A local Cauchy integral formula for slice-regular functions

TL;DR

This paper establishes a local Cauchy integral formula for slice-regular functions in quaternionic space, removing the need for axial symmetry and advancing the understanding of their boundary behavior.

Contribution

It introduces a new local Cauchy-type integral formula for slice-regular functions without axial symmetry, using a decomposition into axially monogenic functions.

Findings

Proves a boundary integral formula for slice-regular functions

Decomposes slice-regular functions into axially monogenic components

Extends integral formulas to non-axially symmetric domains

Abstract

We prove a Cauchy-type integral formula for slice-regular functions where the integration is performed on the boundary of an open subset of the quaternionic space, with no requirement of axial symmetry. In particular, we get a local Cauchy-type integral formula. As a step towards the proof, we provide a decomposition of a slice-regular function as a combination of two axially monogenic functions.