| Preface | p. vii |
| Introduction | p. xvii |
| |
| The Complex Plane and Elementary Functions | p. 1 |
| Complex Numbers | p. 1 |
| Polar Representation | p. 5 |
| Stereographic Projection | p. 11 |
| The Square and Square Root Functions | p. 15 |
| The Exponential Function | p. 19 |
| The Logarithm Function | p. 21 |
| Power Functions and Phase Factors | p. 24 |
| Trigonometric and Hyperbolic Functions | p. 29 |
| Analytic Functions | p. 33 |
| Review of Basic Analysis | p. 33 |
| Analytic Functions | p. 42 |
| The Cauchy-Riemann Equations | p. 46 |
| Inverse Mappings and the Jacobian | p. 51 |
| Harmonic Functions | p. 54 |
| Conformal Mappings | p. 58 |
| Fractional Linear Transformations | p. 63 |
| Line Integrals and Harmonic Functions | p. 70 |
| Line Integrals and Green's Theorem | p. 70 |
| Independence of Path | p. 76 |
| Harmonic Conjugates | p. 83 |
| The Mean Value Property | p. 85 |
| The Maximum Principle | p. 87 |
| Applications to Fluid Dynamics | p. 90 |
| Other Applications to Physics | p. 97 |
| Complex Integration and Analyticity | p. 102 |
| Complex Line Integrals | p. 102 |
| Fundamental Theorem of Calculus for Analytic Functions | p. 107 |
| Cauchy's Theorem | p. 110 |
| The Cauchy Integral Formula | p. 113 |
| Liouville's Theorem | p. 117 |
| Morera's Theorem | p. 119 |
| Goursat's Theorem | p. 123 |
| Complex Notation and Pompeiu's Formula | p. 124 |
| Power Series | p. 130 |
| Infinite Series | p. 130 |
| Sequences and Series of Functions | p. 133 |
| Power Series | p. 138 |
| Power Series Expansion of an Analytic Function | p. 144 |
| Power Series Expansion at Infinity | p. 149 |
| Manipulation of Power Series | p. 151 |
| The Zeros of an Analytic Function | p. 154 |
| Analytic Continuation | p. 158 |
| Laurent Series and Isolated Singularities | p. 165 |
| The Laurent Decomposition | p. 165 |
| Isolated Singularities of an Analytic Function | p. 171 |
| Isolated Singularity at Infinity | p. 178 |
| Partial Fractions Decomposition | p. 179 |
| Periodic Functions | p. 182 |
| Fourier Series | p. 186 |
| The Residue Calculus | p. 195 |
| The Residue Theorem | p. 195 |
| Integrals Featuring Rational Functions | p. 199 |
| Integrals of Trigonometric Functions | p. 203 |
| Integrands with Branch Points | p. 206 |
| Fractional Residues | p. 209 |
| Principal Values | p. 212 |
| Jordan's Lemma | p. 216 |
| Exterior Domains | p. 219 |
| |
| The Logarithmic Integral | p. 224 |
| The Argument Principle | p. 224 |
| Rouché's Theorem | p. 229 |
| Hurwitz's Theorem | p. 231 |
| Open Mapping and Inverse Function Theorems | p. 232 |
| Critical Points | p. 236 |
| Winding Numbers | p. 242 |
| The Jump Theorem for Cauchy Integrals | p. 246 |
| Simply Connected Domains | p. 252 |
| The Schwarz Lemma and Hyperbolic Geometry | p. 260 |
| The Schwarz Lemma | p. 260 |
| Conformal Self-Maps of the Unit Disk | p. 263 |
| Hyperbolic Geometry | p. 266 |
| Harmonic Functions and the Reflection Principle | p. 274 |
| The Poisson Integral Formula | p. 274 |
| Characterization of Harmonic Functions | p. 280 |
| The Schwarz Reflection Principle | p. 282 |
| Conformal Mapping | p. 289 |
| Mappings to the Unit Disk and Upper Half-Plane | p. 289 |
| The Riemann Mapping Theorem | p. 294 |
| The Schwarz-Christoffel Formula | p. 296 |
| Return to Fluid Dynamics | p. 304 |
| Compactness of Families of Functions | p. 306 |
| Proof of the Riemann Mapping Theorem | p. 311 |
| |
| Compact Families of Meromorphic Functions | p. 315 |
| Marty's Theorem | p. 315 |
| Theorems of Montel and Picard | p. 320 |
| Julia Sets | p. 324 |
| Connectedness of Julia Sets | p. 333 |
| The Mandelbrot Set | p. 338 |
| Approximation Theorems | p. 342 |
| Runge's Theorem | p. 342 |
| The Mittag-Leffler Theorem | p. 348 |
| Infinite Products | p. 352 |
| The Weierstrass Product Theorem | p. 358 |
| Some Special Functions | p. 361 |
| The Gamma Function | p. 361 |
| Laplace Transforms | p. 365 |
| The Zeta Function | p. 370 |
| Dirichlet Series | p. 376 |
| The Prime Number Theorem | p. 382 |
| The Dirichlet Problem | p. 390 |
| Green's Formulae | p. 390 |
| Subharmonic Functions | p. 394 |
| Compactness of Families of Harmonic Functions | p. 398 |
| The Perron Method | p. 402 |
| The Riemann Mapping Theorem Revisited | p. 406 |
| Green's Function for Domains with Analytic Boundary | p. 407 |
| Green's Function for General Domains | p. 413 |
| Riemann Surfaces | p. 418 |
| Abstract Riemann Surfaces | p. 418 |
| Harmonic Functions on a Riemann Surface | p. 426 |
| Green's Function of a Surface | p. 429 |
| Symmetry of Green's Function | p. 434 |
| Bipolar Green's Function | p. 436 |
| The Uniformization Theorem | p. 438 |
| Covering Surfaces | p. 441 |
| Hints and Solutions for Selected Exercises | p. 447 |
| References | p. 469 |
| List of Symbols | p. 471 |
| Index | p. 473 |
| Table of Contents provided by Publisher. All Rights Reserved. |
