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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

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

Library of Congress subject headings for this publication: Fourier transformations