Elementary Real and Complex Analysis (Dover Books on Mathematics) by Georgi E. Shilov (PDF)

Description

In this book the renowned Russian mathematician Georgi E. Shilov brings his unique perspective to real and complex analysis, an area of perennial interest in mathematics. Although there are many books available on the topic, the present work is specially designed for undergraduates in mathematics, science and engineering. A high level of mathematical sophistication is not required.The book begins with a systematic study of real numbers, understood to be a set of objects satisfying certain definite axioms. The concepts of a mathematical structure and an isomorphism are introduced in Chapter 2, after a brief digression on set theory, and a proof of the uniqueness of the structure of real numbers is given as an illustration. Two other structures are then introduced, namely n-dimensional space and the field of complex numbers.After a detailed treatment of metric spaces in Chapter 3, a general theory of limits is developed in Chapter 4. Chapter 5 treats some theorems on continuous numerical functions on the real line, and then considers the use of functional equations to introduce the logarithm and the trigonometric functions. Chapter 6 is on infinite series, dealing not only with numerical series but also with series whose terms are vectors and functions (including power series). Chapters 7 and 8 treat differential calculus proper, with Taylor’s series leading to a natural extension of real analysis into the complex domain. Chapter 9 presents the general theory of Riemann integration, together with a number of its applications. Analytic functions are covered in Chapter 10, while Chapter 11 is devoted to improper integrals, and makes full use of the technique of analytic functions.Each chapter includes a set of problems, with selected hints and answers at the end of the book. A wealth of examples and applications can be found throughout the text. Over 340 theorems are fully proved.

User’s Reviews

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⭐This book is a Dover reprint of Shilov’s “Elementary Real and Complex Analysis”. Written in an old-school textbook style, it is not as conversational as some modern texts are, but it does offer numerous explanations here and there. The first nine (out of eleven) chapters are easy to follow. I can read them on train or in some other more distracting environments without difficulties and without needing pencil and paper.Most conventional topics in classical analysis are covered in this book, with the notable exceptions of connectedness of point sets and double sequences/series of numbers or functions. Multivariate calculus is not covered, but there is a chapter on complex analysis.Some symbols and terminologies in this book are old-fashioned. The intersection of two sets G and A is denoted by GA rather than G∩A. Cauchy sequences are called “fundamental sequences”. “Compact” means sequentially compact rather than finite-cover compact. (Finite-cover compactness was referred to as “bicompact” in some older literature, but it is just called a “finite-covering property” in Shilov’s text.) Totally bounded space is called “precompact” (the term “precompact” is not outdated, but it is a bit confusing because it can also mean “relatively compact”). Some important results such as the intermediate value property of derivative or Lebesgue’s criterion of Riemann integrability are relegated to exercises. An unsuspecting reader can easily miss them.To my knowledge, Shilov’s book is the only introductory analysis text that fully justifies that the exponential, logarithmic and circular functions (sine, cosine, etc.) defined in the modern way using power series are identical to those bearing the same names in high-school curricula. It is a long journey that straddles across four chapters and it is the part of the book that I enjoy reading the most. Shilov proves that, if you want some functions called log, sin and cos to enjoy some familiar basic ARITHMETIC properties that we learned in high school (without assuming continuity, differentiability or integrability), such as log(xy)=log(x)+log(y) and sin^2(x)+cos^2(x)=1 (and a few others), these functions must be real analytic and possess unique power series expansions. Conversely, functions defined by those power series do satisfy the properties we laid down before. Therefore these functions are really those we learned in high school.I like how Shilov writes longer proofs. These proofs are often written in ways that are less distracting and highlight the key ideas better. For instance, in a typical proof of the uncountability of the reals using Cantor’s diagonal argument, one needs to resolve the issue that the infinite decimal representation is not unique (e.g. 0.1000… = 0.0999…). However, in Shilov’s proof, this becomes a non-issue because the diagonal is constructed not really by writing down new infinite decimals, but by picking new points in Cantor subdivision. While it is essentially the same diagonal argument, Shilov’s presentation has one less technicality to worry about.We may also compare Shilov’s book with Rudin’s

⭐. In both books, there are proofs for the fundamental theorem of algebra, Riemann rearrangement theorem, Abel’s theorem, L’Hospital rule and Taylor’s formula. Rudin’s proofs are often slicker and I can follow every step in his proofs, but I have troubles to pinpoint the gist of his arguments. To borrow Shilov’s words, “the reader can only take off his hat in silent admiration”. In contrast, I always find the central ideas of Shilov’s proofs easier to grasp.(That said, on the whole, I think Rudin organizes the topics better. But Rudin is really not suitable for novices because its exposition is close to none.)Shilov has made several attempts to streamline the teaching of real analysis. For instance, in chapter 4 (Limits), he proposes a set-theoretic notion called “direction” and uses it to define the concept of limit. His new definition of limit is a genuine generalization of the traditional ones (such as the ε-δ definition), but more restrictive than the filters/nets in general convergence theory (cf. Dixmier’s

⭐). Often in textbooks, the limit of a sequence, the limit of a function at a point and the limit of a function at infinity are formulated differently, but they are now unified. As a result, proofs about basic properties of limits are simplified and feel less repetitive.Unfortunately, these attempts are not thorough. So, his readers might not fully appreciate the merits of these attempts. Shilov’s uses of “direction”, for instance, are mostly confined to chapters 4 and 9. In other chapters, he basically resorts to the traditional definitions of limits. As another example, Shilov tries to introduce the concept of differential rigorously in section 7.3 (along the lines of Fréchet derivative, although this is not made explicit) without using any hand-waving “infinitesimals”. Alas, in section 9.9 (Further Applications of Integration), the ghosts of infinitesimals reappear again.My opinions about the last three chapters are mixed. The presentation in chapter 9 (The Integral) is crystal clear, but the applications to areas and volumes are too restrictive. It is well-known that the general definition of surface area is a hard problem (try to search “Schwarz lantern” on the internet). However, as multivariate calculus is not covered, Shilov can only give special definitions to areas of surfaces of revolution and volumes of solids with known cross-sectional areas. In contrast to the general geometric definition of length of curve he gives in the same chapter, this ad hoc treatment of area and volume is very unsatisfactory.Chapter 10 is about complex analysis. Basic topics up to Laurent series are discussed. I apprepriate that Shilov covers both real and complex analysis in the same book, because I think this is a more holistic approach, but the exposition in chapter 10 is a bit rushed. My opinions may change in future, but at present I don’t recommend anyone to learn complex analysis from it. The proofs in chapters 10 and 11 are more convoluted than those in earlier chapters. A few of them have gaps. There are also occasions in which a proof assumes a stronger condition than stated in the theorem. Fortunately, the stronger condition (e.g. a function is integrable) is always met when Shilov applies the theorem (say, because the function concerned in the application is continuous), but such condition blunders are quite troubling.The final chapter on improper integrals contains some results about the continuity or differentiability of parameter-dependent improper integrals that are rarely seen in other introductory calculus or analysis texts. The chapter ends nicely with a discussion of analytic continuation of Gamma function.Throughout Shilov’s text, there is a “hard analysis” flavour that I find quite intriguing, especially when most other introductory analysis texts written in English are slanted towards the soft analysis side.

⭐This book is not exactly five stars in reality. However, I feel obligated to give it five stars for its effect on my life and educational career. The book covers a lot of good content and does not overlap entirely with what is done in Rudin (from what I hear, the standard Analysis book in the USA). I feel it is better to start with Shilov before going to Rudin’s book. The reason being is that after finishing Shilov’s book (took a long time for me with self study), I read most of Rudin’s book without working it all out with pen and paper. Shilov is a good place to start, but it has some issues that need to be stated before one begins it. I will reveal the issues with the book by telling the story of why I give it five stars… oddly enough.So, why was Shilov worth five stars to me?Let’s start with a short story about me. Many years ago about when I bought this book, I was a physicist in training and I did not like taking someone’s word for anything. In physics, we often proved things with mathematics I knew only how to ‘DO’, but had no idea how to prove that the math we were using was true or valid for the particular case in question (Just look at the Dirac delta). I did not like that situation, so I set out to prove all the mathematics that I used in physics (a journey I have now learned I will never finish, however, I still continue that pursuit). One of the first math books I started reading was this book.Shilov starts from an axiomatic approach to the real numbers, and proves the whole of basic algebra, sequences, sums, and single variable differential and integral calculus (plus, a lot of other stuff). The attempted level of rigor is slightly less than Rudin (which is way less than the Bourbaki collective’s attempt, which itself has its own problems), however, you will find a few typos and, more interestingly, logical errors that are subtle, but glaring once realized. Now, there were only a few proofs that were not fully correct given the level of rigor, yet half of what I learned from Shilov was catching and correcting fallacious proofs. The other half that I learned from Shilov was the approach to proving something: you will learn it in your own way, but it’s usually a clever trick, so look for a good trick that makes all the logic fall into place.Each half I learned from Shilov seems contradictory: if the guys giving bad proofs in the book, how can you be learning what a good proof looks like? With how rigorous most of the book is, I often wonder if the errors in the book are intentional to see if the student is cut out for it. That said, a book with imperfections can’t receive a perfect score. However, I owe this book five stars, because it started me on my journey and it was perfect for what I needed to learn at the point in life I was at. Whether it can do the same for another beginner, I think it is likely. Just remember not every proof is correct in the book, but you’ll figure out which ones are one way or another.

⭐A superb book that is a pleasure to read, with extremely clear presentation, very thorough, and an enormous amount of material covered. There is ample originality here – for example the treatment of limits is refreshing and different to what one normally encounters- but you can also rely on the fact that while nearly 50 years old now, its material still seems extremely well chosen (there is little left out which one would consider elementary analysis, and he does not go off on specialised tangents).The book is not as easy to read for a complete beginner in analysis as some have suggested, but with some familiarity with basic concepts, you will get an awful lot out of it. I should say also that it does not shy away from more advanced concepts. Metric spaces appear in Chapter 2 without much fanfare and by the end of this chapter you find yourself more deeply immersed in topology than you may have expected. Limits and continuity are handled in a generalised way very quickly and what is said everywhere applies to n-dimensional spaces as well as the real line. Line integrals are given a brief but beautiful treatment with the use of the Stieltjes integral in Chapter 9 and complex analysis is in full throttle in Chapter 10 (though it receives brief treatments much earlier).The last three chapters are substantial, with chapter 9 giving a broad 100-page treatment of integration including techniques that are more at home in calculus books (rather than analysis). Chapter 11 continues this after the Complex Analysis Chapter (10).There are many typos which are probably not in the original (a translator would have spotted and corrected them, and judging from other reviews of Silverman’s translations, I’m guessing he introduced them), but that is not a real impediment because the majority are obvious enough. In only two places did I waste over half an hour trying to figure out why a proof did not make sense only to conclude that a mistake (typo?) was the problem. The knowledge that there are such issues, however, can also make you suspicious of valid proofs where steps are skipped by Shilov (not many of these, but there are a few).Overall this is such a superb book that any negative comments must be relegated to minor nuisances for which one is amply compensated by the genius of the author, the supreme clarity of the writing and remarkably orderly presentation of concepts from the simplest to the most demanding.

⭐Hi [updated]Whilst trying to get around my problem in understanding ‘Residues’ whist studying ‘Complex Analysis’ by Stewart, i returned to this book for help. Its true the type-face is small and there are very few graphs in-between the reams of information.The book is written in a semi-formal manner, but the level it’s pitched at is straight-forward if you have read these topics from studies elsewhere. The nice thing about this it helps to compile these topics in my mind from a newer, more formal perspective.The way he writes you can tell he’s an enthusiast for the topic. And its still great value for money!

⭐buon acquisto per il prezzo ma la stampa è migliorabile – dopo un po’, la piccolezza dei caratteri affatica gli occhi e la mente (lo spessore fa sì che la lettura sia appensantita dal lavoro per distinguere alcune lettere)

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⭐As I’ve mentioned when reviewing another of Shilov’s books, this book, too, is great in terms of the clarity and flow of the presentation – Shilov understands the needs of students. Consequently,. the subject is easy to study from this book

⭐本書は二巻からなるMathematical Analysisの第一巻で、内容は微分積分学の初等的なものとなっている。議論は主に1変数函数に限られており、ぶ厚いものを通読したのに、少々不満が残るだろう。ただそういった対象の限定が本書を読みやすいものとしている。議論の順序は必ずしも論理的になっておらず、概念を把握する方を優先している。例えば三角函数などはまずその性質などをいろいろ説明し、かなりうりろの章で定義をする。特徴的なのはdirection(下向き集合とか有向集合と邦訳されている)で極限を統一的に扱っていること。複素函数論においてRiemann面を扱うのは本書の程度としては意外に思う。数式が左寄せしてあるので頁の右側が白くなっているのは慣れるまで違和感が残る。

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