1 Complex Numbers 1
Sums and Products 1
Basic Algebraic Properties 3
Further Algebraic Properties 5
Vectors and Moduli 8
Triangle Inequality 11
Complex Conjugates 14
Exponential Form 17
Products and Powers in Exponential Form 20
Arguments of Products and Quotients 21
Roots of Complex Numbers 25
Examples 28
Regions in the Complex Plane 32

2 Analytic Functions 37
Functions and Mappings 37
The Mapping w = z2 40
Limits 44
Theorems on Limits 47
Limits Involving the Point at Infinity 50
Continuity 52
Derivatives 55
Rules for Differentiation 59
Cauchy-Riemann Equations 62
Examples 64
Sufficient Conditions for Differentiability 65
Polar Coordinates 68
Analytic Functions 72
Further Examples 74
Harmonic Functions 77
Uniquely Determined Analytic Functions 80
Reflection Principle 82

3 Elementary Functions 87
The Exponential Function 87
The Logarithmic Function 90
Examples 92
Branches and Derivatives of Logarithms 93
Some Identities Involving Logarithms 97
The Power Function 100
Examples 101
The Trigonometric Functions sin z and cos z 103
Zeros and Singularities of Trigonometric Functions 105
Hyperbolic Functions 109
Inverse Trigonometric and Hyperbolic Functions 112

4 Integrals 115
Derivatives of Functions w(t) 115
Definite Integrals of Functions w(t) 117
Contours 120
Contour Integrals 125
Some Examples 127
Examples Involving Branch Cuts 131
Upper Bounds for Moduli of Contour Integrals 135
Antiderivatives 140
Proof of the Theorem 144
Cauchy-Goursat Theorem 148
Proof of the Theorem 150
Simply Connected Domains 154
Multiply Connected Domains 156
Cauchy Integral Formula 162
An Extension of the Cauchy Integral Formula 164
Verification of the Extension 166
Some Consequences of the Extension 168
Liouville's Theorem and the Fundamental Theorem of Algebra 172
Maximum Modulus Principle 173

5 Series 179
Convergence of Sequences 179
Convergence of Series 182
Taylor Series 186
Proof of Taylor's Theorem 187
Examples 189
Negative Powers of (z − z0) 193
Laurent Series 197
Proof of Laurent's Theorem 199
Examples 202
Absolute and Uniform Convergence of Power Series 208
Continuity of Sums of Power Series 211
Integration and Differentiation of Power Series 213
Uniqueness of Series Representations 216
Multiplication and Division of Power Series 221

6 Residues and Poles 227
Isolated Singular Points 227
Residues 229
Cauchy's Residue Theorem 233
Residue at Infinity 235
The Three Types of Isolated Singular Points 238
Examples 240
Residues at Poles 242
Examples 244
Zeros of Analytic Functions 248
Zeros and Poles 251
Behavior of Functions Near Isolated Singular Points 255

7 Applications of Residues 259
Evaluation of Improper Integrals 259
Example 262
Improper Integrals from Fourier Analysis 267
Jordan's Lemma 269
An Indented Path 274
An Indentation Around a Branch Point 277
Integration Along a Branch Cut 280
Definite Integrals Involving Sines and Cosines 284
Argument Principle 287
Rouche's Theorem 290
Inverse Laplace Transforms 294

8 Mapping by Elementary 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
Examples 315
Mappings by the Exponential Function 318
Mapping Vertical Line Segments by w = sin z 320
Mapping Horizontal Line Segments by w = sin z 322
Some Related Mappings 324
Mappings by z2 326
Mappings by Branches of z1/2 328
Square Roots of Polynomials 332
Riemann Surfaces 338
Surfaces for Related Functions 341

9 Conformal Mapping 345
Preservation of Angles and Scale Factors 345
Further Examples 348
Local Inverses 350
Harmonic Conjugates 354
Transformations of Harmonic Functions 357
Transformations of Boundary Conditions 360

10 Applications of Conformal Mapping 365
Steady Temperatures 365
Steady Temperatures in a Half Plane 367
A Related Problem 369
Temperatures in a Quadrant 371
Electrostatic Potential 376
Examples 377
Two-Dimensional Fluid Flow 382
The Stream Function 384
Flows Around a Corner and Around a Cylinder 386

11 The Schwarz-Christoffel Transformation 393
Mapping the Real Axis onto a Polygon 393
Schwarz-Christoffel Transformation 395
Triangles and Rectangles 399
Degenerate Polygons 402
Fluid Flow in a Channel through a Slit 407
Flow in a Channel with an Offset 409
Electrostatic Potential about an Edge of a Conducting Plate 412

12 Integral Formulas of the Poisson Type 417
Poisson Integral Formula 417
Dirichlet Problem for a Disk 420
Examples 422
Related Boundary Value Problems 426
Schwarz Integral Formula 428
Dirichlet Problem for a Half Plane 430
Neumann Problems 433

Appendixes 437
Bibliography 437
Table of Transformations of Regions 441
Index 451