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Introduction to Analysis
Synopses & Reviews
KEY BENEFIT:This new book is written in a conversational, accessible style, offering a great deal of examples. It gradually ascends in difficulty to help the student avoid sudden changes in difficulty. Discusses analysis from the start of the book, to avoid unnecessary discussion on real numbers beyond what is immediately needed. Includes simplified and meaningful proofs. Features Exercises and Problems at the end of each chapter as well as Questions at the end of each section with answers at the end of each chapter. Presents analysis in a unified way as the mathematics based on inequalities, estimations, and approximations. For mathematicians.
Book News Annotation:
A textbook based on a one-semester course taught at MIT. Topics include real numbers and monotone sequences, estimations and approximations, limit theorems for sequences, local and global behavior, differentiation, the Riemann integral, infinite sets and the Lebesgue integral, and continuous functions on the plane. The focus is on estimation and approximation rather than on the algebraic or topological aspects of analysis. Appends information about sets, logic, Picards method, applications to differential equations, and ODE solutions.
Annotation c. Book News, Inc., Portland, OR (booknews.com)
Table of Contents
1. Real Numbers and Monotone Sequences.
2. Estimations and Approximations.
3. The Limit of a Sequence.
4. The Error Term.
5. Limit Theorems for Sequences.
6. The Completeness Principle.
7. Infinite Series.
8. Power Series.
9. Functions of One Variable.
10. Local and Global Behavior.
11. Continuity and Limits of Functions.
12. The Intermediate Value Theorem.
13. Continuous Functions on Compact Intervals.
14. Differentiation: Local Properties.
15. Differentiation: Global Properties.
16. Linearization and Convexity.
17. Taylor Approximation.
19. The Riemann Integral.
20. Derivatives and Integrals.
21. Improper Integrals.
22. Sequences and Series of Functions.
23. Infinite Sets and the Lebesgue Integral.
24. Continuous Functions on the Plane.
25. Point-sets in the Plane.
26. Integrals with a Parameter.
27. Differentiating Improper Integrals.
A. Sets, Numbers, and Logic.
B. Quantifiers and Negation.
C. Picard’s Method.
D. Applications to Differential Equations.
E. Existence and Uniqueness of ODE Solutions.
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