Content: Machine generated contents note: 1.1. Brief Historical Introduction --
1.2. Fourier Series and Fourier Transforms --
1.3. Gabor Transforms --
1.4. Basic Concepts and Definitions --
2.1. Introduction --
2.2. Fourier Integral Formulas --
2.3. Definition of the Fourier Transform and Examples --
2.4. Fourier Transforms of Generalized Functions --
2.5. Basic Properties of Fourier Transforms --
2.6. Poisson's Summation Formula --
2.7. Shannon Sampling Theorem --
2.8. Gibbs Phenomenon --
2.9. Heisenberg's Uncertainty Principle --
2.10. Applications of Fourier Transforms to Ordinary Differential Equations --
2.11. Solutions of Integral Equations --
2.12. Solutions of Partial Differential Equations --
2.13. Fourier Cosine and Sine Transforms with Examples --
2.14. Properties of Fourier Cosine and Sine Transforms --
2.15. Applications of Fourier Cosine and Sine Transforms to Partial Differential Equations --
2.16. Evaluation of Definite Integrals --
2.17. Applications of Fourier Transforms in Mathematical Statistics --
2.18. Multiple Fourier Transforms and Their Applications --
2.19. Exercises --
3.1. Introduction --
3.2. Definition of the Laplace Transform and Examples --
3.3. Existence Conditions for the Laplace Transform --
3.4. Basic Properties of Laplace Transforms --
3.5. Convolution Theorem and Properties of Convolution --
3.6. Differentiation and Integration of Laplace Transforms --
3.7. Inverse Laplace Transform and Examples --
3.8. Tauberian Theorems and Watson's Lemma --
3.9. Exercises --
4.1. Introduction --
4.2. Solutions of Ordinary Differential Equations --
4.3. Partial Differential Equations, Initial and Boundary Value Problems --
4.4. Solutions of Integral Equations --
4.5. Solutions of Boundary Value Problems --
4.6. Evaluation of Definite Integrals --
4.7. Solutions of Difference and Differential-Difference Equations --
4.8. Applications of the Joint Laplace and Fourier Transform --
4.9. Summation of Infinite Series --
4.10. Transfer Function and Impulse Response Function of a Linear System --
4.11. Double Laplace Transform, Functional and Partial Differential Equations --
4.12. Exercises --
5.1. Introduction --
5.2. Historical Comments --
5.3. Fractional Derivatives and Integrals --
5.4. Applications of Fractional Calculus --
5.5. Exercises --
6.1. Introduction --
6.2. Laplace Transforms of Fractional Integrals and Fractional Derivatives --
6.3. Fractional Ordinary Differential Equations --
6.4. Fractional Integral Equations --
6.5. Initial Value Problems for Fractional Differential Equations --
6.6. Green's Functions of Fractional Differential Equations --
6.7. Fractional Partial Differential Equations --
6.8. Exercises --
7.1. Introduction --
7.2. Hankel Transform and Examples --
7.3. Operational Properties of the Hankel Transform --
7.4. Applications of Hankel Transforms to Partial Differential Equations --
7.5. Exercises --
8.1. Introduction --
8.2. Definition of the Mellin Transform and Examples --
8.3. Basic Operational Properties of Mellin Transforms --
8.4. Applications of Mellin Transforms --
8.5. Mellin Transforms of the Weyl Fractional Integral and the Weyl Fractional Derivative --
8.6. Application of Mellin Transforms to Summation of Series --
8.7. Generalized Mellin Transforms --
8.8. Exercises --
9.1. Introduction --
9.2. Definition of the Hilbert Transform and Examples --
9.3. Basic Properties of Hilbert Transforms --
9.4. Hilbert Transforms in the Complex Plane --
9.5. Applications of Hilbert Transforms --
9.6. Asymptotic Expansions of One-Sided Hilbert Transforms --
9.7. Definition of the Stieltjes Transform and Examples --
9.8. Basic Operational Properties of Stieltjes Transforms --
9.9. Inversion Theorems for Stieltjes Transforms --
9.10. Applications of Stieltjes Transforms --
9.11. Generalized Stieltjes Transform --
9.12. Basic Properties of the Generalized Stieltjes Transform --
9.13. Exercises --
10.1. Introduction --
10.2. Definitions of the Finite Fourier Sine and Cosine Transforms and Examples --
10.3. Basic Properties of Finite Fourier Sine and Cosine Transforms --
10.4. Applications of Finite Fourier Sine and Cosine Transforms --
10.5. Multiple Finite Fourier Transforms and Their Applications --
10.6. Exercises --
11.1. Introduction --
11.2. Definition of the Finite Laplace Transform and Examples --
11.3. Basic Operational Properties of the Finite Laplace Transform --
11.4. Applications of Finite Laplace Transforms --
11.5. Tauberian Theorems --
11.6. Exercises --
12.1. Introduction --
12.2. Dynamic Linear Systems and Impulse Response --
12.3. Definition of the Z Transform and Examples --
12.4. Basic Operational Properties of Z Transforms --
12.5. Inverse Z Transform and Examples --
12.6. Applications of Z Transforms to Finite Difference Equations --
12.7. Summation of Infinite Series --
12.8. Exercises --
13.1. Introduction --
13.2. Definition of the Finite Hankel Transform and Examples --
13.3. Basic Operational Properties --
13.4. Applications of Finite Hankel Transforms --
13.5. Exercises --
14.1. Introduction --
14.2. Definition of the Legendre Transform and Examples --
14.3. Basic Operational Properties of Legendre Transforms --
14.4. Applications of Legendre Transforms to Boundary Value Problems --
14.5. Exercises --
15.1. Introduction --
15.2. Definition of the Jacobi Transform and Examples --
15.3. Basic Operational Properties --
15.4. Applications of Jacobi Transforms to the Generalized Heat Conduction Problem --
15.5. Gegenbauer Transform and Its Basic Operational Properties --
15.6. Application of the Gegenbauer Transform --
16.1. Introduction --
16.2. Definition of the Laguerre Transform and Examples --
16.3. Basic Operational Properties --
16.4. Applications of Laguerre Transforms --
16.5. Exercises --
17.1. Introduction --
17.2. Definition of the Hermite Transform and Examples --
17.3. Basic Operational Properties --
17.4. Exercises --
18.1. Introduction --
18.2. Radon Transform --
18.3. Properties of the Radon Transform --
18.4. Radon Transform of Derivatives --
18.5. Derivatives of the Radon Transform --
18.6. Convolution Theorem for the Radon Transform --
18.7. Inverse of the Radon Transform and the Parseval Relation --
18.8. Applications of the Radon Transform --
18.9. Exercises --
19.1. Brief Historical Remarks --
19.2. Continuous Wavelet Transforms --
19.3. Discrete Wavelet Transform --
19.4. Examples of Orthonormal Wavelets --
19.5. Exercises --
A-1. Gamma, Beta, and Error Functions --
A-2. Bessel and Airy Functions --
A-3. Legendre and Associated Legendre Functions --
A-4. Jacobi and Gegenbauer Polynomials --
A-5. Laguerre and Associated Laguerre Functions --
A-6. Hermite Polynomials and Weber-Hermite Functions --
A-7. Mittag-Leffler Function --
B-1. Fourier Transforms --
B-2. Fourier Cosine Transforms --
B-3. Fourier Sine Transforms --
B-4. Laplace Transforms --
B-5. Hankel Transforms --
B-6. Mellin Transforms --
B-7. Hilbert Transforms --
B-8. Stieltjes Transforms --
B-9. Finite Fourier Cosine Transforms --
B-10. Finite Fourier Sine Transforms --
B-11. Finite Laplace Transforms --
B-12. Z Transforms --
B-13. Finite Hankel Transforms --
2.19. Exercises --
3.9. Exercises --
4.12. Exercises --
6.8. Exercises --
7.5. Exercises --
8.8. Exercises --
9.13. Exercises --
10.6. Exercises --
11.6. Exercises --
12.8. Exercises --
13.5. Exercises --
16.5. Exercises --
17.4. Exercises --
18.9. Exercises --
19.5. Exercises.