Integral Formulas for k-hypermonogenic Functions in ℝ³

Abstract

We consider harmonic functions with respect to the Laplace–Beltrami operator of the Riemannian metric $ds^{2} = x_{2}^{-2k}(\sum \limits{_{i=0}^{2}}{dx_{i}^{2}})$ and their quaternion function theory in ℝ³. Leutwiler noticed around 1990 that if the usual Euclidean metric is changed to the hyperbolic one, that is k=1, then the power function $(x_{0}+x_{1}e_{1}+x_{2}e_{2})^{n}$, calculated using quaternions, is the conjugate gradient of a hyperbolic harmonic function. We study generalized holomorphic functions, called k-hypermonogenic functions satisfying the modified Dirac equation. Note that 0-hypermonogenic are monogenic and 1-hypermonogenic functions are hypermonogenic defined by H. Leutwiler and the first author.

We prove the Cauchy type integral formulas for k-hypermonogenic where the kernels are calculated using the hyperbolic distance of the Poincará upper half-space model. Earlier these results have been proved for hypermonogenic functions.

Mathematics Subject Classification (2010). Primary 30A05; Secondary 30A45.