4 comments about Continued Fractions.

1. This is Khinchin's classic work, translated from Russian in the 1930's. Although the book is rich with insight and information, Khinchin stays one nautical mile ahead of the reader at all times, the book moves at a truly alarming pace, and the book is not suitable to be used ALONE as an introduction to continued fractions. To supplement this book if this is a first exposure to continued fractions, I would recommend C.D. Old's book, which has many more examples which can be worked through until the reader is comfortable with the topic.

   The book is brilliant and necessary for understanding continued fractions, but can't stand alone without supplemental material unless one is a professional mathematician. Khinchin frequently employs contrapositive proof formats, and there are occasional translation errors from Russian. The errors range from minor (awkward usage) to major (in one place, the translation is "negative" when it should be "non-negative", which confused me for half a day).




2. A Y Khinchin was one of the greatest mathematicians of the first half of the twentieth century. His name is is already well-known to students of probability theory along with A N Kolmogorov and others from the host of important theorems, inequalites, constants named after them. He was also famous as a teacher and communicator. Several of the books he wrote are still in print in English translations, published by Dover. Like William Feller and Richard Feynman he combines a complete mastery of his subject with an ability to explain clearly without sacrificing mathematical rigour.

   In this short book the first two chapters contain a very clear development of the theory of simple continued fractions, culminating in a proof of Lagrange's theorem on the periodicity of the continued fraction representation of quadratic surds. Chapter three presents Khinchins beautiful and original work on the measure theory of continued fractions. The proofs of the theorems in this chapter are also entirely elementary.




3. You won't find many books on such an out-of-fashion theme as continued fractions, will you? Even less on the arithmetic side of the theory. Yes, it's true, many texts on elementary number theory provide a chapter or so about the subject, but if you want to gain a reasonably thorough picture of the field, without dwelling so much on details, you've got to resort to Kinchin's "Continued fractions": readable (no more mathematic needed than basics of analysis), complete (all fundamental conceptual aspects dealt with, included measure theory and implications on irrational numbers), brief (less than a hundred pages with virtually no applications - not even to Pell's equation!) and LIVELY in style.
   All in all a very good start for understanding this profound mathematical tool.



4. A wonderfully written, clear exposition of advanced material which, however, begins simply enough to lure one in.

5 comments about The Elements of Integration and Lebesgue Measure.

1. Measure and Integration is a daunting subject for mathematical neophytes. Bartle's little volume is the right place to start. I first learned measure theory from it 20 years ago and went on to study functional analysis and stochastic approximation.

   I was able to master the material on my own with this book. The problems are at the right level and he begins with the correct level of abstraction. I recommend it over anything else because it is straighforward, clear and focused. Master it then go on to Walter Rudin's Real and Complex Analysis.




2. When I took my first one-semester course on measure and Lebesgue integration my teacher chose Bartle's "The Elements of Integration" as text. After reading many other books on the subject now I'm sure he made a wise decision.

   Assuming almost no strong mathematical background, Bartle is able to build up the basic Lebesgue integral theory introducing the fundamental abstract concepts (sigma-algebra, measurable function, measure space, "almost everywhere", step function, etc.) in such an easy way that the student is not only able to handle them but to UNDERSTAND them.

   From the first part of the book I appreciate specially chapters 6, 7, and 10, on L_p spaces, modes of convergence, and product measures, respectively. These chapters contain the most used results of the basic theory, and they are stated exactly in the way one needs them, making the book very useful for future reference.

   I like the second part very much also, because it stresses the importance of measure theory by itself and not only as a requisite for integration theory. If you are interested in fractal geometry or geometric measure theory you will find chapters 11 to 17 very helpful.

   Since I own this book it has never been lazy in my bookshelf.




3. IF YOU WANT TO UNDERSTAND MEASURE THEORY READ THIS BOOK, MAYBE THE ONLY PROBLEM IS THE LACK OF EXAMPLES BUT THE WAY THAT THE THEORY IS PRESENTED MAKE IT YOUR FIRST CHOICE WHEN YOU TRY TO LEARN MEASURE THEORY.



4. The exposition of integration in this book is the clearest I have read. I also found the chapter on modes of convergence, where it laid out the relationship between things such as L^P-convergence and convergence in measure, to be extremely useful. The second half, where it covers topics like Lebesgue measure, repeats some of the same information from the first part which is a bit iritating if you are reading straight throught, but contains a lot of good information. The book is also quite small making it easy to take with you as a quick reference.
   
   Let me warn you though that this is an introduction to integration and measure _not_ an introduction to real analysis. It does not cover important topics like L^P-approximation, differentiation, etc. For a complete treatment of real analysis, I recommend the books "Lebesgue Integration on Euclidean Space" by Frank Jones and the slightly more abstract "Real and Functional Analysis" by Serge Lange.



5. The book is concise and easy to follow. The author rarely gives lengthy explanations and analogies, but spends the bulk of the book stating solid facts and proofs. I also like the organization of the book. All definitions and theorems are explicitly stated and indexed, not scattered in paragraphs in the body of the text.
   
   The book misses subjects such as complex measures (they are briefly mentioned), the fundamental theorem of calculus under Lebesgue settings, and probability measures, but its ok since the book is an introduction to the subject. A more comprehensive (and harder to read) book is "Real & Complex Analysis" by Walter Rudin. If you are interested in probability, consider Ptrick Billingsley's book "Probability and Measure".

3 comments about Iterative Methods for Sparse Linear Systems, Second Edition.

1. This is a great book for this subject. The book is easy to follow and Saad does a wonderful job of illustrating with examples. This is a great textbook or a book for reference. This book does a particularly good job with Krylov methods and does a reasonable job with preconditioning.



2. This is one of my favorite books in my library on this subject. Also I have used this book for my class as main textbook along with "Iterative Methods for Solving Linear and Nonlinear Equations" by C. T. Kelley , which is another SIAM book.
   Highly recommended.



3. We used this book to prove a theorem in our studies that is directly related to my PhD thesis on spatial data mining and spatial statistics. This book is a master-piece.
   Thanks Dr. Saad.

1 comments about A Guide to Distribution Theory and Fourier Transforms.

1. Distributions are objects most physicists will frequently encounter during their career, but, surprinsingly, the subject is not given the place it deserves in the current ordinary science curriculum.
   I would particularly recommend this book to physics students willing to learn the foundation of distribution theory and its close ties to Fourier transforms. Distribution theory is, basically speaking, a way of making rigorous the operations physicists find Ok to carry on functions, that otherwise wouldn't rigorously make sense. Distribution theory therefore provides a useful way of checking, in the process of a calculation, if it is allowed (according to the extended rules of distribution theory), or if it is definitely dubious (e.g. current distribution theory doesn't provide a mean of making sense of a product of Dirac delta functions, while such expressions sometimes pop out in the context of quantum field theory ; nevertheless, there exist other formal theories, such as Colombo calculus that aim at justifying this ; yet, for some reason, they seem to bear less power than the original distribution theory).
   This work is an easy, gentle, pedagogical piece of mathematical exposition.
   The subject is wonderfully motivated.
   As such, this book is suited to self-study.
   It could also be used as a textbook for an introductory course on the subject, or as an introductory reading to more advanced texts (Aizenman, for instance).
   Highly recommended.

No comments about The Finite Element Method for Solid and Structural Mechanics, Sixth Edition.

4 comments about Numerical Methods.

1. I had previously sat in on a Numerical Analysis class, and I have to say that this book was much better than the one that we were forced to use. The examples were clear, and the text focused much less on proofs and more on how to do problems and calculate their error. The sample programs on disk were also thorough, and included in several different languages (c, fortran, pascal, maple, mathematica, & matlab).



2. In my never-ending quest for the best textbook for each of my courses, I examine a large number of math and computer books. The current book used in my numerical analysis class is "Elementary Numerical Analysis" by Atkinson and Han. My standard criterion for changing is that the new book must be significantly better than my current favorite. By significantly better, I mean that the cost to me when changing is more than offset by the advantages that the new book gives the students.
   While I consider this book on a par with the Atkinson book and maybe even a bit better, it doesn't quite cross the threshold. The coverage is what is ordinarily considered that of a one-semester course in numerical analysis, so in that regard it is still in the pack. A broad spectrum of exercises appears at the ends of the sections and solutions to the odd-numbered ones appear in an appendix.
   I consider this book to be slightly more readable than some of the others that I have seen. Maple is used as the symbolic mathematics package used to illustrate the examples and a CD containing the code used to generate the examples is included. This book is good enough to tempt me to make the switch but not good enough to make me do it.



3. Thanks so much for this book. At my local bookstore I would have had to pay $180. I went online and found a NEW copy for $40 - saving me almost $140. The book is fine, I use it as a textbook for class, and it is very easy to follow.



4. Well this book is good. I don't have the much to say because we didn't really end up utlizing it in the class that I purchased it for. But it's an alright text.

1 comments about Fundamentals of Precalculus.

1. Well, there isn't much I could say about this textbook, but it did do its job. The textbook had examples and explanations for every section, followed by exercises for practice.
   
   I will say though, it was nice that the book is relatively thin compared to other calculus textbooks I saw my friends lugging around. The textbook was precise and to the point.

1 comments about Introduction To Numerical Analysis Using MATLAB with CD-ROM(Mathematics) (Computer Science) (Mathematics).

1. College-level courses strong in numerical analysis as well as those catering to engineers and scientists will appreciate this introduction: the first to present the theory of numerical analysis and the practical justifications of methodology using the latest version of MATLAB. This will also make a fine college-level text for primary or supplemental reading: it provides short programs in MATLAB to be used for scientific applications, surveys MATLAB commands and processes, includes a CD-ROM featuring source code and simulations, and reinforces theory with applications. An outstanding reference.

5 comments about Special Functions & Their Applications.

1. This is a book which I cannot say much about except for the unusual thoroughness, accompanied by detail and depth in treatment of the underlying mathematical properties and applications of Special Functions.

   Lebedev is the quintessential mathematical expert in applying Special Functions to problems in Physics and Engineering, being that he can illustrate all important concepts clearly and umambiguously with carefully prepared diagrams as well as words. I was able to cite the solution of the a problem involving a propagating electromagnetic wave along a transmission line for an important Engineering course project. For such a problem, Lebedev offers a far more detailed and precise solution with given Special Functions than anything I have ever seen in other books of the same nature with the possible exception of a specialized treatise by an MIT EE faculty member on applied electromagnetism. He also comes across as meticulous in derivations of solutions to problems worked out compared to many other authors whose works I have read. This is because he hardly ever skips an important step in deriving a solution for any given problem by leaving it out for the reader's imagination. Yet we know Lebedev as perhaps a mathematician who may not be realistically expected to come up with such complete and exhaustive solutions to practical or real-world problems, worked out with clarity as well as precision and depth. There are numerous other examples which he worked out for different applications (e.g, Legendre's and Laguerre's functions) invariably after he took pains to delineate the various mathematical properties of the Special Functions utilized to obtain the closed-form solutions. He also covers various mathematical functions which may not be as familiar to many engineering practitioners but nonetheless have an important place in applied mathematical analysis. In a sense, he saves them for occasions when we as readers may need to probe further into unfamiliar territory.

   So if you are looking for depth and precision in analysis of physical problems in Engineering and Science, or are trying to cope with reaearch problems in Applied Mathematics, try out this book by Lebedev. It can initially come across as difficult to understand, but Lebedev expects the reader to follow along through diligence. It is almost one of a kind, being that it is very clear and lucid without noticeable loss in depth and mathematical rigor. I highly recommend it because I believe that few other books can even come close in offering good examples in solutions to real-world problems and, at the same time, demonstrate the power of Special Functions in applications. Of course, it is also very inexpensive.




2. Of course this book cannot be compared to " A course of modern analysis ", but as a book in special functions, it have served its purpose. I would also like to make a tribute to Richard Silverman for translating ( not just direct traslating, he translate in a style make it readable to English world. )



3. Yet another excellect translation by Silverman. I've only been in possession of this book for a few days but it's already becoming a favorite mathematics text. Not a pure mathematics text but certainly a very thorough, lucid and most certainly enjoyable discussion of applied mathematics with a particularly engaging discussion of the solution of partial differential equations (Laplacian, Poisson etc.) by means of separation of variables and integral transforms. Along the way it develops the theoretical essentials of gamma functions, exponential integrals, orthogonal polynomials, Bessel functions, spherical harmonics among others. Clearly written with an emphasis on explaining the process of discovering solutions rather than merely presenting particular solutions (though it does have enlightening examples). IMO, well worth the price.



4. Well worth buying, extremely handy, tons of information very much organized for you



5. As the title indicates, the book is designed with the goal of application front and center. That said, it is also important to note that the theoretical background is developed with full mathematical rigor. You can easily see this from the fact that whenever an infinite series is differentiated, its uniform convergence in the region of interest is always established beforehand. And this is just one example.
   
   Now, given the fact that special functions is a vast subject, and the fact that the book is barely 300 pages long, it is obvious that the theoretical coverage, though rigorous, has to be reined in. By this I refer to the fact that most functions are developed from the point of view of series solutions to differential equations, while solution by contour integrals in the plane is basically absent. But then again, it doesn't matter how you develop the functions, the key is to know their properties and be able to apply them. The book will show you just how to do that. HIGHLY RECOMMENDED.
   
   For a more broad-based theoretical coverage, I recommend Whittaker and Watson (but of course), and the book "Special Functions" by X. Z. Wang. These two books complement each other like lovers.

5 comments about Numerical Mathematics and Computing.

1. I had to use this book for an undergraduate Numerical Analysis class. I'm a Computer Science major with a math minor and this is my last semester. I found this book to be horrible when coupled with an instructor that is equally as horrible. The explainations are too short and lack examples, the problems in each chapter are hard to solve based on the chapter's explaination; they seem to deviate far beyond what was explained in the corresponding chapter. There are some formulas and theorem's mentioned that have no examples to show how they work.

   The book is not totally at fault in my case. I also have a horrible instructor and have to rely soley on this book to learn the material. This book just makes it very, very hard to teach myself. My only praise of the book is it's pseudocode for implementing the methods explained. They can easily be used to program them in C++ or other languages.

   Overall the book is very confusing but it is still far better than my instructor who doesn't explain anything or answer questions.




2. The true test of a textbook's value is whether it can be used to learn the material without the benefit of a thorough and clear lecturer. Considering a textbook's value when supplemented with a good professor isn't proper, because the professor can fill in the book's gaps, making it harder to tell whether the book is good or not.
   
   "Numerical Mathematics and Computing" fails miserably at this test of value. The explanations are very short and feel incomplete, leaving students unsure of how to find the correct answers. The examples which are given to clarify the material are few and far between, and good examples are practically non-exsistant. In general, they skip right over the finer details of how to work through problems, and assume the reader understands what's going on. This might work if the student had already been introduced to the material, or if they had a good professor to fill in the gaps, but that shouldn't be assumed. It certainly seems like it was when this book was written.
   
   I would absolutely discourage anyone from getting this book!



3. After two weeks They didn't have a stock of quality so they gave me a discount for any other book and a full refund



4. I was a teaching assistant for an introductory numerical mathematics course which used this text. It's a satisfactory text (nothing special) if you already have a basis in numerical analysis, however students which have no foundation struggle severely.
   
   The problem stems from the fact that the authors, Kincaid and Cheney, first wrote a graduate level numerical analysis text and then they created this text based on the content from the first book. Needless to say, this "introductory" text makes several [invalid] assumptions about the introductory student's abilities.
   
   It's frustrating to see students struggle because numerical analysis is really not that difficult -- but they have to be taught the procedures clearly. This text does not have enough example problems and the ones they included do not describe the steps thoroughly or the logic behind performing them. The text does include a large quantity of homework problems, but the selected answers in the back of the book provide only answers and no explanation of how the answer was arrived at.
   
   Anyways, if you're still going to buy this book its probably becausre you're a student. Hang in there. It's really not that hard but seek help from other textbooks if needed.



5. In my opinion, this is the best numerical analysis textbook.
   Rather than trying to teach and explain everything to the student in detail, it complements the instructor. The idea is that the students learn in class, and use the text book as a reference, and for homeworks. This is a great idea. Unfortunately pretty much all Calculus books try to teach Calculus, but for a regular student, math is very hard to learn from a text-book... A nice instructor, and a clean presentation is a must. I teach the material I see important, the way it makes sense to me. What I need is a book that complements me, not replaces me.