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 Logarithms 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 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
The Transformation w = sin z 318
Mappings by z2 and Branches of z 1/2 324
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