Plane elliptic systems and monogenic functions in symmetric domains. (English) Zbl 0564.30036

Let f be defined in \(\Omega\), open in \({\mathbb{R}}^{m+1}\), and taking values in the complex Clifford algebra over \({\mathbb{R}}^ m\). Such a function f is called (left) monogenic in \(\Omega\) if it is \(C^ 1\) and satisfies \(Df=0\) in \(\Omega\), where D is the generalized Cauchy- Riemannian operator \(D=\sum^{m}_{i=0}e_ i\partial_{x_ i}.\) Let G be a subgroup of \(SO(m+1)\) and let \(\Omega\) be invariant under G. This subgroup acts in a natural way on the space of left monogenic functions in \(\Omega\), yielding a representation of G; splitting up this representation into irreducible parts leads to a generalized Laurent expansion.
In the case where \(G=SO(m)\), \(\Omega\) is axially symmetric and the so- called axial monogenic terms in the Laurent series satisfy special plane elliptic systems. For \(G=SO(m_ 1)\times SO(m_ 2)\) the toral monogenic functions are obtained, while the case \(G=SO(2)\times...\times SO(2)\) provides a new link between monogenic functions and Clifford algebra valued holomorphic functions of several complex variables.
Reviewer: F.Brackx


30G35 Functions of hypercomplex variables and generalized variables
32A30 Other generalizations of function theory of one complex variable
30A05 Monogenic and polygenic functions of one complex variable