Principles of real analysis / S. C. Malik.

Format:

Book

Author/Creator:

Malik, S. C., author.

Language:

English

Subjects (All):

Mathematical analysis.

Functions of real variables.

Numbers, Real.

Physical Description:

1 online resource (392 p.)

Edition:

Second edition.

Place of Publication:

Kent, [England] : New Academic Science Limited, 2013.

Language Note:

English

Summary:

Discusses the theory from its very beginning Foundations have been laid very carefully and the treatment is rigorous, and on modern lines The Riemann integration is treated in full Large number of well graded examples have been given and some of these have been solved.

Contents:

Cover

Preface

Contents

Chapter 1 Real Numbers

1.1 Introduction

1.2 Field Structure and Order Structure

1.3 Bounded and Unbounded Sets: Supremum, Infimum

1.4 Completeness in the Set of Real Numbers

1.5 Absolute Value of a Real Number

Chapter 2 Limit Points: Open and Closed Sets

2.1 Introduction

2.2 Limit Points of a Set

2.3 Closed Sets: Closure of a Set

Chapter 3 Real Sequences

3.1 Functions

3.2 Sequences

3.3 Limit Points of a Sequence

3.4 Convergent Sequences

3.5 Non-Convergent Sequences (Definitions)

3.6 Cauchy's General Principle of Convergence

3.7 Algebra of Sequences

3.8 Some Important Theorems

3.9 Monotonic Sequences

Chapter 4 Infinite Series

4.1 Introduction

4.2 Positive Term Series

4.3 Comparison Tests for Positive Term Series

4.4 Cauchy's Root Test

4.5 D'Alembert's Ratio Test

4.6 Raabe's Test

4.7 Logarithmic Test

4.8 Integral Test

4.9 Gauss's Test

4.10 Series with Arbitrary Terms

Chapter 5 Functions with Interval as Domain (I)

5.1 Limits

5.2 Continuous Functions

5.3 Functions Continuous on Closed Intervals

5.4 Uniform Continuity

Chapter 6 Functions with Interval as Domain (II)

6.1 The Derivative

6.2 Continuous Functions

6.3 Increasing and Decreasing Functions

6.4 Darboux's Theorem

6.5 Rolle's Theorem

6.6 Lagrange's Mean Value Theorem

6.7 Cauchy's Mean Value Theorem

6.8 Higher Order Derivatives

Chapter 7 Applications of Taylor's Theorem

7.1 Extreme Values (Definitions)

7.2 Indeterminate Forms

Chapter 8 Elementary Functions

8.1 Introduction

8.2 Power Series

8.3 Exponential Functions

8.4 Logarithmic Functions (base e)

8.5 Trigonometric Functions

Chapter 9 The Riemann Integral

9.1 Introduction

9.2 Definitions and Existence of the Integral.

9.3 Refinement of Partitions

9.4 Darboux's Theorem

9.5 Conditions of Integrability

9.6 Integrability of the Sum and Difference of Integrable Functions

9.7 The Integral as a Limit of Sums (Riemann Sums)

9.8 Some Integrable Functions

9.9 Integration and Differentiation (The Primitive)

9.10 The Fundamental Theorem of Calculus

9.11 Mean Value Theorems of Integral Calculus

9.12 Integration By Parts

9.13 Change of Variable in an Integral

9.14 Second Mean Value Theorem

Chapter 10 The Riemann-Stieltjes Integral

10.1 Definitions and Existence of the Integral

10.2 A Condition of Integrability

10.3 Some Theorems

10.4 A Definition (Integral as a Limit of Sum)

10.5 Some Important Theorems

Chapter 11 Functions of Several Variables

11.1 Explicit and Implicit Functions

11.2 Continuity

11.3 Partial Derivatives

11.4 Differentiability

11.5 Partial Derivatives of Higher Order

11.6 Differentials of Higher Order

11.7 Functions of Functions

11.8 Change of Variables

11.9 Taylor's Theorem

11.10 Extreme Values: Maxima and Minima

11.11 Functions of Several Variables

Chapter 12 Implicit Functions

12.1 Definition

12.2 Jacobians

12.3 Stationary Values Under Subsidiary Conditions

Appendix I-Theorems on Rearrangement of Terms and Tests for Arbitrary Series

1. Tests for Arbitrary Term Series

2. Rearrangement of Terms

Appendix II-Cantor's Theory of Real Numbers

1. Introduction

2. Sequences of Rational Numbers

3. Real Numbers

4. Addition and Multiplication in R

5. Order in R

6. Real Rational and Irrational Numbers

7. Some Properties of Real Numbers

8. Completeness in R

Bibliography

Index.

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on online resource; title from PDF title page (ebrary, viewed September 8, 2015).

ISBN:

1-78183-049-5

OCLC:

919481164