{"product_id":"fundamental-ideas-of-analysis-isbn-9780471159964","title":"Fundamental Ideas of Analysis","description":"The ideas and methods of mathematics, long central to the physical sciences, now play an increasingly important role in a wide variety of disciplines. Analysis provides theorems that prove that results are true and provides techniques to estimate the errors in approximate calculations. The ideas and methods of analysis play a fundamental role in ordinary differential equations, probability theory, differential geometry, numerical analysis, complex analysis, partial differential equations, as well as in most areas of applied mathematics. \u003cp\u003ePreface\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 1 Preliminaries 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eThe Real Numbers 1\u003c\/p\u003e \u003cp\u003eSets and Functions 6\u003c\/p\u003e \u003cp\u003eCardinality 15\u003c\/p\u003e \u003cp\u003eMethods of Proof 20\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 2 Sequences 27\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eConvergence 27\u003c\/p\u003e \u003cp\u003eLimit Theorems 35\u003c\/p\u003e \u003cp\u003eTwo-state Markov Chains 40\u003c\/p\u003e \u003cp\u003eCauchy Sequences 44\u003c\/p\u003e \u003cp\u003eSupremum and Infimum 52\u003c\/p\u003e \u003cp\u003eThe Bolzano-Weierstrass Theorem 55\u003c\/p\u003e \u003cp\u003eThe Quadratic Map 60\u003c\/p\u003e \u003cp\u003eProjects 68\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 3 The Riemann Integral 73\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eContinuity 73\u003c\/p\u003e \u003cp\u003eContinuous Functions on Closed Intervals 80\u003c\/p\u003e \u003cp\u003eThe Riemann Integral 87\u003c\/p\u003e \u003cp\u003eNumerical Methods 95\u003c\/p\u003e \u003cp\u003eDiscontinuities 103\u003c\/p\u003e \u003cp\u003eImproper Integrals 113\u003c\/p\u003e \u003cp\u003eProjects 119\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 4 Differentiation 121\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eDifferentiable Functions 121\u003c\/p\u003e \u003cp\u003eThe Fundamental Theorem of Calculus  129\u003c\/p\u003e \u003cp\u003eTaylor’s Theorem 134\u003c\/p\u003e \u003cp\u003eNewton’s Method 140\u003c\/p\u003e \u003cp\u003eInverse Functions 147\u003c\/p\u003e \u003cp\u003eFunctions of Two Variables 151\u003c\/p\u003e \u003cp\u003eProjects 159\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 5 Sequences of Functions 163\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003ePointwise and Uniform Convergence 163\u003c\/p\u003e \u003cp\u003eLimit Theorems 169\u003c\/p\u003e \u003cp\u003eThe Supremum Norm 175\u003c\/p\u003e \u003cp\u003eIntegral Equations 183\u003c\/p\u003e \u003cp\u003eThe Calculus of Variations 188\u003c\/p\u003e \u003cp\u003eMetric Spaces 196\u003c\/p\u003e \u003cp\u003eThe Contraction Mapping Principle 203\u003c\/p\u003e \u003cp\u003eNormed Linear Spaces  210\u003c\/p\u003e \u003cp\u003eProjects 219\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 6 Series of Functions 223\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eLim sup and Lim inf 223\u003c\/p\u003e \u003cp\u003eSeries of Real Constants 228\u003c\/p\u003e \u003cp\u003eThe Weierstrass M-test 238\u003c\/p\u003e \u003cp\u003ePower Series 245\u003c\/p\u003e \u003cp\u003eComplex Numbers 252\u003c\/p\u003e \u003cp\u003eInfinite Products and Prime Numbers 260\u003c\/p\u003e \u003cp\u003eProjects 270\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 7 Differential Equations 273\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eLocal Existence 273\u003c\/p\u003e \u003cp\u003eGlobal Existence 283\u003c\/p\u003e \u003cp\u003eThe Error Estimate for Euler’s Method 289\u003c\/p\u003e \u003cp\u003eProjects 296\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 8 Complex Analysis 299\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eAnalytic Functions 299\u003c\/p\u003e \u003cp\u003eIntegration on Paths 305\u003c\/p\u003e \u003cp\u003eCauchy's Theorem 312\u003c\/p\u003e \u003cp\u003eProjects 320\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 9 Fourier Series 323\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eThe Heat Equation 323\u003c\/p\u003e \u003cp\u003eDefinitions and Examples 331\u003c\/p\u003e \u003cp\u003ePointwise Convergence 337\u003c\/p\u003e \u003cp\u003eMean-square Convergence 345\u003c\/p\u003e \u003cp\u003eProjects 355\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 10 Probability Theory 359\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eDiscrete Random Variables 359\u003c\/p\u003e \u003cp\u003eCoding Theory 368\u003c\/p\u003e \u003cp\u003eContinuous Random Variables 376\u003c\/p\u003e \u003cp\u003eThe Variation Metric 386\u003c\/p\u003e \u003cp\u003eProjects 398\u003c\/p\u003e \u003cp\u003eBibliography 403\u003c\/p\u003e \u003cp\u003eSymbol Index 406\u003c\/p\u003e \u003cp\u003eIndex 409\u003c\/p\u003e \u003cp\u003e\u003cb\u003eMichael C. Reed\u003c\/b\u003e received the Bachelor’s degree in mathematics from Yale (1963) and the Ph.D. in mathematics from Stanford (1969). He has worked in many branches of mathematics including functional analysis, mathematical physics, partial differential equations, and the applications of mathematics to human physiology and medicine. He is currently Arts and Sciences Distinguished Professor of Mathematics at Duke University, where he uses Fundamental Ideas of Analysis to teach undergraduate and graduate students.\u003c\/p\u003e","brand":"Wiley","offers":[{"title":"Default Title","offer_id":47989254914277,"sku":"NP9780471159964","price":190.0,"currency_code":"USD","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/1842\/7735\/files\/9780471159964.jpg?v=1761783397","url":"https:\/\/k12savings.com\/products\/fundamental-ideas-of-analysis-isbn-9780471159964","provider":"K12savings","version":"1.0","type":"link"}