Unified Integration (eBook)

Front Cover 1
Unified Integration 4
Copyright Page 5
Contents 6
Preface 10
Acknowledgments 14
Chapter 0. Introduction 16
1. Functions, Intervals, and Limits 16
2. Bounds 21
Chapter I. Elementary Properties of the Integral in One-Dimensional Space 26
1. A Heuristic Approach to the Definition of the Integral 26
2. Definition of the Integral 32
3. Examples 37
4. Existence of .-Fine Partitions 41
5. Elementary Computational Formulas 47
6. Order Properties of the Integral 52
7. Comparison with the Riemann Integral 55
8. Additivity of the Integral 60
9. The Fundamental Theorem of the Calculus, First Part 63
10. The Fundamental Theorem of the Calculus, Second Part 68
11. Substitution and Integration by Parts 72
12. Estimates of Integrals 75
Chapter II. Integration in One-Dimensional Space: Further Development 85
1. A Condition for Integrability 85
2. Absolute Integrability 88
3. Integration of Composite Functions 93
4. The Monotone Convergence Theorem 98
5. Integrals of Products 107
6. Power Series 111
7. “Improper” Integrals 127
8. Examples 131
9. Continuity of the Indefinite Integral: Integration by Parts and Substitution 139
10. The Dominated Convergence Theorem 148
11. Differentiation under the Integration Sign 152
12. Sets of Measure 0 160
13. Approximation by Step-Functions 170
14. Differentiation of Indefinite Integrals 175
15. Calculus of Variations 180
Chapter III. Applications to Differential Equations and to Probability Theory 190
1. Ordinary Differential Equations 190
2. Existence Theorems for Solutions of Differential Equations 195
3. Effects of Change of Data 203
4. Computation of Approximate Solutions 208
5. Linear Differential Equations 215
6. Differentiation of Solutions with Respect to Initial Values and Parameters 220
7. Definition of a More General Integral 228
8. Properties of the Integral 232
9. Densities 237
10. Measurable Functions and Measurable Sets 244
11. Applications to Probability Theory 253
12. Examples 258
Chapter IV. Integration in Spaces of More Than One Dimension 267
1. Notation and Definitions 267
2. Elementary Properties of Intervals and Measure 269
3. Generalizations to Integration in Rr of Theorems in Preceding Chapters 272
4. Iterated Integration 276
5. Change of Variables in Multiple Integrals 288
6. Approximation of Sets by Unions of Intervals and of Integrable Functions by Limits of Step-Functions 307
7. A Second Form of Fubini's Theorem 315
8. Second Form of the Substitution Theorem 319
9. Integration with Respect to Other Measures 323
10. Applications to Probability Theory: Multivariate Distributions 327
11. Independence 333
12. Convolutions 349
13. The Central Limit Theorem 358
14. Distributions in Some Infinite-Dimensional Spaces 365
Chapter V. Line Integrals and Areas of Surfaces 378
1. Geometry in r-Dimensional Space 378
2. Vectors 390
3. Inner Products and Length-Preserving Maps 396
4. Covectors 405
5. Differentiation and Integration of Vector-Valued Functions 410
6. Curves and Their Lengths 416
7. Line Integrals 424
8. The Behavior of Inscribed Polyhedra 437
9. Areas of Surfaces 440
Chapter VI. Vector Spaces, Orthogonal Expansions, and Fourier Transforms 453
1. Complex Vector Spaces 453
2. The Spaces L1 and L2 457
3. Normed Vector Spaces 461
4. Completeness of Spaces L1, L2, L1, and L2 468
5. Hilbert Spaces and Their Geometry 477
6. Approximation by Step-Functions and by Differentiable Functions 488
7. Fourier Series 494
8. Indefinite Integrals and the Weierstrass Approximation Theorem 503
9. Legendre Polynomials 507
10. The Hermite Polynomials and the Hermite Functions 511
11. The Schrödinger Equation for the Harmonic Oscillator 515
12. The Fourier Transform for Certain Smooth Functions 519
13. The Fourier–Plancherel Transform 526
14. The Fourier Transformation and the Fourier–Plancherel Transformation 535
15. Applications to Differential Equations 544
Chapter VII. Measure Theory 550
1. s-Algebras and Measurable Functions 550
2. Definition of the Lebesgue Integral 558
3. Bore1 Sets and Borel-Measurable Functions 571
4. Integration with Respect to Other Functions of Sets 574
5. The Radon-Nikodým Theorem 580
6. Conditional Expectations 592
7. Brownian Motion 606
Index 618
Pure and Applied Mathematics 623