Representation formulas for the general derivatives of the fundamental solution to the Cauchy-Riemann operator in Clifford analysis and applications
type: Item
instance of: scholarly article
title: Representation formulas for the general derivatives of the fundamental solution to the Cauchy-Riemann operator in Clifford analysis and applications
zbMATH Open document ID: 1018.30037
DOI: 10.4171/ZAA/1096
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2003-01-21T00:00:00Z
review text: The authors discusss several essentially different formulas for the general derivatives \(q_n(z)\) of the fundamental solution of the Cauchy-Riemann operator in Clifford analysis. Using Fourier and plane wave decomposition methods, they obtain a compact integral representation formula over a half-space which also lends itself to establish upper bound on the values \(\|q_n(z)\|\). Then they further prove several formulas for \(q_n(z)\) in terms of explicit, non-recurrent finite sums, leading themselves to further representations in fewer hypercomplex variable than used in the recurrence relations. Finally, they apply one of the closed representation formulas for \(q_n(z)\) to express the variant of the Riemann zeta function as a finite sum of real-valued Dirichlet series.
Mathematics Subject Classification ID: 30G35
Mathematics Subject Classification ID: 11M36
zbMATH DE Number: 1827599
zbMATH Keywords: Cauchy-Riemann operator
zbMATH Keywords: fundamental solution
zbMATH Keywords: permutation products
zbMATH Keywords: hypercomplex variable
zbMATH Keywords: Dirichlet series
author: D. Constales
author: R. S. Kraußhar
reviewed by: Li Mingzhong
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.4171/zaa/1096
OpenAlex ID: W2042686367
description: scientific article; zbMATH DE number 1827599