Complex Variables and Applications

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  • Edition: 8th
  • Format: Hardcover
  • Copyright: 2008-01-10
  • Publisher: McGraw-Hill Science/Engineering/Math
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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

Prefacep. x
Complex Numbersp. 1
Sums and Productsp. 1
Basic Algebraic Propertiesp. 3
Further Propertiesp. 5
Vectors and Modulip. 9
Complex Conjugatesp. 13
Exponential Formp. 16
Products and Powers in Exponential Formp. 18
Arguments of Products and Quotientsp. 20
Roots of Complex Numbersp. 24
Examplesp. 27
Regions in the Complex Planep. 31
Analytic Functionsp. 35
Functions of a Complex Variablep. 35
Mappingsp. 38
Mappings by the Exponential Functionp. 42
Limitsp. 45
Theorems on Limitsp. 48
Limits Involving the Point at Infinityp. 50
Continuityp. 53
Derivativesp. 56
Differentiation Formulasp. 60
Cauchy-Riemann Equationsp. 63
Sufficient Conditions for Differentiabilityp. 66
Polar Coordinatesp. 68
Analytic Functionsp. 73
Examplesp. 75
Harmonic Functionsp. 78
Uniquely Determined Analytic Functionsp. 83
Reflection Principlep. 85
Elementary Functionsp. 89
The Exponential Functionp. 89
The Logarithmic Functionp. 93
Branches and Derivatives of Logarithmsp. 95
Some Identities Involving Logarithmsp. 98
Complex Exponentsp. 101
Trigonometric Functionsp. 104
Hyperbolic Functionsp. 109
Inverse Trigonometric and Hyperbolic Functionsp. 112
Integralsp. 117
Derivatives of Functions w(t)p. 117
Definite Integrals of Functions w(t)p. 119
Contoursp. 122
Contour Integralsp. 127
Some Examplesp. 129
Examples with Branch Cutsp. 133
Upper Bounds for Moduli of Contour Integralsp. 137
Antiderivativesp. 142
Proof of the Theoremp. 146
Cauchy-Goursat Theoremp. 150
Proof of the Theoremp. 152
Simply Connected Domainsp. 156
Multiply Connected Domainsp. 158
Cauchy Integral Formulap. 164
An Extension of the Cauchy Integral Formulap. 165
Some Consequences of the Extensionp. 168
Liouville's Theorem and the Fundamental Theorem of Algebrap. 172
Maximum Modulus Principlep. 175
Seriesp. 181
Convergence of Sequencesp. 181
Convergence of Seriesp. 184
Taylor Seriesp. 189
Proof of Taylor's Theoremp. 190
Examplesp. 192
Laurent Seriesp. 197
Proof of Laurent's Theoremp. 199
Examplesp. 202
Absolute and Uniform Convergence of Power Seriesp. 208
Continuity of Sums of Power Seriesp. 211
Integration and Differentiation of Power Seriesp. 213
Uniqueness of Series Representationsp. 217
Multiplication and Division of Power Seriesp. 222
Residues and Polesp. 229
Isolated Singular Pointsp. 229
Residuesp. 231
Cauchy's Residue Theoremp. 234
Residue at Infinityp. 237
The Three Types of Isolated Singular Pointsp. 240
Residues at Polesp. 244
Examplesp. 245
Zeros of Analytic Functionsp. 249
Zeros and Polesp. 252
Behavior of Functions Near Isolated Singular Pointsp. 257
Applications of Residuesp. 261
Evaluation of Improper Integralsp. 261
Examplep. 264
Improper Integrals from Fourier Analysisp. 269
Jordan's Lemmap. 272
Indented Pathsp. 277
An Indentation Around a Branch Pointp. 280
Integration Along a Branch Cutp. 283
Definite Integrals Involving Sines and Cosinesp. 288
Argument Principlep. 291
Rouche's Theoremp. 294
Inverse Laplace Transformsp. 298
Examplesp. 301
Mapping by Elementary Functionsp. 311
Linear Transformationsp. 311
The Transformation w = 1/zp. 313
Mappings by 1/zp. 315
Linear Fractional Transformationsp. 319
An Implicit Formp. 322
Mappings of the Upper Half Planep. 325
The Transformation w = sin zp. 330
Mappings by z[superscript 2] and Branches of z[superscript 1/2]p. 336
Square Roots of Polynomialsp. 341
Riemann Surfacesp. 347
Surfaces for Related Functionsp. 351
Conformal Mappingp. 355
Preservation of Anglesp. 355
Scale Factorsp. 358
Local Inversesp. 360
Harmonic Conjugatesp. 363
Transformations of Harmonic Functionsp. 365
Transformations of Boundary Conditionsp. 367
Applications of Conformal Mappingp. 373
Steady Temperaturesp. 373
Steady Temperatures in a Half Planep. 375
A Related Problemp. 377
Temperatures in a Quadrantp. 379
Electrostatic Potentialp. 385
Potential in a Cylindrical Spacep. 386
Two-Dimensional Fluid Flowp. 391
The Stream Functionp. 393
Flows Around a Corner and Around a Cylinderp. 395
The Schwarz-Christoffel Transformationp. 403
Mapping the Real Axis Onto a Polygonp. 403
Schwarz-Christoffel Transformationp. 405
Triangles and Rectanglesp. 408
Degenerate Polygonsp. 413
Fluid Flow in a Channel Through a Slitp. 417
Flow in a Channel With an Offsetp. 420
Electrostatic Potential About an Edge of a Conducting Platep. 422
Integral Formulas of the Poisson Typep. 429
Poisson Integral Formulap. 429
Dirichlet Problem for a Diskp. 432
Related Boundary Value Problemsp. 437
Schwarz Integral Formulap. 440
Dirichlet Problem for a Half Planep. 441
Neumann Problemsp. 445
Appendixesp. 449
Bibliographyp. 449
Table of Transformations of Regionsp. 452
Indexp. 461
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