In this post, we will see the book A Handbook Of Methods Of Approximate Fourier Transformation And Inversion Of The Laplace Transformation by V. I. Krylov; N. S. Skoblya.
List of symbols 8
Preface 9
Part One
INVERSION OF THE LAPLACE TRANSFORMATION
Chapter 1. Introduction
1.1 Basic concepts in the theory of the Laplace transformation 15
1.2 Complex integral for computing inverse Laplace transforms 22
1.3 Representing functions by the Laplace integral 25
1.4 Ill-conditioning of the problem of computing inverse Laplace transforms 30
Chapter 2. Some Analytical Methods for Computing Inverse Laplace Transforms
2.1 Finding the original function via the inversion formula 32
2.2 Expanding the original function into power series 36
2.3 Expanding the original function into generalized power series 38
Chapter 3. Methods of Numerical Inversion of Laplace Transforms Based on the Use of Special Expansions
3.1 Computing inverse Laplace transforms by polynomials orthogonal on a finite interval 41
3.2 Computing inverse Laplace transforms with the aid of the Fourier sine series 67
3.3 Computing inverse Laplace transforms with the aid
of series in terms of generalized Chebyshev-Laguerre polynomials 70
Chapter 4. Methods of Computing the Mellin Integral with the Aid of Interpolation Quadrature Formulas
4.1 The general theory of interpolation methods 75
4.2 The equal-interval interpolation method 79
4.3 The unequal-interval interpolation method 80
4.4 Other interpolation methods. Using the truncated Taylor series 90
4.5 Some theorems on convergence of interpolation . 92
4.6 Theorems on the convergence of interpolation methods of inversion 105
Chapter 5. Methods of Numerical Inversion of Laplace Trans forms via Quadrature Formulas of Highest Accuracy
5.1 The theory of quadrature formulas 110
5.2 Orthogonal polynomials connected with the quadrature formula of highest accuracy 118
5.3 Methods for computing the coefficients and points of a quadrature formula 136
Chapter 6. Methods of Inverting Laplace Transforms via Quadrature Formulas with Equal Coefficients
6.1 Constructing a computation formula 145
6.2 Remark on the spacing of points 148
Part Two
FOURIER TRANSFORMS AND THEIR APPLICATION TO INVERSION OF LAPLACE TRANSFORMS
Chapter 7. Introduction
7.1 Fourier transforms 150
7.2 Reducing integrals of the Mellin type to the Fourier transformation 157
Chapter 8. Inversion of Laplace Transforms by Means of the Fourier Series
8.1 The case of a rapidly decreasing original function / (x) 159
8.2 The case of rapid decrease of the modulus of the image function F (p) 161
Chapter 9.Interpolation Formulas for Computing Fourier Integrals
9.1 Some preliminary remarks 164
9.2 Algebraic interpolation of the function f (x) 166
9.3. Interpolation by rational functions 201
Chapter 10. Highest-Accuracy Formulas for Computation
10.1 Introduction 233
10.2 Constructing a formula of highest degree of accuracy 236
Part Three
ISOLATING SINGULARITIES OF A FUNCTION IN COMPUTATIONS
Chapter 11. Isolating Singularities of the Image Function F (p)
11.1 Introduction 245
11.2 Removing and weakening the singularities of the image function F (p) 247
11.3 A remark on the increase in the rate of approach to zero of the image function F ( p ) 252
11.4 A table of image functions F (p) and the corresponding original functions / (x) for constructing the singular part of the image function Fx(p) 254
Chapter 12. Isolating Singularities of a Function in the Fourier Transformation
12.1 Removing discontinuities of the first kind 259
12.2 Increasing the rate of approach to zero of the function undergoing transformation 263
Bibliography 266
Index 268
Mir Books 