Levelt A.H.M.  Hypergeometric functions

Название: 
Hypergeometric functions 
Автор: 
Levelt A.H.M. 
Категория: 
Математика

Тип: 
Книга 
Дата: 
30.12.2008 23:14:49 
Скачано: 
37 
Оценка: 

Описание: 
For the first time generalized hypergeometric functions were studied systematically by J. Thomab [4]. He regarded these functions as solutions of an nth order differential equation of Fuchsian type having singularities at the points 0, 1, oo.
Following the method of Riemann's paper E. Gotjrsat [6] defined generalized hypergeometric functions not as solutions of a certain differential equation, but by properties in the large. However, he did not try to determine the monodromy group.
At the end of the last century integral representation of hypergeometric functions were found; they have been extensively used ever since (see e.g. [6] also for references).
In more recent times hypergeometric functions were almost exclusively considered from the point of view of differential equations. Many kinds of special solutions have been obtained, and the relations between them have been deduced analytically. Riemann's program, partly carried out by Goursat, was not completed until now. This slow progress was possibly caused by the difficulties which arise in the case where some differences of exponents are equal to integers. This problem can be solved if one succeeds to characterize the solution space of an analytic differential equation in a neighbourhood of a regular (or weakly) singular point. We shall do this in chapter II, in which a new theory of regular singular points will be given. It is a general theory which is by no means restricted to the hypergeometric differential equation.
The monodromy group of the generalized hypergeometric function can be found in a purely algebraic way, after a certain simple algebraic problem has been solved. Chapter I is devoted to the solution of this problem. Hypergeometric functions will be defined in chapter III under the least possible restrictions on the exponents.
The definition is analogous to Riemann's, which was described above. In the same chapter we shall show that the hypergeometric function satisfies a differential equation, and we shall compute the monodromy group, using the results of chapter I.
Finally, in chapter IV we shall describe some examples of applications. 
Файл: 
428.7 КБ 
Скачать
