Transform Analysis of Generalized FunctionsTransform Analysis of Generalized Functions concentrates on finite parts of integrals, generalized functions and distributions. It gives a unified treatment of the distributional setting with transform analysis, i.e. Fourier, Laplace, Stieltjes, Mellin, Hankel and Bessel Series. Included are accounts of applications of the theory of integral transforms in a distributional setting to the solution of problems arising in mathematical physics. Information on distributional solutions of differential, partial differential equations and integral equations is conveniently collected here. The volume will serve as introductory and reference material for those interested in analysis, applications, physics and engineering. |
Contents
| 1 | |
| 7 | |
| 19 | |
CHAPTER 3 DEFINITION OF DISTRIBUTIONS | 25 |
CHAPTER 4 PROPERTIES OF GENERALIZED FUNCTIONS AND DISTRIBUTIONS | 35 |
CHAPTER 5 OPERATIONS ON GENERALIZED FUNCTIONS AND DISTRIBUTIONS | 47 |
CHAPTER 6 OTHER OPERATIONS ON DISTRIBUTIONS | 77 |
CHAPTER 7 THE FOURIER TRANSFORMATION | 91 |
CHAPTER 9 APPLICATIONS OF THE LAPLACE TRANSFORMATION | 145 |
CHAPTER 10 THE STIELTJES TRANSFORMATION | 207 |
CHAPTER 11 THE MELLIN TRANSFORMATION | 227 |
CHAPTER 12 HANKEL TRANSFORMATION AND BESSEL SERIES | 269 |
BIBLIOGRAPHY | 315 |
| 329 | |
AUTHOR INDEX | 331 |
CHAPTER 8 THE LAPLACE TRANSFORMATION | 107 |
Common terms and phrases
Abelian theorems according analytic continuation analytic function applying base spaces belonging to ID Bessel function bounded support Chapter complex number complex variable Consequently continuous function convergence convolution deduce defined definition denote the space derivative differential equations distributional setting Erdelyi Examples exist a number finite formula Fourier transformation Fourier-Bessel series Fp ſ Fpſ function f(x half-plane Hankel transformation Hence holomorphic infinitely differentiable integer integral equations interval inverse Laplace transformation Lavoine Lemma linear locally summable function lower bound Math Mellin transformation neighbourhood non-negative integer obtain operator plane positive integer preceding sections Problem Proof properties pseudo functions real number Remark ſ f ſ f(x ſ h satisfies Schwartz sequence solution space ID Stieltjes transformation support contained tempered distributions theory yields the result Zemanian zero