Pages: 364

ISBN: 9780819432322

Volume: PM66

- Preface \vii
- Introduction \1
- 1 SPECIAL FUNCTIONS \6
- 1.1 Introduction \6
- 1.2 The Gamma Function \7
- 1.3 The Error Function and Related Functions \16
- 1.4 Bessel Functions \21
- 1.5 Useful Engineering Functions \29
- 2 FOURIER INTEGRALS AND FOURIER TRANSFORMS \37
- 2.1 Introduction \37
- 2.2 Fourier Integral Representations \38
- 2.3 Proof of the Fourier Integral Theorem \47
- 2.4 Fourier Transform Pairs \49
- 2.5 Properties of the Fourier Transform \58
- 2.6 Transforms of More Complicated Functions \65
- 2.7 The Convolution Integrals of Fourier \78
- 2.8 Transforms Involving Generalized Functions \85
- 2.9 Hilbert Transforms \91
- 2.10 Additional Topics \97
- 3 APPLICATIONS INVOLVING FOURIER TRANSFORMS \102
- 3.1 Introduction \102
- 3.2 Boundary Value Problems \103
- 3.3 Heat Conduction in Solids \113
- 3.4 Mechanical Vibrations \125
- 3.5 Potential Theory \131
- 3.6 Hydrodynamics \141
- 3.7 Elasticity in Two Dimensions \151
- 3.8 Probability and Statistics \156
- 4 THE LAPLACE TRANSFORMATION
- 4.1 Introduction \162
- 4.2 The Transforms of Some Typical Functions \164
- 4.3 Basic Operational Properties \170
- 4.4 Transforms of More Complicated Functions \182
- 4.5 The Inverse Laplace Transform \190
- 4.6 Complex Inversion Formula \200
- 4.7 Additional Topics \210
- 5 APPLICATIONS INVOLVING LAPLACE TRANSFORMS \218
- 5.1 Introduction \218
- 5.2 Evaluating Integrals \218
- 5.3 Solutions of ODEs \221
- 5.4 Solutions of PDEs \229
- 5.5 Linear Integral Equations \238
- 6 THE MELLIN TRANSFORM \245
- 6.1 Introduction \245
- 6.2 Evaluation of Mellin Transforms \246
- 6.3 Complex Variable Methods \254
- 6.4 Applications \262
- 6.5 Table of Mellin Transforms \273
- 7 THE HANKEL TRANSFORM \274
- 7.1 Introduction \274
- 7.2 Evaluation of Hankel Transforms \276
- 7.3 Applications \285
- 7.4 Table of Hankel Transforms \290
- 8 FINITE TRANSFORMS \291
- 8.1 Introduction \291
- 8.2 Finite Fourier Transforms \291
- 8.3 Sturm-Liouville Transforms \298
- 8.4 Finite Hankel Transform \303
- 9 DISCRETE TRANSFORMS \310
- 9.1 Introduction \310
- 9.2 Discrete Fourier Transform \311
- 9.3 The Z Transform \321
- 9.4 Difference Equations \330
- 9.5 Table of Z Transforms \333
- BIBLIOGRAPHY \335
- APPENDIX A: REVIEW OF COMPLEX VARIABLES \337
- APPENDIX B: TABLE OF FOURIER TRANSFORMS \340
- APPENDIX C: TABLE OF LAPLACE TRANSFORMS \344
- INDEX \349

### Preface

In recent years, integral transforms have become essential working tools of every engineer and applied scientist. The Laplace transform, which undoubtedly is the most familiar example, is basic to the solution of initial value problems. The Fourier transform, while being suited to solving boundary-value problems, is basic to the frequency spectrum analysis of time-varying waveforms. The purpose of this text is to introduce the use of integral transforms in obtaining solutions to problems governed by ordinary and partial differential equations and certain types of integral equations. Some other applications are also covered where appropriate.

The Laplace and Fourier transforms are by far the most widely used of all integral transforms. For this reason they have been given a more extensive treatment in this book than other integral transforms. However, there are several other integral transforms that also have been used successfully in the solution of certain boundary-value problems and in other applications. Included in this category are Mellin, Hankel, finite, and discrete transforms, which have also been given some discussion here.

The text is directed primarily toward senior and beginning graduate students in engineering sciences, physics, and mathematics who desire a deeper knowledge of transform methods than can be obtained in introductory courses in differential equations and other similar courses. It can also be used as a self- study text for practicing engineers and applied scientists who wish to learn more about the general theory and use of integral transforms. We assume the reader has a basic knowledge of differential equations and contour integration techniques from complex variables. However, most of the material involving complex variables occurs in separate sections so that much of the text can be accessible to those with a minimum background in complex variable methods. As an aid in this regard, we have included a brief appendix relevant to our use of the basic concepts and theory of complex variables in the text. Also, because of the close association of special functions and integral transforms, the first chapter is a short introduction to several of the special functions that arise quite frequently in applications. This is considered an optional chapter for those with some acquaintance with these functions, and thus it is possible to start the text with Chapter 2. Most chapters are independent of one another so that various arrangements of the material are possible.

Applications occur throughout the text and are drawn from the fields of mechanical vibration, heat conduction, potential theory, mechanics of solids and fluids, probability and statistics, and several other areas. A working knowledge in any of these areas is generally sufficient to work the examples and exercises.

In our treatment of integral transforms we have excised formal proofs in several places, but then usually make an appropriate reference for the more formal aspects of the theory. In the applications we often make the assumptions as to the commutability of certain limiting operations, and the derivation of a particular solution sometimes may not be rigorous. However, the approach adopted here is adequate in the usual applications in engineering and applied sciences. We have included a large number of worked examples and exercises to illustrate the versatility and adequacy of this approach in applications to physical problems.

#### Introduction

The classical methods of solution of initial and boundary value problems in physics and engineering sciences have their roots in Fourier's pioneering work. An alternative approach through integral transform methods emerged primarily through Heaviside's efforts on operational techniques. In addition to being of great theoretical interest to mathematicians, integral transform methods have been found to provide easy and effective ways of solving a variety of problems arising in engineering and physical science. The use of an integral transform is somewhat analogous to that of logarithms. That is, a problem involving multiplications or division can be reduced to one involving the simpler processes of addition or subtraction by taking logarithms. After the solution has been obtained in the logarithm domain, the original solution can be recovered by finding an antilogarithm, In the same way, a problem involving derivatives can be reduced to a simpler problem involving only multiplication by polynomials in the transform variable by taking an integral transform, solving the problem in the transform domain, and them finding an inverse transform. Integral transforms arise in a natural way through the principle of linear superposition in constructing integral representations of solutions of linear differential equations.

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