## Complex AnalysisThe book provides an introduction to complex analysis for students with some familiarity with complex numbers from high school. The book consists of three parts. The first part comprises the basic core of a course in complex analysis for junior and senior undergraduates. The second part includes various more specialized topics as the argument principle, the Schwarz lemma and hyperbolic geometry, the Poisson integral, and the Riemann mapping theorem. The third part consists of a selection of topics designed to complete the coverage of all background necessary for passing PhD qualifying exams in complex analysis. Topics selected include Julia sets and the Mandelbrot set, Dirichlet series and the prime number theorem, and the uniformization theorem for Riemann surfaces. The three geometries, spherical, euclidean, and hyperbolic, are stressed. Exercises range from the very simple to the quite challenging, in all chapters. The book is based on lectures given over the years by the author at several places, particularly the Interuniversity Summer School at Perugia (Italy), and also UCLA, Brown University, Valencia (Spain), and La Plata (Argentina). A native of Minnesota, the author did his undergraduate work at Yale University and his graduate work at UC Berkeley. After spending some time at MIT and at the Universidad Nacional de La Plata (Argentina), he joined the faculty at UCLA in 1968. The author has published a number of research articles and several books on functional analysis and analytic function theory. he is currently involved in the California K-12 education scene. |

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### Contents

The Complex Plane and Elementary Functions | 1 |

2 Polar Representation | 5 |

3 Stereographic Projection | 11 |

4 The Square and Square Root Functions | 15 |

5 The Exponential Function | 19 |

6 The Logarithm Function | 21 |

7 Power Functions and Phase Factors | 24 |

8 Trigonometric and Hyperbolic Functions | 29 |

2 Rouches Theorem | 229 |

3 Hurwitzs Theorem | 231 |

4 Open Mapping and Inverse Function Theorems | 232 |

5 Critical Points | 236 |

6 Winding Numbers | 242 |

7 The Jump Theorem for Cauchy Integrals | 246 |

8 Simply Connected Domains | 252 |

The Schwarz Lemma and Hyperbolic Geometry | 260 |

Analytic Functions | 33 |

2 Analytic Functions | 42 |

3 The CauchyRiemann Equations | 46 |

4 Inverse Mappings and the Jacobian | 51 |

5 Harmonic Functions | 54 |

6 Conformal Mappings | 58 |

7 Fractional Linear Transformations | 63 |

Line Integrals and Harmonic Functions | 70 |

2 Independence of Path | 76 |

3 Harmonic Conjugates | 83 |

4 The Mean Value Property | 85 |

5 The Maximum Principle | 87 |

6 Applications to Fluid Dynamics | 90 |

7 Other Applications to Physics | 97 |

Complex Integration and Analyticity | 102 |

2 Fundamental Theorem of Calculus for Analytic Functions | 107 |

3 Cauchys Theorem | 110 |

4 The Cauchy Integral Formula | 113 |

5 Liouvilles Theorem | 117 |

6 Moreras Theorem | 119 |

7 Goursats Theorem | 123 |

8 Complex Notation and Pompeius Formula | 124 |

Power Series | 130 |

2 Sequences and Series of Functions | 133 |

3 Power Series | 138 |

4 Power Series Expansion of an Analytic Function | 144 |

5 Power Series Expansion at Infinity | 149 |

6 Manipulation of Power Series | 151 |

7 The Zeros of an Analytic Function | 154 |

8 Analytic Continuation | 158 |

Laurent Series and Isolated Singularities | 165 |

2 Isolated Singularities of an Analytic Function | 171 |

3 Isolated Singularity at Infinity | 178 |

4 Partial Fractions Decomposition | 179 |

5 Periodic Functions | 182 |

6 Fourier Series | 186 |

The Residue Calculus | 195 |

2 Integrals Featuring Rational Functions | 199 |

3 Integrals of Trigonometric Functions | 203 |

4 Integrands with Branch Points | 206 |

5 Fractional Residues | 209 |

6 Principal Values | 212 |

7 Jordans Lemma | 216 |

8 Exterior Domains | 219 |

The Logarithmic Integral | 224 |

2 Conformal SelfMaps of the Unit Disk | 263 |

3 Hyperbolic Geometry | 266 |

Harmonic Functions and the Reflection Principle | 274 |

2 Characterization of Harmonic Functions | 280 |

3 The Schwarz Reflection Principle | 282 |

Conformal Mapping | 289 |

2 The Riemann Mapping Theorem | 294 |

3 The SchwarzChristoffel Formula | 296 |

4 Return to Fluid Dynamics | 304 |

5 Compactness of Families of Functions | 306 |

6 Proof of the Riemann Mapping Theorem | 311 |

Compact Families of Meromorphic Functions | 315 |

2 Theorems of Montel and Picard | 320 |

3 Julia Sets | 324 |

4 Connectedness of Julia Sets | 333 |

5 The Mandelbrot Set | 338 |

Approximation Theorems | 342 |

2 The MittagLeffler Theorem | 348 |

3 Infinite Products | 352 |

4 The Weierstrass Product Theorem | 358 |

Some Special Functions | 361 |

2 Laplace Transforms | 365 |

3 The Zeta Function | 370 |

4 Dirichlet Series | 376 |

5 The Prime Number Theorem | 382 |

The Dirichlet Problem | 390 |

2 Subharmonic Functions | 394 |

3 Compactness of Families of Harmonic Functions | 398 |

4 The Perron Method | 402 |

5 The Riemann Mapping Theorem Revisited | 406 |

6 Greens Function for Domains with Analytic Boundary | 407 |

7 Greens Function for General Domains | 413 |

Riemann Surfaces | 418 |

2 Harmonic Functions on a Riemann Surface | 426 |

3 Greens Function of a Surface | 429 |

4 Symmetry of Greens Function | 434 |

5 Bipolar Greens Function | 436 |

6 The Uniformization Theorem | 438 |

7 Covering Surfaces | 441 |

Hints and Solutions for Selected Exercises | 447 |

469 | |

List of Symbols | 471 |

473 | |

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### Common terms and phrases

analytic function annulus argz boundary bounded domain branch Cauchy integral Cauchy-Riemann equations closed curve closed path coefficients compact subset complex numbers component conformal map conformal self-map constant continuous function converges normally converges uniformly coordinate disk corresponding critical point decomposition defined derivative differentiable disk centered entire function estimate Example Exercise extended complex plane finite number fixed point fn(z Fourier series fractional linear transformation function f(z Green's function harmonic function Hence Hint hyperbolic identity imaginary axis interval inverse isolated singularity Julia set lemma maximum principle meromorphic function obtain open set open unit disk parameter piecewise smooth point ZQ polynomial power series expansion proof punctured radius of convergence rational function real axis Riemann mapping Riemann surface satisfies Section sector sequence series converges Show simple pole simply connected domain Suppose tends unique unit circle upper half-plane vertical zeros