Noncommutative Cauchy integral formula

TL;DR

This paper extends the classical Cauchy integral formula to slice regular functions on real alternative *-algebras, enabling new local series expansions in noncommutative and nonassociative contexts.

Contribution

It provides the most general Cauchy integral formula for slice regular and C^1 functions on real alternative *-algebras, generalizing holomorphic functions.

Findings

Derived a general Cauchy integral formula for slice regular functions

Established two types of local series expansions for these functions

Extended classical complex analysis results to noncommutative settings

Abstract

The aim of this paper is to provide and prove the most general Cauchy integral formula for slice regular functions and for C^1 functions on a real alternative *-algebra. Slice regular functions represent a generalization of the classical concept of holomorphic function of a complex variable in the noncommutative and nonassociative settings. As an application, we obtain two kinds of local series expansion for slice regular functions.