A Concise Approach to Mathematical Analysis

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Springer Science & Business Media, Jun 27, 2011 - Mathematics - 362 pages
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.
 

Contents

Numbers and Functions
1
Sequences
35
Series
65
Limits and Continuity
95
Differentiation
123
Elements of Integration
145
Sequences and Series of Functions
177
Local Structure on the Real Line
213
Continuous Functions
241
Introduction to the Lebesgue Integral
271
Elements of Fourier Analysis
313
A Appendix
339
B Hints for Selected Exercises
347
Bibliography
361
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