Problems in Real Analysis: Advanced Calculus on the Real Axis

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Springer Science & Business Media, May 29, 2009 - Mathematics - 452 pages
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Problems in Real Analysis: Advanced Calculus on the Real Axis features a comprehensive collection of challenging problems in mathematical analysis that aim to promote creative, non-standard techniques for solving problems. This self-contained text offers a host of new mathematical tools and strategies which develop a connection between analysis and other mathematical disciplines, such as physics and engineering. A broad view of mathematics is presented throughout; the text is excellent for the classroom or self-study. It is intended for undergraduate and graduate students in mathematics, as well as for researchers engaged in the interplay between applied analysis, mathematical physics, and numerical analysis.

Key features:

*Uses competition-inspired problems as a platform for training typical inventive skills;

*Develops basic valuable techniques for solving problems in mathematical analysis on the real axis and provides solid preparation for deeper study of real analysis;

*Includes numerous examples and interesting, valuable historical accounts of ideas and methods in analysis;

*Offers a systematic path to organizing a natural transition that bridges elementary problem-solving activity to independent exploration of new results and properties.

 

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Contents

Sequences
3
12 Introductory Problems
7
13 Recurrent Sequences
18
14 Qualitative Results
30
15 Hardys and Carlemans Inequalities
45
16 Independent Study Problems
51
Series
58
22 Elementary Problems
66
65 Independent Study Problems
285
Inequalities and Extremum Problems
288
72 Elementary Examples
290
73 Jensen Young Holder Minkowski and Beyond
294
74 Optimization Problems
300
75 Qualitative Results
305
76 Independent Study Problems
308
Antiderivatives Riemann Integrability and Applications
311

23 Convergent and Divergent Series
73
24 Infinite Products
86
25 Qualitative Results
89
26 Independent Study Problems
110
Limits of Functions
115
32 Computing Limits
118
33 Qualitative Results
124
34 Independent Study Problems
133
Qualitative Properties of Continuous and Differentiable Functions
136
Continuity
139
42 Elementary Problems
144
43 The Intermediate Value Property
147
44 Types of Discontinuities
151
45 Fixed Points
154
46 Functional Equations and Inequalities
163
47 Qualitative Properties of Continuous Functions
169
48 Independent Study Problems
177
Differentiability
182
52 Introductory Problems
198
53 The Main Theorems
218
54 The Maximum Principle
235
55 Differential Equations and Inequalities
238
56 Independent Study Problems
252
Applications to Convex Functions and Optimization
260
Convex Functions
263
62 Basic Properties of Convex Functions and Applications
265
63 Convexity versus Continuity and Differentiability
273
64 Qualitative Results
278
Antiderivatives
313
82 Elementary Examples
315
83 Existence or Nonexistence of Antiderivatives
317
84 Qualitative Results
319
85 Independent Study Problems
324
Riemann Integrability
325
92 Elementary Examples
329
93 Classes of Riemann Integrable Functions
337
94 Basic Rules for Computing Integrals
339
95 Riemann Iintegrals and Limits
341
96 Qualitative Results
351
97 Independent Study Problems
367
Applications of the Integral Calculus
373
102 Integral Inequalities
374
103 Improper Integrals
390
104 Integrals and Series
402
105 Applications to Geometry
406
106 Independent Study Problems
409
Appendix
415
Basic Elements of Set Theory
417
A2 Finite Countable and Uncountable Sets
418
Topology of the Real Line
419
B2 Some Distinguished Points
420
Glossary
421
References
437
Index
443
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About the author (2009)

Teodora-Liliana Radulescu received her PhD in 2005 from Babes-Bolyai University of Cluj-Napoca, Romania, with a thesis on nonlinear analysis, and she is currently a professor of mathematics at the "Fratii Buzesti" National College in Craiova, Romania. She is a member of the American Mathematical Society and the Romanian Mathematical Society. She is also a reviewer for Mathematical Reviews and Zentralblatt fur Mathematik.

Vicentiu Radulescu received both his PhD and the Habilitation at the Université Pierre et Marie Curie (Paris 6), and he is currently a professor of mathematics at the University of Craiova, Romania and a senior researcher at the Institute of Mathematics "Simion Stoilow" of the Romanian Academy in Bucharest, Romania. He has authored 9 books and over 100 articles.

Titu Andreescu is an associate professor of mathematics at the University of Texas at Dallas. He is also firmly involved in mathematics contests and Olympiads, being the Director of AMC (as appointed by the Mathematical Association of America), Director of MOP, Head Coach of the USA IMO Team and Chairman of the USAMO. He has also authored a large number of books on the topic of problem solving and Olympiad-style mathematics.

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