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Klappentext 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. Zusammenfassung 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. Inhaltsverzeichnis Preface Chapter 1 Preliminaries 1 The Real Numbers 1 Sets and Functions 6 Cardinality 15 Methods of Proof 20 Chapter 2 Sequences 27 Convergence 27 Limit Theorems 35 Two-state Markov Chains 40 Cauchy Sequences 44 Supremum and Infimum 52 The Bolzano-Weierstrass Theorem 55 The Quadratic Map 60 Projects 68 Chapter 3 The Riemann Integral 73 Continuity 73 Continuous Functions on Closed Intervals 80 The Riemann Integral 87 Numerical Methods 95 Discontinuities 103 Improper Integrals 113 Projects 119 Chapter 4 Differentiation 121 Differentiable Functions 121 The Fundamental Theorem of Calculus 129 Taylor's Theorem 134 Newton's Method 140 Inverse Functions 147 Functions of Two Variables 151 Projects 159 Chapter 5 Sequences of Functions 163 Pointwise and Uniform Convergence 163 Limit Theorems 169 The Supremum Norm 175 Integral Equations 183 The Calculus of Variations 188 Metric Spaces 196 The Contraction Mapping Principle 203 Normed Linear Spaces 210 Projects 219 Chapter 6 Series of Functions 223 Lim sup and Lim inf 223 Series of Real Constants 228 The Weierstrass M-test 238 Power Series 245 Complex Numbers 252 Infinite Products and Prime Numbers 260 Projects 270 Chapter 7 Differential Equations 273 Local Existence 273 Global Existence 283 The Error Estimate for Euler's Method 289 Projects 296 Chapter 8 Complex Analysis 299 Analytic Functions 299 Integration on Paths 305 Cauchy's Theorem 312 Projects 320 Chapter 9 Fourier Series 323 The Heat Equation 323 Definitions and Examples 331 Pointwise Convergence 337 Mean-square Convergence 345 Projects 355 Chapter 10 Probability Theory 359 Discrete Random Variables 359 Coding Theory 368 Continuous Random Variables 376 The Variation Metric 386 Projects 398 Bibliography 403 Symbol Index 406 Index 409 ...
