Ebook: A Handbook of Real Variables: With Applications to Differential Equations and Fourier Analysis
Author: Steven G. Krantz (auth.)
- Tags: Functional Analysis, Fourier Analysis, Ordinary Differential Equations, Real Functions
- Year: 2004
- Publisher: Birkhäuser Basel
- Edition: 1
- Language: English
- pdf
The subject of real analysis dates to the mid-nineteenth century - the days of Riemann and Cauchy and Weierstrass. Real analysis grew up as a way to make the calculus rigorous. Today the two subjects are intertwined in most people's minds. Yet calculus is only the first step of a long journey, and real analysis is one of the first great triumphs along that road. In real analysis we learn the rigorous theories of sequences and series, and the profound new insights that these tools make possible. We learn of the completeness of the real number system, and how this property makes the real numbers the natural set of limit points for the rational numbers. We learn of compact sets and uniform convergence. The great classical examples, such as the Weierstrass nowhere-differentiable function and the Cantor set, are part of the bedrock of the subject. Of course complete and rigorous treatments of the derivative and the integral are essential parts of this process. The Weierstrass approximation theorem, the Riemann integral, the Cauchy property for sequences, and many other deep ideas round out the picture of a powerful set of tools.
Content:
Front Matter....Pages i-xiii
Basics....Pages 1-10
Sequences....Pages 11-19
Series....Pages 21-38
The Topology of the Real Line....Pages 39-52
Limits and the Continuity of Functions....Pages 53-69
The Derivative....Pages 71-84
The Integral....Pages 85-101
Sequences and Series of Functions....Pages 103-112
Some Special Functions....Pages 113-138
Advanced Topics....Pages 139-152
Differential Equations....Pages 153-176
Back Matter....Pages 177-201
Content:
Front Matter....Pages i-xiii
Basics....Pages 1-10
Sequences....Pages 11-19
Series....Pages 21-38
The Topology of the Real Line....Pages 39-52
Limits and the Continuity of Functions....Pages 53-69
The Derivative....Pages 71-84
The Integral....Pages 85-101
Sequences and Series of Functions....Pages 103-112
Some Special Functions....Pages 113-138
Advanced Topics....Pages 139-152
Differential Equations....Pages 153-176
Back Matter....Pages 177-201
....