| Contents | 6 |
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| Preface | 9 |
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| Acknowledgements | 13 |
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| Notation | 14 |
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| Basic Results | 16 |
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| 1.1 Introduction | 16 |
| 1.2 Transforms of Elementary Functions | 17 |
| 1.3 Transforms of Derivatives and Integrals | 20 |
| 1.4 Inverse Transforms | 23 |
| 1.5 Convolution | 24 |
| 1.6 The Laplace Transforms of some Special Functions | 26 |
| 1.7 Difference Equations and Delay Differential Equations | 29 |
| 1.8 Multidimensional Laplace T ransforms | 33 |
| Inversion Formulae and Practical Results | 38 |
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| 2.1 The Uniqueness Property | 38 |
| 2.2 The Bromwich Inversion Theorem | 41 |
| 2.3 The Post-Widder Inversion Formula | 52 |
| 2.4 Initial and Final Value Theorems | 54 |
| 2.5 Series and Asymptotic Expansions | 57 |
| 2.6 Parseval's Formulae | 58 |
| The Method of Series Expansion | 60 |
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| 3.1 Expansion as a Power Series | 60 |
| 3.2 Expansion in terms of Orthogonal Polynomials | 64 |
| 3.3 Multi-dimensional Laplace transform inversion | 81 |
| Quadrature Methods | 86 |
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| 4.1 Interpolation and Gaussian type Formulae | 86 |
| 4.2 Evaluation of Trigonometric Integrals | 90 |
| 4.3 Extrapolation Methods | 92 |
| 4.4 Methods using the Fast Fourier Transform ( FFT ) | 96 |
| 4.5 Hartley Transforms | 106 |
| 4.6 Dahlquist's | 106 |
| 4.6 Dahlquist's | 106 |
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| 110 | 106 |
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| 4.7 Inversion of two-dimensional transforms | 115 |
| Rational Approximation Methods | 117 |
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| 5.1 The Laplace Transform is Rational | 117 |
| 5.2 The least squares approach to rational Approximation | 120 |
| 5.3 Pade, Pade-type and Continued Fraction Approximations | 125 |
| 5.4 Multidimensional Laplace Transforms | 133 |
| The Method of Talbot | 135 |
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| 6.1 Early Formulation | 135 |
| 6.2 A more general formulation | 137 |
| 6.3 Choice of Parameters | 139 |
| 6.4 Additional Practicalities | 143 |
| 6.5 Subsequent development of Talbot's method | 144 |
| 6.6 Multi-precision Computation | 152 |
| Methods based on the Post - Widder Inversion Formula | 154 |
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| 7.1 Introduction | 154 |
| 7.2 Methods akin to Post-Widder | 156 |
| 7.3 Inversion of Two-dimensional Transforms | 159 |
| The Method of Regularization | 160 |
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| 8.1 Introduction | 160 |
| 8.2 Fredholm equations of the first kind - theoretical considerations | 161 |
| 8.3 The method of Regularization | 163 |
| 8.4 Application to Laplace Transforms | 164 |
| Survey Results | 169 |
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| 9.1 Cost's Survey | 169 |
| 9.2 The Survey by Davies and Martin | 170 |
| 9.3 Later Surveys | 172 |
| 9.4 Test Transforms | 180 |
| Applications | 181 |
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| 10.1 Application 1. Transient solution for the Batch Service Queue M=MN=1 | 181 |
| 10.2 Application 2. Heat Conduction in a Rod | 190 |
| 10.3 Application 3. Laser Anemometry | 193 |
| 10.4 Application 4. Miscellaneous Quadratures | 200 |
| 10.5 Application 5. Asian Options | 204 |
| Appendix | 209 |
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| 11.1 T able of Laplace T ransforms | 210 |
| 11.2 The Fast Fourier Transform (FFT) | 216 |
| 11.3 Quadrature Rules | 218 |
| 11.4 Extrapolation Techniques | 224 |
| 11.5 Pade Approximation | 232 |
| 11.6 The method of Steepest Descent | 238 |
| 11.7 Gerschgorin's theorems and the Companion Matrix | 239 |
| Bibliography | 242 |
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| Index | 260 |