Problems in Real Analysis

by Radulescu, Teodora-Liliana T.; Radulescu, Vincentiu D.; Andreescu, Titu

• ISBN13: 9780387773780
• ISBN10: 0387773789
• Format: Paperback
• Copyright: 2009-05-12
• Publisher: Springer Verlag

Summary

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.

Author Biography

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 Universit+¬ 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.  

Table of Contents

Foreword	p. vii
Preface	p. ix
Acknowledgments	p. xiii
Abbreviations and Notation	p. xix
Sequences, Series, and Limits
Sequences	p. 3
Main Definitions and Basic Results	p. 3
Introductory Problems	p. 7
Recurrent Sequences	p. 18
Qualitative Results	p. 30
Hardy's and Carleman's Inequalities	p. 45
Independent Study Problems	p. 51
Series	p. 59
Main Definitions and Basic Results	p. 59
Elementary Problems	p. 66
Convergent and Divergent Series	p. 73
Infinite Products	p. 86
Qualitative Results	p. 89
Independent Study Problems	p. 110
Limits of Functions	p. 115
Main Definitions and Basic Results	p. 115
Computing Limits	p. 118
Qualitative Results	p. 124
Independent Study Problems	p. 133
Qualitative Properties of Continuous and Differentiable Functions
Continuity	p. 139
The Concept of Continuity and Basic Properties	p. 139
Elementary Problems	p. 144
The Intermediate Value Property	p. 147
Types of Discontinuities	p. 151
Fixed Points	p. 154
Functional Equations and Inequalities	p. 163
Qualitative Properties of Continuous Functions	p. 169
Independent Study Problems	p. 177
Differentiability	p. 183
The Concept of Derivative and Basic Properties	p. 183
Introductory Problems	p. 198
The Main Theorems	p. 218
The Maximum Principle	p. 235
Differential Equations and Inequalities	p. 238
Independent Study Problems	p. 252
Applications to Convex Functions and Optimization
Convex Functions	p. 263
Main Definitions and Basic Results	p. 263
Basic Properties of Convex Functions and Applications	p. 265
Convexity versus Continuity and Differentiability	p. 273
Qualitative Results	p. 278
Independent Study Problems	p. 285
Inequalities and Extremum Problems	p. 289
Basic Tools	p. 289
Elementary Examples	p. 290
Jensen, Young, Höet;lder, Minkowski, and Beyond	p. 294
Optimization Problems	p. 300
Qualitative Results	p. 305
Independent Study Problems	p. 308
Antiderivatives, Riemann Integrability, and Applications
Antiderivatives	p. 313
Main Definitions and Properties	p. 313
Elementary Examples	p. 315
Existence or Nonexistence of Antiderivatives	p. 317
Qualitative Results	p. 319
Independent Study Problems	p. 324
Riemann Integrability	p. 325
Main Definitions and Properties	p. 325
Elementary Examples	p. 329
Classes of Riemann Integrable Functions	p. 337
Basic Rules for Computing Integrals	p. 339
Riemann Iintegrals and Limits	p. 341
Qualitative Results	p. 351
Independent Study Problems	p. 367
Applications of the Integral Calculus	p. 373
Overview	p. 373
Integral Inequalities	p. 374
Improper Integrals	p. 390
Integrals and Series	p. 402
Applications to Geometry	p. 406
Independent Study Problems	p. 409
Appendix
Basic Elements of Set Theory	p. 417
Direct and Inverse Image of a Set	p. 417
Finite, Countable, and Uncountable Sets	p. 418
Topology of the Real Line	p. 419
Open and Closed Sets	p. 419
Some Distinguished Points	p. 420
Glossary	p. 421
References	p. 437
Index	p. 443
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