1 Complex Numbers 1
Sums and Products 1
Basic Algebraic Properties 3
Further Properties 5
Vectors and Moduli 9
Complex Conjugates 13
Exponential Form 16
Products and Powers in Exponential Form 18
Arguments of Products and Quotients 20
Roots of Complex Numbers 24
Examples 27
Regions in the Complex Plane 31
2 Analytic Functions 35
Functions of a Complex Variable 35
Mappings 38
Mappings by the Exponential Function 42
Limits 45
Theorems on Limits 48
Limits Involving the Point at Infinity 50
Continuity 53
Derivatives 56
Differentiation Formulas 60
Cauchy-Riemann Equations 63
Sufficient Conditions for Differentiability 66
Polar Coordinates 68
Analytic Functions 73
Examples 75
Harmonic Functions 78
Uniquely Determined Analytic Functions 83
Reflection Principle 85
3 Elementary Functions 89
The Exponential Function 89
The Logarithmic Function 93
Branches and Derivatives of Logarithms 95
Some Identities Involving Logarithms 98
Complex Exponents 101
Trigonometric Functions 104
Hyperbolic Functions 109
Inverse Trigonometric and Hyperbolic Functions 112
4 Integrals 117
Derivatives of Functions w（t） 117
Definite Integrals of Functions w（t） 119
Contours 122
Contour Integrals 127
Some Examples 129
Examples with Branch Cuts 133
Upper Bounds for Moduli of Contour Integrals 137
Antiderivatives 142
Proof of the Theorem 146
Cauchy-Goursat Theorem 150
Proof of the Theorem 152
Simply Connected Domains 156
Multiply Connected Domains 158
Cauchy Integral Formula 164
An Extension of the Cauchy Integral Formula 165
Some Consequences of the Extension 168
Liouville’s Theorem and the Fundamental Theorem of Algebra 172
Maximum Modulus Principle 175
5 Series 181
Convergence of Sequences 181
Convergence of Series 184
Taylor Series 189
Proof of Taylor’s Theorem 190
Examples 192
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 217
Multiplication and Division of Power Series 222
6 Residues and Poles 229
Isolated Singular Points 229
Residues 231
Cauchy’s Residue Theorem 234
Residue at Infinity 237
The Three Types of Isolated Singular Points 240
Residues at Poles 244
Examples 245
Zeros of Analytic Functions 249
Zeros and Poles 252
Behavior of Functions Near Isolated Singular Points 257
7 Applications of Residues 261
Evaluation of Improper Integrals 261
Example 264
Improper Integrals from Fourier Analysis 269
Jordan’s Lemma 272
Indented Paths 277
An Indentation Around a Branch Point 280
Integration Along a Branch Cut 283
Definite Integrals Involving Sines and Cosines 288
Argument Principle 291
Rouché’s Theorem 294
Inverse Laplace Transforms 298
Examples 301
8 Mapping by Elementary Functions 311
Linear Transformations 311
The Transformation w＝1/z 313
Mappings by 1/z 315
Linear Fractional Transformations 319
An Implicit Form 322
Mappings of the Upper Half Plane 325
The Transformation w＝sin z 330
Mappings by z2 and Branches of z1/ 2336
Square Roots of Polynomials 341
Riemann Surfaces 347
Surfaces for Related Functions 351
9 Conformal Mapping 355
Preservation of Angles 355
Scale Factors 358
Local Inverses 360
Harmonic Conjugates 363
Transformations of Harmonic Functions 365
Transformations of Boundary Conditions 367
10 Applications of Conformal Mapping 373
Steady Temperatures 373
Steady Temperatures in a Half Plane 375
A Related Problem 377
Temperatures in a Quadrant 379
Electrostatic Potential 385
Potential in a Cylindrical Space 386
Two-Dimensional Fluid Flow 391
The Stream Function 393
Flows Around a Corner and Around a Cylinder 395
11 The Schwarz-Christoffel Transformation 403
Mapping the Real Axis Onto a Polygon 403
Schwarz-Christoffel Transformation 405
Triangles and Rectangles 408
Degenerate Polygons 413
Fluid Flow in a Channel Through a Slit 417
Flow in a Channel With an Offset 420
Electrostatic Potential About an Edge of a Conducting Plate 422
12 Integral Formulas of the Poisson Type 429
Poisson Integral Formula 429
Dirichlet Problem for a Disk 432
Related Boundary Value Problems 437
Schwarz Integral Formula 440
Dirichlet Problem for a Half Plane 441
Neumann Problems 445
Appendixes 449
Bibliography 449
Table of Transformations of Regions 452
Index 461