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This book has been designed with the object of providing tables of integrals of hypergeometric functions of many different types. It is believed that, as a result of a re-ordering of the material, a number of the integrals included here have not previously appeared explicitly in the literature. At the same time, a number of computer programs to evaluate numerically some of the hypergeometric integrals concerned are given. These programs have been set up in such a way that they may be extended to cover wider results with a minimum of essential modification and the international computer language FORTRAN IV has been used so that the programs have the widest possible applicability. It is believed that this is the first time that programs of this type have been included in a book of tables ofintegrals.
Handbooks of mathematical formulae very often include little or no theoretical background to the results listed. In an attempt to fill this gap, the first part of this book consists of seven chapters on the general theory of hypergeometric
integrals. Additionally, an important aspect of this work is the concluding bibliography. The breadth of these aims, combined with the necessity for keeping the size of the book within bounds, has made it necessary to be as concise as is compatible with intelligibility. It may be observed that an important part of the fascination of working with integrals of hypergeometric functions is that the results themselves are of hypergeometric form. The wide variety of applications of hypergeometric integrals is exemplified in Chapter 7 where a number of instances of the appearance of such expressions in statistics and mathematical physics are cited. One reason why such integrals are of importance is that the majority of the classical special functions, such as those of Laguerre, Legendre, Hermite and Bessel, are particular cases of hypergeometric functions.
A hypergeometric integral is any integral which has a hypergeometric function in its integrand. Such hypergeometric integrals have frightened mathematicians for a long time, because they look complicated. This is mainly the fault of the notation which is necessary to exhibit the underlying structure of the integrals. The functions in these integrands range from simple Bessel and Whittaker functions up to generalised hypergeometric functions of several variables with many parameters.
This handbook supplies an ordered list of the main analytical formulae of these integrals, together with some simple computer programs, in FORTRAN, which will evaluate these integrals numerically provided that we keep away from values of the parameters which are near any poles of the integrands.
The book is in two parts. The first part consists of seven chapters which give the general theory of these integrals, and examples of their uses, particularly in the study of statistical distributions, and in various branches of physics. The second part consists of about 100 pages of tables of the formulae and about 50 pages of computer programs to evaluate the integrals. There is also an extensive bibliography.
Preface
Foreword by Dr. L. J. Slater
Theory and Applications
Hypergeometric Functions of One or More Variables Introduction.
The Gauss Hypergeometric Funciion.
The Generalised Hypergeometric Function
Hypergeometric Functions of Two Variables
Multiple Hypergeometric Functions
Integrals of Euler Type
General Euler Integrals
Euler Integrals Associated with the Confluent Hypergeometric Function .
Euler Integrals Associated with the Gauss Function
Definite Integrals and Repeated Integrals
Definite Integrals Involving One Hypergeometric Function
Definite Integrals Associated with the Kummer and Bessel Functions …
Definite Integrals Associated with the Gauss Function . .
Repeated Integrals.
Contour Integrals
Pochhammer Integrals
Barnes Integrals.
Infinite Integrals
Introduction. A Theorem on the Integration of Series over an Infinite Range
Infinite Integrals Associated with the Confluent Hypergeometric Functions
Infinite Integrals as Inverses of Barnes Integrals.
Integrals of Laplace Type
Multiple Integrals
Multiple Euler Integrals
Multiple Laplace Integrals and Hankel Integrals
Multiple Barnes Integrals
Applications
Generalised Univariate Statistical Distributions
Generalised Multivariate Distributions.
The Study of the Angular Displacement of a Shaft.
The Vibration of a Thin Elastic Plate.
Heat Production in a Cylinder
Motion of a Viscous Fluid under a Surface Load
The Time-domain Synthesis of Signals.
A Dirichlet Problem for the Interior of a Unit Sphere
Dual Integral Equations with Legendre Function Kernels
The Linear Flow of Heat in an Anisotropic Solid moving in a Conducting Medium
An Application in the Study of Non-linear Oscillations.
The Laplace Transform of the Product of Several Molecular Integrals
A Problen in Communications Engineering
An application of Repeated Integrals of Bessel Functions
An Application in Combinatorial Analysis
Tables and Computer Programs
Tables of Hypergeometric Integrals
Integrals of Euler Type
Definite Integrals.
Repeated Integrals
Pochhammer Integrals
Integrals with respect to Parameters
Laplace Intergrals
Hankel Loop Integrals
Mellin Integrals…
Multiple Integrals
Computer Programs
Programs for the Evaluation of Euler Integrals
Programs for the Evaluation of Definite Integrals
Programs for the Evaluation of Repeated Integrals
Programs for the Evaluation of Barnes Integrals and Related Integrals
Programs for the Evaluation of Laplace Integrals
Programs for the Evaluation of Mellin Integrals
Programs for the Evaluation of Multiple Integrals
Programs for the Evaluation of Integrels Involving the Modified Bessel
Function of the Second Kind
Programs for the Evaluation of Convolution Integrals.
Selected Bibliography.
Index of Symbols
Subject Index