Hyperbolic Laplacian and Dirichlet Problem in the Bicomplex Context

In this article, we present all the necessary elements for the detailed solutions of the Dirichlet problem for the cases of the –unit disk and for what we will call the semi–positive subspace in (which is the analogous of the half right–side space in complex analysis). The hyperbolic numbers as well as some hyperbolic objects play a crucial role to obtain bicomplex versions of the Poisson's Formulae in both cases.