Mathematical Analysis - Vol 1: Elementary Real And Complex Analysis

By Georgi E. Shilov

Book Overview

Georgi Shilov of Moscow University is one of the most respected of living mathematicians. It is a rather uncommon event when a mathematician as active and as creative as he is prepares a textbook at a relatively elementary level. This one is meant to be used by undergraduates, in most programs in their senior year, and it is written for science and engineering students as well as for mathematics majors. The book fully proves over 340 theorems and probably includes more examples and applications than any comparable text. It is organized into eleven chapters, each of which incorporates a problem set designed to illustrate, amplify, and enrich the material covered (selected hints and answers are given at the end). The starting point of the development is a set of axioms for real numbers (rather than a set of axioms for the natural numbers, as in some other approaches to the subject). The concepts next introduced include, roughly in their order of appearance, set theory, isomorphisms, n-dimensional space, and the field of complex numbers; metric spaces; a general theory of limits that comprises all those considered in analysis, from limits of a numerical sequence to the notions of the derivative and integral; continuous numerical functions on the real line, including the logarithmic, exponential, and the trigonometric functions; the algebra and topology of complex numbers and the fundamental theorem of algebra; infinite series, dealing not only with numerical series but also with those whose terms are vectors and functions (including power series); the differential calculus proper, with Taylor's series leading to a natural extension of real analysis into the complex domain; the general theory of Riemann integration; and, finally, the technique of analytic functions and a consideration of improper integrals that makes full use of the technique that has now been put at the student's disposal.

• Format: Hardcover
• Language: English
• ISBN: 0262191091
• ISBN13: 9780262191098
• Release Date: January 1973
• Publisher: MIT Press
• Length: 496 Pages
• Weight: 2.20 lbs.
• Dimensions: 10.0" x 1.0" x 6.0"

Customer Reviews

An excellent pure maths text.

I purchased this book to study some complex analysis. Being a physicist I would like to brush up on this. The book was completely different to what I expected. Some applications would have been nice, but this text is pure maths. The book is well written, easy to follow and concise. I ended up reading it and gained and appreciation for the thorough consideration of elementary real and complex numbers. Shilov is thorough and avoids making leaps and assertions. This would make the book readable to lower undergraduates. However the significance of some things is not explained, or explained in a very dry manner so people might miss this. I highly recommend this book if you are interested in real and complex analysis from a pure mathematics perspective.

Getting started in math analysis

This book by Shilov covers the fundamentals in beginning analysis(both real and complex). It has in common with Walter Rudin's book (entitled 'Real and Complex Analysis') that it covers both real functions (integration theory and more), as well as Cauchy's theorems for analytic functions. Shilov's book is at an undergraduate level, and it can easily be used for self-study. The Dover edition is affordable. Rudin's book is for the beginning graduate level, and it is widely used in math departments around the world. Both books have stood the test of time. Comparison of Shilov with Rudin: Rudin's 'Real and Complex' has become an institution, and I have to admit I have loved it since I was a student myself, but conventional wisdom will have it that Shilov is a lot gentler on students, and much easier to get started with: It stresses motivation a bit more, the exercises are easier (some of Rudin's exercises are notorious, but I find the challenge charming--not all of my students do though!), and finally Shilov gets to touch upon a few applications; fashionable these days. But that part easily gets dated. I will expect that beginning students will enjoy Shilov's book. Personally, I find that with perseverance, students who keep at it with Rudin's book, will end up with a lot stronger foundation. They are more likely to have proofs in their blood. I guess Shilov can always serve as a leisurely supplementary reading to Rudin. There will never be another book like Rudin's 'Real and Complex', just like there will never be another van Gogh. But the fact that we love van Gogh doesn't prevent us from enjoying other paintings.

A wonderful text -- Highly recommended!

I purchased this book as a reference book for my first analysis course. It is very well written, and easy to follow. Dr. Shilov has a very nice way of organizing this text: He puts all the definitions at the beginning of the chapter and the subsequent sections are results of those definitions. It makes for a very quick reference. His presentation of the included proofs is also very nice. There were several occasions I found myself thumbing through it for a second perspecitve. As far as the actual material presented, Dr. Shilov starts off with funtions of one real variable, then rather quickly generalizes to complex variables and N dimensional functions, so you'll quickly see metric theory and some topology. He does keep in mind this is intended for undergrads and first year grads though.Oh, another nice feature is the price! I'd recommend this book to any math enthusiast as a reference, or to someone going through an early analysis course.

SIMPLIFIED, ACCURATE, AND OBJECT-ORIENTED

Coherent and comprehensive, this "Elementary Real and Complex Analysis" is an emphatic introductory text, which will provide undergraduates with all the guides that they may need.The presentation of this book is such that anybody who is taking Pure and/or Applied Mathematics course would value it. From Analysis Basics to Complex Functions, the authors of this book fulfilled every desire.Worthy of mention is the way they simplified the rather complex Cauchy's Theorem. The same could be said of the chapters covering: Power Series Representations, Topology and Analysis in the Complex Plain, Holomorphic Functions, and Contour Integration.Each topic that appeared in this book received accurate simplification. They are all object-oriented, and were designed to be of great use to self-learners.Highly recommended!

Dr. Shilov could teach complex analysis to a chimpanzee.

The book is one of the finest mathematics texts that I have ever read. That does'nt mean much because I'm 16 years old, and have only been studying advanced math for a little over a year. The treatment was basic enough for someone like me to understand. I strongly recommend it to college students and young people who are interested in mathematics.After introducing concepts from set theory, the foundations of analysis, and the notion of a "mathematical structure," he gives a detailed presentation of limits and series. He also introduces elementary functions in terms of their functional equations. Then he covers differentiation and integration, first of real, and then of complex functions. He uses Taylor series to introduce ideas about complex functions.In short, it is a good book for those who hope to become mathematicians, physicists, or engineers, and have had a few college math courses already.