Cauchy-Green type formulae in Clifford analysis

Ryan, John

doi:10.1090/S0002-9947-1995-1249888-8

Cauchy-Green type formulae in Clifford analysis
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Trans. Amer. Math. Soc. 347 (1995), 1331-1341 Request permission

Abstract:

A Cauchy integral formula is constructed for solutions to the polynomial Dirac equation $({D^k} + \sum \nolimits _{m = 0}^{k - 1} {{b_m}{D^m})f = 0}$, where each ${b_m}$ is a complex number, $D$ is the Dirac operator in ${R^n}$, and $f$ is defined on a domain in $^{{R^n}}$ 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 $(D + \lambda )f = 0$ over bounded domains with piecewise ${C^1}$, or Lipschitz, boundary.

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