A Cauchy integral formula in superspace

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, a purely fermionic Cauchy formula is proved. 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 the even as well 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 proved. © 2009 London Mathematical Society.