1 Complex Numbers 1Sums and Products 1Basic Algebraic Properties 3Further Algebraic Properties 5Vectors and Moduli 8Triangle Inequality 11Complex Conjugates 14Exponential Form 17Products and Powers in Exponential Form 20Arguments of Products and Quotients 21Roots of Complex Numbers 25Examples 28Regions in the Complex Plane 322 Analytic Functions 37Functions and Mappings 37The Mapping w = z2 40Limits 44Theorems on Limits 47Limits Involving the Point at Infinity 50Continuity 52Derivatives 55Rules for Differentiation 59Cauchy-Riemann Equations 62Examples 64Sufficient Conditions for Differentiability 65Polar Coordinates 68Analytic Functions 72Further Examples 74Harmonic Functions 77Uniquely Determined Analytic Functions 80Reflection Principle 823 Elementary Functions 87The Exponential Function 87The Logarithmic Function 90Examples 92Branches and Derivatives of Logarithms 93Some Identities Involving Logarithms 97The Power Function 100Examples 101The Trigonometric Functions sin z and cos z 103Zeros and Singularities of Trigonometric Functions 105Hyperbolic Functions 109Inverse Trigonometric and Hyperbolic Functions 1124 Integrals 115Derivatives of Functions w(t) 115Definite Integrals of Functions w(t) 117Contours 120Contour Integrals 125Some Examples 127Examples Involving Branch Cuts 131Upper Bounds for Moduli of Contour Integrals 135Antiderivatives 140Proof of the Theorem 144Cauchy-Goursat Theorem 148Proof of the Theorem 150Simply Connected Domains 154Multiply Connected Domains 156Cauchy Integral Formula 162An Extension of the Cauchy Integral Formula 164Verification of the Extension 166Some Consequences of the Extension 168Liouville's Theorem and the Fundamental Theorem of Algebra 172Maximum Modulus Principle 1735 Series 179Convergence of Sequences 179Convergence of Series 182Taylor Series 186Proof of Taylor's Theorem 187Examples 189Negative Powers of (z − z0) 193Laurent Series 197Proof of Laurent's Theorem 199Examples 202Absolute and Uniform Convergence of Power Series 208Continuity of Sums of Power Series 211Integration and Differentiation of Power Series 213Uniqueness of Series Representations 216Multiplication and Division of Power Series 2216 Residues and Poles 227Isolated Singular Points 227Residues 229Cauchy's Residue Theorem 233Residue at Infinity 235The Three Types of Isolated Singular Points 238Examples 240Residues at Poles 242Examples 244Zeros of Analytic Functions 248Zeros and Poles 251Behavior of Functions Near Isolated Singular Points 2557 Applications of Residues 259Evaluation of Improper Integrals 259Example 262Improper Integrals from Fourier Analysis 267Jordan's Lemma 269An Indented Path 274An Indentation Around a Branch Point 277Integration Along a Branch Cut 280Definite Integrals Involving Sines and Cosines 284Argument Principle 287Rouche's Theorem 290Inverse Laplace Transforms 2948 Mapping by Elementary Functions 299Linear Transformations 299The Transformation w = 1/z 301Mappings by 1/z 303Linear Fractional Transformations 307An Implicit Form 310Mappings of the Upper Half Plane 313Examples 315Mappings by the Exponential Function 318Mapping Vertical Line Segments by w = sin z 320Mapping Horizontal Line Segments by w = sin z 322Some Related Mappings 324Mappings by z2 326Mappings by Branches of z1/2 328Square Roots of Polynomials 332Riemann Surfaces 338Surfaces for Related Functions 3419 Conformal Mapping 345Preservation of Angles and Scale Factors 345Further Examples 348Local Inverses 350Harmonic Conjugates 354Transformations of Harmonic Functions 357Transformations of Boundary Conditions 36010 Applications of Conformal Mapping 365Steady Temperatures 365Steady Temperatures in a Half Plane 367A Related Problem 369Temperatures in a Quadrant 371Electrostatic Potential 376Examples 377Two-Dimensional Fluid Flow 382The Stream Function 384Flows Around a Corner and Around a Cylinder 38611 The Schwarz-Christoffel Transformation 393Mapping the Real Axis onto a Polygon 393Schwarz-Christoffel Transformation 395Triangles and Rectangles 399Degenerate Polygons 402Fluid Flow in a Channel through a Slit 407Flow in a Channel with an Offset 409Electrostatic Potential about an Edge of a Conducting Plate 41212 Integral Formulas of the Poisson Type 417Poisson Integral Formula 417Dirichlet Problem for a Disk 420Examples 422Related Boundary Value Problems 426Schwarz Integral Formula 428Dirichlet Problem for a Half Plane 430Neumann Problems 433Appendixes 437Bibliography 437Table of Transformations of Regions 441Index 451