# Quaternionic Regularity via Analytic Functional Calculus

## TL;DR

This paper introduces a new intrinsic approach to characterize quaternionic regularity using an analytic functional calculus and Cauchy type transforms on complexified quaternionic functions.

## Contribution

It provides a novel intrinsic characterization of slice quaternionic regularity through an analytic functional calculus framework.

## Key findings

- Characterization of quaternionic regularity via Cauchy type transforms
- Use of analytic stem functions to analyze quaternionic functions
- New intrinsic approach to quaternionic function theory

## Abstract

Denoting by $\mathbb{M}$ the complexification of the quaternionic algebra $\mathbb{H}$, we characterize the family of those $\mathbb{M}$-valued functions, defined on subsets of $\H$, whose values are actually quaternions, using an intrinsic approach. In particular, we show that the slice quaternionic regularity can be characterized via a Cauchy type transform acting on the space of analytic $\mathbb{M}$-valued stem functions.