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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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