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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.
Cover
Preface
Fourier Transforms
Integral Transforms
Fourier Kernels
Fourier's Integral Theorem
Laplace Transform
Foundations of Operator Calculus
Mellin Transform
Multiple Fourier Transforms
Hankel Transforms
Hankel Inversion Theorem
Parseval's Theorem for Hankel Transforms
Hankel Transforms of the Derivatives of a Function
Relation between Hankel Transfonns and Fourier Transfonns
Dual Integral Equations
Finite Transforms
Finite Fourier Transforms
Finite Hankel Transforms
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
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
The Slowing Down of Neutrons in Matter
Fundamental Equations
Age Theory
Diffusion of Thermal Neutrons with Sources Given by the AgeTheory
Exact Solutions of the Transport Equation
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 Coaxal Cylinders
Motion of a Viscous Fluid under a Surface Load
Harmonic Analysis of Nonlinear Viscous Flow
Applications to Atomic and Nuclear Physics
Theory of Radioactive Transformations
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
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
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
Some Properties of Bessel 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 Jn(x)
Approximate Methods of Calculating Integral Transforms
Method of Steepest Descents for Contour Integrals
Numerical Calculations of Fourier Integrals
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