# Special Classes of Monogenic Functions in $$\mathbb {H}$$ H

Special Classes of Monogenic Functions in $$\mathbb {H}$$ H 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 $$\varvec{p}_{k}^{l}(\varvec{x})\,\varvec{q}^{l}(\overline{\varvec{\zeta }})^{-1}$$ p k l ( x ) q l ( ζ ¯ ) - 1 , where $$\varvec{p}_{k}^{l}(\varvec{x})$$ p k l ( x ) and $$\varvec{q}^{l}(\overline{\varvec{\zeta }})$$ q l ( ζ ¯ ) 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Applied Clifford Algebras Springer Journals

# Special Classes of Monogenic Functions in $$\mathbb {H}$$ H

, Volume 28 (3) – Jun 2, 2018
18 pages

/lp/springer_journal/special-classes-of-monogenic-functions-in-mathbb-h-h-nnQaovol1e
Publisher
Springer Journals
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Physics; Mathematical Methods in Physics; Theoretical, Mathematical and Computational Physics; Applications of Mathematics; Physics, general
ISSN
0188-7009
eISSN
1661-4909
D.O.I.
10.1007/s00006-018-0874-7
Publisher site
See Article on Publisher Site

### 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 $$\varvec{p}_{k}^{l}(\varvec{x})\,\varvec{q}^{l}(\overline{\varvec{\zeta }})^{-1}$$ p k l ( x ) q l ( ζ ¯ ) - 1 , where $$\varvec{p}_{k}^{l}(\varvec{x})$$ p k l ( x ) and $$\varvec{q}^{l}(\overline{\varvec{\zeta }})$$ q l ( ζ ¯ ) 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.

### Journal

Advances in Applied Clifford AlgebrasSpringer Journals

Published: Jun 2, 2018

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