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Full Description
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.- Real Numbers.- Subsets of ?.- Variables and Functions.- Sequences.- Definition of a Sequence.- Convergence and Limits.- Subsequences.- Upper and Lower Limits.- Cauchy Criterion.- 3. Series.- Infinite Series.- Conditional Convergence.- Comparison Tests.- Root and Ratio Tests.- Further Tests.- 4. Limits and Continuity.- Limits of Functions.- Continuity of Functions.- Properties of Continuous Functions.- Uniform Continuity.- Differentiation.- Derivatives.- Mean Value Theorem.- L'Hôspital's Rule.- Inverse Function Theorems.- Taylor's Theorem.- Elements of Integration.- Step Functions.- Riemann Integral.- Functions of Bounded Variation.- Riemann-Stieltjes Integral.- Sequences and Series of Functions.- Sequences of Functions.- Series of Functions.- Power Series.- Taylor Series.- Local Structure on the Real Line.- Open and Closed Sets in ?.- Neighborhoods and Interior Points.- Closure Point and Closure.- Completeness and Compactness.- Continuous Functions.- Global Continuity.- Functions Continuous on a Compact Set.- Stone—Weierstrass Theorem.- Fixed-point Theorem.- Ascoli-Arzelà Theorem.- to the Lebesgue Integral.- Null Sets.- Lebesgue Integral.- Improper Integral.- Important Inequalities.- Elements of Fourier Analysis.- Fourier Series.- Convergent Trigonometric Series.- Convergence in 2-mean.- Pointwise Convergence.- A. Appendix.- A.1 Theorems and Proofs.- A.2 Set Notations.- A.3 Cantor's Ternary Set.- A.4 Bernstein's Approximation Theorem.- B. Hints for Selected Exercises.
