Some elements of a theory of multidimensional complex variables. I: General theory (Q910524)

The author considers the algebra over the complex numbers with basis 1, j, \(j^ 2\), assuming \(j^ 3=-1\), generalizations to higher dimensions are given. Relations between the coordinates for different bases are dealt with, especially used are cylindrical and spherical coordinates. Differentiable functions in the usual sense are called analytic, the corresponding Cauchy-Riemann-equations are calculated. These are also considered for the different bases. The real and imaginary parts of such functions are harmonic vectors if written down within the right base. There are no integral theorems (which exist).