Generalized Functions

1. Front Matter

   Pages i-xvii

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2. The Dirac Delta Function and Delta Sequences

   • Ram P. Kanwal

   Pages 1-16

3. The Schwartz-Sobolev Theory of Distributions

   • Ram P. Kanwal

   Pages 17-48

4. Additional Properties of Distributions

   • Ram P. Kanwal

   Pages 49-70

5. Distributions Defined by Divergent Integrals

   • Ram P. Kanwal

   Pages 71-98

6. Distributional Derivatives of Functions with Jump Discontinuities

   • Ram P. Kanwal

   Pages 99-138

7. Tempered Distributions and the Fourier Transform

   • Ram P. Kanwal

   Pages 139-177

8. Direct Products and Convolutions of Distributions

   • Ram P. Kanwal

   Pages 178-216

9. The Laplace Transform

   • Ram P. Kanwal

   Pages 217-227

10. Applications to Ordinary Differential Equations

    • Ram P. Kanwal

    Pages 228-264

11. Applications to Partial Differential Equations

    • Ram P. Kanwal

    Pages 265-311

12. Applications to Boundary Value Problems

    • Ram P. Kanwal

    Pages 312-358

13. Applications to Wave Propagation

    • Ram P. Kanwal

    Pages 359-395

14. Interplay Between Generalized Functions and the Theory of Moments

    • Ram P. Kanwal

    Pages 396-419

15. Linear Systems

    • Ram P. Kanwal

    Pages 420-433

16. Miscellaneous Topics

    • Ram P. Kanwal

    Pages 434-464

17. Back Matter

    Pages 465-476

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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.

• Boundary value problem
• Dirac delta function
• Fourier transform
• Integral equation
• Transformation
• convolution
• mathematical physics
• operator
• wave equation
• partial differential equations

"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

• Department of Mathematics, The Pennsylvania State University, University Park, USA

  Ram P. Kanwal

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