A Course of Modern Analysis (Cambridge Mathematical Library)

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Summary: The Bible of math methods in physics
Review: 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.

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Summary: This book is worth it's weight in gold!!
Review: First I would like to congrulate the 101 years birthday of this superclassic of the classics! If anyone who study mathematics and wants to know how a book can be judged as a classics, this is the one !! The first of edition is in 1902!!!Perhaps no other mathematical textbook is more famous and useful than this one . Think about a book which can survive for almost a century and up to now still no other advanced reference book can replace it. Of course new theories emerged in the last decades and of course would not be included in this book. But the style Whittaker and Watson wrote this book that I have yet to find a similar book!! They wrote this classics in a concise style and prove the theorems in the most rigorous manner. The materials are both useful in pure and applied mathematicians. That is the reason why so many advanced (Noted I use the word advanced ) textbooks today still stated it as the book for further (or ultimate ) reference. I definitely agree with other reader's review on this book that this book is in the hall of fame and will be eternal. Everyone who want to pursue in mathematics MUST own that book. One pity is that we know very little about these two men. As the book only print out the authors' name and even don't even mention which university in which they chaired the professorship. For E. T. Whittaker, fortunately in his another classic book on dynamics, in that book there is a biography ( though just a brief one )written. Unfortunately, for G. N. Watson, even in his classic book on Bessel's function, none of a word about him.I hope in the future, when further reprint edition appears ( it will ), the publisher will provide at least a brief ( or course detailed ones is better ) biography about these two great mathematicians and teachers.( Brilliant mathematicians does not mean they are good teachers,think about the greatest giants like Newton or Gauss,if they were not enthusiastic in passing their knowledge to the next generations,the knowledge would only be lost!!! ) In this aspect,only Euler,the " analysis incarnate ", have done that. P.S., one can find the biographies of them in the "Dictionary of scientic Biography " of these two men.It is worthwhile to look at the biographies of these great teacher!!! Also, I think ,
there is no doubt that this book will survive for another 100 years!!![.]

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Summary: The book on analysis and special functions
Review: 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.

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Summary: All Business Hall of Famer
Review: 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.

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Summary: This book is worth it's weight in gold!!
Review: 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.

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Summary: About the other author, YOU STILL DID NOT BUY IT?
Review: Neville Watson's mother was Mary Justina Griffith, the daughter of the rector of Ardley in Oxfordshire. Neville's father was George Wentworth Watson who was a schoolmaster, but is more famous for his work as a genealogist. He played a large role in the publication of The Complete Peerage, a 13-volume database of the British peerage, generally accepted as the greatest British achievement in the field of genealogy. The first edition was published in London between 1887 and 1898. George and Mary Watson had two children, a boy and a girl, the eldest being Neville.

Neville was educated at St Paul's School in London where he was very fortunate to have the outstanding teacher of mathematics Francis Macaulay. He mixed with equally outstanding pupils, for Littlewood, less than a year older than Watson, was also a pupil at the school. Having won a scholarship to Trinity College, Cambridge, Watson matriculated there in 1904. At this time there were three young fellows of Trinity all of whom had a major influence on Watson's mathematics. They were Whittaker, Barnes, and Hardy. Perhaps the one from this trio who had the greatest influence on him was Whittaker, despite the fact that he left Cambridge in 1906, two years after Watson began his studies there.

Watson graduated as Senior Wrangler in 1907 (meaning that he was ranked in first position among those who were awarded First Class degrees), completing the Mathematical Tripos in the following year in the second division of the First Class. He won a prestigious Smith's Prize in 1909, becoming a Fellow of Trinity College in 1910. This was particularly pleasing to him for he had a great love of his College, and throughout his life he collected prints of the College and of previous Fellows.

After election to his Trinity fellowship, Watson spent four further years in Cambridge before leaving to take up an assistant lectureship in University College, London. From 1918 to 1951 he was Mason Professor of Pure Mathematics at Birmingham. He married Elfrida Gwenfil Lane, the daughter of a farmer from Holbeach in Lincolnshire, in 1925. They had one son.

Watson worked on a wide variety of topics, all within the area of complex variable theory, such as difference equations, differential equations, number theory and special functions. He is best known as a joint author with Whittaker of A Course of Modern Analysis published in 1915. The first edition of the book has only Whittaker as an author. In 1922 Watson published The theory of Bessel functions which was another masterpiece. Titchmarsh wrote of Watson's books (see for example [2]):-

Here one felt was mathematics really happening before one's eyes. ... the older mathematical books were full of mystery and wonder. With Professor Watson we reached the period when the mystery is dispelled though the wonder remains.

One piece of work undertaken by Watson deserves special mention. It involves the problem of wireless waves, which were quickly found to travel long distances despite the fact that theoretically they should not have been able to follow the curvature of the Earth. A mathematical model had been constructed where the Earth was represented by a partially conducting sphere surrounded by an infinite dielectric. Such a model had been used by Macdonald, Rayleigh, Poincaré, Sommerfeld and others. Although Watson was not interested in how best to model the situation, he was, however, very interested in using his expertise to determine mathematical solutions to the given model which others might then check against observations. He obtained solutions to the problem in 1918 which showed conclusively that the model was not a satisfactory one.

In 1902 Heaviside had predicted that there was an conducting layer in the atmosphere which allowed radio waves to follow the Earth's curvature. This layer in the atmosphere, now called the Heaviside layer, was only a conjecture in 1918 but it was suggested to Watson that, having shown the previous model to be wrong, he now look at the model resulting from the postulated Heaviside layer. Watson showed that if the layer was about 100 km above the Earth's surface and it had a certain conductivity, then indeed the solutions obtained closely matched observations. That Heaviside, and Watson, were correct was confirmed in 1923 when the existence of the layer was proved experimentally when radio pulses were transmitted vertically upward and the returning pulses from the reflecting layer were received.

Watson undertook a major project by examining in detail Ramanujan's notebooks, extending his results and supplying proofs. In fact he wrote twenty-five papers relating to results in Ramanujan's notebooks, and he spent many hours making a hand written copy in wonderful script of all the notebooks. He enjoyed numerical calculations and spent many happy hours doing numerical work on his calculating machine.

He was elected to the Royal Society of London in 1919. In 1946 he received the Sylvester Medal of the Royal Society:-

... in recognition of his distinguished contributions to pure mathematics in the field of mathematical analysis and in particular for his work on asymptotic expansion and on general transforms.

Watson was also very active in his support for the London Mathematical Society. He served as secretary from 1919 to 1933, president from 1933 to 1935 and acted as an editor of the Proceedings of the London Mathematical Society until 1946. The Society awarded him their De Morgan Medal in 1947. The Royal Society of Edinburgh elected him to an honorary fellowship.

We find a little of Watson's personality described in [2]:-

He was the university's expert on the timetable; students with unusual combinations of subjects usually had to be referred to him for advice, and for many years after his retirement the dates of the academic year were governed by the "Watsonian cycle". ... He took great trouble with the style of his letters and his conversation and enjoyed finding a pungent phrase to express his points of view or his criticism ... he made no secret of his aversion to cars, telephones, and fountain pens. He loved trains - whose timetables were as familiar to him as those of the university lectures - and unusual stamps.

Article by: J J O'Connor and E F Robertson

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Summary: The DEFINITIVE text for classical Analysis
Review: 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!

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Summary: I love this book
Review: 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.