Special Functions (Encyclopedia of Mathematics and its Applications, Series Number 71) 1st Edition by George E. Andrews (PDF)

Description

Special functions, which include the trigonometric functions, have been used for centuries. Their role in the solution of differential equations was exploited by Newton and Leibniz, and the subject of special functions has been in continuous development ever since. In just the past thirty years several new special functions and applications have been discovered. This treatise presents an overview of the area of special functions, focusing primarily on the hypergeometric functions and the associated hypergeometric series. It includes both important historical results and recent developments and shows how these arise from several areas of mathematics and mathematical physics. Particular emphasis is placed on formulas that can be used in computation. The book begins with a thorough treatment of the gamma and beta functions that are essential to understanding hypergeometric functions. Later chapters discuss Bessel functions, orthogonal polynomials and transformations, the Selberg integral and its applications, spherical harmonics, q-series, partitions, and Bailey chains. This clear, authoritative work will be a lasting reference for students and researchers in number theory, algebra, combinatorics, differential equations, applied mathematics, mathematical computing, and mathematical physics.

User’s Reviews

Editorial Reviews: Review ‘Occasionally there is published a mathematics book that one is compelled to describe as, well, let us say, special. Special Functions is certainly one of those rare books. … this treatise … should become a classic. Every student, user, and researcher in analysis will want to have it close at hand as she/he works.’ The Mathematical Intelligencer‘… the material is written in an excellent manner … I recommend this book warmly as a rich source of information to everybody who is interested in ‘Special Functions’.’ Zentralblatt MATH‘ … this book contains a wealth of fascinating material which is presented in a user-friendly way. If you want to extend your knowledge of special functions, this is a good place to start. Even if your interests are in number theory or combinatorics, there is something for you too … the book can be warmly recommended and should be in all good libraries.’ Adam McBride, The Mathematical Gazette‘… it comes into the range of affordable books that you want to (and probably should have on your desk’. Jean Mawhin, Bulletin of the Belgian Mathematical Society’The book is full of beautiful and interesting formulae, as was always the case with mathematics centred around special functions. It is written in the spirit of the old masters, with mathemtics developed in terms of formulas. There are many historical comments in the book. It can be recommended as a very useful reference.’ European Mathematical Society’… full of beautiful and interesting formulae … It can be recommended as a very useful reference.’ EMS Newsletter‘a very erudite text and reference in special functions’ Allen Stenger, MAA Reviews Book Description An overview of special functions, focusing on the hypergeometric functions and the associated hypergeometric series.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐It is as expected

⭐A good dictionary for the researcher, not proper for the beginners.

⭐Though this book cannot be compared to Whittaker and Watson’s classic book. It comes quite close to it. I just want to comment on the the area covers are too concentrated and the rigorous manner which is the hall mark of ” Modern Analysis ” is lacking. Anyway, this book deserves 5 stars.

⭐Very rare book, difficult to find but found it at an amazing price. Got it and it was as though it was never used and it was in wonderful condition. It also came 2 weeks early, which was a lovely surprise. It even had the author’s signature inside, like it was signed at a book signing or the author himself was selling it.

⭐It is an almost invaluable reference for its topic, although a bit dense and with almost no applications. I always come back to it when trying to understand Bessel functions.

⭐This book is great. It is the best overview I have ever seen of the primary special functions, as seen from a modern viewpoint. Buy it and you will spend many happy hours reading the theorems it contains, and doing the excercizes at the end of each chapter.

⭐I ordered this book through the mail and as soon as I opened it it was a disappointment. The paper quality is average, the typesetting bland and the formulas unappealing, and most importantly, the information incomplete and poorly structured.I had especially been hoping for good coverage of orthogonal polynomials but especially their presentation in this book is very weak: the historical perspective is entirely lacking as is the common theory underlying the classical orthogonal polynomials, and the generating functions which play a crucial role are not even mentioned.In general the book pays no attention to numerical methods for the evaluation of the functions.The scant material on the Gamma and Zeta function shows little insight. There is no mention of the role of the Euler-MacLaurin summation formula to obtain amongst other Stirling’s formula for the Gamma Function.The book is structured as are many mathematics books nowadays: Lemma … Proof etcetera, which is exactly how mathematics books should not be written, in my opinion of course.As a matter of comparison look at H.M. Edwards’s lovely book “Riemann’s Zeta Function” which has nice little chapters, each with an explanatory header, where the material is put in historical context, any difficulties are outlined, and the importance of the result is evaluated.Finally, since this book is part of a series supposed to be some kind of a Mathematics Encyclopedia, I seriously wonder what the purpose is of the exercises or even the proofs.Instead of this hodge-podge of results, one could get for less than half the price the beautiful and insightful book by Hochstadt (The Functions of Mathematical Physics) which covers Orthogonal Polynomials and the Hypergeometric and Confluent Hypergeometric Functions and also Bessel Functions and more… PLUS [Richard R. Silverman’s translation of] N.N. Lebedev (Special Functions & their Applications), which has several chapters on Orthogonal Polynomials.

⭐This book is more advanced than the description leads you to believe. This is not a book for beginners. Some theorems from real analysis and functional analysis not proved are mentioned by name but not even quoted. Not even the introduction tells the reader that real analysis and functional analysis are needed to understand the book. Besides that, the index is very incomplete. It seems to have been prepared by randomly picking pages.

⭐gift

⭐内容が充実で、分かりやすく書かれています。超幾何関数、Bessel, Gamma関数などについて疑問があるとき、この本を参考にしています。参考になる本のひとつです。お勧めです。

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