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A Concise Approach to Mathematical Analysis - Mangatiana A. Robdera

A Concise Approach to Mathematical Analysis

Paperback Published: 3rd December 2002
ISBN: 9781852335526
Number Of Pages: 362

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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.

From the reviews:

THE BULLETIN OF MATHEMATICS BOOKS

"It is written primarily for advanced students of mathematics a" in the 3rd of 4th year of their degree a" 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."

Numbers and Functionsp. 1
Real Numbersp. 1
Subsets of Rp. 15
Variables and Functionsp. 25
Sequencesp. 35
Definition of a Sequencep. 35
Convergence and Limitsp. 39
Subsequencesp. 50
Upper and Lower Limitsp. 53
Cauchy Criterionp. 57
Seriesp. 65
Infinite Seriesp. 65
Conditional Convergencep. 70
Comparison Testsp. 78
Root and Ratio Testsp. 82
Further Testsp. 85
Limits and Continuityp. 95
Limits of Functionsp. 95
Continuity of Functionsp. 104
Properties of Continuous Functionsp. 108
Uniform Continuityp. 112
Differentiationp. 123
Derivativesp. 123
Mean Value Theoremp. 129
L'Hospital's Rulep. 132
Inverse Function Theoremsp. 134
Taylor's Theoremp. 137
Elements of Integrationp. 145
Step Functionsp. 145
Riemann Integralp. 153
Functions of Bounded Variationp. 164
Riemann-Stieltjes Integralp. 168
Sequences and Series of Functionsp. 177
Sequences of Functionsp. 177
Series of Functionsp. 186
Power Seriesp. 194
Taylor Seriesp. 200
Local Structure on the Real Linep. 213
Open and Closed Sets in Rp. 213
Neighborhoods and Interior Pointsp. 221
Closure Point and Closurep. 226
Completeness and Compactnessp. 230
Continuous Functionsp. 241
Global Continuityp. 241
Functions Continuous on a Compact Setp. 245
Stone-Weierstrass Theoremp. 251
Fixed-point Theoremp. 257
Ascoli-Arzela Theoremp. 259
Introduction to the Lebesgue Integralp. 271
Null Setsp. 271
Lebesgue Integralp. 283
Improper Integralp. 296
Important Inequalitiesp. 300
Elements of Fourier Analysisp. 313
Fourier Seriesp. 313
Convergent Trigonometric Seriesp. 319
Convergence in 2-meanp. 322
Pointwise Convergencep. 328
Appendixp. 339
Theorems and Proofsp. 339
Set Notationsp. 341
Cantor's Ternary Setp. 343
Bernstein's Approximation Theoremp. 344
Hints for Selected Exercisesp. 347
Bibliographyp. 361
Indexp. 363
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9781852335526
ISBN-10: 1852335521
Audience: General
Format: Paperback
Language: English
Number Of Pages: 362
Published: 3rd December 2002
Publisher: Springer London Ltd
Country of Publication: GB
Dimensions (cm): 23.11 x 15.49  x 2.29
Weight (kg): 0.54