A 3-Dimensional Bergman Kernel Method with Applications to Rectangular Domains

The construction of reproducing kernel functions is not restricted to real dimension 2. Indeed, the two complex variable case has been already considered by Bergman himself. Results concerning (and restricted to) the construction of Bergman kernel functions in closed form for special domains in the framework of hypercomplex function theory suggest that the so-called Bergman kernel method (BKM) for solving conformal mapping prob-lems, can also be extended to mapping problems in higher dimensions, particularly 3 and 4-dimensional cases. In this talk we revisit the 2-dimensional BKM which is based on the well known re-producing property of the Bergman kernel function. We propose a 3-dimensional BKM-approach and present examples concerned with rectangular domains in R3. The numerical results were obtained by the use of specially developed software packages for quaternions