Bergmansche Integraloperatoren für instationäre Prozesse in der Ebene (Q793211)

The integral operator of Bergman transforms holomorphic functions \(f=f(z)\) into solutions \(u=u(z,z^*)\) of a linear elliptic differential equation of complex variables \[ u_{zz^*}+A(z,z^*)u_ z+B(z,z^*)u_{z^*}+C(z,z^*)u=0. \] This article considers the differential equation (1) \(u_{zz^*}+Su=0\) where \(u=u(z,z^*,t)\) and S is a linear differential operator which depends only on t. A generalization of the integral operator of Bergman is presented which gives a local representation of solutions of (1) in terms of the powers of S. The exact form of the transformation is given for some choices of S which include examples of hyperbolic, parabolic and pseudoparabolic type.