definition of a sequential limit is introduced, one of the first limits that is rigorously proven is that the limit (as \( n \) goes to infinity) of \( x^{n} = 0 \) for each x in \( ]0,1[ \), and this is proven by expressing \( 1/x^{n} \) as \( (1+h)^{n} \) and using the binomial theorem. It’s a clever and enjoyable proof, but the reader still new to proofs would potentially find it daunting, unintuitive, or unable to be adapted to similar limit expressions.
In conclusion, Fundamental Mathematical Analysis is a complete and wide-ranging text that offers a rigorous introduction to the analysis of real-valued, single-variable functions. This text is well organized and peppered with insight and mathematical nuggets. Its high level of rigor might make it challenging to use in a class whose students are not ready to meet the challenge. Nevertheless, I found it an enjoyable read and am confident that a well-prepared student would gain a tremendous deal from this text.
John Ross is an assistant professor of mathematics at Southwestern University