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**Un demi-siècle (1907–1956) de notes communiquées aux Académies de Paris, d’Amsterdam, des Lincei, suivies par des observations et commentaires. I. La variable complexe. II. Le champ réel.**
*(French)*
Zbl 0077.05801

Paris: Gauthier-Villars. 589 p., 99 p. (1957).

The two volumes under review constitute the collection of notes communicated, spread over half a century (1907–1956), to the Academies of Paris, Amsterdam and Lincei by Prof. A. Denjoy, one of the foremost Mathematicians of this century. A collection of 166 notes are listed in chronological order. These are arranged in five sections marked A to E. Each section contains several subsections and each of the latter deal with some well defined topics in the theory of functions of a complex variable or real variable. We give below an idea of the various topics considered in the various subsections and mention a few important results among the author’s contributions to the development of Analysis.

Volume I consists of Section A divided into 8 parts, A-1 to A-8. This volume deals with functions of a complex variable. The topics treated are entire functions of finite order, entire functions of infinite order, functions having a discontinuous set of non-isolated singularities, quasi-analytic functions and series of rational fractions, functions mapping the unit circle into itself, conformal representation of the unit circle, general theory of analytic functions and functions of Minkowski (continuous functions satisfying certain functional equations for convergents of a continued fraction), Fuchsian groups arising from continued fractions and statistical properties of such fractions. We mention a few prominent results in this volume. The earliest result is the famous conjecture that the number of asymptotic values of an entire function of order \(\rho\) cannot exceed \(2\rho\). This conjecture made in 1907 was later proved in 1929 by L. Ahlfors. The next result is the main theorem in the Theory of quasi-analytic functions, now known as the Carleman-Denjoy theorem, that if a function \(f(x)\) in \([a, b]\) has derivatives of all orders, the series \(\sum 1/M_n^{1/n}\) diverges (where \(M_n\) is the lowest upper bound of \(\vert f^{(n)}(x)\vert\) in \([a,b]\), then the vanishing of the function and all its derivatives at some point in the interval implies that \(f\) vanishes in the whole interval. A large number of properties of series of the form \(\sum A_n/(z - a_n)\) and their relation to the notion of quasi-analyticity are to be found in this volume. An elementary proof of Wolff’s theorem that if \(f(z)\) is regular in the unit circle and maps the circle into itself but is not the conformal transform of the circle onto itself then the iterates of \(f\) tend to a constant value in the circle, investigations of the rapidity of approach to the boundary value in conformal representations and examination of the general situations in which the Cauchy integral theorem holds for functions of a complex variable, are some of the other noteworthy contributions of the author contained in this volume.

The second volume deals with functions of real variables. This contains four sections B to E.

The main section B has 8 subsections. The topics dealt with are properties of plane perfect sets, metric properties of sets in finite dimensional Cartesian space, Vitali’s covering theorem, Riemann and Lebesgue integrals, first order derived numbers of functions of one variable, totalisation and calculus of coefficients of trigonometric series. A large number of properties of linear and plane perfect sets are investigated and later used to discuss the properties of the derived numbers of functions of one variable and the formulation of the notion of totalisation applied to derivatives of functions culminating in the definition of the now well-known Denjoy integral. The notion of approximate derivative is introduced and its intimate relation to the Denjoy integral is discussed. These are applied to the problem of trigonometric series generalising known theorems on the behaviour of such series. A discussion of the general situation in which the original Vitali’s covering theorem could be generalised is also found in this section.

Section C contains three subsections dealing with properties of connected and disconnected sets and two proofs of the Jordan curve theorem.

Section D has four subsections. Properties of transfinite numbers form the topic in this section. The author’s work on this topic has now been published under the title “L’énumération transfinie” (Zbl 0049.03503; Zbl 0049.03601; Zbl 0049.03701; Zbl 0056.04702; Zbl 0056.04801) by the same publishers in four books.

Section E contains some contributions of the author to analytical mechanics and probability theory and serves to show the author’s width of interest.

A study of this and other collected works of this famous Mathematician of this century will form an indispensable training for research workers in higher Analysis. The publishers deserve praise for undertaking such publications.

Volume I consists of Section A divided into 8 parts, A-1 to A-8. This volume deals with functions of a complex variable. The topics treated are entire functions of finite order, entire functions of infinite order, functions having a discontinuous set of non-isolated singularities, quasi-analytic functions and series of rational fractions, functions mapping the unit circle into itself, conformal representation of the unit circle, general theory of analytic functions and functions of Minkowski (continuous functions satisfying certain functional equations for convergents of a continued fraction), Fuchsian groups arising from continued fractions and statistical properties of such fractions. We mention a few prominent results in this volume. The earliest result is the famous conjecture that the number of asymptotic values of an entire function of order \(\rho\) cannot exceed \(2\rho\). This conjecture made in 1907 was later proved in 1929 by L. Ahlfors. The next result is the main theorem in the Theory of quasi-analytic functions, now known as the Carleman-Denjoy theorem, that if a function \(f(x)\) in \([a, b]\) has derivatives of all orders, the series \(\sum 1/M_n^{1/n}\) diverges (where \(M_n\) is the lowest upper bound of \(\vert f^{(n)}(x)\vert\) in \([a,b]\), then the vanishing of the function and all its derivatives at some point in the interval implies that \(f\) vanishes in the whole interval. A large number of properties of series of the form \(\sum A_n/(z - a_n)\) and their relation to the notion of quasi-analyticity are to be found in this volume. An elementary proof of Wolff’s theorem that if \(f(z)\) is regular in the unit circle and maps the circle into itself but is not the conformal transform of the circle onto itself then the iterates of \(f\) tend to a constant value in the circle, investigations of the rapidity of approach to the boundary value in conformal representations and examination of the general situations in which the Cauchy integral theorem holds for functions of a complex variable, are some of the other noteworthy contributions of the author contained in this volume.

The second volume deals with functions of real variables. This contains four sections B to E.

The main section B has 8 subsections. The topics dealt with are properties of plane perfect sets, metric properties of sets in finite dimensional Cartesian space, Vitali’s covering theorem, Riemann and Lebesgue integrals, first order derived numbers of functions of one variable, totalisation and calculus of coefficients of trigonometric series. A large number of properties of linear and plane perfect sets are investigated and later used to discuss the properties of the derived numbers of functions of one variable and the formulation of the notion of totalisation applied to derivatives of functions culminating in the definition of the now well-known Denjoy integral. The notion of approximate derivative is introduced and its intimate relation to the Denjoy integral is discussed. These are applied to the problem of trigonometric series generalising known theorems on the behaviour of such series. A discussion of the general situation in which the original Vitali’s covering theorem could be generalised is also found in this section.

Section C contains three subsections dealing with properties of connected and disconnected sets and two proofs of the Jordan curve theorem.

Section D has four subsections. Properties of transfinite numbers form the topic in this section. The author’s work on this topic has now been published under the title “L’énumération transfinie” (Zbl 0049.03503; Zbl 0049.03601; Zbl 0049.03701; Zbl 0056.04702; Zbl 0056.04801) by the same publishers in four books.

Section E contains some contributions of the author to analytical mechanics and probability theory and serves to show the author’s width of interest.

A study of this and other collected works of this famous Mathematician of this century will form an indispensable training for research workers in higher Analysis. The publishers deserve praise for undertaking such publications.

Reviewer: V. Ganapathy Iyer