Real and Complex Analysis (inbunden)
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Format
Inbunden (Hardback)
Språk
Engelska
Antal sidor
432
Utgivningsdatum
1986-09-01
Upplaga
3
Förlag
McGraw Hill
Illustrationer
tabs.
Dimensioner
250 x 170 x 35 mm
Vikt
760 g
Antal komponenter
1
Komponenter
STENCILS
ISBN
9780070542341

Real and Complex Analysis

av Walter Rudin
Inbunden,  Engelska, 1986-09-01
2320
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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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Innehållsförteckning

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 Borel 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 l1

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 Phragmen-Lindelf 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 l

Continuity at the boundary

Conformal ...

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