Fourier Transforms: (Dover Books on Mathema 1.4tics)

Richard A. Silverman's series of translations of outstanding Russian texts and monographs is well known to those in the fields of mathematics, physics, and engineering. This volume is another excellent text from this series, a valuable addition to the English-language literature on Fourier series.This edition is organised into nine well-defined chapters: Trigonometric Fourier Series, Orthogonal Systems, Convergence of Trigonometric Fourier Series, Trigonometric Series with Decreasing Coefficients, Operations on Fourier Series, Summation of Trigonometric Fourier Series, Double Fourier Series and the Fourier Integral, Bessel Functions and Fourier-Bessel Series, and the Eigenfunction Method and its Applications to Mathematical Physics. Every chapter moves clearly from topic to topic and theorem to theorem, with many theorem proofs given. A total of 107 problems will be found at the ends of the chapters, including many specially added to this English-language edition, and answers are given at the end of the text.Richard Silverman's excellent translation makes this book readily accessible to mathematicians and math students as well as workers and students in the fields of physics and engineering. He has also added a bibliography containing suggestions for collateral and supplementary reading.

Table of Contents:
Preface Chapter 1. Fourier Transforms Integral transforms. Fourier kernels. Fourier's integral theorem. Laplace transform. Foundations of operator calculus. Mellin transform. Multiple Fourier transforms Chapter 2. Hankel Transforms Hankel inversion theorem. Parseval's theorem for Hankel transforms. Hankel transforms of the derivatives of a function. Relation between Hankel transforms and Fourier transforms. Dual integral equations Chapter 3. Finite Transforms Finite Fourier transforms. Finite Hankel transforms Chapter 4. The Theory of Vibrations Electrical oscillations in simple circuits. Transverse vibrations of a continuous string. Oscillations of a heavy chain. Transverse oscillations of an elastic beam. Transverse vibrations of a thin membrane. Vibrations of a thin elastic plate. Elastic vibrations of thick cylinders and spheres Chapter 5. The Conduction of Heat in Solids General theory. Conduction of heat when there are no sources present. Two- and three-dimensional boundary value problems. Diffusion of heat in a solid medium which is generating heat Chapter 6. The Slowing Down of Neutrons in Matter Fundamental equations. Age theory. Diffusion of thermal neutrons with sources given by the age theory. Exact solutions of the transport equation Chapter 7. Hydrodynamic Problems Hydrodynamic equations. Irrotational flow of a perfect fluid. Surface waves. Slow motion of a viscous fluid. Motion of a viscous fluid contained between two infinite coaxial cylinders. Motion of a viscous fluid under a surface load. Harmonic analysis of nonlinear viscous flow Chapter 8. Applications to Atomic and Nuclear Physics Theory of radioactive transformations. Van der Waals attraction between spherical particles. Interaction of radiation with an electron. Cascade theory of cosmic ray showers. Distribution of momentum in atomic and molecular systems. Binding energies of the lightest nuclei Chapter 9. Two-Dimensional Stress Systems Equations of motion. Infinite elastic solid with body forces. Application of pressure to the surfaces of a two-dimensional elastic solid. Distribution of stress due to a force in the interior of a semiinfinite elastic medium. Distribution of stress in the neighborhood of a Griffith crack. Indentation problems. Two-dimensional problems in polar coordinates. Dynamical problems Chapter 10. Axially Symmetrical Stress Distributions Equations of equilibrium. Stresses produced by the indentation of the plane surface of a semiinfinite elastic medium by a rigid punch. Application of pressure to the faces of a thick plate. Distribution of stress in the neighborhood of a circular crack in an elastic body. Distribution of stress in a semiinfinite elastic medium due to a torsional displacement of the surface. Stress distribution in a long circular cylinder when a discontinuous pressure is applied to the curved surface Appendix A. Some Properties of Besell Functions Bessel's differential equation. Recurrence relations for Bessel functions of the first kind. Definite integrals involving Bessel functions. Infinite integrals involving Bessel functions. Relation between the Bessel functions and circular functions. Integral expression for the Bessel function J subscript n (x) Appendix B. Approximate Methods of Calculating Integral Transforms Method of steepest descents for contour integrals. Numerical calculations of Fourier integrals Appendix C. Tables of Integral Transforms Fourier transforms. Fourier cosine transforms. Fourier sine transforms. Laplace transforms. Mellin transforms. Hankel transforms. Finite Fourier cosine transforms. Finite Fourier sine transforms. Finite Hankel transforms Index