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-introduction; some analytical methods for computing inverse laplace transforms; methods of numerical inversion of laplace transforms based on the use of special expansions; methods of computing the mellin integral with the aid of interpolation quadrature formulas; methods of numerical inversion of laplace transforms via quadrature formulas of highest accuracy; methods of inverting laplace transforms via quadrature formulas with equal coefficients; fourier transforms and their applications to inversion of laplace transforms; inversion of laplace transforms by means of the fourier series; interpolation formulas for computing fourier integrals, highest accuracy formulas for computation, isolating singularities of the image function F(p); isolating singularities of a function in the fourier transformation-

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 Many thanks to @hawakajhonka for making the books available.