Abstract
From the concept of hypercomplex differentiability (introduced at the end of the 1980’s ([6], [7]) by using local linear approximation properties of monogenic functions) the existence of a monogenic derivative does not directly follow. We show that if some relation between higher order differential forms are introduced then, (as in the complex case) the conjugated Cauchy-Riemann operator again gives the monogenic derivative of a monogenic function in ℝn+1
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References
F. Brackx, R. Delanghe, and F. Sonmmen, Clifford Analysis 76, Pitman, Boston-London-Melbourne, 1982.
H. E. Edwards, Advanced Calculus: A Differential Forms Approach, Third Edition, Birkhäuser, Boston, Basel, Berlin, 1993.
K. Giirlebeck and H. Malonek, A hypercomplex derivative of monogenic functions in ℝn+1 and its applications, Complex Variables Theory Appl., Vol. 39 (1999), 199–228.
K. Giirlebeck and W. Sprößig, Quaternionic and Clifford Calculus for Engineers and Physicists, John Wiley &. Sons, Chichester, 1997.
K. Habetha, Eine bemerkung zur funktionentheory in algebren, in Function Theoretic Methods for Partial Differential Equations, V. E. Meister, N. Weck, and W. L. Wendland, eds., Lecture Notes in Mathematics, Vol. 561, Springer, Berlin, Heidelberg, New York, 1976, 502–509.
H. Malonek, Zum holomorphiebegriff in höheren dimensionen, Habilitation Thesis, PH Halle, 1987.
H. Malonek, A new hypercomplex structure of the Euclidean space ℝm+1 and the concept of hypercomplex differentiability, Complex Variables Theory Appl., Vol. 14 (1990), 25–33.
H. Malonek, Power series representation for monogenic functions in ℝm+1 based on a permutational product, Complex Variables Theory Appl., Vol. 15 (1990), 181–191.
H. Malonek, The concept of hypercomplex differentiability and related differential forms, in Studies in Complex Analysis and Its Applications to Partial Differential Equations 1, R. Kühnau and W. Tutschke, eds., Pitman 256, Longman, 1991, 193–202.
A. S. Mejlihzon, Because of monogeneity of quaternions (in Russian), Doklady Acad. Sc. USSR 59 (1948), 431–434.
I. M. Mitelman and M. V. Shapiro, Differentiation of the MartinelliВochner integrals and the notion of hyperderivability, Math. Nachr. 172 (1995), 211–238.
J. Ryan, Clifford analysis with generalized elliptic and quasi-elliptic functions, Applicable Analysis 13 (1982), 151–171.
F. Sommen, Monogenic differential forms and homology theory, Proc. R. Irish Acad., Vol. 84A No. 2 (1984), 87–109.
F. Sommen, Monogenic differential calculus, Trans. AMS, Vol. 326 No. 2 (1991), 613–632.
W. Sprößig, Über eine mehrdimensionale operatorrechnumg über beschränkten gebieten des Rℝn, Thesis, TH Karl-Marx-Stadt, 1979.
A. Sudbery, Quaternionic analysis, Math. Proc. Cambr. Phil. Soc. 85 (1979), 199–225.
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Malonek, H.R. (2000). Hypercomplex Derivability — The Characterization of Monogenic Functions in ℝn+1 by Their Derivative. 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_15
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DOI: https://doi.org/10.1007/978-1-4612-1374-1_15
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