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Key wordsn u = f . Extending the concept of Neumann functions for the Laplacian to Neumann functions for powers of the Laplacian leads to an explicit representation of the solution to the Neumann-n problem for ∆ n u = f . The representation formula provides the tool to treat more general partial differential equations with leading term ∆ n u in reducing them into some singular integral equations.

In this article, the Dirichlet problem of polyharmonic functions is considered. As well the explicit expression of the unique solution to the simple Dirichlet problem for polyharmonic functions is obtained by using the decomposition of polyharmonic functions and turning the problem into an equivalent Riemann boundary value problem for polyanalytic functions, as the approach to find the kernel functions of the solution for the general Dirichlet problem is given.

Integral representation formulas of Cauchy-Pompeiu type expressing Clifford-algebra-valued functions in domains of R through its boundary values and its first order derivatives in form of the Dirac operator are iterated in order to get higher order Cauchy-Pompeiu formulas. In the most general representation formulas obtained the Dirac operator is replaced by products of powers of the Dirac and the Laplace operator. Boundary values of lower order operators are involved too. In particular the integral operators provide particular solutions to the inhomogeneous equations 8kw = j AkW = g and 8&'w = h. The main subject of this paper is to develop the representation formulas. Properties of the integral operators are not studied here.