* Contains over 100 sophisticated graphics to provide helpful examples and reinforce important concepts

O. Carruth McGehee is the author of An Introduction to Complex Analysis, published by Wiley.

Preface

xiii

Symbols and Terms

xix

Preliminaries

1

(82)

Preview

1

(17)

It Takes Two Harmonic Functions

3

(3)

Heat Flow

6

(3)

A Geometric Rule

9

(1)

Electrostatics

10

(3)

Fluid Flow

13

(1)

One Model, Many Applications

14

(1)

Exercises

15

(3)

Sets, Functions, and Visualization

18

(16)

Terminology and Notation for Sets

18

(2)

Terminology and Notation for Functions

20

(5)

Functions from R to R

25

(2)

Functions from R2 to R

27

(2)

Functions from R2 to R2

29

(1)

Exercises

30

(4)

Structures on R2, and Linear Maps from R2 to R2

34

(17)

The Real Line and the Plane

34

(2)

Polar Coordinates in the Plane

36

(2)

When Is a Mapping M: R2-->R2 Linear?

38

(2)

Visualizing Nonsingular Linear Mappings

40

(4)

The Determinant of a Two-by-Two Matrix

44

(1)

Pure Magnifications, Rotations, and Conjugation

45

(1)

Conformal Linear Mappings

46

(2)

Exercises

48

(3)

Open Sets, Open Mappings, Connected Sets

51

(10)

Distance, Interior, Boundary, Openness

51

(4)

Continuity in Terms of Open Sets

55

(1)

Open Mappings

56

(1)

Connected Sets

57

(1)

Exercises

58

(3)

A Review of Some Calculus

61

(10)

Integration Theory for Real-Valued Functions

61

(2)

Improper Integrals, Principal Values

63

(3)

Partial Derivatives

66

(2)

Divergence and Curl

68

(2)

Exercises

70

(1)

Harmonic Functions

71

(12)

The Geometry of Laplace's Equation

71

(1)

The Geometry of the Cauchy-Riemann Equations

72

(1)

The Mean Value Property

73

(3)

Changing Variables in a Dirichlet or Neumann Problem

76

(1)

Exercises

77

(6)

Basic Tools

83

(104)

The Complex Plane

83

(19)

The Definition of a Field

83

(1)

Complex Multiplication

84

(3)

Powers and Roots

87

(2)

Conjugation

89

(1)

Quotients of Complex Numbers

90

(1)

When Is a Mapping L: C-->C Linear?

91

(1)

Complex Equations for Lines and Circles

92

(1)

The Reciprocal Map, and Reflection in the Unit Circle

93

(3)

Reflections in Lines and Circles

96

(1)

Exercises

97

(5)

Visualizing Powers, Exponential, Logarithm, and Sine

102

(13)

Powers of z

103

(1)

Exponential and Logarithms

104

(2)

Sin z

106

(4)

The Cosine and Sine, and the Hyperbolic Cosine and Sine

110

(1)

Exercises

111

(4)

Differentiability

115

(13)

Differentiability at a Point

115

(4)

Differentiability in the Complex Sense: Holomorphy

119

(3)

Finding Derivatives

122

(2)

Picturing the Local Behavior of Holomorphic Mappings

124

(2)

Exercises

126

(2)

Sequences, Compactness, Convergence

128

(10)

Sequences of Complex Numbers

128

(3)

The Limit Superior of a Sequence of Reals

131

(2)

Implications of Compactness

133

(1)

Sequences of Functions

134

(1)

Exercises

135

(3)

Integrals Over Curves, Paths, and Contours

138

(28)

Integrals of Complex-Valued Functions

138

(1)

Curves

138

(6)

Paths

144

(3)

Pathwise Connected Sets

147

(1)

Independence of Path and Morera's Theorem

148

(2)

Goursat's Lemma

150

(3)

The Winding Number

153

(2)

Green's Theorem

155

(3)

Irrotational and Incompressible Fluid Flow

158

(3)

Contours

161

(1)

Exercises

162

(4)

Power Series

166

(21)

Infinite Series

166

(1)

The Geometric Series

167

(4)

An Improved Root Test

171

(1)

Power Series and the Cauchy-Hadamard Theorem

172

(2)

Uniqueness of the Power Series Representation

174

(4)

Integrals That Give Rise to Power Series

178

(2)

Exercises

180

(7)

The Cauchy Theory

187

(80)

Fundamental Properties of Holomorphic Functions

188

(16)

Integral and Series Representations

188

(5)

Eight Ways to Say ``Holomorphic''

193

(1)

Determinism

193

(3)

Liouville's Theorem

196

(1)

The Fundamental Theorem of Algebra

196

(1)

Subuniform Convergence Preserves Holomorphy

197

(1)

Exercises

198

(6)

Cauchy's Theorem

204

(8)

Cerny's 1976 Proof

205

(3)

Simply Connected Sets

208

(1)

Subuniform Boundedness, Subuniform Convergence

209

(3)

Isolated Singularities

212

(24)

The Laurent Series Representation on an Annulus

212

(4)

Behavior Near an Isolated Singularity in the Plane

216

(3)

Examples: Classifying Singularities, Finding Residues

219

(6)

Behavior Near a Singularity at Infinity

225

(4)

A Digression: Picard's Great Theorem

229

(1)

Exercises

229

(7)

The Residue Theorem and the Argument Principle

236

(15)

Meromorphic Functions and the Extended Plane

236

(3)

The Residue Theorem

239

(3)

Multiplicity and Valence

242

(1)

Valence for a Rational Function

243

(1)

The Argument Principle: Integrals That Count

243

(6)

Exercises

249

(2)

Mapping Properties

251

(9)

Exercises

259

(1)

The Riemann Sphere

260

(7)

Exercises

264

(3)

The Residue Calculus

267

(50)

Integrals of Trigonometric Functions

268

(5)

Exercises

270

(3)

Estimating Complex Integrals

273

(4)

Exercises

276

(1)

Integrals of Rational Functions Over the Line

277

(5)

Exercises

280

(2)

Integrals Involving the Exponential

282

(11)

Integrals Giving Fourier Transforms

286

(4)

Exercises

290

(3)

Integrals Involving a Logarithm

293

(9)

Exercises

301

(1)

Integration on a Riemann Surface

302

(7)

Mellin Transforms

306

(1)

Exercises

307

(2)

The Inverse Laplace Transform

309

(8)

Exercises

315

(2)

Boundary Value Problems

317

(76)

Examples

318

(9)

Easy Problems

318

(5)

The Conformal Mapping Method

323

(3)

Exercises

326

(1)

The Mobius Maps

327

(14)

Exercises

338

(3)

Electric Fields

341

(9)

A Point Charge in 3-Space

341

(1)

Uniform Charge on One or More Long Wires

342

(5)

Examples with Bounded Potentials

347

(3)

Exercises

350

(1)

Steady Flow of a Perfect Fluid

350

(5)

Exercises

354

(1)

Using the Poisson Integral to Obtain Solutions

355

(13)

The Poisson Integral on a Disk

355

(3)

Solutions on the Disk by the Poisson Integral

358

(3)

Geometry of the Poisson Integral

361

(2)

Harmonic Functions and the Mean Value Property

363

(1)

The Neumann Problem on a Disk

364

(1)

The Poisson Integral on a Half-Plane, and on Other Domains

365

(1)

Exercises

366

(2)

When Is the Solution Unique?

368

(2)

Exercises

370

(1)

The Schwarz Reflection Principle

370

(4)

Schwarz-Christoffel Formulas

374

(19)

Triangles

375

(10)

Rectangles and Other Polygons

385

(4)

Generalized Polygons

389

(1)

Exercises

390

(3)

Lagniappe

393

(20)

Dixon's 1971 Proof of Cauchy's Theorem

394

(4)

Runge's Theorem

398

(6)

Exercises

403

(1)

The Riemann Mapping Theorem

404

(2)

Exercises

405

(1)

The Osgood-Taylor-Caratheodory Theorem

406

(7)

References

413

(6)

Index

419

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An Introduction to Complex Analysis

9780471332336

047133233X

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