Customer Reviews

A Course of Modern Analysis

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This book isn't Modern anymore. Thank God! It is certainly the most useful book of mathematics I ever put my hands on. If you read its page of contents, you'll call it prophetic! Every kind of function he studied became important in theoretical physics some time. String theory was started with an amplitude containing only Gamma functions. Renormalization, reborn from the ashes, discovered the Zeta-function (in Whittaker-Watson, for sure), Legendre's less familiar functions were prominent in Regge pole theory (again, the source was Whittaker), and even the Theta functions became important for some field theory skirmishes. You could travel light: Whittaker, Watson, tooth brush, etc. It's not only what there is in it. It's also the fact that it's done better! Consider this: I had once an ugly series to sum up. These were the days before Maple! I couldn't find it anywhere, having looked into immense mathematical tables. I came back to old Whittaker and there it was: in an e! xercise, asking you to prove that the sum of MY series was some function he wrote in all detail! This is Whittaker-Watson. God bless them.

Although I was aware that he'd read other books, and knew much more than is taught here, this was (in my years as his grad student) the only book that I saw Lars Onsager pull off his shelf, well-worn and dog-eared, it was! It's one of the many 'Onsager tales' that circulate among his former students and postdocs that he'd worked through all the problems in this text (just for mental exercise) as undergrad at NTH. One can believe it if one takes the trouble to read his Ph.D. dissertation on weak electrolytes, where a pde is solved exactly by using an 'extremely inventive' method based on complex analysis (the dissertation lies in Yale's Beineke library). I later used the book, along with Stakgold (on boundary-value problems) to teach a first semester grad 'math methods' course to physics and engineering students. I must say that in that time the grad students had no difficulty working the problems, although I certainly did not assign the hardest ones (Tripos...). I usually went as far in series expansions and complex variables as the Mittag-Leffler expansion, spending about a half a semester on W&W before switching to delta functions, boundaty value problems, and Stakgold. Fuch's theorem was covered in the second semester via Bender & Orszag.

The older I get, the more I realise the truth of what my expert colleagues told me a long time ago: there is ONE book on analysis, and it's called Whittaker and Watson. Shame on CUP for reprinting it in less than perfectly top quality. I guess they know that people will always buy it. It is a book that starts from the very basics of real and complex analysis, and moves on to the very depths of classical special functions. It's a joy to read and to teach from. No respectable mathematical physicist can afford not to own a copy. And it's about 1/4 the price of a typical, low level, textbook.

I own the 1940 HB edition (which was itself a reprint). It was terribly hard to track down and I had to pay a fortune for it. Be glad it's now in reprint. This book is probably in more bibliographies than any other in the 20th century mathematics. For that reason alone it's worth every penny. The book is all business with little extraneous comments, applications, or excursions that often make higher mathematics such a joy. That being said, the 608 pages cover a lot of ground which is probably why it is on so many reference lists.

Despite it's fanfare in the mathematic communitiy, the subjects dealt arise from physics and engineering rather than pure mathematics. I don't think there is a chapter without practical application. Unlike many more recent texts on the subject, the authors cover Theta Functions and Elliptic Functions (Jacobian and Weistrass).

This is definitely a Hall of Famer in the Math Universe.

The DEFINITIVE text for classical Analysis

This book is the definitive text in classical Mathematical Analysis. It was first published in 1902 and the fact that it is still in print is testimony to it's wide ranging utility and appeal.

It should be noted that this text is not for those who are new to the rigour of Analysis; its presentation is suitable for a final year undergraduate or for the post-graduate student. More importantly, its wide ranging content of proofs and results would also prove useful to the Physicist.

The first part of the book covers the "essentials" of analysis: continuity, differentiability, summation of series, convergence and uniform convergence, and the theory of the Riemann integral. Subsequent chapters quickly but comprehensively develop the theory of analytic functions, the theorems of Cauchy, Laurent, and Liouville and the calculus of residues. These chapters knit very well into the earlier presentation of the basic processes of analysis! The pleasing thing is that despite the passage of time and the advent of hundreds of books on Complex Variable Theory, Whittaker and Watson's treatment still bears a mark of freshness and rigour.

Also included is a comprehensive treatment of expanding functions in infinite series and asymptotic expansions and summability of series. For completeness, the text also covers the theory of linear differential equations and Fourier series.

The second part of the book is what stands it apart from the rest. The authors provide a comprehensive discussion of the major transcendental functions: Gamma, Zeta, Hypergeometric, Legendre, and Bessel to name the more commonly encountered ones. The treatment is rigorous but the copious number of examples provides opportunity to learn the theory and apply it. Lots of apparently obscure results, many that would be useful in Physics applications, are cited as examples.

The latter chapters presents a treatment of Elliptic, Theta and Mathieu functions.

Overall, Whittaker and Watson will continue to be the guiding light for any serious scholar of classical analysis and an excellent reference point for the solutions to the fundamental equations of Mathematical Physics. Even though I am not a practising Mathematician, I find this a pleasant book to dip into: there's always a little surprise and something new to learn.

This book will live forever!

I decided to purchase this title about three months ago after hearing lots of praise about it on the internet and wanting to learn the subject, and I can now see that this praise was not exaggerated. A hundred years after its first publication, this classic still remains the definitive general reference in the field of special functions and is a very solid textbook in its own right.

The book is split into two main parts: the first consists of short (but detailed) overviews of the various sub-disciplines of analysis from which results are required to develop later results, and the second part is devoted to developing the theories of the various kinds of special functions. The sheer breadth of topics and material that this book covers is utterly incredible. The major topics covered in the first part of the book are convergence theorems, integration-related theories, series expansions of functions and differential/integral equation theories, each of which are split into two or three chapters. The reader is assumed to be familiar with some of the subjects here and these chapters are intended more as a review, but they are still quite self-contained and will also appeal to those who have not encountered the subjects yet. (I am only 16 and know no more than ODEs and a little real analysis, but I learned some material from this)

The second section, which is really the heart of the book, starts off with a detailed treatment of the fundamental gamma and related functions, followed by a chapter on the famous zeta function and its unusual properties. The book then covers the hypergeometric functions - the focus is on the 1F1 and 2F1 types, being ODE solutions - which are perhaps the cornerstone of this field, followed the special cases of Bessel and Legendre functions. There are a number of ways of developing and teaching the ideas regarding these functions; this book mainly uses the differential equation approach, starting by defining these functions as solutions to ODEs and going from there. There is also a chapter on physics applications (using these functions to solve physics equations), which is sure to please the more applied math readers. The next three chapters are devoted to elliptic functions, covering the theta, Jacobi and Weierstrass types. (one chapter on each) The two remaining chapters are on Mathieu functions and ellipsoidal harmonic functions. Along the way, some additional functions are also sometimes mentioned in the problem sets. (barnes G, appell, and a few others) About the only room for improvement here would be some analyses of named integrals (EI, fresnel, etc.) and inverse functions (lambert W log, inverse elliptics, etc.), and perhaps more on multivariable hypergeometrics, but these things are not a big deal considering how much else appears in here, and I have not really seen any book out there that covers these anyway.

Each chapter has several subsections, usually one on each major theorem or property of the function in question, and these consist of the main discussion and proof, a few corollaries, and a couple of exercises that illustrate the usage of the theorem. At the end of the chapter, some more sets of problems are given; these mostly consist of proving identities and formulas involving the functions, so answers are not needed, but it would be nice if there was a showed-work solutions book available for students. The problems themselves are very well designed and some really require the use of novel methods of proof to obtain the result. The language is a bit in the older style with some unconventional spelling and usage, but it does not detract from the subject material at all (actually, I personally liked this form of writing), and the price is about right.

The only real complaint I have with this book has nothing to do with its content; it is the printing quality. The text font is simply too small in a number of places and also sometimes looks "washed out;" while it is still readable, such a classic gem as this definitely deserves a better effort on the publisher's part. (one of CUP's other works on the same subject, Special Functions by Andrews et al, has much better printing, although is not as good as this in other respects)

For those interested in the field of special functions and looking for something to start off with, A Course of Modern Analysis would be, hands down, my first recommendation. You cannot really do much better than this.

If I could, I would give this book ten stars. When I first sat down to read it, I couldn't believe what I was seeing. This is the only book I have ever seen on complex analysis (or any scientific field for that matter) in which the authors cover so much material (everything from residues to integral equations to elliptic functions and MUCH more) and yet manage to make the whole text fit into a framework which is relatively easy to follow, even for someone completely new to complex analysis. Moreover, the majority of the many hundreds of excercizes in this book range from moderately to nail-bitingly hard, and encourage a true understanding of the material being covered. I would reccommend this book for ANYONE who has mastered basic calculus and analysis and wishes to begin learning complex analysis and the theory of special functions. The book's coverage of the following topics is especially noteworthy: The gamma function (the book uses the INFINITE PRODUCT as the basic definition), the hypergeometric function (and the confluent hypergeometric function), bessel functions (a field in which G.N. Watson was a leading expert), and the Weirstrassian and Jacobean elliptic functions and theta functions (I LOVED the intuitive development of the theory of the elliptic functions, which is made to parrallel that of the trigonometric functions, which are of course familiar to the reader). I would ESPECIALLY recommend this book for those pursuing SELF-STUDY (although it is NOT for the mathematically weak-of-heart, but no book on the topic is), as it is quite self-contained and readable for a book on complex analysisis. Once you buy it, you won't even think to complain about the high pricetag, because you will be way too absorbed in the math to think about anything else.

Years ago I discovered this book while studying for my electrodynamics and mechanics comprehensives. What a godsend! If the physics graduate student understands only ten percent of what is in this book he will do fine. Combined with the classical texts on electrodynamics and mechanics I discovered I became truly dangerous in the realm of classical physics. Still am much to the chagrin of my colleagues. Still the best after all these years, I cannot recommend this book too highly.

If you are going to do any Mathematical Physics analysis you should have this for an example and referance.

Please take note: the latest "new" edition of the book was in 1927. That was the fourth edition. Everything since then has been a reprint. You're not getting anything new if you buy a 1997 reprint; even the typeface is the same, only the cover is blue.