A Handbook of Real Variables With Applications to Differential Equat
The subject of real analysis dates to the mid-nineteenth century - the days of Riemann and Cauchy and Weierstrass. Real analysis grew up as a way to make the calculus rigorous. Today the two subjects are intertwined in most people's minds. Yet calculus is
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Steven G. Krantz
A Handbook of Real Variables With Applications to Differential Equations and Fourier Analysis
Springer Science+Business Media, LLC
Steven G. Krantz Department of Mathematics Washington University St. Louis, MO 63130-4899 U.S.A.
Library of Congress Cataloging-in-Publication Data
Krantz, Steven G. (Steven George), 1951A handbook of real variables : with applications to differential equations and Fourier ana1ysis I Steven Krantz. p. cm. Inc1udes bibliographica1 references and index. ISBN 978-1-4612-6409-5 ISBN 978-0-8176-8128-9 (eBook) DOI 10.1007/978-0-8176-8128-9 1. Functions of real variables. 2. Mathematica1 analysis. 1. Title. QA331.5.K72003 515'.8-dc21
2003050248 CIP
AMS Subject Classifications: Primary: 26-00, 26-01; Secondary: 26A03, 26A06, 26A09, 26AI5, 42-01,35-01 ISBN 978-1-4612-6409-5
Printed on acid-free paper.
@2004Springer Science+-Business Media New York Originally published by Birkhliuser Boston in 2004 Softcover reprint ofthe hardcover Ist edition 2004
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Contents Preface
xi
1 Basics 1.1 Sets 1.2 Operations on Sets 1.3 Functions . . . . . 1.4 Operations on Functions 1.5 Number Systems .. . . 1.5.1 The Real Numbers 1.6 Countable and Uncountable Sets
1 4 5
6 6
9
2 Sequences 2.1 Introduction to Sequences. . . . . . . . 2.1.1 The Definition and Convergence 2.1.2 The Cauchy Criterion. 2.1.3 Monotonicity..... 2.1.4 The Pinching Principle 2.1.5 Subsequences..... 2.1.6 The Bolzano-Weierstrass Theorem 2.2 Limsup and Liminf . . . 2.3 Some Special Sequences
11
3
21 21 21 22 23 23 24 24 25 25 27
Series 3.1 Introduction to Series . . . . . . . . . . 3.1.1 The Definition and Convergence 3.1.2 Partial Sums . . . . . 3.2 Elementary Convergence Tests . . . . . 3.2.1 The Comparison Test . . . . . . 3.2.2 The Cauchy Condensation Test 3.2.3 Geometric Series 3.2.4 The Root Test. . . . . . . . . . 3.2.5 The Ratio Test . . . . . . . . . 3.2.6 Root and Ratio Tests for Divergence. 3.3 Advanced Convergence Tests . 3.3.1 Summation by Parts . . . . . . . . .
11 11
12 13 14 14 15 15 17
28 28 v
Contents
VI
3.4
3.5
4
5
3.3.2 Abel's Test . . . . . . . . . . . . . . . 3.3.3 Absolute and Conditional Convergence 3.3.4 Rearrangements of Series Some Particular Series . . . .
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