A handbook of real variables : with appllications to differential equations and Fourier analysis

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. Content Level » Research Keywords » Fourier analysis - ODEs - functional analysis - ksa - real analysis Related subjects » Birkhäuser Mathematics 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. Preface -- Basics -- Sequences -- Series -- The Topology Of The Real Line -- Limits And The Continuity Of Functions -- The Derivative -- The Integral -- Sequences And Series Of Functions -- Some Special Functions -- Advanced Topics -- Differential Equations -- Glossary Of Terms From Real Variable Theory -- List Of Notation -- A Guide To The Literature -- Bibliography -- Index. By Steven G. Krantz. 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. This concise, well-written handbook provides a distillation of real variable theorywith a particular focus on the subject's significant applications to differential equations and Fourier analysis. Ample examples and brief explanations---with very few proofs and little axiomatic machinery---are used to highlight all the major results of real analysis, from the basics of sequences and series to the more advanced concepts of Taylor and Fourier series, Baire Category, and the Weierstrass Approximation Theorem. Replete with realistic, meaningful applications to differential equations, boundary value problems, and Fourier analysis, this unique work is a practical, hands-on manual of real analysis that is ideal for physicists, engineers, economists, and others who wish to use the fruits of real analysis but who do not necessarily have the time to appreciate all of the theory. Valuable as a comprehensive reference, a study guide for students, or a quick review, "A Handbook of Real Variables" will benefit a wide audience. 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 This concise real analysis handbook takes into account the fundamentals of the classical theory of the subject and sheds light on its significant applications to differential equations and Fourier analysis. It de-emphasizes proofs and instead stresses concepts, examples and insights