In his review in National Mathematics Magazine 10 (2) (1935), 69-70, W E Byrne writes:
This book is the result of a course given to junior electrical engineers at the Massachusetts Institute of Technology over a period of years. It is designed to follow ordinary differential equations. ... the book is a welcome arrival in the circle of differential equation texts. All too long has the mathematical passage from the differential equations of circuit theory to the complex numbers used by the electrical engineer been passed over as if it were something mysterious by the writers of elementary electrical engineering texts. Here the issue is met squarely and in addition a glimpse of many other fields of application is given to the future engineer.
This book is unlike any other text on differential equations now in the market. The first seventy-nine pages ... deal with complex numbers, average values, and Fourier series. Towards the end of the book [two chapters] deal with the theory of analytic functions both of real and complex variables, of their expansions in series, and of operations with these series, also with the theory of the convergence of Fourier series. ... It is easy to see that the text is "the outgrowth of a course given for over ten years at the Massachusetts Institute of Technology," as the preface states. All the difficulties usually encountered by students are discussed in this book. ... In fact, in every way this book is admirably suited to its purpose of introducing electrical engineers to differential equations that arise in their field.
In his review in Science 94 (2448) (1941), 518, Richard Courant writes:
Rigour, whatever this word may mean, was one of the great mathematical achievements of the nineteenth century. Only gradually has this tendency penetrated into textbooks. The first great work of this kind, Jordan's "Cours d'analyse," was followed by many others, of which Hardy's "Pure Mathematics" seems to be the foremost in English. Franklin's book is an admirable attempt on a much broader scale to combine rigour with completeness in a volume of modest size. It will appeal to readers who are already well informed but want to revise and to supplement their knowledge in the light of modern precision. Not only are the traditional subjects of a book on advanced calculus covered, but also many more advanced topics are included. There is a section on the Laplace transformation, one on Poisson's sum formula, and a brief exposition of the theory of partial differential equations of the first order. The material is presented in an original way with extraordinary care.
This book is a valuable contribution to the field of advanced calculus. It definitely bridges the gap between formal elementary work and the exacting rigour of modern analysis. ... it would be difficult to find better exposition at this level.
This book is an extraordinarily satisfactory addition to the literature of advanced calculus. This text is indeed a treatise which covers completely the infinitesimal calculus and includes much prerequisite algebra and analysis (and most other concepts) that are needed for geometric and physical applications. The theorems and proofs are given with utmost precision and completeness, and the reader will never have need of another text for the material that is treated here. Of course, this text is not for beginners; the reader must be acquainted with the elementary calculus and he must have some proficiency in its technique in order to follow and appreciate the author's developments and precise deductions. With such preparation the student should obtain in studying this book an excellent mathematical education - even for graduate or research work.
In his review in The Mathematical Gazette 26 (270) (1942), 152-153, Percy Heawood writes:
This booklet explains some of the intricacies of the above problem in a succinct and interesting manner. The main question at issue is whether the divisions of any ordinary map can be properly distinguished by the use of four colours only, so that no two adjacent divisions bear the same colour. And it is still a problem, because the challenge thrown out to mathematicians by Professor Cayley, more than sixty years ago, to give a valid proof that this is so, has not yet been satisfactorily met. An attempt put forward in early days by Mr A B Kempe was found to have a fatal flaw. An extensive literature has since gathered about the subject, which is very adequately treated by Mr Franklin within the compass of thirty-three pages. He gives the most important results, with proofs where they are simple, and references to original papers where they are long and tedious; so that the brochure is very readable.
Before we give extracts from three reviews, we give two quotations from the Preface. Franklin writes that his two principal objectives are:
... first to refresh and improve the reader's technique in applying elementary calculus; second, to present those methods of advanced calculus which are most needed in applied mathematics. ... Each chapter is followed by a number of problems, arranged in an order corresponding to the development of the text. There is a large number, averaging nearly 100 per chapter and they range from routine exercises to elaborate applications to science and engineering.
... The attempt to prove everything often leads to the use of roundabout methods in the beginning of a book. This fixes inefficient techniques in the reader's mind, even when better methods are presented later. In this book the author has given the simplest known method of solving each type of problem the first time it appears, even when this means quoting a theorem from an advanced branch of mathematics without proof. ...
The selection, arrangement and development of material in this book are such as to form a digestible and palatable blending of rigour with intuition, abstract elegance with practical application. It is a distinct contribution to the field.
This book is directed mainly toward the engineering student and towards the mathematics student primarily interested in the physical application of the subject. The author has had long experience with both types of students at the Massachusetts Institute of Technology, and is consequently in an excellent position to know their needs, their likes and dislikes, and their abilities. For such students he has made an ideal selection of material and has used an excellent method of presentation.
As is well-known, this textbook is the second written by Professor Franklin on the subject of Advanced Calculus, the first having been published in 1940 by John Wiley and Sons under the title of "A treatise on advanced calculus." The two books are in some respects different as to content, but in all respects different as to point of view. ... While the first book was a sort of "cours d'Analyse," the new book ... has the purpose of serving the needs of prospective engineers, physicists, and others who may regard mathematics primarily as a tool. ... the new volume may be regarded as one of the best textbooks now available for any advanced calculus course which is intended to be a terminal course in mathematics for engineers, physicists and the like.
In his review in Amer. Math. Monthly 58 (4) (1951), 276-278, J W Green writes:
This is a book for engineers and other technicians. It is designed to show them how to operate with complex quantities and how to solve problems for their solution on the use of Fourier series and integrals, and Laplace transforms. It is vigorous mathematics at its extreme; the emphasis is not on the concepts involved, but on how to use these concepts to work problems. The author makes no attempt to lean on the mathematical rigour, but nevertheless the analysis is done in orderly fashion; the arguments are clear and plausible, and important facts, such as those concerning the convergence of Fourier series are plainly stated without claiming to be proved.
In his review in Science 118 (3067) (1953), 422, M E Shanks writes:
This is a soundly written standard text for a first course in the calculus. The various salient ideas are introduced in an intuitive way but with emphasis on the important concepts. ... The book is written with care and consideration for the student ...
This is a substantial text in elementary calculus. It contains all the topics ordinarily included in a first course, many of them developed more fully than usual, and also a number of additional features. ... The exposition is full and generally very clear; the language is precise and lucid. Much appeal is made to geometric intuition, which is right and proper in a beginning course in calculus. The degree of rigour maintained is probably as high as is practicable at this level. ... The illustrative examples are well chosen and are worked out in clarifying detail. The exercises appear entirely adequate unless one insists on some applications from the social sciences.
In his review in Amer. Math. Monthly 66 (10) (1959), 931-932, Homer Craig writes:
This excellent work on the classical properties of the analytic functions of a complex variable was evidently designed to serve two purposes: (1) to provide mathematics majors at the advanced undergraduate level or the lower graduate levels an introduction to the specific subject treated and perhaps to analysis in general, and (2) to provide for physicists and engineers a basic text and a reference work on the fundamentals of the subject. In the reviewer's opinion it is admirably suited for both of these programmes. The exposition is rigorous but not tiresome, and suitably modern but definitely not pedantic. Apparently, this book was very carefully composed and its structure is such that the beginner can attain a solid knowledge of the subject by through a series of stages (81 articles in all) of fairly uniform and moderate difficulty.
In his review in Amer. Math. Monthly 71 (8) (1964), 940, N D Kazarinoff writes:
The choice of title is accurate: the print is large and uncrowded on pages of medium size, the tersely phrased text is divided into sections of about one page, the total number of pages is small for the ground covered, which is differential and integral calculus of one variable, infinite series, and partial derivatives and multiple integrals ... The book is a challenge to the thick all inclusive texts which dominate today's scene.
In his review in The Mathematical Gazette 50 (372) (1966), 191, D A Quadling writes:
This is much more than just another textbook on "mathematical methods". True, there is plenty of information here about Fourier series, Green's theorem, contour integration, the gamma function, and the rest; but these topics are set firmly in a more fundamental context, and much of the detail is relegated to the very full exercises at the ends of the chapters. The book is in fact a compendium of classical real and complex variable analysis, with applications-perhaps, indeed, one of the last such compendia which will be written. ... One would guess that the book will be valued more as a gazetteer for the graduate mathematician than as a guide-book for the less experienced student; for the pace is fast and the style uncompromising. It is, however, a genuine mine of information, which senior undergraduates could well be encouraged to keep on their shelves, or at least to seek out in faculty and college libraries. One final statistic. This volume contains 3,054 numbered equations. Is this a record?