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This third edition of Generalized Functions expands the treatment of fundamental concepts and theoretical background material and delineates connections to a variety of applications in mathematical physics, elasticity, wave propagation, magnetohydrodynamics, linear systems, probability and statistics, optimal control problems in economics, and more. In applying the powerful tools of generalized functions to better serve the needs of physicists, engineers, and applied mathematicians, this work is quite distinct from other books on the subject.
Key new topics and important features:
- Examination of the Poisson Summation Formula and the concepts of differential forms and the delta distribution on wave fronts
- Enhanced presentation of the Schroedinger, Klein-Gordon, Helmholtz, heat and wave equations
- Exposition driven by additional examples and exercises
- Comprehensive bibliography and index
- Prerequisites: advanced calculus, ordinary and partial differential equations
This second edition of Generalized Functions has been strengthened in many ways. The already extensive set of examples has been expanded. Since the publication of the first edition,there hasbeen tremendous growth inthe subject and Ihave attempted to incorporate some of these new concepts. Accordingly, almost all the chapters have been revised. The bibliography has been enlarged considerably. Some of the material has been reorganized. For example, Chapters 12 and 13 of the first edition have been consolidated into Chapter 12ofthis edition by ajudicious process of elimination and addition of the subject matter. The new Chapter 13 explains the interplay between the theories of moments, asymptotics, and singular perturbations. Similarly, some sections of Chapter 15 have been revised and included in earlier chapters to improve the logical flow of ideas. However, two sections are retained. The section dealing with the application to the probability theory has been revised, and I am thankful to Professor Z. L. Crvenkovic for her help. The new material included in this chapter pertains to the modem topics of periodic distributions and microlocal theory. I have demonstrated through various examples that familiarity with the generalized functions is very helpful for students in physical sciences and technology. For instance, the reader will realize from Chapter 6 how the generalized functions have revolutionized the Fourier analysis which is being used extensively in many fields of scientific activity.
Key new topics and important features:
- Examination of the Poisson Summation Formula and the concepts of differential forms and the delta distribution on wave fronts
- Enhanced presentation of the Schroedinger, Klein-Gordon, Helmholtz, heat and wave equations
- Exposition driven by additional examples and exercises
- Comprehensive bibliography and index
- Prerequisites: advanced calculus, ordinary and partial differential equations
This second edition of Generalized Functions has been strengthened in many ways. The already extensive set of examples has been expanded. Since the publication of the first edition,there hasbeen tremendous growth inthe subject and Ihave attempted to incorporate some of these new concepts. Accordingly, almost all the chapters have been revised. The bibliography has been enlarged considerably. Some of the material has been reorganized. For example, Chapters 12 and 13 of the first edition have been consolidated into Chapter 12ofthis edition by ajudicious process of elimination and addition of the subject matter. The new Chapter 13 explains the interplay between the theories of moments, asymptotics, and singular perturbations. Similarly, some sections of Chapter 15 have been revised and included in earlier chapters to improve the logical flow of ideas. However, two sections are retained. The section dealing with the application to the probability theory has been revised, and I am thankful to Professor Z. L. Crvenkovic for her help. The new material included in this chapter pertains to the modem topics of periodic distributions and microlocal theory. I have demonstrated through various examples that familiarity with the generalized functions is very helpful for students in physical sciences and technology. For instance, the reader will realize from Chapter 6 how the generalized functions have revolutionized the Fourier analysis which is being used extensively in many fields of scientific activity.
"This book on generalized functions is suitable for physicists, engineers and applied mathematicians. The author presents the notion of generalized functions, their properties and their applications for solving ordinary differential equations and partial differential equations. ...
The author demonstrates through various examples that familiarity with generalized functions is very helpful for students in mathematics, physical sciences and technology. The proposed exercises are very good for better understanding of notions and properties presented in the chapters. The book contains new topics and important features."
-Mathematica
"The advantage of this text is in carefully gathered examples explaining how to use corresponding properties.... Even the standard material connecting with partial and ordinary differential equations is rewritten in modern terminology."
-Zentralblatt (Review of a previous edition)
"The author has done an excellent job in presenting examples and in displaying the calculational techniques associated with distributions and the applications. Throughout the book there are a wealth of examples concerning the distributional topics and caluclations introduced and concering the applications, and the examples are presented in detail." ---Zentralblatt (Review of the 1st edition)
"The collaboration of physicists or engineers and mathematics, which is more and more popular and necessary in modern investigations, requires...a common language. The book under review provides this language.... [It] is a well written book, most of the material is accessible to senior undergraduate and graduate students in mathematical, physical and engineering sciences.... [The] book will [also] be useful...for specialists in ODEs, PDEs, functional analysis, [and] physicists, engineers, and lecturers."
-Acta. Sci. Math. (Review of aprevious edition)
"An exceptionally clear exposition... The exercises at the end of each chapter are well-chosen."
-The American Mathematical Monthly (Review of a previous edition)
"This fully revised edition of well-received book expands the treatment of fundamental concepts and theoretical background material delineates connections to a variety of applications in mathematical physics, elasticity, wave propagation, magnetohydrodynamics, linear systems, probability and statistics, optical control problems in economics, and more. It has many new topics and [features] driven by additional examples and exercises. . . It presents a wealth of applications that connot be found in any other single source. the book will be important reading for graduate students in physics and engineering."
--- Educational Book Review
The author demonstrates through various examples that familiarity with generalized functions is very helpful for students in mathematics, physical sciences and technology. The proposed exercises are very good for better understanding of notions and properties presented in the chapters. The book contains new topics and important features."
-Mathematica
"The advantage of this text is in carefully gathered examples explaining how to use corresponding properties.... Even the standard material connecting with partial and ordinary differential equations is rewritten in modern terminology."
-Zentralblatt (Review of a previous edition)
"The author has done an excellent job in presenting examples and in displaying the calculational techniques associated with distributions and the applications. Throughout the book there are a wealth of examples concerning the distributional topics and caluclations introduced and concering the applications, and the examples are presented in detail." ---Zentralblatt (Review of the 1st edition)
"The collaboration of physicists or engineers and mathematics, which is more and more popular and necessary in modern investigations, requires...a common language. The book under review provides this language.... [It] is a well written book, most of the material is accessible to senior undergraduate and graduate students in mathematical, physical and engineering sciences.... [The] book will [also] be useful...for specialists in ODEs, PDEs, functional analysis, [and] physicists, engineers, and lecturers."
-Acta. Sci. Math. (Review of aprevious edition)
"An exceptionally clear exposition... The exercises at the end of each chapter are well-chosen."
-The American Mathematical Monthly (Review of a previous edition)
"This fully revised edition of well-received book expands the treatment of fundamental concepts and theoretical background material delineates connections to a variety of applications in mathematical physics, elasticity, wave propagation, magnetohydrodynamics, linear systems, probability and statistics, optical control problems in economics, and more. It has many new topics and [features] driven by additional examples and exercises. . . It presents a wealth of applications that connot be found in any other single source. the book will be important reading for graduate students in physics and engineering."
--- Educational Book Review