  We give, using higher Bernstein polynomials defined in our paper [2], a
stronger version of our previous result in [1] whose converse is proved in [2]
and we give some complements to the results in [2] which help to compute these
higher order Bernstein polynomials. Then we show some non trivial examples
where we determine the root of the second Bernstein polynomial which is not a
double root of the full Bernstein polynomial and where the main theorem of [2]
applies and localizes where a double pole exists for the meromorphic extension
of the (conjugate) analytic functional given by polar parts of $\omega$ '
$\rightarrow$ |f | 2$\lambda$ f --h $\rho$$\omega$ $\land$ $\omega$' when h
$\in$ N is large enough.
