Product Description

The idea of this book is to give an extensive description of the classical complex analysis, here ''classical'' means roughly that sheaf theoretical and cohomological methods are omitted.

The first four chapters cover the essential core of complex analysis presenting their fundamental results. After this standard material, the authors step forward to elliptic functions and to elliptic modular functions including a taste of all most beautiful results of this field. The book is rounded by applications to analytic number theory including distinguished pearls of this fascinating subject as for instance the Prime Number Theorem. Great importance is attached to completeness, all needed notions are developed, only minimal prerequisites (elementary facts of calculus and algebra) are required.

More than 400 exercises including hints for solutions and many figures make this an attractive, indispensable book for students who would like to have a sound introduction to classical complex analysis.

For the second edition the authors have revised the text carefully.

Customer Reviews (2)

Excellent book/Terrible translation
This book was originally written in German, and the German version is just incredible: a real gem.Good reason to translate it!Unfortunately, this is one of the worst translations from German I have seen.Some of it is just awkward grammar, which the reader may be able to ignore.But, there are also some words and phrases which are translated incorrectly.
For example "Paragraphen" in German does not mean paragraph in English, it means section.But in this book it is translated as paragraph.Try looking for something at the end of a paragraph or in the previous paragraph, when you should actually be looking at the end of the section or in the previous section.An example of this can be found in the explanation of the addition theorem for complex exponents (p. 27).The English text claims there is a remark concerning this at the end of the "paragraph."The paragraph ends and there is no remark.Turn to the end of the section (p. 31) and you will find the remark just above the exercises.
My advice is, if you can read German, get the German version!If you can't read German, you can still get the English version, but you will have to be very patient with the mistakes incurred in the translation (not to be found in the German original). If you own this book, you should systematically go through it and replace "paragraph" everywhere with "section."Most of the other translation mistakes can be figured out by context.

see review
I was truly delighted to find this text. It starts off with ordinary complex analysis at the level of sophomore undergraduate students and proceeds well into graduate-level complex analysis (analytic number theory, elliptic functions, abels theorem, etc). The 'advanced' results are shown using standard methods, so it was a great way for someone who learned the nuts and bolts of contour integration to move into theta functions, the prime number theorem, etc etc...fun stuff. ... Read more

Product Description

Functional Equations and Inequalities with Applications presents a comprehensive, nearly encyclopedic, study of the classical topic of functional equations. Nowadays, the field of functional equations is an ever-growing branch of mathematics with far-reaching applications; it is increasingly used to investigate problems in mathematical analysis, combinatorics, biology, information theory, statistics, physics, the behavioral sciences, and engineering.

This self-contained monograph explores all aspects of functional equations and their applications to related topics, such as differential equations, integral equations, the Laplace transformation, the calculus of finite differences, and many other basic tools in analysis. Each chapter examines a particular family of equations and gives an in-depth study of its applications as well as examples and exercises to support the material.

The book is intended as a reference tool for any student, professional (researcher), or mathematician studying in a field where functional equations can be applied. It can also be used as a primary text in a classroom setting or for self-study. Finally, it could be an inspiring entrée into an active area of mathematical exploration for engineers and other scientists who would benefit from this careful, rigorous exposition.

Product Description
"Chemists familiar with conventional quantum mechanics will applaud and benefit greatly from this particularly instructive, thorough and clearly written exposition of density functional theory: its basis, concepts, terms, implementation, and performance in diverse applications. Users of DFT for structure, energy, and molecular property computations, as well as reaction mechanism studies, are guided to the optimum choices of the most effective methods. Well done!"
Paul von Ragué Schleyer

"A conspicuous hole in the computational chemist's library is nicely filled by this book, which provides a wide-ranging and pragmatic view of the subject.[...It] should justifiably become the favorite text on the subject for practitioners who aim to use DFT to solve chemical problems."
J. F. Stanton, J. Am. Chem. Soc.

"The authors' aim is to guide the chemist through basic theoretical and related technical aspects of DFT at an easy-to-understand theoretical level. They succeed admirably."
P. C. H. Mitchell, Appl. Organomet. Chem.

"The authors have done an excellent service to the chemical community. [...] A Chemist's Guide to Density Functional Theory is exactly what the title suggests. It should be an invaluable source of insight and knowledge for many chemists using DFT approaches to solve chemical problems."
M. Kaupp, Angew. Chem.


... Read more

Customer Reviews (5)

One of the Best books on DFT for Chemists!
This book presents the Density functional Theory (DFT), for Chemists. It is divided in two parts: (A)The Definition of the Model - where the theory is presented; (B) The Performance of the Model - where the applications are explored. If you are interested in Computational Chemistry and want to learnDFT, then this book is for you.

A Chemist's Guide to Density Functional Theory, 2nd Edition
This book is an excellent introduction to density functional theory. And it is not difficult to read straight through.

a more practical book for DFT
Unlike Parr and Yang's Density-Functional Theory of Atoms and Molecules, this book doesn't have many rigorous while lengthy derivations. However, it gives readers a clear clue for DFT ,and most importantly, a way to appreciate this theory as a chemist. I think this book is a perfect complement for Parr and Yang's book.

DFT for Physicists Also!
From a physicist's point of view this book is very clear at explaining Density Functional Theory (DFT). The authors use many chemical examples, but still can be applied to physics. Many physics books on DFT assume the reader knows most of the material so skips many important details that can leave the reader confused. Surprisingly, these chemists spend entire chapters on just about every piece of DFT. They even give many examples, using the Hydrogen molecule as an example quite a few times.

This book is fairly recent, published in 2001. It talks about many DFT codes used today and important functionals such as B3LYP. The book is a little relaxed on the math, so if you are wanting to see some of the detailed math I suggest "Density-Functional Theory of Atoms and Molecules" by Parr & Yang as a good companion book.

DFT for chemists!
Computational and Theoretical chemists concerned with the applications of canonical quantum chemistry (molecular orbital) methods to chemically interesting problems know too well how (computationally) demanding is going beyond the Hartree-Fock (HF) approximation by employing the so called post-HF methods. Hence, very often they must resort on using Density Functional Theory (DFT). Here, however, they need to confront themselves with the terminology invented by their physics collegues: Kohn-Sham orbitals, Fermi hole, local and non-local spin-density functionals, generalized gradient approximation, pseudopotentials, and so forth. Any terminology is associated to a certain model of thought, which requires lot of efforts to be fully comprehended.
The book of Koch and Holthausen represents a praiseworthy attempt of presenting the basic concepts of DFT to research chemists. This 300-pages book is organized in two parts and it contains 13 chapters. Part A is concerned with the definition of the (DFT) model, while Part B discusses the performance of the model in dealing with molecular structures, vibrational frequencies, thermochemical, electrical and magnetic properties, H-bonds, and chemical reactivity. A rich bibliography is appended at the end of the book. Clearly written and logically organized, this book can be considered "THE Chemists's Guide to DFT" and it deserves five stars. ... Read more

Product Description
From the reviews: "A good introduction to a subject important for its capacity to circumvent theoretical and practical obstacles, and therefore particularly prized in the applications of mathematics. The book presents a balanced view of the methods and their usefulness: integrals on the real line and in the complex plane which arise in different contexts, and solutions of differential equations not expressible as integrals. Murray includes both historical remarks and references to sources or other more complete treatments. More useful as a guide for self-study than as a reference work, it is accessible to any upperclass mathematics undergraduate. Some exercises and a short bibliography included. Even with E.T. Copson's Asymptotic Expansions or N.G. de Bruijn's Asymptotic Methods in Analysis (1958), any academic library would do well to have this excellent introduction." (S. Puckette, University of the South) #Choice Sept. 1984#1 ... Read more

Product Description
As Lord Kelvin said, "Fourier's theorem is not only one of themost beautiful results of modern analysis, but it may be said tofurnish an indispensable instrument in the treatment of nearly everyrecondite question in modern physics." It has remained durableknowledge for a century and has extended its applicability to topicsas diverse as medical imaging (CT scanning), the presentation ofimages on screens and their digital transmission, remote sensing,geophysical exploration, and many branches of engineering. Fourier Analysis and Imaging is based on years of teaching acourse at senior or early graduate level and will also be a welcomeaddition to the reference library of those many professionals whosedaily activities involve Fourier analysis in its many guises. ... Read more

Product Description
This book presents, for the first time, a new technology for improving products and innovating new and better products, first developed in Japan by Yoshihiko Sato. Value analysis tear-down combines traditional tear-down with the technologies of value analysis and value engineering. Within a few years of its public announcement in Japan, value analysis tear-down was adopted by all eleven Japanese automobile manufacturers, and many of the Japanese consumer electronics manufacturers. Jerry Kaufman, based in Houston, Texas, is a recognized authority and author on value engineering and value management, and has contributed much that is in these technologies to the process described in this book. The result of his collaboration with Mr. Sato is a process that helps engineers and managers reduce product cost, improve quality, continuously improve existing products, and discover opportunities for innovative change.

The first "how-to-do-it" book in English, it is written specifically for professionals in product engineering, manufacturing engineering, and value engineering; and the managers of these professionals, including plant managers, production managers, manufacturing executives, and research and development executives. It will also be useful to manufacturing, marketing, and management people concerned with product improvement, innovation, and improving their company's competitive position. Value analysis tear-down can be applied in many service and other industries, as well as in manufacturing; wherever there are physical components to be improved or invented. ... Read more

Product Description
This book covers functional analysis and its applications to continuum mechanics. The mathematical material is treated in a non-abstract manner and is fully illuminated by the underlying mechanical ideas. The presentation is concise but complete, and is intended for specialists in continuum mechanics who wish to understand the mathematical underpinnings of the discipline. Graduate students and researchers in mathematics, physics, and engineering will find this book useful. Exercises and examples are included throughout with detailed solutions provided in the appendix. ... Read more

Customer Reviews (1)

Highly recommended
Another superb book by Lebedev and Vorovich (see also their Kluwer book ISBN 1402007566). The authors weave together advanced mathematical and mechanical concepts in a way that should satisfy most mathematically-oriented engineers. Includes a full chapter on nonlinear analysis ... carefully selected material not covered in their earlier book. ... Read more

Product Description
From the Foreword by R.N. Bracewell, Stanford University:

"...can be thoroughly recommended to any reader who is curious about the physical world and the intellectual underpinnings that have lead to our expanding understanding of our physical environment and to our halting steps to control it. Everyone who uses instruments that are based on harmonic analysis will benefit from the clear verbal descriptions that are supplied."

A sweeping exploration of essential concepts and applications in modern mathematics and science through the unifying framework of Fourier analysis! This unique, extensively illustrated book describes the evolution of harmonic analysis, integrating theory and applications in a way that requires only some general mathematical sophistication and knowledge of calculus in certain sections.

Key features:

* Historical sections interwoven with key scientific developments showing how, when, where, and why harmonic analysis evolved* Exposition driven by more than 150 illustrations and numerous examples* Concrete applications of harmonic analysis to signal processing, computerized music, Fourier optics, radio astronomy, crystallography, CT scanning, nuclear magnetic resonance imaging and spectroscopy* Includes a great deal of material not found elsewhere in harmonic analysis books* Accessible to specialists and non-specialists

"The Evolution of Applied Harmonic Analysis" will engage graduate and advanced undergraduate students, researchers, and practitioners in the physical and life sciences, engineering, applied mathematics. ... Read more

Product Description
The transition from studying calculus in high school to studying mathematical analysis in college is notoriously difficult. In this new edition of Numbers and Functions, Dr. Burn invites the student to tackle each of the key concepts, progressing from experience through a structured sequence of several hundred problems to concepts, definitions and proofs of classical real analysis. The problems, with all solutions supplied, draw readers into constructing definitions and theorems. This novel approach to rigorous analysis will enable students to grow in confidence and skill and thus overcome traditional difficulties in learning this subject. ... Read more

Customer Reviews (4)

Best Undergraduate Single Variable Real Analysis Text by Far
This beautiful book is by far the best undergraduate single variable real analysis text I have seen. It covers all the basic topics in impeccable detail. Each chapter opens by listing a few references, labelled "Preliminary", "Concurrent", and "Further" Reading. The main part of each chapter consists of "questions" which guide the student through a complete theoretical development of the material and which the student is invited to work through. This part of the chapter contains definitions, statements of theorems, and informal discussion. This section is followed by a brief summary, outlining the previous material. Next comes a "Historical Note" which is very illuminating and fun to read. The last part of the chapter contains a complete working out of all the "questions". At the end of the book is an extensive bibliography, containing all books mentioned at the beginning of the chapters and many others. There is also an accurate and detailed index.

All in all, the text contains an exhaustive and perspicuous treatment of material which often is presented in a less transparent way in other texts such as Rudin. I also prefer it by far to other excellent recent books such as those by Ross or Abbott. The format engages the reader in a unique way that other books don't.This book was developed for use in the math program at the University of Warwick and as far as I know, it is still in use there. Unfortunately, it is less well known in the US.I cannot recommend this book highly enough. Once you see a copy for yourself, I think you will understand why.

Numbers and Functions, Steps into Analysis
If you follow the study plan laid out in the preface of this book, you'll get abackground in Calculus and Undergraduate Analysis second to none. After this truly thorough coarseof study, 'little Rudin' will presentfew difficulties and 'Big Rudin' will be much more approachable. Depending on your background, you mayhave to buy the references that are conveniently listed at the beginning of each chapter.But the effort and expense is well worthit. If you're just starting out or you're looking for an indepth review of Calculus and basic Analysis, this book is THE best. This book will also benefit those studying applied math and engineering because itprovides the mathematicalbackground for many of the more complicated functions and approximationsused in mathematical applications.
The poor review found above is not reliable. There is abundant background material provided in the excellant (and mostly cheap) references. Each chapter ends with historical insights very helpful in comprehending the material. The whole approach is based the presentation of fundemental concepts and their origins and implications. The title is exactly right. This book combines problem solving with the historical sequence that led tothe development of the major IDEAS behind themathematics. Thus, each problem isa stepping stone to the next problem. Eachsection is a stepping stone to the next section and so on. Also, each problem is broken down into conceptual components that allow the student to understand not only the raw mathematics but also the strategy used to solve the problem. Each problem set is followed by an exposition of the big picture ofjust what concepts are being illucidated.Ilove the very well done and usefulapproach to teaching. If you have no or a weak background in thesubject of Calculus or Analysis, you must do the preliminary reading indicated at the beginning of each chapter. For those with a stronger background the concurrent and further reading suggestions are very helpful though not necessary in all cases. Bepatientand give yourself a chance, it'swell worth it.

Oh MY GOD
Please. As a Math teacher. WOW! I can NOT understand this book. If you are looking for a book to make things more clear and present a review to your current understanding then find another book! This one is way too difficult to read - unless I suppose all you can relate to is numbers and signs, not humans.

Interesting and refreshing approach
I worked through this book several years ago and I remember enjoying its style of pointing out an interesting property of a particular function, and then showing, step by step, that a whole class of functions have thatproperty; that is, the theorems are built up from examples, instead of theother way round. I also think each step was quite manageable - there wereno big gaps where I was left scratching my head not knowing what to do. Itis not meant as a reference book, as you're more likely to findsketchesor hints to parts of proofs, rather than complete proofs. I don't know ifit's ever been used as a text book, but if it were, students couldn't justsit back and absorb knowledge - they would have to figure things out (withthe benefit of lots of pointers to tell them what needs figuring out, andhints about how to do it), which is just what the book is intended toencourage. ... Read more

Product Description
Written in an easy-to-read style, Real Analysis is a comprehensive introduction to this core subject and is ideal for self-study or as a course textbook for first and second-year undergraduates. Combining an informal style with precision mathematics, Real Analysis covers all the key topics with fully worked examples and exercises with solutions. ... Read more

Customer Reviews (3)

The Worst Math Text Ever
This book was the example used by my mathematical writing professor as how NOT to write math texts.The formatting is strange, the theorems and definitions are buried in the text with no delineation whatsoever.There are typos throughout the book, sometimes in the middle of proofs, making the proofs even more difficult to follow.

The lack of exercises and examples are a little frustrating, as well, though this is typical of higher level math texts.I understand that the purpose of this book is to make analysis more accessible for undergraduates, but I feel that this book made the material more difficult than other, more advanced texts.Analysis is a challenging enough subject, and I feel that this book only exacerbates that.

My personal recommendation to anybody starting an analysis course, undergrad or grad is Principles of Mathematical Analysis by Walter Rudin.It's more advanced but far more thorough and illustrative in its proofs.

lacks exercises
As you might already know this book is the basic of analysis, this is literally the introductory book to analysis. It goes into Sequences and Series, Functions and Continuity, Differentiation, Integrations etc (see "contents" for more detail).

The author of this book explicitly says that a prior knowledge on Calculus is expected before reading this book and he really means it. If you did your Calculus I (limit and derivative), Calculus II (integration), and Calculus III (multivariable) a year or two ago, you need to go review at least the topics that are covered in this book. Remember all the convergence/ divergence (Comparison Test, Limit Comparison Test, Ratio Test, Root Test, Lebeniz Test, P-test etc) tests that you did in Calculus II (?) The author manages to squeeze them all in 5 pages or so.

The major drawback of this book (for me) is lack of exercise. There are very few sections which have more than 10 problems, most have 6 or less. Here is a detailed data:

chapter 1 has 6 sections but only 24 problems
chapter 2 has 7 sections but only 44 problems
chapter 3 has 7 sections but only 37 problems
chapter 4 has 5 sections but only 17 problems
chapter 5 has 7 sections but only 27 problems
chapter 6 has 3 sections but only 19 problems
chapter 7 has 3 sections but only 19 problems
chapter 8 has 2 sections but only 1 problem
chapter 9 has 3 sections but only 1 problem

thankfully solutions to all these problems are available in the back of the book. These solutions are worked out well enough.


Another drawback of the book is that it does not inspire you. In most mathematics book (or any other course book for that matter) each chapter begins with a little bit of history, real life application or something that helps connect to the student. This book barely does that. Usually, the section starts with a theorem, followed by a proof or two, and finally two three examples.

Overall imo this book is more of a "study guide" than a textbook itself. The publisher say this book can be used for self study as well as course textbook. I will admit that this is an easy self study book but i would get a more detailed textbook to tag along with this, just in case.

British Style Math Book
This is an introductory text of real analysis and it is kind of British Style (in term of the way they proved the theorems). Also, some advanced topics like "Metric" and "Generalized Riemann Integral" are not covered. If you really want to learn real analysis yourself, try Robert Bartle's "Introduction to Real Analysis", Manfred Stoll's "Introduction to Real Analysis", Apostol's "Mathematical Analysis" and Rudin's "Principle of Mathematical Analysis". Stephen Abbott's "Understanding Analysis" is also an excellent real analysis text. ... Read more

Product Description
This book is an abridged version of the two volumes "Convex Analysis and Minimization Algorithms I and II" (Grundlehren der mathematischen Wissenschaften Vol. 305 and 306). It presents an introduction to the basic concepts in convex analysis and a study of convex minimization problems (with an emphasis on numerical algorithms). The "backbone" of bot volumes was extracted, some material deleted which was deemed too advanced for an introduction, or too closely attached to numerical algorithms. Some exercises were included and finally the index has been considerably enriched, making it an excellent choice for the purpose of learning and teaching. ... Read more

Product Description
Complex analysis is one of the most attractive of all the core topics in an undergraduate mathematics course. Its importance to applications means that it can be studied both from a very pure perspective and a very applied perspective. This book takes account of these varying needs and backgrounds and provides a self-study text for students in mathematics, science and engineering. Beginning with a summary of what the student needs to know at the outset, it covers all the topics likely to feature in a first course in the subject, including: complex numbers differentiation integration Cauchy's theorem and its consequences Laurent series and the residue theorem applications of contour integration conformal mappings and harmonic functions A brief final chapter explains the Riemann hypothesis, the most celebrated of all the unsolved problems in mathematics, and ends with a short descriptive account of iteration, Julia sets and the Mandelbrot set. Clear and careful explanations are backed up with worked examples and more than 100 exercises, for which full solutions are provided. ... Read more

Product Description
Was plane geometry your favourite math course in high school? Did you like proving theorems? Are you sick of memorising integrals? If so, real analysis could be your cup of tea. In contrast to calculus and elementary algebra, it involves neither formula manipulation nor applications to other fields of science. None. It is Pure Mathematics, and it is sure to appeal to the budding pure mathematician. In this new introduction to undergraduate real analysis the author takes a different approach from past studies of the subject, by stressing the importance of pictures in mathematics and hard problems. The exposition is informal and relaxed, with many helpful asides, examples and occasional comments from mathematicians like Dieudonne, Littlewood and Osserman. The author has taught the subject many times over the last 35 years at Berkeley and this book is based on the honours version of this course. The book contains an excellent selection of more than 500 exercises. ... Read more

Customer Reviews (12)

One of the Best Books in Analysis, or even Maths
I dont know how to say how good this book is: it not only teaches us the technical aspects of mathematics, it also teaches us intuitions, ideas behind the proofs, styles, and philosophy.

I would like to also start with a comparison of the classic baby Rudin:

While Rudin's little book is also a real gem, I would say Pugh's belongs to a slightly higher level (due to its problems mainly and topic selection and coverage). Rudin's book could either be used in 1st year one semester as a strong first course in basic analysis (1st 7 chapters) for extremely motivated and hardworking students (such as those at MIT, Harvard, Princeton, and many other good institutions), or ought to be supplemented in an honour's undergrad real analysis (in 2nd year or 3rd year). Its presentation of Lebesgue theory is rather incomplete and no one virtually uses it for lebesgue theory. On the other hand, pugh has a full chapter on it, covering almost all the standard undergrad lebesgue materials.

Pugh's book on the other hand can be the last reading before attempting Folland and Big Rudin. Knowing pugh well and having solved its problems would make Folland and big Rudin not hard, whereas little rudin may not surve this purpose that well. Many Rudin's problems are hard but standard (Prove, Show this. Very few is it true? what about?), whereas Pugh's is more thought provoking (Is it true? What about? What do you think? which mimics a key part in maths research).

Moreover. mathematics is not just about formalism and logic, especially in analysis ang geometry. The ideas and our feeling about how the objects behave are at least equally important. (Anyone can write proofs well with sufficient training; yet not everyone feels that a measurable function is no more complicated than continuous ones in a sense; why lebesgue's definition of length and integral are powerful; weirstrass approximation is as simple as "taking expectations of functionals", etc.) Amazingly Pugh's book trains people to this direction very well.

I prefer Bartle and Sherbert
I used this book in my first Real Analysis course and thoroughly disliked it.I seems that everyone else who reviews this book mentions Rudin.I haven't had a chance to read Rudin yet but I prefer Intro to Real Analysis by Robert G. Bartle and Donald R. Sherbert over this book.Many people like Pugh for it's conversational tone but I found it annoying.This might be a good secondary book but I wouldn't recommend this as your first book in real analysis.Pugh makes the cardinal mistake of mathematicians in introducing the most general case first.The most important thing in mathematics is not the most general case but the process of generalization itself.This is like saying the journey is as important as the destination.To generalize one must start with a specific case and then work, layer by layer, to the most general case.That's one reason I prefer Bartle and Sherbert.It starts with functionsfrom R to R and generalizes from there.It takes up where undergraduate calculus leaves off.I also prefer the exercises in Bartle and Sherbert better. They are challenging without being infuriating. They are still general proofing exercises but are specific enough to deal with specific functions, series, sequences, and so forth.I was also annoyed by the way Pugh qualifies his proofs like Chapter 1 Theorem 2: "Proof Easy" ,or theorem 9: "Proof Tricky!", or Chapter 2 Theorem 10:"Proof, Totally natural!".I feel his language is imprecise and sloppy. I feel the section on cuts is superfluous. It seems that cuts are a lot of work and headaches just to prove that everything I learned in elementary school is correct. I was worried that x+0 didn't really equal x but now with cuts, I can rest assured that it does! Whew, what a relief!The only plus to Pugh is the thorough chapter on metric spaces helps put things into a broader context.All and all I dislike Pugh's book and highly recommend Bartle and Sherbert. as the best introduction to Real Analysis.

Excellent problems and diagrams -- great book
This is an excellent introductory text on real analysis. It is very approachable, and he does a very good job at supplementing the traditional "definition-theorem-proof" style with intuitive explanations and wonderfully descriptive diagrams (the diagrams are one of the strongest points of this book -- and are something that are sadly left out of many otherwise good books on analysis).

My only (minor) complaint is with the layout/formatting of the book -- it is very jumbled together, the typesetting is poor, and it looks like it was printed on a low-resolution $10 printer.

Other than that, it is an excellent companion to a more in-depth/advanced treatment. As far as more "advanced" books go, I would recommend -- Apostol's "Mathematical Analysis" and/or Shilov's "Real and Complex Analysis" -- both of which are incredibly well written and informative.

Pugh is wonderful. Rudin is good too, but both texts working together is the best.
I wish that I had discovered Pugh in my first semester of undergraduate analysis. The assigned text was Rudin and it was a great choice. The exposition there is excellent. The exercises are incredibly well done. Pugh covers just about the same material as Rudin, and in the same rigor, but is more likely to give you paragraphs before and after important theorems/definitions that help to clarify things. I must admit I am not too familiar with the first half of Pugh's text as I didn't discover it until I was well in chapter 10 of Rudin ~~ chapter 5 of Pugh. But, if the first chapters are as good as the fourth and fifth, you can get just as much from Pugh as from Rudin, if not more.

Sometimes, you get a picture (this would have been really helpful back when I was learning what an open cover was). Other times, Pugh actually gives a better presentation. For instance, when discussion the rank theorem, Rudin's statement of it is hard to follow. The proof is about as difficult. Pugh, however, introduces C' equivalence and then gives an alternate statement of the theorem which is much more intuitive. AND some pictures after the proof. Some think having pictures in analysis books is bad--Pugh gives evidence otherwise.

It is difficult to say which text has better exercises as I have not attempted them all. But Pugh definitely has more of them. I think the best thing for any undergraduate to do is to just own both books. Rudin is the standard for a good reason. Pugh's or someone else's exposition may become the standard in the future, but Rudin will always be an excellent reference. Doing Rudin's exercises will help prepare you for your qualifying exams if you ever take them. Pugh has some UC Berkeley good prelim exam questions in his book which prepare you for future math endeavors as well. So I say just buy both. But if you can only buy one.... probably get Pugh because he's cheaper. Or you can get International Edition Rudin for cheaper still.

Brilliant
The style is friendly and fun, and the presentation is really intuitive! My personal favorite! ... Read more

Product Description
The aim of this book is to present a substantial part of matrix analysis that is functional analytic in spirit. Much of this will be of interest to graduate students and research workers in operator theory, operator algebras, mathematical physics and numerical analysis. The book can be used as a basic text for graduate courses on advanced linear algebra and matrix analysis. It can also be used as supplementary text for courses in operator theory and numerical analysis. Among topics covered are the theory of majorization, variational principles for eigenvalues, operator monotone and convex functions, perturbation of matrix functions and matrix inequalities. Much of this is presented for the first time in a unified way in a textbook. The reader will learn several powerful methods and techniques of wide applicability, and see connections with other areas of mathematics. A large selection of matrix inequalities will make this book a valuable reference for students and researchers who are working in numerical analysis, mathematical physics and operator theory. ... Read more

Customer Reviews (3)

Review by me
Nice book. Many useful facts combined in one volume. Real pleasure to read it.
The only drawback is sketchy last chapter (almost no proofs due to the lack of space, I believe).

Fascinating
This book is fascinating!Bhatia has made an excellent selection of topics. It is frequently cited in the quantum information literature, and I assume also in the literature of other research subjects.This is on matrix analysis, and it has the flavor of finite-dimensional functional analysis.It is concise and has a very interesting selection of topics.

I have a few suggested tweaks for future(?) editions or classroom discussions:

Remarks on chapter 2:

The presentation at the beginning of chapter 2 would be more motivated if one operationally defines x to majorize y iff y = Ax for some doubly-stochastic matrix A.Bhatia uses an algebraic definition and then proves the equivalence after six pages later. Immediately giving an unmotivated algebraic condition robs the reader of the chance to discover or prove the condition for himself.

There is a very confusing typo in the proof of theorem II.2.8.The statement

"Let r be the smallest of the positive coordinates of x"

should read

"Let r be the smallest of the positive coordinates of y".

Another small remark: Just after the statement of Corollary II.3.4 Bhatia states that "one part of Theorem II.3.1 and Exercise II.3.2 is subsumed by [Corollary II.3.4]." In fact, they are equivalent!That II.3.1 and II.3.2 imply II.3.4 follows immediately from the following

Observation: If f:R->R and g:R->R are convex and f is monotonically-increasing then f composed with g is convex.

Notes on chapter 4:

It would be nice to have the isomorphism between balls and norms presented, perhaps just as an exercise.Then the reader can get a visual mental picture of the various conditions for a norm to be a symmetric gauge function.It might also be nice to move theorem IV.2.1 to the very beginning of that chapter, so that the reader sees the point of section IV.1 immediately.

A small remark is that the proof of Theorem IV.1.8 is made slightly more transparent by the observation that by Theorem IV.1.6 on has

[Phi(x^p)]^(1/p) = Sup Phi(xz),

where the supremum is over z such that (Phi[z^q])^(1/q)=1.(The Sup is attained when x^p = z^q.)Then Theorem IV.1.8 follows immediately from the triangle inequality and subadditivity of suprema:

[Phi(x+y)^p]^1/p = Sup Phi((x+y)z) <= Sup [Phi(xz)+Phi(yz)] <= Sup Phi(xz) + Sup Phi(yz)

Chapter 5:

Chapter 5 covers some of the most interesting and surprising mathematics I have ever seen.

Remarks:

1. All the regularity needed to classify the matrix monotone functions is already present in the case of 2 x 2 matrix monotone functions.Perhaps concretely classifying them would modularize the parts of a complicated proof, allowing some separation between discussion of operator convexity and monotonicity.(Let f:R->R be non-constant. Then f is 2x2 matrix monotone iff f is differentiable with df/dt>0 everywhere and (df/dt)^(-1/2) concave.Furthermore, the first two estimates of Lemma v.4.1 continue to hold for 2x2 matrix monotone functions.)

2.Theorem V.3.3 has somewhat restrictive assumptions: Let f:R->R be extended to a map on self adjoint matrices using the functional calculus. Then all that is needed to differentiate f(A+tH) at t=0, where A and H are self-adjoint and t is a real parameter, is for f to be differentiable on the spectrum of A.(f could be discontinuous except on spec(A), for example.)

3. I would have liked to have the definition the "second divided difference" of f at the points {a,b,c} to be "the highest-degree-coefficient of the at-most quadratic polynomial P that interpolates f on the set {a,b,c}. When a=b then one choses P such that P'(a)=f'(a) as well. When a=b=c then one also takes P''(a)=f''(a)."This is the point of exercise V.3.7, but it makes for easier reading for the definition to be conceptual and let the exercise be to work out the algebraic consequences.

Furthermore, if desired one can actually avoid this calculation and proceed to the proof of Theorem V.3.10.(Just replace f byinterpolating polynomials and evaluate everything by by algebra. It has the flavor of Feynman diagrams.)

4.In Hansen and Pedersen "Jensen's operator inequality," Bulletin of the London Mathematial Society," 35 pp. 553-564 (2003); arXiv:math.OA/0204049 (2002), the original authors of the non-commutative jensen inequality state

"With hindsight we must admit that we unfortunately proved and used [a different formulation of the noncommutatitve Jensen's inequality].However, this necessitated the further conditions that 0 is an element of I and that f(0) < 0, conditions that have haunted the theory since then."

Bhatia's presentation is somewhat out-of-date because it does not include the more up-to-date Jensen's inequality from the more recent work cited above.(Note that the more recent paper occured after the current 1996 edition of Bhatia was published.)

Furthermore, in the same paper, Hansen and Pedersen also introduce a nice version Jensen's trace inequality.It is the same as their sharper form of Jensen's operator inequality, except that both sides have a trace in front and that the operator convex function f is replaced by an arbitrary (scalar) convex function f:R->R. (f acts on matrices using the functional calculus).In particular, the trace inequality is much simpler to prove and more widely applicable although less powerful.

5. It would be nice in future editions(?) to include a reference to Petz and Nielsen's nice little proof of strong subadditivity of the von Neuman entropy.

Chapter 7:

I would have liked to see section 7.1 replaced with the following theorem statement (very similar to what's already in 7.1), and see it proved without chosing an arbitrary basis.(Using an arbitrary basis makes Bhatia's proof of the C-S theorem a bit messy, but a reformulation avoids that.)

Definition: A unitary map U on a Hilbert space is a planer rotation iff
U restricts to the identity on a subspace P of co-dimension 2, and P is unitarily equivalent to

cos(t) sin(t)
-sin(t) cos(t)

on P.

Theorem: Let E and F be distinct subspaces of the Hilbert space H, with dim E = dim F. Then there exists a set of planer rotations {R_i} with the properties that

1. The two-dimensional rotation subspaces of the R_i are mutually orthogonal and intersect E and F. (In particular, the R_i commute.)

2. Each R rotates by an angle theta in (0, pi/2].

3. E is rotated onto F by the product of the R_i.

Furthermore, the collection of angles theta_i is uniquely determined by E and F, including multiplicity. If the angles theta_i are distinct and strictly less than pi/2 then the corresponding R_i are also uniquely determined.

Further remark on chapter VII: There is an error on page 223.The author states "we have a bijection psi from H tensor H onto L(H), that is linear in the first variable and conjugate linear in the second variable".

This is impossible, since (lamda v) tensor w = w tensor (lamda w).In particular, any map that is linear in the first variable is necessarily linear in the second variable.The practice of introducing a map from H tensor H to L(H) is a cause of much ugly basis-invariance-breaking in quantum information theory and consequently should be discouraged.

An excellent book on this topic!
This book is an expansion of the author's lecture notes "Perturbation Bounds for Matrix Eigenvalues" published in 1987.I have used both versions for my students' projects.The book under review centers aroundthe themes on matrix inequalities and perturbation of eigenvalues andeigenspaces. The first half of the book covers the "classical"material of majorisation and matrix inequalities in a very clear andreadable manner. The second half is a survey of the modern treatment ofperturbation of matrix eigenvalues and eigenspaces. It includes lots ofrecent research results by the author and others within the last ten years. This book has a large collection ofchallenging exercises.It is anexcellent text for a senior undergraduate or graduate course on matrixanalysis. ... Read more

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Product Description
This volume on complex analysis offers an exposition of thetheory of complex analysis via a comprehensive set of examples andexercises. The book is self-contained and the exposition of newnotions and methods is introduced step by step. A minimal amount ofexpository theory is included at the beginning of each section in thePreliminaries, with maximum effort placed on well-selected examplesand exercises capturing the essence of the material. The examplescontain complete solutions and serve as a model for solving similarproblems given in the exercises. The readers are left to find thesolution in the exercises; the answers, and occasionally, some hints,are given. Special sections contain so-called Composite Examples whichconsist of combinations of different types of examples explaining someproblems completely and giving the reader an opportunity to check allhis previously accepted knowledge.
Audience: This volume is intended for undergraduate and graduatestudents in mathematics, physics, technology and economics interestedin complex analysis. ... Read more