调和函数理论 第2版
作者： Axler，S.著
出版时间：2004年版
丛编项： Graduate Texts in Mathematics
内容简介
Harmonic functions：the solutions of Laplace's equation：play a crucial role in many areas of mathematics， physics， and engineering. But learning about them is not always easy. At times the authors have agreed with Lord Kelvin and Peter Tait， who wrote （[18]， Preface） There can be but one opinion as to the beauty and utility of this analysis of Laplace; but the manner in which it has been hitherto presented has seemed repulsive to the ablest mathematicians， and difficult to ordinary mathematical students.
目录
Preface
Acknowledgments
CHAPTER1
BasicPropertiesofHarmonicFunctions
DefinitionsandExamples
InvarianceProperties
TheMean-ValueProperty
TheMaximumPrinciple
ThePoissonKernelfortheBall
TheDirichletProblemfortheBall
ConverseoftheMean-ValueProperty
RealAnalyticityandHomogeneousExpansions
OriginoftheTerm"Harmonic"
Exercises
CHAPTER2
BoundedHarmonicFunctions
Liouvfile'sTheorem
IsolatedSingularities
Cauchy'sEstimates
NormalFamilies
MaximumPrinciples
LimitsAlongRays
BoundedHarmonicFunctionsontheBall
Exercises
CHAPTER3
PositiveHarmonicFunctions
Liouville'sTheorem
Harnack'sInequalityandHarnack'sPrinciple
IsolatedSingularities
PositiveHarmonicFunctionsontheBall
Exercises
CHAPTER4
TheKelvinTransform
InversionintheUnitSphere
MotivationandDefinition
TheKelvinTransformPreservesHarmonicFunctions
Harmonicityatinfinity
TheExteriorDirichletProblem
SyrmnetryandtheSchwarzReflectionPrinciple
Exercises
CHAPTER5
HarmonicPolynomials
PolynomialDecompositions
SphericalHarmonicDecompositionofL2(S)
InnerProductofSphericalHarmonics
SphericalHarmonicsViaDifferentiation
ExplicitBasesofHm(Rn)andHm(S)
ZonalHarmonics
ThePoissonKernelRevisited
AGeometricCharacterizationofZonalHarmbnics
AnExplicitFormulaforZonalHarmonics
Exercises
CHAPTER6
HarmonicHardySpaces
PoissonIntegralsofMeasures
Weak*Convergence
TheSpaceshp(B)
TheHilbertSpaceh2(B)
TheSchwarzLemma
TheFatouTheorem
Exercises
CHAPTER7
HarmonicFunctionsonHalf-Spaces
ThePoissonKernelfortheUpperHalf-Space
TheDirichletProblemfortheUpperHalf-Space
TheHarmonicHardySpaceshP(H)
FromtheBalltotheUpperHalf-Space,andBack
PositiveHarmonicFunctionsontheUpperHalf-Space
NontangentialLimits
TheLocalFatouTheorem
Exercises
CHAPTER8
HarmonicBergmanSpaces
ReproducingKernels
TheReproducingKerneloftheBall
Examplesinbp(B)
TheReproducingKerneloftheUpperHalf-Space
Exercises
CHAPTER9
TheDecompositionTheorem
TheFundamentalSolutionoftheLaplacian
DecompositionofHarmonicFunctions
B6cher'sTheoremRevisited
RemovableSetsforBoundedHarmonicFunctions
TheLogarithmicConjugationTheorem
Exercises
CHAPTER10
AnnularRegions
LaurentSeries
IsolatedSingularities
TheResidueTheorem
ThePoissonKernelforAnnularRegions
Exercises
CHAPTER11
TheDirichletProblemandBoundaryBehavior
TheDirichletProblem
SubharmonicFunctions
ThePerronConstruction
BarrierFunctionsandGeometricCriteriaforSolvability
NonextendabilityResults
Exercises
APPENDIXA
Volume,SurfaceArea,andIntegrationonSpheres
VolumeoftheBallandSurfaceAreaoftheSphere,
SliceIntegrationonSpheres
Exercises
APPENDIXB
HarmonicFunctionTheoryandMathematica
References
SymbolIndex
Index
