Quaternionic ψ-hyperholomorphic functions, singular integral operators and boundary value problems I. ψ-hyperholomorphic function theory

Quaternion-valued functions of 3 and 4 real variables arc 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 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.