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INTEGRAL REPRESENTATIONS IN HERMITEAN CLIFFORD ANALYSIS

• Euclidean Clifford analysis is a higher dimensional function theory offering a refinement 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 operator called the Dirac operator, which factorizes the Laplacian. More recently, Hermitean Clifford analysis has emerged as aEuclidean Clifford analysis is a higher dimensional function theory offering a refinement 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 operator 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 invariant 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 Hermitean 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 complex spinor space. This means that the theory of Hermitean monogenic functions should encompass 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 operators 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.

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Document Type: Conference Proceeding Fred Brackx, Hennie De Schepper, Maria Elena Luna-Elizararras, Michael Shapiro https://doi.org/10.25643/bauhaus-universitaet.2832Cite-Link https://nbn-resolving.org/urn:nbn:de:gbv:wim2-20170314-28326Cite-Link http://euklid.bauing.uni-weimar.de/ikm2009/paper.html 1611-4086 Klaus GürlebeckGND, Carsten KönkeORCiDGND English 2017/03/04 2010/07/14 2017/03/14 Bauhaus-Universität Weimar Bauhaus-Universität Weimar Bauhaus-Universität Weimar / In Zusammenarbeit mit der Bauhaus-Universität Weimar 13 Computer Science Models in Engineering; Multiscale and Multiphysical Models; Scientific Computing Angewandte Informatik; Angewandte Mathematik; Architektur ; Computerunterstütztes Verfahren 000 Informatik, Informationswissenschaft, allgemeine Werke / 000 Informatik, Wissen, Systeme 500 Naturwissenschaften und Mathematik / 510 Mathematik 31 Mathematik / 31.80 Angewandte Mathematik 56 Bauwesen / 56.03 Methoden im Bauingenieurwesen Bauhaus-Universität Weimar / Internationales Kolloquium über Anwendungen der Informatik und Mathematik in Architektur und Bauwesen, IKM, Weimar / Internationales Kolloquium über Anwendungen der Informatik und Mathematik in Architektur und Bauwesen, IKM, Weimar, 18. 2009 Creative Commons 4.0 - Namensnennung-Nicht kommerziell (CC BY-NC 4.0)