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Title: Complex Variables : Introduction and Applications by Mark J. Ablowitz, Athanassios S. Fokas, D. G. Crighton ISBN: 0-521-53429-1 Publisher: Cambridge University Press Pub. Date: 28 April, 2003 Format: Paperback Volumes: 1 List Price(USD): $50.00 |
Average Customer Rating: 5 (6 reviews)
Rating: 5
Summary: Excellent book -- Unique selection of topics
Comment: This is the only textbook that I know that introduces and explains the Hilbert-Riemann problem in a pedagogical way. If anyone knows of any other such book, please tell us. It also deals in a very introductory way with all sort of really nice topics that one cannot find discussed (at a really introductory level) in any similar book: the Painleve property, the classification of singularities, asymptotic expansions, etc, etc. All very powerful applied mathematics.
Rating: 5
Summary: A very good text!
Comment: A very good text!
The best description of this book is that it provides a comprehensive, classical treatment of the subject with a modern touch and serves ideally the needs of anyone studying Complex Analysis.
Starting from the foundations of defining a complex number, through to applications in the evaluation of integrals, the WKB method, Fourier transforms and Riemann-Hilbert problems, the book covers a lot of ground in an easy to follow style. The chapters are long, but logically broken down into digestible sections and interspersed with well illustrated diagrams, numerous worked examples and exercises. The end of chapter exercises provide further opportunity for reinforcing the methods and there's a useful section at the end giving brief hints and answers to selected problems.
Complex Variable analysis is treated from the definition of an analytic function and its relation to the Cauchy-Riemann equations, and in turn their application to an ideal fluid flow. The ideas of multi-valued functions, complex integration, and Cauchy's theorem are excellently treated, as are the consequences: the generalised Cauchy integral formula, the Max-Mod principle, and Liouiville and Morera's theorems.
The rest of the first part of this book, which is essentially pure mathematics, deals with Laurent series, singularities, analytic continuation, the Mittag-Leffler theorem, the ALL IMPORTANT Cauchy Residue Theorem, dealing with branch points, Rouche's theorem, and their application to Fourier transforms.
The second half starts off with perhaps the best I have seen on Conformal Mappings and their application to physical problems in Fluid Mechanics and Electromagnetism. Asymptotic evaluation of integrals covers methods like Watson's lemma, the method of steepest descent, and the WKB method.
A good combination of pure and applied mathematics, though the book avoids either the rigour of classical works such as Whittaker and Watson or the marvellously visual presentation of Tristan Needham.
Highly recommended!
Rating: 5
Summary: The numerous pictures are enough to recommend this text
Comment: This text is distinguished by the numerous diagrams that appear on practically every other page. If you're graphically oriented, like I am, then this itself is enough to recommend this book. Concepts such as branch points and multivalued complex functions are much easier to understand when there is a picture to accompany the concept. The second half of the book is concerned with applications and includes several useful asymptotic methods such as Laplace's integral method. These asymptotic techniques are good for evaluating particularly nasty integrals in which the integrand is really concentrated somewhere in the interval. On the downside, this is not a very formally rigorous book. On the other hand, such formalism is easier to digest once you've seen numerous pictures and examples, in my own opinion.
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Title: Numerical Mathematics by Alfio Quarteroni, Riccardo Sacco, Fausto Saleri ISBN: 0387989595 Publisher: Springer Verlag Pub. Date: 20 April, 2000 List Price(USD): $64.95 |
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Title: Partial Differential Equations : An Introduction by Walter A. Strauss ISBN: 0471548685 Publisher: Wiley Text Books Pub. Date: 03 March, 1992 List Price(USD): $87.95 |
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Title: Visual Complex Analysis by Tristan Needham ISBN: 0198534469 Publisher: Clarendon Pr Pub. Date: June, 2000 List Price(USD): $65.00 |
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Title: Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics by John Derbyshire ISBN: 0309085497 Publisher: Joseph Henry Press Pub. Date: 23 April, 2003 List Price(USD): $27.95 |
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Title: Schaum's Outline of Complex Variables by Murray Spiegel ISBN: 0070602301 Publisher: McGraw-Hill Trade Pub. Date: 01 June, 1968 List Price(USD): $16.95 |
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