Fundamental Mathematical Analysis

Preface......Page 7
Contents......Page 11
1.2 Milestones in the History of Analysis......Page 21
2.1 Natural Numbers and Set Theory......Page 23
2.1.1 Exercises......Page 26
2.2.1 Arithmetic Axioms......Page 27
2.2.2 Axioms of Ordering......Page 29
2.2.4 mathbbQ is Insufficient for Analysis......Page 31
2.2.5 Dedekind Sections......Page 32
2.2.7 Square Root of 2......Page 33
2.2.8 Exercises......Page 34
2.2.9 The Functions Max, Min, and Absolute Value......Page 35
2.2.10 Mathematical Analysis......Page 37
2.2.11 Exercises (cont'd)......Page 38
2.3 Decimal Fractions......Page 40
2.3.1 Practical and Theoretical Meaning of Decimals......Page 41
2.3.2 Algorithm for Decimals......Page 42
2.3.4 Repeating Decimals and Geometric Series......Page 43
2.4.1 Intervals......Page 44
2.4.3 Bounded Subsets of mathbbR......Page 45
2.4.4 Supremum and Infimum......Page 46
2.4.6 Supremum or Maximum?......Page 47
2.4.7 Using Supremum and Infimum to Prove Theorems......Page 48
2.4.9 Exercises (cont'd)......Page 49
2.5 Approximation by Rational Numbers......Page 50
2.5.1 Exercises......Page 51
3.1.1 The Notion of Sequence......Page 54
3.1.2 Defining a Sequence by Induction......Page 56
3.1.3 Infinite Series......Page 57
3.2 Limits......Page 58
3.2.1 Writing the Definition of Limit in English, and in Logic......Page 59
3.2.3 Exercises......Page 61
3.2.4 Free Variables and Bound Variables......Page 62
3.2.5 Proving Things Using the Definition of Limit......Page 63
3.2.6 Denying That limntoinftyan=t......Page 64
3.2.7 Two Fundamental Limits......Page 65
3.2.9 The Limits infty and -infty......Page 66
3.2.10 Exercises (cont'd)......Page 67
3.3 Monotonic Sequences......Page 68
3.3.1 Limits and Inequalities......Page 69
3.3.2 Exercises......Page 70
3.4 Limit Rules......Page 71
3.4.1 Exercises......Page 74
3.5 Limit Points of Sets......Page 76
3.5.1 Weierstrass's Theorem on Limit Points......Page 77
3.5.2 Exercises......Page 78
3.6 Subsequences......Page 79
3.6.1 Exercises......Page 81
3.7 Cauchy's Convergence Principle......Page 82
3.8 Convergence of Series......Page 83
3.8.2 Convergence Tests......Page 84
3.8.3 The Simplest Convergence Tests: Positive Series......Page 85
3.8.4 Geometric Series and D'Alembert's Test......Page 86
3.8.5 Exercises......Page 87
3.8.6 The Series sumn=1infty1/np......Page 89
3.8.7 Telescoping Series......Page 90
3.8.8 Exercises (cont'd)......Page 91
3.9 Decimals Reprised......Page 92
3.10 () Philosophical Implications of Decimals......Page 96
3.10.1 Pointers to Further Study......Page 98
3.11 () Limit Inferior and Limit Superior......Page 99
3.11.1 Exercises......Page 101
3.11.2 Uses of Limit Inferior and Limit Superior......Page 102
3.12 () Continued Fractions......Page 104
3.12.1 Exercises......Page 107
3.12.2 Pointers to Further Study......Page 108
4.1 How Do We Talk About Functions?......Page 109
4.1.1 Examples of Specifying Functions......Page 110
4.2 Continuous Functions......Page 111
4.2.2 Limits of Functions......Page 113
4.2.4 Limit Rules......Page 115
4.2.6 Left and Right Limits......Page 117
4.2.8 The Limits pminfty......Page 119
4.2.9 Exercises (cont'd)......Page 120
4.2.10 Bounded Functions......Page 121
4.2.11 Monotonic Functions......Page 122
4.2.12 Discontinuities of Monotonic Functions......Page 123
4.2.14 Composite Functions......Page 124
4.2.15 Limits of Functions and Limits of Sequences......Page 125
4.2.16 Iterations......Page 126
4.2.17 Exercises (cont'd)......Page 127
4.3.1 The Intermediate Value Theorem......Page 129
4.3.2 Thoughts About the Proof of the Intermediate Value Theorem......Page 130
4.3.4 The Boundedness Theorem......Page 131
4.3.6 The Extreme Value Theorem......Page 133
4.3.7 Using the Extreme Value Theorem......Page 134
4.3.8 Exercises......Page 135
4.4 Inverses of Monotonic Functions......Page 136
4.5 Two Important Technical Propositions......Page 139
4.5.1 The Oscillation of a Function......Page 140
4.5.3 Exercises......Page 142
4.6 () Iterations of Monotonic Functions......Page 143
4.6.2 Pointers to Further Study......Page 145
5.1 The Definition of Derivative......Page 146
5.1.1 Differentiability and Continuity......Page 147
5.1.2 Derivatives of Some Basic Functions......Page 148
5.1.3 Exercises......Page 149
5.2 Differentiation Rules......Page 151
5.2.1 Differentiation of the Power Function......Page 152
5.2.2 The Chain Rule......Page 153
5.2.3 Differentiation of Inverse Functions......Page 155
5.2.5 Exercises......Page 156
5.3.1 Tangent Lines......Page 158
5.3.2 Differential Quotients......Page 159
5.3.3 The Chain Rule and Inverse Functions in Leibniz's Notation......Page 160
5.3.4 Tangents to Plane Curves......Page 161
5.3.5 Exercises......Page 163
5.4.1 Exercises......Page 164
5.5 Significance of the Derivative......Page 166
5.5.1 Maxima and Minima......Page 168
5.5.3 Exercises......Page 169
5.6 The Mean Value Theorem......Page 170
5.6.1 First Consequences of the Mean Value Theorem......Page 171
5.6.2 Exercises......Page 172
5.7.1 Higher Derivatives and Taylor Polynomials......Page 173
5.7.2 Comparison to Taylor's Theorem......Page 174
5.7.4 Geometric Interpretation of the Mean Value Theorem......Page 175
5.7.5 Exercises......Page 177
5.8 L'Hopital's Rule......Page 178
5.8.1 Using L'Hopital's Rule......Page 180
5.8.4 Iterative Use of L'Hopital's Rule: Taylor Polynomials Again......Page 181
5.8.5 Application to Maxima and Minima......Page 183
5.8.6 More on L'Hopital's Rule: The infty/infty Version......Page 184
5.8.7 Exercises......Page 186
5.9 () Multiplicity......Page 188
5.9.1 Exercises......Page 189
5.9.2 Sturm's Theorem......Page 190
5.9.3 Exercises (cont'd)......Page 192
5.10 Convex Functions......Page 193
5.10.1 Tangent Lines and Convexity......Page 195
5.10.2 Inflection Points......Page 197
5.10.3 Exercises......Page 198
5.11 () Jensen's Inequality......Page 201
5.12 () How Fast Do Iterations Converge?......Page 203
5.12.1 The Babylonian Method......Page 206
5.12.2 Newton's Method......Page 207
5.12.3 Exercises......Page 208
5.12.4 Pointers to Further Study......Page 210
6.1 Two Unlike Problems......Page 211
6.2 Defining the Riemann–Darboux Integral......Page 212
6.2.1 Thoughts on the Definition......Page 215
6.3 First Results on Integrability......Page 216
6.3.1 Riemann's Condition......Page 217
6.3.2 Integrability of Continuous Functions and Monotonic Functions......Page 218
6.3.3 Two Simple Integrals Computed......Page 220
6.4 Basic Integration Rules......Page 221
6.4.2 The Integral from a to b......Page 224
6.4.4 Useful Estimates......Page 225
6.4.5 Exercises......Page 227
6.5 The Connection Between Integration and Differentiation......Page 230
6.5.1 Thoughts About the Fundamental Theorem......Page 233
6.5.2 Exercises......Page 235
6.6 () Riemann Sums......Page 236
6.6.1 Things You Can Do with Riemann Sums......Page 240
6.6.3 Pointers to Further Study......Page 241
6.7 () The Arc Length, Volume and Surface of Revolution Integrals......Page 242
6.7.1 Length of Parametric Curves......Page 243
6.7.2 Exercises......Page 244
6.7.3 Volumes and Surfaces of Revolution......Page 245
6.7.4 Exercises (cont'd)......Page 246
6.8 () Approximation by Step Functions......Page 247
6.8.1 Exercises......Page 251
6.8.2 Pointers to Further Study......Page 252
7 The Elementary Transcendental Functions......Page 253
7.1.1 First Steps Towards Defining Sine and Cosine......Page 254
7.1.3 Extending sinx......Page 257
7.1.4 Defining Cosine......Page 258
7.1.6 Addition Rules for Sine and Cosine......Page 259
7.1.8 The Trigonometric Functions tanx, cotx, secx, cscx......Page 260
7.1.9 The Derivatives of arcsinx, arccosx and arctanx......Page 261
7.1.10 Exercises......Page 262
7.2.1 Defining the Natural Logarithm and the Exponential Function......Page 263
7.2.2 Exponentials and Logarithms with Base a......Page 265
7.2.3 The Laws of Logarithms and Exponents......Page 266
7.2.5 Exponential Growth......Page 267
7.2.6 Hyperbolic Functions......Page 268
7.2.8 The Antiderivative int(1/x) dx......Page 269
7.2.9 Exercises......Page 270
7.3 () Defining Transcendental Functions......Page 273
7.3.1 Exercises......Page 275
7.3.2 Pointers to Further Study......Page 278
8.1 Integration by Parts and by Substitution......Page 279
8.1.1 Finding Antiderivatives by Substitution......Page 281
8.1.2 Exercises......Page 282
8.2 Integrating Rational Functions......Page 287
8.2.2 Practicalities......Page 288
8.2.3 Outline of Proof......Page 290
8.2.4 How to Integrate the Fractions......Page 292
8.2.5 Integrating Rational Functions of sinθ and cosθ......Page 293
8.2.6 Further Useful Reduction Formulas......Page 294
8.2.7 Exercises......Page 295
8.3 () Ostrogradski's Method......Page 297
8.4 () Numerical Integration......Page 300
8.4.2 Midpoint Rule......Page 301
8.4.4 Proof of the Error Estimate......Page 302
8.4.5 Exercises......Page 304
8.4.6 Pointers to Further Study......Page 305
9.1 The Complex Number Field......Page 306
9.1.1 Square Roots......Page 308
9.1.2 Modulus and Conjugate......Page 309
9.1.3 Exercises......Page 311
9.2.1 nth Root of a Complex Number......Page 312
9.2.2 Logarithm of a Complex Number......Page 313
9.2.3 Exercises......Page 316
10.1 The Limit of a Complex Sequence......Page 320
10.2.1 Absolutely Convergent Series......Page 322
10.2.3 Extended Forms of the Ratio and Cauchy's Tests......Page 323
10.2.4 Conditional Convergence: Leibniz's Test......Page 324
10.2.5 Rearrangements of Absolutely Convergent Series......Page 326
10.2.6 Exercises......Page 327
10.3 Product of Series......Page 329
10.3.1 Cauchy Product of Series......Page 331
10.3.2 Exercises......Page 332
10.4 () Riemann's Rearrangement Theorem......Page 333
10.5 () Gauss's Test......Page 335
10.5.2 Pointers to Further Study......Page 338
11.1 Problems with Convergence......Page 339
11.2 Pointwise Convergence and Uniform Convergence......Page 341
11.2.1 Cauchy's Principle for Uniform Convergence......Page 342
11.2.3 Uniform Convergence of Series......Page 343
11.2.4 Cauchy's Principle for Uniform Convergence of Function Series......Page 344
11.2.5 Integration and Uniform Convergence......Page 345
11.2.6 Differentiation and Uniform Convergence of Series......Page 346
11.2.7 Exercises......Page 347
11.3 Power Series......Page 350
11.3.1 Radius of Convergence......Page 351
11.3.2 Determining the Radius of Convergence by the Ratio Test......Page 352
11.3.4 The Exponential Series......Page 353
11.3.5 The Number e......Page 355
11.3.6 Differentiating a Power Series......Page 356
11.3.7 Exercises......Page 358
11.4.1 Unification of Exponential and Circular Functions......Page 359
11.4.2 The Binomial Series......Page 360
11.4.3 Series for Arctangent......Page 363
11.4.4 Abel's Lemma and Dirichlet's Test......Page 364
11.4.5 Abel's Theorem on Power Series......Page 366
11.4.6 Exercises......Page 367
11.5 () Summability Theories......Page 374
11.5.1 Abelian Theorems and Tauberian Theorems......Page 375
11.5.2 Exercises......Page 376
11.6 () The Irrationality of e and π......Page 377
11.6.1 Solutions......Page 378
11.7 Taylor Series......Page 380
11.7.1 Taylor's Theorem with Lagrange's Remainder......Page 381
11.7.2 Error Estimates......Page 384
11.7.3 Error Estimates for ln(1+x)......Page 385
11.7.4 Error Estimates for (1+x)a......Page 386
11.7.5 Taylor's Theorem with Cauchy's Remainder......Page 387
11.7.6 Taylor's Theorem with the Integral Form of the Remainder......Page 389
11.7.7 Exercises......Page 391
11.8 () Bernoulli Numbers......Page 395
11.8.1 Computing the Bernoulli Numbers......Page 399
11.8.3 Pointers to Further Study......Page 400
11.9 () Asymptotic Orders of Magnitude......Page 401
11.9.1 Asymptotic Expansions......Page 403
11.10 () Stirling's Approximation......Page 405
11.10.1 Proofs......Page 406
11.10.2 Exercises......Page 408
11.10.3 Pointers to Further Study......Page 409
12.1 Unbounded Domains and Unbounded Integrands......Page 410
12.1.1 Key Examples of Improper Integrals......Page 412
12.1.2 The Comparison Test for Improper Integrals......Page 413
12.1.3 Exercises......Page 414
12.2 Differentiation Under the Integral Sign......Page 416
12.2.1 Exercises......Page 417
12.3 The Maclaurin–Cauchy Theorem......Page 418
12.3.1 The Euler–Mascheroni Constant......Page 420
12.3.2 Exercises......Page 421
12.4 Complex-Valued Integrals......Page 422
12.4.1 Absolutely Convergent Integrals......Page 424
12.4.2 Exercises......Page 425
12.5.1 Fourier Transform......Page 431
12.5.2 Exercises......Page 432
12.5.4 Exercises (cont'd)......Page 433
12.6 () The Gamma Function......Page 435
12.6.1 Exercises......Page 436
12.6.2 Pointers to Further Study......Page 439
Appendix Afterword and Acknowledgements......Page 440
Index......Page 441