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Preface | |

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Complex Numbers | |

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Sums and Products | |

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Basic Algebraic Properties | |

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Further Properties | |

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Vectors and Moduli | |

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Complex Conjugates | |

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Exponential Form | |

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Products and Powers in Exponential Form | |

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Arguments of Products and Quotients | |

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Roots of Complex Numbers | |

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Examples | |

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Regions in the Complex Plane | |

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Analytic Functions | |

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Functions of a Complex Variable | |

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Mappings | |

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Mappings by the Exponential Function | |

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Limits | |

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Theorems on Limits | |

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Limits Involving the Point at Infinity | |

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Continuity | |

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Derivatives | |

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Differentiation Formulas | |

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Cauchy-Riemann Equations | |

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Sufficient Conditions for Differentiability | |

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Polar Coordinates | |

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Analytic Functions | |

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Examples | |

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Harmonic Functions | |

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Uniquely Determined Analytic Functions | |

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Reflection Principle | |

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Elementary Functions | |

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The Exponential Function | |

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The Logarithmic Function | |

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Branches and Derivatives of Logarithms | |

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Some Identities Involving Logarithms | |

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Complex Exponents | |

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Trigonometric Functions | |

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Hyperbolic Functions | |

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Inverse Trigonometric and Hyperbolic Functions | |

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Integrals | |

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Derivatives of Functions w(t) | |

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Definite Integrals of Functions w(t) | |

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Contours | |

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Contour Integrals | |

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Some Examples | |

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Examples with Branch Cuts | |

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Upper Bounds for Moduli of Contour Integrals | |

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Antiderivatives | |

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Proof of the Theorem | |

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Cauchy-Goursat Theorem | |

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Proof of the Theorem | |

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Simply Connected Domains | |

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Multiply Connected Domains | |

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Cauchy Integral Formula | |

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An Extension of the Cauchy Integral Formula | |

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Some Consequences of the Extension | |

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Liouville's Theorem and the Fundamental Theorem of Algebra | |

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Maximum Modulus Principle | |

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Series | |

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Convergence of Sequences | |

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Convergence of Series | |

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Taylor Series | |

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Proof of Taylor's Theorem | |

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Examples | |

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Laurent Series | |

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Proof of Laurent's Theorem | |

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Examples | |

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Absolute and Uniform Convergence of Power Series | |

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Continuity of Sums of Power Series | |

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Integration and Differentiation of Power Series | |

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Uniqueness of Series Representations | |

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Multiplication and Division of Power Series | |

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Residues and Poles | |

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Isolated Singular Points | |

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Residues | |

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Cauchy's Residue Theorem | |

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Residue at Infinity | |

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The Three Types of Isolated Singular Points | |

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Residues at Poles | |

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Examples | |

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Zeros of Analytic Functions | |

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Zeros and Poles | |

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Behavior of Functions Near Isolated Singular Points | |

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Applications of Residues | |

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Evaluation of Improper Integrals | |

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Example | |

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Improper Integrals from Fourier Analysis | |

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Jordan's Lemma | |

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Indented Paths | |

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An Indentation Around a Branch Point | |

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Integration Along a Branch Cut | |

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Definite Integrals Involving Sines and Cosines | |

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Argument Principle | |

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Rouche's Theorem | |

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Inverse Laplace Transforms | |

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Examples | |

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Mapping by Elementary Functions | |

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Linear Transformations | |

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The Transformation w = 1/z | |

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Mappings by 1/z | |

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Linear Fractional Transformations | |

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An Implicit Form | |

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Mappings of the Upper Half Plane | |

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The Transformation w = sin z | |

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Mappings by z[superscript 2] and Branches of z[superscript 1/2] | |

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Square Roots of Polynomials | |

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Riemann Surfaces | |

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Surfaces for Related Functions | |

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Conformal Mapping | |

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Preservation of Angles | |

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Scale Factors | |

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Local Inverses | |

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Harmonic Conjugates | |

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Transformations of Harmonic Functions | |

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Transformations of Boundary Conditions | |

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Applications of Conformal Mapping | |

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Steady Temperatures | |

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Steady Temperatures in a Half Plane | |

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A Related Problem | |

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Temperatures in a Quadrant | |

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Electrostatic Potential | |

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Potential in a Cylindrical Space | |

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Two-Dimensional Fluid Flow | |

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The Stream Function | |

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Flows Around a Corner and Around a Cylinder | |

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The Schwarz-Christoffel Transformation | |

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Mapping the Real Axis Onto a Polygon | |

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Schwarz-Christoffel Transformation | |

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Triangles and Rectangles | |

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Degenerate Polygons | |

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Fluid Flow in a Channel Through a Slit | |

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Flow in a Channel With an Offset | |

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Electrostatic Potential About an Edge of a Conducting Plate | |

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Integral Formulas of the Poisson Type | |

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Poisson Integral Formula | |

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Dirichlet Problem for a Disk | |

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Related Boundary Value Problems | |

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Schwarz Integral Formula | |

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Dirichlet Problem for a Half Plane | |

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Neumann Problems | |

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Appendixes | |

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Bibliography | |

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Table of Transformations of Regions | |

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Index | |