# Francesco Tricomi's books

Giving a list of Francesco Tricomi's books with some information about each is nothing like as simple as it is for most authors. Partly this is the result of the large number of books with various editions, partly because these books have been translated from Italian into several languages, Russian, German, French and English. With apologies to our Russian readers, we have omitted all the Russian translations from the list below. We have ordered them in an attempt to put later editions and translations together, rather than put the books in a strictly chronological order.

Click on a link below to go to the information about that book.

Funzioni Analitiche (1936)

Funzioni Analitiche (2nd ed.) (1946)

Funzioni Ellittiche (1937)

Funzioni Ellittiche (2nd ed.) (1951)

Elliptische Funktionen (1948)

Lezioni di Analisi Matematica (1939)

Esercizi e Complementi di Analisi Matematica. Parte Seconda (1949)

Esercizi e Complementi di Analisi Matematica. Parte Seconda (3rd. ed.) (1960)

Lezioni di Analisi Matematica. Parte Prima (6th ed.) (1948)

Serie Ortogonali di Funzioni (1948)

Vorlesungen über Orthogonalreihen (1955)

Equazioni differenziali (1948)

Equazioni differenziali (2nd. ed.) (1953)

Equazioni differenziali (3rd. ed.) (1961)

Differential Equations (1961)

Lezioni sulle funzioni ipergeometriche confluenti (1952)

Higher Transcendental Functions Vol. I and Vol. II (1953) with Harry Bateman, Arthur Erdelyi, Wilhelm Magnus and Fritz Oberhettinger

Higher Transcendental Functions Vol. III (1955) with Harry Bateman, Arthur Erdelyi, Wilhelm Magnus and Fritz Oberhettinger

Tables of integral transforms. Vol. I. Based, in part, on notes left by Harry Bateman (1954) with Arthur Erdelyi, Wilhelm Magnus and Fritz Oberhettinger

Tables of integral transforms. Vol. II. Based, in part, on notes left by Harry Bateman (1954) with Harry Bateman, Arthur Erdelyi, Wilhelm Magnus and Fritz Oberhettinger

Funzioni ipergeometriche confluenti (1954)

Fonctions Hypergéométriques Confluentes (1960)

Lezioni sulle equazioni a derivate parziali (1954)

Vorlesungen über Orthogonalreihen (1955)

Integral Equations (1957)

Equazioni a derivate parziali (1957)

Aerodinamica Transonica (1962) with C Ferrari

La mia vita di matematico attraverso la cronistoria dei miei lavori (Bibliografia commentata 1916-1967) (1967)

Reportorium der Theorie der Differentialgleichungen (1968)

Istituzioni di analisi superiore (metodi matematici della fisica) (1970)

Click on a link below to go to the information about that book.

Funzioni Analitiche (1936)

Funzioni Analitiche (2nd ed.) (1946)

Funzioni Ellittiche (1937)

Funzioni Ellittiche (2nd ed.) (1951)

Elliptische Funktionen (1948)

Lezioni di Analisi Matematica (1939)

Esercizi e Complementi di Analisi Matematica. Parte Seconda (1949)

Esercizi e Complementi di Analisi Matematica. Parte Seconda (3rd. ed.) (1960)

Lezioni di Analisi Matematica. Parte Prima (6th ed.) (1948)

Serie Ortogonali di Funzioni (1948)

Vorlesungen über Orthogonalreihen (1955)

Equazioni differenziali (1948)

Equazioni differenziali (2nd. ed.) (1953)

Equazioni differenziali (3rd. ed.) (1961)

Differential Equations (1961)

Lezioni sulle funzioni ipergeometriche confluenti (1952)

Higher Transcendental Functions Vol. I and Vol. II (1953) with Harry Bateman, Arthur Erdelyi, Wilhelm Magnus and Fritz Oberhettinger

Higher Transcendental Functions Vol. III (1955) with Harry Bateman, Arthur Erdelyi, Wilhelm Magnus and Fritz Oberhettinger

Tables of integral transforms. Vol. I. Based, in part, on notes left by Harry Bateman (1954) with Arthur Erdelyi, Wilhelm Magnus and Fritz Oberhettinger

Tables of integral transforms. Vol. II. Based, in part, on notes left by Harry Bateman (1954) with Harry Bateman, Arthur Erdelyi, Wilhelm Magnus and Fritz Oberhettinger

Funzioni ipergeometriche confluenti (1954)

Fonctions Hypergéométriques Confluentes (1960)

Lezioni sulle equazioni a derivate parziali (1954)

Vorlesungen über Orthogonalreihen (1955)

Integral Equations (1957)

Equazioni a derivate parziali (1957)

Aerodinamica Transonica (1962) with C Ferrari

La mia vita di matematico attraverso la cronistoria dei miei lavori (Bibliografia commentata 1916-1967) (1967)

Reportorium der Theorie der Differentialgleichungen (1968)

Istituzioni di analisi superiore (metodi matematici della fisica) (1970)

**1. Funzioni Analitiche (1936), by Francesco G Tricomi.**

**1.1. Review by: George E Raynor.**

*Bull. Amer. Math. Soc.*

**44**(1) (9) (1938), 610-611.

This volume has as one of its purposes the presentation of the fundamentals of the theory of functions of a complex variable in sufficient detail to serve for the study of the elliptic functions as treated in

*Funzioni Ellittiche*(1937); and in [it] the author has endeavoured to develop the theory in such a way as to make the applications to particular concrete problems as easy and direct as possible. In this he has succeeded admirably. It is no mean task to select from the wealth of the literature on analytic ... functions the material that would best suit the author's purpose. The selections have been made with discrimination and have been treated skilfully. ... For the technical student wishing to acquire a working knowledge of the fundamentals this book is to be recommended. Furthermore, the student of pure mathematics will find the author's point of view invigorating.

Contents. Chapter 1. Complex numbers, functions, Cauchy-Riemann equations, harmonic functions, conformal mapping, irrotational fluid motion. Chapter 2. Cauchy's integral theorem, residues, Cauchy's integral formula, Dirichlet's problem, mean value theorems. Chapter 3. Infinite series, the Taylor and Laurent expansions, poles and isolated essential singular points. Chapter 4. Analytic continuation, Riemann surfaces, the elementary transcendental functions, Mittag-Leffler's theorem, Weierstrass's theorem.

**2. Funzioni Analitiche (2nd ed.) (1946), by Francesco G Tricomi.**

**2.1. Review by: Walter Strodt.**

*Bull. Amer. Math. Soc.*

**53**(7) (1947), 739-740.

This is a brief sketch of the elementary theory of analytic functions of one complex variable. The author succeeds admirably in presenting a very readable account which makes plain the importance of the concepts, the power of the methods, and the broad outlines of the proofs.

More space than is usual in such a brief account is given to the connections of this theory with the theory of harmonic functions and of contour integration in vector analysis. Nine pages are devoted to simple applications to the theory of hydrodynamics. About six pages are devoted to well-illustrated discussions of the graphical representation of analytic functions by means of level lines of the real and imaginary parts. There are twenty-nine very good diagrams.

Little attention is given to questions of foundations. The reader is apparently assumed to have a working knowledge of real analysis from the point of view of a moderately rigorous upper-college course in the calculus, and the proofs are handled in the spirit of such a course.

The first chapter treats the definitions of continuity and analyticity, with some emphasis on the connection of the latter with the problem of conformal mapping. The second chapter discusses the Cauchy integral formula, with special attention to the relation of the Cauchy integral theorem to Greene theorem; the principle of the maximum is here derived for harmonic functions, and for the modulus of an analytic function. The third chapter considers Taylor's series and Laurent series, and the classification of singularities. The fourth chapter treats the general problem of analytic continuation, the Weierstrass concept of the complete analytic function, and Riemann surfaces; the special topics of Schwarzian reflection, exponential and trigonometric functions, Mittag-Leffler's theorem for meromorphic functions, and the Weierstrass product-theorem for entire functions are considered.

**2.2. Review by: Ralph Philip Boas Jr.**

*Mathematical Reviews*MR0017792

**(8,200i).**

This brief introduction to the theory of functions of a complex variable is little changed from the edition of 1936. The fundamentals of the subject are set forth clearly, under the simplest hypotheses; many geometrical illustrations are given. The first chapter deals with the notion of analytic function, connections with harmonic functions and conformal mapping. The second chapter develops complex integration through the elements of the calculus of residues and Liouville's theorem. The third chapter discusses Taylor and Laurent series and isolated singular points. The fourth chapter deals briefly with analytic continuation, the elementary transcendental functions, and the Mittag-Leffler and Weierstrass representations of meromorphic and entire functions.

**3. Funzioni Ellittiche (1937), by Francesco G Tricomi.**

**3.1. Review by: George E Raynor.**

*Bull. Amer. Math. Soc.*

**44**(1) (9) (1938), 610-611.

In this volume the author has endeavoured to develop the theory in such a way as to make the applications to particular concrete problems as easy and direct as possible. In this he has succeeded admirably. It is no mean task to select from the wealth of the literature on ... elliptic functions the material that would best suit the author's purpose. The selections have been made with discrimination and have been treated skilfully. It is particularly refreshing to read a book on elliptic functions free from the mire of countless formulas that so often clutter books on this subject. For the technical student wishing to acquire a working knowledge of the fundamentals these books are to be recommended. Furthermore, the student of pure mathematics will find the author's point of view invigorating.

Contents. Chapter 1. Historical introduction, periodicity, the Weierstrass $\pi$, $\zeta$, and $\sigma$-functions. Chapter 2. Elliptic integrals. (This is a particularly good treatment of the subject.) Chapter 3. The sn, cn, and dn functions of Jacobi, Jacobi's theta functions. Chapter 4. The $\pi$ functions related algebraically, transformations of elliptic functions, automorphic functions, Landen's transformation. Chapter 5. Applications: rectification of the ellipse and hyperbola, geodesies on an ellipsoid of revolution, loaded elastic columns, simple pendulum, a problem in conformal mapping arising in aerodynamics. Table of principal formulas.

**4. Funzioni Ellittiche (2nd ed.) (1951), by Francesco G Tricomi.**

**4.1. Review by: Zeev Nehari.**

*Mathematical Reviews*MR0051362

**(14,468c).**

The first edition of this text appeared in 1937 and a somewhat augmented German edition, prepared by M Krafft, was published in 1948. While including some additional material, the present edition is in general closer to the original Italian version than to the German edition.

**5. Elliptische Funktionen (1948), by Francesco G Tricomi.**

**5.1. Review by Zeev Nehari.**

*Mathematical Reviews*MR0029001

**(10,532c).**

This is an extremely well written book on the generally useful parts of the theory of elliptic functions. In a subject which, for a century and a half, has been characterised by an overpowering mass of formal relations and identities, the author has successfully avoided succumbing to the temptation of trying to amaze the reader by surprising identities and of including formulas for their sheer beauty. Some other authors have tried to overcome the 'embarras de richesse' by solving an "extremal problem," namely by deriving a maximum of relations in a minimum of space. The author has steered clear of this temptation. The book is well-organised and at no place is coherence sacrificed in favour of a mere display of ingenuity. Derivations which are inessential from the overall point of view of the book are strictly left alone.

Much care has been devoted to clarity of exposition. In contradistinction to a widespread custom in mathematical literature, statements like "it is easily seen" are invariably followed up by full explanations. While it is in the nature of things that the knowledge of some results in the elementary theory of functions of a complex variable has to be assumed, the amount of presupposed knowledge in that field is kept at a minimum; where feasible, such auxiliary results are proved in the text.

A particularly valuable feature of the book is a chapter on applications of elliptic functions which contains material otherwise not easily accessible. It is also worthy of note that the author takes the term "applications" seriously and does not hesitate to carry through the ensuing computations to the last numerical detail. This book can be strongly recommended to the students of both pure and applied mathematics.

**6. Lezioni di Analisi Matematica (1939), by Francesco G Tricomi.**

**6.1. Review by: Einar Hille.**

*Mathematical Reviews*MR0001790

**(1,298f)**.

This is the second volume of a text-book on advanced calculus covering the following topics: (1) The definite integral; upper and lower Riemann integrals, mean value theorems, extensions. (2) Integration in finite terms; rational, algebraic functions. (3) Series expansions and numerical calculations; uniform convergence, power series, Taylor's formula and series, elementary analytic functions, Fourier series, interpolation and numerical integration. (4) Functions of several variables; derivatives, homogeneous functions, total differentials, Taylor's formula and series, implicit functions, Jacobians, functional dependence, maxima and minima, least squares, Schwarz's inequality. (5) Elements of differential geometry; vectors, space curves including length, curvature, torsion, Frenet's formulas, singularities of plane curves, envelopes, surfaces including first fundamental form and curvature. (6) Integration of functions of several variables; geometrical applications, formula of Gauss. (7) Ordinary differential equations; successive approximations, singular solutions, linear equations, integration by series. (8) On partial differential equations and the calculus of variations; elements, equation of Laplace, Green's theorem, vibrating string, fundamental problem of calculus of variations, the brachystochrone and the isoperimetric problem. The presentation is clear and simple but appears to be rigorous.

**6.2. Review by: Amos Hale Black.**

*Amer. Math. Monthly*

**50**(2) (1943), 118.

This is the second of two volumes. The first volume ended with the indefinite integral. The second begins with the definition of the definite integral in the Riemann sense. Applications are made to ordinary algebraic and transcendental functions. Other topics treated are development in series, Taylor's formula, Simpson's rule, and the introduction of Fourier series.

Differentiation of functions of more than one variable is discussed with application to mean value, maximum and minimum, and Taylor's series. Topics in differential geometry include: tangent, length, osculating plane, curvature. and torsion for twisted curves; singularities and envelopes for plane curves; analytical representation and the Dupin indicatrix for surfaces; correspondence between points in two planes.

Multiple integrals are treated with application to areas and volumes.

In the chapter on ordinary differential equations types discussed are first order, linear and linear with constant coefficients, development in series, and some mention of equations of higher order.

Partial differential equations are discussed with their use in Laplace's equation and harmonic functions, Green's theorem, the problem of the vibrating cord, and the fundamental problem in the calculus of variation.

This volume is written in the same clear style as the first volume and, like it, has no additional exercises included although they are available.

**7. Esercizi e Complementi di Analisi Matematica. Parte Seconda (1949), by Francesco G Tricomi.**

**7.1. Review by: Editors.**

*Mathematical Reviews*MR0028903

**(10,516e).**

This supplements the author's

*Lezioni di Analisi Matematica, parte seconda*.

**8. Esercizi e Complementi di Analisi Matematica. Parte Seconda (3rd. ed.) (1960), by Francesco G Tricomi.**

**8.1. Review by: Editors.**

*Mathematical Reviews*MR0122916

**(23 #A248)**.

The first edition (1949) was reviewed [see 7.1]. The present edition is much altered.

**9. Lezioni di Analisi Matematica. Parte Prima (6th ed.) (1948), by Francesco G Tricomi.**

**9.1. Review by: Editors.**

*Mathematical Reviews*MR0025531

**(10,22a).**

This is a slight revision of the 4th edition (1939); the 5th edition (1943) appeared only in lithographed form.

**10. Serie Ortogonali di Funzioni (1948), by Francesco G Tricomi.**

**10.1. Review by: Ralph Philip Boas Jr.**

*Mathematical Reviews*MR0026701

**(10,188e).**

This course of lectures contains a brief introduction to orthogonal functions in general, an account of important topics in Fourier series, and a discussion (nearly half the book) of the classical orthogonal polynomials, in which both properties of the individual polynomials and convergence theorems are given. For the topics included, the treatment is modern and quite thorough.

**11. Vorlesungen über Orthogonalreihen (1955), by Francesco G Tricomi.**

**11.1. Review by: Arthur Erdélyi.**

*Bull. Amer. Math. Soc.*

**67**(5) (1961), 447-449.

This book, like many of its author's other well-known books, originated in courses of lectures given at the University of Turin. It is an enlarged and considerably revised version of a preliminary (mimeographed) Italian edition (

*Serie Ortogonali di Funzioni*, Gheroni, Torino, 1948) which is now out of print. The very competent translation is the work of Dr F Kasch, of Göttingen.

The author's aim is to provide a lucid, comparatively elementary, and highly readable introduction to orthogonal expansions, and in particular to trigonometric series and orthogonal polynomials. In this he succeeds admirably, demanding from the reader little more than a thorough knowledge of advanced calculus (some knowledge of the elementary theory of Lebesgue integrals, and perhaps a little more on infinite series than is contained in some advanced calculus courses). It is not part of the author's plan to replace Zygmund on trigonometrical series, or Szegö on orthogonal polynomials, to aim at encyclopaedic completeness or at penetrating to the most modern parts of the theory; and he valiantly resists the temptation to enter into discussions which, in the framework of this volume, must at best remain sketchy.

If the preceding remarks stress the didactic orientation of the volume under review, they are not intended to convey the impression of a run-of-the-mill textbook. Far from it. As would be expected by the readers of the author's numerous books, individual touches abound, and some original results due to the author himself are incorporated in the presentation, especially in the second half of the book. Here we get a characterisation of the classical orthogonal polynomials by their generalised Rodriguez formulas, and many original results on the asymptotic behaviour and other analytic properties of the classical orthogonal polynomials.

**12. Equazioni differenziali (1948), by Francesco G Tricomi.**

**12.1. Preface.**

A book of this kind can have two distinct and almost irreconcilable aims - either to be a reference book surveying briefly all aspects of the subject and containing an extensive bibliography, or, alternatively, to be a teaching book designed to give the student a clear idea of the problems and methods of the theory of differential equations, which is one of the most important branches of analysis.

This volume has been written with the second of these aims in mind, for there is no lack of good recent reference works. It has grown out of University courses delivered by the author, and it makes no claim to completeness. Only questions which can be treated both with rigour and simplicity are discussed, and the subject matter has been further restricted by excluding those topics which demand mathematical knowledge beyond that of the Honours student.

Limitations of space have compelled me to deal only with ordinary differential equations (partial differential equations are not considered) and to exclude the so-called elementary methods of integration (separation of variables, integration of linear first-order equations, linear equations with constant coefficients, etc.). The actual contents are clearly detailed in the Index; Chapter I is introductory to the following chapters; Chapter II, Chapters III and IV together, Chapter V (the only part of the book which demands some knowledge of the theory of functions of a complex variable) form sections which may be read independently of each other.

Those readers familiar with the principal mathematical interests of the author may be surprised that this volume contains no mention of operational methods, in particular of integration by means of definite integrals. This however would demand more space than is available here, and there are already available the well-known books by Doetsch, Ghizzetti and others, on the use of symbolic methods (i.e. the Laplace transform) for the differential equations of electricity theory, etc.

In presenting the work I have constantly tried to stress the importance in the modern theory of differential equations of reading off directly from the equation the properties of its integrals, in contrast to the earlier aim of integrating the equation explicitly. Difficult cases are hardly ever dealt with in their most general form but in the simplest form possible, so that the fundamental ideas underlying the methods adopted may be most clearly seen.

The reader already expert on this subject will appreciate the usefulness and simplicity of the Prefer change of variable in establishing the existence theorem for eigenvalues (Chapter Ill) and deriving the asymptotic representation of integrals of second-order linear equations (Chapter IV); also the treatment of the characteristics of a first-order equation (Chapter II) which under the less restrictive conditions used here appears for the first time in a textbook.

I would further point out that in the "asymptotic integration" of linear equations by the method of Poincare (Chapter V) I have been able to remove the restriction that the independent variable must tend to infinity through real values; this allows me to obtain the classical asymptotic series for Bessel functions by a procedure which can hardly be improved on.

I hope this book will be found useful, particularly by the students for whom it is intended.

**12.2. Review by: Izaak Opatowski.**

*Science New Series*

**109**(2827) (1949), 236.

This textbook on advanced ordinary differential equations, mainly of linear type, contains the author's improvements of Bendixon's theorem on the shape of solutions, and also of a theorem by Sonin and Pólya on the conditions under which the maxima and minima of a solution of a second order linear differential equation form a steadily increasing or decreasing sequence. Great care was given to general theorems on the location of zeros of the solutions; theorems on the zeros of Bessel functions are derived from much more general ones in a very simple manner.

The study of the boundary value problems of the Sturm-Liouville type is based on an ingenious and quite elementary transformation of Prüfer which reduces the problem to a single differential equation of first order. The author discusses the behaviour of the characteristic values as the differential equation changes as well as relations to integral equations. A chapter on the so-called asymptotic methods includes a brief outline of the Laguerre and Legendre polynomials and illustrates numerically how good the asymptotic approximations under appropriate conditions are. A chapter on differential equations for complex variable includes the customary topics and also methods of solution through divergent series. The book has pronounced traits of originality in the choice of topics and particularly in its manner of presentation. The latter may be exemplified by the introduction of the elliptic functions of Jacobi as an application of a classical theorem on the existence of solutions of differential equations. The clarity of exposition, good printing and appropriate drawings make this book pleasantly readable.

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

*Bull. Amer. Math. Soc.*

**56**(2) (1950), 195-196.

This text contains a somewhat unusual but interesting and well integrated sequence of topics. A Picard type existence theorem is given and a singular point for a first order differential equation is investigated. Most of the discussion is concerned with second order equations or the equivalent systems. For these the linear dependence of solutions, the Wronskian theory, the variation of parameters method, the circuit of a singularity in the complex plane, Fuchs theorem, solution by power series, the Sturm-Liouville theory and the asymptotic behaviour of characteristic functions and characteristic values are given. The last chapter also contains a Cauchy type existence theorem, based on majorants.

There is a valuable emphasis on individual functions, whose properties are derived from the fact that they are solutions of a differential equation. For instance the circular and elliptic functions are treated in this way. The asymptotic behaviour of the Laguerre and Legendre polynomials and the Bessel functions are used to illustrate the characteristic function theory. The hypergeometric series is developed in the last chapter. On the other hand as a matter of policy the usual methods for the integration of first order equations are omitted.

The style is clear and the book should prove a valuable reference. The author claims, quite justly, that this corresponds to a "modern course" in differential equations and there is quite a contrast with the American courses on "methods of solution" and "theory." The pressure from applications and the results of theoretical developments are clearly present. However the existence theory is not the most general possible and the elementary methods and the constant coefficient linear equations are worth considering. The need of two courses seems clear but they should be carefully organised for maximum usefulness.

**12.4. Review by: LeRoy A MacColl.**

*Mathematical Reviews*MR0027919

**(10,375e).**

This is a very well planned and written textbook in ordinary differential equation theory for graduate students. The five chapters into which it is divided are devoted, respectively, to: (1) the fundamental theorems concerning the existence, uniqueness, and continuity properties of solutions; (2) the qualitative properties of the family of characteristics for an equation of the first order; (3) boundary value problems for linear equations of the second order; (4) asymptotic properties of eigenfunctions and eigenvalues for linear equations of the second order; (5) differential equations in the complex domain. The author does not aim at any high degree of completeness, and it is only rarely that he assumes any knowledge on the part of the reader that could not be obtained from an ordinary introductory course in the theory of functions. Within these limits, however, the treatment is sound and remarkably clear. The lucid exposition in the second and third chapters appealed strongly to the reviewer. There are no problems for the reader, but there are numerous illuminating applications of the general theory to some of the particular equations which are important in mathematical physics. There seem to be almost no typographical errors; and the physical appearance of the volume is excellent.

**13. Equazioni differenziali (2nd. ed.) (1953), by Francesco G Tricomi.**

**13.1. Preface.**

Despite the short period which has elapsed since the first edition appeared (1948), this new edition differs in several aspects from the earlier one, mainly because the material of the book deals with one of the most active branches of analysis and within these last years there have been many important contributions in this field.

Several major improvements have been made in the text and many additions to the Bibliography. I would draw attention especially to these features:

- The additions to Chapter 11, in which topological methods are applied to the study of relaxation oscillations and related topics of importance in nonlinear mechanics.

- The deep yet relatively simple discussion in Chapter IV of the asymptotic behaviour of the integrals of the equation $y'' + Q(x)y = 0$ which makes use of recent work of my colleague G Ascoli; this work arose partly from discussion between us regarding this new edition of my book.

- The inclusion in the text and the further development of the general method of treatment of differential equations which I have called "the method of Fubini"; in the earlier edition this work appeared in an appendix.

- The substantial simplification introduced in the determination of the eigenvalues of the Legendre equation in Chapter IV.

F G T.

Turin, October 1952

**13.2. Review by: R T Reid.**

*Bull. Amer. Math. Soc.*

**61**(4) (1955), 371-372.

The revised edition of this book follows very closely the pattern of the first edition, which was reviewed in this

*Bulletin*(vol. 56 (1950) pp. 195-196). The most important cases of inclusion of new material are: (i) Chapter II has been augmented by an introduction to the subject of relaxation oscillations; (ii) Chapter IV has been revised considerably, to provide a more comprehensive treatment of the asymptotic character of solutions of differential equations of the form $y'' + Q(x)y = 0$.

Details of discussion have been altered in various instances, notably in Chapter IV in the treatment of the polynomials of Laguerre and Legendre. Material on the "method of Fubini" that formed an Appendix in the initial edition has been incorporated in Chapter IV; also, a number of new references have been added to the bibliography.

In this new edition the author has produced a commendable improvement of the highly interesting and valuable first edition.

**13.3. Review by: LeRoy A MacColl.**

*Mathematical Reviews*MR0061227

**(15,793a)**.

In this second edition the general plan of the book has been left unchanged, but some additional material has been inserted, and the details of the exposition have been revised in many places. Among the more important changes are the following. (1) The presentation of the Poincaré-Bendixson theory of the qualitative properties of the solutions of a first-order equation has been expanded somewhat, and the applications of that theory to the theory of non-linear vibrations are discussed briefly. (2) Numerous improvements have been made in the chapter dealing with the asymptotic properties of the solutions of linear equations. These include a simpler and deeper treatment of the solutions of the equation $y'' + Q(x)y = 0$, and a revised discussion of the determination of eigenvalues in the case of Legendre's equation. (3) The bibliography has been amplified, and some material which was treated in an appendix in the first edition has been transferred to the main text. All of these changes have been wisely made, and they enhance the value of this already excellent text.

**14. Equazioni differenziali (3rd. ed.) (1961), by Francesco G Tricomi.**

**14.1. Review by: Nicholas D Kazarinoff.**

*Mathematical Reviews*MR0138811 (25 #2254a).

The main changes in this edition are that the theorem of de la Vallée Poussin on the minimal distance between two successive zeros of a second-order equation is improved, a section on asymptotic solutions of differential equations with simple turning points has been added, and the application of the method of successive approximations to linear equations with a non-Fuchsian singular point has been simplified.

**15. Differential Equations (1961), by Francesco G Tricomi.**

**15.1. Review by: Okan Gurel.**

*SIAM Review*

**4**(3) (1962), 269-270.

This English translation by Elizabeth A McHarg corresponds to the third edition of the Italian original. The book contains five chapters:

- The existence and uniqueness theorem ...

- The behaviour of the characteristics of a first-order differential equation ...

- Boundary problems for linear equations of the second order ...

- Asymptotic methods ...

- Differential equations in the complex field ...

**15.2. Review by: John Charles Burkill.**

*The Mathematical Gazette*

**46**(358) (1962), 362-363.

This is a translation of the third Italian edition of Tricomi's book, the first edition dating from 1946. The author is one of the leading analysts of Italy, particularly in the theory of differential and integral equations and of expansions related to them, and his reputation is a guarantee of the quality of the book. His aim was to produce a book for teaching rather than for reference and to give a clear account of selected problems and methods in their most illuminating rather than their most comprehensive form. An honours course in differential equations could well include some of these topics. A more systematic study of them would be, in this country, work for the graduate.

Chapter I contains existence and uniqueness theorems, with an illustration from elliptic functions (a subject now regrettably squeezed out of the education of most young mathematicians). Then follow four substantial chapters each of about 60 pages. Chapter II discusses solutions of the pair of equations

$\Large\frac{dx}{dt}\normalsize = P(x, y), \Large\frac{dy}{dt}\normalsize = Q(x, y)$.

The foundations of this subject were laid by Poincaré, and it is now a field of active research. The next two chapters treat the second-order linear equation. In Chapter III we find the oscillation-theorems and in Chapter IV methods are developed of finding asymptotic expressions for solutions of equations. Chapter V deals with differential equations in the complex field, particularly the Fuchsian theory, with some more results on asymptotic approximation.

**15.3. Review by: Nicholas D Kazarinoff.**

*Mathematical Reviews*MR0138812

**(25 #2254b).**

The main changes in this edition are that the theorem of de la Vallée Poussin on the minimal distance between two successive zeros of a second-order equation is improved, a section on asymptotic solutions of differential equations with simple turning points has been added, and the application of the method of successive approximations to linear equations with a non-Fuchsian singular point has been simplified. ... The English translation (from the third edition) is well done. The fewer number of pages is a welcome feature. This book can now have the wide audience it deserves among English-speaking students.

**16. Lezioni sulle funzioni ipergeometriche confluenti (1952), by Francesco G Tricomi.**

**17.1. Review by: Arthur Erdélyi.**

*Bull. Amer. Math. Soc.*

**60**(1954), 185-189.

Tricomi's book is based on a course of lectures given at the University of Torino, and it is written with that gift for exposition for which its author is so justly famous. The presentation is well planned, leisurely, and detailed, with very few computations "left to the reader" and no vague appeals to "it is easy to see." The book is self-contained, and those parts of mathematical analysis which are needed, and with which the student may not be familiar, form the topics of several digressions. There is no attempt at all-inclusiveness, but the topics which have been selected are discussed rather fully. Books and memoirs are quoted when they are needed in the text: otherwise there is no bibliography, and there is no index. The book naturally favours those parts of the theory of confluent hypergeometric functions, and these are many, to which its distinguished author contributed in recent years. Shortly, this book is an eminently readable introduction to confluent hypergeometric functions which takes the reader, in several directions, to topics of interest in current research.

**16.2. Review by: Arthur Erdélyi.**

*Mathematical Reviews*MR0050064

**(14,269h).**

Confluent hypergeometric series were introduced by Kummer more than a century ago. In 1904, E T Whittaker proposed new definitions and notations which clearly exhibit the symmetry and the transformation properties of confluent hypergeometric functions, and he also showed that a large number of special functions (Bessel functions, Laguerre polynomials, error functions, and many others) are particular confluent hypergeometric functions. Since then a sizable literature has grown up around these functions and more and more applications were found. Several monographs were planned but, as far as the reviewer knows, none appeared in print. Thus the book under review is the first one in the field; and it derives added significance from its author's gift for exposition, and from his many contributions in recent years to the theory of confluent hypergeometric functions.

The book is based on a course of lectures given at the University of Torino, and this origin is noticeable throughout the book. The presentation is well planned, leisurely, detailed (very few computations are "left to the reader"), and self-contained, with several digressions on topics with which the student may not be familiar. There is no attempt at all-inclusiveness but the topics which have been included are discussed rather fully. Books and memoirs are quoted when they are needed in the text; otherwise there is no bibliography, and there is no index. All in all, we have a well written and eminently readable introduction to confluent hypergeometric functions before us which is not, and does not attempt to be, a work of reference. Prerequisites for reading this book are advanced calculus and the more elementary parts of complex function theory.

**17. Higher Transcendental Functions Vol. I and Vol. II (1953), by Harry Bateman, Arthur Erdelyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G Tricomi.**

**17.1. Preface by Arthur Erdelyi.**

The work of which this book is the first volume might be described as an up-to-date version of

*Part II. The Transcendental Functions*of Whittaker and Watson's celebrated 'Modern Analysis'. Bateman (who was a pupil of E T Whittaker) planned his 'Guide to the Function' on a gigantic scale. In addition to a detailed account of the properties of the most important functions, the work was to include the historic origin and definition of, the basic formulas relating to, and a bibliography for all special functions ever invented or investigated. These functions were to be catalogued and classified under twelve different headings according to their definition by power series, generating functions, infinite products, repeated differentiations, indefinite integrals, definite integrals, differential equations, difference equations, functional equations, trigonometric series, series of orthogonal functions, or integral equations. Tables of definite integrals representing each function and numerical tables of a few new functions were to form part of the 'Guide'. An extensive table of definite integrals and a Guide to numerical tables of special functions were planned as companion works.

The great importance of such a work hardly needs emphasis. Bateman's unparalleled knowledge of the mathematical literature, past and present, and his equally exceptional diligence, would have made the book an authoritative account of its vast subject, and in many respects a definitive account; a Greater Oxford Dictionary of special functions.

A realistic appraisal of our abilities and of the time at our disposal led to a drastic revision of Bateman's plans. Only Bateman himself had the erudition to give a reliable and accurate history of special functions, and the manpower available to us was insufficient for the inclusion of all functions. Thus we restricted ourselves to an account (probably far less detailed than that planned by Bateman) of the principal properties of those special functions which we considered the most important ones. The loss thus caused to mathematical scholarship is great, regrettable, and final, but we venture to hope that it will be counterbalanced in some measure by the considerable reduction in size of the book, and by the gain in the clarity of its organisation. We can only hope that although the scope of the present work is much narrower than that envisaged by Bateman, in its humbler sphere the book will be more useful. ...

For the most part we were unable to make extensive use of Bateman's voluminous notes: we found it easier to compile our account of the various functions from our knowledge of these functions supplemented by routine search in the available literature.

**17.2. Review by: Clifford Truesdell.**

*Amer. Math. Monthly*

**61**(8) (1954), 576-578.

A few mathematicians will be disappointed by this work. To them I can remark that, having known Bateman toward the end of his life, I think it unlikely he would have finished either of his planned works no matter how long he had lived. As can be seen from the sequence of books he did finish, in him the characteristic British passion for disorder grew with age to monstrous intensity and the huge task of organisation before actually beginning to write these two works he was reluctant to face. On the other hand, the staff which has written the work under review has taken up a large responsibility, for to them alone was given any opportunity of making use of the great mass of material left by Bateman, an opportunity, as they tell us, they have decided to decline. It seems unfair both to Bateman and to the distinguished authors themselves that Bateman's name, not theirs, appears on the title page, which is cluttered besides with government gobbledegook.

...

The organisation is by functions: gamma, hypergeometric, Legendre, generalisations of the hypergeometric (several types, two chapters), confluent hypergeometric, Bessel, parabolic and paraboloidal, incomplete gamma, orthogonal polynomials, spherical and hyperspherical, orthogonal polynomials in several variables, elliptic. Most of the chapters begin with theory, followed by a table of formulae. There is an attempt to compromise between a pure list of results and an exposition of the subject. More difficult theorems are stated without proof, but when the steps in deriving a formula are fairly easy they are indicated. References tend to be to relatively recent works, especially when there exists a standard treatise for the function in question. There is a brief index of subjects, but no index of names; in the text, the discoverer of a formula is sometimes noted, and if so, accurately, but usually without a reference. There is little if any attempt to provide the reader with methods of attack for new problems. All kinds of readers will welcome heartily the careful and precise statement of range of validity which accompanies every formula.

**17.3. Review by: Edward T Copson.**

*Mathematical Reviews*MR0058756

**(15,419i).**

The late Harry Bateman, during his last years, planned an extensive compilation of the "Special Functions". He intended to investigate and tabulate their properties, the inter-relations between them, their representations in various forms, their macro- and micro-scopic behaviour, and to construct tables of the definite integrals involving them. The whole project was to have been on a gigantic scale; it would have been an authoritative and definitive account of its vast subject.

While much of the material is available, it is not readily accessible, being scattered in books and journals on many fields. The "Guide to the Functions" which Bateman planned would have been invaluable. The project was never completed, and, after his death, the California Institute of Technology and the U.S. Office of Naval Research pooled their resources to continue Bateman's task.

It turned out that no single section of Bateman's work was in a state suitable for immediate publication, and the field was so wide that it appeared essential to narrow it down if anything useful was to be accomplished. It was decided to concentrate on a three-volume work on the Higher transcendental functions (of which the first two are now under review), to be followed by two volumes of tables of integrals. The whole work has been carried out by the staff of the Bateman Manuscript Project, under the directorship of Arthur Erdélyi.

**18. Higher Transcendental Functions Vol. III (1955), by Harry Bateman, Arthur Erdelyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G Tricomi.**

**18.1. Review by: Edward T Copson.**

*Mathematical Reviews*MR0066496

**(16,586c)**.

This is the final volume on the higher transcendental functions prepared at the California Institute of Technology by the staff of the Bateman Manuscript Project; the first two volumes appeared in 1953. Actually, this volume was prepared by Erdélyi after the staff of the project left Pasadena, but the book owes a great deal to Magnus, who continued to work on it.

The topics discussed are automorphic functions, Lamé functions, Mathieu functions and spheroidal and ellipsoidal wave functions, functions of number theory, miscellaneous functions, generating functions. All is done in the thorough well-documented style of the earlier volumes.

**19. Tables of integral transforms. Vol. I. Based, in part, on notes left by Harry Bateman (1954), by Harry Bateman, Arthur Erdelyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G Tricomi.**

**19.1. Review by: Hermann Kober.**

*Mathematical Reviews*MR0061695

**(15,868a)**.

This book is the first volume of two which continue the work begun in "Higher transcendental functions" within the Bateman Manuscript Project. Definite integrals are tabled. The material is immense: a suitable principle of classification of the integrals had to be found. While Bateman had started to arrange them according to their fields of application, in this volume a considerable part of the material is organised in tables of standard transforms: Fourier, Laplace, Mellin transforms and their inversions; while Hankel transforms are left to the second volume. Thus the reader, in order to compute a given definite integral, has to reduce it to one of these standard forms (the authors state that integrals which cannot be simplified in this way are tabled in the second half of volume II). ...

Transforms are stated of algebraic functions, arbitrary powers, step functions, etc., of exponential, trigonometric and related functions, of the Gamma, Error and Legendre functions and orthogonal polynomials, of all types of Bessel and related functions, of hypergeometric and of other higher transcendental functions. A table of the notations and definitions of the latter functions and an index of notations are added. The authors concentrate mostly on integrals involving these functions, as there exist a number of good tables, to which they refer in the introduction, and they state a considerable number of new formulae, particularly on Laplace transforms.

...

This volume contains about 3000 formulas. A very useful book for work both in pure and applied mathematics! The authors have succeeded in systematising the vast subject.

**20. Tables of integral transforms. Vol. II. Based, in part, on notes left by Harry Bateman (1954), by Harry Bateman, Arthur Erdelyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G Tricomi.**

**20.1. Review by: Hermann Kober.**

*Mathematical Reviews*MR0065685

**(16,468c)**.

This is the second of the two volumes of tables of integral transforms. It contains about 2500 formulae. Some of them have been taken from standard literature, some from sources which were not readily accessible; and a great many of them are entirely new. This is not surprising; as even basic facts like the generalisation of the Laplace transformation by the K-transformation and the properties of the latter (see Chapter X) have only been found during the past two decades.

The first part, Chapters VIII-XV, continuing Volume I, contains tables of further integral transforms: Hankel and other transforms for which the kernels are Bessel functions in the widest sense, viz. Y-, K-, H- and Kontorovich-Lebedev transforms; fractional integrals; Stieltjes transforms; and Hilbert transforms. ...

...

The second part of this volume contains various integrals involving higher transcendental functions. Some of them cannot be written as transforms; others were not included in the transform tables and are given here, e.g. formulae in 19.2. The integrals concern orthogonal polynomials, viz. Chebyshev, Legendre, Gegenbauer, Jacobi, Hermite and Laguerre polynomials; the complete and incomplete ì function and related functions; Legendre functions on finite and infinite intervals; various kinds of Bessel functions, and finally of hypergeometric functions. A wealth of formulae!

**21. Funzioni ipergeometriche confluenti (1954), by Francesco G Tricomi.**

**21.1. Review by: Arthur Erdélyi.**

*Bull. Amer. Math. Soc.*

**61**(5) (1955), 456-460.

This book on confluent hypergeometric functions differs quite considerably from the same author's

*Lezioni suite funzioni ipergeometriche confluenti*reviewed recently in this

*Bulletin*(vol. 60 (1954) pp. 185-189), and is to all intents and purposes a new book. Almost the only common points with the

*Lezioni*are: the notation for confluent hypergeometric functions (which differs somewhat from that most commonly adopted), some of the applications discussed in the book, and, of course, the general point of view. In every other respect the differences are much more important than the similarities. The earlier work gave an introduction to confluent hypergeometric functions and was self-contained to a remarkable extent; the present work is a treatise on the subject, demanding from the reader both more knowledge, and more willingness to look up references. The author did not aim at encyclopaedic completeness, yet he endeavoured to give, and fully succeeded in giving, a comprehensive picture of the theory of confluent hypergeometric functions and of their applications in physics, engineering, and probability theory. Readers of earlier books by the same author need not be told that the presentation is excellent, the arrangement well planned, and the style both lucid and readable.

...

... [The author] must be complimented on having produced one of the best monographs of this kind in recent years. All basic results, and many other results, are proved in full detail, some other results are stated with references to the literature where proofs can be found, yet others are merely mentioned, again with references to existing literature. The book is well documented without overwhelming the reader with a multitude of references. Papers are referred to in footnotes throughout the book, and a list of 25 books is given at the end of the volume, where there is also an impressive list of 25 papers by the author on confluent hypergeometric functions and related topics.

**21.2. Review by: Arthur Erdélyi.**

*Mathematical Reviews*MR0076936

**(17,967d)**.

The present book differs considerably from the earlier

*Lezioni sulle funzioni ipergeometriche confluenti*: it is more complete, giving a comprehensive picture of the theory and applications of confluent hypergeometric functions, and assumes more preparation on the part of the reader.

**22. Fonctions Hypergéométriques Confluentes (1960), by Francesco G Tricomi.**

**22.1. Review by: Charles A Swanson.**

*Amer. Math. Monthly*

**68**(1) (1961), 78.

In this abridged version of his monograph

*Funzioni Ipergeometriche Confluenti*(1954; reviewed in Bull. Amer. Math. Soc. 61 (1955), 456-460), the author surveys the more important properties of confluent hypergeometric functions and illustrates through a few well-chosen examples the widespread applicability of these functions. By means of skilful organisation, he is able to include all the dominant features of the parent Italian volume and to bind them together into a cohesive unit. For detailed proofs and for especially involved formulae, however, the reader is often referred to the parent volume or to other easily accessible literature. Not written to be an introduction to the subject (the author has already written an excellent one), nor at the other extreme to be a handbook (an excellent one by Herbert Buchholz is available), this little book very effectively summarises the current state of knowledge of the subject.

**22.2. Review by: Felix M Arscott.**

*Mathematical Reviews*MR0120409

**(22 #11163).**

This is a clear, quite short, but comprehensive account of the confluent hypergeometric equation $zy'' + (c-z)y' - ay = 0$, based on the author's solutions $\phi(a, c; z)$ and $\psi(a, c; z)$ in preference to the Whittaker functions. The notation used is that of Erdélyi et al. in Higher transcendental functions (1953; 1955) and in content the article reviewed is an expanded version of Chapters VI and IX of the former, which were contributed by the author.

Of the four chapters in this article, I and II introduce and describe the solutions $\phi$ and $\psi$, and the advantages gained by choosing these is made clear. Chapter III deals with asymptotic expansions and IV with particular cases and applications. Of this last, the section dealing with the incomplete gamma functions seems to the reviewer to be particularly valuable.

**23. Lezioni sulle equazioni a derivate parziali (1954), by Francesco G Tricomi.**

**23.1. Review by: Richard Bellman.**

*Bull. Amer. Math. Soc.*

**61**(5) (1955), 87-88.

This book furnishes an excellent introduction to the rapidly expanding theory of partial differential equations, written in the author's usual lucid and interesting style.

The work is divided into five parts. The first part, consisting of one hundred and four pages, presents a rapid but thorough summary of classical analytic tools required in the remainder of the book. The theory of integral equations, the gamma function, the hypergeometric function, the Legendre and Bessel functions are all treated. This part is well worth reading on its own.

The second part, consisting of seventy-five pages, is devoted to a discussion of the theory of characteristics for equations of the first and second order. It includes a section devoted to the Hamilton-Jacobi theory and its connection with the calculus of variations.

The third part, one hundred pages, is devoted to equations of hyperbolic type. Various classical approaches, such as those of Laplace and Riemann, are presented, and there is large section on the movement of a compressible fluid.

The fourth part, ninety-five pages, treats the equations of elliptic type. The classical techniques are given, together with a discussion of more modern methods based upon difference equations, and numerical methods such as the "relaxation" method of Southwell. A section on incompressible fluids is included.

The fifth and concluding part is devoted to equations of parabolic type and equations of mixed type. The greater part of this section is concerned with equations of mixed type, a topic investigated in great detail by Tricomi in 1923, and which in recent years has become of great importance in the study of transonic flow. The applications to this theory are given in several sections.

Two very useful features are the ample references to both quite recent and classical papers and the eighty or so representative problems gathered at the ends of the various sections.

The book is particularly to be recommended to anyone intending to work in the mathematical theory of hydrodynamics or aerodynamics.

**23.2. Review by: Edward T Copson.**

*Mathematical Reviews*MR0067293

**(16,703a)**.

This book provides an excellent introduction to the theory of partial differential equations.

The first hundred pages contain an account of the classical analytical methods, including the theory of integral equations and the special functions of analysis, which are needed in the sequel. Ch. II contains, in 75 pages, a discussion of the theory of characteristics for equations of the first and second orders. In Ch. III (about 100 pages), the equation of hyperbolic type is discussed in detail. Equations of elliptic type are dealt with in Ch. IV (again about 100 pages), which includes not only the classical treatment, but also more modern numerical methods. The last chapter (about 100 pages) is devoted to equations of parabolic type and of mixed type.

The book is particularly valuable because of the full references to the literature of the subject and also because of the interesting sets of problems at the ends of each chapter. It is very regrettable that the book has had to be reproduced from typescript and that this has been done in an unsatisfactory manner. It deserves better treatment.

**24. Vorlesungen über Orthogonalreihen (1955), by Francesco G Tricomi.**

**24.1. Review by: Joaquin B Diaz.**

*Quarterly of Applied Mathematics*

**14**(4 (1957), 404.

This is another outstanding text by the author, whose expository skill is well known to mathematicians. The present monograph, as several of Tricomi's monographs, grew out of his lectures on the subject at the University of Turin. The main purpose of the present work is to furnish a rapid introduction to the theory of Fourier series and orthogonal polynomials, and in this the author has succeeded admirably. The arguments of the proofs are clearly presented, and one finds at every turn illuminating comments relating seemingly unrelated (at first glance) portions of the subject matter. Chapter I is devoted to the general theory of orthogonal sets of functions, and covers such topics as approximation in mean square, Fischer-Riesz theorem, and completeness and closure of a set of functions. Of special interest are the criteria of Lauricella, Vitali and Dalzell for the completeness of a set of functions and the proof of the completeness of the set of the trigonometric functions, which is based on Dalzell's criterion. Chapters II and III are concerned with Fourier series, the general theory and the convergence properties, respectively. In chapter II occur Riemann's theorem for the local behaviour of a Fourier series and the convergence criteria of Dirichlet, Dini and Lipschitz. In chapter III one finds the Lusin-Denjoy theorem on absolute convergence of a Fourier series, together with a consideration of Cesàro, Abel and Riemann summability, plus a brief discussion of the Fourier integral theorem. The remaining three chapters are devoted to orthogonal polynomials proper. Chapter IV presents the general properties of orthogonal sets of polynomials. There is a unified treatment of the so-called "classical" orthogonal polynomials, leading to a generalised "Rodriguez formula". Chapter V is dedicated to orthogonal polynomials on a finite interval: Jacobi, Gegenbauer, Tschebyscheff, Legendre, while Chapter VI deals with orthogonal polynomials on an infinite interval: Laguerre, Hermite, Jacobi.

**24.2. Review by: Gábor Szegö.**

*Mathematical Reviews*MR0070746

**(17,30a)**.

The book is essentially a translation of an earlier work of the author published in Italian [

*Serie ortogonali di funzioni*, 1948]. Several new features have been added; this review will point out only a few details. 1) In Ch. 1, the conditions of Lauricella, Vitali, and Dalzell for the completeness of a system of functions deserve interest; they are not too well known. 2) The property of the partial sums of the Fourier series discussed in Ch. 2, p. 75 (cf. also p. 124, footnote) is of course due to L Fejér [Math. Ann. 64, 273–288 (1907), p. 281; M Picone, 1924, is quoted]. 3) Ch. 5 deals with various useful properties of the hypergeometric and confluent hypergeometric functions complementing the material contained in the (somewhat "obsolete") book of the reviewer [Orthogonal polynomials, 1939]. 4) In Ch. 5 (orthogonal polynomials in finite intervals) more references seem desirable, for instance on p. 176 (zeros of Jacobi polynomials). 5) This is better in Ch. 6 (orthogonal polynomials in infinite intervals). The important results of the author on asymptotic formulas and zeros of Laguerre polynomials are only sketched (§ 2); it is only briefly mentioned that the proofs are based on the "method of Fubini". The equiconvergence theorem on Laguerre series due to the reviewer (§6) is only formulated. A more detailed treatment is given to the equiconvergence theorem for Jacobi series (§4); in fact this subject belongs to Ch. 5 since the interval is finite.

All in all the book is a fine didactic accomplishment.

**25. Integral Equations (1957), by Francesco G Tricomi.**

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

*Science New Series*

**127**(3313) (1958), 1494-1495.

The subject matter of this book, as interpreted by the author, is a topic which has become a standard part of everyday analysis, to be used, in particular, in innumerable problems of applied mathematics. A few such problems are treated or alluded to, among them the critical speeds of a rotating shaft, the forced oscillations of finite amplitude for a pendulum, the airfoil equation, the vibrations of a membrane. The book is intentionally short, and yet it covers all the classical types; it presents as simply as possible the essentials of the theories of Volterra, Fredholm, Hilbert, Erhardt Schmidt, Carleman, and Hammerstein. Standard real-variable tools, the Lebesque integral, the $L_{2}$ theory, orthonormal systems, and the transforms of Laplace, Fourier, and Mellin are used systematically but without ostentation, so as not to repel the physicist, engineer, or technician. Topological methods are not used. The style is attractive and is enlivened by some interesting personal comments (concerning Volterra, Fubini, and others).

**25.2. Review by: Joaquin B Diaz.**

*Quarterly of Applied Mathematics*

**17**(1) (1959), 66; 94.

In the opening lines of his preface, the author expresses himself as follows: "One of the first subjects in mathematics to attract my attention was integral equations; yet this book appears after a score of others. Why is this? It is because the writing of a book on integral equations is a rather difficult task, a task for which many years of meditation are necessary. In fact, such a book must satisfy two requirements which are not easily reconciled. In order to facilitate theoretical applications to existence proofs it must present the main results of the theory with adequate generality and in accordance with modern standards of mathematical rigour. On the other hand, it must not be written so abstractly as to repel the physicist, engineer, and technician who certainly need and deserve this mathematical tool." It is the reviewer's firm belief that our gifted author has admirably succeeded in satisfying both requirements, and that this book is destined to take its rightful place among "the score of others" which the mathematical world has already been fortunate to receive from his pen.

**25.3. Review by: Richard Carlton MacCamy.**

*Mathematical Reviews*MR0094665

**(20 #1177).**

In the preface the author points out the difficulties in writing a book on integral equations which can have meaning for the engineer and physicist while still coming to grips with the mathematics involved. In the opinion of the reviewer, Tricomi has had decided success in overcoming these difficulties. In a field which suffers from a lack of readable texts this book is a welcome arrival.

The middle portions of the book, chapters two and three, contain an exposition of classical Fredholm and Hilbert-Schmidt theory. The Fredholm theorem is established (for square integrable kernels) through approximation by degenerate kernels, but a résumé of Fredholm's method, using the Hadamard determinant theorem, is also presented. Chapter three contains a rather detailed discussion of orthonormal sequences and the Hilbert-Schmidt theory for symmetric kernels, and includes a description of the Ritz method. There is a short discussion of positive kernels and Mercer's theorem is proved. Chapter three also contains a concise treatment of the application of integral equations to Sturm-Liouville theory.

The author begins and ends the book with topics which are somewhat less standard. Chapter one is a rather extensive study of Volterra equations. In addition to serving as an introduction to the methods and difficulties in the theory of integral equations, this is a subject of considerable interest in itself, which often is lost in the shadow of Fredholm theory. Some interesting applications are made to differential equations.

Chapter four contains some remarks on singular integral equations, that is, those with kernels which are not square integrable. This chapter should be of particular aid to those who, for either physical or mathematical reasons, wish to read the extensive work of Muskhelishvili et al. The ideas which are essentially function theoretic are clearly presented here without the complications of generality surrounding Muskhelishvili's book "Singular integral equations" (1946). This chapter also includes some bits and pieces from the little understood field of non-linear integral equations.

**25.4. Review by: Albert E Heins.**

*Bull. Amer. Math. Soc.*

**64**(4) (1958), 197-198.

This excellent textbook on integral equations was written to give an adequate introduction of the subject to those who require a knowledge of it in mathematics or in its applications. With a basic knowledge of the theory of the Lebesgue integral and the theory of functions of a complex variable, this book may be read with profit.

The first three chapters are devoted to the so-called regular theory. Here we are given a careful and leisurely discussion of the Volterra equation, the regular Fredholm equation and equations with symmetric kernels. Such a program may sound traditional but an examination of the text proves otherwise. For one thing, the text is liberally peppered with physical problems which lead to such integral equations. Another feature reveals to us the relation between linear differential and integral equations. When further background material in analysis is required, Professor Tricomi supplies the reader with a discussion of the important features and provides references. Many of the historical remarks are of great interest. In short, we have an interesting and sparkling account of the regular theory of linear integral equations which should hold the attention of a serious worker.

The last chapter deals with singular and non-linear integral equations. Integral equations of the convolution type (Abel, Picard and Wiener-Hopf) are merely mentioned since they rightly belong to the realm of Fourier transform theory, a chapter of the subject which the author does not choose to discuss in detail. The main class of singular, linear, integral equations which is treated is of the Cauchy type, and with it, the Hilbert transformation. Carleman's function theoretic method for handling equations of this type is discussed and it is pointed out (at last!) that this work preceded that of Vekua, Mikhlin, etc. We should be thankful to Professor Tricomi for having corrected the misconception which has crept into the literature. The chapter closes with some remarks about non-linear integral equations. A notable feature of this chapter is the care with which one is introduced to the study of singular equations.

There are two appendices, which are intended to round out the text - one on systems of linear equations, the other on Hadamard's theorem on determinants. Some exercises, as well as a small bibliography are provided.

The text may be heartily recommended to pure and applied mathematicians.

**25.5. Publisher's information on the 1985 reprint.**

This classic text on integral equations by the late Professor F G Tricomi, of the Mathematics Faculty of the University of Turin, Italy, presents an authoritative, well-written treatment of the subject at the graduate or advanced undergraduate level. To render the book accessible to as wide an audience as possible, the author has kept the mathematical knowledge required on the part of the reader to a minimum; a solid foundation in differential and integral calculus, together with some knowledge of the theory of functions is sufficient.

The book is divided into four chapters, with two useful appendices, an excellent bibliography, and an index. A section of exercises enables the student to check his progress. Contents include Volterra Equations, Fredholm Equations, Symmetric Kernels and Orthogonal Systems of Functions, Types of Singular or Nonlinear Integral Equations, and more.

Professor Tricomi has presented the principal results of the theory with sufficient generality and mathematical rigour to facilitate theoretical applications. On the other hand, the treatment is not so abstract as to be inaccessible to physicists and engineers who need integral equations as a basic mathematical tool. In fact, most of the material in this book falls into an analytical framework whose content and methods are already traditional.

**26. Equazioni a derivate parziali (1957), by Francesco G Tricomi.**

**26.1. Review by: Albert E Heins.**

*Bull. Amer. Math. Soc.*

**65**(3) (1959), 169-170.

This book gives an interesting introduction to the theory of partial differential equations. It is divided into four chapters which deal with the theory of characteristics (Chapter 1), hyperbolic equations (Chapter 2), elliptic equations (Chapter 3) and equations of the parabolic and mixed type (Chapter 4). In roughly four hundred pages of text the reader is provided with a good and up to date introduction to these topics. It is true that other aspects of these topics might have been considered, but for this level the author is entitled to his tastes. This text provides a healthy balance between the mathematical methods required for the solution of certain partial differential equations and the underlying theory. Numerous problems from classical theoretical physics are discussed and there is an interesting set of problems at the end of each chapter. The written style is superb.

**26.2. Review by: Edward T Copson.**

*Mathematical Reviews*MR0099489

**(20 #5928).**

In 1954, the author published his "

*Lezioni sulle equazioni a derivati parziali*", a valuable book marred by poor reproduction from typescript. The present work, now properly printed, is based on the author's lecture-course at Turin but is not a mere reproduction of the

*Lezioni*. It is a completely rewritten book.

Whilst it does cover much the same ground as the

*Lezioni*, there are certain important charges. The first chapter of the

*Lezioni*, dealing with integral equations and special functions, has been deleted. This is little loss; no one would go to a book on partial differential equations for information on special functions; and no use is now made of the theory of integral equations, because the fundamental existence theorem for harmonic functions is now proved by Perron's method of subharmonic functions.

The first chapter deals with equations of the first order and the theory of characteristics. The Cauchy-Kowalevska theorem is not discussed, because the author does not regard it as providing an effectively useful basis for the general theory of partial differential equations.

Chapter II is concerned with equations of hyperbolic type of order 2. It is noteworthy that the difficult proof, given in the

*Lezioni*, of the existence and uniqueness theorem for the problem of Goursat is greatly simplified by discussing the same problem for a hyperbolic system. The number of independent variables is not restricted to two when the general case involves no greater difficulty.

Chapter III deals with equations of elliptic type, in the main with the potential equation, though there is a short section on biharmonic functions.

Chapter IV discusses equations of parabolic or of mixed type, mainly the equation of heat conduction and Tricomi's equation of mixed type.

The whole book is a thoroughly up-to-date work on the classical approach to the theory of partial differential equations.

**27. Aerodinamica Transonica (1962), by C Ferrari and Francesco G Tricomi.**

**27.1. Review by: Antonio Ferri.**

*Quarterly of Applied Mathematics*

**21**(4) (1964), 362.

The book is one of the mathematical monographs of the Consiglio Nazionale Delle Riche (National Research Council is a Government Institute). It is a unified and coordinated presentation of many of the known problems and of additional original work in transonic flow aerodynamics. The presentation of all of the transonic problems is well organised and given in a systematic way. A special effort has been made to give a consistent presentation of the different approximations used in transonic flow analyses, and special emphasis is given to basic fluid dynamic and mathematical concepts. Therefore, the book must be considered an important contribution to the understanding of transonic aerodynamics.

The book is divided into six chapters, five of which have been written by Professor Ferrari and one by Professor Tricomi. Professor Tricomi also contributed the Appendix on Hypergeometric Functions. The first chapter is a general introduction to the basic equations of compressible flow and of shock waves. Chapter II is a discussion of the mathematical problems and of the approximations of transonic flow analyses. Chapter III is a discussion of mathematical properties of transonic flow solutions and of mixed type equations. Chapters IV, V, VI are related to discussions of applications of transonic flow analysis to specific problems. In this part new material is presented. Chapter IV and V consider the inverse problem. Chapter IV is related to flow inside nozzles. In Chapter V a detailed discussion of the flow field around two-dimensional profiles for subsonic, sonic, and supersonic free stream velocity is presented. Chapter VI considers the "direct problem" and includes two- and three-dimensional flows. Again, subsonic, sonic, and supersonic free stream velocities are considered. In this chapter, approximate solutions related to area rule and law of equivalence are also presented.

The book is a good reference book for aerodynamics and is a good basic book for specialists in transonic flow. The material is clearly presented and well organized. The only comment I have is in relation to the use of the symbols; the same symbols are often used to indicate different quantities and they are not defined in some way in a list of symbols. This requires the careful attention on the part of the reader for interpretation of the equations.

**27.2. Review by: Paul Germain.**

*Mathematical Reviews*MR0143428

**(26 #984)**.

The important work published by the authors provides a valuable synthesis of many works devoted in recent years to the theory of transonic flows. The issues dealt with are generally set out in sufficient detail that it is not necessary to refer to the original works. The whole does not of course cover all the subjects of interest in transonic aerodynamics. But nevertheless, we find in this book the essential methods of study of plane problems.

After a chapter of generalities, the authors obtain the general equations of the hodograph and clearly specify the way in which the law of state of the fluid intervenes, which makes it possible to present immediately several classical approximations by "fictitious fluids." The equations thus obtained, a chapter with a mathematical character allows the presentation of the fundamental results concerning the equations of the mixed type and in particular the equation of Tricomi. The applications which follow are classified as inverse problems and direct problems. The problem of flow around a profile with or without a shock wave is treated at length and gives the opportunity to clearly show both the importance of the work which has been devoted to this question and the difficulties which remain to be overcome. The chapter devoted to direct problems first deals with all exact solutions corresponding to special geometries then approximate methods which have been proposed for the study of transonic plane flows.

This book, although written in a language which is not easily accessible to many scientists, should be recommended to anyone who wants to work in the field of theoretical transonic aerodynamics. Reading it reveals that many questions have undoubtedly not found their definitive answer, despite the abundance of work that the authors have had the merit of gathering here and presenting in a synthetic way to facilitate subsequent work.

**28. La mia vita di matematico attraverso la cronistoria dei miei lavori. (Bibliografia commentata 1916-1967) (1967), by Francesco G Tricomi.**

**28.1. Review by: Hubert C Kennedy.**

*Mathematical Reviews*MR0274255

**(43 #20)**.

In conformity with his thesis that "a scholar's works are his biography", the author has given us an annotated bibliography of his 300 publications in lieu of a biography proper. The narrative part, which is in the form of connecting passages set off by italics, occupies approximately one-sixth of the book. The author's works lie mainly in six areas, as he tells us in the preface: (1) partial differential equations of mixed type, (2) application of analysis to problems of probability and number theory, (3) Laplace and other transforms, and their relation to classic orthogonal polynomials, (4) special functions, especially the confluent hypergeometric functions, (5) mathematical problems of ultrasonic aerodynamics, and (6) numerical analysis. There is an index of 208 names, which follows the praiseworthy practice of including birth and death dates.

The publication of bibliographies, annotated by the authors themselves, is to be encouraged. Too often little or nothing is known of the origin of mathematical works and while, even for non-literary mathematicians, this type of book would not seem too difficult to write, it could be extremely valuable. This one is very well done and could serve as a model. In it the author provides an insight into some unexpected, as well as expected, motivations for his work. His frankness, too, allows us to appreciate the philosophical and didactic position that underlies his mathematical activity. For example, we can trace the current of his antipathy for the "new math" back to at least 1925, with his arrival at the University of Turin and his consequent contact with Peano, whom he sees as a precursor of the "new math".

The author's reputation as an anti-fascist, his curious "pro-Semitism", and a certain distaste for southern Italians make us wish he had expanded this into a full-fledged biography. We hope, however, that his example of a complete annotated bibliography will be followed by many others - at least by those mature mathematicians who can bring to the task, as our author says, "that unbiased judgment that only those who have nothing more to hope or fear for their career may allow themselves".

**29. Reportorium der Theorie der Differentialgleichungen (1968), by Francesco G Tricomi.**

**29.1. Review by: Lamberto Cesari.**

*Mathematical Reviews*MR0232978

**(38 #1301)**.

In this little book the author covers ideas and results from a vast field in ordinary and partial differential equations. One of the reasons for the smallness of the book is that the author has limited himself to initial value problems: thus elliptic problems are not treated, nor are eigenvalue problems. Numerical methods are only sketched. Proofs are given only for the main points, and mostly as illustrations of ideas. The overall result is a very readable and very informative book.

In Chapter 1 on ordinary differential equations the author touches briefly on many topics: direction fields, singular points, stability and instability, the Lyapunov V function, the numerical approach, the theory of linear differential equations, the Fuchs equations, the Gauss and confluent hypergeometric differential equations, and Bessel's equation.

In Chapter 2 on hyperbolic differential equations the author mentions the usual canonical forms, the Cauchy, Darboux and Goursat problems, the Euler-Poisson equations, and wave equations. Huygen's principle is analysed in detail, and Riemann's solutions are given. Methods for the numerical treatment of some of these problems are discussed by the use of lattice systems and differences. The equation of gas dynamics is then discussed: its main properties and the concept of Mach number are described.

In Chapter 3 on parabolic partial differential equations the author covers the heat equation, the problem of the cooling of a slab, the qualitative nature of the solutions, the Green's function method, and the connections with theta functions and with the Gauss error function.

In Chapter 4 on partial differential equations of mixed type the author discusses the equations of transonic gas dynamics, both in the elliptic and in the hyperbolic half-planes, the questions of the existence and uniqueness of regular solutions with suitably prescribed boundary data, the question of expressing solutions to the same equation in terms of hypergeometric and Airy functions, and the main ideas of the Frankl, Laval-Duse, and Timotika-Tamada problems.

Previous books of the author on partial differential equations are his

*Lezioni sulle equazioni a derivate parziali*(1954) and

*Equazioni a derivate parziali*(1958).

**30. Istituzioni di analisi superiore (metodi matematici della fisica) (1970), by Francesco G Tricomi.**

**30.1. Review by: Editors.**

*Mathematical Reviews*MR0354221

**(50 #6703)**.

As would be expected from the author, this is a concise, elegant and lucid treatment of a number of advanced topics in analysis, at a level and in a manner that are as suitable for mathematicians as for physicists. The main topics are functions of a complex variable; Lebesgue integration, the $L^{p}$ spaces and the theory of distributions; integral equations; orthogonal expansions; ordinary differential equations in the real and complex fields; integral transforms; second order partial differential equations.

Last Updated January 2021