LIBRARY OF THE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN F\5 : D5C° 1:E ee: Digitized by the Internet Archive in 2022 with funding from Kahle/Austin Foundation https ://archive.org/details/infinitesimalcal0O000dieu JEAN DIEUDONNE Former Professor at the Faculty of Sciences, Nice Infimitesimal Caleulus HERMANN Publishers in Arts and Science, Paris, France HOUGHTON MIFFLIN COMPANY - BOSTON New York - Atlanta - Geneva, [linois - Dallas - Palo Alto Translated from the original French text Calcul infinitésimal, published by Hermann, Paris, in 1968 in their series Méthodes. Printed in Great Britain by William Clowes & Sons, Limited, London, Beccles and Colchester ISBN: 0-395-12034-9 © HERMANN, PARIS, 1971 All rights reserved. No part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system, without permission in writing from the original publisher. Editors’ Introduction This book containing many of the central ideas of analysis covers the material for a modern junior-level advanced calculus course in American colleges and universities. Stress is placed on the calculational aspects of the subject to develop not only calcu- lational ability but also an understanding of the methods of calculation and of the role of calculation and approximation in analysis. The reader is assumed to be familiar with the rudiments of real analysis, including the basic properties of the real and complex numbers, continuous functions, derivatives, and primitives. These topics are outlined in Chapter o—Jntroduction. The book is roughly divided into three parts. The first part (Chapters I-V) deals with the general question of approximation. Topics include the local question of approximation by asymptotic developments, the global theory of approximation to roots of equations, and uniform convergence. The second part (Chapters VI—X) is an introduction to the theory of functions of a complex variable. The usual local theory of the Cauchy integral, power series development, and a study of singularities and residues is followed by applications of the theory of functions of a complex variable to the theory of approximation and a discussion of conformal mapping. The third part (Chapters XI-XV) is a study of differential equations in the real and complex domains. Primary consideration is given to linear equations. Perturba- tions of linear differential systems are included, and the section concludes with a study of Bessel’s equation. The book can form the basis of an excellent upper division course in real and complex analysis. It provides an excellent background for further study in analysis, and can also be used as a text for a course in mathematical methods in physics or astronomy. It can serve as an excellent review of undergraduate analysis for graduate students. In addition, material can be extracted from the first ten chapters (parti- cularly Chapters VI-X) to form a text for a one-semester junior- or senior-level course in complex analysis. Joun Moors, Princeton University Dae HusemMo ier, Haverford College 7