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
