Book:Ram Prakash Kanwal/Generalized Functions: Theory and Technique/Second Edition

Contents



 * Chapter 1. The Dirac Delta Function and Delta Sequences


 * 1.1. The heaviside function
 * 1.2. The Dirac delta function
 * 1.3. The delta sequences
 * 1.4. A unit dipole
 * 1.5. The heaviside sequences
 * Exercises


 * Chapter 2. The Schwartz-Sobolev Theory of distributions


 * 2.1. Some introductory definitions
 * 2.2. Test functions
 * 2.3. Linear functionals and the Schwartz-Sobolev theory of distributions
 * 2.4. Examples
 * 2.5. Algebraic operations on distributions
 * 2.6. Analytic operations on distributions
 * 2.7. Examples
 * 2.8. The support and singular support of a distribution
 * Exercises


 * Chapter 3. Additional Properties of Distributions


 * 3.1. Transformation properties of the delta distribution
 * 3.2. Convergence of distributions
 * 3.3. Delta sequences with parametric dependence
 * 3.4. Fourier series
 * 3.5. Examples
 * 3.6. The delta function as a Stieltjes integral
 * Exercises


 * Chapter 4. Distributions Defined by Divergent Integrals


 * 4.1. Introduction
 * 4.2. The pseudofunction $\map H x /x^n, n = 1,2,3,\ldots$
 * 4.3. Functions with algebraic singularity of order $m$
 * 4.4. Examples
 * Exercises


 * Chapter 5. Distributional Derivatives of Functions with Jump Discontinuities


 * 5.1. Distributional derivatives in $R_1$
 * 5.2. Moving surfaces of discontinuity in $R_n, n \ge 2$
 * 5.3. Surface distributions
 * 5.4. Various other representations
 * 5.5. First-order distributional derivatives
 * 5.6. Second-order distributional derivatives
 * 5.7. Higher-order distributional derivatives
 * 5.8. The two-dimensional case
 * 5.9. Examples
 * 5.10. The function $\map {Pf} {1/r}$ and its derivatives


 * Chapter 6. Tempered Distributions and the Fourier Transform


 * 6.1. Preliminary concepts
 * 6.2. Distributions of slow growth (tempered distributions)
 * 6.3. The Fourier transform
 * 6.4. Examples
 * Exercises


 * Chapter 7. Direct Products and Convolutions of Distributions


 * 7.1. Definition of the direct product
 * 7.2. The direct product of tempered distributions
 * 7.3. The Fourier transform of the direct product of tempered distributions
 * 7.4. The convolution
 * 7.5. The role of convolution in the regularization of the distributions
 * 7.6. The dual spaces $E$ and $E'$
 * 7.7. Examples
 * 7.8. The Fourier transform of a convolution
 * 7.9. Distributional solutions of integral equations
 * Exercises


 * Chapter 8. The Laplace Transform


 * 8.1. A brief discussion of the classical results
 * 8.2. The Laplace transform distributions
 * 8.3. The Laplace transform of the distributional derivatives and vice versa
 * 8.4. Examples
 * Exercises


 * Chapter 9. Applications to Ordinary Differential Equations


 * 9.1. Ordinary differential operators
 * 9.2. Homogeneous differential equations
 * 9.3. Inhomogeneous differentational equations: the integral of a distribution
 * 9.4. Examples
 * 9.5. Fundamental solutions and Green's functions
 * 9.6. Second-order differential equations with constant coefficients
 * 9.7. Eigenvalue problems
 * 9.8. Second-order differential equations with variable coefficients
 * 9.9. Fourth-order differential equations
 * 9.10. Differential equations of $n$th order
 * 9.11. Ordinary differential equations with singular coefficients
 * Exercises


 * Chapter 10. Applications to Partial Differential Equations


 * 10.1. Introduction
 * 10.2. Classical and generalized solutions
 * 10.3. Fundamental solutions
 * lO.4. The Cauchy-Riemann operator
 * 10.5. The transport operator
 * 10.6. The Laplace operator
 * 10.7. The heat operator
 * 10.8. The Schrödinger operator
 * 10.9. The Helmholtz operator
 * 10.10. The wave operator
 * 10.11. The inhomogeneous wave equation
 * 10.12. The Klein-Gordon operator
 * Exercises


 * Chapter 11. Applications to Boundary Value Problems


 * 11.1. Poisson's equation
 * 11.2. Dumbbell-shaped bodies
 * 11.3. Uniform axial distributions
 * 11.4. Linear axial distributions
 * 11.5. Parabolic axial distributions, $n = 5$
 * 11.6. The fourth-order polynomial distribution, $n = 7$; spheroidal cavities
 * 11.7. The polarization tensor for a spheroid
 * 11.8. The virtual mass tensorfor a spheroid
 * 11.9. The electric and magnetic polarizability tensors
 * 11.10. The distributional approach to scattering theory
 * 11.11. Stokes flow
 * 11.12. Displacement-type boundary value problems in elastostatistics
 * 11.13. The extension to elastodynamics
 * 11.14. Distributions on arbitrary lines
 * 11.15. Distributions on plane curves
 * 11.16. Distributions on a circular disk


 * Chapter 12. Applications to Wave Propagation


 * 12.1. Introduction
 * 12.2. The wave equation
 * 12.3. First-order hyperbolic systems
 * 12.4. Aerodynamic sound generation
 * 12.5. The Rankine-Hugoniot conditions
 * 12.6. Wave fronts that carry infinite singularities
 * 12.7. Kinematics of wave fronts
 * 12.8. Derivation of the transport theorems for wave fronts
 * 12.9. Propagation of wave fronts carrying multilayer densities
 * 12.10. Generalized functions with support on the light cone
 * 12.11. Examples


 * Chapter 13. Interplay Between Generalized Functions and the Theory of Moments


 * 13.1. The theory of moments
 * 13.2. Asymptotic approximation of integrals
 * 13.3. Applications to the singular perturbation theory
 * 13.4. Applications to number theory
 * 13.5. Distributional weight functions for orthogonal polynomials
 * 13.6. Convolution type integral equation revisited
 * 13.7. Further applications


 * Chapter 14. Linear Systems


 * 14.1. Operators
 * 14.2. The step response
 * 14.3. The impulse response
 * 14.4. The response to an arbitrary input
 * 14.5. Generalized functions as impulse response functions
 * 14.6. The transfer function
 * 14.7. Discrete-time systems
 * 14.8. The sampling theorem


 * Chapter 15. Miscellaneous Topics


 * 15.1. Applications to probability and random processes
 * 15.2. Applications to economics
 * 15.3. Periodic distributions
 * 15.4. Applications to microlocal theory



Source work progress
*