18 th International Conference on the Application of Computer Science and Mathematics in Architecture and Civil Engineering K. G¨ urlebeck and C. K ¨ onke (eds.) Weimar, Germany, 07–09 July 2009 INTEGRAL REPRESENTATION FORMULAE IN HERMITEAN CLIFFORD ANALYSIS F. Brackx ∗ , H. De Schepper, M.E. Luna-Elizarrar´ as and M. Shapiro ∗ Clifford Research Group, Faculty of Engineering, Ghent University Galglaan 2, 9000 Gent, Belgium E-mail: fb@cage.ugent.be Keywords: Hermitean Clifford analysis, integral formulae Abstract. Euclidean Clifford analysis is a higher dimensional function theory offering a refine- ment of classical harmonic analysis. The theory is centered around the concept of monogenic functions, i.e. null solutions of a first order vector valued rotation invariant differential op- erator called the Dirac operator, which factorizes the Laplacian. More recently, Hermitean Clifford analysis has emerged as a new and successful branch of Clifford analysis, offering yet a refinement of the Euclidean case; it focusses on the simultaneous null solutions, called Hermitean (or h–) monogenic functions, of two Hermitean Dirac operators which are invari- ant under the action of the unitary group. In Euclidean Clifford analysis, the Clifford–Cauchy integral formula has proven to be a corner stone of the function theory, as is the case for the traditional Cauchy formula for holomorphic functions in the complex plane. Previously, a Her- mitean Clifford–Cauchy integral formula has been established by means of a matrix approach. This formula reduces to the traditional Martinelli–Bochner formula for holomorphic functions of several complex variables when taking functions with values in an appropriate part of com- plex spinor space. This means that the theory of Hermitean monogenic functions should encom- pass also other results of several variable complex analysis as special cases. At present we will elaborate further on the obtained results and refine them, considering fundamental solutions, Borel–Pompeiu representations and the Teoderescu inversion, each of them being developed at different levels, including the global level, handling vector variables, vector differential opera- tors and the Clifford geometric product as well as the blade level were variables and differential operators act by means of the dot and wedge products. A rich world of results reveals itself, indeed including well–known formulae from the theory of several complex variables. 1