COMPLEX ANALYSIS<br>Theodore W. Gamelin<br><br>Contents<br>Preface vii Introduction xvii<br>FIRST PART<br>Chapter I The Complex Plane and Elementary Functions<br>1. Complex Numbers 1<br>2. Polar Representation 5<br>3. Stereographic Projection 11<br>4. The Square and Square Root Functions 15<br>5. The Exponential Function 19<br>6. The Logarithm Function 21<br>7. Power Functions and Phase Factors 24<br>8. Trigonometric and Hyperbolic Functions 29<br>Chapter II Analytic Functions 33<br>1. Review of Basic Analysis 33<br>2. Analytic Functions 42<br>3. The Cauchy-Riemann Equations 46<br>4. Inverse Mappings and the Jacobian 51<br>5. Harmonic Functions 54<br>6. Conformal Mappings 58<br>7. Fractional Linear Transformations 63<br>Chapter III Line Integrals and Harmonic Functions 70<br>1. Line Integrals and Green’s Theorem 70<br>2. Independence of Path 76<br>3. Harmonic Conjugates 83<br>4. The Mean Value Property 85<br>5. The Maximum Principle 87<br>6. Applications to Fluid Dynamics 90<br>7. Other Applications to Physics 97<br>Chapter IV Complex Integration and Analyticity 102<br>1. Complex Line Integrals 102<br>2. Fundamental Theorem of Calculus for Analytic Functions 107<br>3. Cauchy’s Theorem 110<br>4. The Cauchy Integral Formula 113<br>5. Liouville’s Theorem 117<br>6. Morera’s Theorem 119<br>7. Goursat’s Theorem 123<br>8. Complex Notation and Pompeiu’s Formula 124<br>Chapter V Power Series 130<br>1. Infinite Series 130<br>2. Sequences and Series of Functions 133<br>3. Power Series 138<br>4. Power Series Expansion of an Analytic Function 144<br>5. Power Series Expansion at Infinity 149<br>6. Manipulation of Power Series 151<br>7. The Zeros of an Analytic Function 154<br>8. Analytic Continuation 158<br>Chapter VI Laurent Series and Isolated Singularities 165<br>1. The Laurent Decomposition 165<br>2. Isolated Singularities of an Analytic Function 171<br>3. Isolated Singularity at Infinity 178<br>4. Partial Fractions Decomposition 179<br>5. Periodic Functions 182<br>6. Fourier Series 186<br>Chapter VII The Residue Calculus 195<br>1. The Residue Theorem 195<br>2. Integrals Featuring Rational Functions 199<br>3. Integrals of Trigonometric Functions 203<br>4. Integrands with Branch Points 206<br>5. Fractional Residues 209<br>6. Principal Values 212<br>7. Jordan’s Lemma 216<br>8. Exterior Domains 219<br>SECOND PART<br>Chapter VIII The Logarithmic Integral 224<br>1. The Argument Principle 224<br>2. Rouche’s Theorem 229<br>3. Hurwitz’s Theorem 231<br>4. Open Mapping and Inverse Function Theorems 232<br>5. Critical Points 236<br>6. Winding Numbers 242<br>7. The Jump Theorem for Cauchy Integrals 246<br>8. Simply Connected Domains 252<br>Chapter IX The Schwarz Lemma and Hyperbolic Geometry 260<br>1. The Schwarz Lemma 260<br>2. Conformal Self-Maps of the Unit Disk 263<br>3. Hyperbolic Geometry 266<br>Chapter X Harmonic Functions and the Reflection Principle 274<br>1. The Poisson Integral Formula 274<br>2. Characterization of Harmonic Functions 280<br>3. The Schwarz Reflection Principle 282<br>Chapter XI Conformal Mapping 289<br>1. Mappings to the Unit Disk and Upper Half-Plane 289<br>2. The Riemann Mapping Theorem 294<br>3. The Schwarz-Christoffel Formula 296<br>4. Return to Fluid Dynamics 304<br>5. Compactness of Families of Functions 306<br>6. Proof of the Riemann Mapping Theorem 311<br>THIRD PART<br>Chapter XII Compact Families of Meromorphic Functions 315<br>1. Marty’s Theorem 315<br>2. Theorems of Montel and Picard 320<br>3. Julia Sets 324<br>4. Connectedness of Julia Sets 333<br>5. The Mandelbrot Set 338<br>Chapter XIII Approximation Theorems 342<br>1. Runge’s Theorem 342<br>2. The Mittag-Leffler Theorem 348<br>3. Infinite Products 352<br>4. The Weierstrass Product Theorem 358<br>Chapter XIV Some Special Functions 361<br>1. The Gamma Function 361<br>2. Laplace Transforms 365<br>3. The Zeta Function 370<br>4. Dirichlet Series 376<br>5. The Prime Number Theorem 382<br>Chapter XV The Dirichlet Problem 390<br>1. Green’s Formulae 390<br>2. Subharmonic Functions 394<br>3. Compactness of Families of Harmonic Functions 398<br>4. The Perron Method 402<br>5. The Riemann Mapping Theorem Revisited 406<br>6. Green’s Function for Domains with Analytic Boundary 407<br>7. Green’s Function for General Domains 413<br>Chapter XVI Riemann Surfaces 418<br>1. Abstract Riemann Surfaces 418<br>2. Harmonic Functions on a Riemann Surface 426<br>3. Green’s Function of a Surface 429<br>4. Symmetry of Green’s Function 434<br>5. Bipolar Green’s Function 436<br>6. The Uniformization Theorem 438<br>7. Covering Surfaces 441<br>Hints and Solutions for Selected Exercises 447 References 469 List of Symbols 471 Index 473<br>