Real and Complex Analysis - 3rd Revised Edition

Klappentext This is an advanced text for the one- or two-semester course in analysis taught primarily to math, science, computer science, and electrical engineering majors at the junior, senior or graduate level. The basic techniques and theorems of analysis are presented in such a way that the intimate connections between its various branches are strongly emphasized. The traditionally separate subjects of 'real analysis' and 'complex analysis' are thus united in one volume. Some of the basic ideas from functional analysis are also included. This is the only book to take this unique approach. The third edition includes a new chapter on differentiation. Proofs of theorems presented in the book are concise and complete and many challenging exercises appear at the end of each chapter. The book is arranged so that each chapter builds upon the other, giving students a gradual understanding of the subject. This text is part of the Walter Rudin Student Series in Advanced Mathematics. Zusammenfassung Presents the basic techniques and theorems of analysis. This work includes a chapter on differentiation. It presents proofs of theorems and many exercises appear at the end of each chapter. It is arranged so that each chapter builds upon the other! giving students a gradual understanding of the subject. Inhaltsverzeichnis PrefacePrologue: The Exponential FunctionChapter 1: Abstract IntegrationSet-theoretic notations and terminologyThe concept of measurabilitySimple functionsElementary properties of measuresArithmetic in [0, 8]Integration of positive functionsIntegration of complex functionsThe role played by sets of measure zeroExercisesChapter 2: Positive Borel MeasuresVector spacesTopological preliminariesThe Riesz representation theoremRegularity properties of Borel measuresLebesgue measureContinuity properties of measurable functionsExercisesChapter 3: Lp -SpacesConvex functions and inequalitiesThe Lp -spacesApproximation by continuous functionsExercisesChapter 4: Elementary Hilbert Space TheoryInner products and linear functionalsOrthonormal setsTrigonometric seriesExercisesChapter 5: Examples of Banach Space TechniquesBanach spacesConsequences of Baire's theoremFourier series of continuous functionsFourier coefficients of L 1-functionsThe Hahn-Banach theoremAn abstract approach to the Poisson integralExercisesChapter 6: Complex MeasuresTotal variationAbsolute continuityConsequences of the Radon-Nikodym theoremBounded linear functionals on Lp The Riesz representation theoremExercisesChapter 7: DifferentiationDerivatives of measuresThe fundamental theorem of CalculusDifferentiable transformationsExercisesChapter 8: Integration on Product SpacesMeasurability on cartesian productsProduct measuresThe Fubini theoremCompletion of product measuresConvolutionsDistribution functionsExercisesChapter 9: Fourier TransformsFormal propertiesThe inversion theoremThe Plancherel theoremThe Banach algebra L 1ExercisesChapter 10: Elementary Properties of Holomorphic FunctionsComplex differentiationIntegration over pathsThe local Cauchy theoremThe power series representationThe open mapping theoremThe global Cauchy theoremThe calculus of residuesExercisesChapter 11: Harmonic FunctionsThe Cauchy-Riemann equationsThe Poisson integralThe mean value propertyBoundary behavior of Poisson integralsRepresentation theoremsExercisesChapter 12: The Maximum Modulus PrincipleIntroductionThe Schwarz lemmaThe Phragmen-Lindelöf methodAn interpolation theoremA converse of the maximum modulus theoremExercisesChapter 13: Approximation by Rational FunctionsPreparationRunge's theoremThe Mittag-Leffler theoremSimply connected regionsExercisesChapter 14: Conformal MappingPreservation of anglesLinear fractional transformationsNormal familiesThe Riemann mapping theoremThe class L Continuity at the boundaryConformal mapping of an annulusExercisesChapter 15: Zeros of Holomorphic FunctionsInfinite Prod...

Dettagli sul prodotto

• Autori: Walter Rudin
• Editore: Mcgraw Hill Academic
• Lingue: Inglese
• Formato: Copertina rigida
• Pubblicazione: 01.09.1986
• EAN: 9780070542341
• ISBN: 978-0-07-054234-1
• Categorie: Scienze naturali, medicina, informatica, tecnica > Matematica; MATHEMATICS / Mathematical Analysis, MATHEMATICS / Complex Analysis, Complex analysis, complex variables, Real analysis, real variables