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There is a very important case of discontinuity: 1/*t*. It does not belong to
our set of standard discontinuities. It will be useful to expand our set of
discontinuities by introducing the Hilbert transformation:
We see that . But what about *R*_{q,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.

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** Up:** 2: THE STANDARD DISCONTINUITIES
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Stanford Exploration Project

1/13/1998