An Introduction to Mathematical Analysis

An Introduction to Mathematical Analysis

International Series of Monographs on Pure and Applied Mathematics

1st Edition - January 1, 1963

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  • Author: Robert A. Rankin
  • eBook ISBN: 9781483137308

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Description

An Introduction to Mathematical Analysis is an introductory text to mathematical analysis, with emphasis on functions of a single real variable. Topics covered include limits and continuity, differentiability, integration, and convergence of infinite series, along with double series and infinite products. This book is comprised of seven chapters and begins with an overview of fundamental ideas and assumptions relating to the field operations and the ordering of the real numbers, together with mathematical induction and upper and lower bounds of sets of real numbers. The following chapters deal with limits of real functions; differentiability and maxima, minima, and convexity; elementary properties of infinite series; and functions defined by power series. Integration is also considered, paying particular attention to the indefinite integral; interval functions and functions of bounded variation; the Riemann-Stieltjes integral; the Riemann integral; and area and curves. The final chapter is devoted to convergence and uniformity. This monograph is intended for mathematics students.

Table of Contents


  • Preface

    List of Symbols and Notations

    I. Fundamental Ideas and Assumptions

    1. Introduction

    2. Assumptions Relating to the Field Operations

    3. Assumptions Relating to the Ordering of the Real Numbers

    4. Mathematical Induction

    5. Upper and Lower Bounds of Sets of Real Numbers

    6. Functions

    II. Limits and Continuity

    7. Limits of Real Functions on the Positive Integers

    8. Limits of Real Functions of a Real Variable x as x Tends to Infinity

    9. Elementary Topological Ideas

    10. Limits of Real Functions at Finite Points

    11. Continuity

    12. Inverse Functions and Fractional Indices

    III. Differentiability

    13. Derivatives

    14. General Theorems Concerning Real Functions

    15. Maxima, Minima and Convexity

    16. Complex Numbers and Functions

    IV. Infinite Series

    17. Elementary Properties of Infinite Series

    18. Series with Non-Negative Terms

    19. Absolute and Conditional Convergence

    20. The Decimal Notation for Real Numbers

    V. Functions Defined by Power Series

    21. General Theory of Power Series

    22. Real Power Series

    23. The Exponential and Logarithmic Functions

    24. The Trigonometric Functions

    25. The Hyperbolic Functions

    26. Complex Indices

    VI. Integration

    27. The Indefinite Integral

    28. Interval Functions and Functions of Bounded Variation

    29. The Riemann—Stieltjes Integral

    30. The Riemann Integral

    31. Curves

    32. Area

    VII. Convergence and Uniformity

    33. Upper and Lower Limits and Their Applications

    34. Further Convergence Tests for Infinite Series

    35. Uniform Convergence

    36. Improper Integrals

    37. Double Series

    38. Infinite Products

    Hints for Solutions of Exercises

    Index

Product details

  • No. of pages: 624
  • Language: English
  • Copyright: © Pergamon 2013
  • Published: January 1, 1963
  • Imprint: Pergamon
  • eBook ISBN: 9781483137308

About the Author

Robert A. Rankin

About the Editors

I. N. Sneddon

S. Ulam

M. Stark

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