Multiple hypergeometric functions and applications

목차 (Table of Contents)

• CONTENTS
• Preface
• Foreword
• Chapter 1 Single and Double Hypergeometric Functions
• 1.1 Introduction = 13
• 1.2 The Integration of the Hypergeometric Differential Equation = 15
• 1.3 Single Hypergeometric Functions of Higher Order = 19
• 1.4 Appell's Double Hypergeometric Functions = 23
• 1.5 The Kamp$$\acute e$$de F$$\acute e$$riet Function = 29
• 1.6 The Horn Functions = 36
• 1.7 Horn's General Theory of Convergence of Double Hypergeometric Secies = 37
• Chapter 2 The Lauricella Functions
• 2.1 Introduction and Definitions = 41
• 2.2 Convergence of the Lauricella Series = 43
• 2.3 Integral Representations of Euler Type = 48
• 2.4 Integrals of Laplace Type = 49
• 2.5 Integrals of Barnes Type = 51
• 2.6 Integrals with Respect to Parameters = 54
• 2.7 Other Integral Relations = 57
• 2.8 Differential Properties-Addition and Multiplication Theorems = 61
• 2.9 The General Theory of Convergence of Multiple Hypergeometric Series = 65
• 2.10 The Triple Hypergeometric Functions of Lauricella-Saran = 66
• 2.11 An Incomplete $$\mathop F_D^(n)$$Function = 70
• 2.12 Applications = 71
• Chapter 3 Other Hypergeometic Functions of Several Variables
• 3.1 The Srivastava Functions $$H_A$$,$$H_B$$and $$H_C$$ = 74
• 3.2 The Pandey Functions = 75
• 3.3 The Quadruple Hypergeometric Functions = 77
• 3.4 The Multiple Hypergeometric Functions = 89
• 3.5 The Generalised Horn Functions = 97
• 3.6 On the Functions $$\mathop C_n^(k)$$and $$\mathop D_(n)^p,q$$ = 104
• 3.7 Multiple Hypergeometric Functions of Higher Order = 106
• 3.8 Applications = 112
• Chapter 4 Multiple Hypergeometric Transformations
• 4.1 Transformations of the Triple and Quadruple Hypergeometric Functions = 113
• 4.2 Euler Integral Transformations of the Lauricella Functions = 121
• 4.3 Transformations Obtainable from the Pochhammer Integrals = 124
• 4.4 Transformations Obtainable from the Laplace Integrals = 126
• 4.5 Special Transformations of = 127
• 4.6 The Elementary Manipulation of Series = 131
• 4.7 Cases of Reducibility = 132
• 4.8 A Generalisation of Bailey's Theorem = 139
• 4.9 Certain Transformations Involving Multiple Hypergeometric Functions of Higher Order = 141
• Chapter 5 Systems of Partial Differential Equations
• 5.1 The Partial Differential Systems Associated with the Lauricella Functions = 148
• 5.2 The Broad Nature of the General Integrals = 149
• 5.3 The System Associated with $$F_1$$(I) = 152
• 5.4 The System Associated with $$F_1$$(II) = 154
• 5.5 The Application of Contour Integrals = 157
• 5.6 The Integration of the System $$F_1$$(III) = 160
• 5.7 The Integration of the System Associated with $$Φ^2$$ = 163
• 5.8 On the Systems $$\mathop F_D^(3)$$and $$\mathop F_D^(4)$$ = 166
• 5.9 The Integration of the System $$\mathop F_D^(n)$$ = 171
• 5.10 On the System $$\mathop Φ_2^(n)$$ = 173
• Chapter 6 Generating Functions and Recurrence Relations Analytical Continuation
• 6.1 Generating Relations of a General Nature = 179
• 6.2 Applications to Lauricella Polynomials = 189
• 6.3 Generating Relations of the Functions and the Generalised Horn Functions = 191
• 6.4 Recurrence Relations for the Lauricella Functions = 193
• 6.5 The Analytical Continuation of the Functions $$\mathop F_c^(n)$$, $$\mathop F_A^(n)$$and $$\mathop F_B^(n)$$ = 197
• 6.6 The Analytical Continuation of $$\mathop F_D^(n)$$ = 198
• 6.7 The Analytical Continuation of $$\mathop F_D^(3)$$ = 200
• 6.8 Certain Analytical Continuation Formulae for $$\mathop F_D^(n)$$when n is unrestricted = 210
• 6.9 Two Formulae of Toscano = 215
• 6.10 The Function R of Carlson = 216
• Chapter 7 Applications in Statistics
• 7.1 Special Univariate Distributions = 219
• 7.2 Special Multivariate Distributions = 222
• 7.3 Applications to Bayesian Inference = 227
• 7.4 Characteristic Functions = 230
• 7.5 An Application in the Field of Genetics = 233
• 7.6 The Distribution of a Sum of Gamma Variates = 237
• 7.7 Distribution of the Ratio = 237
• Chapter 8 Applications in Physics and Other Fields
• 8.1 Applications to Ordinary Differential Equations = 246
• 8.2 Asymptotic Solutions of Certain Differential Equations = 255
• 8.3 Approximate Solutions of Differential Equations = 260
• 8.4 A Type of Hyperlliptical Integral = 264
• 8.5 Applications to Quantum Theory and Lie Theory = 266
• 8.6 Applications to Dual Integral Equations = 273
• 8.7 Integrals of Several Bessel Functions Applications to the Limiting of Several Sinusoidal Signals with Gaussian Noise = 276
• 8.8 Heat Conduction Applications = 278
• 8.9 An Application in the Theory of Elasticity = 281
• Appendix A Formulae Relayed to Multiple Hypergeometric Functions = 284
• Appendix B Two Useful Computer Programs = 293
• Bibliography = 296
• Index of Applications = 305
• Index of Symbols = 306
• Subject Index = 309