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Some Integral Representations and Singular Integral over Plane in Clifford Analysis

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Abstract

In this paper, Cauchy type integral and singular integral over hyper-complex plane \({\prod}\) are considered. By using a special Möbius transform, an equivalent relation between \({\widehat{H}^\mu}\) class functions over \({\prod}\) and \({H^\mu}\) class functions over the unit sphere is shown. For \({\widehat{H}^\mu}\) class functions over \({\prod}\) , we prove the existence of Cauchy type integral and singular integral over \({\prod}\) . Cauchy integral formulas as well as Poisson integral formulas for monogenic functions in upper-half and lower-half space are given respectively. By using Möbius transform again, the relation between the Cauchy type integrals and the singular integrals over \({\prod}\) and unit sphere is built.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, P.R. China

    Zhang Zhongxiang

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  1. Zhang Zhongxiang
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Correspondence to Zhang Zhongxiang.

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Dedicated to Prof. K. Gürlebeck on the occasion of his 60th birthday

The Project-sponsored by the NNSF for Young Scholars of China (No. 11001206).

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Zhongxiang, Z. Some Integral Representations and Singular Integral over Plane in Clifford Analysis. Adv. Appl. Clifford Algebras 24, 1145–1157 (2014). https://doi.org/10.1007/s00006-014-0498-5

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  • DOI: https://doi.org/10.1007/s00006-014-0498-5

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