The algebraic character of a class of harmonic functions in three variables

Abstract

It is obvious that the coefficients An, of the series development (1) determine completely the function F(r, cos 0, 4). Thus the sequence of coefficients determines the analytic character of F, the locations and nature of singular points, and so on. It seems therefore natural to look for and extract those properties of a sequence of coefficients Any which most readily yield relevant information. In the case of harmonic functions of two variables, the problem reduces immediately to that of detecting the singularities of an analytic function given by its Taylor development. This last mentioned problem, also known as the Hadamard problem, has occupied the attention of mathematicians for a considerable period and a great many results have been obtained. When dealing, however, with harmonic functions of three variables, the problem becomes much more difficult and it seems impossible to apply directly the methods developed in the theory of functions of one complex variable. We have, however, at our disposal the method of integral operators introduced by S. Bergman, which enables us to represent harmonic functions in three variables by means of an integral operator on a function of a complex variable. This representation makes it possible to carry over certain results of the well known theory of functions of one complex variable into the relatively little developed theory of harmonic functions of three variables. The present paper is to be considered as a part of an extended program of study of harmonic functions given by the de-