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Preface | |

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Symbols and Terms | |

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Preliminaries | |

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Preview | |

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It Takes Two Harmonic Functions | |

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Heat Flow | |

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A Geometric Rule | |

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Electrostatics | |

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Fluid Flow | |

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One Model, Many Applications | |

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Exercises | |

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Sets, Functions, and Visualization | |

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Terminology and Notation for Sets | |

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Terminology and Notation for Functions | |

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Functions from R to R | |

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Functions from R[superscript 2] to R | |

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Functions from R[superscript 2] to R[superscript 2] | |

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Exercises | |

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Structures on R[superscript 2], and Linear Maps from R[superscript 2] to R[superscript 2] | |

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The Real Line and the Plane | |

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Polar Coordinates in the Plane | |

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When Is a Mapping M: R[superscript 2] [right arrow] R[superscript 2] Linear? | |

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Visualizing Nonsingular Linear Mappings | |

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The Determinant of a Two-by-Two Matrix | |

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Pure Magnifications, Rotations, and Conjugation | |

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Conformal Linear Mappings | |

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Exercises | |

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Open Sets, Open Mappings, Connected Sets | |

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Distance, Interior, Boundary, Openness | |

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Continuity in Terms of Open Sets | |

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Open Mappings | |

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Connected Sets | |

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Exercises | |

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A Review of Some Calculus | |

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Integration Theory for Real-Valued Functions | |

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Improper Integrals, Principal Values | |

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Partial Derivatives | |

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Divergence and Curl | |

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Exercises | |

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Harmonic Functions | |

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The Geometry of Laplace's Equation | |

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The Geometry of the Cauchy-Riemann Equations | |

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The Mean Value Property | |

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Changing Variables in a Dirichlet or Neumann Problem | |

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Exercises | |

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Basic Tools | |

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The Complex Plane | |

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The Definition of a Field | |

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Complex Multiplication | |

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Powers and Roots | |

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Conjugation | |

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Quotients of Complex Numbers | |

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When Is a Mapping L : C [right arrow] C Linear? | |

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Complex Equations for Lines and Circles | |

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The Reciprocal Map, and Reflection in the Unit Circle | |

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Reflections in Lines and Circles | |

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Exercises | |

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Visualizing Powers, Exponential, Logarithm, and Sine | |

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Powers of z | |

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Exponential and Logarithms | |

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Sin z | |

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The Cosine and Sine, and the Hyperbolic Cosine and Sine | |

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Exercises | |

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Differentiability | |

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Differentiability at a Point | |

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Differentiability in the Complex Sense: Holomorphy | |

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Finding Derivatives | |

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Picturing the Local Behavior of Holomorphic Mappings | |

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Exercises | |

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Sequences, Compactness, Convergence | |

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Sequences of Complex Numbers | |

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The Limit Superior of a Sequence of Reals | |

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Implications of Compactness | |

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Sequences of Functions | |

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Exercises | |

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Integrals Over Curves, Paths, and Contours | |

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Integrals of Complex-Valued Functions | |

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Curves | |

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Paths | |

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Pathwise Connected Sets | |

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Independence of Path and Morera's Theorem | |

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Goursat's Lemma | |

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The Winding Number | |

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Green's Theorem | |

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Irrotational and Incompressible Fluid Flow | |

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Contours | |

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Exercises | |

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Power Series | |

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Infinite Series | |

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The Geometric Series | |

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An Improved Root Test | |

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Power Series and the Cauchy-Hadamard Theorem | |

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Uniqueness of the Power Series Representation | |

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Integrals That Give Rise to Power Series | |

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Exercises | |

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The Cauchy Theory | |

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Fundamental Properties of Holomorphic Functions | |

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Integral and Series Representations | |

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Eight Ways to Say "Holomorphic" | |

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Determinism | |

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Liouville's Theorem | |

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The Fundamental Theorem of Algebra | |

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Subuniform Convergence Preserves Holomorphy | |

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Exercises | |

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Cauchy's Theorem | |

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Cerny's 1976 Proof | |

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Simply Connected Sets | |

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Subuniform Boundedness, Subuniform Convergence | |

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Isolated Singularities | |

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The Laurent Series Representation on an Annulus | |

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Behavior Near an Isolated Singularity in the Plane | |

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Examples: Classifying Singularities, Finding Residues | |

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Behavior Near a Singularity at Infinity | |

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A Digression: Picard's Great Theorem | |

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Exercises | |

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The Residue Theorem and the Argument Principle | |

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Meromorphic Functions and the Extended Plane | |

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The Residue Theorem | |

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Multiplicity and Valence | |

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Valence for a Rational Function | |

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The Argument Principle: Integrals That Count | |

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Exercises | |

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Mapping Properties | |

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Exercises | |

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The Riemann Sphere | |

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Exercises | |

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The Residue Calculus | |

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Integrals of Trigonometric Functions | |

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Exercises | |

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Estimating Complex Integrals | |

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Exercises | |

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Integrals of Rational Functions Over the Line | |

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Exercises | |

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Integrals Involving the Exponential | |

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Integrals Giving Fourier Transforms | |

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Exercises | |

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Integrals Involving a Logarithm | |

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Exercises | |

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Integration on a Riemann Surface | |

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Mellin Transforms | |

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Exercises | |

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The Inverse Laplace Transform | |

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Exercises | |

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Boundary Value Problems | |

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Examples | |

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Easy Problems | |

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The Conformal Mapping Method | |

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Exercises | |

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The Mobius Maps | |

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Exercises | |

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Electric Fields | |

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A Point Charge in 3-Space | |

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Uniform Charge on One or More Long Wires | |

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Examples with Bounded Potentials | |

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Exercises | |

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Steady Flow of a Perfect Fluid | |

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Exercises | |

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Using the Poisson Integral to Obtain Solutions | |

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The Poisson Integral on a Disk | |

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Solutions on the Disk by the Poisson Integral | |

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Geometry of the Poisson Integral | |

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Harmonic Functions and the Mean Value Property | |

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The Neumann Problem on a Disk | |

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The Poisson Integral on a Half-Plane, and on Other Domains | |

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Exercises | |

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When Is the Solution Unique? | |

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Exercises | |

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The Schwarz Reflection Principle | |

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Schwarz-Christoffel Formulas | |

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Triangles | |

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Rectangles and Other Polygons | |

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Generalized Polygons | |

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Exercises | |

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Lagniappe | |

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Dixon's 1971 Proof of Cauchy's Theorem | |

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Runge's Theorem | |

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Exercises | |

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The Riemann Mapping Theorem | |

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Exercises | |

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The Osgood-Taylor-Caratheodory Theorem | |

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References | |

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Index | |