A Concise Approach to Mathematical Analysis

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Springer Science & Business Media, 2003 - Mathematics - 366 pages
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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.
 

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Contents

Numbers and Functions
1
12 Subsets of ℝ
15
13 Variables and Functions
25
Sequences
35
22 Convergence and Limits
39
23 Subsequences
50
24 Upper and Lower Limits
53
25 Cauchy Criterion
57
73 Power Series
194
74 Taylor Series
200
Local Structure on the Real Line
213
82 Neighborhoods and Interior Points
221
83 Closure Point and Closure
226
84 Completeness and Compactness
230
Continuous Functions
241
92 Functions Continuous on a Compact Set
245

Series
65
32 Conditional Convergence
70
33 Comparison Tests Term Comparison Test
78
34 Root and Ratio Tests
82
35 Further Tests
85
Limits and Continuity
95
42 Continuity of Functions
104
43 Properties of Continuous Functions
108
44 Uniform Continuity
112
Differentiation
123
52 Mean Value Theorem
129
53 LHôspitals Rule
132
54 Inverse Function Theorems
134
55 Taylors Theorem
137
Elements of Integration
145
62 Riemann Integral
153
63 Functions of Bounded Variation
164
64 RiemannStieltjes Integral
168
Sequences and Series of Functions
177
72 Series of Functions
186
93 StoneWeierstrass Theorem
251
94 Fixedpoint Theorem
257
95 AscoliArzelā Theorem
259
Introduction to the Lebesgue Integral
271
102 Lebesgue Integral
283
103 Improper Integral
296
104 Important Inequalities
300
Elements of Fourier Analysis
313
112 Convergent Trigonometric Series
319
113 Convergence in 2mean
322
114 Pointwise Convergence
328
Appendix
339
A2 Set Notations
341
A3 Cantors Ternary Set
343
A4 Bernsteins Approximation Theorem
344
Hints for Selected Exercises
347
Bibliography
361
Index
363
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