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…mehr
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.
Steven Krantz, Ph.D., is Chairman of the Mathematics Department at Washington University in St. Louis. An award-winning teacher and author, Dr. Krantz has written more than 45 books on mathematics, including Calculus Demystified, another popular title in this series. He lives in St. Louis, Missouri.
Inhaltsangabe
Basics.- Sets.- Operations on Sets.- Functions.- Operations on Functions.- Number Systems.- Countable and Uncountable Sets.- Sequences.- to Sequences.- Limsup and Liminf.- Some Special Sequences.- Series.- to Series.- Elementary Convergence Tests.- Advanced Convergence Tests.- Some Particular Series.- Operations on Series.- The Topology of the Real Line.- Open and Closed Sets.- Other Distinguished Points.- Bounded Sets.- Compact Sets.- The Cantor Set.- Connected and Disconnected Sets.- Perfect Sets.- Limits and the Continuity of Functions.- Definitions and Basic Properties.- Continuous Functions.- Topological Properties and Continuity.- Classifying Discontinuities and Monotonicity.- The Derivative.- The Concept of Derivative.- The Mean Value Theorem and Applications.- Further Results on the Theory of Differentiation.- The Integral.- The Concept of Integral.- Properties of the Riemann Integral.- Further Results on the Riemann Integral.- Advanced Results on Integration Theory.- Sequences and Series of Functions.- Partial Sums and Pointwise Convergence.- More on Uniform Convergence.- Series of Functions.- The Weierstrass Approximation Theorem.- Some Special Functions.- Power Series.- More on Power Series: Convergence Issues.- The Exponential and Trigonometric Functions.- Logarithms and Powers of Real Numbers.- The Gamma Function and Stirling's Formula.- An Introduction to Fourier Series.- Advanced Topics.- Metric Spaces.- Topology in a Metric Space.- The Baire Category Theorem.- The Ascoli-Arzela Theorem.- Differential Equations.- Picard's Existence and Uniqueness Theorem.- The Method of Characteristics.- Power Series Methods.- Fourier Analytic Methods.- Glossary of Terms from Real Variable Theory.- List of Notation.- Guide to the Literature.
Basics.- Sets.- Operations on Sets.- Functions.- Operations on Functions.- Number Systems.- Countable and Uncountable Sets.- Sequences.- to Sequences.- Limsup and Liminf.- Some Special Sequences.- Series.- to Series.- Elementary Convergence Tests.- Advanced Convergence Tests.- Some Particular Series.- Operations on Series.- The Topology of the Real Line.- Open and Closed Sets.- Other Distinguished Points.- Bounded Sets.- Compact Sets.- The Cantor Set.- Connected and Disconnected Sets.- Perfect Sets.- Limits and the Continuity of Functions.- Definitions and Basic Properties.- Continuous Functions.- Topological Properties and Continuity.- Classifying Discontinuities and Monotonicity.- The Derivative.- The Concept of Derivative.- The Mean Value Theorem and Applications.- Further Results on the Theory of Differentiation.- The Integral.- The Concept of Integral.- Properties of the Riemann Integral.- Further Results on the Riemann Integral.- Advanced Results on Integration Theory.- Sequences and Series of Functions.- Partial Sums and Pointwise Convergence.- More on Uniform Convergence.- Series of Functions.- The Weierstrass Approximation Theorem.- Some Special Functions.- Power Series.- More on Power Series: Convergence Issues.- The Exponential and Trigonometric Functions.- Logarithms and Powers of Real Numbers.- The Gamma Function and Stirling's Formula.- An Introduction to Fourier Series.- Advanced Topics.- Metric Spaces.- Topology in a Metric Space.- The Baire Category Theorem.- The Ascoli-Arzela Theorem.- Differential Equations.- Picard's Existence and Uniqueness Theorem.- The Method of Characteristics.- Power Series Methods.- Fourier Analytic Methods.- Glossary of Terms from Real Variable Theory.- List of Notation.- Guide to the Literature.
Rezensionen
"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)
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