A Course in Complex Analysis From Basic Results to Advanced Topics
This carefully written textbook is an introduction to the beautiful concepts and results of complex analysis. It is intended for international bachelor and master programmes in Germany and throughout Europe; in the Anglo-American system of university educ
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Wolfgang Fischer | Ingo Lieb
A Course in Complex Analysis From Basic Results to Advanced Topics
TEXTBOOK
Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de.
Prof. Dr. Wolfgang Fischer University of Bremen Faculty 3 - Mathematics Bibliothekstraße 1 D-28359 Bremen Germany [email protected] Prof. Dr. Ingo Lieb University of Bonn Mathematical Institute Endenicher Allee 60 D-53115 Bonn Germany [email protected] Large parts of this text are translated from the book Fischer, W./Lieb, I.: Einführung in die Komplexe Analysis, Vieweg+Teubner Verlag, 2010. Translation: Jan Cannizzo
1st Edition 2012 All rights reserved © Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden GmbH 2012 Editorial Office: Ulrike Schmickler-Hirzebruch | Barbara Gerlach Vieweg+Teubner Verlag is a brand of Springer Fachmedien. Springer Fachmedien is part of Springer Science+Business Media. www.viewegteubner.de No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the copyright holder. Registered and/or industrial names, trade names, trade descriptions etc. cited in this publication are part of the law for trade-mark protection and may not be used free in any form or by any means even if this is not specifically marked. Cover design: KünkelLopka Medienentwicklung, Heidelberg Printing company: AZ Druck und Datentechnik, Berlin Printed on acid-free paper Printed in Germany ISBN 978-3-8348-1576-7
Preface Among the most important tools of mathematics are the elementary functions – rational, trigonometric, hyperbolic and exponential functions, logarithms . . . . Their close relation only becomes apparent when one admits complex numbers as their arguments. This leads to developing complex analysis, i.e. calculus with complex numbers: the subject of the present book. Our choice of topics and manner of presentation have been determined by the following considerations: 1. The theory should rapidly lead to a deeper understanding of the elementary functions as well as to new classes of functions (higher functions). We thus present the elementary functions in the first chapter, study their deeper properties in the third chapter, and finally use the powerful methods of complex analysis worked out in chapters II to IV to introduce several non-elementary functions: elliptic functions, the Gamma- and the Zeta-function, and the modular map λ. These chapters (V and VII) contain a proof of the prime number theorem – perhaps the most striking application of complex analysis! – as well as a description of plane cubics in terms of elliptic functions, and a proof of Picard’s theorem on essential singularities. 2. In order to circumvent topological difficulties we start with a local version of Cauchy
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