Softcover reprint of the original 1st ed. 2003, XII, 362p. 47 illus..
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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.
Content Level »Graduate
Keywords »Advanced calculus - Derivative - Fourier series - Mathematical analysis - Mean value theorem - Riemann integral - Taylor series - Taylor's theorem - calculus - compactness - differential equation - fixed-point theorem - improper integral
Numbers and Functions.- Sequences.- Series.- Limits and Continuity.- Differentiation.- Elements of Integration.- Sequences and Series of Functions.- Local Structure on the Real Line.- Continuous Functions.- Introduction to Lebesgue Integral.- Elements of Fourier Analysis.-Appendix.- Hints for Selected Exercises.- Bibliography.- Index