Introduction to real analysis : an educational approach / William C. Bauldry.

Format:

Book

Author/Creator:

Bauldry, William C.

Contributor:

Hazel M. Hussong Fund.

Language:

English

Subjects (All):

Mathematical analysis--Textbooks.

Mathematical analysis.

Functions--Textbooks.

Functions.

Genre:

Textbooks.

Physical Description:

xv, 262 pages : illustrations ; 25 cm

Place of Publication:

Hoboken, N.J. : Wiley, [2009]

Summary:

Bridging the gap between the development and history of real analysis, Introduction to Real Analysis: An Educational Approach presents a comprehensive introduction to real analysis while also offering a survey of the field. With its balance of historical background, key calculus methods, and hands-on applications, this book provides readers with a solid foundation and fundamental understanding of real analysis.

The book begins with an outline of basic calculus, including a close examination of problems illustrating links and potential difficulties. Next, a fluid introduction to real analysis is presented, guiding readers through the basic topology of real numbers, limits, integration, and a series of functions in natural progression. The book moves on to analysis with more rigorous investigations, and the topology of the line is presented along with a discussion of limits and continuity that includes unusual examples in order to direct readers' thinking beyond intuitive reasoning and on to more complex understanding. The dichotomy of pointwise and uniform convergence is then addressed and is followed by differentiation and integration. Riemann-Stieltjes integrals and the Lebesgue measure are also introduced to broaden the presented perspective. The book concludes with a collection of advanced topics that are connected to elementary calculus, such as modeling with logistic functions, numerical quadrature, Fourier series, and special functions.

Detailed appendices outline key definitions and theorems in elementary calculus and also present additional proofs, projects, and sets in real analysis. Each chapter references historical sources on real analysis while also providing proof-oriented exercises and examples that facilitate the development of computational skills. In addition, an extensive bibliography provides additional resources on the topic.

Introduction to Real Analysis: An Educational Approach is an ideal book for upper-undergraduate and graduate-level real analysis courses in the areas of mathematics and education. It is also a valuable reference for educators in the field of applied mathematics.

Contents:

1 Elementary Calculus 1

1.1 Preliminary Concepts 1

1.2 Limits and Continuity 3

1.3 Differentiation 11

1.4 Integration 19

1.5 Sequences and Series of Constants 25

1.6 Power Series and Taylor Series 30

Summary 35

Exercises 36

Interlude: Fermat, Descartes, and the Tangent Problem 42

2 Introduction to Real Analysis 45

2.1 Basic Topology of the Real Numbers 46

2.2 Limits and Continuity 51

2.3 Differentiation 60

2.4 Riemann and Riemann-Stieltjes Integration 71

2.5 Sequences, Series, and Convergence Tests 88

2.6 Pointwise and Uniform Convergence 103

Summary 116

Exercises 117

Interlude: Euler and the "Basel Problem" 122

3 A Brief Introduction to Lebesgue Theory 125

3.1 Lebesgue Measure and Measurable Sets 126

3.2 The Lebesgue Integral 138

3.3 Measure, Integral, and Convergence 155

3.4 Littlewood's Three Principles 165

Summary 165

Exercises 166

Interlude: The Set of Rational Numbers Is Very Large and Very Small 170

4 Special Topics 175

4.1 Modeling with Logistic Functions-Numerical Derivatives 176

4.2 Numerical Quadrature 182

4.3 Fourier Series 195

4.4 Special Functions-The Gamma Function 203

4.5 Calculus Without Limits: Differential Algebra 208

Summary 213

Exercises 213.

Notes:

Includes bibliographical references and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Hazel M. Hussong Fund.

ISBN:

9780470371367

0470371366

OCLC:

310400145

Publisher Number:

99937154172