Integral transforms and their applications. 3rd revised and updated ed.

*(English)*Zbl 1310.44001
Boca Raton, FL: CRC Press (ISBN 978-1-4822-2357-6/hbk). xxvi, 792 p. (2015).

For the first edition see [(1995; Zbl 0920.44001)], and for the second edition [(2006; Zbl 1113.44001)].

From the preface of the present third edition:

1. Chapter 1 on integral transforms has been completely revised and some new material on brief historical introduction to classical and modern integral transforms was added to provide new information about the historical developments of the subject.

2. Chapter 2 on Fourier transforms and their applications has been completely revised and new material added, including new sections on Fourier transforms of generalized functions, the Poisson summation formula, the Gibbs phenomenon, and the Heisenberg uncertainty principle. Many sections have been completely rewritten with numerous new examples of applications and exercises.

3. In view of the major importance and usefulness of the theory of the Laplace transforms, the double Laplace transforms, and their diverse applications, Chapters 3 and 4 have greatly been revised with many new worked-out examples of applications with emphasis on the analysis of mechanical vibrations and electrical networks, systems, and signals, and numerous problems added to the exercises in these chapters.

4. Four entirely new chapters on Radon transforms, and wavelets and wavelet transforms, fractional calculus and its applications to ordinary and partial differential equations have been added to modernize the contents of the book. A new section on the transfer function and the impulse response function with examples of applications was included in Chapters 2 and 4.

5. The book offers a detailed and clear explanation of every concept and method that is introduced, accompanied by carefully selected worked-out examples, with special emphasis being given to those topics in which students experience difficulty.

6. A wide variety of modern examples of applications has been selected from areas of difference, functional, ordinary and partial differential equations, quantum mechanics, integral equations, fluid mechanics and elasticity, mathematical statistics, fractional ordinary and partial differential equations, and special functions.

7. The book is organized with sufficient flexibility in teaching courses to enable instructors to select topics and chapters appropriate to courses of differing lengths, emphases, and levels of difficulty.

8. A wide spectrum of exercises has been carefully chosen and included at the end of each chapter so the reader may further develop both analytical and computational skills in the theory and applications of transform methods and a deeper insight into the subject.

9. Detailed methods of solutions, answers, and hints to an abundance of selected exercises are provided at the end of the book to provide additional help to students. All figures have been redrawn and many new figures have been added for a clear understanding of physical explanations.

10. Standard books have been added to the bibliography to stimulate new interest in future study and research. The index of the book has also been completely revised in order to include a wide variety of topics.

11. The book provides information that puts the reader at the forefront of advanced study and current research.

From the preface of the present third edition:

1. Chapter 1 on integral transforms has been completely revised and some new material on brief historical introduction to classical and modern integral transforms was added to provide new information about the historical developments of the subject.

2. Chapter 2 on Fourier transforms and their applications has been completely revised and new material added, including new sections on Fourier transforms of generalized functions, the Poisson summation formula, the Gibbs phenomenon, and the Heisenberg uncertainty principle. Many sections have been completely rewritten with numerous new examples of applications and exercises.

3. In view of the major importance and usefulness of the theory of the Laplace transforms, the double Laplace transforms, and their diverse applications, Chapters 3 and 4 have greatly been revised with many new worked-out examples of applications with emphasis on the analysis of mechanical vibrations and electrical networks, systems, and signals, and numerous problems added to the exercises in these chapters.

4. Four entirely new chapters on Radon transforms, and wavelets and wavelet transforms, fractional calculus and its applications to ordinary and partial differential equations have been added to modernize the contents of the book. A new section on the transfer function and the impulse response function with examples of applications was included in Chapters 2 and 4.

5. The book offers a detailed and clear explanation of every concept and method that is introduced, accompanied by carefully selected worked-out examples, with special emphasis being given to those topics in which students experience difficulty.

6. A wide variety of modern examples of applications has been selected from areas of difference, functional, ordinary and partial differential equations, quantum mechanics, integral equations, fluid mechanics and elasticity, mathematical statistics, fractional ordinary and partial differential equations, and special functions.

7. The book is organized with sufficient flexibility in teaching courses to enable instructors to select topics and chapters appropriate to courses of differing lengths, emphases, and levels of difficulty.

8. A wide spectrum of exercises has been carefully chosen and included at the end of each chapter so the reader may further develop both analytical and computational skills in the theory and applications of transform methods and a deeper insight into the subject.

9. Detailed methods of solutions, answers, and hints to an abundance of selected exercises are provided at the end of the book to provide additional help to students. All figures have been redrawn and many new figures have been added for a clear understanding of physical explanations.

10. Standard books have been added to the bibliography to stimulate new interest in future study and research. The index of the book has also been completely revised in order to include a wide variety of topics.

11. The book provides information that puts the reader at the forefront of advanced study and current research.

##### MSC:

44-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to integral transforms |

44A10 | Laplace transform |

44A12 | Radon transform |

44A15 | Special integral transforms (Legendre, Hilbert, etc.) |

42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |

42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |

26A33 | Fractional derivatives and integrals |

44A45 | Classical operational calculus |

33-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to special functions |