Integration for engineers and scientists

Cover......Page 1
Title Page......Page 2
Copyright......Page 3
Preface......Page 4
Contents......Page 7
1.1. Introduction......Page 10
1.2. Riemann, Stieltjes and Lesbesgue Integrals......Page 12
1.3. Multiple and Iterated Integrals......Page 17
1.4. Improper and Infinite Integrals......Page 23
1.5. Mean Value Theorems......Page 29
1.6. Inequalities......Page 30
1.7. Indefinite Integrals versus Definite Integrals......Page 34
1.8. Fractional Integration and Differentiation......Page 37
1.9. Line Integrals......Page 38
1.11. Symmetry Arguments......Page 40
Bibliographic Notes and Comments......Page 43
2.1. Introduction......Page 48
2.2. Liouville's Classification of the Elementary Functions......Page 50
2.3. Basic Theorems for Integration in Finite Terms......Page 54
2.4. Practical Integration of Rational Functions......Page 58
2.5. Practical Integration of Algebraic Functions......Page 61
2.6. Elliptic Integrals......Page 66
2.7. Integration of Elementary Transcendental Functions......Page 67
2.8. Symbolic Automatic Integration......Page 72
2.9. Derivation of Integrals from Differential Equations......Page 77
2.10. Approximate Methods......Page 79
2.11. A Practical Example......Page 81
Bibliographic Notes and Comments......Page 84
3.1. Introduction......Page 86
3.2. The Gamma Function......Page 88
3.3. Classical Calculus Methods......Page 93
3.4. Series Methods......Page 99
3.5. Complex Variable Methods......Page 102
3.6. Some General Forms for Definite Integrals......Page 103
3.7. Use of Integral Transforms......Page 105
3.8. Frullanian Integrals......Page 108
3.9. The Willis Expansion......Page 114
3.10. Laplace's Method......Page 117
3.11. Integration By Parts Methods......Page 120
3.12. Concluding Remarks and Examples......Page 124
Bibliographic Notes and Comments......Page 129
4.1. Introduction......Page 134
4.2. Simple Quadrature Formulas with Specified Nodes......Page 136
4.3. Chebyshev's Equal Weight Quadrature Formulas......Page 141
4.4. Gaussian Quadrature......Page 144
4.5. Convergence of Quadrature Formulas......Page 147
4.6. Error Analysis......Page 148
4.7. Compounding and Adaptive Integration......Page 153
4.8. Extrapolation Methods......Page 157
4.9. The Bernstein Quadrature Formula......Page 161
4.10. Monte Carlo Methods......Page 162
4.11. `Best" Quadrature Formulas......Page 163
4.12. Riemann and Riemann-Stieltjes Sums......Page 166
4.13. Integration of Periodic Functions......Page 167
4.14. Improper Integrals......Page 169
4.15. Product Integration......Page 173
4.16. Trigonometric Weight Functions......Page 179
4.17. Integrals Over An Infinite Range......Page 183
4.18. Indefinite Integrals......Page 187
4.19. Multiple Integrals......Page 190
4.20. Linear Integrodifferential Operators......Page 195
Bibliographic Notes and Comments......Page 198
5.1. Introduction......Page 206
5.2. Compound Rules with Correction Terms......Page 207
5.3. Simple Quadrature Rules Using Derivatives......Page 210
5.4. Summation Formulas......Page 214
5.5. The Generalized Midpoint Rule with a Weight Function......Page 215
5.6. Linear Eigenvalue Problems......Page 217
5.7. Boundary-Value Problems......Page 223
Bibliographic Notes and Comments......Page 228
6.1. Introduction......Page 230
6.2. Classification of Integral Equations......Page 231
6.3. Conversion of Differential to Integral Equations......Page 232
6.4. Direct Derivation of Integral Equations......Page 236
6.5. Exact Solution of Integral Equations......Page 240
6.6. Liouville-Neumann Theory......Page 247
6.7. Fredholm Theory......Page 250
6.8. Hilbert-Schmidt Theory......Page 252
6.9. Numerical Solution of Volterra Equations......Page 254
6.10. Numerical Solution of Fredholm Equations......Page 259
6.11. Practical Example......Page 263
Bibliographic Notes and Comments......Page 269
Appendixes......Page 274
APPENDIX 1 List of Doctoral Dissertations on Integration and Integral Equations......Page 276
APPENDIX 2 Integration Functions and Subroutines .......Page 282
APPENDIX 3 Subroutines for Solving Integral Equations......Page 296
AUTHOR INDEX......Page 302
SUBJECT INDEX......Page 308