On some structural sets and a quaternionic (φ,ψ) ‐hyperholomorphic function theory

Quaternionic analysis is regarded as a broadly accepted branch of classical analysis referring to many different types of extensions of the Cauchy-Riemann equations to the quaternion skew field . It relies heavily on results on functions defined on domains in or with values in . This theory is centred around the concept of ψ-hyperholomorphic functions related to a so-called structural set ψ of or respectively. The main goal of this paper is to develop the nucleus of the -hyperholomorphic function theory, i.e., simultaneous null solutions of two Cauchy-Riemann operators associated to a pair of structural sets of . Following a matrix approach, a generalized Borel-Pompeiu formula and the corresponding Plemelj-Sokhotzki formulae are established.

[1]  K. Gürlebeck,et al.  A Quaternionic Beltrami-Type Equation and the Existence of Local Homeomorphic Solutions , 2001 .

[2]  K. Gürlebeck,et al.  Quaternionic Analysis and Elliptic Boundary Value Problems , 1989 .

[3]  Andrew Chang.  Quaternionic analysis , 1979, Mathematical Proceedings of the Cambridge Philosophical Society.

[4]  Nikolai Vasilevski,et al.  Quaternionic ψ-hyperholomorphic functions, singular integral operators and boundary value problems I. ψ-hyperholomorphic function theory , 1995 .

[5]  Run Fueter.  Die Funktionentheorie der DifferentialgleichungenΔu=0 undΔΔu=0 mit vier reellen Variablen , 1934 .

[6]  V. Kravchenko,et al.  Integral Representations For Spatial Models of Mathematical Physics , 1996 .

[7]  濃野 聖晴,et al.  Hyperholomorphic Functions of a Quaternion Variable , 1982 .

[8]  A. I. Kamzolov.  COMMUNICATIONS OF THE MOSCOW MATHEMATICAL SOCIETY: On the Kolmogorov diameters of classes of smooth functions on a sphere , 1989 .

[9]  N. Vasilevski,et al.  On the Bergmann Kernel Function in Hyperholomorphic Analysis , 1997 .

[10]  M. Shapiro,et al.  Differentiation of the Martinelli‐Bochner Integrals and the Notion of Hyperderivability , 1995 .

[11]  M. Shapiro,et al.  On the Π-operator in hyperholomorphic function theory , 1999 .

[12]  Mukhammed-Naser.  Hyperholomorphic functions , 1971 .

[13]  濃野 聖晴.  On the Quaternion Linearization of Laplacian Δ , 1985 .

[14]  M. Shapiro.  Some remarks on generalizations of the one - dimensional complex analysis : hypercomplex approach , 1993 .

[15]  Baruch Schneider,et al.  Boundary value problems for the Cimmino system via quaternionic analysis , 2012, Appl. Math. Comput..

[16]  V. Kravchenko.  Applied quaternionic analysis , 2003 .

[17]  Elliptic, Parabolic and Hyperbolic Analytic Function Theory--0: Geometry of Domains , 2004, math/0410399.