We see that . But what about Rq,1 for q>-1? Unfortunately, the integral:
does not exist because of the non-integrable singularity at the point =0. But we may use the following trick: if q>-1, we apply the operator and obtain the discontinuity R-1. After that we may apply the Hilbert transformation and then the inverse operator .
Remembering the equation (14), we get:
This is a definition of the Hilbert transformation for discontinuities. We define also:
where is the unit operator.