Description
Book SynopsisThe standard topics for a one-term undergraduate real analysis course are covered in this book. In addition, examples are given that show the ways in which real analysis is used in ordinary and partial differential equations, probability theory, numerical analysis, and number theory.
Table of ContentsPreface
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
