Constructing steering-type solutions for higher order Cauchy-Riemann equations in $\mathbb{R}^{m+1}$

TL;DR

This paper develops explicit methods to construct solutions for higher order Cauchy-Riemann equations in higher dimensions, focusing on generating solutions from specific complex functions and analyzing their properties.

Contribution

It introduces a framework for constructing solutions to multidimensional Cauchy-Riemann systems using families of complex functions closed under conjugation and differential operators.

Findings

Solutions include polymonogenic and polyharmonic functions.

Some solutions satisfy homogeneous linear differential equations with hypercomplex derivatives.

Provides explicit construction methods for higher order PDE solutions.

Abstract

The multidimensional Cauchy-Riemann operator provides a framework for studying higher order partial differential equations in $\mathbb{R}^{m+1}$, whose solutions include polymonogenic and polyharmonic functions, among others. In this work, we aim to explicitly construct solutions to such systems, generated from families of complex valued functions which are closed under conjugation and under the action of the complex Cauchy-Riemann operator. Moreover, we prove that precisely some of these solutions also satisfy homogeneous linear differential equations involving the so-called hypercomplex derivative.