Integral transforms and their applications

c5750fm......Page 1
Integral Transforms and Their Applications, Second Edition......Page 2
Preface to the Second Edition......Page 5
Preface to the First Edition......Page 8
About the Authors......Page 12
Contents......Page 14
Answers and Hints to Selected Exercises......Page 0
1.1 Brief Historical Introduction......Page 20
1.2 Basic Concepts and Definitions......Page 25
2.1 Introduction......Page 28
2.2 The Fourier Integral Formulas......Page 29
2.3 Definition of the Fourier Transform and Examples......Page 31
2.4 Fourier Transforms of Generalized Functions......Page 36
2.5 Basic Properties of Fourier Transforms......Page 47
2.6 Poisson’s Summation Formula......Page 56
2.7 The Shannon Sampling Theorem......Page 63
2.8 Gibbs’ Phenomenon......Page 73
2.9 Heisenberg’s Uncertainty Principle......Page 76
2.10 Applications of Fourier Transforms to Ordinary Differential Equations......Page 79
2.11 Solutions of Integral Equations......Page 84
2.12 Solutions of Partial Differential Equations......Page 87
2.13 Fourier Cosine and Sine Transforms with Examples......Page 110
2.14 Properties of Fourier Cosine and Sine Transforms......Page 112
2.15 Applications of Fourier Cosine and Sine Transforms to Partial Differential Equations......Page 115
2.16 Evaluation of Definite Integrals......Page 119
2.17 Applications of Fourier Transforms in Mathematical Statistics......Page 122
2.18 Multiple Fourier Transforms and Their Applications......Page 128
2.19 Exercises......Page 138
3.1 Introduction......Page 152
3.2 Definition of the Laplace Transform and Examples......Page 153
3.3 Existence Conditions for the Laplace Transform......Page 158
3.4 Basic Properties of Laplace Transforms......Page 159
3.5 The Convolution Theorem and Properties of Convolution......Page 164
3.6 Differentiation and Integration of Laplace Transforms......Page 170
3.7 The Inverse Laplace Transform and Examples......Page 173
3.8 Tauberian Theorems and Watson’s Lemma......Page 187
3.9 Exercises......Page 192
4.1 Introduction......Page 200
4.2 Solutions of Ordinary Differential Equations......Page 201
4.3 Partial Differential Equations, Initial and Boundary Value Problems......Page 226
4.4 Solutions of Integral Equations......Page 241
4.5 Solutions of Boundary Value Problems......Page 244
4.6 Evaluation of Definite Integrals......Page 247
4.7 Solutions of Difference and Differential-Difference Equations......Page 249
4.8 Applications of the Joint Laplace and Fourier Transform......Page 256
4.9 Summation of Infinite Series......Page 267
4.10 Transfer Function and Impulse Response Function of a Linear System......Page 270
4.11 Exercises......Page 275
5.1 Introduction......Page 287
5.2 Historical Comments......Page 288
5.3 Fractional Derivatives and Integrals......Page 290
5.4 Applications of Fractional Calculus......Page 297
5.5 Exercises......Page 300
6.1 Introduction......Page 301
6.2 Laplace Transforms of Fractional Integrals and Fractional Derivatives......Page 302
6.3 Fractional Ordinary Differential Equations......Page 305
6.4 Fractional Integral Equations......Page 308
6.5 Initial Value Problems for Fractional Differential Equations......Page 313
6.6 Green’s Functions of Fractional Differential Equations......Page 316
6.7 Fractional Partial Differential Equations......Page 317
6.8 Exercises......Page 330
7.1 Introduction......Page 333
7.2 The Hankel Transform and Examples......Page 334
7.3 Operational Properties of the Hankel Transform......Page 337
7.4 Applications of Hankel Transforms to Partial Differential Equations......Page 340
7.5 Exercises......Page 349
8.1 Introduction......Page 356
8.2 Definition of the Mellin Transform and Examples......Page 357
8.3 Basic Operational Properties of Mellin Transforms......Page 360
8.4 Applications of Mellin Transforms......Page 366
8.5 Mellin Transforms of the Weyl Fractional Integral and the Weyl Fractional Derivative......Page 370
8.6 Application of Mellin Transforms to Summation of Series......Page 375
8.7 Generalized Mellin Transforms......Page 378
8.8 Exercises......Page 382
9.1 Introduction......Page 388
9.2 Definition of the Hilbert Transform and Examples......Page 389
9.3 Basic Properties of Hilbert Transforms......Page 392
9.4 Hilbert Transforms in the Complex Plane......Page 395
9.5 Applications of Hilbert Transforms......Page 397
9.6 Asymptotic Expansions of One-Sided Hilbert Transforms......Page 405
9.7 Definition of the Stieltjes Transform and Examples......Page 408
9.8 Basic Operational Properties of Stieltjes Transforms......Page 411
9.9 Inversion Theorems for Stieltjes Transforms......Page 413
9.10 Applications of Stieltjes Transforms......Page 416
9.11 The Generalized Stieltjes Transform......Page 418
9.12 Basic Properties of the Generalized Stieltjes Transform......Page 420
9.13 Exercises......Page 421
10.1 Introduction......Page 424
10.2 Definitions of the Finite Fourier Sine and Cosine Transforms and Examples......Page 425
10.3 Basic Properties of Finite Fourier Sine and Cosine Transforms......Page 427
10.4 Applications of Finite Fourier Sine and Cosine Transforms......Page 433
10.5 Multiple Finite Fourier Transforms and Their Applications......Page 439
10.6 Exercises......Page 442
11.1 Introduction......Page 446
11.2 Definition of the Finite Laplace Transform and Examples......Page 447
11.3 Basic Operational Properties of the Finite Laplace Transform......Page 453
11.4 Applications of Finite Laplace Transforms......Page 456
11.6 Exercises......Page 460
12.2 Dynamic Linear Systems and Impulse Response......Page 462
12.3 Definition of the Z Transform and Examples......Page 466
12.4 Basic Operational Properties of Z Transforms......Page 470
12.5 The Inverse Z Transform and Examples......Page 476
12.6 Applications of Z Transforms to Finite Difference Equations......Page 480
12.7 Summation of Infinite Series......Page 483
12.8 Exercises......Page 486
13.2 Definition of the Finite Hankel Transform and Examples......Page 490
13.4 Applications of Finite Hankel Transforms......Page 493
13.5 Exercises......Page 498
14.1 Introduction......Page 501
14.2 Definition of the Legendre Transform and Examples......Page 502
14.3 Basic Operational Properties of Legendre Transforms......Page 505
14.4 Applications of Legendre Transforms to Boundary Value Problems......Page 513
14.5 Exercises......Page 514
15.2 Definition of the Jacobi Transform and Examples......Page 517
15.3 Basic Operational Properties......Page 520
15.4 Applications of Jacobi Transforms to the Generalized Heat Conduction Problem......Page 521
15.5 The Gegenbauer Transform and Its Basic Operational Properties......Page 523
15.6 Application of the Gegenbauer Transform......Page 526
16.2 Definition of the Laguerre Transform and Examples......Page 527
16.3 Basic Operational Properties......Page 532
16.4 Applications of Laguerre Transforms......Page 536
16.5 Exercises......Page 539
17.1 Introduction......Page 540
17.2 Definition of the Hermite Transform and Examples......Page 541
17.3 Basic Operational Properties......Page 544
17.4 Exercises......Page 553
18.1 Introduction......Page 554
18.2 The Radon Transform......Page 556
18.3 Properties of the Radon Transform......Page 560
18.4 The Radon Transform of Derivatives......Page 565
18.5 Derivatives of the Radon Transform......Page 566
18.6 Convolution Theorem for the Radon Transform......Page 568
18.7 Inverse of the Radon Transform and the Parseval Relation......Page 569
18.8 Applications of the Radon Transform......Page 575
18.9 Exercises......Page 576
19.1 Brief Historical Remarks......Page 578
19.2 Continuous Wavelet Transforms......Page 580
19.3 The Discrete Wavelet Transform......Page 588
19.4 Examples of Orthonormal Wavelets......Page 590
19.5 Exercises......Page 599
A-1 Gamma, Beta, and Error Functions......Page 602
Legendre Duplication Formula......Page 604
A-2 Bessel and Airy Functions......Page 607
A-3 Legendre and Associated Legendre Functions......Page 613
A-4 Jacobi and Gegenbauer Polynomials......Page 616
A-5 Laguerre and Associated Laguerre Functions......Page 620
A-6 Hermite Polynomials and Weber-Hermite Functions......Page 622
A-7 Mittag Leffler Function......Page 624
TABLE B-1 Fourier Transforms......Page 626
TABLE B-2 Fourier Cosine Transforms......Page 630
TABLE B-3 Fourier Sine Transforms......Page 632
TABLE B-4 Laplace Transforms......Page 634
TABLE B-5 Hankel Transforms......Page 639
TABLE B-6 Mellin Transforms......Page 642
TABLE B-7 Hilbert Transforms......Page 645
TABLE B-8 Stieltjes Transforms......Page 648
TABLE B-9 Finite Fourier Cosine Transforms......Page 651
TABLE B-10 Finite Fourier Sine Transforms......Page 653
TABLE B-11 Finite Laplace Transforms......Page 655
TABLE B-12 Z Transforms......Page 657
TABLE B-13 Finite Hankel Transforms......Page 659
2.19 Exercises......Page 660
3.9 Exercises......Page 666
4.11 Exercises......Page 670
7.5 Exercises......Page 677
8.8 Exercises......Page 678
9.13 Exercises......Page 679
10.6 Exercises......Page 680
12.8 Exercises......Page 682
17.4 Exercises......Page 685
19.5 Exercises......Page 686
Bibliography......Page 688