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1. Complex Numbers 1

Sums and Products 1

Basic Algebraic Properties 3

Further Properties 5

Moduli 8

Complex Conjugates 11

Exponential Form 15

Products and Quotients in Exponential Form 17

Roots of Complex Numbers 22

Examples 25

Regions in the Complex Plane 29

2. Analytic Functions 33

Functions of a Complex Variable 33

Mappings 36

Mappings by the Exponential Function 40

Limits 43

Theorems on Limits 46

Limits Involving the Point at Infinity 48

Continuity 51

Derivatives 54

Differentiation Formulas 57

Cauchy-Riemann Equations 60

Sufficient Conditions for Differentiability 63

Polar Coordinates 65

Analytic Functions 70

Examples 72

Harmonic Functions 75

Uniquely Determined Analytic Functions 80

Reflection Principle 82

3. Elementary Functions 87

The Exponential Function 87

The Logarithmic Function 90

Branches and Derivatives of Logarithms 92

Some Identities Involving Logathms 95

Complex Exponents 97

Trigonometric Functions 100

Hyperbolic Functions 105

Inverse Trigonometric and Hyperbolic Functions 108

4. Integrals 111

Derivatives of Functions w（t） 111

Definite Integrals of Functions w（t） 113

Contours 116

Contour Integrals 122

Examples 124

Upper Bounds for Moduli of Contour Integrals 130

Antiderivatives 135

Examples 138

Cauchy-Goursat Theorem 142

Proof of the Theorem 144

Simply and Multiply Connected Domains 149

Cauchy Integral Formula 157

Derivatives of Analytic Functions 158

Liouville's Theorem and the Fundamental Theorem of Algebra 165

Maximum Modulus Principle 167

5. Series 175

Convergence of Sequences 175

Convergence of Series 178

Taylor Series 182

Examples 185

Laurent Series 190

Examples 195

Absolute and Uniform Convergence of Power Series 200

Continuity of Sums of Power Series 204

Integration and Differentiation of Power Series 206

Uniqueness of Series Representations 210

Multiplication and Division of Power Series 215

6. Residues and Poles 221

Residues 221

Cauchy's Residue Theorem 225

Using a Single Residue 227

The Three Types of Isolated Singular Points 231

Residues at Poles 234

Examples 236

Zeros of Analytic Functions 239

Zeros and Poles 242

Behavior off Near Isolated Singular Points 247

7. Applications of Residues 251

Evaluation of Improper Integrals 251

Example 254

Improper Integrals from Fourier Analysis 259

Jordan's Lemma 262

Indented Paths 267

An Indentation Around a Branch Point 270

Integration Along a Branch Cut 273

Definite Integrals Involving Sines and Cosines 278

Argument Principle 281

Rouche's Theorem 284

Inverse Laplace Transforms 288

Examples 291

8. Mapping by Elementa Functions 299

Linear Transformations 299

The Transformation w = 1/z 301

Mappings by 1/z 303

Linear Fractional Transformations 307

An Implicit Form 310

Mappings of the Upper Half Plane 313

The Transformation w = sin z 318

Mappings by z2 and Branches of z1/2 324

Square Roots of Polynomials 329

Riemann Surfaces 335

Surfaces for Related Functions 338

9. Conformal Mapping 343

Preservation of Angles 343

Scale Factors 346

Local Inverses 348

Harmonic Conjugates 351

Transformations of Harmonic Functions 353

Transformations of Boundary Conditions 355

10. Applications of Conformal Mapping 361

Steady Temperatures 361

Steady Temperatures in a Half Plane 363

A Related Problem 365

Temperatures in a Quadrant 368

Electrostatic Potential 373

Potential in a Cylindrical Space 374

Two-Dimensional Fluid Flow 379

The Stream Function 381

Flows Around a Corner and Around a Cylinder 383

11. The Schwarz-Christoffel Transformation 391

Mapping the Real Axis onto a Polygon 391

Schwarz-Christoffel Transformation 393

Triangles and Rectangles 397

Degenerate Polygons 401

Fluid Flow in a Channel Through a Slit 406

Flow in a Channel with an Offset 408

Electrostatic Potential about an Edge of a Conducting Plate 411

12. Integral Formulas of the Poisson Type 417

Poisson Integral Formula 417

Dirichlet Problem for a Disk 419

Related Boundary Value Problems 423

Schwarz Integral Formula 427

Dirichlet Problem for a Half Plane 429

Neumann Problems 433

Appendixes 437

Bibliography 437

Table of Transformations of Regions 441

Index 451