Lectures On The General Theory Of Integral Functions

LECTURES 0 THE GENERAL THEORY OF INTEGRAL FUNCTIONS ny GECKOES VALIRON ProlVvor in llo University of Strasbourg IHANSIATEf BY E. F. COLU C, WOOD K. . WITH A PREFACE BY W. II. YOUNG Sc. D., Dr. cs Sr. math., D. Sc., F. B. S., Sometime fellow of Pcterhouse, Cambridge, Formerly professor in the Universities of Liverpool, Calcutta, Wales, 7 irr-President de TUnion internationalc dcs mathumaticicns CHELSEA PUBLISHING COMPANY 231 WEST 2 m STREET, NEW YOBK 1, N. Y. 1949 PREFACE These lectures have a twofold interest. Delivered in French by a prominent French scholar to an Hj glish speaking audience consisting mainly of Honours stu dents of whose curriculum the subject matter formed an inte gral part, they are noteworthy as constituting the permanent record of a successful experiment, carried out at the Lniversity College of Wales during the period of reorganisation of its Pure Mathematical Department. But their scientific interest is far greater than this might lead us to suppose. The lectures give us, in the form of a number of elegant and illuminating theorems, the latest word of ma thematical science on the subject of Integral Functions. And they do more. They descend to details, they lake us into the workshop of the working mathematician, they explain to us the nature of his tools, and shew us the vvay to use them while, at the same lime, by the absence of any attempt to conceal Ihe imperfections of the edifice so far conslrucled, they indicate to us the ork still waiting to be done, they inspire us with the desire and furnish us with the means of completing it ourselves. The book will not be found difficult by an earnest student. He cnay hope to master it without any elaborate preliminarypreparation. All that is needed is that he should possess some familiarity vith the more immediate consequences of the fun damental and far reaching theorem due to the genius of Cau chy and known by his name and that he should be at home VI PREFACE. with the concept of number and its extensions such as form to day the foundations of Analysis. For the philosophic mathematician the subject is particu larly Instructive, shewing, as it does, the power of a single fundamental idea, that of generalisation, germinatirig in many minds of Avidely different types of training, and applied to a particular mathematical concept, that of the polynomial. A problem once stated is, as we all know, half solved and the rapidity with which one step in our theory has followed ano ther will be seen by a glance at the bibliography at the end of the volume, in the same connexion the appendices may specially be cited. They contain among other things the com plete solution of questions connected with the inverse function of an integral function, questions raised and only partially answered by Hurwitz in two Notes in the Ccmptes Rendus. They contain also an illustration of the growing influence of the Theory of Sets of Points, not only on the language of the Theory of Integral Functions, but also on its subject matter and we are led to ask whether in the near future the whole trend of research may not here also be in this direction. W.-H. YOUNG. The Athenaeum, London. June, 192,3. AVERTISSEMENT Ce livrc rcproduit les legons que jai cu Ilionneur de fairc aux elevcs de M. Young a PUniversity College of Wales a Aberystwylh en fevrier et mars 1922. Quil me soil permis de renouveler ici expression de ma vivereconnaissance au pro fesseur Young qui ma invite a faire ces lerons qui ma donne les moyens de les publier et qui a bien voulu les presenter au public. Si cet ouvrage rend quelques services aux etudiants, que leur gratitude aille dabord a M. Young qui en a te le pro moleur Mes lemons etaient faites en frangais un jeune mathemati cien de Cambridge, M. Collingwood, me seconda en donnant des explications complementaires a rnes auditeurs et accepta de remplir le role ingrat de traducteur. Je Ten remercie bien vivement...