A Green's Function in the Theory of Functions of Several Complex Variables

Abstract

The material presented here constitutes a sequel to the previous paper of the author entitled new formalism for functions of several complex [2]. In that paper, the emphasis was upon the application of the variational procedure of Lagrange multipliers to minimum problems for multiple integrals of analytic functions, with the Cauchy-Riemann equations treated as differential side conditions. Motivated by the discussions based upon such procedures, we turn here to a more direct development of the theory of boundary value problems associated with the Cauchy-Riemann equations for analytic functions of several complex variables. The chief results center about a new boundary value problem which plays for several variables a role analogous to that of the Dirichlet problem in one variable. This boundary value problem is solved by means of a Dirichlet principle, and we introduce a Green's function in terms of which the solution can be expressed as a boundary integral. A formula giving the Bergman kernel function for several variables [1] in terms of this Green's function is obtained, and we thus generalize known theorems from the theory of functions of one complex variable and the theory of linear elliptic partial differential equations [3]. Similar results are developed for the Szeg6 kernel function in several variables.