Real and Complex Analysis

Hardcover | 1 May 1986 | Edition Number 3

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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.

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Preface Prologue: The Exponential Function
Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, 8] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises
Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises
L p -Spaces Convex functions and inequalities TheL p -spaces Approximation by continuous functions Exercises
Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises
Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients ofL1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises
Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals onL p The Riesz representation theorem Exercises
Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises
Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises
Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebraL1 Exercises
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
Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises
The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelouml;f method An interpolation theorem A converse of the maximum modulus theorem Exercises
Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises
Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The classL Continuity at the boundary Conformal mapping of an annulus Exercises
Zeros of Holomorphic Functions Infinite Products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muuml;ntz-Szas theorem Exercises
Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises
H p -Spaces Subharmonic functions The spacesH p and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises
Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises
Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem
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Real and Complex Analysis

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