A Handbook of Real Variables

1. Front Matter

   Pages i-xiii

   PDF

2. Basics

   • Steven G. Krantz

   Pages 1-10

3. Sequences

   • Steven G. Krantz

   Pages 11-19

4. Series

   • Steven G. Krantz

   Pages 21-38

5. The Topology of the Real Line

   • Steven G. Krantz

   Pages 39-52

6. Limits and the Continuity of Functions

   • Steven G. Krantz

   Pages 53-69

7. The Derivative

   • Steven G. Krantz

   Pages 71-84

8. The Integral

   • Steven G. Krantz

   Pages 85-101

9. Sequences and Series of Functions

   • Steven G. Krantz

   Pages 103-112

10. Some Special Functions

    • Steven G. Krantz

    Pages 113-138

11. Advanced Topics

    • Steven G. Krantz

    Pages 139-152

12. Differential Equations

    • Steven G. Krantz

    Pages 153-176

13. Back Matter

    Pages 177-201

    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.

• Boundary value problem
• Fourier analysis
• Mean value theorem
• ODEs
• functional analysis
• ksa
• real analysis
• ordinary differential equations

"In eleven chapters, Krantz's book succeeds in providing a reference work for 'the working engineer or scientist' encompassing the essence of real analysis...Krantz's book suceeds in providing a reference work for "the working engineer or scientist" encompassing the essence of real analysis. ... True to the idea of a handbook, there are good, but brief, explanations, well-chosen examples, and only a few proofs. In addition to the book's principal audience, students preparing for exams at either the undergraduate or master's level will find this a valuable resource." (MAA Reviews)

"The purpose of this book is to acknowledge that there is a large audience of scientists and others who wish to use the fruits of real analysis, and who are not equipped to stop and appreciate all the theory.  This handbook uses ample examples and brief explanations and must give an opportunity to users of real analysis quickly to look up ideas, without axiomatic machinery and without becoming bogged down in long explanations and proofs. . . This very good written book can be highly recommended to everyone who are teaching or researching in the field of applied mathematics. The book is also of interest to graduate students, researchers in physics, engineering, economics, and other applied sciences." (ZAA)

• Department of Mathematics, Washington University, St. Louis, USA

  Steven G. Krantz

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