Abstract
We consider harmonic functions with respect to the Laplace–Beltrami operator of the Riemannian metric \( ds^{2} = x_{2}^{-2k}(\sum \limits{_{i=0}^{2}}{dx_{i}^{2}}) \) and their quaternion function theory in ℝ3. Leutwiler noticed around 1990 that if the usual Euclidean metric is changed to the hyperbolic one, that is k=1, then the power function \( (x_{0}+x_{1}e_{1}+x_{2}e_{2})^{n} \), calculated using quaternions, is the conjugate gradient of a hyperbolic harmonic function. We study generalized holomorphic functions, called k-hypermonogenic functions satisfying the modified Dirac equation. Note that 0-hypermonogenic are monogenic and 1-hypermonogenic functions are hypermonogenic defined by H. Leutwiler and the first author.
We prove the Cauchy type integral formulas for k-hypermonogenic where the kernels are calculated using the hyperbolic distance of the Poincará upper half-space model. Earlier these results have been proved for hypermonogenic functions.
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Mathematics Subject Classification (2010). Primary 30A05; Secondary 30A45.
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© 2014 Springer International Publishing Switzerland
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Eriksson, SL., Orelma, H., Vieira, N. (2014). Integral Formulas for k-hypermonogenic Functions in ℝ3 . In: Bernstein, S., Kähler, U., Sabadini, I., Sommen, F. (eds) Hypercomplex Analysis: New Perspectives and Applications. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-08771-9_8
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DOI: https://doi.org/10.1007/978-3-319-08771-9_8
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Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-08770-2
Online ISBN: 978-3-319-08771-9
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