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Introduction to Analysis【Arthur P. Mattuck】.pdf
Copyright Page
Contents
Preface
1. Real Numbers and Monotone Sequences
1.1 Introduction. Real numbers.
1.2 Increasing sequences.
1.3 The limit of an increasing sequence.
1.4 Example: the number e.
1.5 Example: the harmonic sum and Euler's number.
1.6 Decreasing sequences. The Completeness Property.
Exercises Ch1
2. Estimations and Approximations
2.1 Introduction. Inequalities.
2.2 Estimations.
2.3 Proving boundedness.
2.4 Absolute values. Estimating size.
2.5 Approximations.
2.6 The terminology "for n large".
Exercises Ch2
3. The Limit of a Sequence
3.1 Definition of limit.
3.2 The uniqueness of limits. The K-ε principle.
3.3 Infinite limits.
3.4 An important limit.
3.5 Writing limit proofs.
3.6 Some limits involving integrals.
3.7 Another limit involving an integral.
Exercises Ch3
4. Error Term Analysis
4.1 The error term.
4.2 The error in the geometric series.
4.3 A sequence converging to √2: Newton's method.
4.4 The sequence of Fibonacci fractions.
Exercises Ch4
5. The Limit Theorems
5.1 Limits of sums, products, and quotients.
5.2 Comparison theorems.
5.3 Location theorems.
5.4 Subsequences. Non-existence of limits.
5.5 Two common mistakes.
Exercises Ch5
6. The Completeness Property
6.1 Introduction. Nested intervals.
6.2 Cluster points of sequences.
6.3 The Bolzano-Weierstrass theorem.
6.4 Cauchy sequences.
6.5 The Completeness Property for sets.
Exercises Ch6
7. Infinite Series
7.1 Series and sequences.
7.2 Elementary convergence tests.
7.3 The convergence of series with negative terms.
7.4 Convergence tests: ratio and n-th root tests.
7.5 The integral and asymptotic comparison tests.
7.6 Series with alternating signs: Cauchy's test.
7.7 Rearranging the terms of a series.
Exercises Ch7
8. Power Series
8.1 Introduction. Radius of convergence.
8.2 Convergence at the endpoints. Abel summation.
8.3 Operations on power series: addition.
8.4 Multiplication of power series.
Exercises Ch8
9. Functions of One Variable
9.1 Functions.
9.2 Algebraic operations on functions.
9.3 Some properties of functions.
9.4 Inverse functions.
9.5 The elementary functions.
Exercises Ch9
10. Local and Global Behavior
10.1 Intervals. Estimating functions.
10.2 Approximating functions.
10.3 Local behavior.
10.4 Local and global properties of functions.
Exercises Ch10
11. Continuity and Limits
11.1 Continuous functions.
11.2 Limits of functions.
11.3 Limit theorems for functions.
11.4 Limits and continuous functions.
11.5 Continuity and sequences.
Exercises Ch11
12. The Intermediate Value Theorem
12.1 The existence of zeros.
12.2 Applications of Bolzano's theorem.
12.3 Graphical continuity.
12.4 Inverse funtions.
Exercises Ch12
13. Continuous Functions on Compact Intervals
13.1 Compact intervals.
13.2 Bounded continous functions.
13.3 Extremal points of continuous functions.
13.4 The mapping viewpoint.
13.5 Uniform continuity.
Exercises Ch13
14. Differentiation: Local Properties
14.1 The Derivative.
14.2 Differentiation formulas.
14.3 Derivatives and Local Properties.
Exercises Ch14
15. Differentiation: Global Properties
15.1 The mean-value theorem.
15.2 Applications of the mean-value theorem.
15.3 Extension of the mean-value theorem.
15.4 L'Hospital's rule for indeterminate forms.
Exercises Ch15
16. Linearization and Convexity
16.1 Linearization.
16.2 Applications to convexity.
Exercises Ch16
17. Taylor Approximation
17.1 Taylor polynomials.
17.2 Taylor's theorem with the Lagrange remainder.
17.3 Estimating error in Taylor approximation.
17.4 Taylor series.
Exercises Ch17
18. Integrability
18.1 Introduction. Partitions.
18.2 Integrability.
18.3 Integrability of monotone and continuous functions.
18.4 Basic properties of integrable functions.
Exercises Ch18
19. The Riemann Integral
19.1 Refinement of partitions.
19.2 Definition of the Riemann integral.
19.3 Riemann sums.
19.4 Basic properties of integrals.
19.5 The interval addition property.
19.6 Piecewise continous and monotone fuctions.
Exercises Ch19
20. Derivatives and Integrals
20.1 First fundamental theorem of calculus.
20.2 Existence and uniqueness of antiderivatives.
20.3 Other relations between derivatives and integrals.
20.4 The logarithm and exponential functions.
20.5 Stirling's formula.
20.6 Growth rate of functions.
Exercises Ch20
21. Improper Integrals
21.1 Basic definitions.
21.2 Comparison theorems.
21.3 The Gamma function.
21.4 Absolute and conditional convergence.
Exercises Ch21
22. Sequences and Series of Functions
22.1 Pointwise and uniform convergence.
22.2 Criteria for uniform convergence.
22.3 Continuity and uniform convergence.
22.4 Integration term-by-term.
22.5 Differentiation term-by-term.
22.6 Power series and analytic functions.
Exercises Ch22
23. Infinite Sets and the Lebesgue Integral
23.1 Introduction. Infinite sets.
23.2 Sets of measure zero.
23.3 Measure zero and Riemann-integrability.
23.4 Lebesgue integration.
Exercises Ch23
24. Continuous Functions on the Plane
24.1 Introduction. Norms and distances in R².
24.2 Convergence of sequences.
24.3 Functions on R².
24.4 Continuous functions.
24.5 Limits and continuity.
24.6 Compact sets in R².
24.7 Continuous functions on compact sets in R².
Exercises Ch24
25. Point Sets in the Plane
25.1 Closed sets in R².
25.2 The Compactness theorem in R².
25.3 Open sets.
Exercises Ch25
26. Integrals with a Parameter
26.1 Integrals depending on a parameter.
26.2 Differentiating under the integral sign.
26.3 Changing the order of integration.
Exercises Ch26
27. Differentiating Improper Integrals
27.1 Introduction.
27.2 Pointwise vs. uniform convergence of integrals.
27.3 Continuity theorem for improper integrals.
27.4 Integrating and differentiating improper integrals.
27.5 Differentiating the Laplace transform.
Exercises Ch27
Appendix A Sets, Numbers, and Logic
A.0 Sets and Numbers.
A.1 If-then statements.
A.2 Contraposition and indirect proof.
A.3 Counterexamples.
A.4 Mathematical induction.
Exercises
Appendix B Quantifiers and Negation
B.1 Introduction. Quantifiers.
B.2 Negation.
B.3 Examples involving functions.
Exercises
Appendix C Picard Iteration
C.1 Introduction.
C.2 The Picard iteration theorems.
C.3 Fixed points.
Exercises
Appendix D Applications to Differential Equations
D.1 Introduction.
D.2 Discreteness of the zeros.
D.3 The alternation of zeros.
D.4 Reduction to normal form.
D.5 Comparison theorem for zeros.
Exercises
Appendix E Existence and Uniqueness Theorems for Solutions to Differential Equations
E.1 Picard’s method of successive approximations.
E.2 Local existence of solutions to y'=f(x,y).
E.3 The uniqueness of solutions.
E.4 Extending the existence and uniqueness theorems.
Exercises
Index
PRENTICE
HALL
ARTHUR MATTUCK
Introduction to Analysis
Arthur Mattuck
Massachusetts Institute of Technology
Prentice Hall
Upper Saddle River, New Jersey 07458
Library of Congress Cataloging-in-Publication Data
Mattuck, Arthur
Introduction to Analysis / Arthur Mattuck
p. cm.
Includes index.
ISBN 0-13-081132-7
1. Mathematical Analysis.
QA300.M23 1999 515-dc21
I. Title
98-25850
CIP
Cover art: Sunrise by Robert Kabak, 1956
Editorial director, Tim Bozik
Editor-in-chief, Jerome Grant
Acquisition editor, George Lobell
Executive managing editor, Kathleen Schiaparelli
Managing editor, Linda Behrens
Editorial assistants, Gale Epps, Nancy Bauer
Assistant VP production/manufacturing, David W. Riccardi
Manufacturing manager,’Trudy Pisciotti
Manufacturing buyer, Alan Fischer
Creative director, Paula Maylahn
Art director, Jane Conte
Marketing manager, Melody Marcus
© 1999 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
All rights reserved. No part of this book may
be reproduced, in any form or by any means,
without permission in writing from the publisher.
Printed in the United States of America
10987654
ISBN 0-13-081132-7
Prentice-Hall International (UK) Limited, London
Prentice-Hall of Australia Pty. Limited, Sydney
Prentice-Hall of Canada, Inc., Toronto
Prentice-Hall Hispanoamericana, S. A., Mexico
Prentice-Hall of India Private Limited, New Delhi
Prentice-Hall of Japan, Inc叫 To妙。
Pearson Education Asia Pte. Ltd., Singapore
Editora Prentice-Hall do Brasil, Ltda., Rio de Janeiro
to the memory of my pα陀ηts
Jacob A. Mattuck
Rae B. Mattuck
αnd my teacher
Arnold Dresden
Contents
Preface
1. Real Numbers and Monotone Sequences 1
Introduction; R启al numbers 1
Increasing sequences 3
1.1
1.2
1.3 Limit of an increasing sequence 4
1.4 Example: the number e 5
1.5 Example: the h缸monic sum and Euler’s number γ8
1.6 Decreasing sequences; Completeness property 10
2. Estimations and Approximations 17
Introduction; Inequalities 17
Estimations 18
Proving boundedness 20
2.1
2.2
2.3
2.4 Absolute values; estimating size 21
2.5 Approximations 24
2.6 The terminology “for n large” 27
3. The Limit of a Sequence 35
Infinite limits 40
3.1 Definition of limit 35
3.2 Uniqueness of limits; the K-e principle 38
3.3
3.4 Limit of αn 42
3.5 Writing limit proofs 43
3.6
3. 7 Another limit involving an integral 45
Some limits involving integrals 44
4. The Error Term 51
The error term 51
4.1
The error in the geometric series; Applications 52
4.2
4.3 A sequence co盯erging to v'2: Newton’s method 53
4.4 The sequence of Fibonacci fractions 56
5. Limit Theorems for Sequences 61
Limits of sums, products，皿d quotients 61
5.1
5.2 Comparison theorems 64
5.3
5.4
5.5
Location theorems 67
Subsequences; Non-existence of limits 68
Two common mistakes 71
Vil
Vlll
Introduction to Analysis
6. The Completeness Property 78
Introduction; Nested intervals 78
6.1
6.2 Cluster points of sequences 80
6.3
6.4 Cauchy sequences 83
6.5 Completeness property for sets 86
The Bolzano-Weierstrass theorem 82
7. Infinite Series 94
Series and sequences 94
Elementary convergence tests 97
The convergence of series with negative terms 100
7.1
7.2
7.3
7.4 Convergence tests: ratio and n ” th root tests 102
7.5
7.6
7. 7 Rearranging the terms of a series 107
The integral and asymptotic comparison tests 104
Series with alternating signs: Cauchy’s test 106
8. Power Series 114
Introduction; Radius of convergence 114
8.1
8.2 Convergence at the endpoints; Abel summation 117
8.3 Operations on power series: addition 119
8.4 Multiplication of power series 120
9. Functions of One Variable 125
Functions 125
9.1
9.2 Algebraic operations on functions 127
9.3
9.4
9.5
Some properties of functions 128
Inverse functions 131
The elementary functions 133
10. Local and Global Behavior 137
Intervals; estimating functions 137
10.1
10.2 Approximating functions 141
10.3 Local behavior 143
10.4 Local and global properties of functions 145
11. Continuity and Limits of Functions 151
11.1 Continuous functions 151
11.2 Limits of functions 155
11.3 Limit theorems for functions 158
11.4 Limits and continuous functions 162
11.5 Continuity and sequences 155
12. The Intermediate Value Theorem 172
12.1 The existence of zeros 172
12.2 Applications of Bolzano’s theorem 175
12.3 Graphical continuity 178
12.4
Inverse functions 179
Contents
IX
13. Continuous Functions on Compact Intervals 185
13.1 Compaεt intervals 185
13.2 Bounded continuous functions 186
13.3 Extremal points of continuous functions 187
13.4 The mapping viewpoint 189
13.5 Uniform continuity 190
14. Differentiation: Local Properties 196
14.1 The derivative 196
14.2 Differentiation formulas 200
14.3 Derivatives and local properties 202
15. Differentiation: Global Properties 210
15.1 The mean-value theorem 210
15.2 Applications of the mean-value theorem 212
15.3 Extension of the mean-value theorem 214
15.4 L’Hospital’s rule for indeterminate forms 215
16. Linearization and Convexity 222
16.1 Linearization 222
16.2 Applications to convexity 225
17. Taylor Approximation 231
17 .1 Taylor polynomials 231
17.2 Taylor’s theorem with Lagrange remainder 233
17.3 Estimating error in Taylor approximation 235
17.4 Taylor series 236
18. Integrability 241
Introduction; Partitions 241
Integrability 242
Integrability of monotone and continuous functions 244
18.1
18.2
18.3
18.4 Basic properties of integrable functions 246
19. The Riemann Integral 251
19.1 Refinement of partitions 251
19.2 Definition of the Riemann integral 253
19.3 Riemann sums 255
19.4 Basic properties of integrals 257
19.5 The interval addition property 258
19.6 Piecewise continuous and monotone functions 260
20. Derivatives and Integrals 269
20.1 The first fundamental theorem of calculus 269
20.2 Existence and uniqueness of antiderivatives 270
20.3 Other relations between derivatives and integrals 274
20.4 The logarithm and exponential functions 276
20.5 Stirling’s formula 278
20.6 Growth rate of functions 280
x
Introduction to Analysis
21. Improper Integrals 290
21.1 Basic definitions 290
21.2 Comparison theorems 292
21.3 The g缸nma function 295
21.4 Absolute and conditional convergence 298
22. Sequences and Series of Functions 305
22.1 Pointwise and uniform convergence 305
22.2 Criteria for uniform convergence 310
22.3 Continuity and uniform convergence 312
22.4
22.5 Differentiation term-by-term 316
22.6 Power series and analytic functions 318
Integration term-by-term 314
23. Infinite Sets and the Lebesgue Integral 329
Introduction; infinite sets 329
23.1
23.2 Sets of measure zero 333
23.3 Measure zero and Riemann-integrability 335
23.4 Lebesgue integration 338
24. Continuous Functions on the Plane 347
Introduction; Norms and distances in JR2 347
24.1
24.2 Convergence of sequences 349
24.3 Functions on 1R2 351
24.4 Continuous functions 352
24.5 Limits and continuity 354
24.6 Compaεt sets in IR.2 355
24. 7 Continuous functions on compact sets in JR2 356
25. Point-sets in the Plane 364
25.1 Closed sets in IR.2 364
25.2 Compactness theorem in JR2 367
25.3 Open sets 368
26. Integrals with a Parameter 375
Integrals depending on a p缸ameter 375
26.1
26.2 Differentiating under the integral sign 377
26.3 Changing the order of integration 380
27. Differentiating Improper Integrals 386
Introduction 386
27.1
27.2 Pointwise vs. uniform convergence of integrals 387
27.3 Continuity theorem for improper integrals 390
27.4
27.5 Differentiating the Laplace tr皿sform 393
Integrating and differentiating improper integrals 391
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