Abstract
Recently, the classical orthogonal function systems of inner and outer monogenic Appell functions were used to find a new class of monogenic functions with (logarithmic) line singularities, which naturally extends the class of outer monogenic Appell functions. Based on these results, this article constructs another novel class of monogenic functions with line singularities, which now also relates the class of inner monogenic Appell functions and thus complements the previously known function classes. It is shown that a subset of the constructed functions are rational monogenic functions of the form \(\varvec{p}_{k}^{l}(\varvec{x})\,\varvec{q}^{l}(\overline{\varvec{\zeta }})^{-1}\), where \(\varvec{p}_{k}^{l}(\varvec{x})\) and \(\varvec{q}^{l}(\overline{\varvec{\zeta }})\) are homogeneous polynomials in the respective variables. Finally, the classical and special classes of monogenic functions are brought together in a unified schematic representation and essential features of the individual function classes and their relationships to each other are shown.
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This article is part of the Topical Collection on Proceedings ICCA 11, Ghent, 2017, edited by Hennie De Schepper, Fred Brackx, Joris van der Jeugt, Frank Sommen, and Hendrik De Bie.
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Bock, S. Special Classes of Monogenic Functions in \(\mathbb {H}\). Adv. Appl. Clifford Algebras 28, 56 (2018). https://doi.org/10.1007/s00006-018-0874-7
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DOI: https://doi.org/10.1007/s00006-018-0874-7