Martinelli–Bochner Type Formulae in Complex Clifford Analysis

Abstract

Various types of solutions of the systems $(D_x+i{D}_y)f=0$ are considered, where $D_x$ and $D_y$ are Dirac type operators in ${\mathbb{R}}^m$. Generalizing the classical Martinelli–Bochner formula for holomorphic functions, such a formula is proved for the $C_1$-solutions of this system. Martinelli–Bochner formulae are also obtained for other overdetermined systems occuring in Clifford analysis.