Clifford Analysis with Indefinite Signature

TL;DR

This paper extends Clifford analysis to indefinite quadratic forms, introducing new monogenic functions and deriving two versions of Cauchy's integral formulas to handle singularities effectively.

Contribution

It develops (p,q)-monogenic functions in indefinite Clifford analysis and presents two novel Cauchy integral formulas inspired by different singularity handling methods.

Findings

Two versions of Cauchy's integral formulas derived

Methods improve handling of singularities in indefinite Clifford analysis

Extension of classical Clifford analysis to indefinite signatures

Abstract

We extend constructions of classical Clifford analysis to the case of indefinite non-degenerate quadratic forms. We define (p,q)-left- and right-monogenic functions by means of Dirac operators that factor a certain wave operator. We prove two different versions of Cauchy's integral formulas for these functions. The two formulas arise from dealing with singularities in distinct ways, and are inspired by the methods of [L, FL]. These results indicate the merit of these methods for dealing with singularities.