AN INTRODUCTION TO COMPLEX ANALYSIS

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1 AN INTRODUCTION TO COMPLEX ANALYSIS O. Carruth McGehee A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York Chichester Weinheim Brisbane Singapore Toronto

Contents Preface Symbols and Terms 1 Preliminaries 1.1 Preview A It Takes Two Harmonie Functions B Heat Flow C A Geometrie Rule D Electrostatics E Fluid Flow F One Model, Many Applications 1.2 1.

2 Contents Preface Symbols and Terms 1 Preliminaries 1.1 Preview A It Takes Two Harmonie Functions B Heat Flow C A Geometrie Rule D Electrostatics E Fluid Flow F One Model, Many Applications Sets, Functions, and Visualization A Terminology and Notation for Sets B Terminology and Notation for Functions C Functions from R to R D Functions from R 2 to R E Functions from R 2 to R 2 Structures on R 2 and Linear Mapsfrom R 2 to R 2 xiii xix vii

3 V/H A The Real Line and the Plane 34 B Polar Coordinates in the Plane 36 C When Is a Mapping M : R 2 -> R 2 Linear? 38 D Visualizing Nonsingular Linear Mappings 40 E The Determinant ofa Two-by-Two Matrix 44 F Pure Magnifications, Rotations, and Conjugation 45 G Conformal Linear Mappings Open Sets, Open Mappings, Connected Sets 51 A Distance, Interior, Boundary, Openness 51 B Continuity in Terms of Open Sets 55 C Open Mappings 56 D Connected Sets A Review ofsome Calculus 61 A Integration Theory for Real-Valued Functions 61 B Improper Integrals, Principal Values 63 C Partial Derivatives 66 D Divergence and Curl Harmonie Functions 71 A The Geometry oflaplace's Equation 71 B The Geometry of the Cauchy-Riemann Equations. 72 C The Mean Value Property 73 D Changing Variables in a Dirichlet or Neumann Problem Basic Tools The Complex Plane 83 A The Definition ofa Field 83 B Complex Multiplication 84 C Powers and Roots 87 D Conjugation 89 E Quotients of Complex Numbers 90 F When Is a Mapping Z,: C -> C Linear? 91 G Complex Equations for Lines and Circles 92

4 ix H The Reciprocal Map, and Reflection in the Unit Circle I Reflections in Lines and Circles Visualizing Powers, Exponential, Logarithm, and Sine A Powers o/z B Exponential and Logarithms C Sin z D The Cosine and Sine, and the Hyperbolic Cosine and Sine Differentiability A Differentiability at a Point B Differentiability in the Complex Sense: Holomorphy C Finding Derivatives D Picturing the Locol Behavior of Holomorphic Mappings Sequences, Compactness, Convergence A Sequences of Complex Numbers B The Limit Superior of a Sequence ofreals C Implications of Compactness D Sequences of Functions Integrals Over Curves, Paths, and Contours A Integrals of Complex-Valued Functions B Curves C Paths D Pathwise Connected Sets E Independence ofpath and Morera 's Theorem F Goursat's Lemma G The Winding Number H Green 's Theorem I Irrotational and lncompressible Fluid Flow J Contours Power Series A Infinite Series

5 X B The Geometrie Series 167 C An Improved Root Test 171 D Power Series and the Cauchy-Hadamard Theorem 172 E Uniqueness of the Power Series Representation 174 F Integrals That Give Rise to Power Series Zauchy Theory Fundamental Properties of Holomorphic Functions A Integral and Series Representations 188 B Eight Ways to Say "Holomorphic" 193 C Determinism 193 D Liouville's Theorem 196 E The Fundamental Theorem of Algebra 196 F Subuniform Convergence Preserves Holomorphy Cauchy's Theorem 204 A Cernfs 1976 Proof 205 B Simply Connected Sets 208 C Subuniform Boundedness, Subuniform Convergence 209 Isolated Singularities 212 A The Laurent Series Representation on an Annulus 212 B Behavior Near an Isolated Singularity in the Plane 216 C Examples: Classifying Singularities, Finding Residues 219 D Behavior Near a Singularity at Infinity 225 E A Digression: Picard's Great Theorem The Residue Theorem and the Argument Principle 236 A Meromorphic Functions and the Extended Plane 236 B The Residue Theorem 239 C Multiplicity and Valence 242 D Valence for a Rational Function 243

3.5 3.6 E The Argument Principle: Integrals That Count Mapping Properties The Riemann Sphere XI 243 249 251 259 260 264 4 The Residue Calculus 4.1 Integrals of Trigonometrie Functions 4.

6 E The Argument Principle: Integrals That Count Mapping Properties The Riemann Sphere XI The Residue Calculus 4.1 Integrals of Trigonometrie Functions 4.2 Estimating Complex Integrals 4.3 Integrals of Rational Functions Over the Line 4.4 Integrals Involving the Exponential A Integrals Giving Fourier Transforms 4.5 Integrals Involving a Logarithm 4.6 Integration on a Riemann Surface A Meilin Transforms 4.7 The Inverse Laplace Transform 5 Boundary Value Problems 5.1 Examples A Easy Problems B The Conformal Mapping Method 5.2 The Möbius Mops 5.3 Electric Fields A A Point Charge in 3-Space B Uniform Charge on One or More Long Wires C Examples with Bounded Potentials 5.4 Steady Flow of a Perfect Fluid

7 XÜ Using the Poisson Integral to Obtain Solutions 355 A The Poisson Integral on a Disk 355 B Solutions on the Disk by the Poisson Integral 358 C Geometry of the Poisson Integral 361 D Harmonie Functions and the Mean Value Property 363 E The Neumann Problem on a Disk 364 F The Poisson Integral on a Half-Plane, and on Other Domains When Is the Solution Unique? The Schwarz Reflection Principle Schwarz-Christoffel Formulas 374 A Triangles' 375 B Rectangles and Other Polygons 385 C Generalized Polygons Lagniappe Dixon's 1971 Proof of Cauchy 's Theorem Runge's Theorem The Riemann Mapping Theorem The Osgood-Taylor-Caratheodory Theorem 406 References 413 Index 419

An Introduction to Complex Function Theory

Bruce P. Palka An Introduction to Complex Function Theory With 138 luustrations Springer 1 Contents Preface vü I The Complex Number System 1 1 The Algebra and Geometry of Complex Numbers 1 1.1 The Field