Noncommutative Cauchy Integral Formula

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.