1.1. Review by: William Prager.
Quarterly of Applied Mathematics 11 (1) (1953), 142.

In addition to an extended bibliography, the book contains photo-offset reproductions of nineteen basic papers on the dynamics of compressible fluids. Obviously, space limitations have excluded some important contributions to the subject; otherwise, the collection illustrates well the rapid development of this field. The bibliography is arranged according to the following headings: Hodograph Method. The Rayleigh-Janzen Method. The Prandtl-Glauert Method. Supersonic Flow. Shock Waves. Boundary Layer. The Oscillating Airfoil and Other Unsteady Flow Phenomena. General.

2.1. From the Preface.

In addition to being a rewarding branch of mathematics in its own right, the theory of functions of a complex variable underlies a large number of enormously powerful techniques which find their application not only in other branches of mathematics but also in the sciences and in engineering. Chapters 1, 2, and 5 of this book provide concisely but honestly the classical aspects of the theory of functions of a complex variable; the rest of the book is devoted to a detailed account of various techniques and the ideas from which they evolve. Many of the illustrative examples are phrased in terms of the physical contexts in which they might arise; however, we have tried to be consistent in including a mathematical statement of each problem to be discussed.

For the acquisition of skill in the use of these techniques, practice is even more important than instruction. Accordingly, we have inserted many exercises, including some which are essential parts of the text. The reader who fails to carry out a substantial number of these exercises will have missed much of the value of this book.

The individual chapters segregate specific topical items, but many readers will find it profitable to study selected parts of Chaps. 3 to 7 as they encounter the related underlying theory in Chap. 2.

We hope that through our presentation the reader will be able to discern the fascination of complex-function theory, recognise its power, and acquire skill in its use.

2.2. Review by: Timothy Y Chow.
SIAM Review 12 (1) (1970), 162-163.

As the title indicates, this book is designed to present the complex function theory and various techniques based on the theory to persons who, according to the statement on the inside jacket, need to know advanced calculus as the only prerequisite.

The book consists of eight chapters. The authors state in the preface that "Chapters 1, 2, and 5 of this book provide concisely but honestly the classical aspects of the theory of functions of a complex variable; the rest of the book is devoted to a detailed account of various techniques and the ideas from which they evolve." Indeed, in the first chapter of 24 pages the authors provide a condensed account of the complex number system, sequences, series (convergence tests and uniform convergence), power series, logarithm and exponential functions, branch cuts and Riemann surfaces. In the second chapter of 52 pages, the topics mentioned are differentiation and integration in the complex plane, Cauchy's theorems, Taylor's and Laurent's series, analytic continuation, harmonic functions, etc. Mostly the material is presented at such a pace that not only proofs but also clarity and precision of statement must be sacrificed.
...
There are hardly any worked-out examples in these two chapters; any intelligent reader who does not wish to be merely told of these facts would find it difficult and time-consuming to learn and understand the theory from this book. Indeed, there is no dearth of much better written texts on complex function theory today.
...
There is some doubt in this reviewer's mind as to whether the authors have presented the theory effectively enough for the beginning reader to learn it and to acquire skill in analysis necessary for the study of the technique parts. Of course, to acquire skill one must do exercises. On pages 22 through 24, after little over a one-page discussion on branch cuts and Riemann surface including the example z^{1/2}, the reader is urged to "pay particular attention" to exercises involving branch cuts, etc., of much more complicated functions. The reader must be very intelligent indeed. However, those who have already had a good background in complex function theory and some skill in analysis (acquired either from this book or otherwise) would find some new and stimulating material in most of the remaining chapters, and would be fascinated with the virtuosity of the authors and the power of the theory as well.
...
Now let us turn to those chapters which deal with the techniques based on the function theory ... Some examples are interesting, the likes of which are rarely found in most textbooks.

2.3. Review by: Ernest C Schlesinger.
American Mathematical Monthly 74 (2) (1967), 221-222.

This book, in spite of its title, is primarily concerned with topics of the technique and application of complex variable theory. It treats conformal mapping, contour integration, some special functions (gamma, hypergeometric, Legendre, and Bessel functions), asymptotic expansions, and transform methods. The final chapter deals with the Wiener-Hopf method and with related material on integral equations. The authors provide many illustrative examples and a well-selected collection of interesting problems. The student who works a substantial number of these will have a good grasp of applied function theory. But he will need an experienced mentor! For the theory is not treated so effectively. The hints for the problems and those parts of the explanations that are "readily seen," are, by no means, always on the surface.

The methods here exposed will be understandable to the user only if he has had previous acquaintance with the underlying justifications, or if his study is directed by someone in a position to clarify and to tighten a good number of the arguments. But the reader with an introductory course in function theory behind him will find here a large variety of uses of this theory. They are explained lucidly, and there are, in most cases, some allusions to the conditions under which the various techniques are applicable.

This work is in no sense a handbook of special tricks; it is a serious text for a substantial year course in the topics that it treats.

2.4. Review by: George Springer.
Mathematical Reviews MR0222256 (36 #5308).

This book is a compromise between theory and applications and the topics selected for it are just those that should make it suitable for a course emphasising applications. The compromise is evident in those places where the authors do not bother to fuss over petty details of rigour. There are numerous examples throughout the text where hypotheses are not made most general or theorems are stated without proof. The saving in space gained in this way enables the authors to include many topics not ordinarily found in complex variables texts and which are important in applied mathematics.

The text is written in a stimulating informal style. Many of the (sometimes non-trivial) steps in proofs or calculations are left to the reader to fill in by himself and there are numerous meaningful exercises, so the reader has ample opportunity to develop his own technique while reading this text. If he reads passively, he will definitely miss much of the development of the subject, for many significant theorems appear only in the problems.
...
For those students who enjoy learning through illustrative examples and problems instead of a more general exposition, this book is ideal, for the authors have included much of their own extensive experience which should find its way through to the conscientious reader.

3.1. From the publisher.

Functions of a complex variable are used to solve applications in various branches of mathematics, science, and engineering. Functions of a complex variable: theory and technique is a book in a special category of influential classics because it is based on the authors' extensive experience in modelling complicated situations and providing analytic solutions. The book makes available to readers a comprehensive range of these analytical techniques based upon complex variable theory. Proficiency in these techniques requires practice. The authors provide many exercises, incorporating them into the body of the text. By completing a substantial number of these exercises, the reader will more fully benefit from this book.

This book is for professionals and students who have advanced knowledge in calculus and who are interested in such subjects as complex variable theory, function theory, mathematical methods, advanced engineering mathematics, and mathematical physics.

4.1. Note.

We have not been able to find reviews of the original text, only the reprint given below.

5.1. From the publisher.

Offers an alternative to the 'rote' approach of presenting standard categories of differential equations accompanied by routine problem sets. The exercises presented amplify and provide perspective for the material, often giving readers opportunity for ingenuity. Little or no previous acquaintance with the subject is required to learn usage of techniques for constructing solutions of differential equations in this reprint volume.

5.2. From the Preface.

The material in this book is not a conventional treatment of ordinary differential equations. It does not contain the collection of proofs commonly displayed as the foundations of the subject, nor does it contain the collection of recipes commonly aimed at the scientist or engineer. Rather, in a way which requires little or no previous acquaintance with the subject, it contains a sequence of heuristic arguments, illustrative examples and exercises, which serve to lead the reader towards the invention, generalisation, and usage of the techniques by which solutions of differential equations can be constructed. Above all, we hope, the reader can gain a perspective concerning the extent to which methods which lead in principle to the solution of a given problem actually lead to a useful description of that solution.

Our purpose is to offer an alternative to the almost "rote" approach, in which the standard categories of differential equations, accompanied by routine problem sets, are systematically listed. We firmly believe that the present approach is one that should be encountered, at least once, by mathematicians, users of mathematics, and those who are merely curious about mathematics; we hope that members of all three sets will find the presentation stimulating.

We consider the exercises to be an essential part of the text. They extend, amplify, and provide perspective for the text material. (In rare cases, the statement of a problem is left deliberately incomplete, so as to give the reader some scope for ingenuity.) If this book is used to accompany a course of lectures, one technique would be to assign text reading and exercises beforehand, to be followed by classroom discussion and amplification. Alternatively, special problem sessions could be included.

In reprinting this book, we have taken the opportunity to correct an (embarrassingly large!) number of misprints, and to clarify certain aspects of the presentation. As in the original printing, the attitudes and approaches in this book remain solely the responsibility of the authors; however, we gratefully acknowledge the efforts of the many colleagues who have struggled to reform our viewpoints. In this connection, we are particularly appreciative of the suggestions provided by Bob Margulies and Frank Fendell, each of whom was kind enough to read portions of the present manuscript.

5.3. Review by: J M Anthony Danby.
SIAM Review 35 (4) (1993), 658-659.

This volume is a new edition of the original text, published in 1968. In these days of drearily identical textbooks, it is good to see one that is unique; its reappearance is welcome.

The book is intended to be a tool for learning; it is assumed that students will fill in details and, at times, make their own discoveries. This accords with the constructivist mood that is finding increasing favour in today's teaching. There are no theorems, no "recipes," and no drill problems. The range of topics considered is extensive. In just over 200 pages, you will meet first- and second-order differential and difference equations, power and asymptotic series, eigenvalue expansions, special functions, (error, gamma, Bessel, Airy, and Legendre), the Laplace transform, variational calculus, solution of partial differential equations using separation of variables, nonlinear differential equations, numerical methods, and singular perturbation methods.

This range of topics would not disgrace a graduate level course, and it is at that level that I feel the text might be most successful. It could be used for a lively honours undergraduate course, but would be unsuitable for a service course. ...

The style of the presentation is discursive. The authors seem to be trying to draw the reader into conversation. They also seem to assume that the reader already shares their interest in the material, and make little attempt to arose it. At times the style becomes somewhat turgid ...

I found the attention to applications of differential equations to be disappointing. ... On the whole, if an instructor is concerned about applications, he will have to supply his own.

In summary, this is a nice book, moderately priced and well worth owning. If I were to use it in a course, I would want a small class of selected students who would work in teams. With unconventional teaching starting to gain favour, an unconventional text is welcome. The publishers have done us a service in reissuing it.

6.1. From the Preface.

This book reflects the authors' experience in teaching partial differential equations, over several years, and at several institutions. The viewpoint is that of the user of mathematics; the emphasis is on the development of perspective and on the acquisition of practical technique.

Illustrative examples chosen from a Dumber of fields serve to motivate the discussion and to suggest directions for generalisation. We have provided a large number of exercises (some with answers) in order to consolidate and extend the text material.

The reader is assumed to have some familiarity with ordinary differential equations of the kind provided by the references listed in the Introduction. Some background in the physical sciences is also assumed, although we have tried to choose examples that are common to a number of fields and which in any event are intuitively straightforward.

Although the attitudes and approaches in this book are solely the responsibility of the authors, we are indebted to a number of our colleagues for useful suggestions and ideas. A note of particular appreciation is due to Carolyn Smith, who patiently and meticulously prepared the successive versions of the manuscript, and to Graham Carey, who critically proof-read most of the final text.

6.2. From the Introduction.

We collect here some formal definitions and notational conventions. Also, we analyse a preliminary example of a partial differential equation in order to point up some of the differences between ordinary and partial differential equations.

The systematic discussion of partial differential equations begins in Chapter 1. We start with the classical second-order equations of diffusion, wave motion, and potential theory and examine the features of each. We then use the ideas of characteristics and canonical forms to show that any second-order linear equation must be one of these three kinds. First-order linear and quasi-linear equations are considered next, and the first half of the book ends with a generalisation of previous results to the case of a larger number of dependent or independent variables, and to sets of equations.

Included in the second half of the book are separate chapters on Green's functions, eigenvalue problems, and a more extensive survey of the theory of characteristics. Much of the emphasis, however, is on practical approximation techniques; attention is directed toward variational methods, perturbations (regular and singular), difference equations, and numerical methods.

6.3. Review by: Editors.
Mathematical Reviews MR0404823 (53 #8623).

This book is based on the authors' lectures on partial differential equations and is strongly oriented towards applications and technique. The authors motivate the study with a number of applications from different areas. Throughout there are many examples and exercises. The reader is assumed to have some familiarity with ordinary differential equations and some background in physical science. Chapter headings include: The diffusion equation, The wave equation, The potential equation, Classification of second order equations, Green's functions, Variational methods, Eigenvalue problems, Finite-difference equations, Numerical methods.

6.4. Review by: Fritz Oberhettinger.
SIAM Review 19 (4) (1977), 748-749.

The coverage may be judged from the list of chapter headings: 1. The diffusion equation, 2. Laplace transform methods, 3. The wave equation, 4. The potential equation, 5. Classification of second order equations, 6. First order equations, 7. Extensions, 8. Perturbations, 9. Green's functions, 10. Variational methods, 11. Eigenvalue problems, 12. More on first-order equations, 13. More on characteristics, 14. Finite-difference equations and numerical methods, 15. Singular perturbation methods.

As the title indicates, this book emphasises the development of techniques and their applications especially concerning problems in mathematical physics. Each chapter contains a large variety of exercises in such fields as diffusion theory, potential theory, wave propagation theory, fluid dynamics etc. The concisiveness and lucidity as well as the material displayed should make this book an attractive text. It can in a way be regarded as a welcome companion of a former publication by these authors (G F Carrier, M Krook, C E Pearson, Functions of a Complex Variable. Theory and Technique, 1966) which this reviewer had the pleasure to use as a text in one of his classes.

6.5. Review by: Roger Howe.
American Scientist 64 (5) (1976), 581.

Partial differential equations are difficult to write a book about. Presenting some chapters of standard material (derivations, separation of variables, series methods, maximum principles) is an obvious beginning, but how can one continue without becoming too specialised, too technical, too theoretical, or all of the above? The authors' answer is to illustrate some important general techniques and to give substantial discussions of first order equations, of characteristics in geometrical, theoretical, and physical terms, and of approximation techniques (including a chapter on singular perturbation theory). It seems a most successful answer. The book can give a very solid introduction to PDE from a practical viewpoint, particularly if the reader will do the many interesting and well-chosen problems (which, the authors state, constitute 72% of the book's value).

In spite of the book's subtitle, the theoretical discussions are geared for the technical man. Rigorous theory and the associated constructs, operators, and distributions are almost entirely missing. The few references to distributions are not satisfying, and the treatment of Green's functions is uninspired. The authors' voice is passive through out, making the style very dry, which detracts from the physical and geometrical discussions but, interestingly, is highly efficient in the sections on methods. Distilled from courses, the book seems well suited for teaching, and I recommend it highly as a text for a well prepared class.

7.1. Review by: Editors.
Mathematical Reviews MR0952148 (89j:35001).

Foreword to the second edition: "The major changes from the first edition are that a new chapter on transform methods has been added, and a section on integral equations has been included in the numerical methods chapter. We have also taken advantage of the opportunity to correct some misprints, to improve some of the phraseology, and to include some additional exercises."