Fundamental theorems for Clifford algebra-valued distributions in elliptic Clifford analysis
type: Item
instance of: scholarly article
title: Fundamental theorems for Clifford algebra-valued distributions in elliptic Clifford analysis
zbMATH Open document ID: 1258.30025
author: Ghislain R. Franssens
published in: Advances in Applied Clifford Algebras
publication date: 2012-04-04T00:00:00Z
review text: By a fundamental theorem in Clifford analysis (CA) one usually means a relation between a pair of Clifford algebra-valued functions, defined over an open subset \(U\) of a base manifold \(X^n\), and expressing a type of integrated reciprocity relation holding over a compact and oriented subset \(\overline{c}\) of \(U\). Such a theorem is constructed as a generalization to Stokes' theorem and can thus be regarded as a far generalization of the concept of partial integration, to be used in CA. In this paper, a fundamental theorem in elliptic Clifford analysis (ECA), with the standard vector Dirac operator, is presented that is valid for Clifford algebra-valued distributions based on \(X^n\). A Clifford algebra-valued distribution based on \(X^n\) is a continuous linear functional, defined on the space of real-valued test functions with compact support in \(X^n\) and taking values in a Clifford algebra. This theorem holds under fairly general conditions on the allowed singularities of the right-hand side distributions and on the region of integration \(\overline{c}\). Next, a specialization of this fundamental theorem is proved that forms the starting point for solving boundary value problems with distributional sources in ECA. Finally, distributional equivalents of the residue theorem, Cauchy's theorem and Cauchy's integral theorem are stated.
reviewed by: Nele De Schepper
Mathematics Subject Classification ID: 30G35
Mathematics Subject Classification ID: 32H30
Mathematics Subject Classification ID: 60E05
zbMATH DE Number: 6021827
zbMATH Keywords: elliptic Clifford analysis
zbMATH Keywords: fundamental theorem
zbMATH Keywords: Dirac operator
zbMATH Keywords: residue theorem
zbMATH Keywords: Cauchy's integral theorem
MaRDI profile type: Publication
full work available at URL: s00006-011-0283-7
OpenAlex ID: W2107163210
cites work: Q3782468
cites work: Q4115333
cites work: Q5332437
DOI: 10.1007/S00006-011-0283-7
description: scientific article; zbMATH DE number 6021827