沃尔特·鲁丁著的《实分析与复分析(英文版原书第3版典藏版)/华章数学原版精品系列》是分析领域内的一部经典著作。毫不夸张地说,掌握了本书,对数学的理解将会上一个新台阶。在第3版中,作者对一些新的课题进行了讨论,并力求全书条理清晰。
本书体例优美,实用性很强,列举的实例简明精彩。无论是实分析部分还是复分析部分,基本上对所有给出的命题都进行了论证。另外,书中还附有大量设计巧妙的习题,通过这些习题可以真实地检测出读者对课程的理解程度,有的还要求对正文中的原理进行论证。
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书名 | 实分析与复分析(英文版原书第3版典藏版)/华章数学原版精品系列 |
分类 | 科学技术-自然科学-数学 |
作者 | (美)沃尔特·鲁丁 |
出版社 | 机械工业出版社 |
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简介 | 内容推荐 沃尔特·鲁丁著的《实分析与复分析(英文版原书第3版典藏版)/华章数学原版精品系列》是分析领域内的一部经典著作。毫不夸张地说,掌握了本书,对数学的理解将会上一个新台阶。在第3版中,作者对一些新的课题进行了讨论,并力求全书条理清晰。 本书体例优美,实用性很强,列举的实例简明精彩。无论是实分析部分还是复分析部分,基本上对所有给出的命题都进行了论证。另外,书中还附有大量设计巧妙的习题,通过这些习题可以真实地检测出读者对课程的理解程度,有的还要求对正文中的原理进行论证。 作者简介 沃尔特·鲁丁(Walter Rudin),1953年于杜克大学获得数学博士学位。曾先后执教于麻省理工学院、罗切斯特大学、威斯康星大学麦迪逊分校、耶鲁大学等。他的主要研究兴趣集中在调和分析和复变函数上。除本书外,他还著有《Functional Analysis》(泛函分析)和《Principles of Mathematical Analysis》(数学分析原理)等其他名著。这些教材已被翻译成十几种语言,在世界各地广泛使用。 目录 Preface Prologue: The Exponential Function Chapter 1 Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ∞] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2 Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borei measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3 LP-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4 Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5 Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6 Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7 Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8 Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9 Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra Lt Exercises Chapter 10 Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11 Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12 The Maximum Modulus Principle Introduction The Schwarz lemma The Phragrnen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13 Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14 Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class y Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15 Zeros of Holomorphic Functions Infinite products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Miintz-Szasz theorem Exercises Chapter 16 Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17 Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18 Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19 Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20 Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and C |
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