Fourier Transforms (Dover Books on Mathematics)

Ian N. Sneddon

ISBN 10: 0486685225 / ISBN 13: 9780486685229

Editore: Dover Publications, 2010

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Dati bibliografici

• Titolo: Fourier Transforms (Dover Books on Mathematics)
• Editore: Dover Publications
• Data di pubblicazione: 2010
• Lingua: Inglese
• ISBN 10: 0486685225
• ISBN 13: 9780486685229
• Rilegatura: Brossura
• Condizione: Good

Descrizione articolo

Riassunto

The purpose of this book is to present the theory of Fourier transforms and related topics in a form suitable for the use of students and research workers interested in the boundary value problems of physics and engineering.
The focus of the book is on applications, rather than on the theory itself; thus, the first three chapters are devoted to a general treatment of the fundamentals, but no attempt is made to present the foundation in their most general form. Instead, the main theorems are established for a certain class of functions which is sufficiently wide to embrace most of those which occur in problems in applied mathematics. The last seven chapters cover the uses of the theory in solving boundary and initial value problems in engineering and physics.
To make the book accessible to undergraduates beginning the study of theoretical physics, no specialized knowledge of physics is assumed, however a good grounding in advanced calculus is a prerequisite. Each chapter begins with a discussion of the physical fundamentals and the derivation of the basic equations. Moreover, the author has taken special pains to include, in the chapters on basic theory, not only the common properties of the Fourier transforms, but also those of the Mellin, Laplace, and Hankel transforms. Finite transforms, dual integral equations, the Wiener-Hopf procedure, and the properties of the Dirac delta function are also considered in some detail. The physical problems included in the text were carefully chosen for their importance and relevance to the topic under discussion.

Contenuti

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