A Concise Approach to Mathematical AnalysisA 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
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 |
361 | |
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Common terms and phrases
absolutely convergent an+1 an+p Cauchy sequence closed interval continuous function converges a.e. converges uniformly countable Definition denoted differentiable disjoint diverges equicontinuous Example exists finite follows Fourier series function ƒ ƒ and g ƒ is continuous ƒ uniformly ƒ x implies ƒ inequality integrable functions Lemma Let ɛ Let f Let ƒ lim ƒ lim inf lim sup mathematical induction n-th N₁ neighborhood nondecreasing nonincreasing notice null set open interval open set partial sums polynomial proof is complete Proof Let Proof Suppose prove rational number real numbers real-valued functions resp Riemann integrable sequence of functions sequence of real series converges Show that ƒ Solution Let step function Suppose that ƒ trigonometric uniform convergence uniformly continuous upper bound value theorem