On the transformation functions from thejm-basis to thez-basis for the unitary representations ofSL2,C

The covariance properties of the transformation functions from thejm-basis to thez-basis are written down and solved explicitly for irreducible unitary representations ofSL2,c belonging to the principal series. The explicit expression of these functions in terms of elementary functions is given. Some interesting completeness and orthogonality properties for the expansion of one- and two-variable complex functions are pointed out.