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A Cauchy--Riemann-Type CK Extension for Monogenic Functions
Abstract
We introduce a generalized Cauchy--Kowalevski extension associated with a biaxial decomposition of ℝm+1, allowing monogenic functions to be reconstructed from data prescribed on the subspace {X∈ℝm+1: z=0} where z=x0+x1e1. The extension admits a natural decomposition into monogenic layers, leading to operators CKz,k. Classes of biaxial monogenic functions generated by these operators are characterized by systems of partial differential equations, and explicit power series solutions are obtained. The action of the operators CKz,k on homogeneous polynomials is then investigated, leading to a class of biaxial monogenic plane waves and the associated system of partial differential equations.
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BibTeX
@misc{pena2026cauchyriemanntype,
title = {{A Cauchy--Riemann-Type CK Extension for Monogenic Functions}},
author = {Dixan Peña Peña},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12939},
eprint = {2609.12939},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12939},
}
RIS
TY - UNPB
TI - A Cauchy--Riemann-Type CK Extension for Monogenic Functions
AU - Dixan Peña Peña
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.12939
UR - https://arxiv.org/abs/2609.12939
AB - We introduce a generalized Cauchy--Kowalevski extension associated with a biaxial decomposition of ℝ^m+1, allowing monogenic functions to be reconstructed from data prescribed on the subspace {X∈ℝ^m+1: z=0} where z=x_0+x_1e_1. The extension admits a natural decomposition into monogenic layers, leading to operators CK_z,k. Classes of biaxial monogenic functions generated by these operators are characterized by systems of partial differential equations, and explicit power series solutions are obtained. The action of the operators CK_z,k on homogeneous polynomials is then investigated, leading to a class of biaxial monogenic plane waves and the associated system of partial differential equations.
ER -