Clifford algebras depending on parameters and their applications to partial differential equations. (English) Zbl 1232.30034
Escassut, A. (ed.) et al., Some topics on value distribution and differentiability in complex and \(p\)-adic analysis. Beijing: Science Press (ISBN 978-7-03-020406-6). Mathematics Monograph Series 11, 430-450 (2008).
Summary: Analyzing the properties of the Clifford algebras used so far, the chapter will introduce more general Clifford algebras in order to cover more general systems of partial differential equations.
The usual Clifford algebras are characterized by the relations \(e^2_i=\mp1\) and \(e_ie_j+e_je_i=0\) for its basic elements, \(i\neq j\). Using Clifford algebras depending on parameters and making use of generalized Cauchy-Riemann operators as well, the present chapter is aimed at describing more general systems of partial differential equations as this is possible in the framework of classical Clifford analysis.
The main result is the construction of a fundamental solution of the Cauchy-Riemann operator in a Clifford algebra whose structure relations are \(e^2_j=-\alpha_j\) and \(e_ie_j+e_je_i=2\gamma_{ij}\). Also a Cauchy-Pompeiu integral formula is obtained in this case.
For the entire collection see [Zbl 1233.30005].
The usual Clifford algebras are characterized by the relations \(e^2_i=\mp1\) and \(e_ie_j+e_je_i=0\) for its basic elements, \(i\neq j\). Using Clifford algebras depending on parameters and making use of generalized Cauchy-Riemann operators as well, the present chapter is aimed at describing more general systems of partial differential equations as this is possible in the framework of classical Clifford analysis.
The main result is the construction of a fundamental solution of the Cauchy-Riemann operator in a Clifford algebra whose structure relations are \(e^2_j=-\alpha_j\) and \(e_ie_j+e_je_i=2\gamma_{ij}\). Also a Cauchy-Pompeiu integral formula is obtained in this case.
For the entire collection see [Zbl 1233.30005].
MSC:
| 30G35 | Functions of hypercomplex variables and generalized variables |
