Elementary Real and Complex Analysis

Title......Page 1
Contents......Page 3
Preface......Page 7
1.1. Set-Theoretic Preliminaries......Page 11
1.2. Axioms for the Real Number System......Page 13
1.4. Consequences of the Multiplication Axioms......Page 15
1.5. Consequences of the Order Axioms......Page 19
1.6. Consequences of the Least Upper Bound Axiom......Page 22
1.7. The Principle of Archimedes and Its Consequences......Page 26
1.8. The Principle of Nested Intervals......Page 31
1.9. The Extended Real Number System......Page 33
2.1. Operations on Sets......Page 37
2.2. Equivalence of Sets......Page 39
2.3. Countable Sets......Page 41
2.4. Uncountable Sets......Page 43
2.5. Mathematical Structures......Page 45
2.6. n-Dimensional Space......Page 49
2.7. Complex Numbers......Page 53
2.8. Functions and Graphs......Page 59
Problems......Page 60
3.1. Definitions and Examples......Page 63
3.2. Open Sets......Page 71
3.3. Convergent Sequences and Homemorphisms......Page 73
3.4. Limit Points......Page 83
3.5. Closed Sets......Page 86
3.6. Dense Sets and Closures......Page 88
3.7. Complete Metric Spaces......Page 90
3.8. Completion of a Metric Space......Page 97
3.9. Compactness......Page 101
Problems......Page 107
4.1. Basic Concepts......Page 109
4.2. Some General Theorems......Page 117
4.3. Limits of Numerical Functions......Page 118
4.4. Upper and Lower Limits......Page 127
4.5. Nondecreasing and Nonincreasing Functions......Page 130
4.6. Limits of Numerical Sequences......Page 133
4.7. Limits of Vector Functions......Page 136
Problems......Page 139
5.1. Continuous Functions on a Metric Space......Page 142
5.2. Continuous Numerical Functions on the Real Line......Page 149
5.3. Monotonic Functions......Page 152
5.4. The Lograithm......Page 155
5.5. The Exponential......Page 159
5.6. Trigonometric Functions......Page 167
5.7. Applications of Trigonometric Functions......Page 175
5.8. Continuous Vector Functions of a Vector Variable......Page 182
5.9. Sequences of Functions......Page 190
Problems......Page 193
6.1. Numerical Series......Page 196
6.2. Absolute and Conditional Convergence......Page 204
6.3. Operations on Series......Page 206
6.4. Series of Vectors......Page 213
6.5. Series of Functions......Page 222
6.6. Power Series......Page 223
Problems......Page 228
7.1. Definitions and Examples......Page 233
7.2. Properties of Differentiable Functions......Page 243
7.3. The Differential......Page 246
7.4. Mean Value Theorems......Page 247
7.5. Concavity and Inflection Points......Page 250
7.6. L\'Hospital\'s Rules......Page 253
Problems......Page 256
8.1. Definitions and Examples......Page 259
8.2. Taylor\'s Formula......Page 261
8.3. More on Concavity and Inflection Points......Page 265
8.4. Another Version of Taylor\'s Formula......Page 267
8.5. Taylor Series......Page 269
8.6. Complex Exponentials and Trigonometric Functions......Page 272
8.7. Hyperbolic Functions......Page 277
Problems......Page 280
9.1. Definitions and Basic Properties......Page 284
9.2. Area and Arc Length......Page 297
9.3. Antiderivatives and Indefinite Integrals......Page 301
9.4. Technique oof Indefinite Integration......Page 306
9.5. Evaluation od Definite Integrals......Page 318
9.6. More on Area......Page 327
9.7. More on Arc Length......Page 338
9.8. Area of a Surface of Revolution......Page 345
9.9. Further Applications of Integration......Page 352
9.10. Integration of Sequences of Functions......Page 359
9.11. Parameter-Dependent Integrals......Page 365
9.12. Line Integrals......Page 371
Problems......Page 380
10.1 Basic Concepts......Page 383
10.2 Line Integrals of Complex Functions......Page 391
10.3. Cauchy\'s Theorem and Its Consequences......Page 399
10.4. Residues and Isolated Singular Points......Page 415
10.5. Mappings and Elementary Functions......Page 426
Problems......Page 437
11.1. Improper Integrals of the First Kind......Page 441
11.2. Convergence of Improper Integrals......Page 448
11.3. Improper Integrals of the Second and Third Kinds......Page 453
11.4. Evaluation of Improper Integrals by Residues......Page 457
11.5. Parameter-Dependent Improper Integrals......Page 467
11.6. The Gramma and Beta Functions......Page 477
Problems......Page 489
Appendix A Elementary Symbolic Logic......Page 492
Appendix B Measure and Integration on a Compact Metric Space......Page 494
Selected Hints and Answers......Page 499
Index......Page 515