##
**Calcul infinitesimal.**
*(French)*
Zbl 0155.10001

Collection méthodes. Paris: Hermann & Cie. 479 p. (1968).

This excellent book is a compendium of the results and techniques of Classical Analysis at the basic level needed as background for pure Mathematicians desiring to study abstractions like Functional Analysis and for Scientists to apply these techniques effectively for their problems. The reviewer fully joins the author in deprecating the modern tendency to look down on Classical Analysis and straightaway plunge into the study of abstractions like Topology and Functional Analysis since the latter is the foundation on which these abstractions are developed. According to the author the techniques of “Infinitesimal Calculus” can be stated as “Upper bounds, Lower bounds and Approximations (Majorer, Minorer, Approcher)” and inequalities are more important than equalities.

There are sixteen chapters, an index, a short bibliography of books and a list of principal formulas. the presentation is lucid and rigorous and is permeated by the author’s deep insight into the subject matter. Results which are not proved are clearly indicated as “assumed”. After some preliminaries in chapter zero, the techniques of “Upper bound, Lower bound” are explained and illustrated in chapter 1, approximations and asymptotic expansions are dealt with in the next two chapters and problems relating integrals involving a parameter in chapter four. Chapter 5 treats uniform approximations including Weierstrass’ approximation theorem. In chapters 6 to 10, elements of complex analysis are treated – analytic functions, Cauchy integral theorem, singular points and residue calculus, applications of analytic functions to problems in approximations and conformal representations. Chapters 11 to 14 are devoted to the theory of ordinary differential equations – Cauchy-Lipschitz method for one and a system of equations of order one, linear differential equations, perturbations of systems of linear differential equations leading to notions of stability of solutions, and second order linear differential equations in detail. Chapter 15 contains an introduction to special functions (Bessel and some related functions). At the end of each chapter there are several problems – most of them standard results with which every analyst must be familiar.

This book and the already well-known book “Foundations of modern analysis” (New York etc.: Academic Press) (1960; Zbl 0100.04201) by the same author should be studied deeply by any one desiring to specialize in recent developments in Topology and Functional Analysis.

There are sixteen chapters, an index, a short bibliography of books and a list of principal formulas. the presentation is lucid and rigorous and is permeated by the author’s deep insight into the subject matter. Results which are not proved are clearly indicated as “assumed”. After some preliminaries in chapter zero, the techniques of “Upper bound, Lower bound” are explained and illustrated in chapter 1, approximations and asymptotic expansions are dealt with in the next two chapters and problems relating integrals involving a parameter in chapter four. Chapter 5 treats uniform approximations including Weierstrass’ approximation theorem. In chapters 6 to 10, elements of complex analysis are treated – analytic functions, Cauchy integral theorem, singular points and residue calculus, applications of analytic functions to problems in approximations and conformal representations. Chapters 11 to 14 are devoted to the theory of ordinary differential equations – Cauchy-Lipschitz method for one and a system of equations of order one, linear differential equations, perturbations of systems of linear differential equations leading to notions of stability of solutions, and second order linear differential equations in detail. Chapter 15 contains an introduction to special functions (Bessel and some related functions). At the end of each chapter there are several problems – most of them standard results with which every analyst must be familiar.

This book and the already well-known book “Foundations of modern analysis” (New York etc.: Academic Press) (1960; Zbl 0100.04201) by the same author should be studied deeply by any one desiring to specialize in recent developments in Topology and Functional Analysis.

Reviewer: V. Ganapathy Iyer

### MSC:

26-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to real functions |

30-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functions of a complex variable |

33-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to special functions |

34-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations |

00A05 | Mathematics in general |