Computational Methods and Function Theory 4 (2004), No. 1, 159--182
Copyright Heldermann Verlag 2004
Heinz Leutwiler
leutwil@mi.uni-erlangen.de , Mathematical Institute, University of Erlangen-Nuremberg, Bismarckstr. 1 1/2, D-91054 Erlangen, Germany.
Peter Zeilinger
zeiling@mi.uni-erlangen.de , Mathematical Institute, University of Erlangen-Nuremberg, Bismarckstr. 1 1/2, D-91054 Erlangen, Germany.
Quaternionic analysis -- and in higher dimensions Clifford analysis -- are well-known extensions of classical complex analysis. A modification of this theory, based on hyperbolic geometry, has recently been developed by the first author [(1) Complex Variables Theory Appl. 17 (1992) 153--171; (2) ibid. 20 (1992) 19--51; (3) Expositiones Math. 14 (1996) 97--123]. A unifying theory, introduced by G. Laville and I. Ramadanoff [Adv. Appl. Clifford Algebras 8 (1998) 321--340], also exists. In this paper we compare these three theories with each other. We thereby mainly focus on the Clifford algebras Cl0,2 and Cl0,3. In case of Cl0,2 » H (the quaternions) we present, for the modified theory, a Cauchy-type formula for the so-called (H)-solutions [see paper (2) of the author cited above], which is based on some recent result of S. L. Eriksson-Bique ["Integral formulas for hypermonogenic functions", to appear].
Keywords: Generalized function theory, quaternions, Dirac operator, hyperbolic metric.
MSC 2000: Primary 30G35; Secondary 30A05, 30F45.
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