Monogenic functions and harmonic vectors

Abstract

We consider special topological vector spaces with a commutative multiplication for some of elements of the spaces and monogenic functions taking values in these spaces. Monogenic functions are understood as continuous and differentiable in the sense of G\^ateaux functions. We describe relations between the mentioned monogenic functions and harmonic vectors in the three-dimensional real space and establish sufficient conditions for infinite monogeneity of functions. Unlike the classical complex analysis, it is done in the case where the validity of the Cauchy integral formula for monogenic functions remains an open problem.

Keywords

• monogenic function
• harmonic function
• harmonic vector
• gateaux derivative
• topological vector space
• commutative multiplication