Integral Transformations, Operational Calculus, and Generalized Functions
It is not the object of the author to present comprehensive cov­ erage of any particular integral transformation or of any particular development of generalized functions, for there are books available in which this is done. Rather, this consists more of an introductory survey in which various ideas are explored. The Laplace transformation is taken as the model type of an integral transformation and a number of its properties are developed; later, the Fourier transformation is introduced. The operational calculus of Mikusinski is presented as a method of introducing generalized functions associated with the Laplace transformation. The construction is analogous to the construction of the rational numbers from the integers. Further on, generalized functions associated with the problem of extension of the Fourier transformation are introduced. This construction is anal­ ogous to the construction of the reals from the rationals by means of Cauchy sequences. A chapter with sections on a variety of transformations is adjoined. Necessary levels of sophistication start low in the first chapter, but they grow considerably in some sections of later chapters. Background needs are stated at the beginnings of each chapter. Many theorems are given without proofs, which seems appropriate for the goals in mind. A selection of references is included. Without showing many of the details of rigor it is hoped that a strong indication is given that a firm mathematical foundation does actu­ ally exist for such entities as the "Dirac delta-function".
1101629207
Integral Transformations, Operational Calculus, and Generalized Functions
It is not the object of the author to present comprehensive cov­ erage of any particular integral transformation or of any particular development of generalized functions, for there are books available in which this is done. Rather, this consists more of an introductory survey in which various ideas are explored. The Laplace transformation is taken as the model type of an integral transformation and a number of its properties are developed; later, the Fourier transformation is introduced. The operational calculus of Mikusinski is presented as a method of introducing generalized functions associated with the Laplace transformation. The construction is analogous to the construction of the rational numbers from the integers. Further on, generalized functions associated with the problem of extension of the Fourier transformation are introduced. This construction is anal­ ogous to the construction of the reals from the rationals by means of Cauchy sequences. A chapter with sections on a variety of transformations is adjoined. Necessary levels of sophistication start low in the first chapter, but they grow considerably in some sections of later chapters. Background needs are stated at the beginnings of each chapter. Many theorems are given without proofs, which seems appropriate for the goals in mind. A selection of references is included. Without showing many of the details of rigor it is hoped that a strong indication is given that a firm mathematical foundation does actu­ ally exist for such entities as the "Dirac delta-function".
54.99 In Stock
Integral Transformations, Operational Calculus, and Generalized Functions

Integral Transformations, Operational Calculus, and Generalized Functions

by R.G. Buschman
Integral Transformations, Operational Calculus, and Generalized Functions

Integral Transformations, Operational Calculus, and Generalized Functions

by R.G. Buschman

Paperback(1996)

$54.99 
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Overview

It is not the object of the author to present comprehensive cov­ erage of any particular integral transformation or of any particular development of generalized functions, for there are books available in which this is done. Rather, this consists more of an introductory survey in which various ideas are explored. The Laplace transformation is taken as the model type of an integral transformation and a number of its properties are developed; later, the Fourier transformation is introduced. The operational calculus of Mikusinski is presented as a method of introducing generalized functions associated with the Laplace transformation. The construction is analogous to the construction of the rational numbers from the integers. Further on, generalized functions associated with the problem of extension of the Fourier transformation are introduced. This construction is anal­ ogous to the construction of the reals from the rationals by means of Cauchy sequences. A chapter with sections on a variety of transformations is adjoined. Necessary levels of sophistication start low in the first chapter, but they grow considerably in some sections of later chapters. Background needs are stated at the beginnings of each chapter. Many theorems are given without proofs, which seems appropriate for the goals in mind. A selection of references is included. Without showing many of the details of rigor it is hoped that a strong indication is given that a firm mathematical foundation does actu­ ally exist for such entities as the "Dirac delta-function".

Product Details

ISBN-13: 9781461285489
Publisher: Springer US
Publication date: 11/26/2013
Series: Mathematics and Its Applications , #377
Edition description: 1996
Pages: 240
Product dimensions: 6.10(w) x 9.25(h) x 0.02(d)

Table of Contents

Preface. 1. Laplace Transformations. 2. Mikusinski Operators. 3. Fourier Transformations. 4. Generalized Functions. 5. Other Transformations. References. Appendices: A. The Titchmarsh Theorem. B. Inversion Integrals. C. Interchange of Order of Infinite Processes. D. Definitions and Properties of Some Special Functions. Tables of Transforms: 1. Laplace. 2. Inverse Laplace. 3. Fourier. 4. Fourier Cosine. 5. Fourier Sine. 6. Mellin. 7. Power Series. 8. Finite Fourier. 9. Finite Fourier Cosine. 10. Finite Fourier Sine. 11. Finite Laplace. 12. Two-Dimensional Laplace. 13. Inverse Two-Dimensional Laplace. Index.
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