A Concise Approach to Mathematical Analysis

By: Mangatiana A. Robdera

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Paperback | 3 December 2002

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384 Pages

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26 x 18 x 2

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A Concise Approach to Mathematical Analysis introduces the undergraduate student to the more abstract concepts of advanced calculus. The main aim of the book is to smooth the transition from the problem-solving approach of standard calculus to the more rigorous approach of proof-writing and a deeper understanding of mathematical analysis. The first half of the textbook deals with the basic foundation of analysis on the real line; the second half introduces more abstract notions in mathematical analysis. Each topic begins with a brief introduction followed by detailed examples. A selection of exercises, ranging from the routine to the more challenging, then gives students the opportunity to practise writing proofs. The book is designed to be accessible to students with appropriate backgrounds from standard calculus courses but with limited or no previous experience in rigorous proofs. It is written primarily for advanced students of mathematics - in the 3rd or 4th year of their degree - who wish to specialise in pure and applied mathematics, but it will also prove useful to students of physics, engineering and computer science who also use advanced mathematical techniques.

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THE BULLETIN OF MATHEMATICS BOOKS

"It is written primarily for advanced students of mathematics - in the 3^rd of 4^th year of their degree - who wish to specialize in pure and applied mathematics, but it will also prove useful to students of physics, engineering and computer science who also use advanced mathematical techniques."

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Numbers and Functions	p. 1
Real Numbers	p. 1
Subsets of R	p. 15
Variables and Functions	p. 25
Sequences	p. 35
Definition of a Sequence	p. 35
Convergence and Limits	p. 39
Subsequences	p. 50
Upper and Lower Limits	p. 53
Cauchy Criterion	p. 57
Series	p. 65
Infinite Series	p. 65
Conditional Convergence	p. 70
Comparison Tests	p. 78
Root and Ratio Tests	p. 82
Further Tests	p. 85
Limits and Continuity	p. 95
Limits of Functions	p. 95
Continuity of Functions	p. 104
Properties of Continuous Functions	p. 108
Uniform Continuity	p. 112
Differentiation	p. 123
Derivatives	p. 123
Mean Value Theorem	p. 129
L'Hospital's Rule	p. 132
Inverse Function Theorems	p. 134
Taylor's Theorem	p. 137
Elements of Integration	p. 145
Step Functions	p. 145
Riemann Integral	p. 153
Functions of Bounded Variation	p. 164
Riemann-Stieltjes Integral	p. 168
Sequences and Series of Functions	p. 177
Sequences of Functions	p. 177
Series of Functions	p. 186
Power Series	p. 194
Taylor Series	p. 200
Local Structure on the Real Line	p. 213
Open and Closed Sets in R	p. 213
Neighborhoods and Interior Points	p. 221
Closure Point and Closure	p. 226
Completeness and Compactness	p. 230
Continuous Functions	p. 241
Global Continuity	p. 241
Functions Continuous on a Compact Set	p. 245
Stone-Weierstrass Theorem	p. 251
Fixed-point Theorem	p. 257
Ascoli-Arzela Theorem	p. 259
Introduction to the Lebesgue Integral	p. 271
Null Sets	p. 271
Lebesgue Integral	p. 283
Improper Integral	p. 296
Important Inequalities	p. 300
Elements of Fourier Analysis	p. 313
Fourier Series	p. 313
Convergent Trigonometric Series	p. 319
Convergence in 2-mean	p. 322
Pointwise Convergence	p. 328
Appendix	p. 339
Theorems and Proofs	p. 339
Set Notations	p. 341
Cantor's Ternary Set	p. 343
Bernstein's Approximation Theorem	p. 344
Hints for Selected Exercises	p. 347
Bibliography	p. 361
Index	p. 363
Table of Contents provided by Ingram. All Rights Reserved.

A Concise Approach to Mathematical Analysis

At a Glance

Paperback

Industry Reviews

Shipping

How to return your order

Defective items

You Can Find This Book In

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