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This concise, well-written handbook provides a distillation of the theory of real variables with a particular focus on the subject's significant applications to differential equations and Fourier analysis. Ideal for the working engineer or scientist, the book uses ample examples and brief explanations--without a lot of proofs or axiomatic machinery--to give the reader quick, easy access to all of the key concepts and touchstone results of real analysis. Topics are systematically developed, beginning with sequences and series, and proceeding to topology, limits, continuity, derivatives, and Riemann integration. In the second half of the work, Taylor series, the Weierstrass Approximation Theorem, Fourier series, the Baire Category Theorem, and the Ascoli-Arzela Theorem are carefully discussed. Picard iteration and differential equations are treated in detail in the final chapter.
Key features:
- Completely self-contained, methodical exposition for the mathematically-inclined researcher; also valuable as a study guide for students
- Realistic, meaningful connections to ordinary differential equations, boundary value problems, and Fourier analysis
- Example-driven, incisive explanations of every important idea, with suitable cross-references for ease of use
- Illuminating applications of many theorems, along with specific how-to hints and suggestions
- Extensive bibliography and index
This unique handbook is a compilation of the major results, techniques, and applications of real analysis; it is a practical manual for physicists, engineers, economists, and others who use the fruits of real analysis but who do not necessarily have the time to appreciate all of the theory. Appropriate as a comprehensive reference or for a quick review, the "Handbook of Real Variables" will benefit a wide audience.
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.
Hinweis: Dieser Artikel kann nur an eine deutsche Lieferadresse ausgeliefert werden.
Key features:
- Completely self-contained, methodical exposition for the mathematically-inclined researcher; also valuable as a study guide for students
- Realistic, meaningful connections to ordinary differential equations, boundary value problems, and Fourier analysis
- Example-driven, incisive explanations of every important idea, with suitable cross-references for ease of use
- Illuminating applications of many theorems, along with specific how-to hints and suggestions
- Extensive bibliography and index
This unique handbook is a compilation of the major results, techniques, and applications of real analysis; it is a practical manual for physicists, engineers, economists, and others who use the fruits of real analysis but who do not necessarily have the time to appreciate all of the theory. Appropriate as a comprehensive reference or for a quick review, the "Handbook of Real Variables" will benefit a wide audience.
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.
Hinweis: Dieser Artikel kann nur an eine deutsche Lieferadresse ausgeliefert werden.
"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)