MATHEMATICAL ANALYSIS

Transcription

1 MATHEMATICAL ANALYSIS S. C. Malik Savita Arora Department of Mathematics S.G.T.B. Khalsa College University of Delhi Delhi, India JOHN WILEY & SONS NEW YORK CHICHESTER BRISBANE TORONTO SINGAPORE

2 Preface to the Second Edition Preface to the First Edition Chapter 1 REAL NUMBERS S!. Introduction 1 2. Field Structure and Order Structure Bounded and Unbounded Sets: Supremum, Infimum 16 4 Completeness in the Set of Real Numbers Absolute Value of a Real Number 27 Chapter 2 OPEN SETS. CLOSED SETS AND COUNTABLE SETS 1. Introduction Limit Points ofa Set Closed Sets : Closure of a Set Countable and Uncountable Sets 49 Chapter 3 REAL SEQUENCES 1. Sequences Limit Points of a Sequence Limits Inferior and Superior Convergent Sequences Non-Convergent Sequences (Definitions) Cauchy's General Principle of Convergence Algebra of Sequences Some Important Theorems Monotonie Sequences 96 Chapter 4 INFINITE SERIES 1. Introduction 109

3 viii 2. Positive Term Series Comparison Tests for Positive Term Series Cauchy's Root Test D'Alembert's Ratio Test Raabe's Test Logarithmic Test Integral Test Gauss's Test Series with Arbitrary Terms Rearrangement of Terms 148 Chapter 5 FUNCTIONS OF A SINGLE VARIABLE (I) Limits Continuous Functions Functions Continuous on Closed Intervals Uniform Continuity 179 Chapter 6 FUNCTIONS OF A SINGLE VARIABLE (II) The Derivative Continuous Functions Increasing and Decreasing Functions Darboux's Theorem Rolle's Theorem Lagrange's Mean Value Theorem Cauchy's Mean Value Theorem Higher Order Derivatives 206 Chapter 7 APPLICATIONS OF TAYLOR'S THEOREM Extreme Values (Definitions) Indeterminate Forms 223 Chapter 8 FUNCTIONS Power Series Exponential Functions Logarithmic Functions Trigonometrie Functions Functional Equations Functions of Bounded Variation Vector-Valued Functions 262

4 ix Chapter 9 THE RIEMANN INTEGRAL Definitions and Existence of the Integral Refinement of Partitions Darboux's Theorem Conditions of Integrability Integrability of the Sum and Difference of Integrable Functions The Integral as a Limit of Sums (Riemann Sums) Some Integrable Functions Integration and Differentiation (The Primitive) The Fundamental Theorem of Calculus Mean Value Theorems of Integral Calculus Integration by Parts Change of Variable in an Integral Second Mean Value Theorem 319 Chapter 10 THE RIEMANN-STIELTJES INTEGRAL Definitions and Existence of the Integral A Condition of Integrability Some Theorems A Definition (Integral as a limit of sum) Some Important Theorems 346 Chapter 11 IMPROPER INTEGRALS Introduction Integration of Unbounded Functions with Finite Limits of Integration 351 b 3. Comparison Tests for Convergence at a of jfdx Infinite Range of Integration Integrand as a Product of Functions 389 Chapter 12 UNIFORM CONVERGENCE Pointwise Convergence Uniform Convergence on an Interval Tests for Uniform Convergence Properties of Uniformly Convergent Sequences and Series The Weierstrass Approximation Theorem 440

5 x Chapter 13 POWER SERIES Generic Term Definition Properties of Functions Expressible as Power Series Abel's Theorem 453 Chapter 14 FOURIER SERIES Trigonometrical Series Some Preliminary Theorems The Main Theorem Intervals Other Than [-:r, TT] 479 Chapter 15 FUNCTIONS OF SEVERAL VARIABLES Explicit and Implicit Functions Continuity Partial Derivatives Differentiability Partial Derivatives of Higher Order Differentials of Higher Order Functions of Functions Change of Variables Taylor's Theorem Extreme Values : Maxima and Minima Functions of Several Variables 554 Chapter 16 IMPLICIT FUNCTIONS Definition Jacobians Stationary Values under Subsidiary Conditions 575 Chapter 17 INTEGRATION ON R Line Integrals Double Integrals Double Integrals Over a Region Green's Theorem Change of Variables 637

6 k xi Chapter 18 INTEGRATION ON R Rectifiable Curves Line Integrals Surfaces Surface Integrals Stokes' Theorem (First generalization of Greerfs Theorem) The Volume of a Cylindrical Solid by Double Integrals Volume Integrals (Triple Integrals) Gauss's Theorem (Divergence Theorem) 708 Chapter 19 METRIC SPACES Definitions and Examples Open and Closed Sets Convergence and Completeness Continuity and Uniform Continuity Compactness Connectedness 800 Chapter 20 THE LEBESGUE INTEGRAL Measurable Sets Sets of Measure Zero Borel Sets Non-Measurable Sets Measurable Functions Measurability of the Sum, Difference, Product and Quotient Measurable Functions Lebesgue Integral Properties of Lebesgue Integral for Bounded Measurable Functions Lebesgue Integral of a Bounded Function Over a Set of Finite Measure Lebesgue Integral for Unbounded Functions The General Integral Some Fundamental Theorems Lebesgue Theorem on Bounded Convergence Integrability and Measurability Lebesgue Integral on Unbounded Sets or Intervals Comparison with Riemann Integral for Unbounded Sets 869

7 xii Appendix I BETA AND GAMMA FUNCTIONS Appendix II CANTOR'S THEORY OF REAL NUMBERS Bibliography Index 1. Sequences of Rational Numbers Real Numbers Addition and Multiplication in R Order in R Real Rational and Irrational Numbers Some Properties of Real Numbers Completeness in R 890