Preface | p. x |

Complex Numbers | p. 1 |

Sums and Products | p. 1 |

Basic Algebraic Properties | p. 3 |

Further Properties | p. 5 |

Vectors and Moduli | p. 9 |

Complex Conjugates | p. 13 |

Exponential Form | p. 16 |

Products and Powers in Exponential Form | p. 18 |

Arguments of Products and Quotients | p. 20 |

Roots of Complex Numbers | p. 24 |

Examples | p. 27 |

Regions in the Complex Plane | p. 31 |

Analytic Functions | p. 35 |

Functions of a Complex Variable | p. 35 |

Mappings | p. 38 |

Mappings by the Exponential Function | p. 42 |

Limits | p. 45 |

Theorems on Limits | p. 48 |

Limits Involving the Point at Infinity | p. 50 |

Continuity | p. 53 |

Derivatives | p. 56 |

Differentiation Formulas | p. 60 |

Cauchy-Riemann Equations | p. 63 |

Sufficient Conditions for Differentiability | p. 66 |

Polar Coordinates | p. 68 |

Analytic Functions | p. 73 |

Examples | p. 75 |

Harmonic Functions | p. 78 |

Uniquely Determined Analytic Functions | p. 83 |

Reflection Principle | p. 85 |

Elementary Functions | p. 89 |

The Exponential Function | p. 89 |

The Logarithmic Function | p. 93 |

Branches and Derivatives of Logarithms | p. 95 |

Some Identities Involving Logarithms | p. 98 |

Complex Exponents | p. 101 |

Trigonometric Functions | p. 104 |

Hyperbolic Functions | p. 109 |

Inverse Trigonometric and Hyperbolic Functions | p. 112 |

Integrals | p. 117 |

Derivatives of Functions w(t) | p. 117 |

Definite Integrals of Functions w(t) | p. 119 |

Contours | p. 122 |

Contour Integrals | p. 127 |

Some Examples | p. 129 |

Examples with Branch Cuts | p. 133 |

Upper Bounds for Moduli of Contour Integrals | p. 137 |

Antiderivatives | p. 142 |

Proof of the Theorem | p. 146 |

Cauchy-Goursat Theorem | p. 150 |

Proof of the Theorem | p. 152 |

Simply Connected Domains | p. 156 |

Multiply Connected Domains | p. 158 |

Cauchy Integral Formula | p. 164 |

An Extension of the Cauchy Integral Formula | p. 165 |

Some Consequences of the Extension | p. 168 |

Liouville's Theorem and the Fundamental Theorem of Algebra | p. 172 |

Maximum Modulus Principle | p. 175 |

Series | p. 181 |

Convergence of Sequences | p. 181 |

Convergence of Series | p. 184 |

Taylor Series | p. 189 |

Proof of Taylor's Theorem | p. 190 |

Examples | p. 192 |

Laurent Series | p. 197 |

Proof of Laurent's Theorem | p. 199 |

Examples | p. 202 |

Absolute and Uniform Convergence of Power Series | p. 208 |

Continuity of Sums of Power Series | p. 211 |

Integration and Differentiation of Power Series | p. 213 |

Uniqueness of Series Representations | p. 217 |

Multiplication and Division of Power Series | p. 222 |

Residues and Poles | p. 229 |

Isolated Singular Points | p. 229 |

Residues | p. 231 |

Cauchy's Residue Theorem | p. 234 |

Residue at Infinity | p. 237 |

The Three Types of Isolated Singular Points | p. 240 |

Residues at Poles | p. 244 |

Examples | p. 245 |

Zeros of Analytic Functions | p. 249 |

Zeros and Poles | p. 252 |

Behavior of Functions Near Isolated Singular Points | p. 257 |

Applications of Residues | p. 261 |

Evaluation of Improper Integrals | p. 261 |

Example | p. 264 |

Improper Integrals from Fourier Analysis | p. 269 |

Jordan's Lemma | p. 272 |

Indented Paths | p. 277 |

An Indentation Around a Branch Point | p. 280 |

Integration Along a Branch Cut | p. 283 |

Definite Integrals Involving Sines and Cosines | p. 288 |

Argument Principle | p. 291 |

Rouche's Theorem | p. 294 |

Inverse Laplace Transforms | p. 298 |

Examples | p. 301 |

Mapping by Elementary Functions | p. 311 |

Linear Transformations | p. 311 |

The Transformation w = 1/z | p. 313 |

Mappings by 1/z | p. 315 |

Linear Fractional Transformations | p. 319 |

An Implicit Form | p. 322 |

Mappings of the Upper Half Plane | p. 325 |

The Transformation w = sin z | p. 330 |

Mappings by z[superscript 2] and Branches of z[superscript 1/2] | p. 336 |

Square Roots of Polynomials | p. 341 |

Riemann Surfaces | p. 347 |

Surfaces for Related Functions | p. 351 |

Conformal Mapping | p. 355 |

Preservation of Angles | p. 355 |

Scale Factors | p. 358 |

Local Inverses | p. 360 |

Harmonic Conjugates | p. 363 |

Transformations of Harmonic Functions | p. 365 |

Transformations of Boundary Conditions | p. 367 |

Applications of Conformal Mapping | p. 373 |

Steady Temperatures | p. 373 |

Steady Temperatures in a Half Plane | p. 375 |

A Related Problem | p. 377 |

Temperatures in a Quadrant | p. 379 |

Electrostatic Potential | p. 385 |

Potential in a Cylindrical Space | p. 386 |

Two-Dimensional Fluid Flow | p. 391 |

The Stream Function | p. 393 |

Flows Around a Corner and Around a Cylinder | p. 395 |

The Schwarz-Christoffel Transformation | p. 403 |

Mapping the Real Axis Onto a Polygon | p. 403 |

Schwarz-Christoffel Transformation | p. 405 |

Triangles and Rectangles | p. 408 |

Degenerate Polygons | p. 413 |

Fluid Flow in a Channel Through a Slit | p. 417 |

Flow in a Channel With an Offset | p. 420 |

Electrostatic Potential About an Edge of a Conducting Plate | p. 422 |

Integral Formulas of the Poisson Type | p. 429 |

Poisson Integral Formula | p. 429 |

Dirichlet Problem for a Disk | p. 432 |

Related Boundary Value Problems | p. 437 |

Schwarz Integral Formula | p. 440 |

Dirichlet Problem for a Half Plane | p. 441 |

Neumann Problems | p. 445 |

Appendixes | p. 449 |

Bibliography | p. 449 |

Table of Transformations of Regions | p. 452 |

Index | p. 461 |

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