On two-sided monogenic functions of axial type

TL;DR

This paper investigates two-sided axially symmetric solutions of a generalized Cauchy-Riemann operator, introducing three methods to generate all polynomial solutions, advancing the understanding of monogenic functions of axial type.

Contribution

It presents three novel methods—Cauchy-Kowalevski extension, plane wave integrals with Funk-Hecke's formula, and primitivation—to construct all polynomial solutions of two-sided axially symmetric monogenic functions.

Findings

All polynomial solutions can be generated using the three proposed methods.

The methods are effective for constructing explicit solutions.

The study enhances the theoretical framework of monogenic functions of axial type.

Abstract

In this paper we study two-sided (left and right) axially symmetric solutions of a generalized Cauchy-Riemann operator. We present three methods to obtain special solutions: via the Cauchy-Kowalevski extension theorem, via plane wave integrals and Funk-Hecke's formula and via primitivation. Each of these methods is effective enough to generate all the polynomial solutions.