Author | Kanwal, Ram P. author |
---|---|
Title | Generalized Functions Theory and Technique [electronic resource] / by Ram P. Kanwal |
Imprint | Boston, MA : Birkhรคuser Boston, 1998 |
Edition | Second Edition |
Connect to | http://dx.doi.org/10.1007/978-1-4684-0035-9 |
Descript | 474 p. online resource |
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 -- 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 -- 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 -- 4. Distributions Defined by Divergent Integrals -- 4.1. Introduction -- 4.2. The pseudofunction H(x)/xn, n = 1, 2, 3,... -- 4.3. Functions with algebraic singularity of order m -- 4.4. Examples -- Exercises -- 5. Distributional Derivatives of Functions with Jump Discontinuities -- 5.1. Distributional derivatives in R1 -- 5.2. Moving surfaces of discontinuity in Rn, n ? 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 Pf(1/r) and its derivatives -- 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 -- 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 -- 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 -- 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 Greenโs 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 nth order -- 9.11. Ordinary differential equations with singular coefficients -- Exercises -- 10. Applications to Partial Differential Equations -- 10.1. Introduction -- 10.2. Classical and generalized solutions -- 10.3. Fundamental solutions -- 10.4. The Cauchy-Riemann operator -- 10.5. The transport operator -- 10.6. The Laplace operator -- 10.7. The heat operator -- 10.8. The Schrรถdinger operator -- 10.9. The Helmholtz operator -- 10.10. The wave operator -- 10.11. The inhomogeneous wave equation -- 10.12. The Klein-Gordon operator -- Exercises -- 11. Applications to Boundary Value Problems -- 11.1. Poissonโs 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 tensor for 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 -- 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 wavefronts -- 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 -- 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 -- 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 -- 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 -- References