Introduction à la théorie des fonctions d'une variable

M., G. B.

doi:10.1038/086172a0

Books Received
Published: 06 April 1911

Introduction à la théorie des fonctions d'une variable

G. B. M.

Nature volume 86, pages 172–173 (1911)Cite this article

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Abstract

THE most interesting chapter in this volume is that which is entitled “Langage geomtrique,” especially from a pedagogic point of view. The main object of the treatise is to deduce everything from purely arithmetical assumptions; but as a practical teacher, Prof. Tannery was well aware of the value of diagrams as an aid to the imagination, or, as he puts it, for purposes of orientation. Consequently he has given a series of quasi-geometrical definitions, by means of which the ordinary formulae and methods of analytical geometry are valid, and may be used practically for constructing diagrams to define boundaries of aggregates, &c. In the ordinary sense, of course, we thus get a locus corresponding to an equation θ(x,y)=o; but in order to emphasise the fact that only arithmetical conditions are really imposed, the author replaces the term “logus”(lieu) by “bond”(lien), and practically confines this to the case where we may put x= θ(t)θ y = φ (t), θ, φ being definite functions for a certain range of the real continuous variable t. The principal results of the chapter are the proof of the existence of simple contours in a plane, which separate it into two distinct continua (this is given after Mr. Ames), and the further conclusion that a domain which is (m+i) times connex can be reduced to two simply connex domains by drawing (m+i) simple curves.

Introduction à la théorie des fonctions d'une variable.

By J. Tannery. Deuxième èdition; tome 2. Intégrates définies, Développements en Série, Langage géométrique, Fonctions de Variables imaginaires. Avec une Note de M. Hadamard. Pp. iv+480. (Paris: A. Hermann et fils, 1910.) Price 15 francs.