Complex Analysis

The book provides an introduction to complex analysis for students with some familiarity with complex numbers from high school. The first part comprises the basic core of a course in complex analysis for junior and senior undergraduates. The second part includes various more specialized topics as the argument principle the Poisson integral, and the Riemann mapping theorem. The third part consists of a selection of topics designed to complete the coverage of all background necessary for passing Ph.D. qualifying exams in complex analysis.

Preface

vii

Introduction

xvii

FIRST PART

The Complex Plane and Elementary Functions

1

(32)

Complex Numbers

1

(4)

Polar Representation

5

(6)

Stereographic Projection

11

(4)

The Square and Square Root Functions

15

(4)

The Exponential Function

19

(2)

The Logarithm Function

21

(3)

Power Functions and Phase Factors

24

(5)

Trigonometric and Hyperbolic Functions

29

(4)

Analytic Functions

33

(37)

Review of Basic Analysis

33

(9)

Analytic Functions

42

(4)

The Cauchy-Riemann Equations

46

(5)

Inverse Mappings and the Jacobian

51

(3)

Harmonic Functions

54

(4)

Conformal Mappings

58

(5)

Fractional Linear Transformations

63

(7)

Line Integrals and Harmonic Functions

70

(32)

Line Integrals and Green's Theorem

70

(6)

Independence of Path

76

(7)

Harmonic Conjugates

83

(2)

The Mean Value Property

85

(2)

The Maximum Principle

87

(3)

Applications to Fluid Dynamics

90

(7)

Other Applications to Physics

97

(5)

Complex Integration and Analyticity

102

(28)

Complex Line Integrals

102

(5)

Fundamental Theorem of Calculus for Analytic Functions

107

(3)

Cauchy's Theorem

110

(3)

The Cauchy Integral Formula

113

(4)

Liouville's Theorem

117

(2)

Morera's Theorem

119

(4)

Goursat's Theorem

123

(1)

Complex Notation and Pompeiu's Formula

124

(6)

Power Series

130

(35)

Infinite Series

130

(3)

Sequences and Series of Functions

133

(5)

Power Series

138

(6)

Power Series Expansion of an Analytic Function

144

(5)

Power Series Expansion at Infinity

149

(2)

Manipulation of Power Series

151

(3)

The Zeros of an Analytic Function

154

(4)

Analytic Continuation

158

(7)

Laurent Series and Isolated Singularities

165

(30)

The Laurent Decomposition

165

(6)

Isolated Singularities of an Analytic Function

171

(7)

Isolated Singularity at Infinity

178

(1)

Partial Fractions Decomposition

179

(3)

Periodic Functions

182

(4)

Fourier Series

186

(9)

The Residue Calculus

195

(29)

The Residue Theorem

195

(4)

Integrals Featuring Rational Functions

199

(4)

Integrals of Trigonometric Functions

203

(3)

Integrands with Branch Points

206

(3)

Fractional Residues

209

(3)

Principal Values

212

(4)

Jordan's Lemma

216

(3)

Exterior Domains

219

(5)

SECOND PART

The Logarithmic Integral

224

(36)

The Argument Principle

224

(5)

Rouche's Theorem

229

(2)

Hurwitz's Theorem

231

(1)

Open Mapping and Inverse Function Theorems

232

(4)

Critical Points

236

(6)

Winding Numbers

242

(4)

The Jump Theorem for Cauchy Integrals

246

(6)

Simply Connected Domains

252

(8)

The Schwarz Lemma and Hyperbolic Geometry

260

(14)

The Schwarz Lemma

260

(3)

Conformal Self-Maps of the Unit Disk

263

(3)

Hyperbolic Geometry

266

(8)

Harmonic Functions and the Reflection Principle

274

(15)

The Poisson Integral Formula

274

(6)

Characterization of Harmonic Functions

280

(2)

The Schwarz Reflection Principle

282

(7)

Conformal Mapping

289

(26)

Mappings to the Unit Disk and Upper Half-Plane

289

(5)

The Riemann Mapping Theorem

294

(2)

The Schwarz-Christoffel Formula

296

(8)

Return to Fluid Dynamics

304

(2)

Compactness of Families of Functions

306

(5)

Proof of the Riemann Mapping Theorem

311

(4)

THIRD PART

Compact Families of Meromorphic Functions

315

(27)

Marty's Theorem

315

(5)

Theorems of Montel and Picard

320

(4)

Julia Sets

324

(9)

Connectedness of Julia Sets

333

(5)

The Mandelbrot Set

338

(4)

Approximation Theorems

342

(19)

Runge's Theorem

342

(6)

The Mittag-Leffler Theorem

348

(4)

Infinite Products

352

(6)

The Weierstrass Product Theorem

358

(3)

Some Special Functions

361

(29)

The Gamma Functions

361

(4)

Laplace Transforms

365

(5)

The Zeta Function

370

(6)

Dirichlet Series

376

(6)

The Prime Number Theorem

382

(8)

The Dirichlet Problem

390

(28)

Green's Formulae

390

(4)

Subharmonic Functions

394

(4)

Compactness of Families of Harmonic Functions

398

(4)

The Perron Method

402

(4)

The Riemann Mapping Theorem Revisited

406

(1)

Green's Function for Domains with Analytic Boundary

407

(6)

Green's Function for General Domains

413

(5)

Riemann Surfaces

418

(29)

Abstract Riemann Surfaces

418

(8)

Harmonic Functions on a Riemann Surface

426

(3)

Green's Function of a Surface

429

(5)

Symmetry of Green's Function

434

(2)

Bipolar Green's Function

436

(2)

The Uniformization Theorem

438

(3)

Covering Surfaces

441

(6)

Hints and Solutions for Selected Exercises

447

(22)

References

469

(2)

List of Symbols

471

(2)

Index

473

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