For two wavenumbers 
and 
I define the corresponding
envelopes 
and 
, the phase functions 
and
, the shifted spectra 
and 
, the odd and even functions
and 
, 
and 
. By definition, there is a simple
relation between 
and 
:
Let us derive the expression of 
:
In the same way it can be shown that:
As a consequence, it is obvious that the envelopes are the same:
Let us show that the two calculi lead to the same phase function:
The previous lines thus demonstrate that the envelope and the phase
function do not depend on the choice of the shifting wavenumber. The derivation is
still valid with 
. In this case 
and:
