Conformal Maps. Part 1

Getting down to a more serious study of conformal maps, we must learn to work with functions defined on open subsets of the Riemann sphere as confidently as we do with functions defined on open subsets of C.

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Serge Lvovski

Principles of Complex Analysis

Moscow Lectures Volume 6 Series Editors Lev D. Beklemishev, Moscow, Russia Vladimir I. Bogachev, Moscow, Russia Boris Feigin, Moscow, Russia Valery Gritsenko, Moscow, Russia Yuly S. Ilyashenko, Moscow, Russia Dmitry B. Kaledin, Moscow, Russia Askold Khovanskii, Moscow, Russia Igor M. Krichever, Moscow, Russia Andrei D. Mironov, Moscow, Russia Victor A. Vassiliev, Moscow, Russia Managing Editor Alexey L. Gorodentsev, Moscow, Russia

More information about this series at http://www.springer.com/series/15875

Serge Lvovski

Principles of Complex Analysis

Serge Lvovski National Research University Higher School of Economics Moscow, Russia Federal Science Center System Research Institute of Russian Academy of Sciences (FGU FNC NIISI RAN) Moscow, Russia

Translated from the Russian by Natalia Tsilevich. Originally published as Принципы комплексного анализа by MCCME, Moscow, 2017.

ISSN 2522-0314 ISSN 2522-0322 (electronic) Moscow Lectures ISBN 978-3-030-59364-3 ISBN 978-3-030-59365-0 (eBook) https://doi.org/10.1007/978-3-030-59365-0 Mathematics Subject Classification (2020): 30-01 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Cover illustration: https://www.istockphoto.com/de/foto/panorama-der-stadt-moskau-gm49008001475024685, with kind permission This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface to the Book Series Moscow Lectures

You hold a volume in a textbook series of Springer Nature dedicated to the Moscow mathematical tradition. Moscow mathematics has very strong and distinctive features. There are several reasons for this, all of which go back to good and bad aspects of Soviet organization of science. In the twentieth century, there was a

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