**1. The Laplace transform (1941), by David Vernon Widder.**

**1.1. Review by: Ralph Philip Boas.**

*The Mathematical Gazette*

**27**(273) (1943), 37-39.

The Laplace transform has been extensively investigated by two classes of people - mathematicians and applied mathematicians. The latter have been chiefly interested in the formal properties of the Laplace transform, which make it useful for obtaining solutions of physical problems; the former have been interested in embedding the formal properties in a mathematically satisfying logical structure. This book was written by a mathematician for other mathematicians, and contains no applications outside pure mathematics. However, it could serve as a useful source in which applied mathematicians might look for the properties which they need to use. The first chapter, which is the most convenient account of Stieltjes integrals yet to have appeared in a book, is also recommended to applied mathematicians. Stieltjes integrals, with their ability to handle both discrete and continuous cases at once, seem admirably suited for use in applied mathematics; however, up to the present time few applied mathematicians seem to have been aware of the potentialities of Stieltjes integrals. In this book the author uses Stieltjes integrals systematically, and is thus able to discuss both classical Laplace transforms and Dirichlet series as cases of the same general theory. ... An experienced analyst will find in this book a large amount of useful material conveniently arranged and concisely expounded; a specialist will observe new theorems and new proofs of old theorems; a beginner will find important classical methods as well as problems at the frontiers of current research. The book contains ample refutation of the opinion, so frequently expressed nowadays, that "classical" analysis is a field in which interesting results are no longer to be expected.

**1.2. Review by: Thomas Arthur Alan Broadbent.**

*Science, New Series* **95** (2473) (1942), 531-532.

The Laplace transformation has many uses in applied mathematics; in pure mathematics it can be used with effect to prove and to discover properties, mainly of formal type, of many special functions f(t), such as the Bessel functions, the Laguerre and Hermite polynomials. But it can also be regarded as a well-defined domain of general function theory, in which general properties of the correlation are studied. The present volume is devoted to this last point of view, and the author completes and very considerably extends the treatment given in an earlier treatise by Doetsch. ... Throughout the book, the analysis is step by step clear and precise. But in a treatise which opens up so many new fields for development perhaps more help could have been given to those students who will want to use the book to prepare themselves to carry its ideas further; this help could have been given by more generous indications of the general lines of attack, particularly where the more difficult theorems are concerned, and also by a more frequent use of special functions to illustrate concretely the bearings of general results. Doetsch's book shows us how effective at times such special illustrations can be. But it is no doubt ungracious to complain that Professor Widder has not given us more, when in 400 well-printed pages he has been able to present us with such a clear and up-to-date account of recent advances in a fascinating field of analysis.

**1.3. Review by: Jacob David Tamarkin.**

*Mathematical Reviews *MR0005923 **(3,232d)**.

The present book is a significant contribution to a field of analysis whose importance becomes more and more obvious. Except for a recent book by Doetsch [Theorie und Anwendungen der Laplace-Transformation, Springer, Berlin, 1937], there are no other books on the theory of Laplace transforms. The present book, the material of which overlaps but little with that of Doetsch, will be particularly welcomed by anyone who needs a general introduction to this fascinating subject. It is very clearly written and can be easily understood by a student who has a knowledge of principles of functions of a single real or a single complex variable. At the same time it contains a considerable amount of new material resulting from researches of the author and of R P Boas.

**1.4. Review by: Francis Joseph Murray.**

*Bull. Amer. Math. Soc.* **48** (9.1) (1942), 642-646.

The book is certainly enjoyable and interesting. The style is clear, there are few typographical errors and the subject matter is increasingly impressive as one reads on. The applications are particularly striking. In some cases, it is not clear why so many proofs of the same theorem are given and a guide to a reader who might be interested in any one of the many specific results would be valuable. However, it would be quite easy to use various topics treated in the book in a course, whose main interest is not integral operators. One might mention, the Riemann-Stieltjes integral, functions of bounded variation, methods of summation of series, positive definite series, the moment problems, Bernstein's theorem, the Tauberian theorems, the prime number theorem, the Laguerre polynomials, the notion of a positive definite kernel of an integral equation, and the specific integral equations mentioned. Thus the author has presented us with a treatise on a branch of analysis of great importance and whose applications are of wide interest. The book is extremely satisfactory, when concerned with either its principal topics or the other related developments and one is confident that it will have a most valuable effect both on research and graduate study.

**2. Advanced Calculus (1947), by David Vernon Widder.**

**2.1. Review by: Harrie Stewart Wilson Massey.**

*Science Progress (1933-)*

**36**(142) (1948), 342.

It is rare nowadays for a book to appear on advanced calculus. The usual course in mathematics takes the student through elementary calculus only, the rest of his time being occupied in discussion of the philosophy rather than the technique of analysis. This may be appropriate for the pure mathematician, but it often leaves the applied mathematician ill-equipped with technique. There is much of interest and use in this book. The treatment of partial differentiation and its applications is very much fuller than in most available texts. Many students become familiar with the more theoretical aspects of functions of two or more variables without being capable of manipulating partial derivatives or evaluating a double integral. Reference to the examples discussed in this book would help to remedy this unfortunate situation - the chapters on Multiple Integrals, and on Line and Surface Integrals are also both sound and applicable. Unusual features are the presence of chapters on Stieltjes Integrals and on the Laplace Transform and its applications. The subject-matter of these chapters, as indeed of the book as a whole, is well chosen and should provide valuable material for both pure and applied mathematicians. In addition to the above-mentioned chapters, there are others dealing with Vectors, Differential Geometry, Limits and Indeterminate Forms, Infinite Series, Convergence of Improper Integrals, The Gamma Function and Fourier Series.

**2.2. Review by: Hubert Stanley Wall.**

*Mathematics Magazine* **22** (3) (1949), 159-161.

According to the author' s preface, "This book is designed for students who have had a course in elementary calculus covering the work of three or four semesters. However, it is arranged in such a way that it may also be used to advantage by students with somewhat less preparation." The chapter headings are as follows: 1. Partial differentiation; II. Vectors; III. Differential Geometry; IV. Applications of partial differentiation; V. Stieltjes Integral; VI. Multiple Integrals; VII. Line and Surface Integrals; VIII. Limits and Indeterminate Forms; IX. Infinite Series; X. Convergence of Improper Integrals; XI. The Gamma Function. Evaluation of Definite Integrals; XII. Fourier Series; XIII. The Laplace Transform; XIV. Applications of the Laplace Transform.

**2.3. Review by: Reuben Louis Goodstein.**

*The Mathematical Gazette ***31** (297) (1947), 298-300.

The student of applied mathematics who learns his Advanced Calculus from this book will not only find in Analysis a superlative instrument, but will gain a very deep appreciation of the beauty in a structure of pure reason. A graceful and smooth flowing style, a mastery of clear exposition, generous and expansive lay-out and printing all combine to make the reader's lot a happy one. Advanced Calculus is not intended for the mathematical specialist in training, for there is no emphasis on generality. Each theorem is established under the simplest conditions adequate for application. Hypotheses and conclusions are numbered and listed, and what is proved, and what assumed are apparent at a glance. The treatment and development follow orthodox lines, but the emphasis on a continuous derivative effects considerable simplification.

**2.4. Review by: Norman Levinson.**

*Mathematical Reviews* MR0021051 **(9,16b)**.

The most noticeable feature of this book is the extreme precision in the statement of theorems and their proofs. Unlike most advanced calculus books, this one is entirely satisfactory as regards rigor. Moreover, it is much richer in mathematical content than most advanced calculus books. The chapter headings are: Partial differentiation, Vectors, Differential geometry, Applications of partial differentiation, Stieltjes integral, Multiple integrals, Line and surface integrals, Limits and indeterminate forms, Infinite series, Convergence of improper integrals, The gamma function, Evaluation of definite integrals, Fourier series, The Laplace transform, Applications of the Laplace transform.

**3. The Convolution Transform (1955), by I I Hirschman and D V Widder.**

**3.1. Review by: Laurence Chisholm Young.**

*Science, New Series*

**123**(3195) (1956), 512-513.

The title of this book brings to mind general researches on groups, function-spaces, Laurent Schwartz distributions, and, of course, it includes after suitable changes of variables, the Laplace transform, to which one of the authors has already devoted a well-known and much prized treatise. However, it is said that the author of an equally prized book on the good city of Boston - a city for which I have a special affection - once collaborated on a further volume dealing with the United States, and that his friends were relieved to find that this further and somewhat shorter volume limited itself to those parts of the United States that could be reached in an hour or so from Boston on foot. The heart of the present book is again the Laplace transform, for it turns convolutions into ordinary products, and many readers will indeed be relieved to find that the book limits itself to material accessible, from fundamentals exposed in the earlier treatise, by pedestrian methods of real and complex variable theory available in Titschmarsh. The limitations that the authors impose on the notion of convolution are drastic. ... In spite of these drastic limitations, no doubt due to the author's desire to spare the reader, the book is undoubtedly an excellent complement to the earlier one on the Laplace Transform, and, as the authors state, some of the earlier results can now be better understood as special cases of the newer developments.

**4. Advanced Calculus, Second Edition (1961), by David Vernon Widder.**

**4.1. Review by: Earl LaFon.**

*Amer. Math. Monthly*

**69**(6) (1962), 578.

The new edition of Widder's Advanced Calculus continues in his clear, precise style. The examples bring out the exact content of the theorems, as they should. General theoretical concepts are frequently used before they are proved. For example, in the chapter on Stieltjes Integrals, the existence theorem is stated and used early , but the proof is left until the last section. More real variable theory is included than the chapter titles suggest. The coverage of Laplace transforms (in real variables) is by far the most complete which this reviewer has ever seen in an advanced calculus book. In addition to expected topics, this book presents the Heine-Borel theorem, Schwartz's example obtaining other than surface area from the limiting area of inscribed polyhedra, an analytic proof of the invariance of the inner and outer products of vectors, developable surfaces, Cesaro summability, Stirling's formula, infinite products, Fourier series, the bilateral Laplace transformation, and a treatment of the plucked string. The material is carefully selected and well treated. It would be difficult for most classes to cover all the material in this book in two semesters.

**5. An Introduction to Transform Theory (1971), by D V Widder.**

**5.1. Preface.**

This book is essentially compiled from notes on lectures given by the author at Harvard University in a half-course on transform theory. It was attended chiefly by seniors and first-year graduate students, and only a basic knowledge of real and complex function theory was assumed. The book is designed to touch on a variety of the most fundamental aspects of the theory rather than to strive for encyclopaedic coverage of any part. We hope that it will be useful to a student who is sampling various kinds of mathematics before settling on a direction for his own research.

The text begins with a rapid introduction of the use of Laplace integrals for solving differential equations. Although emphasis throughout is on the theoretical rather than on the applied side of the subject, any student of transform theory will wish to be cognizant of this most important application.

The basic properties of Laplace integrals can be conjectured by analogy from those of Dirichlet series. Consequently our theory begins with a chapter on such series. Since this "discrete" transform does not present some of the complications of the continuous, or integral, transform, it offers good introductory material. The most famous Dirichlet series is probably the one defining the zeta-function of Riemann. It is also the simplest in some ways since all the coefficients are unity. Yet it remains an enigma in that its zeros have not yet been completely located. Its tremendous influence on mathematics over the years almost makes its study obligatory for all mathematicians and certainly for students of analysis and number theory. Its basic properties, especially those needed later, are collected in Chapter 3.

Chapter 4 gives a proof of the prime number theorem, as one important application of Dirichlet series. To understand it the reader need have no previous knowledge of number theory. The material begins with Chebyshev's derivation of the order of magnitude of the nth prime although this is unnecessary for the main theorem. But this historical approach serves to give an introduction to the methods of number theory to familiarize the student with the number theoretical functions involved and to give him a better appreciation of the final result.

Although Dirichlet series form ideal introductory material, the student who wishes to immerse himself at maximum speed into the theory of integral transforms may omit Chapters 2-4, and proceed directly to the rest of the book. Chapter 5 sets forth the classic results about Laplace and Stieltjes transforms. The following chapter takes up the more recent inversions of these transforms, after first developing the Laplace asymptotic method. The latter is an indispensable tool for analysts and applied mathematicians.

In Chapter 7 a very rapid approach to the convolution transform is to be found. This basically subsumes earlier results and should serve to solidify the reader's understanding. The reason for the success of the earlier inversion formulas becomes apparent as they are recaptured in this more general setting.

Chapter 8 endeavours to introduce the reader to Tauberian theorems rapidly and simply. Two approaches are taken: one, via the general Tauberian theorem of N Wiener [1933], the other through Karamata's specialized method. The former is for general kernels but is restricted to two-sided Tauberian conditions, the latter is for special kernels but permits the more general one-sided conditions. It is noteworthy that no use of Fourier analysis is made. This is avoided by our introduction of the uniqueness class U, to which the kernels here considered are already known to belong. The classic series theorems of Hardy and Littlewood are extracted as special cases.

We hope that the final chapter will prove intriguing to the reader, perhaps stimulating him to investigate more general results in the same direction. We present here amusing algorithms for the inversion, by series, of two special transforms. But the method is general, as the author has shown.

Exercises appear at the ends of chapters, some with answers. They are usually simple, intended to help the reader to test and to solidify his mastery of the text.

Theorems are generally stated in the same systematic and compact style used by the author in his "Advanced Calculus," The few logical symbols employed to accomplish this are for the most part self-explanatory, but a few are explained parenthetically when introduced for the first time.

**5.2. Review by: Isidore I Hirschman.**

*American Scientist* **62** (1) (1974), 120.

The present volume can be described as an expansion to book length of the sort of outstanding exposition found in certain articles in American Mathematical Monthly. Although it makes no unusual demands on the reader (routine courses in complex analysis and measure theory are sufficient), it is rigorous and complete and so ingeniously organized that it affords an exciting tour in which a selection of beautiful and varied results are exhibited and at the same time shown to be in some sense facets of a single theory. A very partial list of the sights offered on the tour includes introductions to the Riemann zeta function, general Tauberian theorems, the prime number theorem, completely monotonic functions, the Hausdorff moment problem, and convolution transforms generated by entire functions of Laguerre Polyá type. This book can be recommended without reserve to at least two classes of readers - those who are interested in other areas of mathematics but would like at least a minimal acquaintance with this one, and those who feel some attraction to this area but would like to orient themselves somewhat before plunging in. It can be read alone, or it can be used with great advantage as a textbook for a one-semester course. In fact, this is how it came to be written.

**5.3. Review by: Isidore I Hirschman.**

*SIAM Review* **15** (2.1) (1973), 396-397.

This volume, which grew out of courses offered by its author at Harvard, is a progression of linked mathematical themes centred about the Laplace transform. Each subject leads naturally into the next and no subject is abandoned before it has yielded some deep and exciting result. It is written in a beautiful and lucid style, and with an elegance usually associated with more abstract branches of mathematics. The pre-requisites are modest - a familiarity with basic real and complex variables.

**6. The Heat Equation (1975), by D V Widder.**

**6.1. Preface.**

This book is designed for students who have had no previous knowledge of the theory of heat conduction nor indeed of the general theory of partial differential equations. On the other hand, a degree of mathematical sophistication is assumed in that the reader is expected to be familiar with the basic results of the theory of functions of a complex variable, Laplace transform theory, and the standard working tools involving Lebesgue integration. It should be understandable to beginning graduate students or to advanced undergraduates.

The heat equation is derived in Chapters I and XII as a consequence of two basic postulates, easily accepted from physical experience. From this point on, the theorems and results are logical consequences of the heat equation. If the conclusions are at variance with physical facts, and they are slightly so, the fault must be traced to the postulates. For example, the equation forces the conclusion that "action at a distance" is possible. That is, heat introduced at any point on a linear bar raises temperature *instantaneously *at remote portions of the bar. This scandalizes reason and contradicts experiment, so that we must conclude that the postulates are only approximations to the physical situation. But it has also been evident since Fourier's time that they are *good *approximations.

The early chapters develop a theory of the integral transforms that are needed for the integral representations of solutions of the heat equation. Results that are needed here about the theta-functions of Jacobi are proved in Chapter V. Transforms for which theta-functions are kernels are used for solving boundary-value problems for the finite bar. No previous knowledge of theta-functions is assumed. At only one point is an unproved formula about them employed, and even here a second approach to the desired result avoids use of that one formula.

Much of the material in Chapters VIII-XIV is based on the author's own research, but it is presented in simplified form. The emphasis is on the expansion of solutions of the heat equation into infinite series. Here the analogies from complex analysis of series developments of analytic functions are very revealing. These are pointed out in detail in Chapter XI. In the final chapter the essential results from four research papers are given simplified proof.

All the material could probably be presented in a half course. More realistically, Chapters V, VI, XII, XIII, and those parts of Chapters VII and VIII dealing with the finite rod, could be omitted. These could be replaced by classic boundary-value problems.

Theorems are generally slated in the same systematic and compact style used by the author in "Advanced Calculus" and in "An Introduction to Transform Theory." The few logical symbols needed to accomplish this are for the most part self-explanatory, but a few are explained parenthetically when introduced.

**6.2. Review by: Isidore I Hirschman.**

*American Scientist* **65** (3) (1977), 377.

This book is a study in depth of the partial differential equation of heat conduction ^{∂2u}/_{∂x2} = ^{∂u}/_{∂t} on intervals of the line (including, of course, the line itself). It is, as has been pointed out in the comprehensive review by R. P. Boas (Bull. Am. Math. Soc. 82, Sept. 1976), a treatment of the heat equation in many respects parallel to the earlier and long continuing study of the two-dimensional Laplace equation in the disk. The richness and variety of the results obtained are quite astonishing, and the clarity and elegance of their presentation are what we have learned to expect and admire in Widder's books (and papers).

**6.3. Review by: Deborah Tepper Haimo.**

*SIAM Review* **19** (2) (1977), 364-365.

The heat equation has been the subject of extensive investigation since the time of Fourier. For almost half a century, major contributions in its study have been made by David V Widder. The fourteen chapters in his new book, The Heat Equation, incorporate many of these results in simplified version and provide an essentially self-contained, readable, and up-to-date account of the theory. ... Books on partial differential equations and boundary value problems generally include some discussion of the heat equation, but no book exclusively devoted to a study of that equation has appeared since the 1948 treatise by Carslaw-Jaeger. That rather encyclopaedic reference with its emphasis on the physical applications differs markedly from the Widder book. The latter is devoted to a classical and analytic development of the fundamentals of the theory as a basis for the presentation of some recent research results, for the most part the author's own. Directed toward advanced undergraduates and beginning graduate students, the book avoids a functional analytic approach. Although no exercises are included, skilfully selected examples are used to illustrate the theory and to emphasize limitations of conclusions. Written in the carefully precise style which is the author's hallmark, the book affords an overview of a line of contemporary research on the classical heat equation.

**6.4. Review by: Gerald G Bilodeau.**

*Mathematical Reviews* MR0466967 **(57 #6840)**.

This book is a self-contained development of the mathematical theory of the heat equation. Although there is a short chapter on higher dimensions, it is fair to say that the book is concerned exclusively with the heat equation in one space variable, *u*_{t} = *u*_{xx,}where *u* = *u*(*x*, *t*) is called a temperature function. Moreover, in spite of the title, this is a book for students of mathematics by one of the outstanding mathematical contributors to the theory. Writing in the precise and lucid style typical of him, the author creates a structure of results, using only the familiar classical methods from real variables (including the Lebesgue integral) and complex variables along with very few facts from the theory of the Laplace transform. The final product is astonishing not only in numbers and variety of results but also in the easy manner in which the reader is led to rather profound theorems. ... Finally, we close this review with a general comment. There are no problems in this book so that it would not serve easily as a textbook. Moreover, the subject matter is indeed specialized; but the student who studies well the material in this book will have received an education in the methods and spirit of classical analysis from an acknowledged master of the field, which will serve him well in his probable career as a teacher of mathematics.