Abstract
Additive decompositions of harmonic functions play an important role in function theory, and in its application to the solution of partial differential equations. One of the most fundamental results is the decomposition of a harmonic function into a sum of a holomorphic and an anti-holomorphic function. This decomposition can be generalized to the case of quaternion valued harmonic function, where the summands are then monogenic and anti-monogenic, respectively. For paravector-valued functions, sometimes called \(\mathcal{A}\)-valued functions, this decomposition is not possible. In previous articles it was shown that a harmonic function can be represented by a linear combination of monogenic, anti-monogenic and ψ-hyperholomorphic functions, where ψ = {1,e 2, − e 1} is the structural set. In this paper we will study the question of such a representation for a general structural set ψ.
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Gürlebeck, K., Nguyen, H.M., Legatiuk, D. (2014). An Additive Decomposition of Harmonic Functions in \(\rm{I\!R}^{3}\) . In: Murgante, B., et al. Computational Science and Its Applications – ICCSA 2014. ICCSA 2014. Lecture Notes in Computer Science, vol 8579. Springer, Cham. https://doi.org/10.1007/978-3-319-09144-0_14
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DOI: https://doi.org/10.1007/978-3-319-09144-0_14
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