Complex Variables - Springer

1 S. PonnusamyHerb SilvermanComplex Variableswith ApplicationsBirkh auserBoston Basel BerlinS. PonnusamyIndian Institute of Technology, MadrasDepartment of MathematicsChennai, 600 036 IndiaHerb SilvermanCollege of CharlestonDepartment of MathematicsCharleston, SC design by Alex Subject Classification (2000): 11A06, 11M41, 30-XX, 32-XX (primary); 26 Axx, 40 Axx,26 Bxx, 33 Bxx, 26 Cxx, 28 Cxx, 31 Axx, 35 Axx, 37F10, 45E05, 76M40 (secondary)Library of Congress Control Number:2006927602 ISBN-10: 0-8176-4457-1eISBN: 0-8176-4513-6 ISBN-13: 978-0-8176-4457-4 Printed on acid-free 2006 Birkh auser BostonAll rights reserved. This work may not be translated or copied in whole or in part without the writ-ten permission of the publisher (Birkh auser Boston, c/o Springer Science+Business Media LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews orscholarly analysis.

2 Use in connection with any form of information storage and retrieval, electronicadaptation, computer software, or by similar or dissimilar methodology now known or hereafter de-veloped is use in this publication of trade names, trademarks, service marks and similar terms, even if theyare not identified as such, is not to be taken as an expression of opinion as to whether or not they aresubject to proprietary in the United States of America.(TXQ/MP) my father, Saminathan Pillai S. PonnusamyTo my wife, Sharon Fratepietro Herb SilvermanPrefaceThe student, who seems to be engulfed in our culture of specialization, tooquickly feels the necessity to establish an area of special interest.

3 In keepingwith this spirit, academic bureaucracy has often forced us into a compart-mentalization of courses, which pretend that linear algebra is disjoint frommodern algebra, that probability and statistics can easily be separated, andeven that advanced calculus does not build from elementary book is written from the point of view that there is an interdepen-dence between real and Complex Variables that should be explored at ev-ery opportunity. Sometimes we will discuss a concept in real Variables andthen generalize to one in Complex Variables . Other times we will begin witha problem in Complex Variables and reduce it to one in real Variables .

4 Bothmethods generalization and specialization are worthy of careful expect Complex numbers to be difficult to comprehend and imag-inary units to be shrouded in mystery. Hopefully, by staying close to thereal field, we shall overcome this regrettable terminology that has been thrustupon us. The authors wish to create a spiraling effect that will first enablethe reader to draw from his or her knowledge of advanced calculus in order todemystify Complex Variables , and then use this newly acquired understandingof Complex Variables to master some of the elements of advanced will also compare, whenever possible, the analytic and geometric char-acter of a concept.

5 This naturally leads us to a discussion of rigor . Thecurrent trend seems to be that anything analytic is rigorous and anythinggeometric is not. This dichotomy moves some authors to strive for rigor atthe expense of rich geometric meaning, and other authors to endeavor to be intuitive by discussing a concept geometrically without shedding any ana-lytic light on it. Rigor, as the authors see it, is useful only insofar as it clarifiesrather than confounds. For this reason, geometry will be utilized to illustrateanalytic concepts, and analysis will be employed to unravel geometric notions,without regard to which approach is the more , in an attempt to motivate, a discussion precedes a , in an attempt to illuminate, remarks about key steps and possibleimplications follow a theorem.

6 No apologies are made for this lack of tersenesssurrounding difficult theorems. While brevity may be the soul of wit, it is notthe soul of insight into delicate mathematical concepts. In recognition of theprimary importance of observing relationships between different approaches,some theorems are proved in several different ways. In this book, travelingquickly to the frontiers of mathematical knowledge plays a secondary role tothe careful examination of the road taken and alternative routes that lead word should be said about the questions at the end of each section. Theauthors feel deeply that mathematics should be questioned not only for itsinternal logic and consistency, but for the reasons we are led where we the conclusion seem reasonable ?

7 Did we expect it? Did the steps seemnatural or artificial? Can we re-prove the result a different way? Can we stateintuitively what we have proved? Can we draw a picture?1 Questions , as used at the end of each section, cannot easily be catego-rized. Some questions are simple and some are quite challenging; some arespecific and some are vague; some have one possible answer and some havemany; some are concerned with what has been proved and some foreshadowwhat will be proved. Do all these questions have anything in common? are all meant to help the student think, understand, create, and ques-tion. It is hoped that the questions will also be helpful to the teacher, whomay want to incorporate some of them into his or her need be said about the exercises at the end of each section becauseexercises have always received more favorable publicity than have often the difference between a question and an exercise is a matter ofterminology.

8 The abundance of exercises should help to give the student agood indication of how well the material in the section has been prerequisite is at least a shaky knowledge of advanced calculus. Thefirst nine chapters present a solid foundation for an introduction to complexvariables. The last four chapters go into more advanced topics in some detail,in order to provide the groundwork necessary for students who wish to pursuefurther the general theory of Complex this book is to be used as a one-semester course, Chapters 5, 6, 7,8, and 9 should constitute the core. Chapter 1 can be covered rapidly, andthe concepts in Chapter 2 need be introduced only when applicable in latterchapters.

9 Chapter 3 may be omitted entirely, and the mapping properties inChapter 4 may be wanted to write a mathematics book that omitted the word trivial .Unfortunately, the Riemann hypothesis, stated on the last page of the text,1 For an excellent little book elaborating on the relationship between questioningand creative thinking, see G. Polya,How to Solve It,second edition, PrincetonUniversity press, Princeton, New Jersey, not have been mentioned without invoking the standard terminologydealing with thetrivial zerosof the Riemann zeta function. But the spirit, ifnot the letter, of this desire has been fulfilled. Detailed explanations, remarks,worked-out examples and insights are plentiful.

10 The teacher should be able toleave sections for the student to read on his/her own; in fact, this book mightserve as a self-study teacher s manual containing more detailed hints and solutions to ques-tions and exercises is available. The interested teacher may contact us bye-mail and receive a pdf wish to express our thanks to the Center for Continuing Educationat the Indian Institute of Technology Madras, India, for its support in thepreparation of the , we thank Ann Kostant, Executive Editor, Birkh auser, who hasbeen most helpful to the authors through her quick and efficient responsesthroughout the preparation of this PonnusamyIIT Madras, IndiaHerb SilvermanJune 2005 College of Charleston, vii1 Algebraic and Geometric The Complex Field.