On the representation of Bergman–Vekua-operators for three-dimensional equations

Abstract

The integral operators of S. Bergman and I. N. Vekua transform holomorphic functions of a complex variable into solutions of linear partial differential equations of elliptic type in the plane. Generalizing these methods we find integral transforms for the solution of partial differential equations of various type (e.g. parabolic, elliptic, pseudoparabolic) with three independent variables. The transforms associate holomorphic functions of two variables and the mentioned solution. We give (for an example) an explicit representation of the kernel of these transforms (by a sum of a Duhamel product and Cauchy integrals) which generalizes recent results of D. L. Colton and R. P. Gilbert.