A family of Cauchy-Riemann type operators

A natural question is whether and in which sense the denition of a holomorphic function depends on the choice of the two vectors {1, i} that form a basis of C over R. In fact these two vectors determine both the form of the Cauchy-Riemann operator, and the splitting of a holomorphic function in its harmonic real and imaginary components. In this paper we consider the basis {1, exp(i heta) } of C over R, and define as heta-holomorphic the functions that belong to the kernel of a Cauchy-Riemann type operator determined by this basis. We study properties of these functions, and discuss the relation between them and classical holomorphic functions. This analysis will lead us to discover the special role that heta= pi/2 plays, that renders the theory of holomorphic functions special among this family of theories.