Generalized partial-slice monogenic functions

TL;DR

This paper introduces a new unified function theory that encompasses both monogenic and slice monogenic functions, establishing fundamental properties and invariance under conformal transformations.

Contribution

It develops a comprehensive framework that generalizes existing theories, providing new tools and results for analyzing these functions.

Findings

Proves the identity theorem and representation formula.

Establishes Cauchy and integral formulas for the new class.

Shows conformal invariance under specific Möbius transformations.

Abstract

The two function theories of monogenic and of slice monogenic functions have been extensively studied in the literature and were developed independently; the relations between them, e.g. via Fueter mapping and Radon transform, have been studied. The main purpose of this article is to describe a new function theory which includes both of them as special cases. This theory allows to prove nice properties such as the identity theorem, a Representation Formula, the Cauchy (and Cauchy-Pompeiu) integral formula, the maximum modulus principle, a version of the Taylor and Laurent series expansions. As a complement, we shall also offer two approaches to these functions via slice functions and via global differential operators. In addition, we discuss the conformal invariance property under a proper group of M\"{o}bius transformations preserving the partial symmetry of the involved domains.