Slice monogenic functions

TL;DR

This paper introduces a new class of monogenic functions in Clifford algebra, extending polynomials and power series, and establishes fundamental results like a Cauchy integral formula and zeroes analysis.

Contribution

It defines monogenicity for functions on +1 with values in Clifford algebra, inspired by recent work, and proves core properties including a Cauchy integral formula.

Findings

Defined monogenic functions in Clifford algebra context

Proved a Cauchy integral formula for these functions

Analyzed zeros of polynomials and power series

Abstract

In this paper we offer a definition of monogenicity for functions defined on $\rr^{n+1}$ with values in the Clifford algebra $\rr_n$ following an idea inspired by the recent papers \cite{gs}, \cite{advances}. This new class of monogenic functions contains the polynomials (and, more in general, power series) with coefficients in the Clifford algebra $\rr_n$. We will prove a Cauchy integral formula as well as some of its consequences. Finally, we deal with the zeroes of some polynomials and power series.