H-Transforms: Theory and Applications

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Format: Hardcover
Pub. Date: 2004-03-17
Publisher(s): CRC Press
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Summary

H-Transforms: Theory and Applications presents a unified approach to the study of a wide class of integral transforms containing H- functions as kernels-or H-Transforms-and their applications. It provides a general introduction to the theory of integral transforms and details the existence, representation, expansion, and properties of H-transforms. Discussions also include applications of integral transforms with kernels involving the Meijer G-transform and special functions of hypergeometric and Bessel type. This book will not only appeal to postgraduates and researchers in pure and applied, but also to specialists in physics, mechanics, and engineering.

Author Biography

Megumi Saigo is a Professor in the Department of Mathematics at the University of Fukuoka, Japan.

Table of Contents

Preface ix
Definition, Representations and Expansions of the H-Function
1(30)
Definition of the H-Function
1(2)
Existence and Representations
3(2)
Explicit Power Series Expansions
5(2)
Explicit Power-Logarithmic Series Expansions
7(2)
Algebraic Asymptotic Expansions at Infinity
9(3)
Exponential Asymptotic Expansions at Infinity in the Case Δ > 0, a* = 0
12(5)
Exponential Asymptotic Expansions at Infinity in the Case n = 0
17(2)
Algebraic Asymptotic Expansions at Zero
19(2)
Exponential Asymptotic Expansions at Zero in the Case Δ < 0, a* = 0
21(3)
Exponential Asymptotic Expansions at Zero in the Case m = 0
24(1)
Bibliographical Remarks and Additional Information on Chapter 1
25(6)
Properties of the H-Function
31(40)
Elementary Properties
31(2)
Differentiation Formulas
33(3)
Recurrence Relations and Expansion Formulas
36(5)
Multiplication and Transformation Formulas
41(2)
Mellin and Laplace Transforms of the H-Function
43(5)
Hankel Transforms of the H-Function
48(3)
Fractional Integration and Differentiation of the H-Function
51(5)
Integral Formulas Involving the H-Function
56(6)
Special Cases of the H-Function
62(5)
Bibliographical Remarks and Additional Information on Chapter 2
67(4)
H-Transform on the Space Lν,2
71(22)
The H-Transform and the Space Lν,τ
71(1)
The Mellin Transform on Lν,τ
72(2)
Some Auxiliary Operators
74(3)
Integral Representations for the H-Function
77(5)
Lν,2-Theory of the General Integral Transform
82(4)
Lν,2-Theory of the H-Transform
86(4)
Bibliographical Remarks and Additional Information on Chapter 3
90(3)
H-Transform on the Space Lν,τ
93(40)
Lν,τ-Theory of the H-Transform When a* = Δ = 0 and Re(μ) = 0
93(4)
Lν,τ-Theory of the H-Transform When a* = Δ = 0 and Re(μ) < 0
97(3)
Lν,τ-Theory of the H-Transform When a* = 0, Δ > 0
100(4)
Lν,τ-Theory of the H-Transform When a* = 0, Δ < 0
104(3)
Lν,τ-Theory of the H-Transform When a* > 0
107(1)
Boundedness and Range of the H-Transform When a*1 > 0 and a*2 > 0
108(4)
Boundedness and Range of the H-Transform When a* > 0 and a*1 = 0 or a*2 = 0
112(3)
Boundedness and Range of the H-Transform When a* > 0 and a*1 < 0 or a*2 < 0
115(3)
Inversion of the H-Transform When Δ = 0
118(3)
Inversion of the H-Transform When Δ ≠ 0
121(5)
Bibliographical Remarks and Additional Information on Chapter 4
126(7)
Modified H-Transforms on the Space Lν,τ
133(32)
Modified H-Transforms
133(1)
H1-Transform on the Space Lν,τ
134(6)
H2-Transform on the Space Lν,τ
140(5)
Hσ,κ-Transform on the Space Lν,τ
145(5)
H1σ,κ-Transform on the Space Lν,τ
150(5)
H2σ,κ-Transform on the Space Lν,τ
155(5)
Bibliographical Remarks and Additional Information on Chapter 5
160(5)
G-Transform and Modified G-Transforms on the Space Lν,τ
165(38)
G-Transform on the Space Lν,τ
165(8)
Modified G-Transforms
173(2)
G1-Transform on the Space Lν,τ
175(4)
G2-Transform on the Space Lν,τ
179(4)
Gσ,κ-Transform on the Space Lν,τ
183(5)
G1σ,κ-Transform on the Space Lν,τ
188(5)
G2σ,κ-Transform on the Space Lν,τ
193(4)
Bibliographical Remarks and Additional Information on Chapter 6
197(6)
Hypergeometric Type Integral Transforms on the Space Lν,τ
203(60)
Laplace Type Transforms
203(3)
Meijer and Varma Integral Transforms
206(6)
Generalized Whittaker Transforms
212(4)
Dγ-Transforms
216(3)
1F2-Transforms
219(4)
2F1-Transforms
223(4)
2F1-Transforms
227(6)
Modified 2F1-Transforms
233(5)
The Generalized Stieltjes Transform
238(2)
pFq-Transform
240(6)
The Wright Transform
246(5)
Bibliographical Remarks and Additional Information on Chapter 7
251(12)
Bessel Type Integral Transforms on the Space Lν,τ
263(94)
The Hankel Transform
263(7)
Fourier Cosine and Sine Transforms
270(2)
Even and Odd Hilbert Transforms
272(4)
The Extended Hankel Transform
276(4)
The Hankel Type Transform
280(5)
Hankel-Schwartz and Hankel-Clifford Transforms
285(4)
The Transform Yη
289(10)
The Struve Transform
299(8)
The Meijer kη-Transform
307(6)
Bessel Type Transforms
313(8)
The Modified Bessel Type Transform
321(4)
The Generalized Hardy-Titchmarsh Transform
325(12)
The Lommel-Maitland Transform
337(5)
Bibliographical Remarks and Additional Information on Chapter 8
342(15)
Bibliography 357(20)
Subject Index 377(4)
Author Index 381(4)
Symbol Index 385

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