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Quaternionic \(\psi\)-hyperholomorphic functions, singular integral operators and boundary value problems. I: \(\psi\)-hyperholomorphic functions theory. (English) Zbl 0847.30033

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

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
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