Integral Transforms and Their Applications, Second Edition

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CRC Press, Oct 11, 2006 - Mathematics - 722 pages
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Keeping the style, content, and focus that made the first edition a bestseller, Integral Transforms and their Applications, Second Edition stresses the development of analytical skills rather than the importance of more abstract formulation. The authors provide a working knowledge of the analytical methods required in pure and applied mathematics, physics, and engineering. The second edition includes many new applications, exercises, comments, and observations with some sections entirely rewritten. It contains more than 500 worked examples and exercises with answers as well as hints to selected exercises.

The most significant changes in the second edition include:

  • New chapters on fractional calculus and its applications to ordinary and partial differential equations, wavelets and wavelet transformations, and Radon transform
  • Revised chapter on Fourier transforms, including new sections on Fourier transforms of generalized functions, Poissons summation formula, Gibbs phenomenon, and Heisenbergs uncertainty principle
  • A wide variety of applications has been selected from areas of ordinary and partial differential equations, integral equations, fluid mechanics and elasticity, mathematical statistics, fractional ordinary and partial differential equations, and special functions
  • A broad spectrum of exercises at the end of each chapter further develops analytical skills in the theory and applications of transform methods and a deeper insight into the subject

    A systematic mathematical treatment of the theory and method of integral transforms, the book provides a clear understanding of the subject and its varied applications in mathematics, applied mathematics, physical sciences, and engineering.
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    Contents

    Integral Transforms
    1
    12 Basic Concepts and Definitions
    6
    Fourier Transforms and Their Applications
    9
    22 The Fourier Integral Formulas
    10
    23 Definition of the Fourier Transform and Examples
    12
    24 Fourier Transforms of Generalized Functions
    17
    25 Basic Properties of Fourier Transforms
    28
    26 Poissons Summation Formula
    37
    104 Applications of Finite Fourier Sine and Cosine Transforms
    416
    105 Multiple Finite Fourier Transforms and Their Applications
    422
    106 Exercises
    425
    Finite Laplace Transforms
    429
    112 Definition of the Finite Laplace Transform and Examples
    430
    113 Basic Operational Properties of the Finite Laplace Transform
    436
    114 Applications of Finite Laplace Transforms
    439
    115 Tauberian Theorems
    443

    27 The Shannon Sampling Theorem
    44
    28 Gibbs Phenomenon
    54
    29 Heisenbergs Uncertainty Principle
    57
    210 Applications of Fourier Transforms to Ordinary Differential Equations
    60
    211 Solutions of Integral Equations
    65
    212 Solutions of Partial Differential Equations
    68
    213 Fourier Cosine and Sine Transforms with Examples
    91
    214 Properties of Fourier Cosine and Sine Transforms
    93
    215 Applications of Fourier Cosine and Sine Transforms to Partial Differential Equations
    96
    216 Evaluation of Definite Integrals
    100
    217 Applications of Fourier Transforms in Mathematical Statistics
    103
    218 Multiple Fourier Transforms and Their Applications
    109
    219 Exercises
    119
    Laplace Transforms and Their Basic Properties
    133
    32 Definition of the Laplace Transform and Examples
    134
    33 Existence Conditions for the Laplace Transform
    139
    34 Basic Properties of Laplace Transforms
    140
    35 The Convolution Theorem and Properties of Convolution
    145
    36 Differentiation and Integration of Laplace Transforms
    151
    37 The Inverse Laplace Transform and Examples
    154
    38 Tauberian Theorems and Watsons Lemma
    168
    39 Exercises
    173
    Applications of Laplace Transforms
    181
    42 Solutions of Ordinary Differential Equations
    182
    43 Partial Differential Equations Initial and Boundary Value Problems
    207
    44 Solutions of Integral Equations
    222
    45 Solutions of Boundary Value Problems
    225
    46 Evaluation of Definite Integrals
    228
    47 Solutions of Difference and DifferentialDifference E quations
    230
    48 Applications of the Joint Laplace and Fourier Transform
    237
    49 Summation of Infinite Series
    248
    410 Transfer Function and Impulse Response Function of a Linear System
    251
    411 Exercises
    256
    Fractional Calculus and Its Applications
    269
    52 Historical Comments
    270
    53 Fractional Derivatives and Integrals
    272
    54 Applications of Fractional Calculus
    279
    55 Exercise
    282
    Applications of Integral Transforms to Fractional Differential and Integral Equations
    283
    62 Laplace Transforms of Fractional Integrals and Fractional Derivatives
    284
    63 Fractional Ordinary Differential Equations
    287
    64 Fractional Integral Equations
    290
    65 Initial Value Problems for Fractional Differential Equations
    295
    66 Greens Functions of Fractional Differential Equations
    298
    67 Fractional Partial Differential Equations
    299
    68 Exercises
    312
    Hankel Transforms and Their Applications
    315
    72 The Hankel Transform and Examples
    316
    73 Operational Properties of the Hankel Transform
    319
    74 Applications of Hankel Transforms to Partial Differential Equations
    322
    75 Exercises
    331
    Mellin Transforms and Their Applications
    339
    82 Definition of the Mellin Transform and Examples
    340
    83 Basic Operational Properties of Mellin Transforms
    343
    84 Applications of Mellin Transforms
    349
    85 Mellin Transforms of the Weyl Fractional Integral and the Weyl Fractional Derivative
    353
    86 Application of Mellin Transforms to Summation of Series
    358
    87 Generalized Mellin Transforms
    361
    88 Exercises
    365
    Hilbert and Stieltjes Transforms
    371
    92 Definition of the Hilbert Transform and Examples
    372
    93 Basic Properties of Hilbert Transforms
    375
    94 Hilbert Transforms in the Complex Plane
    378
    95 Applications of Hilbert Transforms
    380
    96 Asymptotic Expansions of OneSided Hilbert Transforms
    388
    97 Definition of the Stieltjes Transform and Examples
    391
    98 Basic Operational Properties of Stieltjes Transforms
    394
    99 Inversion Theorems for Stieltjes Transforms
    396
    910 Applications of Stieltjes Transforms
    399
    911 The Generalized Stieltjes Transform
    401
    912 Basic Properties of the Generalized Stieltjes Transform
    403
    913 Exercises
    404
    Finite Fourier Sine and Cosine Transforms
    407
    102 Definitions of the Finite Fourier Sine and Cosine Transforms and Examples
    408
    103 Basic Properties of Finite Fourier Sine and Cosine Transforms
    410
    Z Transforms
    445
    123 Definition of the Z Transform and Examples
    449
    124 Basic Operational Properties of Z Transforms
    453
    125 The Inverse Z Transform and Examples
    459
    126 Applications of Z Transforms to Finite Difference Equations
    463
    127 Summation of Infinite Series
    466
    128 Exercises
    469
    Finite Hankel Transforms
    473
    133 Basic Operational Properties
    476
    135 Exercises
    481
    Legendre Transforms
    485
    142 Definition of the Legendre Transform and Examples
    486
    143 Basic Operational Properties of Legendre Transforms
    489
    144 Applications of Legendre Transforms to Boundary Value Problems
    497
    145 Exercises
    498
    Jacobi and Gegenbauer Transforms
    501
    153 Basic Operational Properties
    504
    154 Applications of Jacobi Transforms to the Generalized Heat Conduction Problem
    505
    155 The Gegenbauer Transform and Its Basic Operational Properties
    507
    156 Application of the Gegenbauer Transform
    510
    Laguerre Transforms
    511
    163 Basic Operational Properties
    516
    164 Applications of Laguerre Transforms
    520
    165 Exercises
    523
    Hermite Transforms
    525
    172 Definition of the Hermite Transform and Examples
    526
    173 Basic Operational Properties
    529
    174 Exercises
    538
    The Radon Transform and Its Applications
    539
    182 The Radon Transform
    541
    183 Properties of the Radon Transform
    545
    184 The Radon Transform of Derivatives
    550
    185 Derivatives of the Radon Transform
    551
    186 Convolution Theorem for the Radon Transform
    553
    187 Inverse of the Radon Transform and the Parseval Relation
    554
    188 Applications of the Radon Transform
    560
    189 Exercises
    561
    Wavelets and Wavelet Transforms
    563
    192 Continuous Wavelet Transforms
    565
    193 The Discrete Wavelet Transform
    573
    194 Examples of Orthonormal Wavelets
    575
    195 Exercises
    584
    Some Special Functions and Their Properties
    587
    A2 Bessel and Airy Functions
    592
    A3 Legendre and Associated Legendre Functions
    598
    A4 Jacobi and Gegenbauer Polynomials
    601
    A5 Laguerre and Associated Laguerre Functions
    605
    A6 Hermite Polynomials and WeberHermite Functions
    607
    A7 Mittag Leffler Function
    609
    Tables of Integral Transforms
    611
    TABLE B2 Fourier Cosine Transforms
    615
    TABLE B3 Fourier Sine Transforms
    617
    TABLE B4 Laplace Transforms
    619
    TABLE B5 Hankel Transforms
    624
    TABLE B6 Mellin Transforms
    627
    TABLE B7 Hilbert Transforms
    630
    TABLE B8 Stieltjes Transforms
    633
    TABLE B9 Finite Fourier Cosine Transforms
    636
    TABLE B10 Finite Fourier Sine Transforms
    638
    TABLE B11 Finite Laplace Transforms
    640
    TABLE B12 Z Transforms
    642
    TABLE B13 Finite Hankel Transforms
    644
    Answers and Hints to Selected Exercises
    645
    39 Exercises
    651
    411 Exercises
    655
    68 Exercises
    662
    88 Exercises
    663
    913 Exercises
    664
    106 Exercises
    665
    116 Exercises
    667
    135 Exercises
    670
    189 Exercises
    671
    Bibliography
    673
    Index
    689
    Copyright

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