Examples and Theorems in Analysis

by Peter Walker

Overview

Examples and Theorems in Analysis takes a unique and very practical approach to mathematical analysis. It makes the subject more accessible by giving the examples equal status with the theorems. The results are introduced and motivated by reference to examples which illustrate their use, and further examples then show how far the assumptions may be relaxed before the result fails.

A number of applications show what the subject is about and what can be done with it; the applications in Fourier theory, distributions and asymptotics show how the results may be put to use. Exercises at the end of each chapter, of varying levels of difficulty, develop new ideas and present open problems.

Written primarily for first- and second-year undergraduates in mathematics, this book features a host of diverse and interesting examples, making it an entertaining and stimulating companion that will also be accessible to students of statistics, computer science and engineering, as well as to professionals in these fields.

Product Details

ISBN-13:	9781852334932
Publisher:	Springer London
Publication date:	11/18/2003
Edition description:	Softcover reprint of the original 1st ed. 2004
Pages:	287
Product dimensions:	7.01(w) x 9.25(h) x 0.28(d)

Table of Contents

1. Sequences.- 1.1 Examples, Formulae and Recursion.- 1.2 Monotone and Bounded Sequences.- 1.3 Convergence.- 1.4 Subsequences.- 1.5 Cauchy Sequences.- Exercises.- 2. Functions and Continuity.- 2.1 Examples.- 2.2 Monotone and Bounded Functions.- 2.3 Limits and Continuity.- 2.4 Bounds and Intermediate Values.- 2.5 Inverse Functions.- 2.6 Recursive Limits and Iteration.- 2.7 One-Sided and Infinite Limits. Regulated Functions.- 2.8 Countability.- Exercises.- 3. Differentiation.- 3.1 Differentiable Functions.- 3.2 The Significance of the Derivative.- 3.3 Rules for Differentiation.- 3.4 Mean Value Theorems and Estimation.- 3.5 More on Iteration.- 3.6 Optimisation.- Exercises.- 4. Constructive Integration.- 4.1 Step Functions.- 4.2 The Integral of a Regulated function.- 4.3 Integration and Differentiation.- 4.4 Applications.- 4.5 Further Mean Value Theorems.- Exercises.- 5. Improper Integrals.- 5.1 Improper Integrals on an Interval.- 5.2 Improper Integrals at Infinity.- 5.3 The Gamma function.- Exercises.- 6. Series.- 6.1 Convergence.- 6.2 Series with Positive Terms.- 6.3 Series with Arbitrary Terms.- 6.4 Power Series.- 6.5 Exponential and Trigonometric Functions.- 6.6 Sequences and Series of Functions.- 6.7 Infinite Products.- Exercises.- 7. Applications.- 7.1 Fourier Series.- 7.2 Fourier Integrals.- 7.3 Distributions.- 7.4 Asymptotics.- 7.5 Exercises.- A. Fubini’s Theorem.- B. Hints and Solutions for Exercises.

Editorial Reviews

From the reviews:

"This book is a unique and very practical contribution to the teaching of calculus … . The aim of this book is to try to give the subject concreteness and immediacy by giving the well-chosen examples equal status with the theorems. … this excellent book is written primarily for first- and second-year undergraduates in mathematics; but it will also be of interest to students of statistics, computer science and engineering … . We warmly recommend it as an entertaining and stimulating companion …" (Ferenc Móricz, Acta Scientiarum Mathematicarum, 71, 2005)

"This book takes a unique and very practical approach to mathematical analysis. It makes the subject more accessible by giving the examples equal status with the theorems. … A number of applications show what the subject is about, and what can be done with it … . Exercises at the end of each chapter, of varying levels of difficulty, develop new ideas and present open problems." (L'enseignement mathématique, 50:1-2, 2004)

"The author presents a book on analysis in which theorems and examples are equally important. … It is a good textbook for students … to obtain a more complete picture of the material and to master basic methods of work in mathematical analysis." (EMS Newsletter, March, 2005)

From the Publisher