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Preface Symbols and Terms Preliminaries 1.1 Preview A It Takes Two Harmonic Functions B Heat Flow C A Geometric Rule D Electrostatics E Fluid Flow F One Model Many Applications Exercises 1.2 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 R2 to R E Functions from R2 to R2 Exercises 1.3 Structures on R2, and Linear Maps from R2 to R2 A The Real Line and the Plane B Polar Coordinates in the Plane C When Is a Mapping M : R2 ¡ú R2 Linear? D Visualizing Nonsingular Linear Mappings E The Determinant of a Two-by-Two Matrix F Pure Magnifications, Rotations, and Conjugation G Conformal Linear Mappings Exercises 1.4 Open Sets, Open Mappings, Connected Sets A Distance, Interior, Boundary, Openness B Continuity in Terms of Open Sets C Open Mappings D Connected Sets Exercises 1.5 A Review of Some Calculus A Integration Theory for Real- Valued Functions B Improper Integrals, Principal Values C Partial Derivatives D Divergence and Curl Exercises 1.6 Harmonic Functions A The Geometry of Laplace's Equation B The Geometry of the Cauchy-Riemann Equations C The Mean Value Property D Changing Variables in a Dirichtet or Neumann Problem Exercises 2 Basic Tools 2.1 The Complex Plane A The Definition of a Field B Complex Multiplication C Powers and Roots D Conjugation E Quotients of Complex Numbers F When Is a Mapping L : C ¡ú C Linear? G Complex Equations for Lines and Circles H The Reciprocal Map, and Reflection in the Unit Circle I Reflections in Lines and Circles Exercises 2.2 Visualizing Powers, Exponential, Logarithm, and Sine A Powers ofz B Exponential and Logarithms C Sin z D The Cosine and Sine, and the Hyperbolic Cosine and Sine Exercises 2.3 Differentiability A Differentiability at a Point B Differentiability in the Complex Sense: Holomorphy C Finding Derivatives D Picturing the Local Behavior of Holomorphic Mappings Exercises 2.4 Sequences, Compactness, Convergence A Sequences of Complex Numbers B The Limit Superior of a Sequence of Reals C Implications of Compactness D Sequences of Functions Exercises 2.5 Integrals Over Curves, Paths, and Contours A Integrals of Complex-Valued Functions B Curves C Paths D Pathwise Connected Sets E Independence of Path and Morera's Theorem F Goursat's Lemma G The Winding Number H Green's Theorem I Irrotational and Incompressible Fluid Flow J Contours Exercises 2.6 Power Series A Infinite Series B The Geometric Series C An Improved Root Test D Power Series and the Cauchy-Hadamard Theorem E Uniqueness of the Power Series Representation F Integrals That Give Rise to Power Series Exercises 3 The Cauchy Theory 3.1 Fundamental Properties of Holomorphic Functions A Integral and Series Representations B Eight Ways to Say "'Holomorphic" C Determinism D Liouville's Theorem E The Fundamental Theorem of Algebra F Subuniform Convergence Preserves Holomorphy Exercises 3.2 Cauchy's Theorem A Cerny's 1976 Proof B Simply Connected Sets C Subuniform Boundedness, Subuniform Convergence 3.3 lsolated Singularities A The Laurent Series Representation on an Annulus B Behavior Near an Isolated Singularity in the Plane C Examples: Classifying Singularities, Finding Residues D Behavior Near a Singularity at Infinity E A Digression: Picard'sGreat Theorem Exercises 3.4 The Residue Theorem and the Argument Principle A Meromorphic Functions and the Extended Plane B The Residue Theorem C Multiplicity and Valence D Valence.for a Rational Function E The Argument Principle: Integrals That Count Exercises 3.5 Mapping Properties Exercises 3.6 The Riemann Sphere Exercises 4 The Residue Calculus 4.1 Integrals of Trigonometric Functions Over a Compact lnterval Exercises 4.2 Estimating Complex Integrals Exercises 4.3 Integrals of Rational Functions Over the Line Exercises 4.4 Integrals Involving the Exponential A Integrals Giving Fourier Transforms Exercises 4.5 Integrals Involving a Logarithm Exercises 4.6 Integration on a Riemann Surface A Mellin Transforms Exercises 4. 7 The Complex Inversion Formula for the Laplace Transform Exercises 5 Boundary Value Problems 5.1 Examples A Easy Problems B The Conformal Mapping Method Exercises 5.2 The Mobius Maps Exercises 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 Exercises 5.4 Steady Flow of a Perfect Fluid Exercises 5.5 Using the Poisson Integral to Obtain Solutions A The Poisson Integral on a Disk B Solutions on the Disk by the Poisson Integral C Geometry of the Poisson Integral D Harmonic Functions and the Mean Value Property E The Neumann Problem on a Disk F The Poisson Integral on a Half-Plane, and on Other Domains Exercises 5.6 When Is the Solution Unique? Exercises 5.7 The Schwarz Reflection Principle 5.8 Schwarz-Christoffel Formulas A Triangles B Rectangles and Other Polygons C Generalized Polygons Exercises 6 Lagniappe 6.1 Dixon's 1971ProofofCauchy's Theorem 6.2 Runge's Theorem Exercises 6.3 The Riemann Mapping Theorem Exercises 6.4 The Osgood-Taylor-Carath~odory Theorem References Index