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Title: A Course of Modern Analysis by E. T. Whittaker, G. N. Watson ISBN: 0-521-58807-3 Publisher: Cambridge University Press Pub. Date: 13 September, 1996 Format: Paperback Volumes: 1 List Price(USD): $60.00 |
Average Customer Rating: 5 (13 reviews)
Rating: 5
Summary: The book on analysis and special functions
Comment: 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.
Rating: 5
Summary: A true classic of classics indeed...
Comment: 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.
Rating: 5
Summary: About the other author, YOU STILL DID NOT BUY IT?
Comment: 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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Title: Special Functions by N. N. Lebedev, Richard Silverman ISBN: 0486606244 Publisher: Dover Pubns Pub. Date: 01 June, 1972 List Price(USD): $14.95 |
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Title: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies by E. T. Whittaker ISBN: 0521358833 Publisher: Cambridge University Press Pub. Date: 15 December, 1988 List Price(USD): $53.00 |
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Title: Riemann's Zeta Function by Harold M. Edwards ISBN: 0486417409 Publisher: Dover Pubns Pub. Date: 13 June, 2001 List Price(USD): $14.95 |
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Title: Special Functions by George E. Andrews, Richard Askey, Ranjan Roy, G.-C. Rota ISBN: 0521789885 Publisher: Cambridge University Press Pub. Date: 15 February, 2001 List Price(USD): $45.00 |
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Title: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Table by Milton Abramowitz, Irene A. Stegun ISBN: 0486612724 Publisher: Dover Pubns Pub. Date: 01 June, 1965 List Price(USD): $34.95 |
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