Abstract
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 in [18], [19] and [20]. A unifying theory, introduced by G. Laville and I. Ramadanoff in [16], also exists. In this paper we compare these three theories with each other. We thereby mainly focus on the Clifford algebras 
and 
. In case of 
we present, for the modified theory, a Cauchy-type formula for the so-called (H)-solutions (see [19]), which is based on some recent result of S. L. Eriksson-Bique [3].
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References
N. Aronszajn, T. Creese, L. Lipkin, Polyharmonic Functions, Clarendon Press, Oxford, 1983.
F. Brackx, R. Delanghe, R. Sommen, Clifford Analysis, Pitman, Boston-London-Melbourne, 1982.
S. L. Eriksson-Bique, Integral formulas for hypermonogenic functions, to appear.
S. L. Eriksson-Bique, On modified Clifford analysis, Complex Variables 45 (2001), 11–32.
—, A correspondence of hyperholomorphic and monogenic functions in ℝ4, in: F. Brackx et al. (eds), Clifford Analysis and its Applications, Kluwer Acad. Publ. (2001), 71–80.
S. L. Eriksson-Bique and H. Leutwiler, Hypermonogenic functions, in: Clifford Algebras and their Applications in Mathematical Physics, Vol. 2, Birkhäuser, Boston 2000, 287–302.
S. L. Eriksson-Bique and H. Leutwiler, On modified quaternionic analysis in ℝ3, Arch. Math. 70 (1998), 228–234.
—, Hypermonogenic functions and their Cauchy-type theorems, to appear.
S. L. Eriksson-Bique and H. Leutwiler, Hyperholomorphic functions, Comput. Methods Funct. Theory 1 2001) No.1, 179–192.
R. Fueter, Die Funktionentheorie der Differentialgleichungen Δu = 0 und ΔΔu = 0 mit vier reellen Variablen, Comment. Math. Helv. 7 (1934/35), 307–330.
T. Hempfling, Multinomials in Modified Clifford Analysis, C.R. Math. Rep. Acad. Sci. Canada, No.18 (2,3) (1996), 99–102.
—, Aspects of Modified Clifford Analysis, Symposium ’Analytical and Numerical Methods in Quaternionic and Clifford Analysis’ (Seiffen) (K. Gürlebeck, W. Sprössig, ed.), June 1996, 49–59.
T Hempfling (1998) The Dirac Operator in \({\rm R}_{+}^{d+1}\) with Hyperbolic Metric and Modified Clifford Analysis J Ryan D Struppa J Ryan D Struppa (Eds) Dirac Operators in Analysis 95–108
T. Hempfling and H. Leutwiler, Modified quaternionic analysis in ℝ4, in: V. Dietrich et al. (eds), Clifford Algebras and their Applications in Mathematical Physics, Kluwer Acad. Publ. (1998), 227–237.
W. Hengartner and H. Leutwiler, Hyperholomorphic functions in ℝ3, Advances in Clifford Algebras 11(S1) (2001), 247–259.
G. Laville and I. Ramadanoff, Holomorphic Cliffordian functions, Advances in Applied Clifford Algebras 8 No.2 (1998), 321–340.
G. Laville and E. Lehman, Analytic Cliffordian functions, to appear.
H. Leutwiler, Modified Clifford analysis, Complex Variables, Theory Appl. 17 (1992), 153–171.
H. Leutwiler, Modified quaternionic analysis in ℝ3, Complex Variables, Theory Appl. 20 (1992), 19–51.
H. Leutwiler, Rudiments of a function theory in ℝ3, Expositiones Math. 14 (1996), 97–123.
—, Quaternionic analysis in ℝ3 versus its hyperbolic modification, in: F. Brackx et al. (eds), Clifford Analysis and its Applications, Kluwer Acad. Publ. (2001), 193–211.
H. Maaß, Automorphe Funktionen von mehreren Veränderlichen und Dirichletsche Reihen, Abh. Math. Sem. Univ. Hamburg 16, 3/4 (1949), 72–100.
A. Sudbery, Quaternionic analysis, Math. Proc. Camb. Philos. Soc. 85 (1979), 199–225.
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Leutwiler, H., Zeilinger, P. On Quaternionic Analysis and its Modifications. Comput. Methods Funct. Theory 4, 159–182 (2004). https://doi.org/10.1007/BF03321063
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DOI: https://doi.org/10.1007/BF03321063