Second Edition

Donald Sarason

A M E R I C A N M A T H E M A T I C A L S O C I E T Y

Complex Function Theory

Donald Sarason

A M E R I C A N M A T H E M A T I C A L S O C I E T Y

Complex Function Theory Second Edition

http://dx.doi.org/10.1090/mbk/049

2000 Mathematics Subject Classification. Primary 30–01.

Front Cover: The figure on the front cover is courtesy of Andrew D. Hwang.

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Sarason, Donald.Complex function theory / Donald Sarason.— 2nd ed.

p. cm.Includes index.ISBN-13: 978-0-8218-4428-1 (alk. paper)ISBN-10: 0-8218-4428-8 (alk. paper)1. Functions of complex variables. I. Title.

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Contents

Preface to the Second Edition ix

Preface to the First Edition xi

Chapter I. Complex Numbers 1

§I.1. Definition of C 2

§I.2. Field Axioms 2

§I.3. Embedding of R in C. The Imaginary Unit 3

§I.4. Geometric Representation 3

§I.5. Triangle Inequality 4

§I.6. Parallelogram Equality 5

§I.7. Plane Geometry via Complex Numbers 5

§I.8. C as a Metric Space 6

§I.9. Polar Form 6

§I.10. De Moivre’s Formula 7

§I.11. Roots 8

§I.12. Stereographic Projection 9

§I.13. Spherical Metric 10

§I.14. Extended Complex Plane 11

Chapter II. Complex Differentiation 13

§II.1. Definition of the Derivative 13

§II.2. Restatement in Terms of Linear Approximation 14

§II.3. Immediate Consequences 14

iii

iv Contents

§II.4. Polynomials and Rational Functions 15§II.5. Comparison Between Differentiability in the Real and

Complex Senses 15§II.6. Cauchy-Riemann Equations 16§II.7. Sufficient Condition for Differentiability 17§II.8. Holomorphic Functions 17§II.9. Complex Partial Differential Operators 18§II.10. Picturing a Holomorphic Function 19§II.11. Curves in C 20§II.12. Conformality 21§II.13. Conformal Implies Holomorphic 22§II.14. Harmonic Functions 23§II.15. Holomorphic Implies Harmonic 24§II.16. Harmonic Conjugates 24

Chapter III. Linear-Fractional Transformations 27§III.1. Complex projective space 27§III.2. Linear-fractional transformations 28§III.3. Conformality 29§III.4. Fixed points 29§III.5. Three-fold transitivity 29§III.6. Factorization 30§III.7. Clircles 31§III.8. Preservation of clircles 31§III.9. Analyzing a linear-fractional transformation—an example 32

Chapter IV. Elementary Functions 35§IV.1. Definition of ez 35§IV.2. Law of Exponents 36§IV.3. ez is holomorphic 36§IV.4. Periodicity 37§IV.5. ez as a map 37§IV.6. Hyperbolic functions 38§IV.7. Zeros of coshz and sinhz. 38§IV.8. Trigonometric functions 39§IV.9. Logarithms 40§IV.10. Branches of arg z and log z. 40

Contents v

§IV.11. log z as a holomorphic function 41§IV.12. Logarithms of holomorphic functions 42

§IV.13. Roots 43

§IV.14. Inverses of holomorphic functions 43

§IV.15. Inverse trigonometric functions 44§IV.16. Powers 45

§IV.17. Analytic continuation and Riemann surfaces 45

Chapter V. Power Series 49

§V.1. Infinite Series 49§V.2. Necessary Condition for Convergence 49

§V.3. Geometric Series 50

§V.4. Triangle Inequality for Series 50

§V.5. Absolute Convergence 50§V.6. Sequences of Functions 51

§V.7. Series of Functions 51

§V.8. Power Series 52

§V.9. Region of Convergence 53§V.10. Radius of Convergence 54

§V.11. Limits Superior 54

§V.12. Cauchy-Hadamard Theorem 55

§V.13. Ratio Test 56§V.14. Examples 57

§V.15. Differentiation of Power Series 58

§V.16. Examples 60

§V.17. Cauchy Product 61§V.18. Division of Power Series 63

Chapter VI. Complex Integration 65

§VI.1. Riemann Integral for Complex-Valued Functions 65§VI.2. Fundamental Theorem of Calculus 66

§VI.3. Triangle Inequality for Integration 66

§VI.4. Arc Length 67

§VI.5. The Complex Integral 67§VI.6. Integral of a Derivative 68

§VI.7. An Example 68

vi Contents

§VI.8. Reparametrization 69§VI.9. The Reverse of a Curve 70

§VI.10. Estimate of the Integral 71

§VI.11. Integral of a Limit 71

§VI.12. An Example 71

Chapter VII. Core Versions of Cauchy’s Theorem, and Consequences 75

§VII.1. Cauchy’s Theorem for a Triangle 75

§VII.2. Cauchy’s Theorem for a Convex Region 78

§VII.3. Existence of a Primitive 78§VII.4. More Definite Integrals 79

§VII.5. Cauchy’s Formula for a Circle 79

§VII.6. Mean Value Property 81

§VII.7. Cauchy Integrals 82§VII.8. Implications for Holomorphic Functions 83

§VII.9. Cauchy Product 84

§VII.10. Converse of Goursat’s Lemma 85

§VII.11. Liouville’s Theorem 86§VII.12. Fundamental Theorem of Algebra 86

§VII.13. Zeros of Holomorphic Functions 87

§VII.14. The Identity Theorem 89

§VII.15. Weierstrass Convergence Theorem 89§VII.16. Maximum Modulus Principle 90

§VII.17. Schwarz’s Lemma 91

§VII.18. Existence of Harmonic Conjugates 93

§VII.19. Infinite Differentiability of Harmonic Functions 94§VII.20. Mean Value Property for Harmonic Functions 94

§VII.21. Identity Theorem for Harmonic Functions 94

§VII.22. Maximum Principle for Harmonic Functions 95§VII.23. Harmonic Functions in Higher Dimensions 95

Chapter VIII. Laurent Series and Isolated Singularities 97

§VIII.1. Simple Examples 97

§VIII.2. Laurent Series 98§VIII.3. Cauchy Integral Near ∞ 99

§VIII.4. Cauchy’s Theorem for Two Concentric Circles 100

Contents vii

§VIII.5. Cauchy’s Formula for an Annulus 101§VIII.6. Existence of Laurent Series Representations 101

§VIII.7. Isolated Singularities 102

§VIII.8. Criterion for a Removable Singularity 105

§VIII.9. Criterion for a Pole 105§VIII.10. Casorati-Weierstrass Theorem 106

§VIII.11. Picard’s Theorem 106

§VIII.12. Residues 106

Chapter IX. Cauchy’s Theorem 109§IX.1. Continuous Logarithms 109

§IX.2. Piecewise C1 Case 110

§IX.3. Increments in the Logarithm and Argument Along a Curve 110

§IX.4. Winding Number 111§IX.5. Case of a Piecewise-C1 Curve 111

§IX.6. Contours 113

§IX.7. Winding Numbers of Contours 114

§IX.8. Separation Lemma 115§IX.9. Addendum to the Separation Lemma 117

§IX.10. Cauchy’s Theorem 118

§IX.11. Homotopy 119

§IX.12. Continuous Logarithms—2-D Version 119§IX.13. Homotopy and Winding Numbers 120

§IX.14. Homotopy Version of Cauchy’s Theorem 121

§IX.15. Runge’s Approximation Theorem 121

§IX.16. Second Proof of Cauchy’s Theorem 122§IX.17. Sharpened Form of Runge’s Theorem 123

Chapter X. Further Development of Basic Complex Function 125

§X.1. Simply Connected Domains 125§X.2. Winding Number Criterion 126

§X.3. Cauchy’s Theorem for Simply Connected Domains 126

§X.4. Existence of Primitives 127

§X.5. Existence of Logarithms 127§X.6. Existence of Harmonic Conjugates 128

§X.7. Simple Connectivity and Homotopy 128

viii Contents

§X.8. The Residue Theorem 129§X.9. Cauchy’s Formula 130§X.10. More Definite Integrals 130§X.11. The Argument Principle 137§X.12. Rouche’s Theorem 138§X.13. The Local Mapping Theorem 140§X.14. Consequences of the Local Mapping Theorem 140§X.15. Inverses 141§X.16. Conformal Equivalence 141§X.17. The Riemann Mapping Theorem 142§X.18. An Extremal Property of Riemann Maps 143§X.19. Stieltjes-Osgood Theorem 144§X.20. Proof of the Riemann Mapping Theorem 146§X.21. Simple Connectivity Again 148

Appendix 1. Sufficient condition for differentiability 151

Appendix 2. Two instances of the chain rule 153

Appendix 3. Groups, and linear-fractional transformations 155

Appendix 4. Differentiation under the integral sign 157

References 159

Index 161

Preface to the SecondEdition

As is usual in a second edition, various minor flaws that had crept intothe first edition have been corrected. Certain topics are now presented moreclearly, it is hoped, after being rewritten and/or reorganized. The treatmenthas been expanded only slightly: there is now a section on division of powerseries, and a brief discussion of homotopy. (The latter topic was relegatedto a couple of exercises in the first edition.) Four appendices have beenadded; they contain needed background which, experience has shown, is notpossessed nowadays by all students taking introductory complex analysis.

In this edition, the numbers of certain exercises are preceded by anasterisk. The asterisk indicates that the exercise will be referred to later inthe text. In many cases the result established in the exercise will be neededas part of a proof.

I am indebted to a number of students for detecting minor errors inthe first edition, and to Robert Burckel and Bjorn Poonen for their valuablecomments. Special thanks go to George Bergman and his eagle eye. George,while teaching from the first edition, read it carefully and provided a longlist of suggested improvements, both in exposition and in typography. I owemy colleague Henry Helson, the publisher of the first edition, thanks forencouraging me to publish these Notes in the first place, and for his manykindnesses during our forty-plus years together at Berkeley.

ix

x Preface to the Second Edition

The figures from the first edition have been redrawn by Andrew D.Hwang, whose generous help is greatly appreciated. I am indebted to Ed-ward Dunne and his AMS colleagues for the patient and professional waythey shepherded my manuscript into print.

Finally, as always, I am deeply grateful to my wife, Mary Jennings, forher constant support, in particular, for applying her TEXnical skills to thisvolume.

Berkeley, CaliforniaJanuary 26, 2007

Preface to the FirstEdition

These are the notes for a one-semester introductory course in the theory offunctions of a complex variable. The aim of the notes is to help studentsof mathematics and related sciences acquire a basic understanding of thesubject, as a preparation for pursuing it at a higher level or for employingit in other areas. The approach is standard and somewhat old-fashioned.

The user of the notes is assumed to have a thorough grounding in basicreal analysis, as one can obtain, for example, from the book of W. Rudincited in the list of references. Notions like metric, open set, closed set, inte-rior, boundary, limit point, and uniform convergence are employed withoutexplanation. Especially important are the notions of a connected set andof the connected components of a set. Basic notions from abstract algebraalso occur now and then. These are all concepts that ordinarily are familiarto students by the time they reach complex function theory.

As these notes are a rather bare-bones introduction to a vast subject, thestudent or instructor who uses them may well wish to supplement them withother references. The notes owe a great deal to the book by L. V. Ahlforsand to the book by S. Saks and A. Zygmund, which, together with theteaching of George Piranian, were largely responsible for my own love affairwith the subject. Several other excellent books are mentioned in the list ofreferences.

The first edition was published by Henry Helson under the title Notes on Complex FunctionTheory.

xi

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1

xii Preface to the First Edition

The notes contain only a handful of pictures, not enough to do justiceto the strong geometric component of complex function theory. The user isadvised to make his or her own sketches as an aid to visualization. Thanksgo to Andrew Hwang for drawing the pictures.

The approach in these notes to Cauchy’s theorem, the central theo-rem of the subject, follows the one used by Ahlfors, attributed by him toA. F. Beardon. An alternative approach based on Runge’s approximationtheorem, adapted from Saks and Zygmund, is also presented.

The terminology used in the notes is for the most part standard. Twoexceptions need mention. Some authors use the term “region” specifically torefer to an open connected subset of the plane. Here the term is used, fromtime to time, in a less formal way. On the other hand, the term “contour”is used in the notes in a specific way not employed by other authors.

I wish to thank my Berkeley Math H185 class in the Spring Semester,1994, for pointing out a number of corrections to the prepublication versionof the notes, and my wife, Mary Jennings, who read the first draft of thenotes and helped me to anticipate some of the questions students might raiseas they work through this material. She also typed the manuscript. I deeplyappreciate her assistance and support. The notes are dedicated to her.

Berkeley, CaliforniaJune 8, 1994

References

1. Ahlfors, L. V., Complex Analysis, McGraw-Hill, 1979.

2. Axler, S., Bourdon, P., and Ramey, W., Harmonic Function Theory, Springer-Verlag,1992.

3. Burckel, R. B., An Introduction to Classical Complex Analysis, Vol. 1, Birkhauser,1979.

4. Conway, J. B., Functions of One Complex Variable, Springer-Verlag, 1978.

5. Marsden, J. E., and Hoffman, M. J., Basic Complex Analysis, W. H. Freeman, 1987.

6. Newman, M. H. A., Elements of the Topology of Plane Sets of Points, Cambridge, 1961.

7. Rudin, W., Principles of Mathematical Analysis, McGraw-Hill, 1976.

8. Saks, S., and Zygmund, A., Analytic Functions, Elsevier, 1971.

159

Index

absolute convergence, 50

absolute value, 3

analytic, 17

analytic continuation, 45

arc length, 67

argument, 6

argument principle, 137, 138

Bernoulli numbers, 104

bijection, 28

branch

function, 44

of arg z, 40

of log z, 40

of a holomorphic function, 42

of an inverse, 44

of the logarithm, 42

Burckel, R. B., 85

Caratheodory, C., 143

Casorati-Weierstrass theorem, 106

Cauchy integral, 83, 99

Cauchy product, 61, 84

Cauchy’s formula, 130

for a circle, 79

Cauchy’s formula for an annulus, 101

Cauchy’s theorem, 75, 109, 118, 121, 122

for a convex region, 78

for a triangle, 75

for simply connected domains, 126

for two concentric circles, 100

Cauchy-Hadamard theorem, 55

Cauchy-Riemann equations, 16

in polar coordinates, 16

chain rule, 14, 153

clircle, 31

complex conjugate, 4

complex integral, 67

complex number system, 1

complex partial differential operators, 18

complex plane, 3

complex projective space, 27

conformal automorphism, 142

conformal equivalence, 141

conformal map, 22

contour, 113

simple, 114

convergence

local uniform, 51

of an infinite series, 49

uniform, 51

cross ratio, 30, 32

curve, 20

differentiable, 20

regular, 20

De Moivre’s formula, 7

differentation of power series, 58

differentiable

in the complex sense, 13

in the real sense, 15, 151

differentiation under the integral sign, 157

division of power series, 63

domain, 125

elementary function, 35

entire function, 86

Euler, L., 35

exponential function (ez), 35

extended complex plane, 11

Fejer, L., 143

161

162 Index

field axioms, 2fundamental theorem of algebra, 1, 86fundamental theorem of calculus, 66

Gauss, C. F., 1

general linear group for dimension two, 29geometric function theory, 148geometric series, 50Goursat’s lemma, 75, 85

Goursat, E., 75group, 155

harmonic conjugate, 24, 93, 128harmonic function, 23

Herglotz’s formula, 137holomorphic, 17homothetic map, 19homotopy, 119

Hurwitz’s theorem, 139hyperbolic functions, 38

identity theorem, 89identity theorem for harmonic functions, 94

imaginary axis, 3imaginary part, 4imaginary unit, 3increment in the argument, 110

increment in the logarithm, 110index (of a point with respect to a curve),

111inverse of a holomorphic function, 43, 141inverse trigonometric functions, 44

inversion, 30isolated singularity, 102

essential, 102pole, 102

removable, 102isomorphism, 155

Koebe, P., 143

Laplace’s equation, 23, 25in polar coordinates, 24

Laurent series, 98annulus of convergence, 98convergence, 98

principal part, 102law of exponents, 35, 36, 62limit superior, 54linear-fractional transformation, 27, 155

conjugate, 31factorization, 30fixed points, 29preservation of clircles, 31

three-fold transitivity, 29Liouville’s theorem, 86local mapping theorem, 140

locally uniformly bounded, 144logarithm, 40

of a continuous function, 109, 119

of a holomorphic function, 42, 127logarithmic derivative, 42

maximum modulus principle, 90

maximum principle for harmonic functions,95

mean value propertyfor harmonic functions, 94for holomorphic functions, 81

meromorphic function, 138modulus, 3Morera’s theorem, 85Morera, G., 85

multiplicity of a zero, 87multiply connected domain, 128

normal family, 144

orderof a pole, 103of a zero, 87

parallelogram equality, 5partial fraction decomposition, 104partial sum, 49

Picard’s theorem, 106Picard, E., 106Pick’s lemma, 92

piecewise C1, 66piecewise continuous, 65point at infinity, 11Poisson’s formula, 137

polar form, 6pole, 102

order, 103simple, 103

polynomials, 15potential theory, 143power series, 52

center, 52coefficient, 52

primitive, 78, 127principle value of arg, 6

product rule, 14purely imaginary number, 3

quotient rule, 14

radius of convergence, 54ratio test, 56rational functions, 15

real axis, 3real part, 4reparametrization, 69

Index 163

residue, 106residue theorem, 113, 129reverse of a curve, 70Riemann, 47Riemann integral for complex-valued

functions, 65Riemann map, 143Riemann mapping theorem, 142, 146Riemann sphere, 11Riemann surface, 45Riemann, G. F. B., 11Riesz, F., 143roots, 43Rouche’s theorem, 138Runge’s approximation theorem, 121, 123

Schmiegungsverfahren, 147Schwarz’s lemma, 91separation lemma, 115simple connectivity and homotopy, 128, 148simply connected domain, 125

spherical metric, 10stereographic projection, 9Stieltjes-Osgood theorem, 144sum rule, 14

tangent direction, 21Toeplitz matrix, 63triangle inequality, 4, 50triangle inequality for integration, 66trigonometric functions, 39

uniformly Cauchy, 51univalent, 44

Vitali’s theorem, 146

Weierstrass convergence theorem, 89Weierstrass, K., 121winding number, 111, 114

zero (of a holomorphic function)order, 87simple, 88

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Complex Function Theory is a concise and rigorous introduction to the theory of functions of a complex variable. Written in a classical style, it is in the spirit of the books by Ahlfors and by Saks and Zygmund. Being designed for a one-semester course, it is much shorter than many of the standard texts. Sarason covers the basic material through Cauchy’s theorem and applications, plus the Riemann mapping theorem. It is suitable for either an introductory graduate course or an undergraduate course for students with adequate preparation.

The fi rst edition was published with the title Notes on Complex Function Theory.