Mathai / Saxena / Haubold The H-Function

Theory and Applications
1. Auflage 2009
ISBN: 978-1-4419-0916-9
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 268 Seiten

TheH-function or popularly known in the literature as Fox'sH-function has recently found applications in a large variety of problems connected with reaction, diffusion, reaction-diffusion, engineering and communication, fractional differ- tial and integral equations, many areas of theoretical physics, statistical distribution theory, etc. One of the standard books and most cited book on the topic is the 1978 book of Mathai and Saxena. Since then, the subject has grown a lot, mainly in the elds of applications. Due to popular demand, the authors were requested to - grade and bring out a revised edition of the 1978 book. It was decided to bring out a new book, mostly dealing with recent applications in statistical distributions, pa- way models, nonextensive statistical mechanics, astrophysics problems, fractional calculus, etc. and to make use of the expertise of Hans J. Haubold in astrophysics area also. It was decided to con ne the discussion toH-function of one scalar variable only. Matrix variable cases and many variable cases are not discussed in detail, but an insight into these areas is given. When going from one variable to many variables, there is nothing called a unique bivariate or multivariate analogue of a givenfunction. Whatever be the criteria used, there may be manydifferentfunctions quali ed to be bivariate or multivariate analogues of a given univariate function. Some of the bivariate and multivariateH-functions, currently in the literature, are also questioned by many authors.

Autoren/Hrsg.

Mathai, A. M.

Saxena, Ram Kishore

Haubold, Hans J.

Inhaltsverzeichnis

1;Preface;7
2;Contents;9
3;1 On the H-Function With Applications;13
3.1;1.1 A Brief Historical Background;13
3.2;1.2 The H-Function;14
3.3;1.3 Illustrative Examples;19
3.4;1.4 Some Identities of the H-Function;23
3.4.1;1.4.1 Derivatives of the H-Function;25
3.5;1.5 Recurrence Relations for the H-Function;28
3.6;1.6 Expansion Formulae for the H-Function;29
3.7;1.7 Asymptotic Expansions;31
3.8;1.8 Some Special Cases of the H-Function;33
3.8.1;1.8.1 Some Commonly Used Special Cases of the H-Function;38
3.9;1.9 Generalized Wright Functions;41
3.9.1;1.9.1 Existence Conditions;42
3.9.2;1.9.2 Representation of Generalized Wright Function;43
4;2 H-Function in Science and Engineering;56
4.1;2.1 Integrals Involving H-Functions;56
4.2;2.2 Integral Transforms of the H-Function;56
4.2.1;2.2.1 Mellin Transform;56
4.2.2;2.2.2 Illustrative Examples;57
4.2.3;2.2.3 Mellin Transform of the H-Function;58
4.2.4;2.2.4 Mellin Transform of the G-Function;59
4.2.5;2.2.5 Mellin Transform of the Wright Function;59
4.2.6;2.2.6 Laplace Transform;59
4.2.7;2.2.7 Illustrative Examples;60
4.2.8;2.2.8 Laplace Transform of the H-Function;61
4.2.9;2.2.9 Inverse Laplace Transform of the H-Function;62
4.2.10;2.2.10 Laplace Transform of the G-Function;63
4.2.11;2.2.11 K-Transform;64
4.2.12;2.2.12 K-Transform of the H-Function;65
4.2.13;2.2.13 Varma Transform;66
4.2.14;2.2.14 Varma Transform of the H-Function;66
4.2.15;2.2.15 Hankel Transform;67
4.2.16;2.2.16 Hankel Transform of the H-Function;68
4.2.17;2.2.17 Euler Transform of the H-Function;69
4.3;2.3 Mellin Transform of the Product of Two H-Functions;71
4.3.1;2.3.1 Eulerian Integrals for the H-Function;71
4.3.2;2.3.2 Fractional Integration of a H-Function;73
4.4;2.4 H-Function and Exponential Functions;78
4.5;2.5 Legendre Function and the H-Function;80
4.6;2.6 Generalized Laguerre Polynomials;82
5;3 Fractional Calculus;86
5.1;3.1 Introduction;86
5.2;3.2 A Brief Historical Background;87
5.3;3.3 Fractional Integrals;88
5.3.1;3.3.1 Riemann–Liouville Fractional Integrals;90
5.3.2;3.3.2 Basic Properties of Fractional Integrals;90
5.3.3;3.3.3 Illustrative Examples;92
5.4;3.4 Riemann–Liouville Fractional Derivatives;94
5.4.1;3.4.1 Illustrative Examples;99
5.5;3.5 The Weyl Integral;102
5.5.1;3.5.1 Basic Properties of Weyl Integrals;102
5.5.2;3.5.2 Illustrative Examples ;103
5.6;3.6 Laplace Transform;105
5.6.1;3.6.1 Laplace Transform of Fractional Integrals;105
5.6.2;3.6.2 Laplace Transform of Fractional Derivatives;105
5.6.3;3.6.3 Laplace Transform of Caputo Derivative;106
5.7;3.7 Mellin Transforms ;107
5.7.1;3.7.1 Mellin Transform of the nth Derivative;108
5.7.2;3.7.2 Illustrative Examples;108
5.8;3.8 Kober Operators;109
5.8.1;3.8.1 Erdélyi–Kober Operators ;109
5.9;3.9 Generalized Kober Operators;112
5.10;3.10 Saigo Operators;114
5.10.1;3.10.1 Relations Among the Operators;117
5.10.2;3.10.2 Power Function Formulae ;117
5.10.3;3.10.3 Mellin Transform of Saigo Operators;119
5.10.4;3.10.4 Representation of Saigo Operators;119
5.11;3.11 Multiple Erdélyi–Kober Operators;124
5.11.1;3.11.1 A Mellin Transform;125
5.11.2;3.11.2 Properties of the Operators;126
5.11.3;3.11.3 Mellin Transform of a Generalized Operator;127
6;4 Applications in Statistics;129
6.1;4.1 Introduction;129
6.2;4.2 General Structures;129
6.2.1;4.2.1 Product of Type-1 Beta Random Variables;131
6.2.2;4.2.2 Real Scalar Type-2 Beta Structure;134
6.2.3;4.2.3 A More General Structure;135
6.3;4.3 A Pathway Model;137
6.3.1;4.3.1 Independent Variables Obeying a Pathway Model;138
6.4;4.4 A Versatile Integral;141
6.4.1;4.4.1 Case of <1 or <1;143
6.4.2;4.4.2 Some Practical Situations;146
7;5 Functions of Matrix Argument;149
7.1;5.1 Introduction;149
7.2;5.2 Exponential Function of Matrix Argument;150
7.3;5.3 Jacobians of Matrix Transformations;153
7.4;5.4 Jacobians in Nonlinear Transformations;156
7.5;5.5 The Binomial Function;159
7.6;5.6 Hypergeometric Function and M-transforms;161
7.7;5.7 Meijer's G-Function of Matrix Argument;164
7.7.1;5.7.1 Some Special Cases;165
8;6 Applications in Astrophysics Problems;169
8.1;6.1 Introduction;169
8.2;6.2 Analytic Solar Model;169
8.3;6.3 Thermonuclear Reaction Rates;173
8.4;6.4 Gravitational Instability Problem;175
8.5;6.5 Generalized Entropies in Astrophysics Problems;178
8.5.1;6.5.1 Generalizations of Shannon Entropy;179
8.6;6.6 Input–Output Analysis;181
8.7;6.7 Application to Kinetic Equations;183
8.8;6.8 Fickean Diffusion;184
8.8.1;6.8.1 Application to Time-Fractional Diffusion;185
8.9;6.9 Application to Space-Fractional Diffusion;187
8.10;6.10 Application to Fractional Diffusion Equation;188
8.10.1;6.10.1 Series Representation of the Solution;190
8.11;6.11 Application to Generalized Reaction-Diffusion Model;192
8.11.1;6.11.1 Motivation;192
8.11.2;6.11.2 Mathematical Prerequisites;193
8.11.3;6.11.3 Fractional Reaction–Diffusion Equation;195
8.11.4;6.11.4 Some Special Cases;196
8.11.5;6.11.5 Fractional Order Moments;199
8.11.6;6.11.6 Some Further Applications;200
8.11.7;6.11.7 Background;201
8.11.8;6.11.8 Unified Fractional Reaction–Diffusion Equation;202
8.11.9;6.11.9 Some Special Cases;203
8.11.10;6.11.10 More Special Cases;208
9;Appendix;214
9.1;A.1 H-Function of Several Complex Variables;214
9.2;A.2 Kampé de Fériet Function and Lauricella Functions;216
9.2.1;A.2.1 Kampé de Fériet Series in the Generalized Form;216
9.2.2;A.2.2 Generalized Lauricella Function;217
9.3;A.3 Appell Series;220
9.3.1;A.3.1 Confluent Hypergeometric Function of Two Variables;221
9.4;A.4 Lauricella Functions of Several Variables;222
9.4.1;A.4.1 Confluent form of Lauricella Series;224
9.5;A.5 The Generalized H-Function (The -Function);224
9.5.1;A.5.1 Special Cases of -Function;225
9.6;A.6 Representation of an H-Function in Computable Form;227
9.7;A.7 Further Generalizations of the H-Function;228
10;Bibliography;230
11;Glossary of Symbols;267
12;Author Index;269
13;Subject Index;275