Abstract
The complex derivative serves as one of the definitions for holomorphic functions but also as an important characteristic of the latter having algebraic, topologic and analytic aspects. The goal of the paper is to explain that in the framework of quaternionic and Clifford analyses there exists the hyperderivative of a hyperholomorphic function which extends to the corresponding situations a series of fundamental properties of its complex antecedent.
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References
Buff J.J., Characterization of analytic function of a quaternion variable. Pi Mu Epsilon J. (1973), 387–392
Gürlebeck K., Malonek H.R.: A Hypercomplex Derivative of Monogenic Functions in \({\mathbb{R}^{n+1}}\) and its Applications. Complex Variables 39, 199–228 (1999)
Gürlebeck K., Sprößig W.: Quaternionic analysis and elliptic boundary value problems. Akademie-Verlag, Berlin (1989)
Gürlebeck K., Sprössig W. Quaternionic and Clifford Calculus for Physicists and Engineers. John Wiley and Sons, 1997.
K. Gürlebeck, K. Habetha, W. Sprössig. Holomorphic Functions in the Plane and ndimensional Space. Birkhäuser, 2008.
K. Habetha, Function Theory in Algebras. Complex Analysis, Methods, Trends, and Applications; Akad. Verlag, Berlin (1983), 225–237.
Haefeli H.: Hypercomplex Differential. Comment. Math. Helv. 20, 382–420 (1947)
R. W. Hamilton, Elements of Quaternions. Longmans Green, 1886, reprinted by Chelsea, New York, 1969.
V.V. Kravchenko, M.V. Shapiro, Integral representations for spatial models of mathematical physics. Addison-Wesley-Longman, Pitman Research Notes in Mathematics 351, 1996.
N. M. Krylov. Dokl. AN SSSR 55 \({\sharp}\) 9 (1947), 799–800.
Lugojan S.: Quaternionic Derivability. An. Univ. din Timisoara, Seria Stiinte Matematice XXIX(2-3), 175–190 (1991)
Lugojan S.: Quaternionic Derivability. Semin. geom. Si topol/Univ. Timisoara 105, 1–22 (1992)
M. E. Luna-Elizarrarás, M.A. Macás-Cedeño, M. Shapiro, Hyperderivatives in Clifford analysis and some applications to the Cliffordian Cauchy-type integrals. Hypercomplex Analysis. Series: Trends in Mathematics. Sabadini, Irene; Shapiro, Michael; Sommen, Frank (Eds.) 2009, VI, 289 p.,
M. E. Luna Elizarrarás, M. A. Macías Cedeño, M. Shapiro, On the hyperderivatives of Dirac-hyperholomorphic functions of Clifford analysis. Accepted to appear in proceedings IWOTA 2009.
M. E. Luna Elizarrarás, M. A. Macías Cedeño, M. Shapiro. On the hyperderivatives of Moisil-Théodoresco hyperholomorphic functions. Proceedings of ISAAC Conference 2009.
M. S. Marinov, Meilikhson type theorems, I. Anal. Univ. din Timisoara, vol. XXXIV, f. 1 Ser. Mat.-Informatica (1966), 95–110.
A. S. Meilikhson. Dokl. AN SSSR 59 \({\sharp}\) 3 (1948), 431–434.
Mitelman I.M., Shapiro M.: Differentiation of the Martinelli-Bochner Integrals and the Notion of Hyperderivability. Math. Nachr. 172, 211–238 (1995)
Snyder H.H.: An introduction to theories of regular functions on linear associative algebras. Lect. Notes Pure Appl. Math. 68, 75–93 (1982)
M. V. Shapiro, N. L. Vasilevski, Quaternionic ψ-Hyperholomorphic Functions, Singular Integral Operators and Boundary Value Problems I. ψ-Hyperholomorphic Function Theory. Complex Variables, 27, 1995, 17-46.
Sudbery A.: Quaternionic Analysis. Math. Proc. Camb. Phil. Soc. 85, 199–225 (1979)
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The research was partially supported by CONACYT projects as well as by Instituto Politécnico Nacional in the framework of COFAA and SIP programs.
Lecture held by M. Shapiro in the Seminario Matematico e Fisico di Milano on Sept. 9, 2010
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Luna-Elizarrarás, M.E., Shapiro, M. A Survey on the (Hyper-) Derivatives in Complex, Quaternionic and Clifford Analysis. Milan J. Math. 79, 521–542 (2011). https://doi.org/10.1007/s00032-011-0169-0
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DOI: https://doi.org/10.1007/s00032-011-0169-0