Complex Variables: Introduction and Applications (Cambridge Texts in Applied Mathematics)

This text offers a broad coverage of the subject, from fundamental properties of complex numbers, analytic functions, and singularities to more advanced topics such as conformal mapping and Riemann-Hilbert problems. Although individuals interested in pure mathematics may find some of the proofs insufficiently rigorous, those using the book as a reference for engineering or scientific problems may find the text too rigid. Overall, however, the authors have done an excellent job balancing the subject matter and successfully achieving their goal of, when necessary, "sacrificing a rigorous axiomatic development with a logical development based upon suitable assumptions."

Although the mathematical development of the text is clear, concise, and easy to follow, many of the applied examples, such as those for uniform flow in section 2.1, would benefit from further physical insight. Individuals already familiar with physics will have no difficulty following many of the examples, and extending them to other situations. Those less grounded in the physical sciences, however, may find the starting equations for some of the examples to be less than intuitive. Though additional explanations would increase the book's already substantial heft, the change would benefit many readers.

It is a joy to read a well-written technical book with almost no typographical or technical errors. Except for minor (and easily recognizable) typographical errors such as that in equation 2.2.12b, the book is nearly flawless. This leaves the student free to concentrate on learning the material unencumbered by worries about the text's accuracy.

The index is nicely composed, complete, and accurate. This makes the book particularly useful as a reference. Typically, the reader will have little trouble using the index to go directly to the pages of most interest and applicability regarding the subject of inquiry. It would be nice to see a more complete bibliography, as well as a summary of common symbols. Especially useful would be a summary of some of the more important equations (such as Green's theorem, Cauchy's theorem, the Fourier transform, the Helmholtz equation, etc.) derived or demonstrated in the book. A list of important equations, particularly, would improve the book's utilization as a desk reference.

For the student, the text presents answers to odd-numbered questions in the back of the book. For the most part, the text presents only the answers, but occasionally the authors provide additional insight into the problem's solution, as in section 5.2. This will be useful for those engaged in independent study.

Overall, this is an excellent text, and one of the most complete and well-written books on complex variables I have seen. I highly recommend it to anyone interested in the subject, and have placed it prominently upon my reference bookshelf.


The numerous pictures are enough to recommend this text   June 7, 2001
UNPINGCO (Los Angeles, CA)
15 out of 15 found this review helpful

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.


A very good text!   November 29, 2002
JAY JINA (United Kingdom)
13 out of 14 found this review helpful

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!


Excellent book -- Unique selection of topics   March 8, 2003
10 out of 10 found this review helpful

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.


Covers everything from i to Riemann-Hilbert Problems   April 13, 1998
staker@wilma.colorado.edu (Boulder, Colorado)
7 out of 9 found this review helpful

Great text! Starts with the very basics, including complex aglegra to complex integration. A new topic is usually presented from expanding a previous one, and is immediately supported by examples or solved problems. Most odd numbered problems have answers in back. Author is an excellent Professor at UC of Boulder, and his text is used by both applied math and engineering students.