zbMATH — the first resource for mathematics

Hypermonogenic functions. (English) Zbl 0965.30020
Ryan, John (ed.) et al., Clifford algebras and their applications in mathematical physics. Papers of the 5th international conference, Ixtapa-Zihuatanejo, Mexico, June 27-July 4, 1999. Volume 2: Clifford analysis. Boston, MA: Birkhäuser. Prog. Phys. 19, 287-302 (2000).
One possible way to get a function theory in higher dimensions is to use the Clifford algebra $$Cl_n$$ and to study solutions of $$D_nf=0$$, where $$D_n=\sum_{k=0}^n e_k\partial_{x_k}$$ is the Dirac operator. These solutions are called left monogenic functions. In comparison with classical function theory and their well-known approach in higher dimensions there appear disadvantages as e.g. the fact that the power function $$x^m$$ is not left monogenic, where $$x=\sum^n_{k=0}x_ke_k$$. For this reason the authors introduce a modified Dirac operator $M_n:=D_N +{n-1\over x_n}Q_{n-1}'$ and call $$f$$ a (left) hypermonogenic function if $$M_nf=0$$ in some open subset $$\Omega\setminus \{x:x_n =0\} \subset \mathbb{R}^{n+1}$$. If a hypermonogenic function $$f=\sum^n_{l=0} f_le_l$$ with real functions $$f_l$$, then $$f$$ is called a $$H^n$$-solution. The authors study properties of hypermonogenic functions, $$H_n$$-solutions, respectively. Especially, all power functions $$x^m$$ are $$H_n$$-solutions. It would be interesting to understand how this fact can be used to create a function-theoretical approach for hypermonogenic functions or to complete a function theory for monogenic functions.
For the entire collection see [Zbl 0945.00020].

MSC:
 30G35 Functions of hypercomplex variables and generalized variables