On Modified Biquaternionic Analysis in ℂ³

Abstract

Identify ℂ³ with the set of reduced biquaternions of the form z = z ₀+ z ₁e₁ + z ₂ e ₂, where z _i ∈ ℂ. For Ω⊂ ℂ³, let ω: Ω → ℂ³ be a function ω = ω₀+ ω₁ e ₁+ ω₂ e ₂. We consider the solutions of Our system is a modification of a system introduced by Li Yucheng and Qiao Yuying and a complexification of a system introduced by H. Leutwiler. These M_k-solutions are connected with k-hyperbolic harmonic functions h by

$$w={\partial h\over \partial {\bar z}_{o}}-{\partial h\over \partial {\bar z}_{1}}{e_{1}}-{\partial h\over \partial {\bar z}_{2}}{e_{2}}$$

where h satisfies

$$z_{2}\triangle h- 4k{\partial h\over \partial {\bar z}_{2}}=0$$

The k-hyperbolic harmonic functions are also connected to polyharmonic ones, Δ^k h is harmonic.

We find basic properties for M_k-solutions and k-hyperbolic harmonic functions and examine their bases.