Graduate Texts in Mathematics 245 Graduate Texts in Mathematics Series Editors: Sheldon Axler San Francisco State University, San Francisco, CA, USA Kenneth Ribet University of California, Berkeley, CA, USA Advisory Board: Colin Adams, Williams College, Williamstown, MA, USA Alejandro Adem, University of British Columbia, Vancouver, BC, Canada Ruth Charney, Brandeis University, Waltham, MA, USA Irene M. Gamba, The University of Texas at Austin, Austin, TX, USA Roger E. Howe, Yale University, New Haven, CT, USA David Jerison, Massachusetts Institute of Technology, Cambridge, MA, USA Jeffrey C. Lagarias, University of Michigan, Ann Arbor, MI, USA Jill Pipher, Brown University, Providence, RI, USA Fadil Santosa, University of Minnesota, Minneapolis, MN, USA Amie Wilkinson, University of Chicago, Chicago, IL, USA Graduate Texts in Mathematics bridge the gap between passive study and creative understanding, offering graduate-level introductions to advanced topics in mathematics. The volumes are carefully written as teaching aids and highlight characteristic features of the theory. Although these books are frequently used as textbooks in graduate courses, they are also suitable for individual study. For further volumes: http://www.springer.com/series/136 Rub´ı E. Rodr´ıguez • Irwin Kra • Jane P. Gilman Complex Analysis In the Spirit of Lipman Bers Second Edition 123 Rub´ı E. Rodr´ıguez Irwin Kra Facultad de Matematicas´ Department of Mathematics Pontificia Universidad Catolica´ de Chile State University of New York at Stony Brook Santiago, Chile Stony Brook, NY, USA Jane P. Gilman Department of Mathematics and Computer Science Rutgers University Newark, NJ, USA ISSN 0072-5285 ISBN 978-1-4419-7322-1 ISBN 978-1-4419-7323-8 (eBook) DOI 10.1007/978-1-4419-7323-8 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2012950351 © Springer Science+Business Media New York 2007, 2013 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. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. 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. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) For Victor, Eleanor and Bob and to the memory of Mary and Lipman Bers Preface to Second Edition The second edition contains significant new material in several new sections. We have also expanded sections from the first edition (improving, we expect, the exposition throughout) and included new figures and exercises for added clarity, as well as, of course, corrected errors and typos in the previous version. Among the most important changes are: • We have expanded and clarified several sections from the first edition. • We have significantly enlarged the exercise sections. Some of the problems are routine, others challenging, and some require knowledge of other subjects usually covered in various first-year graduate courses. The problems are listed in more or less random order as far as their difficulty. • In both editions of this text, we use the approach to integration based on differential forms. In an alternative approach differential forms are a by-product of work on integration of functions motivated by ideas from standard treatments of integral calculus. That is the approach that Bers took in courses that he taught; it is also the approach used by Ahlfors. Of course, either of the two approaches are equally valid and lead to the same major results. In this second edition, we provide an appendix that outlines this alternative path to the main results. • New sections on Perron’s method for solving the Dirichlet problem, Green’s function, an alternative proof of the Riemann mapping theorem, and a description of the divisor of a bounded analytic function on the disc via infinite Blaschke products are included. • We prove the Bers theorem on isomorphisms between rings of holomorphic functions on plane domains. • A section on historical references prepared by Ranjan Roy has been added. In addition, the following items related to our work might be of interest to those reading this volume. • An answer manual for the exercises prepared by Vamsi Pritham Pingali is available, to instructors using the book for a course, from the publisher. • An electronic version of the book is available from the publisher. vii viii Preface to Second Edition • One of the authors (IK) has created and maintains a section about the book on his web site. It contains, among other things, updated information and errata. http://www.math.sunysb.edu/irwin/bookcxinfo.html The other authors may also have information about the book on their web sites. It is our pleasure to thank Ranjan Roy for producing and allowing us to include in this volume his historical note. We are grateful to our colleagues and students who pointed out places for improvement in the first edition and in drafts of the second one. Among them: Bill Abikoff, Robert Burckel, Eduardo Friedman, Bryna Kra, Peter Landweber, Howard Masur, Sudeb Mitra, Lee Mosher, Robert Sczech, and Jacob Sturm. It is still true, of course, that errors and shortcomings may remain in the final version of this edition and these are entirely our responsibility. Spring 2012 New York, NY, USA Irwin Kra Santiago, Chile Rub´ı E. Rodr´ıguez Newark, NJ, USA Jane P. Gilman Preface to First Edition This book presents fundamental material that should be part of the education of every practicing mathematician. This material will also be of interest to computer scientists, physicists and engineers. Because complex analysis has been used by generations of practicing mathemati- cians working in a number of different fields, the basic results have been developed and redeveloped from a number of different perspectives. We are not wedded to any one viewpoint. Rather we will try to exploit the richness of the development and explain and interpret standard definitions and results using the most convenient tools from analysis, geometry and algebra. Complex analysis has connections and applications to many other subjects in mathematics, both classical and modern, and to other sciences. It is an area where the classical and the modern techniques meet and benefit from each other. We will try to illustrate this in the applications we give. Complex analysis is the study of complex valued functions of a complex variable and its initial task is to extend the concept of differentiability from real valued functions of a real variable to these functions. A complex valued function of a complex variable that is differentiable is termed analytic,andthefirstpartofthis book is a study of the many equivalent ways of understanding the concept of analyticity. The equivalent ways of formulating the concept of an analytic function are summarized in what we term the fundamental theorem for functions of a complex variable. In dedicating the first part of this book to the very precise goal of stating and proving the fundamental theorem we follow a path in the tradition of Lipman Bers from whom we learned the subject. In the second part of the text we then proceed to the leisurely exploration of interesting consequences and applications of the fundamental theorem. We are grateful to Lipman Bers for introducing us to the beauty of the subject. The book is an outgrowth of notes from Bers’s original lectures. Versions of these notes have been used by us at our respective home institutions, some for more than 20 years, as well as by others at various universities. We are grateful to many colleagues and students who read and commented on these notes. Our interaction ix x Preface to First Edition with them helped shape this book. We tried to follow all useful advice and correct, of course, any mistakes or shortcomings they identified. Those that remain are entirely our responsibility. Newark, NJ, USA Jane P. Gilman New York, NY, USA Irwin Kra Santiago, Chile Rub´ı E. Rodr´ıguez Acknowledgement The first author was supported in part by Fondecyt Grant # 1100767. The third author was supported in part by grants from the National Security Agency, from the Rutgers University Research Foundation, and from Yale University while a visiting Fellow. xi Contents 1 The Fundamental Theorem in Complex Function Theory ............ 1 1.1 Some Motivation ..................................................... 1 1.1.1 Where Do Series Converge? ............................... 1 1.1.2 A Problem on Partitions .................................... 2 1.1.3 Evaluation of Definite Real Integrals ...................... 3 1.2 The Fundamental Theorem of Complex Function Theory ........