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On operators and elementary functions in Clifford analysis. (English) Zbl 0962.30026
This is an interesting article regarding regular functions in Clifford analysis. Elements from the Clifford algebra of the form \(x_0 + \underline{x}\) with \(x_0 \in\mathbb{R}\) and \(\underline{x} \in\mathbb{R}^n\) are called paravectors. For these paravectors as arguments the classical elementary functions are defined and Moivre’s formulas are shown. Then the Cauchy-Fueter operator is decomposed into scalar and spherical terms, and some relations for these operators and paravector valued functions are proved. In the last chapter the Fueter map for regular functions \(\tau_n = \kappa_n\Delta^{\frac{n-1}{2}}\) where \(\kappa_n\) is a normalization factor, is used to get representations and properties for the elementary functions.

MSC:
30G35 Functions of hypercomplex variables and generalized variables
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[1] Brackx, F., Delanghe, R. and F. Sommen: Clifford Analysis. Boston - London - Mel- bourne: Pitman 1982. · Zbl 0529.30001
[2] Common, A. K. and F. Sommen: Axial monogenic functions from holomorphic functions. J. Math. Anal. App!. 179 (1993), 610 - 628. · Zbl 0802.30001 · doi:10.1006/jmaa.1993.1372
[3] De Craaf, J.: Note on the use of spherical vectorfields in Clifford analysis. In: Clifford Al- gebras and their Applications in Mathematical Physics (eds.: F. Brackx et al.). Dordrecht: Kluwer Acad. PubI. 1993, pp. 91 - 100.
[4] Delanghe, R., Sommen, F. and V. Soucek: Clifford Algebra and Spinor- Valued Functions. Dordrecht: Kluwer Acad. Pub!. 1992. · Zbl 0747.53001
[5] Fueter, R.: Analytische Theore einer Quaternionenvarzablen. Comment. Math. Helv. 4 (1932), 9 - 20. · Zbl 0005.07101 · doi:10.1007/BF01202702 · eudml:138569
[6] [7] Fueter, R.: Reguldre Funktionen einer Quaternionenvariablen. Zurich: Universität 1949.
[7] Gürlebeck, K. and W. Sprössig: Quaternionic Calculus for Physicists and Engineers (Mathematical Methods in Practice: Vol. 1). Chichester: John Wiley & Sons 1997. · Zbl 0897.30023
[8] Imaeda, M.: On regular functions of a power-associative hypercomplex variable. In: Proc. Clifford Algebras and their Applications in Mathematical Physics, Canterbury 1985 (eds.: J. S. R. Chisholm and A. K. Common; D. Reidel NATO ASI Series C: Vol. 183). Dordrecht - Boston: Reidel 1986, pp. 565 - 572. · Zbl 0611.30038
[9] Jank, C. and F. Sommen: Clifford analysis, biaxial symmetry and pseudoanalytic func- tions. Complex variables, Theory and Applications 13 (1990), 195 - 212. · Zbl 0703.30044
[10] Lawrentjew, M. A. and B. W. Schabat: Methoden der kornplexen Funktionentheorie. Berlin: Dt. Verlag Wiss. 1965.
[11] Leutwiler, H.: Modified Clifford analysis. Complex Variables, Theory and Appl. 17 (1992), 153 - 171. · Zbl 0758.30037 · doi:10.1080/17476939208814508
[12] Leutwiler, H.: Modified quaternionic analysis in R 3 . Complex Variables 20 (1992), 19 -51. (14] Leutwiler, II.: More on modified quaternionic analysis in R 3 . Forum Math. 7 (1995), 279 - 305. S - · Zbl 0768.30037
[13] Lounesto, P. and P. Bergh: Axially symmetric vector fields and their complex potentials. Complex Variables, Theory and Appl. 2 (1983), 139 - 150. · Zbl 0562.30036
[14] Mari nov, M. S.: On the S-regular functions. J. Nat. Geom. 7 (1995), 21 - 44.
[15] Nono, K.: Regularity of functions with values in Clifford algebras based on a generalized axially symmetric potential theory operator. In: Clifford Algebras and Their Applications in Mathematical Physics (Fundamental Theories of Physics: Vol. 55; eds.: F. Brackx et al.). Dordrecht: Kluwer Acad. PubI. 1993, pp. 159 - 166. · Zbl 0839.31005
[16] Qian, T.: Generalization of Fueter’ s result to R”. Rend. Mat. Acc. Lincei 8 (1997)9, 111 - 117. · Zbl 0909.30036 · eudml:244304
[17] Sce, M.: Osservazioni suite serie di potenze nei moduli quadtratici. Atti Acc. Lincei Rend. fis. 23 (1957), 220 - 225. · Zbl 0084.28302
[18] Som men, F.: A product and an exponential function in hypercomplex function theory. AppI. Anal. 12 (1981), 13 - 26. · Zbl 0454.30039 · doi:10.1080/00036818108839345
[19] Sommen, F.: Special functions in Clifford analysis and axial symmetry. .J. Math. Anal: AppI.: 130 (1988), 100 - 133. · Zbl 0634.30042 · doi:10.1016/0022-247X(88)90389-7
[20] Sudbery, A.: Quaternionic analysis. Proc. Camb. Phil. Soc. 85 (1979), 199 - 225.: · Zbl 0399.30038 · doi:10.1017/S0305004100055638
[21] Van Acker, N.: Clifford differentiaaloperatoren en randwaardetheorie van de nuloplossin- gen ervan op de sfer en de Lie-sfer. Proefschrift. Gent: Rijksuniversiteit 1992.
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