Volume Cauchy formulas for slice functions on real associative *-algebras

TL;DR

This paper develops volume Cauchy integral formulas for slice functions on real associative *-algebras, generalizing classical formulas to quaternionic and Clifford algebra settings.

Contribution

It introduces a family of volume Cauchy formulas for slice and slice regular functions on real associative *-algebras, extending existing integral formulas to new algebraic contexts.

Findings

Derived volume Cauchy formulas for quaternionic slice functions

Extended Cauchy formulas to Clifford algebra slice monogenic functions

Unified framework for integral formulas across different algebraic structures

Abstract

We introduce a family of Cauchy integral formulas for slice and slice regular functions on a real associative *-algebra. For each suitable choice of a real vector subspace of the algebra, a different formula is given, in which the domains of integration are subsets of the subspace. In particular, in the quaternionic case, we get a volume Cauchy formula. In the Clifford algebra case, the choice of the paravector subspace R^(n+1) gives a volume Cauchy formula for slice monogenic functions.