# Complex Analysis

Springer Science & Business Media, Jul 17, 2003 - Mathematics - 478 pages
The 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 References 469 List of Symbols 471 Index 473 Copyright