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A Handbook Of Methods Of Approximate Fourier Transformation And Inversion Of The Laplace Transformation

by: V. I. Krylov; N. S. Skoblya

Publication date: 1977

Topics: mathematics handbook, Laplace transform, Fourier integrals, Fourier transforms, approximate methods, numerical methods, interpolation, Chebyshev-Laguerre polynomials, Mellin Integral, Quadrature Formulas

Publisher: Mir Publishers

Collection: mir-titles; additional_collections

Language: English

Harmonic analysis and the Laplace transformation are tools that are frequently used to solve a wide range of theoretical and applied problems. This text, contains most of the familiar methods of approximate inversion of the Laplace transformation and calculation of Fourier integrals. This book is designed for scientists and engineers that have to deal with the theory and applications of the Laplace transform and Fourier integrals. It will be a useful handbook in every computer centre and designing bureau.

This text is aimed at a broad category of readers engaged in the theory of the Laplace transformation or its scientific and technical applications. For this reason, the authors did not strive for particular brevity in presentation and attempt ed to make the text accessible to nonmathematicians as well. It is assumed the reader has a basic knowledge of analysis and the theory of functions of a complex variable as given in any extended college course of mathematics.

Translated from the Russian by George Yankovsky

First published 1977

Revised from the 1974 Russian edition

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

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