Brown-Churchill-Complex Variables and Application 8th edition

482 Pages·2008·4.41 MB·English

by  James Brown

COMPLEX VARIABLES
AND APPLICATIONS
Eighth Edition
James Ward Brown
ProfessorofMathematics
The UniversityofMichigan–Dearborn
Ruel V. Churchill
LateProfessorofMathematics
TheUniversityofMichigan
COMPLEXVARIABLESANDAPPLICATIONS,EIGHTHEDITION
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ABOUT THE AUTHORS
JAMES WARD BROWN is Professor of Mathematics at The University of
Michigan–Dearborn.HeearnedhisA.B.inphysicsfromHarvardUniversityandhis
A.M. and Ph.D. in mathematics from The University of Michigan in Ann Arbor,
where he was an Institute of Science and Technology Predoctoral Fellow. He is
coauthor with Dr. Churchill of Fourier Series and Boundary Value Problems, now
in its seventh edition. He has received a research grant from the National Science
Foundation as well as a Distinguished Faculty Award from the Michigan Associa-
tionofGoverningBoardsofCollegesandUniversities.Dr.BrownislistedinWho’s
Whoin the World.
RUELV. CHURCHILL was,atthe time of his deathin1987, ProfessorEmeritus
ofMathematicsatTheUniversityofMichigan,wherehebeganteachingin1922.He
receivedhisB.S.inphysicsfromtheUniversityofChicagoandhisM.S.inphysics
and Ph.D. in mathematics from The University of Michigan. He was coauthor with
Dr. Brown of Fourier Series and Boundary Value Problems, a classic text that he
firstwrotealmost70yearsago.HewasalsotheauthorofOperationalMathematics.
Dr. Churchill held various offices in the Mathematical Association of America and
in other mathematical societies and councils.
iii
TotheMemoryofMyFather
GeorgeH.Brown
andofMyLong-TimeFriendandCoauthor
RuelV.Churchill
TheseDistinguishedMenofScienceforYearsInfluenced
TheCareersofManyPeople,IncludingMyself.
JWB
CONTENTS
Preface x
1 Complex Numbers 1
Sumsand Products 1
BasicAlgebraicProperties 3
FurtherProperties 5
Vectorsand Moduli 9
ComplexConjugates 13
ExponentialForm 16
Productsand PowersinExponentialForm 18
Argumentsof Productsand Quotients 20
RootsofComplexNumbers 24
Examples 27
RegionsintheComplexPlane 31
2 Analytic Functions 35
Functionsof aComplexVariable 35
Mappings 38
MappingsbytheExponentialFunction 42
Limits 45
Theoremson Limits 48
v
vi contents
LimitsInvolvingthePointat Infinity 50
Continuity 53
Derivatives 56
DifferentiationFormulas 60
Cauchy–RiemannEquations 63
Sufficient ConditionsforDifferentiability 66
Polar Coordinates 68
AnalyticFunctions 73
Examples 75
HarmonicFunctions 78
UniquelyDetermined AnalyticFunctions 83
Reflection Principle 85
3 Elementary Functions 89
The ExponentialFunction 89
The LogarithmicFunction 93
Branchesand DerivativesofLogarithms 95
Some IdentitiesInvolvingLogarithms 98
ComplexExponents 101
TrigonometricFunctions 104
HyperbolicFunctions 109
InverseTrigonometricandHyperbolicFunctions 112
4 Integrals 117
DerivativesofFunctionsw(t) 117
DefiniteIntegralsof Functionsw(t) 119
Contours 122
ContourIntegrals 127
Some Examples 129
Examples withBranchCuts 133
Upper BoundsforModuliofContourIntegrals 137
Antiderivatives 142
Proofof theTheorem 146
Cauchy–GoursatTheorem 150
Proofof theTheorem 152
contents vii
SimplyConnectedDomains 156
MultiplyConnectedDomains 158
CauchyIntegralFormula 164
AnExtensionoftheCauchyIntegralFormula 165
SomeConsequencesoftheExtension 168
Liouville’sTheoremand theFundamentalTheorem ofAlgebra 172
MaximumModulusPrinciple 175
5 Series 181
ConvergenceofSequences 181
ConvergenceofSeries 184
TaylorSeries 189
ProofofTaylor’sTheorem 190
Examples 192
Laurent Series 197
ProofofLaurent’sTheorem 199
Examples 202
Absoluteand UniformConvergenceofPower Series 208
Continuityof SumsofPower Series 211
IntegrationandDifferentiationof PowerSeries 213
UniquenessofSeriesRepresentations 217
Multiplicationand Divisionof PowerSeries 222
6 Residues and Poles 229
IsolatedSingularPoints 229
Residues 231
Cauchy’sResidueTheorem 234
Residueat Infinity 237
TheThree TypesofIsolatedSingularPoints 240
Residuesat Poles 244
Examples 245
ZerosofAnalyticFunctions 249
Zerosand Poles 252
BehaviorofFunctionsNear IsolatedSingularPoints 257
viii contents
7 Applications of Residues 261
EvaluationofImproperIntegrals 261
Example 264
ImproperIntegralsfromFourierAnalysis 269
Jordan’sLemma 272
IndentedPaths 277
An IndentationAroundaBranchPoint 280
IntegrationAlonga BranchCut 283
DefiniteIntegralsInvolvingSinesand Cosines 288
ArgumentPrinciple 291
Rouche´’sTheorem 294
InverseLaplace Transforms 298
Examples 301
8 Mapping by Elementary Functions 311
Linear Transformations 311
The Transformationw=1/z 313
Mappingsby1/z 315
Linear FractionalTransformations 319
An ImplicitForm 322
MappingsoftheUpperHalfPlane 325
The Transformationw=sinz 330
Mappingsbyz2 and Branchesofz1/2 336
SquareRootsofPolynomials 341
Riemann Surfaces 347
Surfaces forRelatedFunctions 351
9 Conformal Mapping 355
Preservationof Angles 355
Scale Factors 358
Local Inverses 360
HarmonicConjugates 363
TransformationsofHarmonicFunctions 365
TransformationsofBoundaryConditions 367
contents ix
10 Applications of Conformal Mapping 373
Steady Temperatures 373
Steady TemperaturesinaHalfPlane 375
A RelatedProblem 377
TemperaturesinaQuadrant 379
ElectrostaticPotential 385
Potentialina CylindricalSpace 386
Two-DimensionalFluidFlow 391
TheStreamFunction 393
FlowsAroundaCornerandArounda Cylinder 395
11 The Schwarz–Christoffel Transformation 403
MappingtheReal AxisOntoa Polygon 403
Schwarz–ChristoffelTransformation 405
Trianglesand Rectangles 408
DegeneratePolygons 413
FluidFlow inaChannelThroughaSlit 417
FlowinaChannelWithan Offset 420
ElectrostaticPotentialAboutanEdgeof aConductingPlate 422
12 Integral Formulas of the Poisson Type 429
PoissonIntegralFormula 429
DirichletProblemfora Disk 432
RelatedBoundaryValueProblems 437
SchwarzIntegralFormula 440
DirichletProblemfora HalfPlane 441
Neumann Problems 445
Appendixes 449
Bibliography 449
TableofTransformationsof Regions 452
Index 461