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Bulletin of the London Mathematical Society Advance Access published online on May 22, 2009
Bulletin of the London Mathematical Society, doi:10.1112/blms/bdp045
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© 2009 London Mathematical Society

A Cauchy integral formula in superspace

H. De Bie and F. Sommen

Clifford Research Group
Department of Mathematical Analysis
Faculty of Engineering
Ghent University
Galglaan 2
9000 Gent
Belgium
fs@cage.ugent.be

Received 15 September 2008. Revision received 18 February 2009.

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.

2000 Mathematics Subject Classification 30G35 (primary), 58C50 (secondary).

The first author is supported by a Ph.D. Fellowship of the Research Foundation, Flanders (FWO).


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