Cauchy-Green type formulae in Clifford analysis

Abstract

A Cauchy integral formula is constructed for solutions to the polynomial Dirac equation ( D k + ∑ m = 0 k − 1 b m D m ) f = 0 ({D^k} + \sum \nolimits _{m = 0}^{k - 1} {{b_m}{D^m})f = 0} , where each b m {b_m} is a complex number, D D is the Dirac operator in R n {R^n} , and f f is defined on a domain in R n ^{{R^n}} and takes values in a complex Clifford algebra. Some basic properties for the solutions to this equation, arising from the integral formula, are described, including an approximation theorem. We also introduce a Bergman kernel for square integrable solutions to ( D + λ ) f = 0 (D + \lambda )f = 0 over bounded domains with piecewise C 1 {C^1} , or Lipschitz, boundary.