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S. Ponnusamy Herb Silverman Complex Variables with Applications Birkhaüser Boston • Basel • Berlin S.Ponnusamy HerbSilverman IndianInstituteofTechnology,Madras CollegeofCharleston DepartmentofMathematics DepartmentofMathematics Chennai,600036 Charleston,SC29424 India U.S.A. CoverdesignbyAlexGerasev. MathematicsSubjectClassification(2000):11A06,11M41,30-XX,32-XX(primary);26Axx,40Axx, 26Bxx,33Bxx,26Cxx,28Cxx,31Axx,35Axx,37F10,45E05,76M40(secondary) LibraryofCongressControlNumber:2006927602 ISBN-10:0-8176-4457-1 eISBN:0-8176-4513-6 ISBN-13:978-0-8176-4457-4 Printedonacid-freepaper. (cid:2)c2006BirkhaüserBoston Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewrit- tenpermissionofthepublisher(BirkhaüserBoston,c/oSpringerScience+BusinessMediaLLC,233 SpringStreet,NewYork,NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsor scholarlyanalysis.Useinconnectionwithanyformofinformationstorageandretrieval,electronic adaptation,computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterde- velopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,evenifthey arenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyare subjecttoproprietaryrights. PrintedintheUnitedStatesofAmerica. (TXQ/MP) 987654321 www.birkhauser.com To my father, Saminathan Pillai —S. Ponnusamy To my wife, Sharon Fratepietro —Herb Silverman Preface The student, who seems to be engulfed in our culture of specialization, too quicklyfeelsthenecessitytoestablishan“area”ofspecialinterest.Inkeeping with this spirit, academic bureaucracy has often forced us into a compart- mentalization of courses, which pretend that linear algebra is disjoint from modern algebra, that probability and statistics can easily be separated, and even that advanced calculus does not build from elementary calculus. This 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 and then generalize to one in complex variables. Other times we will begin with a problem in complex variables and reduce it to one in real variables. Both methods—generalization and specialization—are worthy of careful considera- tion. We expect “complex” numbers to be difficult to comprehend and “imag- inary” units to be shrouded in mystery. Hopefully, by staying close to the realfield,weshallovercomethisregrettableterminologythathasbeenthrust upon us. The authors wish to create a spiraling effect that will first enable thereadertodrawfromhisorherknowledgeofadvancedcalculusinorderto demystifycomplexvariables,andthenusethisnewlyacquiredunderstanding of complex variables to master some of the elements of advanced calculus. Wewillalsocompare,wheneverpossible,theanalyticandgeometricchar- acter of a concept. This naturally leads us to a discussion of “rigor”. The current trend seems to be that anything analytic is rigorous and anything geometric is not. This dichotomy moves some authors to strive for “rigor” at the expense of rich geometric meaning, and other authors to endeavor to be “intuitive” by discussing a concept geometrically without shedding any ana- lyticlightonit.Rigor,astheauthorsseeit,isusefulonlyinsofarasitclarifies rather than confounds. For this reason, geometry will be utilized to illustrate analyticconcepts,andanalysiswillbeemployedtounravelgeometricnotions, without regard to which approach is the more rigorous. viii Preface Sometimes, in an attempt to motivate, a discussion precedes a theorem. Sometimes,inanattempttoilluminate,remarksaboutkeystepsandpossible implicationsfollowatheorem.Noapologiesaremadeforthislackofterseness surroundingdifficulttheorems.Whilebrevitymaybethesoulofwit,itisnot the soul of insight into delicate mathematical concepts. In recognition of the primary importance of observing relationships between different approaches, some theorems are proved in several different ways. In this book, traveling quickly to the frontiers of mathematical knowledge plays a secondary role to the careful examination of the road taken and alternative routes that lead to the same destination. Awordshouldbesaidaboutthequestionsattheendofeachsection.The authors feel deeply that mathematics should be questioned—not only for its internal logic and consistency, but for the reasons we are led where we are. Doestheconclusionseem“reasonable”?Didweexpectit?Didthestepsseem naturalorartificial?Canwere-provetheresultadifferentway?Canwestate intuitively 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 are specific and some are vague; some have one possible answer and some have many; some are concerned with what has been proved and some foreshadow what will be proved. Do all these questions have anything in common? Yes. They are all meant to help the student think, understand, create, and question. It is hoped that the questions will also be helpful to the teacher, who may want to incorporate some of them into his or her lectures. Less need be said about the exercises at the end of each section because exercises have always received more favorable publicity than have questions. Very often the difference between a question and an exercise is a matter of terminology. The abundance of exercises should help to give the student a good indication of how well the material in the section has been understood. The prerequisite is at least a shaky knowledge of advanced calculus. The first nine chapters present a solid foundation for an introduction to complex variables. The last four chapters go into more advanced topics in some detail, inordertoprovidethegroundworknecessaryforstudentswhowishtopursue further the general theory of complex analysis. If 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, and the concepts in Chapter 2 need be introduced only when applicable in latter chapters. Chapter 3 may be omitted entirely, and the mapping properties in Chapter 4 may be omitted. We 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 questioning and creative thinking, see G. Polya, How to Solve It, second edition, Princeton University press, Princeton, New Jersey, 1957. Preface ix could not have been mentioned without invoking the standard terminology dealing with the trivial zeros of the Riemann zeta function. But the spirit, if nottheletter,ofthisdesirehasbeenfulfilled.Detailedexplanations,remarks, worked-outexamplesandinsightsareplentiful.Theteachershouldbeableto leavesectionsforthestudenttoreadonhis/herown;infact,thisbookmight serve as a self-study text. A teacher’s manual containing more detailed hints and solutions to questions and exercises is available. The interested teacher may contact us by e-mail and receive a pdf version. We wish to express our thanks to the Center for Continuing Education at the Indian Institute of Technology Madras, India, for its support in the preparation of the manuscript. Finally, we thank Ann Kostant, Executive Editor, Birkhaüser, who has been most helpful to the authors through her quick and efficient responses throughout the preparation of this manuscript. S. Ponnusamy IIT Madras, India Herb Silverman June 2005 College of Charleston, USA Contents Preface ........................................................ vii 1 Algebraic and Geometric Preliminaries .................... 1 1.1 The Complex Field ...................................... 1 1.2 Rectangular Representation............................... 5 1.3 Polar Representation..................................... 15 2 Topological and Analytic Preliminaries .................... 25 2.1 Point Sets in the Plane................................... 25 2.2 Sequences .............................................. 32 2.3 Compactness............................................ 39 2.4 Stereographic Projection ................................. 44 2.5 Continuity.............................................. 48 3 Bilinear Transformations and Mappings ................... 61 3.1 Basic Mappings ......................................... 61 3.2 Linear Fractional Transformations ......................... 66 3.3 Other Mappings......................................... 85 4 Elementary Functions ..................................... 91 4.1 The Exponential Function ................................ 91 4.2 Mapping Properties......................................100 4.3 The Logarithmic Function ................................108 4.4 Complex Exponents .....................................114 5 Analytic Functions ........................................121 5.1 Cauchy–Riemann Equation ...............................121 5.2 Analyticity .............................................130 5.3 Harmonic Functions .....................................141 xii Contents 6 Power Series...............................................153 6.1 Sequences Revisited......................................153 6.2 Uniform Convergence ....................................164 6.3 Maclaurin and Taylor Series ..............................173 6.4 Operations on Power Series ...............................186 7 Complex Integration and Cauchy’s Theorem...............195 7.1 Curves .................................................195 7.2 Parameterizations .......................................207 7.3 Line Integrals ...........................................217 7.4 Cauchy’s Theorem.......................................226 8 Applications of Cauchy’s Theorem.........................243 8.1 Cauchy’s Integral Formula................................243 8.2 Cauchy’s Inequality and Applications ......................263 8.3 Maximum Modulus Theorem .............................275 9 Laurent Series and the Residue Theorem ..................285 9.1 Laurent Series ..........................................285 9.2 Classification of Singularities..............................293 9.3 Evaluation of Real Integrals ..............................308 9.4 Argument Principle......................................331 10 Harmonic Functions .......................................349 10.1 Comparison with Analytic Functions.......................349 10.2 Poisson Integral Formula .................................358 10.3 Positive Harmonic Functions..............................371 11 Conformal Mapping and the Riemann Mapping Theorem..379 11.1 Conformal Mappings.....................................379 11.2 Normal Families.........................................390 11.3 Riemann Mapping Theorem ..............................395 11.4 The Class S ............................................405 12 Entire and Meromorphic Functions ........................411 12.1 Infinite Products ........................................411 12.2 Weierstrass’ Product Theorem ............................422 12.3 Mittag-Leffler Theorem ..................................437 13 Analytic Continuation .....................................445 13.1 Basic Concepts..........................................445 13.2 Special Functions........................................458 References and Further Reading...............................473 Index of Special Notations.....................................475 Contents xiii Index..........................................................479 Hints for Selected Questions and Exercises ....................485