*Lezioni di analisi matematica: redatte per uso degli studenti*(2 volumes) (1929), by G Sansone and

*Lezioni sulla teoria delle funzioni di una variabile complessa*(2 volumes) (1947), by G Sansone both ran to several editions. An extract of a review of the fourth edition to the first of these is given below while extracts of reviews of an English translation of the second is given below. We make no attempt to list all of Sansone's books.

**1. Moderna teoria delle funzioni di variabile reale. I: Aggregati, analisi delle funzioni, integrazione, derivazione. II: Sviluppi in serie di funzioni ortogonali (1936), by G Vitali and G Sansone.**

**1.1. Review by: Enea Bortolotti.**

JfM 62.0189.01.

The first volume of this work is due to the deceased G Vitali; it contains a self-contained and unified description, largely new, both in terms of form and content, of the principles of the modern theory of set theory, and the modern theory of the functions of a real variable. ... On the death of the author (1932), the volume was almost completed; G Sansone, with sincere accuracy and conscientious fidelity to the train of thought and the form of representation chosen by the author, has completed the work and arranged its publication. The improvements and additions of Sansone are very pleasing to the harmonious structure of the author; they contribute to making the work a reliable and easy guide for one who wants to know the most beautiful and the highest theories of the modern analysis of the real variable, as well as for a suitable introduction to the subject matter of which Sansone gives a concise and clear representation in the second volume: the theory of the developments in series of orthogonal functions, and especially in the series of Legendre, of Chebyshev- Laguerre, and from Chebyshev-Hermite polynomials. To these theories, which are of great current importance for various branches of applied mathematics, Sansone and his colleagues and students have provided valuable contributions. Together with some of these results, the author uses a large number of results drawn from a careful and thorough investigation of the enormous literature on this subject (until 1934), and merges it into a representation of remarkable clarity and elegance, full of useful simplifications and new points of view and evidence.

**2. Lezioni di analisi matematica redatte per uso degli studenti. I (4th edition) (1940), by G Sansone.**

**2.1. Review by: H Geppert.**

JfM 66.0193.01.

The first volume of the lectures of the author deals with the fundamentals of analysis, algebra and the differential calculus in one variable. He distinguishes his work from other textbooks by an unusual abundance and penetration of the treated material. Contents: 1. Combinatorics. Matrices and determinants. Linear equations. 2. The real numbers are established entirely by Dedekind's section. 3. The complex numbers. 4. Basic principles of set theory. 5. Numbers and limits. 6. Theory of series. 7. Functions, their limits; continuity (very rigorous). 8. Derivatives of functions of a single variable; the derivation of the surface of a trapezoid is also discussed here, and the derivation of complex functions is defined. 9. Mean value theorem. Differentiation of series. 10. Taylor's Series. Here, the various forms of the remainder are discussed (e.g. a theorem of S Bernstein). 11. Analytical and geometrical applications of Taylor's Theorem. 12. Basic Properties of polynomials of one variable. For the fundamental theorem of algebra, the author (unfortunately) uses d'Alembert's proof. Bézout's theorem is unsatisfactory because of the lack of agreements about multiplicity and the role of the infinite. - Now follows a second, more deeply penetrating part. 13. Infinite products. Double rows. Infinite determinants and the solution of infinite linear normal equations with infinitely many unknowns with vanishing and non-vanishing determinant. Continued fractions; applying them to the proof that the cardinality of the points of a square coincides with that of a line; infinite continued fractions. 14. Interpolation. Bernoulli numbers and polynomials. 15. Real functions. Fluctuation. Semicontinuity. Upper and lower, left and right derivatives. Functions with Lipschitz condition. Upper and lower limit. Cauchy-Hadamard's theorem. 16. Completion of the theory of series. Theorem by Riemann-Dini. Cesàro and Hölder sums. Theorem of Toeplitz. 17. Symmetrical functions of the roots of an algebraic equation. Form of resultant. Tschirnhaus transformations. 18. Theorem on the location of the roots. Equations of third and fourth degrees. 19. Theorems on symmetrical determinants; the principal minor; Hadamard's determinant. 20. Linear substitutions. Quadratic forms. Invariants.

**3. Equazioni differenziali nel campo reale (2 volumes) (1941), by G Sansone.**

**3.1. Review by: Oskar Perron.**

JfM 67.0306.01

An authoritative work which does not stop with the elementary methods of integration, but which develops the general principles and modern theories with great completeness and thoroughness. The main weight is on linear differential equations to which more than half the total work is devoted. But also the most varied proofs of existence and uniqueness are developed according to the most diverse methods, mostly, of course, by successive approximations, but, for example, the theory of integral equations and the direct methods of variational calculus are also used.

**4. Moderna Teoria delle Funzioni di Variabile Reale (2nd edition) Volume I Aggregati, analisi delle funzioni, integrazione, derivazione (1943), by G Vitali and G Sansone.**

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

*Mathematical Reviews*MR0015141.

The first edition, edited by Sansone from a posthumous manuscript of Vitali, appeared in 1935. The second edition is substantially the same as the first, somewhat amplified in a few places. It gives a clear and concise account, along classical lines, of the one-dimensional Lebesgue integral and the necessary preparatory topics from the theory of functions of a real variable. The five chapters deal, respectively, with sets and transfinite numbers; measure of linear sets; measurable functions, functions of bounded variation; integration of measurable functions, integration of series; differentiation of integrals and of functions of bounded variation.

**5. Moderna Teoria delle Funzioni di Variabile Reale (2nd edition) Volume II Sviluppi in serie di funzioni ortogonali (1946), by G Vitali and G Sansone.**

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

*Mathematical Reviews*MR0015535.

This volume, by Sansone, is subtitled 'Sviluppi in Serie di Funzionali Ortogonali'. The topics discussed are, by chapters: (1) Developments in orthogonal functions, introduction to Hilbert space; (2) Fourier series; (3) Series of Legendre polynomials and of spherical harmonics; (4) Laguerre and Hermite series; (5) Approximation and interpolation; (6) Stieltjes integrals (including a discussion of Fourier transforms of distribution functions; this chapter has little connection with the rest of the book). The edition of 1935 was considerably shorter. ... The book would be suitable as a text for a second course in functions of a real variable. The exposition is clear and rigorous, and original in many places. The point of view is modern, the Lebesgue integral being used throughout.

**6. Equazioni Differenziali nel Campo Reale (2nd edition) Volume I (1948), by G Sansone.**

**6.1. Review by: L A MacColl**.

*Mathematical Reviews*MR0026731.

This is the first volume of what is evidently to be a very substantial and valuable treatise on differential equations, with the chief emphasis placed on the real domain. The present volume is devoted exclusively to ordinary differential equations. The contents are divided into six chapters, as follows. (I) Fundamental theorems concerning the existence, uniqueness, and continuity properties of solutions in the real domain. (II) General properties of systems of linear differential equations. (III) Fundamental theorems concerning differential equations in the complex domain, with particular emphasis on linear equations of the second order. (IV) Boundary value problems for an equation of the second order. (V) Boundary value problems for equations of higher orders. (VI) Linear differential equations with periodic coefficients.

**7. Equazioni Differenziali nel Campo Reale (2nd edition) Volume II (1949), by G Sansone.**

**7.1. Review by: L A MacColl.**

*Mathematical Reviews*MR0030663.

The first volume of this treatise has been reviewed previously. In this concluding volume the treatment of ordinary differential equations in the real domain is continued in six chapters, devoted, respectively, to: the asymptotic properties of solutions; existence and uniqueness of solutions in the large; singular points and singular solutions; operational methods for solving linear equations; numerical, graphical, and mechanical methods of solution; discussions of some important particular differential equations. In general the high standards which were set in the first volume have been fully maintained.

**8. Equazioni differenziali non lineari (1956), by G Sansone and R Conti.**

**8.1. Review by: G Stampacchia.**

*Mathematical Reviews*MR0088607.

In the present volume, mostly by geometric means, the asymptotic properties of the solutions of [certain] systems ordinary differential equation systems of the normal type [are studied]. These equations model many problems suggested by applied sciences corresponding to mechanical systems with only one degree of freedom. ... Many of the results appear in book form for the first time.

**9. Orthogonal functions (1959), by G Sansone.**

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

*Mathematical Reviews*MR0103368.

This is a translation of the first four chapters of the third edition (1952) of Sansone's second volume of Vitali's '*Moderna teoria delle funzioni di variabile reale'*, dealing with general theorems, Fourier series, series of Legendre polynomials and spherical harmonics, and series of Laguerre and Hermite functions.

**10. Lectures on the Theory of Functions of a Complex Variable. Vol. 1 (1960), by G Sansone and J Gerretsen.**

**10.1. Chapter headings.**

1. Holomorphic functions. Power series as holomorphic functions. Elementary functions. (40 pages)

2. Cauchy's integral theorem and its corollaries. Expansion in Taylor series. (81 pages)

3. Regular and singular points. Residues. Zeros. (63 pages)

4. Weierstrass's factorization of integral functions. Cauchy's expansion in partial fractions. Mittag-Leffler's problem. (60 pages)

5. Elliptic functions. (73 pages)

6. Integral functions of finite order. (42 pages)

7. Dirichlet series, the zeta function of Riemann. The Laplace integral. (69 pages)

8. Summability of power series outside the circle of convergence. Sum formulas. Asymptotic series. (64 pages).

**10.2. Review by Allen Shields.**

*Bull. Amer. Math. Soc.* **69** (1963), 39-40.

In 1947 Professor Sansone published his *Lezioni nulla teoria dette funzioni di una variabile complessa *in two volumes of 359 and 564 pages respectively. The present volume is a completely new text, based on the Italian edition. ... This first volume avoids the geometric point of view entirely. No mention is made of conformal mapping (more correctly: it is mentioned once on p. 5). This is all postponed to a second volume which will also treat multi-valued functions. Following Artin the proof of the Cauchy integral theorem is based on the notion of winding numbers. Elliptic functions are treated at length - but here again the geometric (and group-theoretic) point of view is avoided, and is promised for the second volume. The book is very well written and the reader is led to many beautiful and classical results. The authors' viewpoint is classical: they wish "to present some masterpieces of mathematical thinking and to make these accessible to a rather wide circle of interested readers in not too pedantic a way." In the reviewer's opinion they have succeeded well in this.

**10.3. Review by: R. L. G.**

*The Mathematical Gazette* **45** (353) (1961), 267.

This very attractive text is based on an earlier Italian language treatise by Sansone. The first volume, on holomorphic functions, discusses Cauchy's theorem, singularities and residues, elliptic functions, the Riemann Zeta function, and summability of power series outside the circle of convergence. Cauchy's theorem is proved for closed curves in an open set of differentiability (Jordan's theorem is not used or proved). Following Artin the theory of singular points is developed without the use of Laurent series. Picard's theorem is proved only for integral functions of finite order, the general case being postponed to the second volume.

**10.4. Review by: W J Thron.**

*Mathematical Reviews* MR0113988.

This is the first volume of a text based on the two-volume work, in Italian, published by the first author in 1947. The subjects considered in the book are largely of a formal nature. The more geometric aspects of holomorphic functions (including such topics as analytic continuation, Riemann surfaces, conformal mapping) are not taken up in this volume at all, but will presumably be treated in a proposed second part. By assuming knowledge of elementary analysis, by not devoting any space to the axiomatic foundations of the subject, and by omitting details of proofs, the authors have managed to present a large number of theorems and formulas, going well beyond what is to be found even in the more voluminous of recent texts on complex variables. Many types of expansions for holomorphic functions are given, and special functions and classes of functions are discussed at some length.

**10.5. Review by: Hiroshi Yamauchi.**

*Amer. Math. Monthly* **68** (5) (1961), 518-519.

This readable book is the first part of a two-volume English edition of Sansone's '*Lezioni sulla Teoria delle Funzioni di una Variabile Complessa'* (1947) with a new text containing revisions and rearrangement of the Italian work. It starts from first principles but does not emphasise logical foundations. The book is primarily designed for beginners interested in a detailed account of special topics of the classical theory. No attempt is made to present a scholarly work and the reader will notice that a bibliography is not provided. Among the topics discussed are: power series, elementary functions, Cauchy integral theorem, residue theory, Weierstrass factor theorem, Mittag-Leffler theorem, elliptic functions, integral functions of finite order, Dirichlet series, Riemann zeta function, Laplace integral and asymptotic series.

**11. Qualitative Theorie nichtlinearer Differentialgleichungen (1963), by R Reissig, G Sansone and R Conti.**

**11.1. Review by: J P LaSalle.**

*Mathematical Reviews*MR0158121.

This book overlaps and at the same time complements the earlier volume by Sansone and Conti [*Equazioni differenziali non lineari*]. The two volumes together provide an extensive compilation of qualitative studies of second-order non-linear differential equations. The first third of this book introduces the classical results of Liapunov, Poincaré, Bendixson and Birkhoff, and the more recent method of Wazewski. The latter two-thirds of the book is devoted to second-order equations (one degree of freedom).

**12. Nonlinear Differential Equations (2nd edition) (1964), by G Sansone and R Conti.**

**12.1. Chapter headings.**

1. General Theorems about Solutions of Differential Systems.

2. Particular Plane Autonomous Systems.

3. The Singularities of Briot-Bouquet.

4. Plane Autonomous Systems.

5. Autonomous Plane Systems with Perturbations.

6. On Some Autonomous Systems with One Degree of Freedom.

7. Nonautonomous Systems with One Degree of Freedom.

8. Linear Systems.

9. Stability.

**12.2. Review by: John A Nohel.**

*SIAM Review* **9** (2) (1967), 258-260.

The English edition of this outstanding book is not merely a translation from the Italian; the authors have used this occasion to revise and complement each chapter with notes, exercises, interesting remarks, as well as bibliographical data of each topic. In fact, the detailed and up to date (through 1960) bibliography at the end of each chapter, and occasionally at the end of important sections, is an outstanding feature of the book.

**12.3. Review by: R A Struble.**

*Science, New Series* **148** (3677) (1965), 1583.

This book is concerned with the mathematical analysis of certain ordinary nonlinear differential equations. The treatment throughout is rigorous and remarkably clear, and it provides for a systematic study of some important problems. It is not so much a book on the theory of nonlinear equations as it is a detailed working out of the theory in special circumstances. An introductory chapter that includes a few traditional results on the existence, uniqueness, extension, and differentiability of solutions for arbitrary systems precedes five chapters (400 pages) devoted solely to two-dimensional systems. This is the most complete treatment of these systems published in any single volume and includes many results heretofore available only in research journals. ... The volume is attractive, well written, and carefully translated. Each chapter is followed by an extensive bibliography, which substantiates, extends, and in general relates to many of the topics covered. This makes the book valuable as a reference source as well as appropriate for use as a textbook for a graduate course in differential equations.

**12.4. Review by: F M Arscott.**

*The Mathematical Gazette* **50** (371) (1966), 77.

Of books on differential equations there seems to be no end, and not all of them contribute much to the sum of human happiness, but here is one to welcome unreservedly. The field of non-linear differential equations is frightening. By contrast with the fertility of the physical world in spawning non-linear problems, the achievements of mathematicians in solving them seem infinitesimal, despite the contributions of giants from Poincaré onwards. The field is so broad, and much of the existing literature so specialised, that the mathematician wishing to fish in this pond has little idea where the big fish lurk. Besides straightforward "unsolved" non-linear equations there are the fields of existence theory, stability and numerical techniques, each having strong overtones of practical applicability, and to any one of which a mathematician could devote half a lifetime. Meanwhile the non-mathematician who needs information on a particular topic could seek it a long time without success in the present literature. This book, however, will go far to meet current needs, even though it is concerned only with ordinary differential equations. The Italian edition of 1956 is its basis, but complementary sections have been added to many chapters, taking account of developments up to 1961. The exposition is lucid and the translation excellent; no book on so advanced a topic could be easy reading but it is hard to see how the treatment of this text could be improved.

**12.5. Review by: Richard Bellman.**

*Mathematics of Computation* **19** (92) (1965), 699.

This is a superb book devoted to the classical and modern theory of linear and nonlinear ordinary differential equations. It covers existence and uniqueness theorems, stability theory, perturbation techniques, asymptotic behaviour, periodic solutions, and Briot-Bouquet theory, with encyclopaedic thoroughness and in careful detail. Perhaps most valuable is the way in which ideas and concepts are illustrated by means of specific examples. An almost complete set of references to important papers in the field is given. Students in mathematics, engineering, and physics will find this book of great value, and it will be equally useful to research workers. The authors have written a beautiful and lucid exposition of this area of analysis which can be used as a basis for a variety of different courses. It is unreservedly recommended.

**13. Non-linear Differential Equations of Higher Order (1974), by R Reissig, G Sansone and R Conti.**

**13.1. Review by: A M Fink.**

*SIAM Review*

**18**(1) (1976), 143-144.

This is a translation of a book that originally appeared in German. It is a welcome addition to the literature on boundedness and stability of higher order equations. The authors cite the hope that their book will bring attention to papers which are not well known for linguistic or other reasons. Indeed, the rather specialised topics covered by this book have not received much attention from the English-speaking mathematical community. In fact, less than twenty percent of the bibliographic items appeared in their original form in English. The book quite naturally splits into three parts. The first two chapters deal with generalities that are used in the subsequent discussions. They are concerned with the study of the qualitative behaviour of differential equations through the use of comparison theorems and the method of Lypanov functions. Much of the discussion is standard fare, although a significant portion is not. The middle four chapters form the second part and deal with several classes of third and fourth order equations. Much detailed constructions of comparison equations and Lypanov functions are included. The last two chapters form the most interesting section of the book. The first of these is concerned with nonlinear equations with separated variables. ... the last chapter is a very well written exposition of the nth order Lur'e problem. ... The detailed computations should be accessible to engineers and mathematicians alike. A secondary strength of the book is the abundance of well worked out examples of Lypanov functions.

**13.2. Review by: George R Sell.**

*Bull. Amer. Math. Soc.* **82** (2) (1976), 198-207.

There have been so many developments within the last twenty years that today it would be almost impossible to write a comprehensive treatment of the subject. Since the appearance of Cesari's book [*Asymptotic behavior and stability problems in ordinary differential equations *(1959)]* *one finds, for the most part, special purpose books which deal with selective topics in the qualitative theory. The recent book by R Reissig, G Sansone, and R Conti on *Nonlinear differential equations of higher order *is such a book. This book presents the general theory of Lyapunov functions (as well as the related subject of comparison theorems) as applied in the study of stability, boundedness, and periodicity. The general discussion of nth order differential systems is a standard treatment which one would expect to see in a book on Lyapunov stability theory. The unique feature of this book is an extensive treatment of third and fourth order differential equations. ... this book contains an essentially complete discussion of the status of our knowledge of the qualitative behavior of solutions of third and fourth order equations. It is a good account of the subject, but it is intended for specialists only. The final chapter of the book should be of interest to researchers in control theory. It consists of a rather complete discussion of the Lure problem together with the two techniques for solving this problem, viz. the technique of Lyapunov functions and a technique due to Popov which is based on Laplace transform methods. The book concludes with a brief discussion comparing these two techniques.