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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 | |