1.1. Contents.

Die Gammafunktion; Die hypergeometrische Funktion; Zylinderfunktionen; Kugelfunktionen; Orthogonale Polynome; Die konfluente hypergeometrische Funktion und ihre Spezialfälle; Elliptische Integrale, Thetafunktionen und elliptische Funktionen; Integraltransformationen und Integralumkehrungen; Koordinatentransformationen.

1.2. From the Preface.

The increasing use of mathematical tools in the physical and technical literature requires a constant reference in extensive, sometimes difficult to access, works of the mathematical literature and in numerous individual works. There are, however, a very large number of results, in particular of formulas which can be represented in a small space and which account for a great part of what is always needed, which must always be consulted with difficulty. The authors hope that the treatise presented by them will be useful in giving the properties of a number of special functions. In order to avoid disruption of the text by too many references, compilations of the abbreviations used and the various function symbols as well as the bibliography have been added at the end of the book.

2.1. Contents.

The gamma function and related functions; The hypergeometric function; Bessel functions; Legendre functions; Orthogonal polynomials; Rummer's function; Whittaker functions; Parabolic cylinder functions and parabolic functions; The incomplete gamma function and special cases; Elliptic integrals, theta functions and elliptic functions; Integral transforms; Transformation of systems of coordinates.

2.2. From the Preface.

We hope that these additions will be useful and yet not too numerous for the purpose of locating with ease any particular result. Compared to the first two (German) editions a change has taken place as far as the list of references is concerned. They are generally restricted to books and monographs and accommodated at the end of each individual chapter. Occasional references to papers follow those results to which they apply. The authors felt a certain justification for this change. At the time of the appearance of the previous edition nearly twenty years ago much of the material was scattered over a number of single contributions. Since then most of it has been included in books and monographs with quite exhaustive bibliographies.

2.3. From the Publisher of a 2013 edition.

This is a new and enlarged English edition of the book which, under the title "Formeln und Sätze für die speziellen Funktionen der mathematischen Physik" appeared in German in 1946. Much of the material (part of it unpublished) did not appear in the earlier editions. We hope that these additions will be useful and yet not too numerous for the purpose of locating with ease any particular result. Compared to the first two (German) editions a change has taken place as far as the list of references is concerned. They are generally restricted to books and monographs and accommodated at the end of each individual chapter. Occasional references to papers follow those results to which they apply. The authors felt a certain justification for this change. At the time of the appearance of the previous edition nearly twenty years ago much of the material was scattered over a number of single contributions. Since then most of it has been included in books and monographs with quite exhaustive bibliographies.

3.1. No details available at present.

4.1. Dedication.

This work is dedicated to the memory of Harry Bateman as a tribute to the imagination which led him to undertake a project of this magnitude, and the scholarly dedication which inspired him to carry it so far toward completion.

4.2. From the Preface.

The late Professor Harry Bateman of the California Institute of Technology was one of those rare scientists who, responding to the interplay between mathematical analysis and physical understanding, made outstanding contributions to American applied mathematics. His contributions to aero- and fluid mechanics, to electro-magnetic theory, to thermodynamics, to geophysics, and to a host of other fields in which his adroit mathematical skills were applied, resulted in significant advances in these fields. During his last years he had embarked upon a project whose successful completion, he believed, would prove of great value to scientists in all fields. He planned an extensive compilation of "special functions" - solutions of a wide class of mathematically and physically relevant functional equations. He intended to investigate and to tabulate properties of such functions, interrelations between such functions, their representations in various forms, their macro- and microscopic behaviour, and to construct tables of important definite integrals involving such functions.

It is true that much of this material was already in existence. However, anyone who has been faced with the task of handling and discussing and understanding in detail the solution to an applied problem which is described by a differential equation is painfully familiar with the disproportionately large amount of scattered research on special functions one must wade through in the hope of extracting the desired information. Professor Bateman was eminently qualified to embark on such a compilation, for he was unusually familiar - and systematically so - with existing mathematical literature on the subject; he was exceptionally adept in mathematical analysis; and he was ever conscious of the needs of the scientist who must so often use these functions. When his death cut short his work, the California Institute of Technology, in recognition of one of its great scientists, and the Office of Naval Research, in recognition of the extremely useful service such a compilation could render to both basic and applied science, pooled their efforts to continue the task initiated by Professor Bateman.

In 1948 arrangements were completed between the California Institute of Technology and the Office of Naval Research to employ at the California Institute of Technology four mathematical analysts of international reputation to complete Professor Bateman's work: Professors Arthur Erdélyi of the University of Edinburgh; Wilhelm Magnus of the University of Göttingen; Fritz Oberhettinger of the University of Mainz; and Francesco Tricomi of the University of Torino. It was not long after this team began work that it became apparent that not only would Professor Bateman's original project find its completion in their unusually competent hands, but that the activities of such a group would lead to significant mathematical investigations and advances in the general field of mathematical analysis, as well as in the more particular field of special functions. The present compendia bear undeniable witness to the success of the undertaking.

5.1. Dedication.

This work is dedicated to the memory of Harry Bateman as a tribute to the imagination which led him to undertake a project of this magnitude, and the scholarly dedication which inspired him to carry it so far toward completion.

5.2. From the Preface.

The present volume is the first of two which are intended as companions and sequel to our Higher Transcendental Functions. Volume I of that work contains the Preface and Foreword to the whole series, describing the history and the aims of the so-called Bateman Manuscript Project.

A considerable proportion of the tremendous amount of material collected by the late Professor Harry Bateman concerns definite integrals. The organization and presentation of this material is a very difficult task to which Bateman devoted considerable attention. It is fairly clear that the arrangement used in shorter tables of integrals is not very suitable for a collection about three times the size of Bierens de Haan, and the circumstance that a considerable proportion of these integrals involves higher transcendental functions with their manifold and not always highly standardized notations, does not make this task easier.

Eventually, Bateman decided to break up his integral tables into several more or less self-contained parts, classifying integrals according to their fields of application. A collection of integrals occurring in the theory of axially symmetric potentials was prepared, and other similar collections were to follow. Clearly such a plan involves a generous amount of duplication if the resulting tables are to he self-contained, but it also has great advantages from the user's point of view.

In planning our work on definite integrals, we were in the fortunate position of being able to restrict its scope. In recent years several excellent tables of integrals of elementary functions appeared, the most easily available ones being those by W Meyer zur Capellen, and by W Grabner and N Hofreiter. We also learned through the courtesy of the authors that a handbook of elliptic integrals by P F Byrd and M D Friedman is in preparation (and will have been published before this volume appears). In the assumption that our tables would he used in conjunction with other existing tables, we decided to concentrate mostly on integrals involving higher transcendental functions. We list no double integrals, and, except in the case of inverse transforms no contour integrals.

We adopted Bateman's idea of breaking up the tables into several, more or less self contained, parts; but we modified his principle of subdivision. We found that much of our material could be organized in tables of integral transforms, and accordingly the present volume, and about one half of volume II of our tables, consists of tables of integral transforms; those of our integrals which have not been classified as integral transforms being contained in the second half of volume II. We hope that this division will be found useful. Integral transforms have become an extensively used tool, and their practical application depends largely on tables of transform pairs. Laplace transforms are almost unique in that several up-to-date and thoroughly satisfactory tables of such transforms are available. For Fourier transforms there is an excellent collection of integrals, but it was compiled in 1931, and newer editions do not include additional material. For Hankel and Mellin transforms, and other integral transforms, we know of no extensive tables. In addition to the well-known transforms we give tables of integral transforms whose kernel is a Bessel function of the second kind, a modified Bessel function, Struve function, and the like, partly because some of these transforms are useful in solving certain boundary value problems, or certain integral equations, and partly because they afford a convenient classification of integrals.

6.1. For details.

See 5.1 "Dedication" and 5.2 "From the Preface" of Vol. I above.

7.1. From the Preface.

This is the last of the volumes prepared by the staff of the Bateman Manuscript Project, an enterprise whose origin and aims were described in the prefatory material to the first volume. There are altogether three volumes of Higher Transcendental Functions supplemented by two volumes of Tables of Integral Transforms. The present volume contains chapters on automorphic functions, Lamé and Mathieu functions, spheroidal and ellipsoidal wave functions, functions occurring in number theory and some other functions; and there is also a chapter on generating functions. The volume was prepared after the staff of the Bateman Manuscript Project left Pasadena, but Professor Magnus continued working on Chapters XIV, XVII, XIX after he joined the staff of New York University.

8.1. Contents.

Introduction to groups; Group axioms; Examples of groups; Multiplication table of a group; Generators of a group; Graph of a group; Definition of a group by generators and relations; Subgroups; Mappings; Permutation groups; Normal subgroups; The Quaternion group; Symmetric and alternating groups; Path groups; Groups and wallpaper designs; Group of the dodecahedron and the icosahedron.

8.2. From the Mathematical Association of America.

The abstract nature of group theory makes its exposition, at an elementary level, difficult. The authors of the present volume have overcome this obstacle by leading the reader slowly from the concrete to the abstract, from the simple to the complex, employing effectively graphs or Cayley diagrams to help the student visualize some of the structural properties of groups. Among the concrete examples of groups, the authors include groups of congruence motions and groups of permutations. A conscientious reader will acquire a good intuitive grasp of this powerful subject.

8.3. Review by: Albert A Mullin.
Mathematics Magazine 39 (4) (1966), 242-243.

Bringing the flavour of modern abstract algebra to beginning undergraduate students is the primary concern of this brief, but well conceived, monograph. Implicit in its goals is the establishment of relations between abstract mathematical structures and numbers. For the sake of concreteness the authors adopt the notion of a group as the paradigm par excellence for algebraic structures. In order to make good use of the student's natural intuition the writers employ the usual four axioms for a group involving left and right identities together with left and right inverses. Thus, at a slight cost of conciseness and possibly elegance, they may well have purchased better understanding of group theory for a wider audience than mathematics majors; i.e., they invoke here, as well as elsewhere, the useful pedagogical principle of the conservation of logic and psychology. The book is replete with well chosen elementary examples of groups from both geometrical and arithmetical contexts. Special emphasis is given to finite groups, i.e., finite sets which satisfy the group axioms. No attempt is made to relax any of the axioms in order to study more general algebra structures. Graphs are introduced as pictorial models for various kinds of groups, thereby tapping the topological intuition of the student.

8.4. Review by: Walter Ledermann.
The Mathematical Gazette 50 (374) (1966), 429.

This is an attractively written and truly elementary introduction to the theory of groups. In the first few chapters the reader is gently guided towards the group concept and, although the pace quickens somewhat in the later parts of the book, the authors consistently avoid the rigour of full generality and abstraction when they feel that an intuitive understanding of the ideas can be attained without it. The distinctive feature of the book, however, is the emphasis on geometry, both as motivation and illustration of the general theory and as a tool for studying the structure of individual groups by means of its graph or Cayley diagram. For those to whom geometry is the gateway to mathematics - I am doubtful whether this is human nature or just a consequence of our Greek inheritance - the geometrical approach to group theory will be most welcome. Group pictures are pretty, and they would be prettier still if they were executed in different colours, as was originally suggested by Cayley. They afford an excellent introduction to the notions of a free group, generators and relations, the fundamental group of a manifold and, as a special bonus, the symmetry groups of "Wallpaper designs."

9.1. from the Preface.

The term "Hill's equation" is a convenient abbreviation defining the class of homogeneous, linear, second-order differential equations with real, periodic coefficients. Although such differential equations have been investigated before the publication of the memoir on the motion of the lunar perigee by G W Hill in 1877, it seems to be justified to give his name to this whole class of differential equations in view of the important and lasting contributions made by Hill to their theory. There exist hundreds of applications of Hill's equation to problems in engineering and physics, including problems in mechanics, astronomy, the theory of electric circuits, of the electric conductivity of metals, and of the cyclotron. The fundamental importance of Hill's equation for stability problems was established by Lyapunov in 1907. It may be appropriate to remark here that the theory of Hill's equation reveals the occurrence of a surprising phenomenon which can be described in rather simple terms. If a force varying periodically with time acts on a mass in such a manner that the force tends to move the mass back into a position of equilibrium in proportion to the dislocation of the mass, one might expect the mass to be confined to a neighbourhood of the position of equilibrium. In particular, once the force is strong enough to achieve this effect, one would expect a stronger force to be even more efficient for this purpose. Such, however, need not be the case. In fact, an increase of the restraining force may cause the mass to oscillate with wider and wider amplitudes. The theory of the intervals of instability of Hill's equation provides the precise description of this phenomenon.

In order to describe the purpose, scope, and character of this tract, it may be said that we intend to provide orientation but had to abstain from giving full information. The elementary facts of the theory are proved in full except for those which belong to the general theory of linear differential equations. These are merely stated, and a reference is given. The mathematical tools used in the proofs are of a rather limited nature. We apply the result from the theory of entire analytic functions which states that a function of order of growth 1/2 has infinitely many zeros, and we use the Riemann-Lebesgue Theorem, but nothing more sophisticated than these fairly elementary facts. Results which can be arrived at only through very lengthy proofs have been stated but the proofs have been replaced by references. Also, we did not try to duplicate the monographs by Starzinskii and Krein which deal with closely related subjects. References to these surveys are given wherever this seemed to be appropriate.

Our monograph consists of two parts, the first of which (Chapters I and II) deals with the basic theory whereas the second part (Chapters III-VIII) contains details, refinements, and special cases. As for the applications, we hope to present eventually as a separate book a survey of several hundred papers in which Hill's equation is used as a mathematical tool. To include the applications here would have more than doubled the size of the book. The same would have happened if we had tried to treat Hill's equation merely as a special case of a system of linear homogeneous differential equations with periodic coefficients. A glance at the literature quoted in the monograph by Cesari will confirm this statement.

We have tried to give credit wherever it belongs. but it is almost an a priori certainty that we did not succeed here. We regret any omissions and misplacements of references which may have occurred.

9.2. Review by: Harry Hochstadt.
Mathematical Reviews MR0197830 (33 #5991).

This book presents a relatively complete and self-contained account of the theory of Hill's equation. ... the book provides a complete and up-to-date survey of known results regarding the discriminant. Many special equations are discussed in detail. It would have been impossible to prove all stated results in detail, but complete references are provided. The authors promise a companion volume in the future, which will devote itself to the many applications of Hill's equation in the physical sciences.

9.3. From the Publisher of the 2013 reprint.

The hundreds of applications of Hill's equation in engineering and physics range from mechanics and astronomy to electric circuits, electric conductivity of metals, and the theory of the cyclotron. New applications are continually being discovered and theoretical advances made since Lyapunov established the equation's fundamental importance for stability problems in 1907. Brief but thorough, this volume offers engineers and mathematicians a complete orientation to the subject. "Hill's equation" connotes the class of homogeneous, linear, second order differential equations with real, periodic coefficients. This two part treatment encompasses the most pertinent, necessary information; only the theory's elementary facts are proved in full, with minimal use of sophisticated mathematics. Part I explains the basic theory: Floquet's theorem, characteristic values and intervals of stability, analytic properties of the discriminant, infinite determinants, asymptotic behaviour of the characteristic values, theorems of Lyapunov and Borg, and related topics. Part II examines numerous details: elementary formulas, oscillatory solutions, intervals of stability and instability, discriminant, coexistence, and examples. Particular attention is given to stability problems and to the question of coexistence of periodic solutions. Although intended for professional mathematicians and engineers, the volume is written so clearly and vigorously that it can be recommended for graduate students and advanced undergraduates.

10.1. From the Preface.

This book contains an exposition of those parts of group theory which arise from the presentation of groups in terms of generators and defining relations. Groups appear naturally in this form in certain topological problems, and the first serious contributions to this part of group theory were made by Poincaré, Dehn, Tietze, and other topologists. The name "Combinatorial Group Theory" refers to the frequent occurrence of combinatorial methods, which seem to be characteristic of this discipline.

The book is meant to be used as a textbook for beginning graduate students who are acquainted with the elements of group theory and linear algebra. The first two chapters are of a fairly elementary nature, and a particularly large number of exercises were included in these parts. The exercises are not always easy ones, but the hints given are usually broad enough to make them so. Some interesting results have been presented in the form of exercises; the text proper does not make use of these results except where specifically indicated. (It is a good idea for the reader to examine the exercises even if he does not wish to attempt them.)

There is not very much overlapping of the topics presented here with those treated in the books on group theory by A Kurosh and by Marshall Hall, Jr. The subjects of Nielsen Transformations (Chapter 3), Free and Amalgamated Products (Chapter 4), and Commutator Calculus (Chapter 5) are treated here in a more detailed fashion than in the works of Kurosh and of Hall.

All theorems which are labelled with a number are proved in full. However, we have stated some advanced results without proof, whenever the original proofs were long and could not be amalgamated with the main body of the text. Such results are stated either as theorems labelled with the name of the author (e.g., Grushko's Theorem) or with a letter and number (e.g., Theorems N1 to N13 on Nielsen transformations, or T1 to T5 on topological aspects).

We have tried to give references to relevant papers and monographs in the later parts of the book (after the first two chapters). Usually. such references are collected at the end of each section under the heading "References and Remarks."

The sixth (and last) chapter contains a brief survey of some recent developments. It is hardly necessary to say that we could not even try to give a complete account. We are painfully aware of the many gaps. Some methods and results, as well as references, may have escaped our attention altogether.

Over the years, we have received suggestions and criticisms from many mathematicians, and we owe much to comments from our colleagues as well as from our students. We also wish to acknowledge the help given 10 us by the National Science Foundation which, through several grants given to New York University and Adelphi University, facilitated the cooperation of the authors.

This book is dedicated to the memory of Max Dehn. We believe this to be more than an acknowledgment of a personal indebtedness by one of the authors [Wilhelm Magnus] who was Dehn's student. The stimulating effect of Dehn's ideas on presentation theory was propagated not only through his publications, but also through talks and personal contacts; it has been much greater than can be documented by his papers. Dehn pointed out the importance of fully invariant subgroups in 1923 in a talk (which was mimeographed and widely circulated but never published). His insistence on the importance of the word problem, which he formulated more than fifty years ago, has by now been vindicated beyond all expectations.

10.2. Review by: Paul Moritz Cohn.
Amer. Math. Monthly 74 (5) (1967), 611-612.

One of the subtleties which the beginning student of group theory has to master is this: When a group is given by generators and defining relations, there is no obvious way of telling when different expressions represent the same element (even the elements represented by different generators need not be distinct). This is essentially the word problem and from the work of Novikov, Boone and Britton we know that groups with unsolvable word problem exist. But from the point of view of the working group theorist, and even more, of the general mathematician using groups, it is more important to have a good knowledge of methods of solving the word problem. The book under review is devoted to a study of such methods. ... Throughout the book the emphasis is on techniques, rather than general results and this outlook is supported by the very substantial collection of exercises, many of interest in their own right. The tone is elementary throughout; the authors are at pains to make themselves understood and at no point do they presuppose any specific knowledge. Of course to profit from the book the reader should have some acquaintance with group theory as well as a liking for combinatorial methods. Given these prerequisites (as well as pencil, paper and patience), the reader will acquire a valuable tool for handling groups, and a most useful collection of specimens.

10.3. Review by: Graham Higman.
Mathematical Reviews MR0207802 (34 #7617).

This book is an excellent and detailed account, with many examples, of some aspects of group theory closely connected with generators and relations. The approach throughout is very concrete. ... this book covers a section of group theory not adequately treated elsewhere in the literature. The treatment is very thorough, almost everything being written out in explicit detail. If this makes for a heavy book, materially, for a comparatively small corner of mathematics, it must be said also that it makes the book a delight to read, and easy to find one's way around. In spite of the very numerous problems, the reviewer sees this rather as a reference book than as a textbook, but for anyone with any interest in infinite groups, it will be indispensable.

10.4. From the Publisher of the 2004 reprint.

A seminal, much-cited account of combinatorial group theory - co-authored by a distinguished teacher of mathematics and a pair of his colleagues - this text for graduate students features numerous helpful exercises. The book begins with a fairly elementary exposition of basic concepts and a discussion of factor groups and subgroups. The topics of Nielsen transformations, free and amalgamated products, and commutator calculus receive detailed treatment. The concluding chapter surveys word, conjugacy, and related problems; adjunction and embedding problems; varieties of groups; products of groups; and residual and Hopfian properties. In addition to the exercises, which appear throughout the text, supplementary materials include an extensive bibliography of important books and monographs, as well as a list of theorems, corollaries, and definitions and a list of symbols and abbreviations.

11.1. The Preface.

11.2. Review by: H S M Coxeter.
American Scientist 63 (5) (1975), 588-589.

Almost every mathematician, at some stage of his career, must have flipped the pages of Fricke and Klein, 'Vorlesungen über die Theorie der automorphen Funktionen', or of Klein and Fricke, 'Vorlesungen über die Theorie der elliptischen Modulfunctionen', admiring the drawings and wondering what they signify. Magnus's book reproduces many of them and tells the whole story lucidly, beginning with the theory of homographies in the inversive plane and the corresponding motions in the hyperbolic plane. Many of Klein's statements were vague or obscure; these are now made precise, and the subject has been brought up-to-date by the inclusion of modern methods discovered by A M Macbeath, J Mennicke, B H Neumann, and others, including the author himself. ... It should appeal to students and teachers as well as to professional mathematicians.

11.3. Review by: Èrnest Borisovich Vinberg.
Mathematical Reviews MR0352287 (50 #4774).

This is an elementary book on discrete groups of Möbius transformations. It does not contain a systematic account of the subject, but rather a collection of theorems, examples and figures illustrating different aspects of the theory. Most of them are inspired by the work of F Klein and R Fricke. Some of the examples are taken from almost inaccessible sources. There are numerous historical remarks and references to contemporary works.

12.1. From the Introduction.

Combinatorial group theory may be characterized as the theory of groups which are given by generators and defining relations or, as we would say today, by a presentation. Of course, this is not a complete definition of the field and we shall not even try to give one. But at least for Part I it is a fully adequate description of the subject we plan to deal with.

The first problem facing the historian is that of finding a starting point. In our case, this was rather easy. The paper by Walther von Dyck [1882] is the first paper in which generators and defining relations are not only introduced as new concepts, but are also used effectively for mathematical research. It is most likely that at least the germs of the ideas introduced by Dyck can be traced to earlier authors. A thorough study of the emergence of the various aspects of the concept of a group was conducted by H Wussing (1969).

Starting with a report on Dyck's first paper and an analysis of its contents, our book describes the unfolding of combinatorial group theory, its concepts, problems, results, and its relation with other fields, in particular, with topology. Here we encounter the second difficulty facing every historian. It is impossible to write a universal history. In the case of mathematics, it is even impossible to explain the technical terms appearing in the influencing disciplines in any but the most superficial manner. We have met the difficulties arising here with a compromise, i.e., something which, by its very nature, cannot be entirely satisfactory.

12.2. Review by: Ramaiyengar Sridharan.
Indian Journal of History of Science 19 (3) (1984), 293-294.

The statement that the modern scientist wants to know more and more about less and less is not far from truth even in a subject like Mathematics which is becoming unwieldy and fragmented. Granting that it is impossible for an individual mathematician to have a profound grasp of extensive areas in mathematics, one can at least ask for a history of ideas in a rather specific area, a history which not merely gives credit to the individual contributors, but places these ideas in their proper perspective and relevance. It was however a moot question for the reviewer whether such a specialised topic like "combinatorial group theory" deserved a history (or as the authors of the book term "a case study in the history of ideas"). This doubt with which the reviewer began to read this book disappeared as soon as he became aware of the wealth of information contained in it. Just to quote one example: - It is sheer delight to read about the "history" of the principal ideal theorem and how it led to the conjecture and the solution of the so-called class-field-tower problem. Such examples abound in this book. In spite of the rather unattractive title, this book is very much worth going through for the valuable insight it offers into the history of group theory and the light it throws on the lives of various mathematicians who played a decisive role in the development of ideas in group theory and related fields.

12.3. Review by: Robert Bryce.
Mathematical Reviews MR0680777 (85c:01001).

Combinatorial group theory is that part of group theory which deals with groups given by presentations, that is by systems of generators and defining relations. Its origins are different from those of other parts of group theory; whereas, for example, finite groups arise first in the developing theory of equations in the first half of the nineteenth century, groups given by presentations arise from the developing topological approach to the study of geometry in the second half of the nineteenth century. Other classes of groups characterized by some special property or mode of definition have their own history. Of course concepts and methods of proof which arise in one of these areas have found fruitful application in others but most group-theoretical study can be described accurately enough as belonging to "combinatorial group theory" or to "permutation groups", etc. Thus the authors discuss general group theory only to the extent that it has influenced, or itself been influenced by, combinatorial group theory. Combinatorial group theory is still one of the most important links between geometry and algebra. The book is divided into two parts by the year 1918, not just because of the hiatus of World War I, but also because that year "marked a change in the nature of research in combinatorial group theory".