Hyperbolic function theory in R3 Sirkka-Liisa Eriksson-Bique

We consider the Poincare model of a hyperbolic geometry in R3(ie., the metric is ds2 = dx 2+dy2+dt2 t2) and Möbius transformations. We study also an old problem how to extend one variable complex func-tion theory to higher dimensions. We denote by H the real associative algebra of quaternions generated by e1,e2 satisfying e21 = e 2 2 = 1 and e1e2 = e2e1. In 1992 Leutwiler noticed that xm (x 2 H) is a conjugate gradient of a harmonic function with respect to a hyper-bolic metric. We study an extension of solutions of Leutwiler, called hyperholomorphic. Hyperholomorphic functions satisfy the equatio