Complex Variables and Applications
by Brown, James; Churchill, RuelEdition: 8th
Format: Hardcover
Pub. Date: 2008-01-10
Publisher(s): McGraw-Hill Science/Engineering/Math
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Summary
These classic textbooks, specializing in the techniques and applications of advanced mathematics to physical science and engineering, have endured as perennial standards for more than 60 years. The latest editions preserve the hallmark features that made Brown and Churchill a household name in advanced mathematics education-clear and concise exposition, interesting examples, and accessible level-while adding new enhancements, improved organization, and more modern examples and applications to serve another generation of students.
Author Biography
James Ward Brown is Professor of Mathematics at The University of Michigan-Dearborn.
Table of Contents
| 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 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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