Shapiro, M. V.; Vasilevski, N. L. Quaternionic \(\psi\)-hyperholomorphic functions, singular integral operators and boundary value problems. I: \(\psi\)-hyperholomorphic functions theory. (English) Zbl 0847.30033 Complex Variables, Theory Appl. 27, No. 1, 17-46 (1995). Quaternion-valued functions of 3 and 4 real variables are studied which form exact analogues of the class of all holomorphic (or anti-holomorphic) functions and which are called hyperholomorphic. The set of such analogues is parametrizable by the group \(O_3(\mathbb{R})\) or \(O^+_3(\mathbb{R})\) depending on the number of variables. Connections between various classes of hyperholomorphic functions are established, and some basic facts of the function theory are given. Properties of the singular integral operator with the quaternionic Cauchy kernel are treated and commented in detail. Reviewer: B.Fauser Cited in 3 ReviewsCited in 58 Documents MSC: 30G35 Functions of hypercomplex variables and generalized variables 32A30 Other generalizations of function theory of one complex variable 31B10 Integral representations, integral operators, integral equations methods in higher dimensions 15A66 Clifford algebras, spinors Keywords:functions of hypercomplex variables; quaternions; singular integral operators; monogenetic functions; Clifford algebras; Cauchy kernel PDFBibTeX XMLCite \textit{M. V. Shapiro} and \textit{N. L. Vasilevski}, Complex Variables, Theory Appl. 27, No. 1, 17--46 (1995; Zbl 0847.30033) Full Text: DOI