A Cauchy integral formula in superspace

TL;DR

This paper extends hypercomplex function theory in superspace by deriving a Cauchy integral formula that combines fermionic and bosonic cases, using Clifford analysis techniques.

Contribution

It introduces a unified Cauchy integral formula in superspace, integrating fermionic and bosonic components within a Clifford algebra framework.

Findings

Derived a fermionic Cauchy formula.

Combined fermionic and bosonic formulas for the general case.

Proved a Cauchy-Pompeiu formula and a representation formula for monogenic functions.

Abstract

In previous work the framework for a hypercomplex function theory in superspace was established and amply investigated. In this paper a Cauchy integral formula is obtained in this new framework by exploiting techniques from orthogonal Clifford analysis. After introducing Clifford algebra valued surface- and volume-elements first a purely fermionic Cauchy formula is proven. Combining this formula with the already well-known bosonic Cauchy formula yields the general case. Here the integration over the boundary of a supermanifold is an integration over as well the even as the odd boundary (in a formal way). Finally, some additional results such as a Cauchy-Pompeiu formula and a representation formula for monogenic functions are proven.