| Preface to First Edition | p. ix |
| Preface to Second Edition | p. xi |
| Introduction | |
| Why Numerical Integration? | p. 1 |
| Formal Differentiation and Integration on Computers | p. 3 |
| Numerical Integration and Its Appeal in Mathematics | p. 4 |
| Limitations of Numerical Integration | p. 5 |
| The Riemann Integral | p. 7 |
| Improper Integrals | p. 10 |
| The Riemann Integral in Higher Dimensions | p. 17 |
| More General Integrals | p. 20 |
| The Smoothness of Functions and Approximate Integration | p. 20 |
| Weight Functions | p. 21 |
| Some Useful Formulas | p. 22 |
| Orthogonal Polynomials | p. 28 |
| Short Guide to the Orthogonal Polynomials | p. 33 |
| Some Sets of Polynomials Orthogonal over Figures in the Complex Plane | p. 42 |
| Extrapolation and Speed-Up | p. 43 |
| Numerical Integration and the Numerical Solution of Integral Equations | p. 48 |
| Approximate Integration Over a Finite Interval | |
| Primitive Rules | p. 51 |
| Simpson's Rule | p. 57 |
| Nonequally Spaced Abscissas | p. 60 |
| Compound Rules | p. 70 |
| Integration Formulas of Interpolatory Type | p. 74 |
| Integration Formulas of Open Type | p. 92 |
| Integration Rules of Gauss Type | p. 95 |
| Integration Rules Using Derivative Data | p. 132 |
| Integration of Periodic Functions | p. 134 |
| Integration of Rapidly Oscillatory Functions | p. 146 |
| Contour Integrals | p. 168 |
| Improper Integrals (Finite Interval) | p. 172 |
| Indefinite Integration | p. 190 |
| Approximate Integration Over Infinite Intervals | |
| Change of Variable | p. 199 |
| Proceeding to the Limit | p. 202 |
| Truncation of the Infinite Interval | p. 205 |
| Primitive Rules for the Infinite Interval | p. 207 |
| Formulas of Interpolatory Type | p. 219 |
| Gaussian Formulas for the Infinite Interval | p. 222 |
| Convergence of Formulas of Gauss Type for Singly and Doubly Infinite Intervals | p. 227 |
| Oscillatory Integrands | p. 230 |
| The Fourier Transform | p. 236 |
| The Laplace Transform and Its Numerical Inversion | p. 264 |
| Error Analysis | |
| Types of Errors | p. 271 |
| Roundoff Error for a Fixed Integration Rule | p. 272 |
| Truncation Error | p. 285 |
| Special Devices | p. 295 |
| Error Estimates through Differences | p. 297 |
| Error Estimates through the Theory of Analytic Functions | p. 300 |
| Application of Functional Analysis to Numerical Integration | p. 317 |
| Errors for Integrands with Low Continuity | p. 332 |
| Practical Error Estimation | p. 336 |
| Approximate Integration in Two or More Dimensions | |
| Introduction | p. 344 |
| Some Elementary Multiple Integrals over Standard Regions | p. 346 |
| Change of Order of Integration | p. 348 |
| Change of Variables | p. 348 |
| Decomposition into Elementary Regions | p. 350 |
| Cartesian Products and Product Rules | p. 354 |
| Rules Exact for Monomials | p. 363 |
| Compound Rules | p. 379 |
| Multiple Integration by Sampling | p. 384 |
| The Present State of the Art | p. 415 |
| Automatic Integration | |
| The Goals of Automatic Integration | p. 418 |
| Some Automatic Integrators | p. 425 |
| Romberg Integration | p. 434 |
| Automatic Integration Using Tschebyscheff Polynomials | p. 446 |
| Automatic Integration in Several Variables | p. 450 |
| Concluding Remarks | p. 461 |
| On the Practical Evaluation of Integrals | p. 463 |
| Fortran Programs | p. 480 |
| Bibliography of Algol, Fortran, and PL/I Procedures | p. 509 |
| Bibliography of Tables | p. 518 |
| Bibliography of Books and Articles | p. 524 |
| Index | p. 605 |
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