Extended Complex Number Analysis and Conformal-like Transformations

In a previous paper, we considered a possible extension of complex numbers, as well as its connected trigonometry. We now examine, the properties of complex analysis that can be transposed to these numbers: this goes from analyticity conditions to the residue theorem. We then show that the associated conformal symmetry is infinite and leads to n (n is the dimension of the algebra) copies of the Virasoro algebra, and define, with an appropriate prolongation of the function z ↦ 1/z, the analogue of the global conformal group. These algebras could be used for the description of scale invariant systems in more than two dimensions. We also give a Zn-graded product adapted to these numbers.