Generalized partial-slice monogenic functions

Xu, Zhenghua; Sabadini, Irene

doi:10.1090/tran/9356

Generalized partial-slice monogenic functions
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by Zhenghua Xu and Irene Sabadini;

Trans. Amer. Math. Soc. 378 (2025), 851-883

DOI: https://doi.org/10.1090/tran/9356

Published electronically: December 27, 2024

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Abstract:

The two function theories of monogenic and of slice monogenic functions have been extensively studied in the literature and were developed independently; the relations between them, e.g. via Fueter mapping and Radon transform, have been studied. The main purpose of this article is to describe a new function theory which includes both of them as special cases. This theory allows to prove nice properties such as the identity theorem, a Representation Formula, the Cauchy (and Cauchy-Pompeiu) integral formula, the maximum modulus principle, a version of the Taylor series and Laurent series expansions. As a complement, we shall also offer two approaches to these functions via generalized partial-slice functions and via global differential operators. In addition, we discuss the conformal invariance property under a proper group of Möbius transformations preserving the partial symmetry of the involved domains.

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