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Integral formulas for k-hypermonogenic functions in \(\mathbb R^3\). (English) Zbl 1314.30098
Bernstein, Swanhild (ed.) et al., Hypercomplex analysis: new perspectives and applications. Selected papers presented at the session on Clifford and quaternionic analysis at the 9th congress of the International Society for Analysis, its Applications, and Computation (ISAAC), Krakow, Poland, August 5–9, 2013. New York, NY: Birkhäuser/Springer (ISBN 978-3-319-08770-2/hbk; 978-3-319-08771-9/ebook). Trends in Mathematics, 119-132 (2014).
Harmonic functions with respect to the Laplace-Beltrami operator in the Riemanian metric \(ds^{2} \equiv x^{-2k}(\sum_{i=0}^{2}dx_{i}^{2})\) are considered with their quaternion function theory in \(\mathbb{R}^{3}\). The authors study generalized holomorphic functions, called \(k\)-hypermonogenic functions, satisfying a modified Dirac equation. In [S.-L. Eriksson, in: More progresses in analysis. Proceedings of the 5th international ISAAC congress, Catania, Italy, July 25–30, 2005. Hackensack, NJ: World Scientific. 1051–1064 (2009; Zbl 1185.30053)] a Cauchy-type theorem for \(k\)-hypermonogenic functions was proved, where the kernels have a complicated form.
As the main result of the present article, a different way is presented to compute these kernels by an explicit integral formula in \(\mathbb{R}^{3}\). Note that 0-hypermonogenic functions are monogenic and 1-hypermonogenic functions are hypermonogenic as introduced in [S.-L. Eriksson and H. Leutwiler, Adv. Appl. Clifford Algebr. 19, No. 2, 269–282 (2009; Zbl 1172.30027)]. In future articles, the integral operators will be studied together with their generalization to higher dimensions.
For the entire collection see [Zbl 1300.30002].

MSC:
30G35 Functions of hypercomplex variables and generalized variables
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