Abstract
Using the Clifford algebra C n , with generators \( {e_{1}}, \ldots ,{e_{n}},e_{i}^{2} = - 1, \) the Cauchy-Riemann operator in ℝn+1 is defined as \( \bar{\partial } = \frac{\partial }{{\partial xo}} + \sum\limits_{{i = 1}}^{n} {\frac{\partial }{{\partial {x_{i}}}}{e_{i}}} , \) where \( x = {x_{0}} + \sum\limits_{{i = 1}}^{n} {{x_{i}}{e_{i}}} , \) is a paravector. We consider Fueter-type paravector functions f = u( x o, p) + I( x) v( x o, p), where \( {p^{2}} = \sum\limits_{{i = 1}}^{n} {x_{i}^{2}} ,I\left( x \right): = \frac{1}{p} = \sum\limits_{{i = 1}}^{n} {{x_{i}}{e_{i}}} , \) and u, v are real-valued. The equation \(\bar{\partial }f = 0\) splits into two parts. One of them depends only on xo,p. This leads to a system of partial differential equations which coincides with the system defining hypermonogenic functions. These functions arise for example as solutions of the Dirac equation in the upper half space ℝ n+1+ endowed with the Poincaré metric.
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References
F. Brackx, R. Delanghe, and F. Sommen, Clifford Analysis, Pitman, London, 1983.
P. Cerejeiras, Decomposition of analytic hyperbolically harmonic functions, Proc. of the 4th Conf. on Clifford Algebras and Their Appl. in Math. Phys., Aachen, Germany, Kluwer, May 1996, 45–52.
J. Cnops, Hurwitz pairs and applications of Мöbius transformations, Habilitation Thesis, Univ. Gent., 1994.
S. -L. Ericsson-Bisque, Comparison of quaternionic analysis and modified Clifford analysis, Dirac Operators in Analysis, J. Ryan and D. Struppa, eds., Pitman Research Notes in Math. 394 (1998), 109–121.
S. -L. Ericsson-Bique and H. Leutwiler, On modified quaternionic analysis in ℝ3,Arch. Math. 70 No. 3, (1998), 228–234.
R. Fueter, Die funktionentheorie der differentialgleichungen Δμ = 0 and ΔΔμ = 0 mit vier reellen Variablen, Comment. Math. Helv.No. 7, (1934/35), 307–330.
J. E. Gilbert and M. A. M. Murray, Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge University Press, 1991.
K. Giirlebeck and W. Sprössig, Quaternionic and Clifford Calculus for Physicists and Engineers, J. Wiley, 1996.
Th. Hempfling, Quaternionale analysis in ℝ4, Diplomarbeit, Univ. Erlangen-Nürnberg, February 1993.
Th. Hempfling, Aspects of modified Clifford analysis, symposium Analytical and Numerical Methods in Quaternionic and Clifford Analysis, K. Gürlebeck and W. Sprössig, eds., Seiffen, June 1996, 49–59.
Th. Hempfling, Multinomials in modified Clifford analysis, C. R. Math. Rep. Acad. Sci. (2,3) No. 18, Canada, 1996, 99–102.
Th. Hempfling, Beiträge zur modifizierten Clifford analysis, Ph.D. Thesis, Univ. Erlangen-Nürnberg, August 1997.
Th. Hempfling, The Dirac operator in ℝd+1yperbolic metric and modified Clifford analysis, in Dirac Operators in Analysis, J. Ryan and D. Struppa, eds., Pitman Research Notes in Math. 394, 1998, 95–108.
Th. Hempfling and H. Leutwiler, Modified quaternionic analysis in ℝ4, Proc. of the 4th Conf. on Clifford Algebras and Their Appl. in Math. Phys., Aachen, Germany, Kluwer, May 1996, 227–238.
H. Leutwiler, Modified Clifford analysis, Complex Variables Theory Appl. No. 17 (1992), 153–171.
H. Leutwiler, Modified quaternionic analysis in ℝ3, Complex Variables Theory Appl. No. 20 (1992), 19–51.
H. Leutwiler, Rudiments of a function theory in ℝ3, Exposition. Math. No. 14 (1996), 97–123.
M. Riesz, Sur les fonctions conjugées, Math. Z. 27 (1927), 218–244.
M. Riesz, Clifford numbers and spinors, Lecture Series No. 38, Institute for Physical Science and Technology, Maryland, 1958.
W. Sprössig, On operators and elementary functions in Clifford analysis, ZAA, 18 No. 2 (1999), 349–360.
E. M. Stein and G. Weiss, Generalization of the Cauchy-Riemann equations and representations of the rotation group, Amer. J. Math. 90 (1968), 163–196.
F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Scott, Foresman and Co., Glenview, Illinois, London, 1971.
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Hempfling, T. (2000). On the Radial Part of the Cauchy-Riemann Operator. In: Ryan, J., Sprößig, W. (eds) Clifford Algebras and their Applications in Mathematical Physics. Progress in Physics, vol 19. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1374-1_14
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DOI: https://doi.org/10.1007/978-1-4612-1374-1_14
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