Syllabus (s.c.malik-mathematical Analysis)

MATHEMATICAL ANALYSIS

S. C. Malik Savita Arora Department of Mathematics S.G.T.B. Khalsa College University of Delhi Delhi, India

JOHN NEW YORK

WILEY

CHICHESTER

&

SONS

BRISBANE TORONTO

SINGAPORE

Contents Preface to the Second Edition Preface to the First Edition Chapter 1

REAL NUMBERS

S

!. 2. 3.

Introduction 1 Field Structure and Order Structure 11 Bounded and Unbounded Sets: Supremum, Infimum 16 4 Completeness in the Set of Real Numbers 5. Absolute Value of a Real Number 27 Chapter 2

19

OPEN SETS. CLOSED SETS AND COUNTABLE SETS 1. Introduction 33 2. Limit Points o f a Set 38 3. Closed Sets : Closure of a Set 42 4. Countable and Uncountable Sets 49

Chapter 3

REAL SEQUENCES 1. Sequences 53 2. Limit Points of a Sequence 56 3. Limits Inferior and Superior 59 4. Convergent Sequences 66 5. Non-Convergent Sequences (Definitions) 68 6. Cauchy's General Principle of Convergence 73 7. Algebra of Sequences 75 8. Some Important Theorems 85 9. Monotonie Sequences 96

Chapter 4

INFINITE SERIES 1.

Introduction

109

viii

Contents 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Chapter 5

Positive Term Series 114 Comparison Tests for Positive Term Series Cauchy's Root Test 124 D'Alembert's Ratio Test 125 Raabe's Test 127 Logarithmic Test 131 Integral Test 132 Gauss's Test 135 Series with Arbitrary Terms 139 Rearrangement of Terms 148

FUNCTIONS OF A SINGLE VARIABLE (I) 1. Limits 154 2. Continuous Functions 165 3. Functions Continuous on Closed Intervals 4. Uniform Continuity 179

Chapter 6

FUNCTIONS OF A SINGLE VARIABLE (II) 1. 23. 4. 5. 6. 7. 8.

Chapter 7

APPLICATIONS OF TAYLOR'S THEOREM

174

185

216

216

FUNCTIONS 1. 2. 3. 4. 5. 6. 7.

154

The Derivative 185 Continuous Functions 188 Increasing and Decreasing Functions 191 Darboux's Theorem 194 Rolle's Theorem 195 Lagrange's Mean Value Theorem 196 Cauchy's Mean Value Theorem 198 Higher Order Derivatives 206

1. Extreme Values (Definitions) 2. Indeterminate Forms 223 Chapter 8

118

Power Series 236 Exponential Functions 238 Logarithmic Functions 240 Trigonometrie Functions 243 Functional Equations 249 Functions of Bounded Variation Vector-Valued Functions 262

236

251

Contents Chapter 9

ix 270

THE RIEMANN INTEGRAL 1. Definitions and Existence of the Integral 270 2. Refinement of Partitions 277 3. Darboux's Theorem 280 4. Conditions of Integrability 281 5. Integrability of the Sum and Difference of Integrable Functions 284 6. The Integral as a Limit of Sums (Riemann Sums) 293 7. Some Integrable Functions 300 8. Integration and Differentiation (The Primitive) 304 9. The Fundamental Theorem of Calculus 306 10. Mean Value Theorems of Integral Calculus 311 11. Integration by Parts 316 12. Change of Variable in an Integral 318 13. Second Mean Value Theorem 319

Chapter 10 THE RIEMANN-STIELTJES INTEGRAL

330

1. Definitions and Existence of the Integral 330 2. A Condition of Integrability 333 3. Some Theorems 334 4. A Definition (Integral as a limit of sum) 338 5. Some Important Theorems 346 Chapter 11 IMPROPER INTEGRALS

351

1. Introduction 351 2. Integration of Unbounded Functions with Finite Limits of Integration 351 b

3.

Comparison Tests for Convergence at a of

4. 5.

Infinite Range of Integration 370 Integrand as a Product of Functions

jfdx

355

389

Chapter 12 UNIFORM CONVERGENCE 1. Pointwise Convergence 404 2. Uniform Convergence on an Interval 406 3. Tests for Uniform Convergence 412 4. Properties of Uniformly Convergent Sequences and Series 422 5. The Weierstrass Approximation Theorem 440

404

x

Contents

Chapter 13 POWER SERIES

440

1. Generic Term 446 2. Definition 446 3. Properties of Functions Expressible as Power Series 450 4. Abel's Theorem 453 Chapter 14 FOURIER SERIES

463

1. Trigonometrical Series 463 2. Some Preliminary Theorems 465 3. The Main Theorem 471 4. Intervals Other Than [-:r, TT] 479 Chapter 15 FUNCTIONS OF SEVERAL VARIABLES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

492

Explicit and Implicit Functions 492 Continuity 501 Partial Derivatives 505 Differentiability 509 Partial Derivatives of Higher Order 517 Differentials of Higher Order 524 Functions of Functions 526 Change of Variables 533 Taylor's Theorem 544 Extreme Values : Maxima and Minima 548 Functions of Several Variables 554

Chapter 16 IMPLICIT FUNCTIONS

562

1. Definition 562 2. Jacobians 567 3. Stationary Values under Subsidiary Conditions 575 Chapter 17 INTEGRATION ON R2 1. 2. 3. 4. 5.

Line Integrals 588 Double Integrals 596 Double Integrals Over a Region Green's Theorem 629 Change of Variables 637

588

618

Contents Chapter 18 INTEGRATION ON R3

xi 652

1. Rectifiable Curves 652 2. Line Integrals 657 3. Surfaces 662 4. Surface Integrals 670 5. Stokes' Theorem (First generalization of Greerfs Theorem) 687 6. The Volume of a Cylindrical Solid by Double Integrals 692 7. Volume Integrals (Triple Integrals) 698 8. Gauss's Theorem (Divergence Theorem) 708 Chapter 19 METRIC SPACES 1. 2. 3. 4. 5. 6.

726

Definitions and Examples 726 Open and Closed Sets 737 Convergence and Completeness 758 Continuity and Uniform Continuity 768 Compactness 781 Connectedness 800

Chapter 20 THE LEBESGUE INTEGRAL

811

1. Measurable Sets 811 2. Sets of Measure Zero 820 3. Borel Sets 824 4. Non-Measurable Sets 824 5. Measurable Functions 828 6. Measurability of the Sum, Difference, Product and Quotient Measurable Functions 831 7. Lebesgue Integral 836 8. Properties of Lebesgue Integral for Bounded Measurable Functions 839 9. Lebesgue Integral of a Bounded Function Over a Set of Finite Measure 845 10. Lebesgue Integral for Unbounded Functions 850 11. The General Integral 853 12. Some Fundamental Theorems 853 13. Lebesgue Theorem on Bounded Convergence 857 14. Integrability and Measurability 859 15. Lebesgue Integral on Unbounded Sets or Intervals 869 16. Comparison with Riemann Integral for Unbounded Sets 869

k

xii

Contents

Appendix I BETA AND GAMMA FUNCTIONS Appendix II CANTOR'S THEORY OF REAL NUMBERS 1. 2. 3. 4. 5. 6. 7. Bibliography Index

Sequences of Rational Numbers 879 Real Numbers 881 Addition and Multiplication in R 882 Order in R 885 Real Rational and Irrational Numbers 888 Some Properties of Real Numbers 888 Completeness in R 890